Bispectrum_monograph.pdf

Bispectrum-based methods and algorithms
for radar, telecommunication signal processing and
digital image reconstruction
Totsky A.V., Lukin V.V., Zelensky A.A., Astola J.T.,
Egiazarian K.O., Khlopov G.I., Morozov V.Ye., Kurbatov I.V.,
Molchanov P.A., Roenko A.A., Fevralev D.V.

Contents
Introduction 1
1. Some properties of bispectrum analysis in application to digital signal processing 3
1.1. Properties of cumulant and moment functions 3
1.2. Triple correlation functions and bispectra 7
1.3. Bispectral density estimation techniques 11
1.4. Bispectrum-based algorithms in filtering and signal reconstruction applications 14
1.5. Reduction of waveform distortions in bispectrum-based signal reconstruction systems 31
1.6. Conclusions 43
2. Combined bispectrum-filtering techniques for unknown signal shape estimation in noise 45
2.1. 1-D non-adaptive filtering of signal Fourier spectrum estimates recovered by bispectrum
estimation in Gaussian and non-Gaussian noise environment 46
2.2. Smoothing the noisy bimagnitude and biphase or real and imaginary parts of bispectrum
estimates by using non-adaptive 2-D linear and nonlinear filtering 53
2.3. Novel approaches developed for improving bispectrum estimates 71
2.4. Adaptive 1-D filtering in bispectrum-based signal reconstruction 98
2.5. Conclusions 106
3. Bispectrum-based image reconstruction techniques
by using tapering pre-distortion inserted in image rows 107
3.1. Noisy and jittered image reconstruction by using additive pre-distortion 107
3.2. Bispectrum-based image reconstruction by using multiplicative pre-distortions 116
3.3. Search of the optimal parameters of additive and
multiplicative pre-distortion functions 119
4. Signal detection in Gaussian noise by using third-order test statistics 129

4.1. Detection of deterministic signals observed in additive Gaussian noise based on likelihood ratio
criterion by using third-order statistic 129
4.2. Application of triple correlation and bispectrum for interference immunity improvement in
digital telecommunications systems 138
5. Bispectrum estimation in radar applications 149
5.1. Radar range profile estimation of naval objects 149
5.2. Reduction of aspect dependent speckle distortions in aerial HRRP 160
5.3. Time-frequency analysis of backscattered signals in ground surveillance Doppler radar 169
5.4. Parametrical short-time bispectral estimation for time-frequency analysis of multi-frequency
signals 187
6. Conclusions 193
References 195

PREFACE
This work is based on results obtained during seven-year period of cooperative research of the
Dept of Transmitters, Receivers and Signal Processing, National Aerospace University, Kharkov,
Ukraine, and the Institute for Signal Processing, Tampere University of Technology, Finland. Starting
from 2006, the Institute of Radiophysics and Electronics, Ukrainian National Academy of Science,
Kharkov, Ukraine has joined this research.
The goal of this research work has been the theoretical and experimental development of
bispectrum-based techniques and algorithms for digital processing of signals and images. The basic
application is radars of different types intended for detection and automatic recognition of aerial and
naval targets, surveillance in clutter. Other applications like communications and data compression are
briefly considered as well.
Statistical characteristics of bispectrum estimates are studied for Gaussian and non-Gaussian
environments. On this basis, novel methods for improving bispectrum estimates and bispectrum-based
reconstruction of unknown signal waveform from an observed set of realizations are proposed and
investigated. Useful properties of bispectrum are revisited and it is demonstrated how these properties
can be exploited in particular applications. The cases of rather small input signal-to-noise ratios are
paid special attention. The results of numerical simulations are given. At the same time, for several
applications like naval radar, ground surveillance radar, etc., the real life experimental data have been
processed and presented.
To our opinion, the obtained results and designed techniques can be useful for automatic target
recognition systems, radar security systems, sonars, communications that operate in noise channels.

Introduction
Higher order statistic analysis has attracted the attention of many researchers in 80 th of the pre-
vious century as an alternative to conventional spectral-correlation approach [1–4]. A.W. Lohmann
and B. Wirnitzer (see, e.g., [1]) addressed optical and astronomical applications of triple correlation.
Their work stimulated considering other applications like sonars, biomedical engineering, non-
destructive control, radars, communications, etc. that have been generalized in [2] by C.L. Nikias and
M.R. Raghuveer. Theoretical aspects were summarized by J.M. Mendel [3]. Recently the amount of
publications dealing with higher order statistics increased radically [4].
Despite of this progress, for almost twenty years researchers were basically dealing with the
theoretical aspects of bispectrum estimation. This is explained by the extensive computations required
for processing of multidimensional data. With increased computational efficiency and memory cha-
racteristics of computers, the interest to practical applications of bispectrum-based processing has in-
creased.
This is connected with known inherent advantages of higher-order statistics and bispectrum that
radically differ them from conventional spectrum-correlation estimation. These advantages are signal
shift invariance, immunity to zero mean noise with symmetric probability density function, preserva-
tion of signal phase spectrum, etc.
However, many both theoretical and practical questions remained unclear. They are:
- The statistical properties of bispectrum and triple correlation estimates have not been analyzed
in detail, effective ways for their improvement have not been thoroughly studied yet.
- The phase wrapping problem still exists and it influences the quality of the signal waveform
reconstructed from bispectrum.
- The performance of bispectrum-based processing is not thoroughly investigated for such prac-
tically important situations like small input signal-to-noise ratios (SNR) and a small number of
observed realizations.
These problems are addressed below in this work which contains a systematic description of re-
sults presented in our recent publications. The work contains five Chapters. Chapter 1 gives theoreti-
cal background and deals with basic properties of bispectrum and triple correlation function and accu-
racy of bispectrum estimation. Some particular aspects like phase unwrapping are discussed.
Chapter 2 is devoted to combined bispectrum-filtering techniques that exploit positive features
of bispectrum and filtering, linear and non-linear, non-adaptive and adaptive. Non-Gaussianity and
1

non-stationarity of fluctuations in bispectral domain induced by leakage of input noise is demonstrat-
ed. This serves the purpose of designing novel adaptive filters suitable for this application.
Reconstruction of images distorted by jitter effects and noise is considered in Chapter 3. As
shown, bispectrum based processing can be useful for this case under the condition that pre-distortions
are inserted in row images.
New material dealing with applying bispectrum for data coding in telecommunication systems is
presented in Chapter 4. The results of decoding probability analysis are presented and discussed for
complex noise environments.
Radar applications of bispectrum are considered in Chapter 5. It contains experimental results
for coastal naval, surveillance and aerial target detection radars.
The author come from three groups of researchers. The first group represents the Department of
Transmitters, Receivers and Signal Processing, National Aerospace University, Kharkov, Ukraine.
This group includes Associate Professor Totsky A.V., Professors Zelensky A.A. and Lukin V.V., Drs
Roenko A.A., Fevralev D.V., Kurbatov I.V. and undergraduate student Molchanov P.A. The scientists
of the Institute of Signal Processing, Tampere University of Technology, Finland, namely, Professors
J. Astola and K. Egiazarian contributed to this research. We would also like to thank our co-authors
Prof. Katkovnik V.Ya. and Dr Paliy D. for fruitful cooperation at one stage of research. Professor
Khlopov G.I. and Dr. Morozov V.Ye. represent the Institute of Radiophysics and Electronics, Ukrai-
nian National Academy of Science. They mainly participated in radar design and manufacturing as
well as in organizing and carrying out radar experiments.
The cooperation resulted in preparation of more than thirty papers in 2002-2008 given in the
reference list.
2

1. Some properties of bispectrum analysis in application to
1.1. Properties of cumulant and moment functions
digital signal processing
Conventional power or energy spectral density estimation is a well-known and widely used
technique for random signal analysis in frequency domain. Averaged Fourier magnitude spectrum
density does not contain any information about behavior of a process under study in the frequency
domain for centered random processes since the spectral components are statistically independent in
different observed process realizations. In this case, one estimates the energy distribution of statistical-
ly independent spectral components since the energy does not depend on the phase relationships for
separate frequencies. Indeed, for the processes containing independent spectral components, the ener-
gy spectrum estimate is the exhaustive characteristic conventionally used in spectral analysis of such
processes.
In several practical applications of signal processing, an analyzed process can contain correlated
spectral components observed in different realizations. Study of these correlation relationships can
give us very useful information for correct understanding, analysis and description of physical effects
that cause a given process. Note that such information about correlation relationships is irretrievably
lost in conventional energy spectrum estimate.
Cumulant functions (higher-order correlation functions and spectra) estimation can serve as a
very useful and promising tool for signal analysis and processing. Such approach has several impor-
tant and attractive advantages comparing to above mentioned energy spectrum estimation. These ad-
vantages are listed below.
First, consider mathematical description of cumulant spectra for a real-valued stationary discrete
process {x(i), i=0,1,2,…}. The joint cumulants c τ , τ , ..., τ ) of r-th order can be defined as
( x 1 2 r − 1
⎡∂ ln Θ(
ω , ω ,..., ω ) ⎤
r
( r)
=
x
( τ , τ ,..., τ ) = −
x 1 2 r − 1
r
⎢
⎣
1 2
∂ω1∂ω2... ∂ω
r
r
⎥
⎦ω1
= ω2 = ... = ωr
= 0
c c j
3
, (1.1.1)
where Θ(ω1, ω2,…, ωr) = x is the multidimensional characteristic
function; ω1, ω2,…, ωr are the frequencies; j = − 1 ; x denotes ensemble average procedure; and
τ1, τ2,…, τr-1 are the integer-valued shifts.

The cumulants (1.1.1) serve as the characteristics of the probability distribution and they can be
represented by the following coefficients in Taylor series for the function lnΘ(ω) in the neighborhood
of the point of origin
∞ ( r )
cx
r
ln Θ ( ω ) = ∑ ( jω)
. (1.1.2)
r!
r=
1
The joint moments m ( τ τ ..., τ ) for a stationary process {x(i), i=0,1,2,…} differ from the
x 1, 2, r − 1,
cumulants (1.1.1) as follows
m = m ( τ , τ ,..., τ ) = x( i) x( i + τ ) x( i + τ )... x( i + τ ) =
( r )
x x 1 2 r−1 1 2 r−1
x
= −
∂ Θ(
ω , ω ,..., ω )
r
k
j [ 1 2 r
∂ω1∂ω2... ∂ωr
] ω ω ω
1= 2 = ... = r = 0
4
. (1.1.3)
The joint moments (1.1.3) can be defined by the expansion coefficients of the characteristic
function Θ(ω) in Taylor series in the neighborhood of the point of origin as
( r)
m
( ω ) 1 x
∑ ( j
r 1 r!
∞
= +
=
r
)
Θ ω . (1.1.4)
The relationship between the joint cumulants (1.1.1) and the joint moments (1.1.3) in the origin
(the integer-valued shifts are τ1=τ2=… τr-1=0) can be defined by the formulas
( 3)
x
( 1)
x
( 1)
x
m c = = ) (i x ,
( 2)
x
( 2)
x
( ) 2
m(
1)
c = m − , (1.1.5)
( 3)
x
−
x
( ) 3
( 1)
m
( 1)
( 2)
3mx mx
2 x
m c = + ,
c
( 4)
For the case of a zero-mean process, i.e., for
the following form
x
2
( ) ( ) 4
( 1)
2 ( 2)
( 1)
m m −6
m
( 4)
( 2)
( 1)
( 3)
m 3⎜
⎛m
⎟
⎞
= − −4m
x mx
+ 12 x x x ,
x ⎝ x ⎠
…………………………………………………….
( 1)
x
( 1)
x
m c = = 0,
( 2)
( 2)
c mx
2
( 1)
x
m = x (i)
= 0, the formulas (1.1.5) transform to
2
= = x(
i)
= σ , (1.1.6)
x
( 3)
x
( 3)
x
x ( i)
m c = = ,
3

2
( 4)
( 4)
( 2)
4
2
c = m ⎛ ⎞ ( ) 3⎛
− ⎜m
⎟ = x i − ⎜σ
x x
3 ⎞
⎟ ,
x ⎝ ⎠
⎝ ⎠
……………………………………………………….
where σ 2 is the variance of a process considered.
Let us consider a real-valued discrete process {x(i), i=0,1,2,…,I–1} with the zero mean, i.e.,
x ( i)
= 0 . The relationship between the moment and cumulant functions for this zero-mean process is
defined by the following formulas
( 2)
( 2)
x( i)
x(
i + k)
= mx
( k)
= cx
( k)
, (1.1.7а)
( 3)
( 3)
x( i)
x(
i + k)
x(
i + l)
= mx
( k,
l)
= cx
( k,
l)
, (1.1.7b)
( 4 )
( 4)
( 2)
( 2)
x(
i)
x(
i + k)
x(
i + l)
x(
i + m)
= mx
( k,
l,
m)
= cx
( k,
l,
m)
+ cx
( k)
cx
( m − l)
+
, (1.1.7c)
( 2)
( 2)
( 2)
( 2)
+ cx
( k)
cx
( m − k)
+ cx
( m)
cx
( l − k)
where k, l and m are the shift indices.
The formula (1.1.7a) describes the relationships between the second-order statistics and it de-
fines conventional autocorrelation function. Note that second-order moment and cumulant functions
are equal to each other.
The formula (1.1.7b) describes the relationships between the third-order statistics and it defines
triple autocorrelation function. It should be noted that the third-order moments and cumulants are
equal to each other.
According to the formula (1.1.7b) defining the relationship between the fourth-order statistics,
the fourth-order moment function is not equal to the fourth-order cumulant function.
Spectral density of r-th order (polispectrum or cumulant spectrum) Сx(ω1, ω2,…, ωr-1) of a ran-
dom process {x(i)} can be defined by the following multidimensional Fourier transform of the r-th or-
( r)
x ( 1 2 r −1
der cumulant c τ , τ ,..., τ ) as
+∞ +∞
( r)
( r)
Cx
( ω1,
ω2,
..., ωr
− 1,
) = ∑ ... ∑cx
( τ1, τ 2,...,
τ r −1)
exp[ − j(
ω1τ
+ ω2τ
+ ... + ω 1 )]
1 2 r − τ . (1.1.8)
r − 1
τ = −∞ τ = −∞
1 r −1
The generalized formula (1.1.8) permits to define the energy spectrum Рx(ω) (r = 2), the bispec-
trum Вx(ω1, ω2) (r = 3) and the trispectrum Тx(ω1, ω2, ω3) (r = 4), respectively, in the form
+∞
(2)
Рx(ω) = ∑ cx ( l)exp[ − j( ωl)]
, (1.1.9а)
l=−∞
5
2

+∞
+∞
∑ ∑
( 3)
Вx(ω1, ω2) = c l , l ) exp[ − j(
ω l + ω l )] , (1.1.9b)
x
l1= −∞ l2=
−∞
+∞
+∞
∑ ∑ ∑
( 1 2
1 1 2 2
+∞
( 4)
Тx(ω1, ω2, ω3) = c l , l , l ) exp[ − j(
ω l + ω l + ω l )] . (1.1.9c)
x
l1 = −∞ l2 = −∞ l3
= −∞
6
( 1 2 3
1 1 2 2 3 3
The equations (1.1.9a-b) contain the cumulant functions the properties of which are interesting
and worth considering in detail.
1) If xi, i = 1, 2,…, K is a sequence of random variables and αi = 1, 2,…, K are some constant val-
ues, then
2) Permutation property for random variables
K ⎛ ⎞
с(α1x1, α2x2,…,αКxК) = ⎜
⎜∏=
α i ⎟
⎟c(
x1,
x2,...,
xK
) . (1.1.10)
⎝ i 1 ⎠
where (i1, i2,…, iK) is the index permutation (1, 2,…,K).
с(x1, x2,…xК) = с(xi1, xi2,…,xiK), (1.1.11)
3) Additivity property of the cumulants in their arguments
с(x + y, z1, z2,…, zК) = с(x, z1, z2,…, zК) + с(y, z1, z2,…, zК), (1.1.12)
which signifies that the cumulant of the sum of arguments is equal to the sum of the cumulants of
the separate arguments.
4) If α is a constant value then
с(α + x1, x2,…, xК) = с(x1, x2,…, xК). (1.1.13)
5) In the case when the random variables xi, i = 1, 2,…, K and yi, i = 1, 2,…,K are independent, we
have
с(x1 + y1 , x2 + y2,…, xК+ yK) = с(x1, x2,…, xК) + с(y1, y2,…, yK). (1.1.14)
Assume that an observed process is z(i) = x(i) + n(i), i = 1, 2,…, K, and x(i) and n(i) are inde-
pendent processes. According to the property (1.1.14), one can obtain
(K )
(K )
(K )
c z (l1, l2,…, lК-1) = c x (l1, l2,…, lК-1) + c n (l1, l2,…, lК-1). (1.1.15)
If one of the process, for example, n(i) is Gaussian, then under condition of К ≥ 3,
lК-1) = 0 we obtain
(K )
c n (l1, l2,…,

(K )
(K )
c z (l1, l2,…, lК-1) = c x (l1, l2,…, lК-1). (1.1.16)
The last expression (1.1.16) demonstrates important insensitivity property to the additive Gaus-
sian noise for the cumulants the order of which is equal or more than three. From the practical point of
view of signal processing in additive Gaussian noise environment, cumulant estimates permit to sepa-
rate non-Gaussian signal from additive Gaussian noise and, hence, to increase SNR ratio.
Below we will pay attention to the third-order statistics for their application in digital signal and
image processing. For this purpose, first, we will consider the main properties of triple correlation and
bispectrum and techniques used for their estimation.
1.2. Triple correlation functions and bispectra
One of the main motivations for using bispectrum analysis in signal processing application is the
following. Bispectrum density estimate or third-order cumulant spectrum estimate, in opposite to the
energy spectrum estimate, allow not only to describe statistical characteristics of an observed process
more correctly but also to define the presence of spectral component correlation relationships and to
reconstruct phase relationships existing between the Fourier spectrum components. Therefore, the
main difference of bispectrum from energy spectrum is in preservation of phase information and pos-
sibility of this information reconstruction. Already only this promising peculiarity of bispectrum con-
tributed to wide usage of the bispectrum analysis and estimation techniques in digital signal
processing. Permanent growing of the interest in bispectrum analysis is accompanied by appearance of
a great number of papers. In this connection, let us mention the fundamental review papers [1–3] and
the paper [4] containing a classification list of more than 1700 papers dedicated to bispectrum analysis
in different applications.
Consider the advantages of bispectrum analysis comparing to energy spectrum analysis more in
detail. One of the most promising bispectrum property usually used for extraction a signal from Gaus-
sian noise in digital measurement systems is the tendency to zero the bispectrum of an interference
that has a symmetrical probability density function (PDF). This property provides robustness of the
bispectrum-based filtering techniques regarding to additive Gaussian interference in radar [5–8], as-
tronomy [9–11], underwater acoustic [12–14], and biomedical [15, 16] signal processing systems as
well as in optical [17] and digital [18–20] image processing systems.
Bispectrum analysis can serve as a sensitive and precise tool permitting to define and measure
the deviation of the observed process from Gaussian distribution, i.e., to estimate non-Gaussianity.
7

This property seems to be very useful in noisy-like process in machine diagnostics systems [21], un-
derwater acoustic systems [12], nondestructive monitoring [22], and biomedical diagnostics [16].
Let us consider the bispectrum properties for a real-valued stationary discrete process {x (m) (i)}
given by finite sample number i =0, 1, 2,…,I–1 in temporal or spatial domain and observed with a fi-
nite set of m=1, 2,…, M independent realizations x (m) (i).
Conventional autocorrelation function Rx(k) that is second-order statistic and a function of one
variable can be written as
I −1
( m)
( m)
Rx
( k)
= ∑ [ x ( i)
− E][
x ( i + k)
− E]
, (1.2.1)
i=
0
where k= –I+1,…,I–1 is the temporal or spatial shift index; ∞ denotes ensemble averaging for
infinite realization number, i.e. for M→ ∞;
I −1
∑
2
( m)
2
R ( 0)
= σ = [ x ( i)
− E]
is the variance.
x
x
i=
0
∞
8
∞
−1
1 ( )
= ∑ ( )
I
m
E x i is the mean value;
I
Autocorrelation function Rx(k) (1.2.1) possesses the well-known symmetry property
Rx x
i=
0
( k)
= R ( −k)
. (1.2.2)
According to the Wiener-Khinchin theorem, the spectral density Px(p) is defined in the form of
the following discrete direct Fourier transform (DFT)
or by
k
= ∑ +∞ =
P ( p)
R ( k)
exp( − j2πkp)
, (1.2.3)
x
k = −∞
x
( ) * ( m)
P ( p)
= Ẋ
m
( p)
Ẋ
( p)
, (1.2.4)
x
( )
( )
where p=–I+1,…,I+1 is the frequency sample index; Ẋ
m
m
( p)
x ( i)
exp( − j2πip)
is the discrete
∞
I 1
= ∑ −
Fourier transform for m-th realization; * denotes complex conjugation.
It should be stressed one more time that Fourier phase spectrum information is lost in the spec-
tral density (1.2.4) due to multiplication of the complex conjugated functions.
Triple autocorrelation function (TAF) Rx(k,l) that is third-order statistic and a function of two va-
riables can be represented in the form of
I −1
( m)
( m)
( m)
Rx
( k,
l)
= ∑ [ x ( i)
− E][
x ( i + k)
− E][
x ( i + l)
− E]
, (1.2.5)
i=
0
i=
0
∞
∞

where k= –I+1,…,I–1 and l= –I+1,…,I–1 are the independent shift indices.
Note that TAF (1.2.5) possesses the following symmetry property [2] that is very useful for sig-
nal processing practice
Rx x
x
x
x
( k,
l)
= R ( l,
k)
= R ( l − k,
−k)
= R ( k − l,
−l)
= R ( −k,
l − k)
. (1.2.6)
According the definitions given in [1, 2], bispectrum is the DFT of TAF. Unlike the real-valued
spectral density (1.2.3) and (1.2.4), bispectrum (or bispectrum density) is the complex-valued function
Ḃ
x ( p,
q ) of two independent frequencies p and q that can written as the following 2-D discrete DFT
of TAF (1.2.5)
or as
I −1
I −1
Ḃ
( p,
q)
R ( k,
l)
exp[ − j2π
( kp + lq)]
, (1.2.7а)
x
= ∑ ∑
x
k = −I
+ 1 l=
−I
+ 1
( ) ( ) * ( m)
( ) ( ) ( )
Ḃ
( p,
q)
= Ẋ
m
( p)
Ẋ
m
( q)
Ẋ
( p + q)
= Ẋ
m
( p)
Ẋ
m m
( q)
Ẋ
( − p − q)
, (2.1.7b)
x
where ̇ ( p,
q)
= Ḃ
( p,
q)
exp[ jγ
( p,
q)]
; ̇ ( p,
q)
and γ ( p,
q)
are the magnitude bispectrum
Bx x
x
B x
(bimagnitude) and phase bispectrum (biphase), respectively p= –I+1,…,I–1 and q= –I+1,…,I–1 are
the frequency indices.
Comparing the spectral (1.2.4) and bispectral (1.2.7b) densities allows noting that spectral densi-
ty Px(p) is the ensemble averaging performed for the multiplication of two complex conjugated func-
tions corresponding to the same frequency value p and bispectral density Ḃ
x ( p,
q ) is the ensemble
averaging carried out by multiplication of three complex-valued functions corresponding to different
frequency values p, q and –p–q.
mean
Illustrations and comments of the main properties of bispectrum are listed below.
1. TAF and bispectrum are zero-valued functions for stationary Gaussian process with zero
9
∞
x
Rx(m,n)=0, ̇ ( p,
q)
= 0 . (1.2.8)
B x
2. TAF and bispectrum are equal to zero for deterministic signals having zero asymmetry. For
example, TAF and bispectrum are of zero values for a simple harmonic oscillation x( i)
= A0
cos( 2πfi)
(f denotes the frequency in this example). However, when a little signal nonlinear distortions appear or
when a constant component is observed in the signal, TAF and bispectrum become non-zero-valued
functions. This property can serve as a very sensitive tool for detection and measurement of signal
nonlinear distortions.
∞

3. Bispectrum is the periodical function with period equal to 2π
̇ ( p,
q)
= Ḃ
( p + 2π
, q + 2π
) . (1.2.9)
B x
4. Bispectral density has the following symmetry property [2]
*
*
Ḃ
( p,
q)
= Ḃ
( q,
p)
= Ḃ
( − p,
−q)
= Ḃ
( −q,
− p)
=
. (1.2.10)
= Ḃ
( − p − q,
q)
= Ḃ
( p,
− p − q)
= Ḃ
( − p − q,
p)
= Ḃ
( q,
− p − q)
Symmetry relationships (1.2.10) define bispectrum in the limits given by a hexagon in the bifre-
quency domain.
Analysis of the expressions (1.2.10) shows that bispectrum of a real-valued process can be de-
fined completely just only within the limits of the main triangular domain given by the inequalities
q≥0, p≥q, p+q≤I–1. (1.2.11)
It is sufficiently to use the symmetry relationships (1.2.10) and (1.2.11) for computation of bis-
pectrum values in all other parts of hexagonal domain. Note that the conditions (1.2.11) limiting the
total bispectrum sample number by the main triangular domain allow essential decreasing the re-
quirements to the PC memory and reducing the processing time in practical computations.
5. Unlike power spectral density containing only information about signal magnitude Fourier
spectrum, the bispectral density permits to preserve both information about magnitude and phase
Fourier spectra. This property follows from the equations (1.2.7a) and (1.2.7b).
6. Important invariance property of bispectrum to a signal temporal (or spatial) shift follows
from (1.2.7b). This property can be explained with the following formula
Ḃ
( p,
q)
Ẋ
( p)
Ẋ
( q)
Ẋ
x τ = τ τ τ ( − p − q)
=
, (1.2.12)
= Ẋ
( p)
Ẋ
( q)
Ẋ
( − p − q)
exp( − j2πτp)
exp( − j2πτq)
exp[ − j2πτ
( − p − q)]
= Ḃ
( p,
q)
where X ( p)
= X ( p)
exp( − j2πτ
)
̇ ̇ is the Fourier transform of a process shifted by a τ value in tempor-
τ
al domain. It follows from the expression (1.2.12) that both for the original process x (m) (i) and for its
copy x (m) (i-τ) shifted by τ, the corresponding bispectra are congruent. This important property seems
to be useful for solving the jittered signal reconstruction problems. At the same time, however, this
bispectrum invariance property (1.2.12) can cause the problems in some bispectrum-based signal
processing algorithms. Particularly, undesirable image row shifts can arise with using bispectrum-
based image reconstruction techniques (these techniques will be considered below in detail).
7. Very often in practice one can study behavior of a wideband process passing through a device
having square-law characteristic. In this case, sum or difference frequency components can appear at
the non-linear device output. Quadratic phase coupling phenomenon can be present at the non-linear
10
x

device output. It is necessary in many practically important cases to detect and estimate just those
spectral components that contain quadratic coupling. Since the phase relationships are lost in power
spectral density, it is impossible to extract information about phase coupling from power spectral den-
sity. Meanwhile, it is possible to do that using bispectrum.
This property can be explained by using the following example. Let us consider two processes
x i)
= cos( 2π
f i + ϕ ) + cos( 2πf
i + ϕ ) + cos( 2πf
i + ϕ ) , (1.2.13а)
1(
1 1
2 2
3 3
x i)
= cos( 2π
f i + ϕ ) + cos( 2πf
i + ϕ ) + cos[ 2πf
i + ( ϕ + ϕ )]
(1.2.13b)
2 ( 1 1
2 2
3 1 2
where f3=f1+f2 are the frequencies and φ1, φ2, φ3 are the initial phases that are supposed to be indepen-
dent random values having uniform distribution law and that change within the limits of [0, 2π].
Assume that that the frequency f3 in (1.2.13а) is an independent component and an initial phase
φ3 corresponding to this frequency is an independent random value. Assume also that the frequency f3
in (1.2.13b) is the result of quadratic phase coupling. It is evident that the power spectrum densities
for these two considered processes are of the same value, i.e. Px1(p)= Px2(p). At the same time, magni-
tude bispectrum of the process (1.2.13а) tends to be zero, i.e. ̇ ( p,
q)
= 0 , and the magnitude bis-
pectrum for the process (1.2.13b) is of non-zero value, i.e. ̇ ( p,
q)
≠ 0 . Therefore, bispectrum can
serve as very sensitive indicator for detection of phase coupling. This peculiarity will be considered
below in detail for application in ground surveillance radars.
After brief consideration of the main properties of TAF and bispectrum that will be used in the
next chapters, we pass to the description of the modern bispectral density estimation techniques.
1.3. Bispectral density estimation techniques
Note that the existing bispectrum-based approaches used in digital signal processing can be di-
vided into non-parametric and parametric, as well as into direct and indirect techniques [2, 23]. The
main attention in this work will be paid to non-parametric bispectrum estimation and some our results
11
Bx 2
Bx1 concerning parametric bispectral estimation will be represented below.
First, we start with consideration of non-parametric indirect bispectral density estimation tech-
nique [2] containing the following sequence of data processing procedures.
1. Computation of a set of M sample TAF estimates
ˆ ( m)
( , )
R k l for each m-th observed realiza-
tion given by a sequence of real-valued samples {x (m) (0), x (m) (1), x (m) (2),…, x (m) (I–1)}, m=1, 2,…,M
observed at the input of measurement system as

Rˆ
( m)
x
( k,
l)
I 1
= ∑ −
i=
0
x
( m)
( i)
x
12
( m)
( i + k)
x
( m)
( i + l)
. (1.3.1)
It is supposed that a considered process in (1.3.1) {x (m) (i)} has zero-mean, i.e. Ex = 0.
2. Ensemble averaging the sample estimates (1.3.1) performed for M realizations to obtain the
smoothed TAF estimate Rˆ M ( k,
l)
in the form of
m
Rˆ
1 ( )
M ( k,
l)
= Rˆ
x ( k,
l)
. (1.3.2)
M
M
∑
m=
1
Note that for an ergodic process the ensemble averaging procedure (1.3.2) can be replaced by
the averaging procedure performed in temporal domain, i.e. by dividing the observed sequence {x (m)
(i)} into M separate segments.
3. Bispectrum density computation in the form of a complex-valued array
jγˆ
( p,
q)
B
ˆ̇
( , )
ˆ
x ind
p q = Ḃ
( p,
q)
e
by using direct Fourier transform of Rˆ ( k,
l)
x ind
x ind
M (1.3.2) as
I 1 I 1 − −
Ḃ
ˆ
ˆ
x ind ( p,
q)
= ∑∑ RM
( k,
l)
W ( k,
l)
exp[ − j2π
( kp + lq)]
, (1.3.3)
k = 0 l=
0
where Ḃ ˆ
( p,
q)
and γ ˆ ( p,
q)
are the magnitude and phase bispectral estimates, respectively;
x ind
x ind
W(k,l) is the weighting window function that is desirable for improving the accuracy in bispectral es-
timate. A weighting window function optimization is the separate problem considered, for example, in
[24, 25] and the readers interested in more detailed information can be addressed to these papers.
Non-parametric direct bispectral density estimation technique comparing to the indirect one has
better processing speed, first, due to using FFT algorithm and, second, due to excluding the computa-
tionally intensive procedure necessary for computation of the TAF (1.3.1). The direct technique con-
tains the following set of processing procedures.
1. Direct Fourier transform performed for each realization {x (m) (i)} of the observed process
I 1
= ∑ −
( )
( )
Ẋ
m
m
( p)
x ( i)
exp( − j2πip)
, p=0, 1, 2,…,I–1. (1.3.4)
i=
0
2. Computation of the m-th sample bispectrum estimate by using the following triple product of
complex-valued functions (1.3.4) for the frequencies p, q and p+q as
ˆ )
x dir
( )
( ) ( ) *(
Ḃ
m
m
m
m
( p,
q)
= Ẋ
( p)
Ẋ
( q)
Ẋ
( p + q)
, m=1,2,…,M. (1.3.5)
3. Averaging the sample estimates (1.3.5) for the ensemble of M realizations to obtain smoothed
estimate ˆ
Ḃ ( p, q)
in the form of
x dir

m
Ḃ
ˆ 1
( )
x dir ( p,
q)
= Ḃ
ˆ
x dir ( p,
q)
. (1.3.6)
M
13
M
∑
m=
1
It should be stressed that bispectral density estimates formed by indirect and direct techniques
differ from each other. These estimates are congruent in the case of W(k,l) = 1 in (1.3.3).
It was shown in [26] by Brillinger that for unlimited sample number I of an input sequence or
for an unlimited realization number M, bispectrum estimates obtained by indirect and direct tech-
niques converge in average to the true bispectral density, i.e. these estimates are asymptotically un-
biased and consistent
̇ ˆ
( p,
q)
= Ḃ
ˆ
( p,
q)
≅ Ḃ
( p,
q),
I,
M → ∞ . (1.3.7)
Bx dir
x ind
x
The following asymptotical expressions are represented in [2] for bispectrum density estimate
variances for indirect and direct techniques, respectively
and
ˆ
ˆ
V
var{Re Ḃ x ind ( p,
q)}
= var{Im Ḃ
x ind ( p,
q)}
≅
P(
p)
P(
q)
P(
p + q)
(1.3.8а)
2
( 2L
+ 1)
M
1
var{Re Ḃ ˆ
ˆ
x dir ( p,
q)}
= var{ImḂ
x dir ( p,
q)}
≅ P(
p)
P(
q)
P(
p + q)
, (1.3.8b)
M
where real and imaginary bispectrum density parts are denoted by Re and Im, respectively;
V
=
L
∑
L
∑
k = −L
l=
−L
2
W ( k,
l)
is the energy parameter corresponding to the weighting window; L =I–1 is the
weighting window length; P(…) is the process power density; var{…} is the estimate variance.
Note that for Dirichlet window W(k,l)=1, and V/(2L+1) 2 =1 in (1.3.8a). Because of this, the va-
riances (1.3.8a) and (1.3.8b) become the same.
It should be especially noted that according to the asymptotic expressions (1.3.8a) and (1.3.8b) it
is necessary to observe a very large number of process realizations M for obtaining an unbiased bis-
pectrum density estimate with small estimated variance.
However, in signal processing practice, a number M is not too large. Therefore, one of the most
important problems is improving the bispectrum density estimates obtained for a limited realization
number in noise environment.
Some new approaches to solving this problem will be considered and the obtained results will
be discussed below.

1.4. Bispectrum-based algorithms in filtering and signal reconstruction applications
A typical problem arising in aforementioned signal processing systems is the process estimation
problem or filtering problem that comes to the reconstruction of unknown signal shape (waveform) at
the filter output in interference environment with high accuracy in statistical sense.
Attraction of the bispectrum-based techniques in application to filtering and signal reconstruc-
tion problems is, first of all, in the high accuracy of non-Gaussian signal shape reconstruction in addi-
tive Gaussian and, in general, symmetric PDF noise environment for low input SNR.
In this subsection, we pay attention to conventional bispectrum-based algorithms used for signal
reconstruction from additive mixture of information signal and noise.
We will study the following typical in practice signal and noise model observed at the digital fil-
tering system input. Let us consider a real-valued deterministic 1-D signal s(i) given by an uniform set
of samples, i=0,1,…,I-1. Assume that M realizations of the randomly shifted signal are observed.
Thus, each m-th (m=1, 2,…, M) realization x (m) (i) could be presented as
x (m) (i) = s(i – τ (m) ) + n (m) (i), (1.4.1)
where n (m) (i) is the m-th realization of the stationary additive white Gaussian noise (AWGN) with zero
mean value and sample variance σ (m)2 ; τ (m) is the random integer-valued shift of the deterministic
signal s(i). AWGN is assumed to be uncorrelated with the original signal s(i). The signal waveform is
supposed to be unchanged for all realizations. The signal's TAF is assumed to be a priori non-zero.
The observation model (1.4.1) is rather typical for many applications [5,14,27,28,29]. In
particular, x (m) (i) can describe the noisy output of a high resolution radar system that forms a target
range profile s(i). The temporal shifts τ (m) can arise in practice due to the target’s random motion from
one scan to another or random vibration of radar platform. Statistical characteristics of τ (m) (PDF,
variance, etc.) depend upon many physical factors. They are a priori unknown and different for vari-
ous practical applications. However, it is realistic to observe deviations of τ (m) comparable to the total
available signal length. Similar properties of signal and noise can be observed for an active sonar
operating in the pulse mode.
According to the observation equation (1.4.1), filtering problem is formulated as follows: to es-
timate a priori unknown signal shape s(i) in AWGN environment. The conventional bispectrum-based
approach to signal shape reconstruction from noisy mutually shifted realizations [9, 10] includes the
following three main stages:
1) obtaining an estimate of signal bispectrum (bispectral density);
14

2) recovery of the signal Fourier spectrum (magnitude and phase spectra) from the signal bis-
pectrum estimate;
3) signal shape reconstruction from the obtained estimate of the signal Fourier spectrum by in-
verse Fourier transform.
Existing approaches are based on the following pair of equations that relate the signal bispectrum
Ḃ s ( p, q) = Ḃ s( p, q) exp[ jγ ( p, q)]
and the signal Fourier spectrum Ṡ ( p) = Ṡ ( p) exp[ jϕ( p)]
that was used first for 1-D image reconstruction in astronomy [9, 10]
*
B ̇
s ( p,
q ) = Ṡ
( p ) Ṡ
( q ) Ṡ
( p + q ) , (1.4.2)
γ s ( p,
q ) = ϕ(
p ) + ϕ(
q ) − ϕ(
p + q ) , [-π, π] (1.4.3)
where φ(…) denotes signal phase Fourier spectrum.
Note that equations (1.4.2) and (1.4.3) are true for Hermitian conjugated real-valued signal
*
Fourier spectrum, i.e. for S( p ) = S ( − p ) ̇ ̇ and φ(p) = – φ(–p).
A recursive algorithm for signal Fourier spectrum reconstruction by using bispectral density es-
timation [10] is based on the equations (1.4.2) and (1.4.3) and it can be represented in the form of
ˆ ϕ ( + q)
= ˆ ϕ(
p)
+ ˆ ϕ(
q)
− ˆ γ ( p,
q),
p = 0,...,
I −1;
0 ≤ q ≤ p;
p + q ≤ I −1
, (1.4.4)
p s
Ḃ
ˆ
s ( p,
q)
ˆ
Ṡ
( p + q)
= , p = 0,...,
I −1;
0 ≤ q ≤ p;
p + q ≤ I −1
, (1.4.5)
ˆ ˆ
Ṡ
( p)
Ṡ
( q)
where Ḃ ˆ
s ( p,
q)
and γ ˆ s ( p,
q)
are the sampled estimates of the magnitude Ḃ s ( p,
q)
and phase
γ s ( p,
q)
bispectra, respectively, computed for a finite number of realizations; S(...) ˆ̇ and ˆϕ (...) are
the signal magnitude and phase Fourier spectrum estimates reconstructed from the magnitude and
phase bispectrum estimates, respectively.
The following sequence of processing steps is used in [10] for recursive computation of signal
phase and magnitude Fourier spectrum estimates recovered from phase and magnitude bispectrum es-
timates
15

S
ˆ̇
( 0)
=
ˆ ϕ(
0)
=
ˆ ϕ(
1)
= ˆ ϕ(
1)
+ ˆ ϕ(
0)
− ˆ γ ( 1,
0),
ˆ ϕ(
2)
= 2 ˆ ϕ(
1)
− ˆ γ ( 1,
1),
ˆ ϕ(
3)
= ˆ ϕ(
2)
+ ˆ ϕ(
1)
− ˆ γ ( 2,
1),
ˆ ϕ(
4)
= 2 ˆ ϕ(
2)
− ˆ γ ( 2,
2),
ˆ ϕ(
4)
= ˆ ϕ(
3)
+ ˆ ϕ(
1)
− ˆ γ ( 3,
1),
ˆ ϕ(
5)
= ˆ ϕ(
3)
+ ˆ ϕ(
2)
− ˆ γ
ˆ ϕ(
6)
= ˆ ϕ(
3)
+ ˆ ϕ(
3)
− ˆ γ ( 3,
3),
ˆ ϕ(
5)
= ˆ ϕ(
4)
+ ˆ ϕ(
1)
− ˆ γ ( 4,
1),
.......... .......... .......... .......... ...,
ˆ ϕ(
3
S
ˆ̇
( 4)
= B
ˆ̇
ˆ
Ṡ
( 5)
= B
ˆ̇
0,
S
ˆ̇
( 1)
= B
ˆ̇
( 1,
0)
ˆ
Ṡ
( 2)
= B
ˆ̇
( 1,
1)
S
ˆ̇
( 3)
= B
ˆ̇
( 2,
1)
S
ˆ̇
( 3)
= B
ˆ̇
( 3,
0)
S
ˆ̇
( 4)
= B
ˆ̇
( 3,
1)
S
ˆ̇
( 6)
= B
ˆ̇
( 3,
3)
−1)
= ˆ ϕ(
I −1)
+ ˆ ϕ(
0)
− ˆ γ ( I −1,
0)
16
( 3,
2),
I s
B
ˆ̇
s
s
s
s
s
s
s
s
s
( 0,
0)
,
( 2,
2)
( 3,
2)
s
( S
ˆ̇
( 1)
S
ˆ̇
( 0)
),
s
( S
ˆ̇
( 2)
S
ˆ̇
( 1)
),
( S
ˆ̇
( 3)
S
ˆ̇
( 1)
),
s
ˆ 2
( Ṡ
( 1)
) ,
ˆ 2
( Ṡ
( 2)
) ,
( S
ˆ̇
( 3)
S
ˆ̇
( 0)
),
ˆ ˆ
( Ṡ
( 3)
Ṡ
( 2)
),
ˆ 2
( Ṡ
( 3)
) ,
s
s
s
s
s
s
, (1.4.6)
.......... .......... .......... .......... .......,
. (1.4.7)
ˆ
ˆ ˆ
Ṡ
( I −1)
= B
ˆ̇
( I −1,
0)
( Ṡ
( I −1)
Ṡ
( 0)
)
We would like to emphasize the following important peculiarities of the recursive algorithms
(1.4.6) and (1.4.7).
1) It is seen from (1.4.6) that the first phase sample ˆϕ ( 1)
is undetermined. This peculiarity re-
sults from the above considered bispectrum invariance property to signal translation (1.2.12). Due to
this uncertainty, the first phase sample value is usually supposed to be set ˆϕ ( 1)
=0. Neglecting the li-

near phase caused by original signal shift leads to phase error just in the first step of recursive
processing. This phase error can be substantial or non-substantial in different signal processing appli-
cations and this peculiarity will be discussed later. Now we only note that if the value ˆϕ ( 1)
differs by
2π, phase distortions do not appear in the reconstructed signal. Otherwise, systematic phase error
caused by arbitrary choise of ˆϕ ( 1)
=0 is accumulated step by step in recursive procedure.
2) The sample values of ˆ γ ( p,
q)
computed usually by using standard softwares within the
s
limits of the principal value in arc tangent function as ˆ γ ( , q)
= Arc tan{Im Ḃ
ˆ
( p,
q)
/ Re B
ˆ̇
( p,
q)}
17
s
p s
s
are of ambiguous values in the sense that any quantity multipled by 2π added to ˆ γ ( p,
q)
in an
arbitrary sample does not change its value. Therefore, bispectrum phase computations are
accompained by phase wrapping (or phase function discontinuity) and, hence, computed phase
bispectrum differs from true function that has continious behavior.
3) Due to the bispectrum symmetry property (1.2.10), only the samples located within the limits
of the principal triangular domain (see inequalities (1.2.11)) are necessary for signal Fourier spectrum
reconstruction.
4) Each signal Fourier phase (1.4.6) and each magnitude (1.4.7) sample is represented in the
recursive steps by (p –1)/2 times for odd p and p/2 times for even p. This redundancy in some cases
can provide improving bispectrum estimate by exploiting the averaging of the corresponding redun-
dant samples. Note that in order to avoid the accumulation of phase wrapping errors, it is reasonable
to perform averaging of the values in the form of exp[jφ(p)]. After performing the averaging, the
phase and magnitude values have to be placed in the consequent recursive step.
5) Since correlation coefficients between the mentioned redundant samples are not equal to zero
values and in each concrete case their values depend upon signal and noise characteristics, the effi-
ciency of noise smoothing by using redundant data averaging depends upon their correlatedness.
6) Bispectrum estimate improving by averaging the redundant data differs in different parts of
principal triangular domain in bifrequency plane. For low frequencies (for small p and q), noise
smoothing is worse comparing to high frequencies because for larger p and q more averaged values
participate in averaging procedure.
In order to be able to recover a signal waveform from its bispectrum with maximal accuracy, it
is necessary to obtain such bispectrum estimate that is as close as possible to the true signal bispec-
trum. In other words, the accuracy of bispectrum-based signal waveform reconstruction largely de-
pends on the accuracy of the bispectrum estimate.
Because efficiency of bispectrum-based signal waveform reconstruction algorithms directly de-
s

pends upon quality of bispectrum estimate, it is a very important problem to analyze bispectrum esti-
mation accuracy. Theoretically, it has been demonstrated in [2] that under the assumption that appro-
priate windowing is accomplished, the conventional direct bispectrum estimate (1.3.6) is unbiased and
consistent asymptotically, i.e. for M → ∞ .
However, in the majority of practical applications we can only operate with short data blocks (I
is usually not more than several hundred samples) and the number of available realizations M is essen-
tially limited by the restricted time interval given for data measurements. It should be especially noted
that in many important cases, for example, for bispectrum-based signal processing in radar applica-
tions, low input SNRs could cause an intolerably large variance in bispectrum estimate in respect to
what is required for satisfactory performance of a signal reconstruction system.
Thus, one of the aims of this subsection is the study of the performance of bispectral estimator
in typical practical signal processing situations with short data blocks, limited sample size and low
input SNRs.
According to the observation equation (1.4.1) and direct bispectrum estimation technique (see
the formulas (1.3.4 – 6)), noisy bispectrum estimate can be written as [30]
ˆ * * j2 m( p q) *
j2 mq
Ḃ x( p, q) Ṡ ( p) Ṡ ( q) Ṡ ( p q) Ṡ ( p) Ṡ ( q) E[ Ṅ − πτ +
m( p q) e ] Ṡ ( p) Ṡ πτ
( p q) E[ Ṅ m(
q) e ]
= + + + + + +
+ ̇ ̇ + ̇ + ̇ ̇ ̇ + + ̇ ̇ ̇ + + , (1.4.8)
* j2πτ mp S( q) S ( p q) E[ Nm( p) e ]
* − j2πτ mp S( p) E[ Nm( q) Nm( p q) e ]
*
− j2πτ mq
S( q) E[ Nm( p) Nm( p q) e ]
*
Ṡ j2 m(
p q)
*
( p q) E[
Ṅ m( p) Ṅ πτ +
m( q) e ] + E[ Ṅ m( p) Ṅ m( q) Ṅ m( p+ q)] = Ḃ s( p, q) + Ḃ err ( p, q)
+ +
where ∑ −
2πip
I 1 − j
N ( p)
= n ( i)
e I
m
m
i = 0
̇ is DFT of the m-th noise realization; ˆ *
Ḃ s ( p,
q)
= Ṡ
( p)
Ṡ
( q)
Ṡ
( p+
q)
and
ˆ̇
( p,
q)
are the true signal bispectrum and error (noise and random shift induced) component
B err
distorting the true signal bispectrum, respectively; S(p) ̇ is the complex-valued Fourier spectrum of
true signal s(i).
Analyzing the formula (1.4.8), we see that:
- the first three terms of the error component ˆ̇
( p,
q)
, in fact, tend to zero asymptotically, i.e.
for sample size М→∞, under assumption that AWGN {nm(i)} in (1.4.1) has zero mean;
- the last term of the error component ˆ̇
( p,
q)
is the third-order statistic that also
18
B err
B err
asymtotically (for М→∞) tends to zero because additive noise is assumed to have a
symmetrical PDF;

- the forth, fifth, and sixth terms of the error component ˆ̇
( p,
q)
depend multiplicatively on
the true signal Fourier spectrum S(p) ̇ and these terms mainly cause the bias and contribute
considerably to the total variance of the bispectrum estimator.
TAF estimate obtained by using indirect technique (see the formulas (1.3.1 – 3)) can be
represented in the form of
Rˆ M(
k, l) = ˆ(
m) R ( k, l) ( m)
= R ( k, l) + n ( i) [ R ( k) + R ( l)
+
s s s
, (1.4.9)
ˆ(2) ˆ(2) ˆ(2) + R ( k + l)] + s ( i)[ R ( k) + R ( l) + R ( k + l)] + Rˆ ( k, l)
s
M nM nM nM nM
where Rs ( k,
l)
is the TAF of the original signal s(i); R (...) is the signal autoccorrelation function;
s
ˆ ( 2)
R n M (...) is the noise autocorrelation estimate; s ( i ) is the signal mean value; ˆ ( m)
R ( k,
l)
M n M is the
noise TAF estimate.
Analysis of (1.4.9) shows that the estimate Rˆ M ( k,
l)
is equal to signal TAF Rs ( k,
l)
if and only
if the three following conditions are satisfied simultaneously
( )
n m
( i)
=0; s ( i ) =0; R ( k,
l ) ˆ n M =0. (1.4.10)
It has been demonstrated in [29] that the estimates (1.4.8) and (1.4.9) are asymptotically un-
biased and consistent within the principal triangular symmetry domain (1.2.11) if and only if
19
B err
S( 0)
= s(
i)
= 0
̇ . (1.4.11)
Therefore, the necessary condition for obtaining unbiased bispectrum estimate is that a signal
mean is zero according to (1.4.11). It means that the mean value of the true signal s(i) in (1.4.1) should
be zero.
Moreover, according to [29], even if the signal has a nonzero mean, the bispectrum estimate can
be asymptotically unbiased for all frequencies except the bispectral samples belonging to the frequen-
cy axes and diagonal in the principal triangular domain in bifrequency plane (p, q):
p =0 or q =0 or p+q = 0 or p=q or p+q=I/2. (1.4.12)
However, in our opinion, this statement is correct only asymptotically, i.e. for М→∞ and for in-
put SNR>>1. Indeed, in the majority of the papers dedicated to the problems of bispectrum-based de-
terministic signal reconstruction the latter assumption SNR>>1 is often used. Nevertheless, in practice
this assumption is not always true. Thus, the task of quantitative investigation of the accuracy of the

ispectrum estimator in processing of zero-mean and nonzero-mean deterministic signals under low
input SNRs, limited sample size M and short data blocks is of paramount interest and importance.
By using the definition of the variance var( ż ) of complex random variables [31]
var( ż ) = E(
ż
− E(
ż
) ) , (1.4.13a)
where ż is an arbitrary complex random variable, a formula for the variance of the bispectrum esti-
mate (1.4.8) has been obtained in [30] as
var{ B
ˆ̇
( p,
q)}
= E{
B
ˆ̇
( p,
q)
− E[
B
ˆ̇
( p,
q)]
} =
x
2
2
= E{Re
B
ˆ̇
( p,
q)
− E{Re
B
ˆ̇
( p,
q)}}
+ E{Im
B
ˆ̇
( p,
q)
− E{Im
B
ˆ̇
( p,
q)}}
20
2
2
= E{
Re B
ˆ̇
( p,
q)
− E{Re
B
ˆ̇
( p,
q)}
+ j{Im
B
ˆ̇
( p,
q)
− E{Im
B
ˆ̇
( p,
q)}}
} = , (1.4.13b)
x
x
x
x
x
x
where Re{…} and Im{…} are the real and imaginary parts of bispectrum estimate, respectively; E de-
notes expectation procedure.
It is seen from (1.4.13) that overall variance of bispectrum estimate is
x
x
var{
ˆ̇
( p,
q)}
= var{Re{ B
ˆ̇
( p,
q)}}
+ var{Im{ Ḃ
ˆ
( p,
q)}}
. (1.4.14)
Bx x
x
Thus, the total variance of bispectrum estimates of an arbitrary signal corrupted by AWGN can
be written as the sum of variances of real and imaginary parts of the bispectrum estimate.
After substitution of the function ˆ̇
( p,
q)
from (1.4.8) to (1.4.14), each of two terms in (1.4.14)
B err
will contain the sum of 55 terms. It is difficult to analyze them analytically. Because of that, below,
the statistical analysis of bispectrum estimator performance will be given using computer simulations
results presented in [30].
In the paper [29], the problem of bispectral reconstruction of deterministic signals embedded in
AWGN was considered. Unlike the algorithm (1.4.7) considered above, the algorithm described in
[29] has the following form
1
M
M I −1
ˆ ( m)
Ṡ (0) = x ( i), p = 0, q = 0,
∑∑
m= 1 i=
0
x
x
2

1
3 6
⎡
B
ˆ̇ (1,1)
ˆ
s Ḃ
⎤
(3,1)
ˆ ⎢ s ⎥
Ṡ (1) = ⎢
ˆ
(2,1)
ˆ
⎥
⎢ Ḃ s Ḃ
s (2, 2) ⎥
⎢⎣ ⎥⎦
, p = 1,..., I −1; 1 ≤ q ≤ p; p + q ≤ I −1,
S
ˆ̇ (2) = B
ˆ̇
(1,1)
2
S
ˆ̇
(1) ,
s
.................................,
ˆ ˆ ˆ ˆ
Ṡ ( I − 1) = Ḃ ( I − 2,1) Ṡ ( I − 2) Ṡ
(1)
s
21
. (1.4.15)
The authors of the cited paper [29], guided by results of theoretical analysis, affirm that bispectrum
estimates for signals with non-zero mean values become asymptotically unbiased by removing
the bispectrum samples belonging to the diagonal axis and frequency axes in the principal triangular
region of the support of the bispectrum. Unfortunately, one can reach this ideal result only if the data
length used for the bispectrum estimation is infinite and spectral leakage is absent.
The second approach to improve a bispectrum estimator that is commonly employed in bispectrum-based
signal processing is subtraction of a signal mean value (DC component) from an observed
input sequence (see, for example, [2, 15, 29]). This way of improving the bispectrum estimate theoretically
permits to exclude the distorting influence of the signal dependent terms in bispectrum estimate.
However, spectral leakage does not allow removing completely the influence of these signal dependent
terms.
Windowing, well-known in spectral analysis to decrease spectral leakage, has also been theoretically
studied in the estimation of third-order statistics [2, 25, 26] because it seems to be an effective
way for improvement of bispectrum estimate. However, for many practical applications, particularly,
for bispectrum-based reconstruction of deterministic signals imbedded in AWGN of large intensity,
the performance of those techniques has not been investigated yet.
Therefore, statistical investigation of the aforementioned methods will permit to assess bispectrum
estimator performance achievable in realistic cases.
Simple test signals s(i) in the form of single pulse with three different typical in signal
processing waveforms: rectangular, triangular and Gaussian have been used in computer simulations
[30]. The pulse lengths have been varied and the maximal amplitude for all types of pulses has been
set to unity. AWGN with different variances and random signal shifts τm of different deviation values
are considered in statistical computer simulations.
Total (TOSD) and truncated (TRSD) standard deviations of bispectrum estimator calculated

within the triangular symmetry domain G (see formula (1.2.11)) of bispectrum estimate have been
computed and analyzed.
TOSD is defined in the form of
TOSD = ∑∑ var{ B
ˆ
( p,
q)}
, p,
q ∈G
̇ . (1.4.16)
p q
x
To analyze the contribution to the TOSD (1.4.16) of the bispectrum samples belonging to the
axes and diagonal, the TRSD has been introduced. We define TRSD in the form which is similar to
(1.4.16) but excluding the bispectrum samples that are located on the aforementioned axes p=0 and
q=0 and on the diagonal strips p=q and p+q = I/2.
The test digital signal s(i) chosen for studying the bispectrum estimator performance is simu-
lated as the sequence of non-negative real values (i=0,1,…,64). The random signal shift τm is sup-
posed to be of uniform distribution. Maximum deviation in our computer simulations was equal to
τmmax = 10 samples. AWGN has been modeled as a zero mean WGN with the specified standard devi-
ation σn (it was constant within one set of experiments but varied from one set to another to get the
dependence on noise standard deviation).
First, we analyze the behavior of the function ( , q)
var{ B
ˆ
( p,
q)}
̇
σ =
for different number of
22
p x
realizations M for the signals s(i) of rectangular and triangular waveforms and for different standard
deviations of AWGN. Statistical stability has been observed starting with M=50.
Fig. 1.4.1. Plot of SD = σ ( p,
q)
computed for
T=2, σn = 0.1, M=100 and non-zero signal DC
component.
Fig. 1.4.2. Plot of SD = σ ( p,
q)
computed for
T=2, σn = 0.5, M=100 and non-zero signal DC
component.

Fig. 1.4.3. Plot of SD = σ ( p,
q)
computed for
T=10, σn = 0.1, and M=100.
23
Fig. 1.4.4. Plot of SD = σ ( p,
q)
computed for
T=10, σn = 0.5, and M=100.
The illustrative examples of behavior of σ ( p,
q)
for the test pulse signal of rectangular wave-
form and non-zero signal DC component, different lengths T and for M=100 Monte Carlo runs are
represented in Figures 1.4.1 – 4 by 3-D graphs.
The graphs obtained for the test signal with subtracted DC component are shown in Figures
1.4.5–8 (the rest of conditions employed in derivations are the same as in Figures 1.4.1–4).
Note that the graphs in Figures 1.4.1–8 are plotted only for the bispectral samples belonging to
the principal triangular domain of symmetry given by inequalities (1.2.11).
Fig. 1.4.5. Plot of SD = σ ( p,
q)
computed for
T=2, σn = 0.1, and M=100.
Fig. 1.4.6. Plot of SD = σ ( p,
q)
computed for
T=2, σn = 0.5, and M=100.

Fig. 1.4.7. Plot of SD = σ ( p,
q)
computed for
T=10, σn = 0.1, M=100 and subtracted DC com-
lowing.
ponent.
24
Fig. 1.4.8. Plot of SD = σ ( p,
q)
computed for
T=10, σn = 0.5, M=100 and subtracted DC com-
ponent.
Comparative analysis of the graphs plotted in Figures 1.4.1 – 1.4.8 allows concluding the fol-
1. According to the formula (1.4.8), the largest values of the functions σ ( p,
q)
are concentrated
on the frequency axis q=0 and on two strips that limit the considered principal triangular domain on
the bifrequency plane (p,q).
2. Behavior of the function σ ( p,
q)
is definitely depends on the form of the signal magnitude
Fourier spectrum. This is clearly visible for low standard deviations σn of AWGN, when the values of
σ ( p,
q)
are in strict correlation with the magnitude of the Fourier spectrum:
- the functions σ ( p,
q)
decay slowly from their maximums that are concentrated at low fre-
quencies for short pulse length of T=2 samples that corresponds to spread signal magnitude Fourier
spectrum S(p)=Tsinc(2πpT/I) (see Figures 1.4.1 and 1.4.5);
- decaying of the functions σ ( p,
q)
becomes more rapid for the wider pulse length of T=10
samples and side lobes of the magnitude Fourier spectrum become apparent in the graphs (see Figures
1.4.3 and 1.4.7).
3. Spectral leakage is evidently observed in the graphs of the functions σ ( p,
q)
and it becomes
more visible for large standard deviations σn of AWGN. Increasing of the leakage for larger σn is

clearly seen from the corresponding comparison of the pairs of the plots in Fig. 1.4.1 and Fig. 1.4.2;
Fig. 1.4.3 and Fig. 1.4.4; Fig. 1.4.5 and Fig. 1.4.6 and Fig. 1.4.7 and Fig. 1.4.8, respectively.
4. Subtraction of DC component from the original signal leads to decreasing the maximum of
the function σ ( p,
q)
. It is clearly seen from the corresponding comparative analysis of the pairs of the
graphs in Fig. 1.4.1 and Fig. 1.4.5; Fig. 1.4.2 and Fig. 1.4.6; Fig. 1.4.3 and Fig. 1.4.7 and Fig. 1.4.4
and Fig. 1.4.8.
Though subtraction of DC component from the original signal allows slightly improving bispec-
trum estimate, the latter still remains quite distorted in the practical case of limited data blocks length I
and sample size M that are available for averaging. Hence, in practice, the above-mentioned DC com-
ponent subtraction proposed in [2, 15, 29] does not lead to essential improvement of bispectrum esti-
mator performance.
5. It should be stressed, that since the main part of energy of the spectrum of the test signal
Tsinc(2πpT/I) is concentrated within the interval of [–1/T, 1/T], the SNR in this frequency band is also
maximal. Hence, the contribution of the errors caused by the fifth, sixth, seventh, and eighth term in
(1.4.8) is not too crucial for large SNRs from the point of view of distortions in bispectrum-based sig-
nal waveform reconstruction. This statement is confirmed by the results obtained in [32, 33].
6. One more important feature of bispectrum estimator should be noted. It concerns the inva-
riance of a bispectrum estimate with respect to signal translations. The terms of the form of
25
e
− j2π pτ
m
that appear in (1.4.8) due to the random signal shifts τm of the true signal, fortunately, do not cause
decreasing the bispectrum estimator accuracy.
The results obtained for the signals of rectangular, triangular and Gaussian waveforms, maxi-
mum signal shift deviation τm=10 samples and non-zero DC component are demonstrated by Tables
1.4.1– 3, respectively.
The lengths of pulse of triangular and Gaussian waveforms were selected accordingly to the
length of the pulse of rectangular waveform to preserve the same signal power for correct comparative
analysis.
The influence of the signal DC component on the bispectrum estimator accuracy has been also
studied for the three above mentioned test signal waveforms with subtracted DC component.
Computer simulation results for this case are presented below in Tables 1.4.4 – 6 for rectangu-
lar, triangular and Gaussian signal waveforms, respectively.

Table 1.4.1. The results obtained for rectangular signal waveform (M=100 Monte Carlo runs,
TOSD
TRSD
Т
non-zero DC component).
26
σn
0.1 0.2 0.3 0.4 0.5
2 896 2.51·10 3 5.87·10 3 1.09·10 4 1.85·10 4
6 1.84·10 3 5.03·10 3 1.02·10 4 1.71·10 4 2.60·10 4
10 2.63·10 3 6.39·10 3 1.26·10 4 2.05·10 4 3.33·10 4
2 760 2.17·10 3 5.04·10 3 9.42·10 3 1.60·10 4
6 1.34·10 3 3.79·10 3 7.89·10 3 1.36·10 4 2.13·10 4
10 1.66·10 3 4.22·10 3 8.62·10 3 1.49·10 4 2.54·10 4
Table 1.4.2. The results obtained for triangular signal waveform (M=100 Monte Carlo runs,
TOSD
TRSD
Т
non-zero DC component).
σn
0.1 0.2 0.3 0.4 0.5
4 694 2.15·10 3 5.39·10 3 1.09·10 4 1.85·10 4
12 1.06·10 3 3.42·10 3 7.33·10 3 1.39·10 4 2.41·10 4
20 1.30·10 3 4.07·10 3 9.26·10 3 1.59·10 4 2.60·10 4
4 578 1.84·10 3 4.6·10 3 9.40·10 3 1.60·10 4
12 685 2.38·10 3 5.35·10 3 1.10·10 4 1.93·10 4
20 680 2.37·10 3 5.86·10 3 1.09·10 4 1.92·10 4
Table 1.4.3. The results obtained for Gaussian signal waveform of
TOSD
TRSD
Monte Carlo runs, non-zero DC component).
α
σn
0.1 0.2 0.3 0.4 0.5
0.78 665 2.09·10 3
s(
i)
= e
5.3·10 3 1.08·10 4 1.84·10 4
0.09 1.12·10 3 3.54·10 3 7.49·10 3 1.41·10 4 2.43·10 4
0.03 1.43·10 3 4.39·10 3 9.44·10 3 1.73·10 4 2.89·10 4
0.78 552 1.78·10 3 4.52·10 3 9.30·10 3 1.58·10 4
0.09 728 2.50·10 3 5.70·10 3 1.13·10 4 1.95·10 4
0.03 737 2.63·10 3 6.03·10 3 1.20·10 4 2.08·10 4
2
2
−α
( i−
I / 2−1)
(M=100

Table 1.4.4. The results obtained for rectangular signal waveform (M=100 Monte Carlo runs;
TOSD
TRSD
Т
subtracted DC component).
27
σn
0.1 0.2 0.3 0.4 0.5
2 810 2.27·10 3 5.47·10 3 1.03·10 4 1.98·10 4
6 1.57·10 3 4.18·10 3 8.61·10 3 1.45·10 4 2.54·10 4
10 1.97·10 3 5.1·10 3 9.83·10 3 1.76·10 4 2.84·10 4
2 738 2.05·10 3 4.81·10 3 8.97·10 3 1.74·10 4
6 1.34·10 3 3.68·10 3 7.54·10 3 1.28·10 4 2.20·10 4
10 1.66·10 3 4.31·10 3 8.35·10 3 1.50·10 4 2.50·10 4
Table 1.4.5. The results obtained for triangular signal waveform (M=100 Monte Carlo runs;
TOSD
TRSD
Т
subtracted DC component).
σn
0.1 0.2 0.3 0.4 0.5
4 623 2.00·10 3 4.76·10 3 1.04·10 4 1.83·10 4
12 854 2.78·10 3 6.53·10 3 1.29·10 4 2.17·10 4
20 861 2.87·10 3 7.04·10 3 1.31·10 4 2.34·10 4
4 566 1.78·10 3 4.25·10 3 9.10·10 3 1.60·10 4
12 710 2.34·10 3 5.60·10 3 1.10·10 4 1.90·10 4
20 663 2.34·10 3 5.82·10 3 1.10·10 4 1.92·10 4
Table 1.4.6. The results obtained for Gaussian signal waveform of (M=100 Monte Carlo runs,
TOSD
TRSD
α
subtracted DC component).
σn
0.1 0.2 0.3 0.4 0.5
0.78 610 1.99·10 3 5.11·10 3 1.05·10 4 1.80·10 4
0.09 887 2.97·10 3 6.47·10 3 1.27·10 4 2.24·10 4
0.03 938 3.18·10 3 7.25·10 3 1.39·10 4 2.47·10 4
0.78 552 1.78·10 3 4.52·10 3 9.30·10 3 1.58·10 4
0.09 728 2.50·10 3 5.70·10 3 1.12·10 4 2.08·10 4
0.03 737 2.63·10 3 6.03·10 3 1.20·10 4 2.08·10 4

Comparative analysis of the results represented in Tables 1.4.1 – 6 demonstrates the following:
1. Removing the bispectrum samples located on the boundaries of above-mentioned triangular
domain allows decreasing TRSD in comparison to TOSD. This coincides with conclusions in [29].
We have analyzed the ratio of TOSD to TRSD (RTT). RTT grows with increasing of signal length and
it becomes considerably larger than unity for small standard deviations σn of AWGN.
Maximum values of RTT reached approximately 90% as seen in Tables 1.4.2 and 1.4.3 for tri-
angular and Gaussian signal waveforms of maximal length and for σn = 0.1. But for low input SNR
the expedience of removal data on aforementioned borders in bispectrum-based signal waveform re-
construction is doubtful since significant part of signal Fourier spectrum is rejected.
2. RTT decreases with increasing of σn since the fluctuation errors in bispectrum estimates dis-
tributed on the whole triangular bispectrum domain become prevailing.
3. The method of improving bispectrum estimate by subtraction of signal DC component, in
general, leads to decreasing TOSD and TRSD as seen from the corresponding comparison of Tables
1.4.1 and 1.4.4 (rectangular signal waveform), Tables 1.4.2 and 1.4.5 (triangular signal waveform),
and Tables 1.4.3 and 1.4.6 (Gaussian signal waveform).
However, this method of improving bispectrum estimator performance works well for small σn
values and RTT approaches to unity with σn increasing.
4. The RTT value for T increasing grows more slowly for zero DC component signal bispectrum
estimate in comparison to non-zero DC component signal bispectrum estimate (compare the corres-
ponding results in Tables 1.4.1, 2 and 3, and the results in Tables 1.4.4 – 6). Moreover, RTT corres-
ponding to the zero signal DC components is less in comparison to RTT for non-zero DC component
signals.
To verify expected decreasing of spectral leakage, the windowing in temporal domain has been
performed for improvement of bispectrum estimate. For this reason, typical Hann window of the form
has been used in computer simulations.
w(i)=1/2[1–cos(2π(i–I/2–1)/I)], (1.4.17)
Let us consider the results of computer simulations for bispectrum estimate obtained for multiply-
ing the observed process by the window (1.4.17). These results are represented in Table 1.4.7.
As seen from comparison of the results in Tables 1.4.1 and 1.4.7, for the considered signal and
noise parameters the Hann window is able to only slightly improve bispectrum estimate. This im-
provement can reach approximately 6% for small standard deviations σn and it is practically negligible
28

for large σn values. Thus, the statement about improvement of bispectrum estimate by windowing [2]
is absolutely correct, but it is not reasonably large for the considered range of signal and noise para-
meters.
Table 1.4.7. The results obtained for rectangular signal waveform weighed by the Hann window
(M=100 Monte Carlo runs; non-zero signal DC component).
TOSD
TRSD
Т
29
σn
0.1 0.2 0.3 0.4 0.5
2 849 2.49·10 3 5.85·10 3 1.05·10 4 2.01·10 4
6 1.89·10 3 4.83·10 3 9.17·10 3 1.69·10 4 2.71·10 4
10 2.50·10 3 6.31·10 3 1.22·10 4 2.03·10 4 3.2·10 4
2 728 2.12·10 3 5.03·10 3 9.06·10 3 1.74·10 4
6 1.39·10 3 3.66·10 3 7.13·10 3 1.34·10 4 2.24·10 4
10 1.58·10 3 4.18·10 3 8.48·10 3 1.44·10 4 2.40·10 4
Thus, the result of studying the accuracy of the bispectrum estimator obtained for three afore-
mentioned methods and for realistic cases are the following.
1. Although the subtraction of DC component from the original signal slightly improves the bis-
pectrum estimate, the latter still remains severely distorted due to finite data block length, limited
number of observed realizations and spectral leakage.
2. Improving bispectrum estimate by removing the bispectrum samples belonging to borders of
the principal triangular region and by subtracting the DC signal component have shown relatively
good effectiveness only for low intensity AWGN. However, spectral leakage severely restricts effi-
ciency of this approach.
3. Conventional windowing of input data leads to rather small improvement of bispectrum esti-
mator performance for high intensity AWGN.
It will be demonstrated below in the next Chapter of the work, that the use of smoothing the bis-
pectrum estimates by nonlinear filters and robust approaches to processing a set of realizations seems
to be a more efficient way for improving the accuracy of a bispectrum estimator especially for small
input SNR.
In bispectrum-based signal reconstruction technique proposed in [34], least-square method and
all bispectrum samples belonging to the hexagonal domain are used. For solving the problem of un-
known signal phase Fourier spectrum reconstruction by using phase bispectrum estimate, equation
(1.4.3) in [34] is represented in the following matrix form

where φ is the vector of unknown Fourier phase values equal to
Aφ=γ , (1.4.18)
φ=(φ1, φ2, φ3,…, φN) T , (1.4.19)
N is the total number of unknown phases; T denotes transpose procedure; γ is the vector of phase bis-
pectrum values defined as
γ = (γ1,1, γ1,2,…, γ1,N-1, γ2,2, γ2,3,…,γ2,N-2, …, γN/2,N/2) Т . (1.4.20)
The matrix of the coefficients A in (1.4.18) can be expressed as
⎡2
⎢
⎢
1
⎢1
⎢
⎢.
⎢.
⎢
А= ⎢1
⎢0
⎢
⎢0
⎢
.
⎢
⎢.
⎢
⎣.
−1
1
0
.
.
0
2
1
.
.
.
0
−1
.
.
0
0 −1
1
.
.
.
0
0
1 −1
.
.
0
0 −1
.
.
.
30
0
0
0
.
.
0
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1
0
0 ⎤
0
⎥
⎥
0 ⎥
⎥
. ⎥
. ⎥
⎥
−1⎥
. (1.4.21)
0 ⎥
⎥
. ⎥
.
⎥
⎥
0 ⎥
⎥
−1⎦
Dimension of the matrix in (1.4.21) is (N/2) 2 ×(N-1) for even and [(N-1)(N+1)]/4× (N-1) for odd
N values, respectively.
For signal phase Fourier spectrum reconstruction it is proposed in [34] to solve the following
equation by the least-square method
Т −1
Т
ϕϕϕϕ = ( А А) А γ . (1.4.22)
It should be noted that two typical obstacles occur with solving the equation (1.4.22) by using
the least-square method. The first one is performing of the complicated operations for large dimension
matrix inversion. The second one is the ambiguity of the obtained solution and as a result, large phase
errors arise caused by above noted phase wrapping.
According to phase unwrapping algorithm proposed in [35], phase bispectrum is defined in the
following form
γ = γ ′ + 2π , (1.4.23)
i,
j
i,
j
where γ ′ i, j is the principal argument value in the complex-valued bispectrum given within the interval
of [0,2π) and k i,
j is the integer value that determines the number of bispectrum phase wrapping in an
k i,
j

arbitrary point on the bispectral plane (i, j).
where γ ’ =
According to this approach, matrix equation (1.4.18) can be transformed to the form of
Aφ=γ ’ +2πk , (1.4.24)
' ' ' ' ' '
' T
γ 1,
1,
γ 1,
2 , ..., γ 1,
N − 1,
γ 2,
2 , γ 2,
3,...,
γ 2,
N −2
,..., γ N / 2,
N / 2 ) is the vector of principal values in the
(
phase bispectrum; the vector k = (k1,1, k1,2,…, k1,N-1, k2,2, k2,3,…,k2,N-2, …, kN/2,N/2) Т .
dition
In addition to the equation (1.4.24), a new matrix C is formed to accomplish the following con-
СА = 0. (1.4.25)
By multiplying two parts in the equation (1.4.24) with the matrix C, one can obtain
САφ = С(γ ’ +2πk), (1.4.26)
Сk = (-1/2 π) С γ ’ . (1.4.27)
Solving the latter equation with respect to k and substitution of the solution to (1.4.24) allows
performing phase bispectrum unwrapping [35].
The main drawback of the algorithm proposed in [35] is its high sensitivity to noise. Due to a
noise influence, real-valued quantities arise in the right part in the equation (1.4.27) and it is necessary
to approximate them to the nearest integers. Phase errors arise as the result.
Thus, the problem of phase wrapping is of great importance and the next subsection will be ded-
icated to some approaches proposed by us to avoid this problem and to decrease phase errors in bis-
pectrum-based signal reconstruction techniques.
1.5. Reduction of waveform distortions in bispectrum-based signal reconstruction systems
A new approach suggested in [36, 37] and based on utilization of continuous sine and cosine
functions instead of conventional discontinuous bispectrum phase arc tangent function in signal recon-
struction systems is considered in this subsection. First, we discuss the problem in deterministic
statement, and after that we pay attention to noise influence for the approach proposed.
In addition to the bispectrum-based signal reconstruction algorithms considered in the previous
subsections, the most authors directly or indirectly stereotypically use phase information in their ap-
proaches [10, 29, 38–41].
The authors of the cited papers assume that it is necessary to compute the phase bispectrum for
signal reconstruction directly [10, 29, 38]; or to restore a signal without direct consideration of phase
bispectrum but using phase information inherent in signal log-bispectrum (bicepstrum) [39] or diffe-
rential cepstrum [40]; or to recover the phase of a system transfer function from any pair of consecu-
31

tive phase bispectrum slices [41]. Note that, on one hand, errorless recovery of phase information is
possible only under certain conditions and constraints imposed on the processed signals. On the other
hand, the errors that arise in phase recovery may provoke significant distortions in reconstructed sig-
nals. For example, 2-D (see references [10, 29, 38]) or 1-D (see nonparametric algorithm described in
[39] and [41]) phase unwrapping is required for reconstruction of the true phase of a signal Fourier
spectrum. Effectiveness of the parametric algorithm [39] largely depends on the location of signal ze-
ros and poles towards the unit circle on z-plane. It does not allow having zeros or poles on the unit
circle since the parametric algorithm [39] is based on the cepstral coefficients that are not defined in
this case. The additional a priori information in the form of signal power cepstrum, as well as a priori
knowledge of signal magnitude to transform the processed sequence to minimum/maximum phase
signal are necessary for successful operation of the parametric algorithm [39]. The bispectrum itera-
tive reconstruction algorithm [39] also operates under certain conditions such as a phase of bispectrum
is supposed to be known and, hence, differences of cepstrum coefficients are also known. However,
high convergence rate in iterative algorithm [39] is possible when a priori information about the sig-
nal is available. Strict requirements must be imposed on differentiability of z-transform of a processed
sequence that cannot have zeros on the unit circle in algorithm [40] using differential cepstrum.
In our opinion, although phase information can be interesting for a number of applications, its
direct measurement is unnecessary for solving bispectrum-based signal reconstruction problem. In this
subsection we describe a new approach to solving this problem without addressing to direct phase cal-
culations. Instead of traditional approaches and their above-mentioned intrinsic limitations, we pro-
pose to use recursive signal reconstruction procedures for quadrature components of both complex-
valued normalized bispectrum and signal Fourier spectrum.
According to the direct method (see (1.3.4 – 1.3.6)) of bispectral density estimation, for a real-
valued, deterministic, discrete-time and time-limited signal xi (i=0, 1,…, I–1, and, traditionally, unita-
ry and uniform temporal sampling is assumed) its bispectrum B p,
q can be defined as the following
complex-valued function
where
X
B
p,
q
jβ
p,
q
= X X X *
+ = B + jB = B e , (1.5.1)
p
q
jϕ
p
Re
Im
p = X p e is the discrete Fourier transform of xi; B p,q
and p,q
parts, respectively;
p,
q
p
q
Re
p,
q
32
Im
p,
q
p,
q
B are the real and imaginary
Im
Re 2 Im 2
B p,
q
= ( B ) ( B ) is the magnitude bispectrum; p,
q
p,
q
p,
q = arctan
Re
B p,
q
B +
β is the
phase bispectrum determined within the principal interval of inverse tangent function, i.e.,

β p , q ∈ ( −π
, π]
; ϕ p and X p are the signal phase and magnitude Fourier spectra, respectively.
To recover the magnitude p Xˆ and phase ϕˆ p of the Fourier spectrum of the signal from its
measured bispectrum (1.5.1), it is common to use recursive solution of the equations (1.4.4) and
(1.4.5).
Since the procedure (1.4.5) usually permits to recover a magnitude Fourier spectrum without
any distortions, we are focusing on the typical phase spectrum recovery procedure (1.4.4) that will be
compared below to the proposed algorithm.
The recursive strategy proposed in [10] and repeated later by many authors for signal Fourier
phase spectrum recovery by the procedure (1.4.4) is implemented practically for bispectrum samples
that belong to the principal bispectrum triangular domain. Note that two different possible sequences
of recursive steps are possible. The first one is parallel to the frequency axis p = 0 and the second one
is parallel to the axis q = 0. These two different data processing ways are represented by the following
corresponding two sets of equations
ˆ ϕ = −β
2
3
4
5
4
5
2
3
4
N −1
2
1,
1
ˆ ϕ = ˆ ϕ − β
ˆ ϕ = ˆ ϕ − β
ˆ ϕ = ˆ ϕ − β
2
2
N −1
2
1,
2
1,
3
1,
4
.......... .......... ....
ˆ ϕ = ˆ ϕ − β
N
ˆ ϕ = 2 ˆ ϕ − β
3
1,
N −1
2,
2
ˆ ϕ = ˆ ϕ + ˆ ϕ − β
N −2
2,
3
.......... .......... .......... ...
ˆ ϕ = ˆ ϕ + ˆ ϕ − β
N
N + 1
2
2,
N −2
.......... .......... .......... ......
ˆ ϕ = ˆ ϕ + ˆ ϕ − β
N
N −1
N + 1
,
2 2
, (1.5.2a)
33
ˆ ϕ = −β
2
3
ˆ ϕ
4
4
5
ˆ ϕ
2
3
= ˆ ϕ
N −1
2
4
= ˆ ϕ
1,
1
ˆ ϕ = ˆ ϕ − β
2
ˆ ϕ = ˆ ϕ − β
ˆ ϕ = ˆ ϕ − β
N −1
1,
2
ˆ ϕ = 2 ˆ ϕ − β
2,
2
1,
3
.......... .......... ....
N
+ ˆ ϕ
1,
4
N + 1
2
.......... .......... .......... ...
N
− β
1,
N −1
− β
N −1
N + 1
,
2 2
. (1.5.2b)
It was shown in [10] that the procedures (1.5.2a) and (1.5.2b) may produce different results at
the reconstruction system output in case of noise presence.
Note that the behavior of the function β p, q in (1.5.2a) and (1.5.2b) is characterized by disconti-
nuities (wrappings) in – π and π radians that may cause the abruptions in the recovered signal phase
Fourier spectrum ϕˆ p+ q . Note that the behavior (steepness and oscillation frequency) of the functions
Re
p,q
B and
Im
p,q
B along the frequency axes p = 0 and q = 0 may vary considerably depending on a
processed signal waveform. For fixed uniform sampling interval ∆p = ∆q = 1/I in the bispectrum plane

(p, q), sampling errors of the function β p, q are dependent on its behavior that is usually unknown in
practice. Thus, for fixed sampling interval that is commonly limited by computer processing rate and
memory capacity and under a priori unknown Nyquist frequency, one should expect the distortions of
Fourier phase spectrum recovered by procedures (1.5.2a) or (1.5.2b).
Let us investigate the accuracy of reconstructed signal that is achieved by the conventional algo-
rithms utilizing direct phase measurements. As a typical example, we consider the algorithm devel-
oped in the paper [10]. For this purpose, we introduce the following quantitative measure of distortion
for the reconstructed waveform
1
ˆ
0
min{
}
1
∑
0
−
I −
∑ xi−t
− xi
i=
δ =
, (1.5.3)
t I
x
i
i =
where minimization over t =0, 1,…, I –1 is used to take into account possible temporal shift in the re-
constructed signal xˆ i .
We now pay attention to the methodical distortions that are typical for conventional algorithm
[10] and its implementation in a number of papers.
First, we consider the problem of reconstruction of the single pulse signals of rectangular shape
with different pulse lengths ∆t observed in the fixed limited sample grid, for example, of I = 256 sam-
ples. Let us study the test signal centered with respect to the point of origin. In this case, distortion
values calculated according to (1.5.3) for the procedures (1.5.2a) and (1.5.2b) depending on the pulse
lengths ∆t are represented in Table 1.5.1.
It should be stressed, that the procedures (1.5.2a) and (1.5.2b) cause the same distortions and
analysis of the distortion values given in Table 1.5.1 demonstrates that δ tends to increase with ∆t in-
creasing.
Table 1.5.1. Waveform distortions δ calculated for the procedures (1.5.2a) and (1.5.2b).
∆t 9 13 17 21 25 29 33 37 41 45 49
δ 0.0002 0.003 0.017 0.059 0.105 0.145 0.097 0.173 0.156 0.204 0.192
As a typical example, the signal of the length of ∆t = 37 samples and the amplitude of A = 1
reconstructed by the procedure (1.5.2a) or (1.5.2b) is shown in Fig. 1.5.1. It is clearly seen from Fig.
1.5.1 that the original rectangular signal waveform is sufficiently distorted.
34

Signal amplitude
1,4
1,2
1
0,8
0,6
0,4
0,2
0
-0,2
-0,4
129
138
147
156
165
174
183
192
201
210
219
228
237
246
35
255
8
17
26
Sample number
Fig. 1.5.1. Signal reconstructed by the procedure (1.5.2a) or (1.5.2b), δ = 0.173.
Let us consider now the second type of the test signal given in the form of two short rectangular
pulses. Suppose that pulses have the same length of ∆t1 = ∆t2 = 3 samples, different amplitudes equal
to A1 = 1 and A2 = 4, and mutual pulse center shift equal to 7 samples.
The two-pulse signal reconstructed by the procedure (1.5.2a) or (1.5.2b) is shown in Fig. 1.5.2.
As it is seen from Fig. 1.5.2, the reconstructed signal waveform is sufficiently distorted. Note that the
distortions (1.5.3) computed for this two-pulse test signal reconstructed by the procedure (1.5.2a) and
(1.5.2b) are equal to approximately the same value δ ≈ 0.6515.
Signal amplitude
5
4
3
2
1
0
-1
129 138 147 156 165 174 183 192 201 210 219 228 237 246 255 8 17 26 35 44 53 62 71 80 89 98 107 116 125
Sample number
Fig. 1.5.2. Signal reconstructed by the procedure (1.5.2a) or (1.5.2b), δ ≈ 0.6515.
Thus, in the cases in which bispectral data are available on a sparse set of points and a signal
35
44
53
62
71
80
89
98
107
116
125

waveform is a priori unknown, the conventional algorithm [10] may provoke large distortions in the
reconstructed signal shape.
Below we propose a new approach that allows sufficient decreasing the distortions in the recon-
structed signal waveform. The main idea of our approach is to avoid the discontinuous bispectrum
phase function β p, q from the signal waveform recovery procedure. We propose: first, to replace the
function p, q
β with smooth and continuous functions as cos β p, q and sin β p, q , and, second, to recover
cos ϕˆ p+ q and sin ϕˆ p+ q instead of p+ q
or (1.5.2b).
as
ϕˆ in the conventional reconstruction recursive procedure (1.5.2a)
The proposed algorithm is based on rewriting the equation (1.5.1) using normalized bispectrum
B
B
p,
q
p,
q
j sin (cos ˆ j sin ˆ )(cos ˆ j sin ˆ )(cos ˆ j sin ˆ )
+
+ =
+ = ϕ
ϕ ϕ ϕ ϕ ϕ
β
β . (1.5.4)
cos p,
q
p,
q
p
36
p
q
q
p+
q −
From equation (1.5.4), the recovery procedures for cosine and sine of signal Fourier phase from
cosine and sine of phase bispectrum can be written as
cos ˆ ϕ = (cos ˆ ϕ cos ˆ ϕ − sin ˆ ϕ sin ˆ ϕ ) cos β + (cos ˆ ϕ sin ˆ ϕ + sin ˆ ϕ cos ˆ ϕ ) sin β
p + q
p q p q p,
q
p q p q p,
q
sin ˆ ϕ = (cos ˆ ϕ sin ˆ ϕ + sin ˆ ϕ cos ˆ ϕ ) cos β − (cos ˆ ϕ cos ˆ ϕ − sin ˆ ϕ sin ˆ ϕ ) sin β
p + q
p q p q p,
q
p q p q p,
q
p+
q
, (1.5.5a)
. (1.5.5b)
Just as in the conventional approach (see equations (1.5.2a) or (1.5.2b)), we propose below two
different sequences of recursive steps. The first one, in accordance with (1.5.5a) and (1.5.5b) (parallel
to the frequency axis p = 0), can be defined as
⎧cos
ˆ ϕ = ˆ
N (cosϕ
⎪
⎨
⎪sin
ˆ ϕ = (sin ˆ
N ϕ N
⎪⎩
N −1
2
−1
2
cos ˆ ϕ
cos ˆ ϕ
N + 1
2
N + 1
2
− sin ˆ ϕ
+ cos ˆ ϕ
⎪⎧
cos ˆ ϕ = ˆ
N cosϕ
N
⎨
⎪⎩
sin ˆ ϕ = sin ˆ
N ϕ N
N −1
2
N −1
2
sin ˆ ϕ
sin ˆ ϕ
N + 1
2
N + 1
2
⎪⎧
cos ˆ ϕ 2 = cos β1,
1
⎨
,
⎪⎩
sin ˆ ϕ 2 = −sin
β1,
1
………………………,
) cos β
) cos β
−1
−1
cos β
cos β
N −1
N + 1
,
2 2
N −1
N + 1
,
2 2
1,
N −1
1,
N −1
+ (sin ˆ ϕ
+ (sin ˆ ϕ
+ sin ˆ ϕ
− cos ˆ ϕ
N −1
2
N −1
2
N −1
N −1
cos ˆ ϕ
sin ˆ ϕ
sin β
sin β
N + 1
2
N + 1
2
1,
N −1,
1,
N −1
+ cos ˆ ϕ
− cos ˆ ϕ
N −1
2
N −1
2
. (1.5.6)
……………………………,
sin ˆ ϕ
cos ˆ ϕ
N + 1
2
N + 1
2
) sin β
) sin β
N −1
N + 1
,
2 2
N −1
N + 1
,
2 2
Since the sequence of recursive steps in the direction parallel to the axis q=0 differs from (1.5.6)
.

only by succession of the equations, we omit the equations corresponding to the sequence of recursive
steps parallel to the frequency axis q = 0. See the reference on how the samples are actually collected
from the main bispectrum triangular domain in [10] where detailed illustrations of recursive strategy
are given.
To compare the performance of the conventional algorithm employing the procedures (1.5.2a)
or (1.5.2b) and the proposed algorithm using (1.5.6), we begin with example of single pulse signals
with different pulse lengths ∆t and centered at the origin.
As a comparative example, the signal of the length of ∆t = 37 samples and amplitude of A = 1
reconstructed by algorithm (8) is shown in Fig. 1.5.3. It is clearly seen from Fig. 1.5.3 that the original
rectangular signal waveform is not distorted.
Note that both proposed procedures (parallel to the frequency axis p=0 or q=0) give errorless re-
sults for other considered pulse lengths ∆t.
Signal amplitude
1.2
1
0.8
0.6
0.4
0.2
0
129
143
157
171
185
199
213
227
241
255
37
13
27
Sample number
Fig. 1.5.3. Signal reconstructed by the proposed procedure (1.5.6), δ = 0.
In Fig. 1.5.4, the two-pulse signal (pulse lengths ∆t1 = ∆t2 = 3 samples, pulse amplitudes A1 = 1
and A2 = 4 and the mutual center shift are the same as in Fig. 1.5.2) recovered by the proposed proce-
dures is shown. Comparison of the two-pulse signal waveform reconstructed by conventional proce-
dure (1.5.2a) or (1.5.2b) (see Fig. 1.5.2) and by the proposed procedures (see Fig. 1.5.4) permits to
notice significant enhancement of the reconstructed signal quality due to application of the proposed
algorithm.
41
55
69
83
97
111
125

Signal amplitude
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
-0,5
129
137
145
153
161
169
177
185
193
201
209
217
225
233
241
249
1
9
38
17
25
Sample number
Fig. 1.5.4. Signal reconstructed by the proposed procedure (1.5.6), δ = 0.0335.
The main distinction of our approach from the known approaches is in abandonment of recovery
of phase information, because the phase is not interesting itself for signal reconstruction from bispec-
trum. In our opinion, traditional computation of bispectrum phase is unnecessary since we need only
the normalized bispectrum for solving a reconstruction problem. The use of the normalized bispec-
trum is free from typical constraints imposed on the processed signals and gives significant reduction
of reconstructed signal distortions.
The proposed approach gives us a possibility to decrease algorithm sensitivity to sampling rate
in bispectrum domain that is an important feature for reconstruction of an unknown signal waveform.
The sampling frequency in our approach can be chosen approximately by Nyquist criterion and no
extension of data file by zero padding is required for sampling errors decreasing. Hence, the proposed
algorithm is computationally effective and one of its main advantages is simplicity.
Now we pass to a study of noise immunity of the algorithm (1.5.6) in a bispectrum-based signal
reconstruction system operating under influence of both AWGN and mixture of AWGN and impul-
sive noise [37]. The following set of values has been computed for comparative performance analysis
33
41
of the proposed algorithm (1.5.6) and conventional algorithm [10].
2
1) The sampling variance σ inp , i.e. variance calculated by practically limited number of realiza-
tions M and SNRinp assessed at the input of the signal reconstruction system and, respectively, com-
puted as
49
57
I −1
2
2 1 ( m)
σ inp = ∑[ x ( i)
− s(
i)]
, (1.5.7)
I −1
i=
0
M
65
73
81
89
97
105
113
121

SNR
P
inp = s
2 , (1.5.8)
σ inp
where M denotes expectation computed over M recorded realizations; ∑ −1
1 2
= [ ( ) − ]
I
P s s i ms
is the
I
power of original signal s(i); ∑ −1
1
= ( )
I
m s s i .
I
i=
0
2) The sampling variance at the output of the signal reconstruction system
2
I−1 I−1
where σ = min ∑ ( sˆ ( i) −s( i −t) ) − ∑ ( sˆ ( i) −s( i −t)
)
out k k k
i= 0 i=
0
t
K
2 2
out out
K k = 1
39
i=
0
_ 1
σ = ∑ σ k , (1.5.9)
2
1 ⎡ 1
⎤
I ⎢
⎣ I
⎥
; )
⎦ ( ˆ i sk is the reconstructed signal waveform estimate
for k-th experiment (k = 1,2,…, K); K is the number of the repetitions executed in statistical
investigations for obtaining reliable estimate; t is the signal shift index (t = 0, 1,…, I–1) introduced
taking into consideration the well-known bispectrum invariance property for the signal translation.
3) The SNR out at the output of the signal reconstruction system
SNR
out
_ 2
= P /σ out . (1.5.10)
s
4) The criterion ε which demonstrates the improvement of SNR at the signal reconstruction sys-
tem output comparing to its input
SNR
ε = out . (1.5.11)
SNR inp
Again we consider a test signal s(i) (i=1, 2,…, 256) given in the form of two pulses of rectangu-
lar shape and different amplitudes. The two-pulse signal location in the temporal axis has been
changed randomly from one m-th observed realization to another. The total number of realizations is
equal to M = 200 in our simulations. The signal random shift deviation was equal to τ (m) = 40 samples.
2
Two kinds of interferences have been studied. The first, AWGN of variance ofσ inp , has been
added to each observed realization. The second, original signal has been corrupted by additive mixture
of AWGN and impulsive noise. Impulsive noise component has been generated by the given pulse
amplitudes and probabilities of appearance the pulses. The so called “salt and pepper” model of im-
pulsive noise has been utilized. The noise pulse amplitudes were equal to Apos = 2 and varying proba-
bility of appearance the pulses was given by the value P.
The test pulse signal length has been varied over the wide limits from 3 to 35 samples. It makes

possible to investigate bispectrum-based signal reconstruction performance depending on the pulse
signal length in noise environment. Two examples of the test signal reconstructed in AWGN envi-
2
ronment ( σ inp =0.3) by the technique proposed in [37] and the conventional technique [10] are demon-
strated in Figures 1.5.5 and 1.5.6.
(a) (b)
Fig. 1.5.5. Signal reconstructed (the length of each signal pulse is equal to 15 samples): (a) the
technique [10], ε = 2.63. (b) the technique proposed in [37], ε = 4.99.
(a) (b)
Fig. 1.5.6. Signal reconstructed (the length of each signal pulse is equal to 35 samples): (a) the
technique [10], ε = 1.26. (b) the technique proposed in [37], ε = 3.91.
It is clearly seen from Figures 1.5.5a and 1.5.6a that the original signal rectangular waveform is
distorted considerably for the signals reconstructed by the technique [10] due to the above mentioned
phase wrapping. Note that the level of distortions increases with increasing of the pulse signal length
since the number of phase «π-jumps» in biphase estimate increases with increasing of signal pulse
40

length. Because of this, the reconstructed signal waveform distortions increase.
At the same time, the proposed technique using the continuous sine and cosine functions pre-
serves the original rectangular signal waveform (see Figures 1.5.5b and 1.5.6b).
Comparison of the values ε that characterize the signal reconstruction performance including
both improvement of signal-to-noise ratio at the signal reconstruction system output in comparison to
its input and preservation of the original signal waveform permits to note that the technique proposed
provides better performance comparing to the conventional technique.
The plots of improvement parameter ε as a function of signal pulse length computed for tech-
nique [10] (Bartelt, Lohmann and Wirnitzer – BLW technique) and for the technique suggested in [37]
are shown in Figures 1.5.7 and 1.5.8, respectively.
2
Fig. 1.5.7. Improvement ε as a function of pulse signal length, σ inp =0.3.
41

2
Fig. 1.5.8. Improvement ε as a function of pulse signal length, σ inp = 1.
The plots in Figures 1.5.7 and 1.5.8 demonstrate the preference of the technique proposed com-
paring to the BLW technique almost in the total range of the pulse signal lengths studied.
The graphs illustrating the behavior of the improvement value ε as a function of pulse signal
length for additive mixture of AWGN and impulsive noise are shown in the Figures 1.5.9 and 1.5.10.
2
Fig. 1.5.9. Improvement ε as a function of pulse signal length, σ inp = 0.3; P = 5%.
42

2
Fig. 1.5.10. Improvement ε as a function of pulse signal length, σ inp = 0.3; P = 30%.
Thus, it can be concluded that technique proposed provides better performance comparing to
BLW technique apart from the very short pulse signal length.
1.6. Conclusions
The analysis carried out above shows that the bispectrum and bispectrum-based processing pos-
sess several important features. They are high immunity to noise which is zero mean and has symme-
tric PDF, insensitivity to random shifts of a signal component in observed realizations, ability to pre-
serve phase information and phase coupling between signal frequencies, etc. These useful properties
have been already exploited in abovementioned applications and they open new perspectives. Howev-
er, it is not always clear how positive features of bispectrum-based processing can be exploited in a
particular application.
However, bispectrum properties’ study is far away from completeness. First, there are several
ways of bispectrum estimation. In next Chapters we prefer to rely on direct technique of bispectrum
estimation defined by expressions (1.3.4) – (1.3.6). This way is rather simple and fast since it allows
exploiting FFT algorithms. Besides, below we will not use any windowing of input data since the ef-
fect of windowing on bispectrum estimation accuracy is not sufficient (see subsection 1.4).
43

Second, noise present in observed input realizations leaks to bispectrum domain and makes bis-
pectrum estimate noisy. Moreover, even if noise in observed input realizations is pure additive, zero
mean, i.i.d. and Gaussian, noise in bispectral domain (in real and imaginary parts of bispectrum esti-
mate) occurs to be of quite complex statistical properties (see expression (1.4.8) and investigation re-
sults presented in subsection 1.4). If input SNR is low and a number of observed realizations M is
small, the obtained estimate of bispectrum can be rather noisy. This noise leaks to the reconstructed
signal shape estimate via nonlinear procedures of signal spectrum recovery from bispectrum estimate
and further signal reconstruction. Thus, improvement of bispectrum estimate is an important task.
Third, although there exists the conventional procedure of signal spectrum recovery from bis-
pectrum estimate [10] and it performs well enough, there are also some alternatives and they can be
used as well (see subsection 1.5).
Fourth, bispectrum based processing does not produce perfect reconstruction of a signal wave-
form due to uncertainties in setting the phase sample ˆϕ ( 1)
(see subsection 1.4). Besides, reconstructed
signal estimates are “centered” with respect to the origin (see examples in Figures 1.5.3, 1.5.6). This
should be taken into account in practical applications and in selection of quantitative criteria to
characterize accuracy of signal waveform reconstruction.
Fifth, although usually bispectrum based processing is applied for zero mean and symmetric
PDF noise that affects an input signal, recent results [32] show that bispectrum based reconstruction
can be carried out also for non-Gaussian and mixed noise environments. However, this problem is not
well studied yet.
Sixth, bispectral based processing operates with 2-D functions (bispectrum or TAF estimates)
even if an input signal is a 1-D function, i.e., dimensionality of processed data increases. Therefore,
special attention should be paid to computational efficiency of bispectrum-based techniques.
Finally, bispectrum-based technique performance should be compared to that of other
techniques which could be applied for solving particular practical tasks in order to draw final
conclusions.
44

2. Combined bispectrum-filtering techniques for unknown
signal shape estimation in noise
In Chapter 1 we have already mentioned that the useful property of bispectrum-based processing
is its ability to carry out quasi-coherent accumulation of mutually shifted (with random and a priori
unknown shift) realizations of noisy signals where a signal component shape is unknown but the same
for all realizations of an observed set. A general assumption concerning the signal component is that
its shape is the same or almost the same for all observed realizations. This assumption, e.g., holds for
target range profile estimates obtained in a set of sequential scans following each other rather fre-
quently under condition that a target aspect angle does not change from one scan to another. An ex-
ample of such range profile estimates for a missile model [50] is presented in Fig. 2.1. As seen, these
range profile estimates are practically the same. A simple model of such range profiles could be two
rather short pulses with different amplitudes.
Analysis of the presented range profiles also shows another peculiarity of observed signal. As
seen, there are such sample minimal and maximal indices imin and imax that s(i)=0 for i=0,…,imin and
i=imax,…,I–1. Moreover, imax – imin is considerably less than I. These properties will be later used in
simulations in this Chapter.
Note that high resolution radar range profiles estimates can be used for air target type classifica-
tion. As shown in [50], classification accuracy depends upon many factors but one of the basic ones is
noise intensity in an obtained estimate of radar range signature. Moreover, it is also mentioned that for
target type determination it is desirable to center an estimate with respect to origin. Then a classifier,
e.g., that one based on neural network is able to perform better. Thus, it is possible to assume that the
better (less noisy) a range profile estimate is, the better probability of correct classification of air tar-
get type can be provided.
Within the bispectrum based data processing intended on unknown signal shape reconstruction
this means the following. First, better estimate of a signal waveform can be obtained if a better esti-
mate of a signal Fourier spectrum recovered from bispectrum is provided. Second, better estimate of a
signal Fourier spectrum recovered from bispectrum can be provided if a better estimate of bispectrum
is produced. Thus, it is reasonable to apply any methods that provide better estimates of bispectrum
and/or signal Fourier spectrum recovered from bispectrum.
One typical approach to obtaining better estimates of 1-D or 2-D processes is their filtering. Till
the moment, a great number of various filters has been proposed and used in numerous applications.
45

Thus, it is a problem to find (select) or to design a proper filter for a given application. Commonly the
situation is such that more a priori information about properties of signal component and noise is
available, more effective and efficient filters can be found or designed. Because of this, below we
consider a possibility to combine bispectrum base processing with filtering (linear and nonlinear) and
other approaches to improvement of signal bispectrum and spectrum estimation.
One more problem is that both bispectrum and noisy signal Fourier spectrum recovered from
bispectrum are complex-valued 2-D and 1-D processes, respectively. This opens a question of what
could be a proper way of filtering. For example, filtering can be applied to magnitude and phase func-
tions or to real and imaginary parts. The materials given in this Chapter partly address all these issues.
S
0.15
0.1
0.05
0.15
0.1
0.05
0
0
50 100 150 200 250
i
0.15
0.05
46
0.1
а b
50 100 150 200 250
0
0.15
0.1
0.05
c d
0
50 100 150 200 250
50 100 150 200 250
Fig. 2.1 Centered range profiles of a model missile for aspect angles 180˚ ( а), 175˚ (b),
174.95˚ (c), and 174.90˚ (d) in the case of input n oise absence.
2.1. 1-D non-adaptive filtering of signal Fourier spectrum estimates recovered by bispec-
trum estimation in Gausian and non-Gaussian noise environment
As already said, in opposite to conventional correlation and power spectral analysis, the main
benefits of bispectrum-based approach are the preservation of complex-valued signal Fourier spec-

trum, invariance property to random signal translation and rather sufficient suppression of additive
noise with symmetrical PDF shape. Due to these advantages, bispectrum-based signal processing
techniques allow quasi-coherent accumulating the observed noisy signals for recovering the estimates
of original signal shape in the case of signal shifts that randomly vary from one to another realization
recorded. However, reconstruction of signal waveform from noisy bispectrum estimate in case of a
practically limited ensemble realizations measured under low input SNRs leads to considerable errors
at signal reconstruction system and its performance becomes unsatisfactory. It should be especially
stressed, that the performance of bispectrum-based signal reconstruction techniques in case of influ-
ence of non-Gaussian noise, for instance, real-life mixture of AWGN and impulsive noise is poorly
studied and not considered in literature.
In this Chapter we describe and analyze such combined bispectrum-filtering signal reconstruc-
tion techniques [8, 32, 33, 42–49] that provide suppression of both AWGN and mixture of AWGN
and impulsive noise. We also focus on the low input SNR values because this case is the most inter-
esting and important for practice.
In practice, one often observes several realizations of a noisy signal of a fixed or almost constant
shape but with random mutual shifts. This occurs frequently in high-resolution radar systems when the
radar target signatures and range profiles [50, 51] are estimated (see above). These effects may be due
to the influence of external and internal AWGN, the heterogeneity of the propagation medium, ran-
dom vibrations of the radar platform or aerial random target motion.
In such situations the use of temporal accumulation of the recorded realizations for interference
suppression is ineffective and it smears the reconstructed signal waveform because of uncompensated
random shifts. Other, more complicated, methods involving estimation of the mutual shifts of the sig-
nal can be applied. However, in the case of a low input SNR, the estimates of mutual shifts of noisy
signals are poor [52], which also leads to distortions in the reconstructed signal waveforms. Note that,
for radar and sonar applications, low (below unity) input SNR is quite typical, and then it is especially
important to have as good reconstruction performance as possible.
Let us use signal and noise model in the form of observation equation (1.4.1) that is rather typi-
cal for radar applications.
As a typical scenario for radar applications, we consider the test signal given in the form of two
short pulses of rectangular shape. Such a signal with a pulse length of 3 samples, pulse amplitudes
equal to 2 and 6, and mutual pulse shift equal to 5 samples is shown in Fig. 2.1.1. Note that in radar
applications, prolonged objects are usually modeled as several backscattering centers [50, 51]. The
corresponding echo signal envelopes recorded for an arbitrary aspect angle have the form of several
47

pulses with different amplitudes. In the simplest case, but without losing a generality, a two-point
backscattered model can be used and considered as the test signal. A noisy realization of this test sig-
nal obtained for input SNR inp =0.46 is presented in Fig. 2.1.2. It can be seen from Fig. 2.1.2 that for
such a low SNR inp value, it is quite hard (if not impossible) to recognize the echo-signal waveform in
order to perform object classification.
s(i)
6
5
4
3
2
1
0
50 100 150 200 250
Figure 2.1.1. Noise-free original test signal.
i
48
x(i) 6
5
4
3
2
1
0
−1
−2
−3
50 100 150 200 250
Figure 2.1.2. Signal and AWGN mixture.
In general, the conventional bispectrum signal reconstruction permits to suppress the AWGN pro-
vided that a number M of mutually shifted realizations is large. However, in many practical cases, the
ensemble input averaged SNR inp value is low and a small number of observed realizations M is availa-
ble. In such situations, the conventional bispectrum signal waveform reconstruction [10] leads to large
errors in reconstructed signal shape.
Let us first consider a simplest case of M=1. Then a signal shape estimate can be obtained by fil-
tering an observed realization of noisy signal. At the same time, filtering can be also applied in spec-
tral domain. In case of using bispectrum-based processing such filtering can be applied to signal spec-
trum recovered from bispectrum.
For suppression of both AWGN and impulsive noise, the noise smoothing procedure of signal
Fourier spectrum estimate recovered from noisy bispectrum estimate performed by using of either li-
near or nonlinear filters [53] has been proposed by us in [42]. In fact, that was the first attempt of
combining the bispectrum based reconstruction of a signal shape with linear and nonlinear filtering.
The simplest sliding window linear and nonlinear filters are the mean and median filters, respectively
[53]. Thus, let us first consider just these very simple filters.
The proposed smoothing procedure [33, 42] of amplitude ̇ ˆ
( r)
and phase ˆ ϕ ( r)
signal
Fourier spectrum estimates performed by linear mean or nonlinear median filters can be respectively
written as
S bisp
bisp
i

S ̇ ˆ 1 ˆ
bisp−
mean ( r)
=
Ṡ
bisp ( r + n)
, (2.1.1)
2N
+ 1
N
∑
n=
− N
1
ˆ ϕ bisp−
mean ( r ) = ˆ ϕ bisp ( r + n)
, (2.1.2)
2N
+ 1
N
∑
n=
− N
ˆ̇ ˆ
( r)
= MED{
Ṡ
( k)
}, k = r − N,...,
r + N , (2.1.3)
Sbisp − median
bisp
ˆ ϕ ( ) = MED{
ˆ ϕ ( k)}
, k = r − N,...,
r + N , (2.1.4)
bisp − median
r bisp
where (2N+1) is the scanning window size of mean or median filter; MED{…} denotes the median
operation.
For performance comparison of different filters we have used the set of parameters given in
Chapter 1 and described by the formulas (1.5.7 – 11).
Five types of different signal processing procedures were studied in our numerical simulations:
Filter #1 – the conventional BLW bispectral filter [10]; Filter #2 – the conventional (nonlinear) me-
dian filter with different scanning window size (2N+1) [53] applied to noisy signal realization; Filter
#3 – the conventional (linear) mean filter with different scanning window size (2N+1) [53] applied to
noisy signal realization; Filter #4 – the proposed combined bispectral-median filter using (2.1.3) and
(2.1.4); Filter #5 – the proposed combined bispectral-mean filter using (2.1.1) – (2.1.2).
In Tables 2.1.1 and 2.1.2, the results of computer simulation for Gaussian and non-Gaussian
noise environment are presented, respectively. The input Gaussian noise variances
49
2
σ inp are equal to
0.3; 0.5 and 1.0 in Table 2.1.1. For impulsive “salt and pepper” noise the following parameters were
chosen: positive and negative impulsive noise amplitudes were fixed to Apoz=2 and Aneg=0,
respectively; the probabilities of negative Pneg and positive Ppos noise were equal to 5%, 10% and 30
% (Pneg=Ppos and total probability Pimp=Pneg+Ppos, see Table 2.1.2; the input Gaussian noise variance
2
σ inp was fixed to 0.3).
Analysis of the results presented in Table 2.1.1 demonstrates the following.
(a) The median filter (Filter #2) provides the best results (the largest ε ) for minimum consi-
dered window size value 2N+1=3 and maximum SNR inp . But with SNR inp decreasing the parameter
ε for the median filter decreases.

Table 2.1.1. Results obtained for AWGN environment.
Filter#
SNR inp SNR out
ε
SNR inp SNR out
Window size 2N+1=3
50
ε
SNR inp SNR out
1 3.166 5.898 1.863 1.907 3.989 2.092 0.947 1.948 2.056
2 3.166 16.295 5.147 1.907 9.783 5.130 0.947 4.775 5.042
3 3.166 8.991 2.840 1.907 6.844 3.598 0.947 4.363 4.607
4 3.166 7.887 2.491 1.907 5.343 2.802 0.947 2.644 2.792
5 3.166 8.368 2.643 1.907 5.767 3.024 0.947 2.829 2.987
Window size 2N+1=5
1 3.238 5.343 1.650 1.979 3.699 1.869 0.939 1.838 1.957
2 3.238 2.885 0.891 1.979 2.579 1.303 0.939 2.339 2.491
3 3.238 4.815 1.487 1.979 4.643 2.346 0.939 3.721 3.963
4 3.238 7.324 2.262 1.979 5.917 2.990 0.939 3.816 3.393
5 3.238 8.085 2.497 1.979 7.239 3.658 0.939 3.617 3.852
Window size 2N+1=7
1 3.142 5.458 1.737 1.945 3.390 1.743 0.993 2.038 2.052
2 3.142 2.825 0.899 1.945 2.729 1.403 0.993 2.414 2.431
3 3.142 4.037 1.285 1.945 6.331 3.255 0.993 3.490 3.515
4 3.142 9.605 3.057 1.945 5.934 3.051 0.993 4.027 4.055
5 3.142 11.063 3.521 1.945 6.518 3.351 0.993 4.611 4.644
Window size 2N+1=9
1 3.245 6.146 1.894 1.908 4.449 2.332 0.953 2.038 2.139
2 3.245 2.794 0.861 1.908 2.707 1.419 0.953 2.492 2.615
3 3.245 3.657 1.127 1.908 3.612 1.893 0.953 3.628 3.429
4 3.245 10.731 3.307 1.908 9.626 5.045 0.953 4.380 4.596
5 3.245 12.72 3.920 1.908 11.675 6.119 0.953 4.927 5.170
ε

Table 2.1.2. Results obtained for mixture of AWGN (
Filter# SNR inp SNR out
ε
SNR inp SNR out
Window size 2N+1=3
51
2
σ inp = 0.3) and impulsive noise (Ppoz=Pneg).
ε
SNR inp SNR out
1 2.030 3.110 1.532 1.553 2.729 1.757 0.991 1.794 1.810
2 2.030 10.990 5.414 1.553 7.530 4.849 0.991 3.112 3.140
3 2.030 6.703 3.302 1.553 5.404 3.480 0.991 3.540 3.572
4 2.030 4.413 2.174 1.553 4.319 2.781 0.991 2.910 2.936
5 2.030 4.498 2.211 1.553 4.338 2.793 0.991 2.478 2.501
Window size 2N+1=5
1 2.008 3.032 1.510 1.767 2.070 1.774 0.910 1.675 1.841
2 2.008 2.885 1.437 1.767 3.193 1.807 0.910 2.640 2.901
3 2.008 4.371 2.177 1.767 4.158 2.353 0.910 3.049 3.353
4 2.008 5.012 2.496 1.767 5.660 3.203 0.910 3.274 3.598
5 2.008 5.012 2.496 1.767 6.121 3.464 0.910 3.226 3.545
Window size 2N+1=7
1 1.950 3.050 1.564 1.709 3.114 1.822 1.077 2.070 1.922
2 1.950 2.808 1.440 1.709 2.883 1.687 1.077 2.477 2.300
3 1.950 3.793 1.945 1.709 3.857 2.257 1.077 3.180 2.953
4 1.950 5.146 2.639 1.709 5.952 3.483 1.077 4.056 4.184
5 1.950 5.581 2.862 1.709 6.953 3.858 1.077 4.962 4.607
Window size 2N+1=9
1 1.871 3.314 1.771 1.749 3.232 1.848 1.074 1.939 1.805
2 1.871 2.702 1.444 1.749 2.739 1.566 1.074 2.442 2.274
3 1.871 3.246 1.735 1.749 3.446 1.970 1.074 2.900 2.701
4 1.871 6.971 3.726 1.749 6.576 3.760 1.074 4.039 3.761
5 1.871 7.475 3.995 1.749 7.132 4.078 1.074 4.219 3.928
ε

(c) The combined bispectral-median/mean filters (Filters ##4 and 5) lose to the median and
mean filters in case of window size 2N+1=3 and maximum value SNR inp . With increase of window
size 2N+1 the median and mean filter performance reduces but for the combined bispectral-
median/mean filters their performance improves. In particular, the maximum ε reaches 6.119 for the
combined bispectral-mean filter (Filter#5) for window size of 2N+1=9.
(d) Thus, the proposed combined bispectral-mean/median method occurs to be more effective
than conventional filters for suppression of AWGN when SNR inp values are small.
Analysis of numerical simulation results presented in Table 2.1.2 demonstrates the following.
(a) In general, all filters provide worse suppression of mixed noise than for AWGN (see Table
2.1.1 for comparing).
(b) The median filter (Filter #2) provides the best performance for minimum window size val-
ue 2N+1=3 and maximum SNR inp value. But with decreasing of SNR inp the parameter ε for median
filter decreases faster comparing to the results presented in Table 2.1.1.
(c) The mean filter (Filter #3) provides worse mixed noise suppression than the median filter for
minimum window size value 2N+1=3 and for large SNR inp . However, with SNR inp decreasing the
mean filter parameter ε increases at the same time.
(d) The combined bispectral-median/mean filters (Filters ##4 and 5) work worse than the me-
dian and mean filters for minimum window size 2N+1=3 and for large values of SNR inp . With win-
dow size increasing and for SNR inp decreasing, both the median and mean filter parameters ε de-
crease but the combined bispectral-median/mean filter performance remains the same or even im-
proves.
(e) Thus, the proposed combined bispectral-mean/median filters [33, 42] imply rather good sta-
bility to influence of intensive mixed noise and they occur to be useful for non-Gaussian noise sup-
pression when SNRinp values are small.
Numerical simulation results demonstrate promising potential of the proposed combined
processing for suppression of both intensive AWGN and mixed non-Gaussian noise. In doing so, the
proposed technique incorporates the advantages of both bispectral processing and linear or non-linear
filtering. Note that the advantages of the combined technique [42] appear themselves when the signal
is corrupted by intensive AWGN.
52

2.2. Smoothing the noisy bimagnitude and biphase or real and imaginary parts of bispectrum
estimates by using non-adaptive 2-D linear and nonlinear filtering
Another approach suggested for improving bispectrum-based signal reconstruction performance
is considered in papers [8, 33]. The main idea of this approach is in smoothing the noisy bispectrum
estimates by using 2-D linear and nonlinear filtering.
First, let us study by visual analysis the distortions of bispectrum estimate for above considered
test signal shown in Fig. 2.1.1. The magnitude and phase bispectra of this noise-free signal are shown
in Figures 2.2.1 and 2.2.2, respectively.
Obviously, the magnitude bispectrum is a rather smooth function that does not contain disconti-
nuities. The basic information maximums of the magnitude bispectrum are concentrated in the domain
of low frequencies (central part of bifrequency plane) and phase bispectrum function contains the
number of discontinuities (phase wrapping).
Fig. 2.2.1. Noise-free bimagnitude as a function
of two frequencies.
Fig. 2.2.2. Noise-free biphase as a function of
two frequencies.
Examples of noisy bispectrum magnitude and phase estimates ensemble averaged for M=256 rea-
lizations and computed for SNR inp =0.5 are demonstrated in Figures 2.2.3 and 2.2.4, respectively. It is
clearly seen that despite the relatively large realization number of bispectrum estimates (M=256) par-
ticipating in ensemble averaging, bimagnitude and biphase functions are severely destroyed by
AWGN present in the original process that leaks into bispectral domain.
53

Fig. 2.2.3. Noisy bimagnitude as a
function of two frequencies.
54
Fig. 2.2.4. 3-Noisy biphase as a
function of two frequencies.
Our strategy suggested in [33] is based on the reduction of the reconstructed signal estimation
errors by smoothing the noisy bispectrum estimates with 2-D non-adaptive linear or nonlinear filters
before signal Fourier spectrum recovery. It was proposed to improve the bispectrum estimate by either
smoothing the phase and magnitude bispectra or filtering the real and imaginary parts of bispectrum
estimate.
Our preliminary investigations [42] have demonstrated that the smoothing of the TAF estimates
is inefficient for short pulse signals considered. The reason is that the main lobe width of the TAF es-
timate in the considered case is small and, thus, the nonlinear/linear filters not only reduce noise but
also deteriorate useful information features of these triple correlation estimates.
On the contrary, for short pulse signals the corresponding bimagnitude
ˆ̇
( p,
q)
and biphase
Bs ind
ˆ γ ( p,
q)
are the relatively slowly varying functions in bifrequency plane (subscript ind denote the
s ind
computation by using indirect technique (see (1.3.1 – 1.3.3)). Therefore, it can be expected that effec-
tive noise suppression can be provided by linear or nonlinear filters if they are applied to the functions
̇ ˆ
( p,
q)
and ˆ γ ( p,
q)
or to filtering the real Re{ ˆ̇
( p,
q)}
and imaginary Im{ ˆ̇
( p,
q)}
Bs ind
s ind
Bs ind
Bs ind
parts of bispectrum estimate.
The smoothed bimagnitude
~
ˆ̇
~
( p,
q)
and biphase ˆ γ ( p,
q)
estimates or the filtered real
Bs ind
R
~
e{
̇ ˆ
( p,
q)}
and imaginary m { ˆ ( , )}
~ I ̇ p q components obtained at the output of linear (for ex-
Bs ind
Bs ind
ample, mean) or nonlinear (for instance, median) filter can be, respectively, written as follows
s ind

~
̇ ˆ
~
( p,
q)
, ˆ γ ( p,
q)
= MEAN / MED{
B
ˆ̇
( p,
q)
}, { ˆ γ ( p,
q)}
, (2.2.1)
BMean / Median
Mean / Median
s ind
s ind
Re ̃ {
ˆ
( , )},Im {
ˆ
/
Ḃ Mean Median sind
p q ̃
/
Ḃ Mean Median sind
( p, q)}
=
= MEAN / MED{Re{ Ḃ ˆ
( p, q)}},{Im{ Ḃ ˆ
sind sind
( p, q)}}
, (2.2.2)
where MEAN{…} denotes the 2-D mean operation, for example,
̇ 1
)} ; MED{…} denotes the 2-D median
N N
m Mean{
B
ˆ
s ind ( p,
q)}
= ∑ ∑ Im{ B
ˆ
s ind ( p + n , q + n
( 2N
+ 1)
n1
= − N n2
= − N
~
I ̇
2
1 2
operation, for example, ~
e {
ˆ̇
( p,
q)}
= MED{Re{
B
ˆ̇
( p , )}}, p1=p-N,…,p+N; q1=q-
R Median Bs ind
s ind 1 q1
N,…,q+N; (2N+1)(2N+1) is the scanning 2-D window size (WS).
Note that the selection of the standard median and mean filters [53] is rather arbitrary because
we study the possibility of bispectral estimator improving by typical linear and non-linear filters. To
obtain best possible performance one should optimize the filter for this particular application.
Below we will compare the performance of the proposed approach with two conventional ap-
proaches: 1) the approach of Bartelt et al [10] (BLW technique or T#1); 2) the approach of Sundara-
moorthy et al [18] (referred as SRD technique or T#2).
Our combined approach leads to the following eight different combined bispectrum-filtering
signal reconstruction techniques: T#3 based on BLW and bimagnitude/biphase smoothing by 2-D
mean filter; T#4 that uses BLW and bimagnitude/biphase smoothing by 2-D median filter; T#5 based
on SRD and bimagnitude/biphase smoothing by mean filter; T#6 that implies the SRD and bimagni-
tude/biphase smoothing by median filter; T#7 that uses the BLW and smoothing of real/imaginary
bispectrum estimate components by mean filter; T#8 based on BLW and smoothing of real/imaginary
bispectrum estimate components by median filter; T#9 based on SRD and smoothing of
real/imaginary bispectrum estimate components by mean filter; T#10 that uses SRD and the smooth-
ing of real/imaginary bispectrum estimate components by median filter.
In addition to the set of parameters (1.5.7 – 1.5.11) serving for performance evaluation for the
combined bispectrum-filtering signal reconstruction techniques, we also introduce the MS bias
2
δ outT
# 1...
T # 10 of the reconstructed signal that determines integrated dynamic errors
1 I
∑ −1
2
2
δ out T # 1...
T # 10 = [ s(
i)
− sˆ
T # 1...#
10 ( i)]
. (2.2.3)
I i=
0
55

Fig. 2.2.5. Results for T#1 and
T #7.
Fig. 2.2.8. Results for T#2 and
T #9.
Fig. 2.2.6. Results for T#2 and T
#5.
Fig. 2.2.9. Results for T#2 and
T#10.
56
Fig. 2.2.7. Results for T#2 and T
#6.
Fig. 2.2.10. Results for T#1 and
T#3.

Output variance
0,014
0,012
0,01
0,008
0,006
0,004
0,002
0
30 15 10 5 3 1
Probability on the input, %
T#2 T#5 WS: 3x3
T#5 WS: 5x5 T#5 WS: 7x7
Fig. 2.2.11. Results for T #2 and T
Output variance
0,014
0,012
0,01
0,008
0,006
0,004
0,002
0
#5.
30 15 10 5 3 1
Probability on the input, %
T#2 T#10 WS: 3x3
T#10 WS: 5x5 T#10 WS: 7x7
Fig. 2.2.14. Results for T#2 and
T#10.
Output variance
0,014
0,012
0,01
0,008
0,006
0,004
0,002
0
30 15 10 5 3 1
Probability on the input, %
T2# T#6 WS: 3x3
T#6 WS: 5x5 T#6 WS: 7x7
Fig. 2.2.12. Results for T#2 and
T#6.
MS bias
0,05
0,045
0,04
0,035
0,03
0,025
0,02
0,015
0,01
0,005
0
0,5 1 2 5 10 20
57
Input SNR
T#1 T#7 WS: 3x3
T#7 WS: 5x5 T#7 WS: 7x7
Fig. 2.2.15. Results for T#1 and
T#7.
Output variance
0,014
0,012
0,01
0,008
0,006
0,004
0,002
0
30 15 10 5 3 1
Probability on the input, %
T#2 T#9 WS: 3x3
T#9 WS: 5x5 T#9 WS: 7x7
Fig. 2.2.13. Results for T#2 and
MS bias
0,07
0,06
0,05
0,04
0,03
0,02
0,01
0
T#9.
0,5 1 2 5 10 20
Input SNR
T#1 T#8 W S: 3x3
T#8 W S: 5x5 T#8 W S: 7x7
Fig. 2.2.16. Results for T#1 and
T#8.

MS bias
0,06
0,05
0,04
0,03
0,02
0,01
0
0,5 1 2 5 10 20
Input SNR
T#2 T#9 WS: 3x3
T#9 WS: 5x5 T#9 WS: 7x7
Fig. 2.2.17. Results for T#2 and
T#9.
The values of
2
2
σ out T # 1...
T # 10 (1.5.9) and out T # 1...
T # 10
58
MS bias
0,06
0,05
0,04
0,03
0,02
0,01
0
0,5 1 2 5 10 20
Input SNR
T#2 T#10 WS: 3x3
T#10 WS 5x5 T#10 WS 7x7
Fig. 2.2.18. Results for T#2 and T#10.
δ (2.2.3) as the functions of SNR inp (1.5.2) or
probability of impulsive noise Ppos=Pneg computed for K=30 experiments are shown in Figures 2.2.5 –
2.2.18. The results are obtained for different scanning WS. The input AWGN variances
2
σ inp (2.2.13)
for M=256 are equal to 1.12; 0.56; 0.28; 0.11; 0.056, and 0.028. They correspond to the SNR inp equal
to 0.5; 1.0; 2.0; 5.0; 10.0 and 20.0, respectively. The impulsive noise parameters are: pulse magni-
tudes Apos=Aneg=1; Ppos=Pneg=30%; 15%; 10%; 5%; 3% and 1%; total spike probability Pimp=Ppos+Pneg.
The analysis of the plots in Figures 2.2.5 – 2.2.18 permits to conclude the following.
2
(a) The variances σ decrease with SNR inp increasing (see Figures 2.2.5 – 9) for
the case of AWGN.
out T # # 1−3,
5,
6,
7,
9,
10
2
(b) The output variances σ also reduce with decreasing of Ppos=Pneg (see Figures
2.2.10 – 18).
out T # # 1,
2,
5,
6,
9,
10
(c) The advantages of the proposed approach in comparison to the conventional bispectral tech-
niques are especially well observed for low SNR inp values and for high probabilities of spikes
Ppos=Pneg, i.e. for the situations which are the most complicated and for which the improvement of
reconstruction system performance is extremely required. The output variances as well as the slope of
output variance variations depend on the signal reconstruction technique as well as on the WS.

(d) Great improvement of signal reconstruction fidelity due to effective AWGN suppression was
achieved for the proposed T#7 in comparison to the conventional T#1. In this case the output variance
has been decreased by more than 3 times for SNR inp =0.5 due to smoothing the real and imaginary
bispectrum estimate components by 2-D mean filter with the scanning WS=7x7 (see Fig. 2.2.5); the
nonlinear processing using smaller WS (5x5 and 3x3) also provide the output variance decreasing in
comparison to the conventional T#1 that is rather large just for relatively small SNR inp .
(e) For the case of AWGN, the absolute values of output variance that have been observed for
the conventional T#2 are larger than the corresponding ones for conventional T#1. This conclusion
follows from comparison of the solid curves in Figures 2.2.5 and 2.2.6.
(f) Radical improvement of signal reconstruction fidelity for the case of AWGN is achieved for
the proposed T#9 and T#10 in comparison to the conventional T#2. The output variances have been
decreased by more than 7 times due to smoothing the real and imaginary components by 2-D mean
(see Fig. 2.2.8) and median (see Fig. 2.2.9) filters with WS=7x7 when SNR inp is small, for instance,
SNR inp =0.5; the application of the filters with the smaller WS also results in output variance reduc-
tion; the results obtained for the standard mean filter (for fixed scanning WS) are a little bit better than
for median filter.
(g) The proposed T#9 and T#10 that perform smoothing the real and imaginary bispectrum es-
timate components (see Figures 2.2.8 and 2.2.9, respectively) ensure better suppression of AWGN in
comparison to T#5 and T#6 that operate with smoothing the phase and magnitude bispectrum esti-
mates (see Figures 2.2.6 and 2.2.7, respectively).
(h) For the impulsive noise case, the maximum output variance values that have been obtained
for the conventional T#1 are larger than the corresponding values for the conventional T#2 (see Fig-
ures 2.2.10 – 12). In other words, the conventional T#2 is more robust to impulsive noise influence
than the conventional T#1. The rapid increasing of output variance observed for T#1 (Fig. 2.2.10) in
the case of rather large probability of impulsive noise (15%) can be explained by non-zero probability
of occurrence of abnormal outputs (failures) of signal reconstruction system (similar phenomena are
described in the paper [15]). For such outputs the fluctuation errors for some experiments in (2.2.15)
drastically differ from the values typical for normal experiments. Thus, in case of abnormal experi-
2
ment presence among K experiments, the value σ greatly increases.
outT
# 1
(i) The best result in the sense of impulsive noise removal was achieved for the proposed T#9.
The output variance has decreased by approximately 7 times when the mean filter with WS= 7x7 was
applied for the case of maximum impulsive noise probabilities Ppos= Pneg= 30% (see Fig. 2.2.13).
59

(k) The reconstructed signal estimates were found biased for all techniques investigated both for
Gaussian and non-Gaussian noise environments. But the maximum MS bias values normalized by the
signal power Ps=0.558 are not larger than 4.5% for the proposed T##7 – 10 (see the Figures 2.2.15 –
18) with employing the scanning WSs larger than 3x3 (the case of AWGN). At the same time, the
maximum MS biases normalized by Ps are not larger than 10% for the proposed T##7 – 10 for using
the maximum WS=7x7 for the case of impulsive noise. In general, the presence of impulsive noise,
especially with high probability, can lead to considerable MS bias.
(l) The MS bias values were found to be always less for the conventional T#1 and T#2 in com-
parison to the proposed techniques only for the case of impulsive noise.
(m) For the case of AWGN the MS bias decreases quasi-monotonically with SNR inp increasing
both for the conventional T#1 and the proposed T#7 and T#8 (see Figures 2.2.15 and 16) as well as
for the conventional T#2 and the proposed T#9 and T#10 (see Figures 2.2.17 and 18) for the smallest
WS=3x3.
(n) The MS bias values have approximately the constant values over the entire examined range
of SNR inp variation for both the proposed T##7, 8 (see Figures 2.2.15 and 16 except the smallest scan-
ning WS of 3x3 samples) and T##9, 10 (see Figures 2.2.17 and 18 except WS=3x3). Thus, the scan-
ning WS increasing does not lead to sufficient bias of the reconstructed signal estimates for the case of
AWGN.
(o) The numerical experiments have been performed for different filter scanning WSs ranging
from 3x3 to 11x11 samples, but practically for all the proposed techniques for the considered test sig-
nal model the best results have been obtained for the WS=7x7. However, this does not mean that this
choice could be the best for other test signals and types of filters that can be examined for the particu-
lar application.
Note that consideration of only mean and median smoothing filters in our numerical experi-
ments in [33] is not exhaustive and served only the purpose to demonstrate that the proposed approach
is promising and challenging.
It has been shown using different sets of simulation examples that the techniques proposed in
[33] may yield a significantly better waveform reconstruction than the conventional bispectrum-based
techniques. It should be especially noted that the proposed combined bispectrum-filtering techniques
are robust not only to random signal shifts and additive Gaussian noise but to impulsive noise as well.
The statistical experiments obtained for variety of test cases show that the filtering of real and
imaginary components of noisy bispectrum estimates is more efficient than the processing of bimagni-
60

tude and biphase estimates. One reason is that real and imaginary components of noisy bispectrum
estimates are more “suitable” 2-D processes for filtering than bimagnitude and biphase estimates since
the latter ones contain discontinuities that are difficult to preserve.
The selection of the appropriate technique and its parameters among the proposed ones in [33]
depends upon particular application, noise environment properties, criterions used and priority of re-
quirements to signal reconstruction system accuracy.
For solving the problems of signal classification and object identification, resolution and recog-
nition, it is more important to take into account the output variance rather than MS bias. For the case
of signal reconstruction with the aim of further estimation of its parameters, both MS bias and output
variance should be considered. In our opinion, the proposed techniques [33] can be useful for solving
aforementioned tasks in practical applications in high resolution radar range profiles estimation in the
cases of a priori unknown signal and noise autocorrelation and cross-correlation characteristics.
Note that the performance of the combined bispectral-filtering methods depends upon the fol-
lowing factors:
a) the statistical properties of noise leaked into bispectral domain from signal temporal domain;
b) the peculiarities of behavior of information components of bispectrum estimate that are dependent
on the parameters of original signal and noise mixture and these properties are often a priori unknown;
c) the type and parameters of 2-D filter applied for bispectrum estimate smoothing.
In such situation it is intuitively clear that the best performance could be provided in case of ap-
plication of adaptive 2-D filters or, at least, adaptive selection of the best non-adaptive filter from a set
of available ones (filter bank) with taking into account the available information about signal and
noise properties that is commonly limited. However, to ensure an opportunity for such adaptation,
quite intensive study and analysis of different filter performance for wide variety of possible signals
and noise environment should be carried out. So, our attempts below can be considered as initial steps
intended for solving this complicated task.
First, we have started with studying the performance of two other types of nonlinear non-
adaptive filters [43]. Among a large number of modern nonlinear filters the K-nearest neighbor
(KNN) filter [54] and the FIR-median hybrid (FMH) filter [53] have been selected for solving a prob-
lem of signal reconstruction by bispectrum estimation.
The output of KNN filter is given as the mean value of the KNN (1≤ KNN ≤NxN) samples whose
values are the closest to the value of the central sample of the filter window with the size of NxN sam-
ples. The output of FMH filter is defined as the median value of the set of FIR subfilters. The scan-
61

ning WS of KNN filter can be different while FMH filter has only the version with the fixed WS of
5x5 samples.
The two types of scanning window filters listed above do not presume the availability of a priori
information about noise type (additive, multiplicative, etc) and characteristics (for example, variance)
like some other types of filters do (for instance, the standard sigma or the local statistic Lee filters).
This is, at the same time, good and bad, since, on one hand, the selected two filters can be easily ap-
plied to processing the noisy bispectrum estimates without making up preliminary analysis of their
statistical properties. On the other hand, one can expect better performance for the case of more so-
phisticated filter application that take into account the characteristics of noise in bispectrum estimates
to be processed. But this is a direction of further investigations in the next subsections of this work.
~
The smoothed bimagnitude ˆ
( p,
q )
~
̇ and biphase ˆ γ ( p , q ) or the filtered real R
~
e {
ˆ
( p,
q )}
B s
62
s
̇ and
imaginary m {
ˆ
( , )}
~
I ̇ p q parts of bispectrum estimate computed at the output of the KNN or FMH fil-
B s
ters are, respectively, represented in [43] as
~
̇ ˆ
( p,
q)
= KNN{
B
ˆ̇
( p,
q)}
, (2.2.4)
BKNN s
~
ˆ γ ( , q)
= KNN{
ˆ γ ( p,
q)}
, (2.2.5)
KNN
p s
~
ˆ̇
( p,
q)
= FMH{
Ḃ
ˆ
( p,
q)},
(2.2.6)
BFMH s
~
ˆ γ ( , q)
= FMH{
ˆ γ ( p,
q)}
, (2.2.7)
FMH
p s
R
~
e {
ˆ̇
( p,
q)}
= KNN{Re[
B
ˆ̇
( p,
q)]}
, (2.2.8)
KNN
Bs s
m {
ˆ
( , )} {Im[
ˆ
( , )]}
~
I ̇ p q = KNN Ḃ
p q , (2.2.9)
KNN
Bs s
R
~
e {
ˆ̇
( p,
q)}
= FMH{Re[
B
ˆ̇
( p,
q)]},
(2.2.10)
FMH
Bs s
m {
ˆ
( , )} {Im[
ˆ
( , )]},
~
I ̇ p q = FMH Ḃ
p q
(2.2.11)
FMH
Bs s
where KNN{…} denotes the filtering procedure performed by the 2-D KNN filter; FMH{…} denotes
the filtering procedure for the 2-D FMH filter.
The five following algorithms are investigated and compared to each other in [43]:
a) the conventional BLW technique (Technique #1);
b) the proposed combined technique based on BLW technique and the bimagnitude/biphase smooth-
B s

ing by the 2-D KNN filter (2.2.4 – 5) (Technique #2);
c) the combined technique that uses BLW technique and the bimagnitude/biphase smoothing by the 2-
D FMH filter (2.2.6 – 7) (Technique #3);
d) the combined technique based on BLW technique and the real and imaginary bispectral estimate
components smoothing by the 2-D KNN filter (2.2.8 – 9) (Technique #4);
e) the combined technique based on BLW technique and the smoothing of real and imaginary bispec-
trum estimate components by 2-D MFH filter (2.2.10 – 11) (Technique #5).
To analyze and compare the signal reconstruction performance, we propose to follow two ap-
proaches. The first one assumes visual express analysis of the reconstructed signal plots. The second
one implies more detailed quantitative comparative analysis of the graphs demonstrated the fluctuative
and bias errors at the output of reconstruction system as well as the analysis of estimation consistency.
The reconstructed signal plots obtained for 256 Monte Carlo runs performed for AWGN and
mixed AWGN and impulsive noise are represented in Figures 2.2.19 – 24.
The signal reconstructed by conventional Technique #1 for the case of AWGN is shown in Fig.
2.2.19. The identification of the small intensity object (small pulse magnitude) from Fig. 2.2.19 by
visual analysis is impossible.
As clearly seen from Fig. 2.20, the reliable identification of the small intensity object is pro-
vided by using the proposed Technique #5.
The signal reconstructed in case of the mixed noise by Technique #1 is shown in Fig. 2.2.21.
It should be stressed, that the spikes are “smeared-out” in Fig. 2.2.21 and reasonably good per-
formance of the conventional Technique #1 is achieved.
Fig. 2.2.19. Signal reconstructed
by Technique #1 (AWGN,
SNR inp =0.6).
Fig. 2.2.20. Signal reconstructed
by Technique #5 (AWGN,
SNR inp =0.6).
63
Fig. 2.2.21. Signal reconstructed
by Technique #1 (mixed noise,
2
σ inp =0.4; Ppos = Pneg =12%).

Fig. 2.2.22. Signal reconstructed
by Technique #5 (mixed noise,
2
σ inp =0.4, Ppos = Pneg =12%).
Fig. 2.2.23. Signal reconstructed
by Technique #2 for NxN=7x7;
KNN=24 samples (mixed noise,
Ppos = Pneg =12%;
2
σ inp =0.4).
64
Fig. 2.2.24. Signal reconstructed
by Technique #4 for NxN=7x7;
KNN=24 samples (mixed noise,
Ppos = Pneg =12%;
2
σ inp =0.4).
The smoothing of the real and imaginary bispectrum parts by 2-D FMH nonlinear filter permits
to provide better performance of the proposed combined bispectrum-filtering signal reconstruction
Technique # 5 (see Fig. 2.2.22) than for the conventional Technique #1 (see Fig. 2.2.21) in the sense
of better mixed noise suppression in the reconstructed signal.
Better efficiency of the proposed Technique #4 (real and imaginary bispectrum parts smoothing)
with respect to the proposed Technique # 2 (bimagnitude and biphase filtering) in the case of KNN
filter application is demonstrated for the case of mixed noise (compare the Figures 2.2.23 and 2.2.24).
For assessing the reconstructed signal estimation consistency, the output system ensemble aver-
2
2
aged fluctuation variances σ (1.5.9) as well as the MS bias δ (2.2.3) for the Techniques
outT # 1,
4,
5
outT # 1,
4,
5
##1, 4 and 5 were calculated depending on the number of Monte Carlo runs (M=1, 10, 50, 100, 256
2
2
and 512 realizations). While calculating the statistical parameters σ and δ , each set of
M realizations was repeated K=30 times.
outT # 1,
4,
5
outT # 1,
4,
5
The plots of output variances and MS bias depending upon observed realization number M for
the case of AWGN are represented in Figures 2.2.25 and 2.2.26, respectively ( σ =0.8=const and the
maximal random signal shift deviation of 24 samples were modeled). The plots of output variances
and MS bias depending upon M for the case of mixed white AWGN ( σ =0.4=const and the maximal
random signal shift deviation of 24 samples were modeled) and impulsive noise (Ppos=Pneg=6%) are
given in Figures 2.2.27 and 2.2.28, respectively.
2
inp
2
inp

Output variance
0.6
0.5
0.4
0.3
0.2
0.1
0
1 10 50 100 256 512
Number of realizations
T#1 T#4
T#5
Fig. 2.2.25. Output variance as a function of reali-
zation number M (AWGN).
Output variance
0.3
0.25
0.2
0.15
0.1
0.05
0
1 10 50 100 256 512
Number of realizations
T#1 T#4
T#5
Fig. 2.2.27. Output variance as a function of reali-
zation number M (mixed noise).
65
Bias
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
1 10 50 100 256 512
Number of realizations
T#1 T#4
T#5
Fig. 2.2.26. MS bias as a function of realization
number M (AWGN).
Bias
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
1 10 50 100 256 512
Number of realizations
T#1 T#4
T#5
Fig. 2.2.28. MS bias as a function of realization
number M (mixed noise).
The analysis of these plots shows that the output variance values (see Figures 2.2.25 and 2.2.27)
gradually approach to zero values, i.e., tend to be strictly consistent in mathematical sense. This means
that for both conventional Technique #1 and the proposed Techniques ## 4 and 5 (as well as for Tech-
niques ## 2 and 3 for which the plots have similar behavior) there exists some M0 that for M≥M0 the
2
random fluctuation errors σ can be considered negligibly small. Then for practical applications
outT # 1,
4,
5
it is possible to find such M0 that satisfies some reasonable preset requirements to the reconstruction
system accuracy.
MS bias errors for the proposed bispectrum-filtering Techniques ## 4 and 5 are practically the
same as for the conventional Technique #1 (see Figures 2.2.26 and 2.2.28 for comparison) for realiza-
tions number M≥50 for AWGN (see Fig. 2.2.26) and they slightly differ for the case of mixed noise
(see Fig. 2.2.28).

2
2
In Tables 2.2.1 and 2.2.2 the values of σ and δ depending on the
outT # 1...
5
the case of Gaussian and on the probability Ppos = Pneg (
66
outT # 1...
5
2
inp
2
σ inp / SNR inp for
σ =0.4 was fixed value in case of mixed
noise) for the case of mixed noise are represented, respectively. The scanning window size for differ-
ent filters was fixed to NxN=5x5, for KNN filter KNN=12. These data have been obtained for fixed
number of M=256 realizations and K=30 experiments.
Table 2.2.1. Numerical simulation results for the case of AWGN.
2
σ inp /
SNR inp
Technique #1 Technique #2 Technique #3 Technique #4 Technique #5
σ
2
outT # 1
σ
2
2
δ outT # 1 outT # 2
δ
2
outT # 2
σ
2
outT # 3
3.2/0.3 0.0363 0.0322 0.0435 0.0264 0.0285 0.0306 0.0248 0.0259 0.0189 0.0344
1.6/0.6 0.0294 0.0164 0.0186 0.0154 0.0110 0.0197 0.0100 0.0136 0.00885 0.0211
0.8/1.2 0.0134 0.00846 0.00609 0.00855 0.00516 0.0124 0.00395 0.00712 0.00352 0.0128
0.4/2.3 0.0108 0.00510 0.00335 0.00540 0.00226 0.00733 0.00251 0.00520 0.00193 0.00763
Table 2.2.2. Numerical simulation results for the case of mixed Gaussian ( σ =0.4=const) and impul-
sive noise.
Ppos=Pneg
%
δ
2
outT # 3
σ
2
outT # 4
δ
2
inp
2
outT # 4
σ
2
outT # 5
Technique #1 Technique #2 Technique #3 Technique #4 Technique #5
σ
2
outT # 1
σ
2
2
δ outT # 1 outT # 2
δ
2
outT # 2
σ
2
outT # 3
24 0.00496 0.0123 0.00358 0.0194 0.0025 0.0328 0.00272 0.019 0.00238 0.0329
12 0.00451 0.0103 0.00341 0.0143 0.00232 0.0253 0.00227 0.0134 0.00201 0.0250
6 0.00375 0.0087 0.00326 0.00937 0.00217 0.0185 0.00195 0.00927 0.00172 0.0178
3 0.00308 0.00746 0.00432 0.00750 0.00226 0.0140 0.00229 0.00683 0.00184 0.0134
δ
2
outT # 3
σ
2
outT # 4
δ
2
outT # 4
σ
2
outT # 5
δ
2
outT # 5
2
As seen from Table 2.2.1 the fluctuation error σ decreases by approximately 3 times with
increasing of inp
increasing of SNR inp .
outT # 1
2
SNR by 8 times. At the same time, the MS bias out T # 1
δ
2
outT # 5
δ decreases proportionally to
2
The fluctuation error σ decreases by approximately 13 times with increasing of SNR inp by
2
8 times. The MS bias out T # 2
outT # 2
2
δ is about the same level as out T # 1
KNN=12 produces small dynamic errors in comparison to FMH filter.
The fluctuation errors
2
outT # 3
δ . The KNN filter with NxN=5x5 and
σ are less than for the Technique #1, and the MS bias values
2
δ out T # 3 are only slightly worse than for Technique #1 because any smoothing filter, clearly, always
introduces some additional dynamic errors.
It should be stressed, that the proposed smoothing of real and imaginary bispectral estimation
components by 2-D KNN or FMH filters (compare the data for the Techniques ## 4 and 5 to the data

for all other Techniques) provide the best results in the sense of producing smaller fluctuation errors.
At the same time, the MS bias values for the Technique #5 are the worst among the considered Tech-
niques.
For low SNR inp ( SNR inp =0.3 and 0.6), the MS bias values are minimal in the case of the KNN
filter application (see the data for Technique# 2).
For the case of mixed noise, the minimum MS bias values are observed for the conventional
Technique #1 (see Table 2.2.2). The smoothing of real and imaginary bispectrum components by 2-D
FMH nonlinear filter (see data in Table 2.2.2 obtained for the proposed Technique #5) still provides
the minimum fluctuation errors in comparison to Techniques ## 1 – 4. As one can see, the fluctuation
2
errors σ tend to decreasing when the values Ppos=Pneg reduce. But for minimal considered value
outT # 2−5
Ppos=Pneg=3% these errors slightly increase. This peculiarity, to our opinion, can be caused, on one
hand, due to complicated contribution (“re-distribution”) of Gaussian and non-Gaussian noise compo-
nents in the bispectrum estimate and, on the other hand, by individual peculiarities of the investigated
types of nonlinear filters that smooth Gaussian and impulsive noise components in different manner.
This problem is one more distinctive subject of our investigations and will be discussed below.
It should be stressed, that the efficiency of nonlinear filtering applied in bispectral domain large-
ly depends on the statistical property of complex-valued bispectrum estimate and, first of all, on its
PDF shape and stationarity in bifrequency plane. Note that this property has not been studied yet and
not considered in the literature. Because of this, we have paid attention to the problem of statistical
assessing the behavior of complex-valued noisy bispectrum estimate in our papers [8, 45].
First, let us address the error (noise induced) component ˆ̇
( p,
q)
in bispectrum estimate (1.4.8)
containing six signal-dependent and one non-signal-dependent terms. Analysis of these error compo-
nent caused by noise leakage from signal reconstruction system input performed in [8] permits to note
the following important peculiarities.
1) With increasing realization number M participating in ensemble averaging, the last non-
signal-dependent term
E Ṅ p Ṅ q Ṅ p q in formula (1.4.8) asymptotically tends to zero if
*
[ m( ) m( ) m(
+ )]
AWGN {n (m) (i), i=0,1,…,I-1; m=1, 2,…, M} in (1.4.1) is supposed to be of zero mean. It has been
demonstrated in [8] that the real and imaginary components of this complex-valued random process
are not Gaussian (see the histogram in Fig. 2.2.29.a) although for large M it approximates Guassian
PDF according to the central limit theorem (see the histogram in Fig. 2.2.29,b). These histograms have
been computed for SNRinp=0, when only the last term in (1.4.8) is of non-zero value.
67
B err

Fig. 2.2.29a. Histogram of bispectrum real part values for signal-absence case, M =1.
2.2.29b. Histogram of bispectrum real part values for signal-absence case, M =50.
2) Since the considered AWGN is of zero mean, hence, for large M the terms
E Ṅ p q e E Ṅ q e E Ṅ p e .
*
− j2 πτm ( p+ q) j2πτ mq j2πτ mp
[ m( + ) ] = [ m( ) ] = [ m(
) ] →0
3) The rest of noise induced terms in (1.4.8) are the “complex signal Fourier spectrum
(m)
depending” and “random shift ( τ ) depending”. In other words, they have signal dependent proper-
ties and result in presence of multiplicative behavior component in the computed bispectral estimates.
More detailed analysis of bispectrum statistical characteristics has been carried out in our paper
[45]. In [45] we studied the following noisy bispectrum estimate component computed for m-th arbi-
trary realization observed
( m) ( m)
( m) ( m)* − jτ p * ( m) jτ q
n ( , ) ( ) ( ) ( ) ( ) ( ) ( )
B p q = S p S q N p + q e + S p S p + q N q e +
( m) ( m)
* ( m) j p ( m) ( m)* − j p
τ τ
+ S( q) S ( p + q) N ( p) e + S( p) N ( q) N ( p + q) e +
( m) ( m)
( m) ( m)* − j q * ( m) ( m) j ( p+ q)
τ τ
+ S( q) N ( p) N ( p + q) e + S ( p + q) N ( p) N ( q) e +
+
Consider the last term
( m) ( m)
N p N q
( m)*
( ) ( ) N ( p + q)
( m) ( m) ( m)*
N p N q N p q
68
(2.2.12)
( ) ( ) ( + ) in (2.2.12) to assess its statistical properties.

( m) ( m) ( m)
Let us use the following notations: N ( p) = N ( p) + jN ( p)
,
( m) ( m) ( m)
Re Im
Re Im
N ( q) = N ( q) + jN ( q)
, and N ( p + q) = N ( p + q) − jN ( p + q)
, where all Re and Im
( m)* ( m) ( m)
Re Im
noise components are zero mean and Gaussian. Then we obtain
( m) ( m) ( m)* ( m) ( m) ( m)
Re{ N ( p) N ( q) N ( p + q)} = N ( p) N ( q) N ( p + q)
+
Re Re Re
( m) ( m) ( m) ( m) ( m) ( m)
+ N ( p) N ( q) N ( p + q) + N ( p) N ( q) N ( p + q)
−
Re Im Im Im Re Im
( m) ( m) ( m)
− N ( p) N ( q) N ( p + q)
Im Im Re
( m) ( m) ( m)* ( m) ( m) ( m)
Im{ N ( p) N ( q) N ( p + q)} = N ( p) N ( q) N ( p + q)
+
Im Im Im
( m) ( m) ( m) ( m) ( m) ( m)
+ N ( p) N ( q) N ( p + q) + N ( p) N ( q) N ( p + q)
−
Im Re Re Re Im Re
( m) ( m) ( m)
− N ( p) N ( q) N ( p + q)
Re Re Im
69
, (2.2.13)
. (2.2.14)
Both Re and Im components in (2.2.13) and (2.2.14) contain the sums of four terms that are the
products of three zero mean Gaussian random variables. Our attempts to evaluate analytically the PDF
of the eight triple product random values like
N ( p) N ( q) N ( p + q)
failed and we had to
( m) ( m) ( m)
Re/ Im Re/ Im Re/ Im
employ numerical simulation for this purpose. It has been clearly demonstrated that the PDF of such
random variables is non-Gaussian. An example of the histogram obtained in [45] is shown in Fig.
2.2.30. The estimated curtosis values are about 7…7.5 and this also strongly evidences in favor of
non-Gaussianity of PDF.
Both equations (2.2.13) and (2.2.14) contain the sums of four random variables with non-
Gaussian distributions and the Re and Im components of the term
( m) ( m) ( m)*
N ( p) N ( q) N ( p q)
+ are
characterized by non-Gaussian PDFs. According to our numerical simulations [45], the estimated
curtosis values for Re and Im components of this term are within the limits of 2…5, i.e., they are
positive, and this confirms aforementioned properties of its PDF.
Moreover, let us demonstrate that the fourth, fifth and sixth terms in (2.2.12) are also
characterized by non-Gaussian behavior of Re and Im component distributions. For this purpuse let us
analyze in detail one of these terms that includes the product of two noise DFTs, for example, the
fourth term
( m) ( m)* − jτ ( m)
p
S( p) N ( q) N ( p q) e
+ . Its Re and Im parts can be expressed, respectively, as [45]

Re[
( m)
( m)
*
S(
p)
N ( q)
N ( p
j ( m)
+
− τ p
+ q)
e ]=
+
( m)
( m)
( m)
( m)
= N ( q)
N ( p + q){
SRe(
p)
cosτ
p+
SIm(
p)
sinτ
p}
+
Re Re
( m)
( m)
( m)
( m)
N ( q)
N ( p + q){
SRe(
p)
cosτ
p+
SIm(
p)
sinτ
p}
+
Im Im
, (2.2.15a)
( m)
( m)
( m)
( m)
N ( q)
N ( p + q){
SRe(
p)
sinτ
p−
SIm(
p)
cosτ
p}
+
Re Im
( m)
( m)
( m)
( m)
+ N ( q)
N ( p + q){
SRe(
p)
sinτ
p−
SIm(
p)
cosτ
p}
Im Re
( m)
( m)
( m)
( m)
= N ( q)
N ( p + q){
SIm(
p)
cosτ
p−
SRe(
p)
sinτ
p}
+
Re Re
( m)
( m)
( m)
( m)
( m)
( m)
* − jτ(
m)
+ N ( q)
N ( p + q){
SIm(
p)
cosτ
p−
SRe(
p)
sinτ
p}
−
p Im Im
Im[ S(
p)
N ( q)
N ( p + q)
e ]=
, (2.2.15b)
( m)
( m)
( m)
( m)
− N ( q)
N ( p + q){
SIm(
p)
sinτ
p+
SRe(
p)
cosτ
p}
+
Im Re
( m)
( m)
( m)
( m)
+ N ( q)
N ( p + q){
SIm(
p)
sinτ
p+
SRe(
p)
cosτ
p}
Re Im
( m)
( m)
where S (...) and S (...) are the Re and Im parts of the signal Fourier spectrum, respectively.
Re
Im
Fig. 2.2.30. Histogram of the noise induced terms in ((2.2.13)) and (2.2.14).
Now let us study Re (2.2.15a) and Im (2.2.15b) component distributions. For this purposis let us
consider PDFs f1(x) and f1(y) of two independent Gaussian variables x and y expressed as
f ( x)
= 1
2
1 x
⋅exp( − ) , f ( y)
= 2
1
2π ⋅σ
2σ
X
X
2
1 y
⋅exp( − ) 2
2π ⋅σ
2σ
Y
Y
where σX and σY are the standard deviations of the corresponding random variables.
The PDF g(z) of the product xy =z can be written in the form [55] as
∞
70
, (2.2.16)
( ) 1 ( ) 2 ( ) dx
g z = f x ⋅ f y . (2.2.17)
∫
x
−∞
After mathematical transformations, the formula (2.2.17) can be represented finally as

where C =
1 .
πσ σ
2 X Y
1 z
g( z) = C ⋅ ⋅ exp( − ) , (2.2.18)
z σ σ
It is seen from the formula (2.2.18) that the PDFs of products like ( q)
N ( p q)
71
X Y
( m)
( m)
N + are non-
Re Re
Gaussian and heavy-tailed. Both expressions (2.2.15a) and (2.2.15b) contain four such terms multip-
( m )
( m)
lied by the corresponding factors like [ S ( p)
cosτ
p + S ( p)
sinτ
p]
that do not change the form of PDF
Re
( m)
Re
( m)
( m)
(2.2.18). Although four random variables like N ( q)
N ( p + q){
S ( p)
cosτ
p + S ( p)
sinτ
p}
are summed
up, the PDFs of the fourth, fifth and sixth terms in Eq. (2.2.12) are still characterized by non-Gaussian
distributions of Re and Im components.
Im
The first, second and third terms in (2.2.12) possesses complex-valued Gaussian behavior.
However, due to non-Gaussianity of the latter four terms, the PDFs of Re and Im of B ( p,
q)
are, in
general, non-Gaussian. Only if the input SNR is large, i.e., if S(..) >> σBN
(where σ BN is the Re or Im root
(m)
n
mean square of B (...) ), the PDF of Re or Im of B (...) part approaches to Gaussian.
(m)
n
It should be stressed that under such conditions conventional bispectrum estimation techniques
[2] based on ensemble averaging procedure (1.3.6) perform in optimal manner.
However, in the case of low input SNR, as it can be seen from the considerations above, the
( m)
term ( p,
q)
n
B in (2.2.12) becomes non-Gaussian and this provokes considerable errors in bispectral es-
timate. Note that this peculiarity has not been discussed in literature and first it has been considered in
our paper [45].
Therefore, due to non-Gaussianity for low SNR and non-stationarity of real and imaginary bis-
pectrum estimate parts, another methods for enhancement of bispectrum estimates should be used and
they can be expected to be more efficient in the sense of obtaining better bispectrum estimates and,
consequently, better performance of signal reconstruction systems. Several novel approaches proposed
by us in [44 – 48] and dedicated to improving bispectrum estimates will be described in the next sub-
section.
2.3. Novel approaches developed for improving bispectrum estimates
First, we begin with describing an approach based on processing real and imaginary bispectrum
estimate parts by using vector filters [44]. The motivation for using vector filters is that they can
( m)
Re
Re
Im
( m)
n

process multichannel data and real and imaginary bispectrum estimate parts can be considered as two-
channel 2-D data (images).
Since statistical characteristics of noise in (2.2.12) depend upon original signal Fourier spectrum
and it is usually supposed unknown, then it is difficult to apply the vector filters that take into account
statistical characteristics of noise in processed data. Therefore, we have at disposal only quite a li-
mited set of vector filters that can be applied. Among them we have considered the standard vector
median filter (denoted as VMF) [57]. Then the vector sample is
i
{ Re( Ḃ ˆ
x ( pl
, q j )); Im( B
ˆ̇
x ( pl
, q j )) } , i = 1,...,
NxN
x =
and p varies from l
2 / ) 1 − ( N − p to + ( N −1)
/ 2
from q − ( N −1)
/ 2 to − ( N + 1)
/ 2
pectrum estimate sample.
72
p , q varies
j
q for the scanning 2-D window of size NxN centered on the (p,q)-th bis-
Two types of directional filters have been used in our computer simulations [44]. The selection
of directional vector filters is motivated by desire to preserve phase information in bispectral data and,
as known, directional vector filters often perform this task (in the sense of color preservation in RGB
images) better than conventional vector filters [56].
The basic idea of vector directional filters is to order vector samples in accordance to the dis-
tance measure that is equal to the angle between the vectors in the scanning window. According to this
definition, vector directional median can be calculated as
N N
⎧
xvdm = ⎨∑ A( xvdm, xk ) ≤ ∑ A( xm, x k ) ;
⎩ k= 1 k=
1
⎫
m = 1, N; vdm∈1, N ,
⎬
⎭
(2.3.1)
A x , x denotes the absolute value of angle between vectors xi and xk. To achieve computa-
where ( )
i
k
tional efficiency improvement of directional ordering of vector samples we propose to use a distance
measure that is proportional but not equal to the angles between vector samples. Such measure can be
calculated as a distance between vectors normalized on their lengths
where ⋅ is either L 1 or
D
ph
k , m
=
x x , (2.3.2)
N
k m
∑ −
m= 1 xk
xm
2
L 2 norm, x denotes the length of vector x.
Despite vector directional median filters use only directional information for ordering vector
samples, the processing also changes the amplitude of filtered vectors. Posing the condition that filter
output should not change the amplitude of central window sample and using the distance measure
(2.3.2) for ordering we propose the following definition of median phase filter (MPF)
mpf = c ⋅ x
x x
x
p
p
, (2.3.3)
where xc is the window central sample; xp corresponds to vector median calculated from direction or-

dered samples as
x
ph ph { D , D , ; m 1, N; p 1, N}
= ≤ = ∈
p p k m k
73
. (2.3.4)
Applying the same approach to vector α-trimmed filter [57] we can define α-trimmed phase fil-
ter (TPF) as
apf = c ⋅ x
x x
x
pa
pa
, (2.3.5)
where xpa is the α-trimmed mean output computed as
N (1 −α
)
1
( k )
x pa = ⋅ ∑ X .
p
N(1
−α
) k=
1
(2.3.6)
(k )
ph
where X is the k-th vector sample ordered according to distance D in (2.3.2). A parameter α per-
p
mits to vary the filter noise suppression and robust properties similarly to the vector α-trimmed filter.
Below we consider the case α=0.5. It is worth noting that for all aforementioned vector filters their
performance depends upon the scanning window size and the selected norm denoted as L 1 and
k, m
Table 2.3.1. Computer simulation results [44].
Filter type Input Output MS Bias
SNR variance
VMF, norm 0.25 0.0724 0.0633
L1 0.5 0.0285 0.0226
1.0 0.0132 0.0170
2.0 0.0072 0.0129
5.0 0.0039 0.0104
MPF, norm L1 0.25 0.0257 0.0298
0.5 0.0094 0.0256
1.0 0.0046 0.0225
2.0 0.0031 0.0215
5.0 0.0030 0.0111
MPF, norm
L2 2
0.25 0.0371 0.0573
0.5 0.0174 0.0391
1.0 0.0114 0.0307
2.0 0.0031 0.0173
5.0 0.0030 0.0162
TPF, norm L1 0.25 0.0280 0.0373
0.5 0.0089 0.0228
1.0 0.0039 0.0195
2.0 0.0020 0.0178
5.0 0.0015 0.0174
TPF, norm L2 2
0.25 0.0314 0.0417
0.5 0.0119 0.0317
1.0 0.0061 0.0213
2.0 0.0045 0.0164
5.0 0.0026 0.0107
BLW 0.25 0.0343 0.0632
0.5 0.0152 0.0381
1.0 0.0077 0.0243
2.0 0.0054 0.0155
5.0 0.0022 0.0102
2
L 2 .

For quantitative performance evaluation of the proposed methods, the fluctuation variance and
MS bias computed in the form of (2.2.15) and (2.2.16), respectively, have been used as criterions.
Note that the signal estimates reconstructed by noisy bispectrum estimates are usually biased.
The ideal main goal is to minimize both fluctuation variance and MS bias at the same time or, at least,
to maximally reduce them. However, commonly this is a conflicting problem.
Table 2.3.1 contains the results obtained for a wide range of input SNR variation [44]. All three
vector filters have the same scanning WS of 5x5 samples. The test signal consists of two short pulses
with rectangular shapes and different amplitudes. In our simulations the observed process sample
number I=256, the realization number M=256, the experiment repetition number K=30.
The analysis of the computer simulation results given in Table 2.3.1 demonstrates the following.
1. The application of the considered vector filters can lead to both improvement and worsening
the signal reconstruction system performance. In particular, the VMF application commonly results in
2
larger output variance σ out than for conventional BLW method. At the same time, the bispectrum
2
processing by VMF can produce smaller δ out than the conventional BLW technique.
2. The results obtained for the MPF (L1 norm) are almost always better than for the BLW tech-
nique, both smaller fluctuation error 2
out
σ and MS bias out
norm for the MPF and TPF produces better results than in the case of
74
2
δ are provided. In general, the use of 1
L
2
L 2 norm used. This is, proba-
bly, because the vector filters based on L 1 norm distort details in images (here in bispectrum real and
imaginary parts) in less degree. Besides, its use in the considered case can be motivated by the fact
that noise in bispectrum estimates can be non-Gaussian (see previous subsection 2.2).
3. TPF possesses “intermediate place” properties between VMF and MPF. The application of
TPF to the bispectrun estimation results in some improvement of system performance in comparison
to the BLW technique for the cases of low input SNR.
Summarizing the results obtained for vector filters in combined bispectrum-filtering framework,
two conclusions can be drawn.
First, there is no improvement due to using vector filters in comparison to separate processing of
bispectrum real and imaginary components by nonlinear filters. This is not surprising since vector fil-
ters produce benefits in comparison to their scalar counterparts if there is essential correlation of mul-
tichannel data components [56, 57]. However, bispectrum real and imaginary components are not
strongly correlated.
Second, the obtained results show that non-adaptive filters (either scalar or vector) are unable to

produce stable improvement of combined bispectrum-filtering processing in comparison to the con-
ventional BLW technique.
Since real and imaginary bispectrum components becomes non-Gaussian and non-stationary, it
can be expected that this peculiarity is also true for the real and imaginary parts of signal Fourier spec-
trum estimates recovered from bispectrum. The corresponding investigations have been carried out in
[46]. From the earlier experience, it can be expected that the statistical properties of leaked noise de-
pend upon many factors like an original signal shape, the number of realizations, variance of input
noise, etc.
Based on the results of these investigations, the selection of appropriate filtering technique be-
comes more purposeful and effective. It was demonstrated in [46] that the modified adaptive discrete
cosine transform (DCT) based filters can be rather successfully used for improving the signal Fourier
spectrum estimate recovered from bispectrum. Besides, we have come to the conclusion that it is ne-
cessary to study the bispectrum-based processing performance for a wider set of test signals. This is
needed to be sure that the designed methods are effective for different characteristics of input signals.
Three test signals s(i) in the form of two pulses with rectangular shapes, various lengths ∆t1,2,3,
different amplitudes of A1 = 2 and A2 = 6 and the mutual pulse shift that was the same for all three test
signals and equal to ∆t12=5 samples have been studied in simulations [46] (see Figures 2.3.1a, 2.3.2a
and 2.3.3a). The pulse length ∆t has been varied. It was equal to ∆t1 = 3 (signal #1), ∆t2 = 7 (signal #2)
and ∆t3 = 11 (signal #3) samples. The test signal powers P s are of the values of 0.46, 1.06 and 1.60,
respectively.
s(i) 6
4
2
0
( m)
x ( i)
50 100 150 200 250 i
50 100 150 200 250 i
a b
Fig.2.3.1. The noise-free test signal #1 (∆t1 = 3) (a) and a noisy realization x (m) (i), SNR inp =0.46 (b).
75
4
2
0
-2

s(i) 6
4
2
0
50 100 150 200 250 i
a
( m)
x ( i)
6
76
4
2
0
-2
50 100 150 200 250 i
Fig.2.3.2. The noise-free test signal #2 (∆t2 = 7) (a) and a noisy realization x (m) (i), SNR inp =1.05 (b).
s(i)
6
4
2
0
50 100 150 200 250 i
a
( m)
x ( i)
6
4
2
0
-2
b
50 100 150 200 250 i
Fig.2.3.3. The noise-free test signal #3 (∆t3 =11) (a) and a noisy realizations x (m) (i), SNR inp =1.60 (b).
The original test signals were simulated as the sequences of non-negative real values generated
on the array of I=256 samples. We have simulated M =200 realizations (Monte Carlo runs) for each k-
th statistical experiment, k=1, 2…, K. The corresponding noisy and randomly shifted realizations x (m) (i)
for the test signals obtained for different input SNRs ( SNR inp ) are shown in Figures 2.3.1b, 2.3.2b and
2.3.3b, respectively.
ˆ
As the first step in analysis of statistical characteristics of noise leaked in Re{ S bisp(
r ) }
ˆ { bisp }
b
̇ and
Im S ̇ ( r ) , the following variance estimates were calculated for noisy signal Fourier spectra recov-
ered from bispectrum in the form of
σ
σ
2
Re Ens
2
Im Ens
1
where { } { }
K
ˆ ˆ k
Re S bisp( r ) = ∑ Re S bisp(
r )
{ } { } 2
ˆ k
ˆ̇
Bisp
Bisp
K 1
( r ) =
⎡ ⎤
∑ Re S ̇ ( r ) − Re S ( r ) , (2.3.7a)
K ⎢ ⎥
−1 k = 1 ⎣ ⎦
{ } { } 2
ˆ k
S
ˆ
bisp( r ) S bisp(
r )
K 1 ⎡ ⎤
( r ) = ∑ Im ̇ − Im ̇ , (2.3.7b)
K ⎢ ⎥
−1 k = 1 ⎣ ⎦
̇ ̇ 1
,
K { } { }
k = 1
1
K
ˆ ˆ k
Im S ̇
bisp( r ) = Im S ̇ ∑ bisp(
r ) . As seen, the estimates
K k =
have been obtained for different frequencies defined by their indices r.

To determine whether or not leaked noise obeys Gaussian distribution, the analysis of the func-
ˆ k
Re S ̇ ˆ k
bisp(
r ) and Im{ S ̇
bisp(
r ) } has been performed using W-criterion of Wilcoxon (W-test)
tions { }
[55] for each r-th frequency (r=0,1,…,I–1) and for K=30.
ˆ k
The plots of realizations of Re{ S bisp(
r ) }
̇ 2
and variance σ Re Ens(
r ) (2.3.7a) are shown in Figures
ˆ k
2.3.4a, 6a and 8a for all three considered test signals. The plots of realizations of Im{ S bisp(
r ) }
2
variance σ Im Ens(
r ) (2.3.7b) are represented in Figures 2.3.5a, 2.3.7a and 2.3.9a, respectively.
77
̇ and
The W-test results are illustrated in Figures 2.3.4b–9b. For the frequencies for which noise has
been decided to obey Gaussian distribution W(r)=1. In the opposite case, W(r)=0.
Figures 2.3.4 and 2.3.5 correspond to the test signal #1 (Fig. 2.3.1), Figures 2.3.6 and 2.3.7 re-
late to the test signal #2 (Fig. 2.3.2) and Figures 2.3.8 and 2.3.9 – to the test signal #3 (Fig. 2.3.3).
150
100
50
0
1
W(r)
ˆ k
Fig. 2.3.4. Signal #1 (∆t=3): Re{ S bisp(
r ) }
0
ˆ̇
{ Bisp }
Re S ( r )
2
σ Re Ens( r )
50 100 150 200 250
a
r
50 100 150 200 250
b
r
̇ 2
and σ Re Ens(
r ) (a) and the W-test results obtained for
SNR inp =0.46 (b).

60
40
20
0
-20
1
W(r)
ˆ k
Fig. 2.3.5. Signal #1 (∆t=3): Im{ S bisp(
r ) }
0
60
50
40
30
20
10
0
1
W(r)
ˆ k
Fig. 2.3.6. Signal #2 (∆t=7): Re{ S bisp(
r ) }
0
50 100 a 150 200 250
r
50 100 150 200 250
b
r
78
ˆ̇
{ Bisp }
Im S ( r )
2
σ Im Ens( r )
̇ 2
and σ Im Ens(
r ) (a) and the W-test results obtained for
SNR inp =0.46 (b).
ˆ̇
{ Bisp }
Re S ( r )
2
σ Re Ens( r )
50 100 150 200 250
a
r
50 100 150 200 250
b
r
̇ 2
and σ Re Ens(
r ) (a) and the W-test results obtained for
SNR inp =1.05 (b).
The analysis of the plots in Figures 2.3.4 – 2.3.9 shows that noise component is of non-
stationary behavior in the sense that noise variance obviously depends on frequency. Furthermore,
Gaussian distribution is observed mainly in low frequency domain (the sample number r=129 corres-
ponds to zero frequency in Figures 2.3.4b–2.3.9b).

Moreover, it is seen from the W-test plots (see Figures 2.3.4b – 9b) that the number of frequen-
cies that obey Gaussian distribution
30
20
10
0
-10
-20
-30
1
W(r)
0
I
N = ∑ W ( r)
depends on both signal length and SNR inp .
W
r=
0
Fig. 2.3.7. Signal #2 (∆t=7): { bisp }
80
60
40
20
0
1
W(r)
ˆ k
Fig. 2.3.8. Signal #3 (∆t=11): Re{ S bisp(
r ) }
0
50 100 a 150 200 250
r
50 100
b
150 200 250 r
ˆ k
Im S ̇ ( r )
2
and σ Im Ens(
r ) (a) and the W-test results obtained for
SNR inp =1.05 (b).
79
ˆ̇
{ Bisp }
Im S ( r )
2
σ Im Ens( r )
ˆ̇
{ Bisp }
Re S ( r )
2
σ Re Ens( r )
50 100 a 150 200 250 r
50 100 150 200 250
b
r
̇ 2
and σ Re Ens(
r ) (a) and the W-test results obtained for
SNR inp =1.60 (b).

40
30
20
10
0
-10
-20
-30
-40
1
W(r)
ˆ k
Fig. 2.3.9. Signal #3 (∆t=11): Im{ S bisp(
r ) }
0
50 100 a 150 200 250 r
50 100 150 200 250
b
r
80
ˆ̇
{ Bisp }
Im S ( r )
2
σ Im Ens( r )
̇ 2
and σ Im Ens(
r ) (a) and the W-test results obtained for
SNR inp =1.60 (b).
As can be seen from Table 2.3.2, the values NW tend to decreasing when SNR inp becomes small-
er, i.e. the noise component becomes “more non-Gaussian”. Note that the same tendency to declina-
tion from Gaussianity has been observed for real and imaginary parts of bispectrum estimates [45].
Thus, the complex-valued Fourier spectrum recovered from bispectrum estimate is mainly distorted
by non-stationary and non-Gaussian noise. This conclusion gives us an opportunity to restrict the class
of filters that are applicable for filtering the considered processes.
Table 2.3.2. Value Nw as a function of signal length and SNR inp .
N
N
N
N
N
N
Signal # Signal #1 (∆t=3)
SNR inp 1.53 0.92 0.46 0.23 0.15
ˆ̇
( { bisp } )
W Re S ( r )
ˆ̇
( { bisp } )
W Im S ( r )
101 83 55 43 37
58 49 54 38 48
Signal # Signal #2 (∆t=7)
SNR inp 3,50 2,10 1,05 0,53 0,35
ˆ̇
( { bisp } )
W Re S ( r )
ˆ̇
( { bisp } )
W Im S ( r )
179 129 113 77 74
170 134 106 92 82
Signal # Signal #3 (∆t=11)
SNR inp 3,20 1,60 0,80 0,53 0,32
ˆ̇
( { bisp } )
W Re S ( r )
ˆ̇
( { bisp } )
W Im S ( r )
189 155 119 125 96
181 144 117 104 80

The aforementioned properties (possible non-Gaussianity) of noise show that it is not worth ap-
ˆ
Re S ̇ ˆ
bisp(
r ) and Im{ S ̇
bisp(
r ) } . At the same time, for
plying conventional linear filters for processing { }
numerous nonlinear filters, there is always a problem what among them to select and with what para-
meters [53]. Therefore, here we run into not typical and rather complicated situation. Noise is not
purely additive and not purely multiplicative, but it is frequency dependent.
Probably, there are several ways out and below we consider only one of them. The basic atten-
tion is paid to the DCT-based filtering techniques. Note that the filters based on orthogonal transforms
(wavelets, DCT) have been originally designed for removal of additive Gaussian noise with a priori
known variance that is constant for entire processed signal [58, 59].
However, later DCT-based filter modifications for data dependent noise suppression have been
proposed in [59]. Moreover, in [60] it has been demonstrated that orthogonal transform based filtering
can be rather successfully applied in the case a noise PDF is not strictly Gaussian but quite close to it.
In particular, electromyographic noise (that is non-Gaussian) was removed from electrocardiograms
by means of wavelet based denoising [60]. This was the basis for our trial to apply DCT-based filter-
ing in our case [46]. Note that below we concentrate on considering spatially invariant DCT filtering
in blocks which is more time consuming but more efficient than filtering in non-overlapping blocks.
For applying DCT-based filters, one should a priori know noise variance (standard deviation)
for threshold setting. Since in our case noise variance is a priori unknown, we have to somehow esti-
mate it. This can be accomplished in two ways. First, one can estimate some average noise variance
2 ˆσ ent or standard deviation ˆσ ent for entire signal to be processed and to use this estimate for setting a
constant threshold for all blocks (this type of DCT filtering will be further denoted as DCTc).
Another approach is to introduce the following modification into DCT-filter (and to make it lo-
cally adaptive). Let us for each position of a block obtain a local estimate of noise variance and then
use it for threshold setting. One way to obtain such local estimate for each block is [60]
{ }
ˆ σ = 1. 483⋅ med W( l ) , l = 1,...,N b , (2.3.8)
where med{} means the sample median, W(l) is the l-th DCT spectral coefficient, N b denotes the
block size. For a given r the corresponding value ˆ σ ( r ) is derived for the block that includes the data
samples with the indices from r − N / 2 + 1 to r + N / 2 (this locally adaptive type of DCT filtering will
be referred to as DCTla).
b
b
81

On one side, the accuracy of the estimate (2.3.8), in general, reduces with decreasing N b . On the
other hand, if noise is essentially of non-stationary behavior, the estimate
from the values of local variance of noise.
20
15
10
5
0
-5
-10
-15
-20
ˆ
Fig. 2.3.10. Noisy realization of Im{ S bisp(
r ) }
50 100 150 200 250 r
82
ˆ̇
{ Bisp }
Im S ( r )
ˆ σ ( r )
2 ˆσ ent can considerably differ
̇ for the test signal #1 and the local estimates ˆ σ ( r ) .
Recall that DCT filtering is applied separately for processing real and imaginary parts of signal
Fourier spectrum recovered from bispectrum. Hard thresholding [58 – 60] has been employed. For
DCTc the threshold was set as βσ ˆ ent , and for DCTla the threshold was set as βσ ˆ ( r ) individually for
each block. Recall that in DCT and wavelet-based filtering the factor β determines the denoising algo-
rithm properties and it is commonly recommended to apply β varying within the limits of 2.0…4.0.
Increasing of β commonly leads to better noise suppression but it results at the same time in worse
preservation of details and discontinuities.
ˆ
Particular example of computation Im{ S bisp(
r ) }
that the estimates ˆ σ ( r ) correlate with intensity of noise fluctuations.
̇ represented in Fig. 2.3.10 [46] demonstrates
It is seen from Fig. 2.3.10 that the most intensive noise fluctuations are observed in the neigh-
borhoods of r equal to 47 and 210 as well as in the neighborhoods of r equal to 17 and 240 sample
numbers. And the estimates ˆ σ ( r ) are the largest just in those neighborhoods (note that
ˆ σ( r ) = ˆ σ(
257 − r ) for I=256).
One more question in DCT-based filtering is the selection of the block size N b . In our simula-
tions [46] we used two values, namely, Nb = 16 and Nb = 32.
mance.
The set of parameters (1.5.7 – 1.5.11) is used for evaluation the proposed technique perfor-
The numerical simulation results for all three aforementioned test signals and for several values
of SNR inp are presented in Tables 2.3.3, 2.3.4, and 2.3.5, respectively.

Table 2.3.3. Simulation results for the test signal #1 using constant threshold with β =4.
SNR inp 1.53 0.92 0.46 0.15 0.09
εCONV 9.07 12.80 16.80 14.17 11.01
εDCT c 16 9.63 14.18 21.68 48.93 65.31
εDCT la16 10.38 15.29 23.50 52.18 65.01
εDCT c 32 9.71 14.40 22.30 50.77 71.66
εDCT la32 10.35 15.28 23.53 56.89 74.47
Table 2.3.4. Simulation results for the test signal #2 and different threshold values.
SNR inp ββββ 3.50 2.10 1.05 0.53 0.21
εCONV – 9.42 11.25 16.30 14.75 10.72
2.7 10.75 13.52 24.53 31.68 56.43
3.2 10.92 13.73 25.25 33.40 61.51
εDCT la16
εDCT la32
4.0 11.24 14.13 26.12 35.28 65.05
2.7 10.71 13.38 24.07 31.19 60.83
3.2 10.90 13.57 24.80 32.75 66.00
4.0 11.15 13.86 25.45 34.50 67.18
Table 2.3.5. Simulation results for the test signal #3 and different threshold values.
SNR inp ββββ 3.20 1.60 0.80 0.53 0.32
εCONV – 17.61 16.12 15.84 13.57 10.35
2.7 23.24 24.90 37.94 46.15 50.31
3.2 23.74 25.46 39.79 49.22 54.23
εDCTla16
εDCTla32
4.0 24.14 26.06 41.31 50.42 55.73
2.7 23.03 24.47 38.33 47.02 53.70
3.2 23.23 25.14 40.85 50.27 58.12
4.0 23.34 25.85 42.01 51.67 59.58
The filter MSE performance criterion ε for the conventional bispectrum technique is denoted as
εCONV and for the proposed bispectrum filtering method as εDCT16 and εDCT32 depending upon N b . The
analysis of the data in Tables 2.3.3 – 2.3.5 allows us to conclude the following.
1) The application of the DCT-based filtering for all considered situations (test signals and SNR inp
values) produces better SNR out and, hence, larger ε in comparison to conventional technique [10].
2) The maximal benefit due to application of the combined bispectrum-filtering techniques is
observed for rather small SNR inp and this is very important for practice; in such cases SNR out for the
83

proposed combined techniques empolying the DCT-based filtering can be more than by six times
larger than SNR out for the conventional method.
3) The results for Nb = 16 and Nb = 32 are practically the same; because of this, it seems reasona-
ble to apply DCT-based filters with Nb = 16 since in this case data processing can be performed faster.
4) The DCT-based filtering with local adaptation of the set threshold produces better results than
the DCT-based filtering with the fixed threshold (see data in Table 2.3.3); similar tendencies have
been also observed for other test signals.
5) For the considered application, setting the parameter β equal to 4 practically always is the op-
timum; this is confirmed by data presented in Tables 2.3.4 and 2.3.5 where the results for three differ-
ent values of β are presented.
One more advantage of the DCT-based filtering application within the framework of combined
bispectrum-filtering reconstruction is that it provides improvement of signal waveform estimation ac-
curacy irrespectively to what is this waveform. Note that for the bispectrum-filtering techniques con-
sidered in [42] the application of some linear or nonlinear filters led to improvement of signal wave-
form estimation while the use of some other filters or the same filters but with other scanning win-
dows resulted in degradation of such estimates. And the problem was to ensure robustness of bispec-
trum-filtering techniques in wide sense, i.e. to provide enhancement of reconstruction for a wide set of
unknown signal waveforms and ranges of SNR inp variation.
ˆ
To demonstrate the performance of the DCT-based filtering, the plot of Im{ S bisp(
r ) }
84
̇ for one
experiment before filtering (dotted curve) and the corresponding plot after DCT-based filter (DCTla,
Nb = 16, β =4, solid curve) are shown in Fig. 2.3.11.
ˆ
Fig. 2.3.11. Im{ S bisp(
r ) }
20
15
10
5
0
-5
-10
-15
-20
before filtering
after DCT-based filtering
50 100 150 200 250
̇ for one experiment before filtering (dashed line) and after DCT-based filter
with Nb = 16 (solid line), the test signal # 1.
r

Sˆ ( i)
6
5
4
3
2
1
0
-1
50 100 150 200 250
i
Fig. 2.3.12. Signal reconstructed by the conventional
BLW technique for inp
SNR =0.15 (test signal # 1).
Sˆ ( i)
6
5
4
3
2
1
0
-1
50 100 150 200 250 i
Fig. 2.3.13. Signal waveform reconstructed by the proposed bispectrum-filtering method for
SNR inp =0.15 (test signal # 1).
As seen from Fig. 2.3.11, the noise is considerably suppressed and the information features are
preserved well. This leads to improving the signal waveform reconstruction. To prove this, the signal
reconstructed by the conventional BLW technique [10] is given in Fig. 2.3.12. The signal waveform
reconstructed by the proposed DCT-based bispectrum-filtering method is represented in Fig. 2.3.13.
Obviously, the residual noise fluctuations in this plot are noticeably less intensive and two rectangular
shape pulses can be more easily identified. Note that residual distortions in the central part of the plot
can be explained by phase wrapping influence.
Thus, the main attractive benefits obtained with using the DCT-based filtering for bispectrum-
based signal reconstruction [46] are the following. First, the best performance improvement is reached
for small input SNR values. Second, for a rather wide set of the test signals that participated in com-
puter simulation the performance improvement is provided without solving the task of proper (optim-
85

al) selection of the filter type and/or filter parameters. Finally, it has been demonstrated that the DCT-
based filtering can be modified to perform well enough in case of non-stationary noise whose variance
is not a constant value for entire observed process to be filtered.
Robust and adaptive technique for improving the accuracy of noisy bispectrum estimated has
been also proposed in our paper [47]. Several different approaches for processing noisy bispectrum
data have been considered and compared. They are the following.
Recall that the conventional bispectrum estimation [2] is based on ensemble averaging of sam-
ple bispectrum estimates by an ensemble of M realizations (see 1.3.6) to obtain a smoothed estimate
⌢
̇ ( p,
q)
in the form
B X
⌢
B ̇
X
⌢
= Ḃ
mean(
p,
q)
=
⌢
( m)
Ḃ
( p,
q)
,
M
(2.3.9)
where ... denotes the ensemble averaging by M realizations.
M
However, bispectrum estimate can be also derived as [45]
⌢
̇ ( p,
q)
= r
⌢
⌢
( p,
q)
+ ji
( p,
q)
, (2.3.10)
Bmed med
med
where
ˆ ( m)
rˆ
med ( p,
q)
= med{Re[
Ḃ
( p,
q)],
m ∈[
1;
M ]} , ˆ
ˆ ( m)
i ( p,
q)
med{Im[
B ( p,
q)],
m [ 1;
M ]}
m
med = ̇ ∈ .
m
Similarly to sample median, any other estimate can be applied. For example, the bispectrum es-
timate obtained using Hodges-Lehmann [53] estimate can be written as
⌢ ⌢
Ḃ HL(
p,
q)
= r
⌢
HL(
p,
q)
+ jiHL
( p,
q)
, (2.3.11)
where
⌢ ⌢
ˆ ( m)
rˆ
HL ( p,
q)
= med{Re[
Ḃ
( p,
q)],
m ∈ [ 1;
M ]}; (Re[ Ḃ ( m) ( p, q)] + Re[ Ḃ ( M + 1 −m)
( p, q)]) / 2, m ∈[1;
M / 2]} ,
m
⌢ ⌢
ˆ
ˆ ( m)
iHL
( p,
q)
= med{Im[
Ḃ
( p,
q)],
m ∈ [ 1;
M ]}; (Im[ Ḃ ( m) ( p, q)] + Im[ Ḃ ( M + 1 −m)
( p, q)]) / 2, m ∈[1;
M / 2]} ,
m
⌢
and the notation Ḃ ( m)
( p, q)
is used for the m-th order statistic in the processed data sample [53].
Recently, we have introduced a novel simple and efficient robust adaptive estimate [61]. This
estimate is worth applying for processing samples of data having heavy tail PDFs and symmetrical
with respect to their mean. For bispectrum data processing the algorithm [47] is
⌢
⌢̇ ⎧Ḃ
( p,
q),
if K ∈ [ ψ ; +∞)
B ( p,
q)
= ⎨ ⌢
HL P
adapt
,
⎩Ḃ
med ( p,
q),
if K
P
∈ ( 0;
ψ )
(2.3.12)
where K P is the robust estimate of PDF kurtosis obtained on basis of percentiles
( K P = 0. 5(
X 75 − X 25)
/( X 90 − X10
) where X denotes the q-th percentile); ψ is the threshold (its quasi op-
q
timal value is equal to 0.2). The parameter K P in (2.3.12) serves as adaptation parameter for hard
switching of sample median and Hodges-Lehmann estimate (see details in [61]).
As a test signal, we have used two rectangular shape pulses with the amplitudes of 2 and 6. The
86

lengths of both pulses δp were equal, but three different values of δp have been considered in our ex-
periments, namely, 3, 7, and 11 samples. The distance between two rectangular pulses was the same in
all experiments and equal to 5 samples. Such test signals have been used to model different high reso-
lution radar range profiles and to verify the applicability of the considered methods for processing
various types of signals [47].
The performances of the following six bispectrum-based methods have been studied and com-
pared in [47].
(Mean).
1) Conventional method of signal waveform reconstruction [2, 10] by using (2.3.9)
2) Method of signal waveform estimation that uses the sample median (2.3.10) for obtain-
ing the bispectrum estimate for M realizations (Median).
3) The proposed method based on adaptive robust estimate (2.3.10) and (2.3.11) (AE).
4) Combined method on basis of (2.3.9) and post-processing of Re and Im parts of
⌢
filt
S ̇
bisp(
r ) by the DCT-based filter described above (Mean+DCT), the recommended value β = 4 was
used in our computer simulations.
5) Combined technique that employs bispectrum estimate (2.3.10) and DCT-based filter-
⌢
filt
ing of of Re and Im components of S ̇
bisp(
r ) (Median+DCT).
6) Adaptive combined method by using (2.3.11) and (2.3.12) followed by post-processing of Re
⌢
filt
and Im components of S ̇
bisp(
r ) by DCT based filter (AE+DCT).
Thus, for the last two methods we have combined two approaches for improving the bispectrum
estimate, namely, its forming using robust estimates and the obtained estimate filtering.
Output variance
1,0000
0,1000
0,0100
0,3 0,5 1 2 3
Input noise variance
Mean Median AE
Mean+DCT Median+DCT AE+DCT
Fig. 2.3.14. Output MSE as a function of the input noise variance, δp=3, M =64.
87

variance
The MSE
O u tp u t varian ce
1,0000
0,1000
0,0100
0,3 0,5 1 2 3 5
Input nois e variance
Mean Median AE
Mean+DCT Median+DCT AE+DCT
Fig. 2.3.15. Output MSE as a function of the input noise variance, δp=7, M =64.
Output variance
1,0000
0,1000
0,0100
0,3 0,5 1 2 3 5
Input noise variance
Mean Median AE
Mean+DCT Median+DCT AE+DCT
Fig. 2.3.16. Output MSE as a function of the input noise variance, δp=11, M=64.
Output variance
1,000
0,100
0,010
0,3 0,5 1 2 3 5
Input noise variance
Mean Median AE Mean+DCT Median+DCT AE+DCT
Fig. 2.3.17.Output MSE as a function of the input noise variance, δp=7, M=128.
2
σ out values (denoted as the output variance in the graphs) as functions of input noise
2
σ n are shown in Figures 2.3.14 – 2.3.17. The plots in Figures 2.3.14 – 2.3.16 correspond to
realization number of M=64 and the plots in Fig. 2.3.17 relate to M=128.
88

Analysis of the plots in Figures 2.3.11 – 14 allows concluding the following.
1. The use of the adaptive robust estimate (2.3.11) and (2.3.12) provides “robustness” of bis-
pectrum-based processing (method 3) in wide sense, for small
tional method 1 while for large
mance as the method 2 based on median estimate.
89
2
σ n it performs like conven-
2
σ n (small input SNR) the method 3 has the same perfor-
2. The use of the robust estimates (methods 2 and 3) instead of conventional estimate (2.3.9)
produces sufficient reduction of
2
σ out for large
2
σ n (small input SNRs); the values of
2
σ out for
the methods 2 and 3 can be up to 30% smaller than for the conventional bispectrum-based
method 1; this property is important for practical applications.
3. The DCT-based filtering (methods 4 – 6) provides even more radical reduction of
2
σ out in
comparison to the methods 1 – 3 and this improvement of performance is the largest for
small input SNRs.
4. The performance of the method 6 is either the best or close to the best reachable among the
considered methods for entire range of
2
σ n variation and for all test signals.
5. As can be expected, the use of larger number of accumulated realizations M leads to better
performance (compare the corresponding plots in Figures 2.3.17 and 2.3.15) but in practice
the number of observed realizations is limited.
6. All the tendencies observed for smaller M (see the items 1 – 4) are also valid for larger M.
Thus, the main benefits of the adaptive methods proposed in [47] are their applicability and high
performance for a wide range of input SNRs (especially for the situations when input SNR is low) and
different a priori unknown signal waveforms.
Another combined bispectrum-filtering technique proposed in our paper [48] is based on the 2-D
DCT data processing, evaluation of the optimal parameters of the corresponding filters, estimation of
the bounds that can be reached in the improvement of the output SNR and analysis of the conditions
under which maximum improvement of SNR takes place.
According to results obtained for several bispectrum-based approaches considered above in this
Chapter, the separate filtering of real and imaginary parts of bispectrum estimate provides sufficiently
better results comparing to other bispectrum-based signal reconstruction approaches.
However, the following question arises: what filters are reasonable to employ in this case? The
point is that the noise leaking to bispectrum estimate is neither of additive nor of multiplicative nature
in strict sense. Saying more exactly, this noise possesses signal (bispectrum) depending properties (see

[45]). Note that noise leaks into bispectrum domain from the input noisy signal and is subjected to
several nonlinear transformations. Characteristics of the noise in the bispectrum domain depend not
only on the variance of input noise but on the shape of signal Fourier spectrum.
For illustration of noise distortions leaked in bispectrum estimate, the graphs of the functions
⌢ ⌢
Re{B(p,q)} ̇ and Im{B(p,q)} ̇ corrupted by noise are plotted in Figures 2.3.18 and 2.3.19, respectively, for
a single pulse signal of rectangular shape and pulse length of ∆t = 7. Only a quarter of the bifrequency
plane is shown in these Figures due to the symmetry property of bispectrum (1.2.10) [2].
3
2
1
0
x 10 5
20
40
⌢
Fig. 2.3.18. Noisy Re{B(p,q)} ̇ as a function of two frequencies computed for ∆t = 7, SNR inp =0.35.
3
2
1
0
-1
-2
x 10 4
20
40
⌢
Fig. 2.3.19 Noisy Im{B(p,q)} ̇ as a function of two frequencies computed for ∆t = 7, SNR inp =0.35.
90
60
60
It has been demonstrated in [62] that for majority of frequencies in bifrequency plane, induced
noise possesses non-Gaussian PDF with zero mean and unknown variance.
Such properties of noise cause problems in selecting a proper filter for smoothing 2-D arrays
⌢ ⌢
Re{B(p,q)} ̇ and Im{B(p,q)} ̇ . One more peculiarity is that the behaviour of information component of
these functions is a priori unknown. Thus, it is difficult to recommend some non-linear non-adaptive
20
20
40
40
60
60

filter [53] and to properly select its parameters (scanning window size, trimming factors for α-trimmed
filters, etc.).
It is known (see [59, 63]) that the DCT-based filtering of images is commonly carried out in
square shape blocks. Processing both with overlapping or non-overlapping blocks is possible. The best
performance of the algorithm can be obtained for full overlapping of blocks. But this requires more
time for computations. Full overlapping means that each consequent block is shifted by only one sam-
ple with respect to the previous one. Respectively, the worst performance but minimal time consump-
tion due to sufficient decreasing of summation and multiplication operations are observed for non-
overlapping blocks. This mode corresponds to the block mutual shift equal to the block size. However,
the so-called blocking artifacts can arise in the case of non-overlapping block filtering [59].
For noise variance estimation that is usually necessary for operation of the DCT-based filters,
there exist two possible approaches. According to the first one, the averaged variance of the observed
process can be defined for total process and used for calculation of the constant-valued threshold for
all blocks.
The second alternative way is to estimate local variance inside each block and to calculate on its
basis the local threshold value for each given block. It makes the DCT-based filtering locally adaptive.
It was demonstrated in [46] that the DCT-based filtering with adaptively adjusted threshold is
more efficient procedure. Let us modify the 1-D adaptive DCT-based filtering technique to the case of
processing 2-D arrays of bispectrum samples.
Similarly to 1-D signal case, let us estimate the standard deviation (SD) for each 2-D block as
⌢
σ = 1, 483⋅ med{| W (x, y) |} , (2.3.13)
p,q pq
where med {…} is the median value of the sample; Wpq(x,y) denotes the DCT coefficients for the
block with coordinates (p0=p– b N
2
b +1, p– N
2 +2,…
the block side size (each 2-D block contains NbxNb samples).
p
91
N
b
b
+ , q0=q –-
2
N
2
b +1, q– N
2 +2,…
Thus, the proposed in [48] filtering procedure includes the following steps:
q
N
b
+ ); Nb is
- calculation of DCT spectral samples of for real and imaginary parts of the bispectrum estimate
⌢
B(p,q) ̇ containing NbxNb samples in each block;
- estimation of the local SD according to (2.3.13);
- calculation of threshold β⋅σ p,q
⌢
, where β defines smoothing properties of the filter. Usually the
latter value is within the interval 2…4 [58 – 60, 63]. Note that noise filtering improves with β
increasing but it leads to worse detail preserving;
2

- zeroing of the samples with absolute values are smaller than the threshold β⋅σ p,q
⌢
- carrying out inverse DCT;
- joint processing (averaging) of block outputs for all pixels if overlapping blocks are used.
The following four different methods were studied in computer simulations performed and dis-
cussed in [48]:
1) the traditional bispectrum-based signal reconstruction technique [2, 10] (Technique #1);
⌢
2) the combined bispectrum-filtering signal reconstruction technique with filtering of B(p,q) ̇ by the
standard median filter with the sliding window size of 5x5 [53] (Technique #2). In fact, this tech-
nique has been analyzed earlier in subsection 2.2 and below it is used for comparison purposes;
⌢
3) the combined bispectrum-filtering technique with filtering of B(p,q) ̇ by the α-trimmed filter with
the sliding window size of 5x5 and trimming of 5 maximum and 5 minimum values [53] (Tech-
nique #3);
⌢
4) the proposed technique with the DCT-based filtering of B(p,q) ̇ (Technique #4).
For the latter Technique the block size was of 8x8 pixels and three different values of the para-
meter β have been considered. Filtering was carried out both with partial and full block overlapping.
Note that the best results have been obtained for full overlapping of blocks. In case of partial overlap-
ping with the shift equal to Nb/2, the provided ε values were smaller by 3…15 % than in case of full
overlapping. Therefore, below only the data for DCT-based filtering with full overlapping are pre-
sented.
Three test signals of the same type (denoted below as signals ##1, 2, and 3) containing two posi-
tive-valued pulses of rectangular shape and amplitudes of 2 and 6 were used for computer simulation.
The pulse lengths were selected equal to ∆t1,2,3 = 3, 7, and 11 samples for the test signals ##1, 2, and
3, respectively. The interval between two pulses was fixed and it was equal to 5 samples. The total
length of each sequence is of I = 256 samples.
The results of computer simulations obtained in [48] are given in Tables 2.3.6 – 8.
Table 2.3.6. The values ε obtained for the test signal #1.
SNR inp
Technique
1.53 0.92 0.46 0.23 0.15
1 9.07 12.80 16.80 17.07 14.17
2 26.11 33.04 34.45 39.54 46.15
3 13.74 17.78 27.42 38.58 62.72
β =2.7 11.22 15.19 24.93 42.67 68.55
4
β =3.2 11.71 19.11 27.99 45.77 66.38
β =3.7 12.02 14.34 28.19 51.10 89.25
92
;

Table 2.3.7. The values ε obtained for the test signal #2.
SNR inp
Technique
3.50 2.10 1.05 0.53 0.35
1 9.42 11.25 16.30 14.75 13.50
2 5.99 9.80 17.73 28.36 32.38
3 5.91 10.34 20.46 34.09 37.97
β =2.7 10.19 14.34 27.17 48.25 62.44
4
β =3.2 11.06 14.05 27.84 60.85 72.23
β =3.7 11.54 13.93 28.39 54.95 64.35
Table 2.3.8. The values ε obtained for the test signal #3.
SNR inp
Technique
3.20 1.60 0.80 0.53 0.32
1 17.61 16.12 15.84 13.57 10.35
2 18.88 27.04 35.92 38.26 33.87
3 13.53 22.20 33.65 38.08 36.60
β =2.7 19.72 30.50 44.42 50.00 47.16
4
β =3.2 20.82 30.94 42.92 54.95 51.80
β =3.7 21.53 32.08 44.61 53.71 53.51
The non-adaptive Techniques ## 2 and 3 comparing to the Technique #1 can provide both im-
proving and worsening the SNRout (increasing or decreasing of ε). This depends on the test signal pa-
rameters and SNR inp . For example, for the test signal #1 the Techniques # 2 occurs to be very effective
for SNR inp =1.53. But the same Technique # 2 is characterized by the minimal ε among the considered
bispectrum-filtering methods for SNR inp =0.15 (see Table 2.3.6). Moreover, the use of non-adaptive
Techniques ## 2 and 3 can even lead to ε decreasing in comparison to traditional bispectrum Tech-
nique # 1. For instance, this happens for the test signal #2 when SNR inp are rather large (3 – 50 and
2.10, see Table 2.3.7).
These results confirm the existence of problems in selection of proper non-adaptive filters to be
applied within combined bispectrum-filtering framework. Therefore, practical use of the Techniques
##2 and 3 is limited.
The Technique #4 outperforms the Techniques 2 and 3 in most practically important cases. The
exclusion is the signal #1 for which the Techniques ##2 and 3 can provide larger values of ε for rela-
tively large SNR inp . The latter peculiarity can be explained by smoothness of the processed function.
Note that the proposed adaptive Technique #4 is the most effective (among the considered tech-
niques) in cases of low SNR inp for all test signals. The optimal value of the parameter β is recom-
93

mended to be within interval from 3.2 to 3.7.
Fig. 2.3.20.
Fig. 2.3.21.
2
1.5
1
0.5
0
filt
Re{ }
x 10 5
20
40
94
60
⌢
B(p,q) ̇ processed by adaptive DCT-based filter for ∆t = 7, SNR inp = 0.35, β = 3.7.
3
2
1
0
-1
-2
x 10 4
20
40
60
⌢
filt
Im{B(p,q) ̇ } processed by adaptive DCT-based filter for ∆t = 7, SNR inp = 0.35, β = 3.7.
Visual comparison of the real and imaginary parts of the bispectrum estimate before (Figures
2.3.15 and 2.3.16) and after (Figures 2.3.17 and 2.3.18) filtering allows getting imagination about the
efficiency of the proposed Technique #4. It is evident that the use of the proposed adaptive 2-D DCT-
based filter provides noise suppression in the real and imaginary parts of bispectrum estimate.
20
20
40
40
60
60

⌢
s(i)
Fig. 2.3.22. The reconstructed signal s(i)
7
6
5
4
3
2
1
0
-1
50 100 150 200 250 i
⌢
obtained by the Technique #1, ∆t = 7, SNR inp = 0.35.
Significant residual fluctuations are observed in the signal reconstructed by the conventional
Technique #1 (see Fig. 2.3.22). Technique #3 (see Fig. 2.3.23) causes rather large distortions but, at
the same time, it provides better noise suppression than Technique #1.
In turn, the proposed Technique #4 provides only negligible noise presence and small distortions
in the reconstructed signal (see Fig. 2.3.24).
⌢
s(i)
Fig. 2.3.23. The reconstructed signal s(i)
⌢
s(i)
Fig. 2.3.21. The reconstructed signal s(i)
7
6
5
4
3
2
1
0
-1
7
6
5
4
3
2
1
0
-1
50 100 150 200 250 i
⌢
obtained by the Technique #3, ∆t = 7, SNR inp = 0.35.
50 100 150 200 250 i
⌢
SNR inp = 0.35, β = 3.7.
95
obtained by the Technique #4, ∆t = 7,
The presented results demonstrate good performance of the proposed technique [48] within the
total range of the considered input SNRs and test signals. This technique provides improvement of the

SNRout by 8 dB comparing to the conventional bispectrum technique [10] in the most important in
practice cases of low input SNRs.
As it was supposed, the use of the non-adaptive algorithms seems to be unsuitable in general.
However, in some cases they are able to give good results (see, for example, the data in Table 2.3.6
obtained by Technique #2). In turn, adaptive procedures are able to change their parameters depending
on the situation at hand (behavior of the processed 2-D functions and noise level).
The comparison of the values of ε has been also carried out for the proposed Technique #4 and
the techniques earlier developed in [46]. The results are approximately of the same value for rather
large input SNRs. However, the proposed Technique #4 provides the values ε that are larger by ap-
proximately 10…20% than adaptive technique considered in [46]. Recall that the difference is in us-
ing either the 2-D DCT-based filtering for the Technique #4 or the 1-D DCT-based filtering for the
methods in [46]. Certainly, the 2-D DCT-based filtering requires larger processing time. But within
entire framework of the combined bispectrum-filtering processing it does not lead to considerable in-
creasing of total computation time.
The latter numerical simulation results show that the use of adaptive filtering within combined
bispectrum-filtering approach leads to obvious benefits. However, the use of the designed local adap-
tive DCT-based filters is not the only opportunity. Thus, in this Section we consider and carry out
brief performance analysis for three other techniques.
First, in paper [64] a method of 2-D adaptive filtering based on the so-called Z-parameter has
been designed. This method (further Technique # 5) is based on the use of preliminary nonlinear filter,
calculation of Z-parameter for each sliding window position, and hard-switching between two or sev-
eral nonlinear filter outputs according to results of Z-parameter comparison to the corresponding thre-
shold. One advantage of this filter is that noise type (additive, multiplicative, mixed) should not be
known a priori. For the considered application, we used the α-trimmed filter with the sliding window
size of 5x5 and trimming of 5 maximum and 5 minimum values as a preliminary filter. Besides, this
filter as well as the standard 5x5 median filter were used as component filters (switching between their
outputs was performed). The threshold was equal to 0.4.
Second, we considered an opportunity of applying the standard sigma filter [65] where for each
given sliding window position a local standard deviation is estimated according to (2.3.13). Then, us-
ing this estimate, 2σ-neighborhood is determined and averaging of pixel values that belong to this
neighborhood for given sliding window position are carried out. This method is referred below as
Technique # 6.
Third, it is possible to apply a two-stage filtering procedure. At the first stage, Technique # 4
96

(β=3.2) is applied and then its output is subjected to post-processing by the 3x3 center weighted me-
dian filter with the center weight of 3. The goal of such post-processing is to remove spiky values re-
tained by the DCT-based filter. This approach is noted as Technique # 7. The obtained simulation re-
sults for the Techniques # 5, 6, and 7 for all three test signals are presented below in Tables 2.3.9, 10,
and 11, respectively.
Table 2.3.9. The values ε calculated for the test signal #1.
SNR inp
Technique
1.53 0.92 0.46 0.23 0.15
5 20.34 22.77 29.31 42.87 49.92
6 14.26 14.06 26.27 30.73 35.13
7 10.97 17.99 36.41 40.31 61.96
Table 2.3.10. The values ε calculated for the test signal #2.
SNR inp
Technique
3.50 2.10 1.05 0.53 0.35
5 33.55 38.12 40.45 45.17 43.82
6 12.71 19.57 34.26 29.14 28.14
7 11.61 14.64 28.71 60.84 73.16
Table 2.3.11. The values ε calculated for the test signal #3.
SNR inp
Technique
3.20 1.60 0.80 0.53 0.32
5 17.20 24.78 39.13 46.08 39.52
6 9.83 21.92 29.77 27.80 21.61
7 21.05 31.39 41.93 55.37 52.51
Comparing the data in Tables 2.3.9, 10, and 11 to the corresponding data in Tables 2.3.6, 7, and
8 as well as analyzing these data within each Table, it is possible to conclude the following. The
Technique # 6 is able to produce some performance improvement in comparison to the conventional
BLW method (Technique # 1) but among the considered combined bispectrum-filtering techniques the
Technique # 6 is surely not the best. The use of adaptive nonlinear filter (Technique # 5) instead of
non-adaptive ones (Technique # 2 and Technique # 3) can be expedient. The Technique # 5 either
provides the largest values of ε (like, e.g., for the test signals #2 and #3, see the data in Tables 2.3.10
and 2.3.11and compare them to the corresponding data in Tables 2.3.7 and 2.3.8) or appropriately
large values of ε.
Selection the Technique # 7 instead of the Technique # 4 does not seem reasonable. Really, in
majority of the considered situations the Technique # 7 produces practically the same values of ε as
the Technique # 4 does. Since the Technique # 7 is slightly more complicated than the Technique # 4,
97

there are no reasons of applying the Technique # 7.
Thus, the developed approach (Technique # 4) provides the best signal reconstruction system
performance comparing to the non-adaptive techniques in most cases. It is shown that the maximum
improvement has been obtained for small input SNR. The novelty of the approach proposed in [48] is
in the local estimation of standard deviations inside each block as well as in the corresponding calcu-
lation of hard threshold for DCT-based filter.
2.4. Adaptive 1-D filtering in bispectrum-based signal reconstruction
Detailed performance study of both non-adaptive and adaptive filtering based on so-called Z-
parameter and LPA-ICI in bispectral-based signal reconstruction in noise has been performed in our
paper [49]. Below we pay attention to the results obtained with application of 1-D adaptive filtering of
the data in bispectrum-based signal reconstruction system.
The 1-D local adaptive filters (LAFs) perform in such a manner that for each estimation sample
the window size is selected to suppress the random noise as much as possible and to preserve the sig-
nal features.
One additional motivation to use LAFs [64, 66 – 69] is their ability to operate under a priori un-
known noise variance and/or in the cases of non-stationary noise. This is important since the noise
components in the Fourier spectrum real and imaginary parts recovered by bispectrum estimation have
frequency-dependent statistical properties (see previous subsection).
Z
A LAF described in [64, 66] is a hard switching filter which output U ( r ) is computed as
f1
⎧U ( r) if Z( r) ≥ a2
Z ⎪ f2
U ( r) = ⎨U
( r) if a1 ≤ Z( r) < a2
, (2.4.1)
⎪ f3
⎪⎩ U ( r) if Z( r) < a1
f1
f 2 f 3 where U ( r)
, U ( r)
and U ( r)
are the outputs of the three non-adaptive filters f1, f2 and f3, and a1
and a2 are the thresholds (see the block scheme in Fig. 2.4.1).
98

Fig. 2.4.1. Block scheme of local adaptive filters based on Z-parameter.
Z-parameter in (2.4.1) is calculated as
r+
( N −1)
/ 2
f
p
∑
j=
r−(
N p−1)
/ 2
99
r+
( N −1)
/ 2
p
∑
f
Z ( r)
= ( U ( j)
−U
( j))
U ( j)
−U
( j)
, (2.4.2)
j=
r−(
N p−1)
/ 2
where U ( j ) denotes the input signal, U ( j)
f
is the output of the nonlinear pre-filter [66], for example,
the α-trimmed or Wilcoxon filter. The α-trimmed filter with Np =9 and the trimming parameter Nα = 2
has been employed in [66], where Nα is the number of the values rejected after sample data sorting in
the scanning window.
Z-parameter values in (2.4.2) are quite close to zero if a current r-th sample belongs to a con-
stant or linearly changing fragment of the input signal U ( j ) . Then, if | Z( r) | < a1
, a filter that possesses
good noise suppression properties (f3) should be applied.
On the contrary, large absolute values of Z-parameter ( | Z( r) | ≥ a2
) are observed for fragments
containing discontinuities for which one needs to apply a signal shape preserving filter (f1). Note that
the maximal value of | Z( r ) | is equal to unity.
For intermediate values of |Z(r)|, a 1 ≤ | Z( r)
| < a 2 , the output of the local adaptive filter is assigned
to the output of f2.
In general, the performance of the LAF from [66] depends upon the set of filters f1, f2 and f3 as
well as on thresholds a1 and a2 that are used. In what follows, we consider possible sets of filters f1, f2
and f3. As for the thresholds, they are fixed (a1=0.2 and a2=0.4) according to the recommendations
given in [66].

Two typical LAFs given by (2.4.1) and (2.4.2) have been considered in [49]. The first one is
nonlinear LAF (ADAPT-NLIN) for which the following switch filters have been used: a median filter
with a scanning window size of 5 samples (f1), a α-trimmed filter with N=9, Nα=2 (f2), and a α-
trimmed filter with N=13, Nα =3 (f3). The second one is linear LAF (ADAPT-LIN) that consists of the
following switch filters: a mean filter with a window size of N=5 samples (f1), a mean filter with N=9
(f2), and a mean filter with N =13 (f3).
Another approach to adaptive filtering is the local polynomial approximation (LPA) with use of
the intersection of confidence intervals (ICI) rule proposed in [67 – 72]. The LPA is a tool for linear
filter design. In particular, the zero-order polynomial approximation is used in order to obtain the
scanning window mean filters. LPA also allows one to apply higher-order polynomial approximations,
which can be useful provided that the filtered signal is smooth enough. We will pay attention only to
the second-order polynomial approximation.
The ICI rule is an algorithm for adaptive window size selection. The idea of this approach is as
follows. The algorithm searches for a largest local vicinity of the point of estimation where the LPA
assumption fits well to the data. The estimates Uˆ h ( r)
(the filtered estimates of the bispectrum real
Re{ Sˆ bisp ( r)}
or imaginary Im{ Sˆ bisp ( r)}
parts for the case considered) are calculated for a grid of window
sizes (scales) , ,..., } h h h H h = ∈ , where J h h h < < < ..., . The adaptive scale is defined as the largest
{ 1 2 J
1
2
+
h of those windows in the set H whose estimate does not differ significantly from the estimators cor-
responding to the smaller window sizes. More precisely, a sequence of confidence intervals
( ) ( )
⎡ ⎤
⎢ ˆ ˆ
j h σ j ˆ h σ ⎥
⎢ U j
ˆ
h U ⎥
j h
⎣ j ⎦
D = U r − Γ , U r + Γ is determined, where Γ > 0 is a threshold parameter and σ ˆ is a
Uh
j
standard deviation of estimate. The ICI rule can be stated as follows: consider the intersection of con-
j
fidence intervals I j = ∩ D
i=
1 i and let j + be the largest of the indices j for which I j is non-empty. Then
the optimal scale h + + is defined as h = h + and, as a result, the optimal scale estimate is equal to Uˆ + ( r)
.
j
h
Theoretical analysis provided in [71] shows that this adaptive scale gives the best possible point-
wise mean-squared error. In practice this means that adaptively, for each sample, ICI allows the max-
imum degree of smoothing, stopping before over-smoothing begins [70].
The threshold parameter Γ is a key element of the algorithm. Too large a value of this parameter
leads to signal over-smoothing and too small a value leads to under-smoothing. A reasonable value to
preserve the signal and remove the noise as much as possible is somewhere in between.
Optimal values of Γ can be derived from some heuristic and theoretical considerations (see, e.g.
[67, 68, 70, 72]). We prefer to treat the threshold Γ as a fixed design parameter in our investigations.
100

This approach has been developed for linear estimates [67, 68, 70] as well as for nonlinear (me-
dian) filters [69]. For filtering, we use symmetric U h and non-symmetric (left U
sym
h and right U
left
h ) right
windowed estimates, where h sym , h left and h right are the window sizes of the corresponding estimates.
The ICI rule gives the adaptive window size for each of these estimates. Let us denote the correspond-
ing optimal estimates by ˆ + ( r)
, ) ( ˆ + r and ) ( ˆ r . The final estimate is calculated as the
U
h sym
U
h left
U
h right
+
weighted mean with the combined final LPA estimate U ˆ ( r)
produced in the form of
Uˆ ( r) = λ ( r) U + ( r) + λ ( r) U + ( r) + λ ( r) U + ( r)
, (2.4.3)
left h right
left h sym
right hsym
−2 −2 −2
σ left ( r) σ right ( r) σ sym ( r)
left ( r) = , ( ) , ( )
2 2 2 right r = r =
− − − −2 −2 −2 sym −2 −2 −2
σ left + σ right + σ sym σ left + σ right + σ sym σleft + σ right + σ sym
λ λ λ
,
( r) ( r) ( r) ( r) ( r) ( r) ( r) ( r) ( r)
where σ left , σ right , and σ sym are the standard deviations of the estimates. Thus, the inverse variances are
used as the weights of the partial symmetric, left and right estimates for fusing in the final one.
Similarly, the adaptive window size median [69] (Median-ICI) is introduced in the form of the
symmetric and non-symmetric left and right estimates as
Uˆ
h
Uˆ
h
Uˆ
h
sym
left
right
( r)
= median{
Uinp
( r + n)},
n = −(
h −1)
/ 2,...,
( h −1)
/ 2,
( r)
= median{
Uinp
( r + n)},
n = 0,...,
( h −1)
/ 2,
non −symmetrical
right median
101
symmetrical
median
( r)
= median{
Uinp
( r + n)},
n = −(
h −1)
/ 2,...,
0,
non−
symmetrical
left median , (2.4.4)
where Uinp ˆ is the input process and h ∈ H . The ICI rule makes these median filters be data adaptive
with the varying window size optimizing each of these symmetric and non-symmetric estimates. The
final estimate is found again in the weighted mean form (2.4.3).
We apply these adaptive linear and median filters to the real and imaginary parts of the recov-
ered Fourier spectra. The following parameters of the ICI rule are used. The window sizes are H = {1,
5 ,7, 11, 21, 43, 61 samples} for the linear LPA filters of order 2 and H = {1, 5 , 11, 21, 43} for the
median filters. In the experiments, the threshold parameter Γ is fixed to be 1.5 for both the cases.
The two-pulse test signal has been employed in computer experiments with pulse width of ∆t=7.
The values of SNR inp are varied from 0.35 to 3.5. The criterion χ is introduced additionally to criteria
(1.5.7 – 1.5.11) for filtering performance estimation as
χ = ε , (2.4.5)
ε
0

where ε0 is the parameter calculated according to (1.5.11) for the conventional BLW technique [10]
(denoted as Technique 1, below) for a given SNR inp .
The plots of χ (2.4.5) as the functions of SNR inp are presented in Fig. 2.4.2. The results are given
for the following techniques:
A) The conventional technique 1;
B) The combined technique that employs smoothing the real and imaginary parts of signal Fourier
spectrum recovered from bispectrum by α-trimmed filter with Np = 9, Nα = 2;
C) The same filtering as in B) performed by α-trimmed filter with Np = 13, Nα = 3;
D) The combined technique based on adaptive filtering ADAPT-NLIN;
E) The combined signal reconstruction technique using adaptive filtering ADAPT-LIN;
F) Technique 4 (median filter) with Np =5;
G) Technique 5 (mean filter) with Np =5;
H) Technique 5 (mean filter) with N p=9;
I) Technique 5 (mean filter) with N p=13.
Figure 2.4.2. Criterion χ as the function of inp
SNR for different techniques. The curves A-I correspond
to the following techniques: A – to the conventional Technique 1; B – to the combined
technique that employs smoothing by α-trimmed filter with Np = 9, Nα = 2; C – to the
combined technique that uses smoothing by α-trimmed filter with Np = 13, Nα = 3; D – to the
combined technique based on adaptive filtering ADAPT-NLIN; E – to the combined signal
reconstruction with ADAPT-LIN; F – to the Technique 4 (median filter) with Np =5; G – to
the Technique 5 (mean filter) with Np=5; H – to the Technique 5 (mean filter) with Np=9; I –
to the Technique 5 (mean filter) with Np=13.
102

The plots in Fig. 2.4.2a give us the possibility to compare the performance of the adaptive filter
ADAPT-NLIN and the corresponding switch nonlinear filters. Similarly, Fig. 2.4.2b gives the same
opportunity for comparative analysis in the case of ADAPT-LIN that is based on the mean filters with
different scanning window sizes [49].
The analysis of these data allows us to conclude the following.
- The two types of adaptive filters considered (see curves D and E) show good performance and
they lose only a little to the best among the non-adaptive (switch) filters for a wide range of values of
SNR inp ( SNR inp 2). This
can be explained by the worse dynamic properties of linear filters in comparison to the nonlinear ones.
- The efficiency of both the adaptive and non-adaptive techniques is the largest for low SNR inp
(which is important and useful for practical applications).
- Note that in some cases the use of the combined methods can lead to a worse performance with
respect to the conventional technique. For example, the use of the mean filter with Np=13 (see curve I
in Fig. 2.4.2b) produces χ 1.5, this is due to the large distortions introduced by the
mean filter with rather large Np. The goal of using adaptive filtering is to avoid such situations for the
considered application of combined bispectrum-filtering data processing.
The reconstructed test signal plots (example of pulse length of ∆t=3) obtained for the conven-
tional [10] and the proposed combined technique ADAPT-LIN [49] are shown in Figs. 2.4.3a and
2.4.3b, respectively. It can be seen that the noise in the reconstructed signal is sufficiently suppressed
in the case of filtering the real and imaginary parts of the recovered Fourier spectrum separately.
For comparison purpose we have also considered another method to reconstruct the signal wave-
form from a set of mutually shifted noisy realizations. For this method, the first realization x (1) (i) in
(1.4.1) was used as a reference one. Then, for all the other observed realizations x (m) (i) (m=2,…,M),
cross-correlation functions for the given m-th realization and the reference realization have been cal-
culated. The position of the global maximum of each cross-correlation function has been determined
and used as an estimate of the signal random shift τm. After this, cyclic shifting of the realizations
x (m) (i) (m=2,…,M) has been performed by τm. Finally, averaging of all the M aligned realizations has
been carried out.
Resulting output is presented in Fig. 2.4.3c. As can be seen, due to the errors in the mutual shift
estimation, waveform pulses that are originally rectangular become smeared. Their destroyed ampli-
tudes (about 1 and 3) are considerably smaller than the original ones (2 and 6).
103

s(i) ^
7
6
5
4
3
2
1
0
−1
50 100 150 200 250
i
^
s(i)
(a) (b)
s(i) ^
7
6
5
4
3
2
1
0
-1
50 100 150 200 250
(c)
104
7
6
5
4
3
2
1
0
50 100 150 200 250
Fig. 2.4.3. Reconstructed signal obtained by: the conventional [10] (a), the proposed method [49] (b),
and the method based on cross-correlation and realization alignment (∆t=3 and the minimal considered
SNR inp =0.15) (c).
The numerical simulation results obtained in [49] for different three test signals (all in the form
of two rectangular pulses with ∆t12=5 and the amplitudes А1=2, А2=6, but with different ∆t) are given
in Table 2.4.1 for the suggested combined bispectrum-filtering techniques.
For the conventional Technique 1 [10], the notation NONE is used in Table 2.4.1.
i
i

Table 2.4.1. Numerical simulation results obtained [49] for adaptive filters.
Technique SNR inp ε SNR inp ε SNR inp ε SNR inp ε SNR inp ε
NONE
9.07
12.80
Test signal 1: ∆t =3
16.80
17.07
14.17
ADAPT-NLIN 9.92 14.50 20.75 27.94 30.99
ADAPT-LIN 1.53 10.18 0.92 15.09 0,46 22.40 0.23 34.63 0.15 42.52
LPA-ICI 9.39 13.77 20.06 28.74 34.95
Median-ICI 9.08 12.94 18.06 22.01 22.21
NONE
9.42
11.25
Test signal 2: ∆t =7
16.30
14.75
13.50
ADAPT-NLIN 10.19 12.52 21.11 23.38 27.67
ADAPT-LIN 3.49 9.88 2.09 12.31 1.05 22.24 0.52 28.30 0.35 36.85
LPA-ICI 10.02 12.66 21.61 25.43 33.68
Median-ICI 8.85 10.87 16.39 18.29 20.38
NONE
17.61
16.12
Test signal 3: ∆t =11
15.84
13.57
10.35
ADAPT-NLIN 15.79 18.23 24.95 27.55 26.55
ADAPT-LIN 3.20 11.83 1.60 16.29 0.80 26.38 0.53 33.43 0.32 37.25
LPA-ICI 20.73 21.96 32.29 38.95 44.41
Median-ICI 15.37 16.14 19.38 20.08 20.54
It should be noted, that there is no obvious favorite among these techniques. For the short time
pulses (∆t = 3), the filter ADAPT-LIN provides the best performance, i.e. the largest values of ε have
been reached. The results for ADAPT-NLIN are better than those for LPA-ICI if SNR inp is rather large
and vice versa. For the test signal with ∆t =7, ADAPT-LIN is the best for small SNR inp = 0.35 but it is
the worst for SNR inp = 3.49 and 2.09. Note that for the test signals with ∆t =3 and 7, all the three adap-
tive filters produce improvement for entire range of SNR inp considered.
However, for the test signal length of ∆t =11 samples, the improvement is observed for only
small SNR inp . Both ADAPT-LIN and ADAPT-NLIN produce a reduction in ε in comparison to the
conventional technique (NONE) for SNR inp =3.2. At the same time, LPA-ICI still performs well. It
ensures improvement for the entire range of SNR inp considered.
Since the signal and properties of the recovered spectrum are a priori unknown, the advantage
of LPA-ICI is that it produces improvement for wider conditions (possible variations of the signal
waveform and SNR inp ). Median-ICI always produces worse results than LPA-ICI.
Thus, use of local adaptive filters instead of non-adaptive ones is one solution to improve the
combined bispectrum-filtering performance. It is shown [49] that the adaptive filters based on Z-
parameter and ICI rule provide stable and considerable benefits of the combined bispectrum-filtering
techniques in comparison to the conventional bispectrum based processing. The improvement pro-
105

vided is the largest for low (around and less than unity) input SNR.
2.5. Conclusions
It has been shown that there a quite many different approaches to combined bispectrum-filtering
processing of mutually shifted noisy realizations with the purpose of signal shape reconstruction.
Analysis of statistical properties of noise leaked to bispectrum estimates and signal Fourier spectrum
estimates shows that these properties are rather complicated. This makes problematic selection of fil-
tering methods able to perform well in conditions of complex-valued non-stationary and non-Gaussian
noise when a priori information about signal component is limited. Several different approaches have
been studied by us during recent years. The basic conclusions are the following:
1) It is expedient to carry out separate filtering of real and imaginary parts of complex va-
lued bispectrum or signal Fourier spectrum recovered from bispectrum.
2) Adaptive nonlinear filters are, in general, able to provide better performance than linear
and non-adaptive ones. Several novel adaptive 1-D and 2-D filters well suited for the considered ap-
plication have been proposed recently. Within combined bispectrum-filtering framework, these filters
are able to provide performance improvement for a wide range of input SNR and typical signal shapes
and parameters.
3) The largest benefit due to combining the bispectrum approach with nonlinear and adap-
tive filtering is commonly provided for low input SNR and/or a limited number of processed realiza-
tions. This is very important for practice since performance improvement is a crucial task just for such
conditions.
4) Analysis carried out by numerical simulations has allowed giving practical recommen-
dations concerning filter parameters like the parameter β for adaptive DCT based filters.
106

3. Bispectrum-based image reconstruction techniques by us-
ing tapering pre-distortion inserted in image rows
3.1. Noisy and jittered image reconstruction by using additive pre-distortion
A lot of attention in the literature has been recently paid to the problems of 1-D unknown wave-
form signal reconstruction from bispectrum estimates (see Chapter 2 and references there). It is well-
known that the information about signal shape resides primarily in the signal phase Fourier spectrum
and not in the magnitude Fourier spectrum or power spectrum. Since bispectrum estimation provides
preservation the signal Fourier phase as well as low sensitivity to AWGN and signal translation inva-
riance, it is natural to expect promising results in cases of bispectrum-based 2-D image processing. In
fact, several bispectrum-based image reconstruction algorithms have been proposed recently [18, 73 –
75]. However, the existing approaches usually deal with the following restrictions: (1) due to signal
shift invariance property, image row Fourier spectrum recovered from bispectrum corresponds to a
circularly shifted row that might cause image distortions and result in problem of image row align-
ment; (2) signal phase Fourier spectrum recovery from bispectrum argument provides correct results
only when bispectrum and Fourier phase values are within the principal phase value interval limited
by (-π, +π), otherwise phase discontinuities (caused by phase wrapping), phase ambiguity and image
reconstruction errors arise; (3) phase unwrapping technique can lead to significant distortions in the
presence of AWGN.
Note that phase wrapping is the most difficult problem to be solved between the listed restric-
tions and phase unwrapping may provoke very large errors in the cases of large phase fluctuations and
noise influence. Because of this, it is preferable to avoid completely phase unwrapping procedure. No
phase unwrapping is required and principal arguments of the phase bispectrum are necessary for im-
age reconstruction technique [75] based on the derivation of log-magnitude estimates. However, the
main requirement of the approach suggested in [75] is to have no zeros for signal z-transform on the
unit circle nor in complex conjugate pairs. The last requirement is right only for limited minimum
phase signal class and image rows usually belong to non-minimum phase signals.
To alleviate above mentioned drawbacks, a novel approach to bispectrum-based digital image
reconstruction has been proposed in our papers [76 – 80]. The goal of the approach suggested in [76 –
80] is twofold. First, we offer the correction of random shifts of image rows by computing the differ-
ences between the adjacent row cross-correlation maximum coordinates that correspond to jittery im-
age and centre of gravity values of the adjacent row cross-correlation functions reconstructed from
107

ispectrum estimates. Second, to reduce phase errors caused by rapid phase changes, to avoid phase
unwrapping and, hence, to obtain correct reconstructed Fourier phase values, we propose to insert pre-
distortions added to every processed 1-D row image after jitter removal.
Let us consider the approach based on insertion additive pre-distortions placed in each image
row in the form of large amplitude δ-impulses [76, 78]. We assume that an unknown object depicted
in a digital image is corrupted by AWGN and jitter. Each k-th (k=1, 2, 3,…, I) image row is a real-
( m)
valued non-negative sequence { x ( i)}
(i=1, 2, 3,…, I) that is observed at the digital reconstruction and
k
object recognition system input as the following m-th (m=1, 2, 3,…, M) realization
( m) ( m) ( m)
k
( i) =
k
( i − τ k
) +
k
( i)
, (3.1.1)
x s n
(m)
where τ is a random shift of the original real-valued deterministic non-minimum phase signal (i)
k
(original, i.e. not distorted by AWGN and jitter image of k-th arbitrary row), for which bispectrum is
( )
supposed to be non-zero; n ( i)
m is the m-th realization of AWGN with specified variance
k
AWGN is assumed to be uncorrelated to a priori unknown signals { (i)
are random and caused by jitter variables with uniform distribution.
108
s k
2
σ m .
τ }
(m)
s }. We also assume that { k k
In practice, relative random displacement between adjacent rows (jitter) can be provoked by sto-
chastic properties of radio telecommunication channel, mechanical raster scanning system errors as
well as by data digitizing from a noisy analog image. In the latter case, synchronization pulses are cor-
rupted by AWGN affecting the loss of “lock” in digitizing device (see, for example, [81]). It should be
noted, that large jitter may be one of the main restrictions in high speed digital telecommunication
systems.
Notice that the considered model (3.1.1) is more complicated comparing to the existing models
described in [18, 73]. In these papers, the total sequence of 16256 samples (original 2-D image was of
size 127x128 pixels) was randomly placed repeatedly in a 1-D noisy frame of 16384 samples in order
to simulate jitter interference. However, an important aspect of the problem of adjacent rows de-
jittering was not treated yet. Furthermore, images restored by the approach suggested in [18, 73] are
circularly shifted and these images need manual realignment that is a quite time consuming process.
Note that this is practically inappropriate for automatic pattern recognition systems. Moreover, in [18]
and [73] only tone interference in the form of five sinusoidal signals with random phases added to
each realizations was considered.
To alleviate these shortcomings and restrictions, a novel approach to image reconstruction in jit-
ter and AWGN, as well is impulsive noise environment is considered below.

The proposed image reconstruction technique includes the following set of processing stages
and steps [76, 78].
Stage 1. Jitter removal by correlation and bispectrum-based image row processing.
Step 1.1. The computation of cross-correlation function estimates between two adjacent jittered and
noisy image rows (3.1.1) in the form of
( m)
k,
k + 1
109
I
∑
i=
1
m
Rˆ
( )
( l)
= x ( i)
x ( i − l)
, (3.1.2)
(
where l is the delay index and the total number of the functions R ( l)
m
k k + is equal to I-1 for each m-th
realization in (3.1.1).
k
( m)
k + 1
ˆ )
, 1
Step 1.2. Evaluation and storage the coordinates of the maximums of the functions (3.1.2) as
{ R ˆ
}
( m)
( m)
k , k + 1(
l)}
max ⇒ { lmax
k jittered . (3.1.3)
Step 1.3. Bispectrum reconstruction (BR) of the cross-correlation function estimates R ˆ
k k ( l)}
BR
{ , + 1 by
using ensemble averaging of the M bispectrum estimates of the set (3.1.2) and the conventional BLW
algorithm [10].
Step 1.4. Evaluation and storage of the centers of gravity (CG) { l CG k } BR of the BR cross-correlation
estimates as follows
{ Rˆ ( l)} ⇒ { l } . (3.1.4)
k , k+ 1 BR CG k BR
The steps 1.3 and 1.4 need more detailed explanation. Since BR always provokes the suppres-
sion of the linear Fourier phase factor, bispectrum-based reconstructed rows will be always centered.
According to the previously mentioned translation invariance property (1.2.12), the position of a
row reconstructed from bispectrum is determined by CG value. Hence, the centering of the functions
{ R ˆ
k,
k+
1
( l)}
BR according to the values { l
CG k
}
BR (see formula (3.1.4)) always takes place. Therefore,
after BR of the correlation estimates { R ˆ
k,
k+
1
( l)}
BR , they will be centered according to the row image
CGs defined as
CG
I
∑
l=
1
k = I
∑
l=
1
lRˆ
Rˆ
k , k + 1
k , k + 1
( l)
. (3.1.5)
( l)
Due to the low sensitivity of the bispectrum-based signal recconstruction techniques to AWGN,
the latter influences quite feebly on the CG values. Therefore, the CGs (3.1.5) can be selected as a
reliable reference values for the proposed jitter removal in AWGN environment.

Step 1.5. Jittered rows’ alignment procedure carried out on the basis of the following differences
rewritten as
∆ = { l } −{
l } . (3.1.6)
( m)
k
110
( m)
max k
After jitter removal by insertion the differences (3.1.6) in (3.1.1), the latter expression can be
jittered
CG k
( m)
( m)
x ( i)
s ( i)
+ n
k
( i)
k corrected k
BR
≅ . (3.1.7)
Stage 2. Bispectrum-based image reconstruction with insertion the pre-distortions in the image rows.
Step 2.1. Addition to the primary function (3.1.7) of some secondary function for which its Fourier
spectrum pronouncedly has no zeros and, hence, the total magnitude Fourier spectrum does not con-
tain zeros. As an simplest example, just two δ-impulses (here δ(…) is the Kronecker delta function)
can be chosen. These δ-impulses are placed in the first and the last samples of each image row (3.1.7).
After jitter removal and insertion of such pre-distortions, the modified (pre-distorted) image row can
be written as
( m) ( m)
f ( i) = A δ ( i − 1) + x ( i) + A δ ( i − I ) , (3.1.8)
k 0 k corrected 0
where A0 is the pre-distortion function amplitude that must satisfy the following important condition
to provide absence the zeroes in the Fourier spectrum
A
0
>>
I
∑
i=
1
x
( m)
k
( i)
. (3.1.9)
Due to satisfaction of the last condition (3.1.9), the modified image rows (3.1.8) belong to the
minimum phase signal class. The proposed transform to minimum phase signal space lets us to avoid
the above noted phase wrapping. Hence, recovery of the image row phase Fourier spectrum from
bispectrum estimate can be obtained uniquely.
At the same time, it should be stressed that CGs of the row images (3.1.8) occur to be of
approximately fixed values due to:
- satisfaction of the condition (3.1.9);
- quite high robustness of bispectrum-based reconstruction technique to the AWGN influence.
Therefore, in accordance to the proposed stategy, the CGs obtained for different rows take
approximately the same values.
Step 2.2. Computation the bispectrum estimates of the row images (3.1.8) by direct technique (see Eq.
(1.3.4 –1.3.6)).
Step 2.3. Image row phase and magnitude Fourier spectra recovery from bispectrum estimates
by using the conventional BLW recursive algorithm [10]. In contrast to the iterative reconstruction

technique [75] that is quite complicated and seriously limited by unpredictability of its convergence,
the BLW algorithm [10] seems us more simple and reliable. It should be especially emphasized that
the recursive algorithm works perfectly if the processed signal is of minimum phase.
Step 2.4. Row by row 2-D image reconstruction by inverse Fourier transform of the complex valued
row Fourier spectra.
Now let us pay attention to the example of reconstruction the 8-bit test image “Barbara” of di-
mensions of IxI = 256x256 pixels.
The corruption has been simulated in the following way: AWGN with zero mean and with fixed
sample variance of
2
(m)
σ m =100 was added independently to each image row. Random shift
k
111
τ was
supposed to be uniformly distributed with fixed maximum deviation of ±20 pixels. Every k-th (k=1, 2,
3,…, 256) image row of the original image was randomly circularly shifted to simulate the jitter influ-
ence. The pre-distortion δ-impulse amplitudes were equal to A0=50000 to avoid phase wrapping relia-
bly, on one hand, and to provide CGs of different lines to be fixed values, on the other hand. Only
small number of the observed realizations (M =5 frames) were used in our numerical simulations.
In Fig.3.1.1, the noise- and jitter-free original test image is shown.
Fig. 3.1.1. The original noise- and jitter-free
test image.
riance of
Fig. 3.1.2. Image corrupted by AWGN and jitter.
One m-th (m=1, 2,…, 5) realization of original image corrupted by AWGN with sample va-
2
σ m =100 and with row jitter with maximum deviation of ±20 pixels is represented in Fig.
3.1.2. As it is evident from Fig. 3.1.2, the image is completely destroyed by distortions and it is practi-
cally impossible to recognize an unknown object in this image.

(
In accordance to the proposed approach, the cross-correlation function estimates R ( l)
m
112
ˆ )
k , k + 1 (see
Eq. 3.1.2) between two neighbor jittered and noisy image rows were computed. The cross-correlation
estimates are represented in Fig. 3.1.3 as the set of 255 1-D functions (“rows”). Note that lighter color
corresponds to larger values.
Fig. 3.1.3. The cross-correlation function esti-
mate derived between neighbor jittery and noi-
sy image rows.
Fig. 3.1.4. The cross-correlation function esti-
mate recovered from noisy bispectrum esti-
mate.
As can be seen from Fig. 3.1.3, the maximums of the cross-correlation functions are distributed
randomly “from row to row” due to jitter influence.
For jitter removal and 2-D image true alignment we need to derive the differences
∆ (see Eq.
3.1.6) that have been computed according to the CG values of the above mentioned cross-correlation
estimates recovered from bispectrum estimate.
The cross-correlation function estimates recovered by conventional recursive BLW algorithm
[10] are represented in Fig. 3.1.4.
It is seen clearly from Fig. 3.1.4 that the BR cross-correlation function maximums have
“ranged” in vertical white “column”. This “column” will serve us as the quite reliable reference to de-
rive the correction differences (3.1.6).
To illustrate typical behavior of an original (noise- and jitter-free) 1-D line image and its magni-
tude and phase Fourier spectra, the plots of a 1-D line image and its magnitude and phase Fourier
spectra are shown in the Figures 3.1.5 – 3.1.7, respectively. Because of the presence of several zeroes
in the magnitude Fourier spectrum (see Fig. 3.1.6), the Fourier phase changes very rapidly and phase
(m)
k

wrapping is clearly seen in Fig. 3.1.7. Such Fourier phase behavior provokes phase ambiguity problem
and causes the phase errors in image recovery.
Fig. 3.1.5. An arbitrary image row of the origi-
nal image represented in Fig. 3.1.1. (no pre-
distortion).
Fig. 3.1.7. Image row phase Fourier spectrum
corresponding to original image row in Fig.
3.1.5. (no pre-distortion).
Fig. 3.1.6. The magnitude Fourier spectrum of
the image row in Fig. 3.1.5.
Fig. 3.1.8. Pre-distorted image row.
That is why, in order to avoid phase ambiguity problem and to decrease phase errors at the same
time, the use of pre-distortions were proposed by us. In simplest case, the pre-distortions just in the
form of power δ-impulses (see Eq. 3.1.8) have been employed.
An arbitrary selected pre-distorted original image row, its magnitude and phase Fourier spectra
are demonstrated in Figures 3.1.8 – 3.1.10, respectively.
113

Fig. 3.1.9. Magnitude Fourier spectrum of the
row in Fig. 3.1.8.
Fig. 3.1.11. Magnitude bispectrum of a noisy
and jittered image row (no pre-distortion).
Fig. 3.1.10. Phase Fourier spectrum of the row
in Fig. 3.1.8.
Fig. 3.1.12. Phase bispectrum of a noisy and
jittered image row (no pre-distortion).
Since the amplitudes A0 of the pre-distortion δ-impulses were selected in accordance to the con-
dition (3.1.9), each image row will have no zeroes for magnitude Fourier spectrum (see Fig. 3.1.9) and
no phase wrapping for phase Fourier spectrum (see Fig. 3.1.10).
Hence, due to the insertion of pre-distortions, it is possible to uniquely recover image row phase
Fourier spectrum. In this case, the recursive algorithm BLW [10] works accurately in the sense of
unique phase recovery, decreasing phase errors and suppression of AWGN.
114

Fig. 3.1.13. Magnitude bispectrum estimate of
a noisy and jittered image row (pre-distortion
is insertrd).
Fig. 3.1.14. bispectrum estimate of a noisy and
jittered image row (pre-distortion is inserted).
Fig. 3.1.15. Image reconstructed by the proposed technique.
To prove this important statement, the plots of the image row magnitude and phase bispectra
without and with pre-distortions are represented in Figures 3.1.11 – 3.1.14, respectively.
From these plots, one can see that magnitude bispectrum corresponding to the image row includ-
ing pre-distortions (see Fig. 3.1.13) does not have zeroes in opposite to the magnitude bispectrum
shown in Fig. 3.1.11 (pre-distortions are absent). The phase wrapping values that are seen well in Fig.
3.1.12 in the total bispectrum domain are “pushed off” from the limits of the principal calculation tri-
angle domain on the bispectrum plane in Fig. 3.1.14. Hence, due to the proposed insertion of pre-
distortions, each processed image row has been transformed to the minimum phase signal space.
115

Fig. 3.1.15 demonstrates the test image reconstructed by the proposed technique performed with
row by row. As can be seen, the reconstructed image is cleaner than the distorted version shown in
Fig. 3.1.2. The object presented in Fig. 3.1.15 can be recognized.
However, there are some artifacts in the reconstructed image: ringing at the left and right ends
caused by the inserted pre-distortions. This ringing can be attributed to the Gibbs phenomenon.
3.2. Bispectrum-based image reconstruction by using multiplicative pre-distortions
Despite solving the problem of recognition of unknown object, image reconstruction technique
[76, 78] considered in the previous subsection results in arising distortions that are mainly concen-
trated at the leftmost and rightmost pixels of each reconstructed image row. These errors appear due to
difficulties of additive pre-distortion compensation at the final stage of image reconstruction as well as
due to spectral leakage.
In this subsection, we consider the technique [77, 79] that provides three benefits: jitter removal,
spectral leakage decreasing, and phase ambiguity avoidance.
The technique suggested in [77, 79] contains the following processing stages and steps.
Stage 1. De-jittering of adjacent image rows.
(
Step 1.1. Estimation of the sampled cross-correlation function R ( l)
m
cent jittery and noisy image rows (3.1.1) according to the procedure (3.1.2).
116
ˆ )
k , k + 1 calculated for each two adja-
Step 1.2. Evaluation and storage of the maximum coordinates { l max k'}
jit of the functions (3.1.2) ac-
cording to (3.1.3).
Step 1.3. Computations of the jitter corrections
(m)
k
( m)
∆ by using (3.1.4).
Step 1.4. Jittery rows alignment by (3.1.4) and obtaining the de-jittered rows in the form of (3.1.5).
Stage 2. Spectral leakage decreasing and Fourier phase spectrum discontinuity avoidance.
Step 2.1. Multiplication of the de-jittered functions (3.1.5) by some pre-distortion function whose
Fourier spectrum pronouncedly has no zeros and, hence, the total function magnitude Fourier spec-
trum does not contain zeros. As such pre-distortion, tapering Gaussian shape function has been se-
lected in the simplest case. Pre-distorted image row then can be expressed as
( m) ( m)
k
( i) pr
( i) k cor
( i)
f = w x , i∈[1,L] (3.2.1)
( m) ( m)
k
( I − i + 1) =
pr
( i) k cor
( I − i + 1) ,
f w x
where wpr(i) is the pre-distortion tapering function defined by

w
pr
( i)
2
[ ( L−i
)]
= e
µ
117
, (3.2.2)
where variables L> ∑
=
I ( m)
{ w ( i)}
x i
pr
i 1
k
( ) , k=1, 2, 3,…, I. (3.2.3)
max
Step 2.2. Computation of the sampled m-th bispectrum estimates according to
ˆ ( m)
( m)
( m)
B ( p,
q)
= [ X k cor ( p)
⊗W
pr ( p)
] [ X k cor ( q)
⊗W
( q)
]
f pr
k
( m)
*
*
[ X ( p + q)
⊗W
( p + q)
]
k cor
pr
, (3.2.4)
(m)
where X (...) and Wr(…) are the direct discrete Fourier transforms of the functions (3.1.5) and
k cor
(3.2.2), respectively; ⊗ denotes the convolution; p=1, 2, 3,…, I and q=1, 2, 3,…, I are the independent
spatial frequency indices.
Notice that the role of the tapering pre-distortion (3.2.2) is threefold:
- to obtain an improved bispectrum estimate (3.2.4) due to spectral leakage decrease in the sense of
bias decrease;
- to eliminate phase wrapping by transform of image rows to the maximum-phase signals;
- to fix the coordinate of each k-th image row CG and, hence, to perform automatic image rows
alignment after bicpectrum image row reconstruction.
Stage 3. Bispectrum-based image row reconstruction.
Step 3.1. Image row phase and magnitude Fourier spectra recovery from bispectrum estimates
(3.2.4) by the conventional BLW algorithm [10].
Step 3.2. Image row reconstruction by discrete inverse Fourier transform of the image row Fourier
spectrum recovered from the bispectrum estimate.
Step 3.3. Compensation of the pre-distortions (3.2.2) by multiplying the reconstructed image rows
by the function inverse to (3.2.2).
Test images “Barbara” and “Letters” (TICSP means Tampere International Center for Signal
Processing) participated in computer simulations are shown in Figures 3.2.1 and 3.2.2.

Fig. 3.2.1. The original noise- and jitter-free
test image “Barbara”.
Fig. 3.2.2. The original noise- and jitter-free
118
test image “Letters”.
The first test object (“Barbara”) and the second one (“Letters”) corrupted by AWGN with fixed
variance of 100 and jitter are shown in Fig. 3.2.3 and Fig. 3.2.4, respectively. As can be seen from Fig.
3.2.3 and Fig. 3.2.4, the images are completely concealed by jitter and AWGN and it is impossible to
recognize visually an a priori unknown object (to percept this image).
Fig. 3.2.3. Image “Barbara” corrupted by
AWGN and jitter.
Fig. 3.2.4. Image “Letters” corrupted by
AWGN and jitter.
Figures 3.2.5 and 3.2.6 illustrate the images reconstructed by the proposed technique for L=32
pixels and µ = 0.065 that corresponds to { w ( i)}
=14787 (see the condition (3.2.3)).
pr max
Fig. 3.2.5. Reconstructed object “Barbara”. Fig. 3.2.6. Reconstructed object “Letters”.

As can be seen, the reconstructed images are sufficiently cleaner than the distorted ones in Fig-
ures 3.2.3 and 3.2.4.
The reconstructed objects demonstrated in Figures 3.2.5 and 3.2.6 can be confidently recognized
despite the slightly jagged vertical image edges.
Unfortunately, there are the distortions in the reconstructed images in the form of little circular
shifts of some image rows. These distortions are caused due to the centering of the bispectrum recon-
structed k-th image row with respect to the CGk coordinate. If original image row was sk(i), then the
reconstructed image row sˆ k ( i)
is shifted with respect to CGk coordinate, i.e. sˆ k ( i)
= sk ( i − CGk
) .
Hence, despite satisfaction of the condition (3.2.3) as a whole, the CGk coordinates of some pre-
distorted image rows may be slightly shifted (jagged) with respect to the central image row pixel. The
residual jags of CGk coordinate can be explained by large gradient of intensities in different image
rows.
3.3. Search of the optimal parameters of additive and multiplicative pre-distortion functions
The problem of searching the optimal parameters of additive and multiplicative pre-distortion
functions that are used for reconstruction of digital images corrupted by jitter and mixture of additive
Gaussian and impulsive noise has been considered in [80].
Note that in practice it is really impossible to assess uniquely the performance for existing image
reconstruction algorithms. In real-life situation it is possible to estimate approximately the enhance-
ment of a reconstructed image only for uniform image segments by analysis the interference variance
σ 2 computed for corrupted image and residual interference variance σ 2 res estimated in the reconstructed
image. However, in practice, due to the problem of selecting some really uniform image test seg-
ments, the estimates σ 2 and σ 2 res can significantly differ from the corresponding true values.
Therefore, it is reasonable to use the standard test images owing a priori known structure for as-
sessing true performance of the image reconstruction algorithms. The interferences are usually in-
serted artificially in these segments for standard image testing.
Usually, investigated images can contain both flat segments and different heterogeneities. The
flat image segments contain the pixel intensities close to a local mean value corresponding to large-
scale homogeneous image formations. The heterogeneities contain both small-sized or point-like ob-
jects and one-dimensional prolonged objects. Because of this, we have selected several standard dif-
ferent test images for statistical investigations. They are the following: “Baboon” is the high-textured
test image containing number of small-sized objects (details) and abrupt intensity changes (edges)
119

(Fig. 3.3.1a); “Barbara” is the test image containing a variety of objects with linear shape and several
homogeneous segments (Fig. 3.3.1b), and “Lenna” is the test image containing homogeneous objects
having sharp edges and a small number of textured segments (Fig. 3.3.1c).
a b c
Fig. 3.3.1. Original test images:(а) Baboon; (b) Barbara; (c) – Lenna.
Original 8-bit test images shown in Fig. 3.3.1 have been artificially corrupted by additive mix-
ture of Gaussian noise with zero mean and different variances equal to 70, 100 and 130, as well by
impulsive noise of uniform probability density function within the interval of [0, 255] and with differ-
ent probabilities of pulse appearance equal to 1%, 3% and 5%. Random deviations of adjacent image
rows (jitter) have been equal to ±10, ±30 and ±50 pixels.
The test images corrupted by jitter of deviation up to ±30 pixels are shown in Fig. 3.3.2. Note
that recognition the original images of Baboon, Barbara and Lenna is impossible by visual inspection
the images represented in Fig. 3.3.2.
Our goal is to reconstruct these distorted images and to compare the effectiveness of the pro-
posed image reconstruction techniques.
a b c
Fig. 3.3.2. Original test images distorted by jitter: (а) Baboon; (b) Barbara; (c) Lenna.
120

The following parameters computed for estimation of reconstructed image quality and compara-
tive analysis of performances of two different image reconstruction techniques [76, 78] and [77, 79]
considered in previous subsections have been used in our computer simulations:
a) the fluctuation variance
2
σ inp for the distorted image
2
inp =
1 I J
( m)
2
∑ ∑[
x ( i,
j)
− s(
i,
j)]
IJ −1
i=
1 j=
1
σ , (3.3.1)
where M1 is the number of statistically independent image frames participating in ensemble averag-
( )
ing denoted in formula (3.3.1) by ; x ( i,
j)
m
is the m-th realization (m = 1, 2, …, M1) of an
arbitrary (i, j)-th pixel intensity in the distorted image;
b) the input signal-noise-ratio SNR inp computed as
1 −1
−1
= 0 = 0
I J
IJ i j
SNRinp
Ps
2
inp
where P s = ∑ ∑ [ s(
i,
j)
− E]
, and E = ∑ ∑s(
i,
j)
;
c) the fluctuation variance
( )
s m
2
121
M1
= , (3.3.2)
σ
1 −1
−1
= 0 = 0
I J
IJ i j
2
σ out computed for the reconstructed image as
2 ⎪⎧
1 I J
( m)
2⎪⎫
out = min⎨
∑∑[
sˆ
( i,
j)
−s(
i −t,
j)]
⎬
⎪⎩
IJ −1i=
1j= 1
⎪⎭ M1
σ , (3.3.3)
where ˆ ( i,
j)
and s (i, j) are the (i, j)-th pixel intensity in the reconstructed and original images,
respectively; t is the shift placed in formula (3.3.3) according to the bispectrum invariance property
with respect to signal shift;
d) the output signal-to-noise ratio SNR out computed for reconstructed image as
out
s P
SNR
= ; (3.3.4)
σ
e) the parameter ε of the reconstructed image which allows assessing the improvement of SNR in
the reconstructed image
2
out
= SNRout
SNRinp
ε . (3.3.5)
The problem of estimation an optimal δ-impulse amplitudes in predistortions (3.1.8) from the
point of view of the best reconstructed image quality will be considered in this subsection. The recon-

structed image quality has been studied for different δ-impulse pre-distortion amplitude values by
computing the accuracy parameter ε (3.3.5).
The results of computer simulations obtained with additive pre-distortion function in jitter and
additive Gaussian interference environment are shown in Fig. 3.3.3.
a b c
Fig. 3.3.3. Examples of the reconstructed test images that have been artificially corrupted by jitter with deviation
of ±30 pixels and Gaussian noise with variance of 130:
а – the pre-distortion δ-impulse amplitude is equal to 21000;
b – the pre-distortion δ-impulse amplitude is equal to 49000;
c – the pre-distortion δ-impulse amplitude is equal to 44000.
ε
Pre-distortion amplitude
Only jitter
× × × Jitter + Gaussian noise with variance of 70
Jitter + Gaussian noise with variance of 100
Jitter + Gaussian noise with variance of 130
+ + + Jitter + Gaussian noise with variance of 100 + impulsive noise of 3%
Fig. 3.3.4. Parameter ε as a function of pre-distortion δ-impulse amplitudes for the image Lenna.
Comparison of the distorted test images shown in Fig. 3.3.2 and the corresponding reconstructed
122

images in Fig. 3.3.3 permits concluding that, in opposite to the corrupted images, reliable object rec-
ognition is possible by visual inspection of the images illustrated in Fig. 3.3.3.
The results of computing the parameter ε (3.3.5)) are demonstrated in Figures 3.3.4 – 3.3.6.
ε
Pre-distortion amplitude
Only jitter
× × × Jitter + Gaussian noise with variance of 70
Jitter + Gaussian noise with variance of 100
Jitter + Gaussian noise with variance of 130
+ + + Jitter + Gaussian noise with variance of 100 + impulsive noise of 3%
Fig. 3.3.5. Parameter ε as a function of pre-distortion δ-impulse amplitudes for the image Barbara.
ε
Pre-distortion amplitude
Only jitter
× × × Jitter + Gaussian noise with variance of 70
Jitter + Gaussian noise with variance of 100
Jitter + Gaussian noise with variance of 130
+ + + Jitter + Gaussian noise with variance of 100 + impulsive noise of 3%
Fig. 3.3.6. Parameter ε as a function of pre-distortion δ-impulse amplitudes for the image Baboon.
123

Analysis of the graphs plotted in Figures 3.3.4 – 3.3.6 permits to note the following:
- the dependence of the reconstructed image efficiency on the pre-distortion δ-impulse ampli-
tude is of non-linear behavior;
- small fluctuations of ε value are observed nearby the global maximum;
- the curves in Figures 3.3.4 and 3.3.5 are bi-modal, i.e. the parameter ε is small both for too
small and too large pre-distortion amplitudes. The curves in Fig. 3.3.6 have one global
maximum;
- the optimum pre-distortion δ-impulse amplitudes are of values equal to approximately
45000 for smooth Lenna and Barbara images, however, the optimum amplitude is equal to
20000 for high-textured baboon image;
- the parameter ε reduces if additive noise variance increases, there are optimal pre-distortion
δ-impulse amplitudes, but the reconstructed image quality is approximately the same in cas-
es of non-optimal selection.
The parameter ε decreases, on the average, by 0.00025 due to influence of small intensity im-
pulse noise (with the probability of 3 %). However, the parameter ε decreases more considerably (by
about 0.042) if impulse noise probability increases to 5%.
Thus, computer simulation results demonstrated in Figures 3.3.4 – 3.3.6 indicate good efficiency
of the approach suggested for solving the unknown object recognition problems under joint influence
of heavy jitter, additive Gaussian and impulsive noise.
To estimate an optimal multiplicative pre-distortion function shape (3.2.2) from the point of
view of maximum value (3.3.5), statistical investigations have been carried out for the test image re-
construction. The values of L and µ in (3.3.2) have been varied within wide limits.
The computer simulations results are presented in Figures 3.3.7 – 3.3.12.
Analysis of the graphs plotted in Figures 3.3.7, 3.3.9, and 3.3.11 shows that ε value depends
non-linearly upon the varied L and µ parameters. Clear extremum of ε is observed for the optimal L
and µ values.
Behavior of ε value depending on the optimal L and µ parameters (see the Figures 3.3.8, 3.3.10,
and 3.3.12) is close to exponential. Though the curves plotted in Figures 3.3.8, 3.3.10, and 3.3.12 look
similarly, the function ε(L, µ) values differ for each particular image.
It should be stressed, that ε value decreases on the average by 0.5 and it decreases slightly under
the influence of impulsive noise. Therefore, image reconstruction technique by using multiplicative
pre-distortion is robust to impulsive noise of small level.
124

Fig. 3.3.7. Parameter ε as a function of L and µ obtained for Lenna image corrupted by jitter with
maximal deviation of ± 30 pixels and mixture of additive Gaussian noise with variance of 100 and
impulsive noise with probability of 3 %.
0,8
0,4
0,1
0,6
0,2
µ
L
20 40 60 80 100 120
Fig. 3.3.8. The curve illustrating optimal selection of the parameters L and µ obtained according to
the data represented in Fig. 3.3.7.
125
µ
L
ε
1,5
1,0
0,5

Fig. 3.3.9. Parameter ε as a function of L and µ obtained for Barbara image corrupted by jitter with
maximal deviation of ± 30 pixels and mixture of additive Gaussian noise with variance of 130 and
0,8
0,4
0,1
0,6
0,2
L
impulsive noise with probability of 5 %.
µ
Fig. 3.3.10. The curve illustrating optimal selection of the parameters L and µ obtained according
to the data represented in Fig. 3.3.9.
126
ε 2
20 40 60 80 100 120
µ
L
1,5
1
0,5

Fig. 3.3.11. Parameter ε as a function of L and µ obtained for Baboon image corrupted by jitter
with maximal deviation of ± 30 pixels and mixture of additive Gaussian noise with variance of 130
0,8
0,4
0,1
0,6
0,2
L
and impulsive noise with probability of 5 %.
µ
20 40 60 80 100 120
Fig. 3.3.12. The curve illustrating optimal selection the parameters L and µ obtained according to
the data represented in Fig. 3.3.11.
127
ε
L
µ
2
1,5
1
0,5

Analysis of the represented results permits to note that the proposed technique removes jitter
distortions quite effectively. However, different artifacts are observed in the reconstructed images in
the form of some brightness distortions.
Thus, computer simulation results demonstrate quite good performance for reliable unknown
object recognition under influence of heavy jitter. However, object recognition reliability decreases
due to presence of additive Gaussian and impulsive noise.
Accuracy of bispectral density estimation improves with increasing the observed image frame
number. Therefore, one may expect improving the image reconstruction performance.
128

4. Signal detection in Gaussian noise by using third-order test
statistics
4.1. Detection of derministic signals observed in additive Gaussian noise based on likelihood ra-
tio criterion by using third-order statistic
Matched filtering is often used in practice tool for known waveform signal detection in additive
Gaussian noise in modern telecommunication [82] and radar [83] systems. Under influence of AWGN
or in case of known interference spectral density which is necessary for performing denoising proce-
dure, as well as by carrying out synchronization in a receiver, the matched filter (MF) is an optimal
device in the sense of minimizing false alarm probability of signal detection and maximizing output
SNR.
Note that optimal processing is performed usually by the second-order statistic estimate in the
form of correlation integral calculated for a noisy received signal and reference signal and comparison
the correlation integral to a threshold [82, 83]. A decision about presence of desirable signal is per-
formed if the threshold is exceeded.
In this Chapter, we consider a new approach to solving the problem of known waveform signal
detection in AWGN by using third-order statistic formed at the MF output [84]. An approach sug-
gested in [84] possesses some attractive and promising benefits comparing to the traditional approach
based on using second-order statistics. These benefits are in higher noise immunity to AWGN and
non-sensitivity to random signal shifts.
Let us consider a third-order test statistic estimated for solving the problem of deterministic sig-
nal detection in AWGN. Assume that a discrete temporal process
served at the MF input in the form of set of M realizations as
( m) ( m) ( m)
x i s i τ n i
129
{
( m) I −1
x ( i)}
i=
0 , m = 1, 2,…,M is ob-
( ) = ( − ) + ( ) , (4.1.1)
where s(i) is an a priori known deterministic signal; i=0,1,…,I-1 is the temporal sample index; τ (m) is
the integer-valued signal shift that value randomly varies from one realization to another; n (m) (i) is the
m-th realization of AWGN with zero mean.
A signal waveform s(i) is assumed to be unchangeable for all M realizations observed. Signal
and noise are supposed to be statistically independent in the observation (4.1.1).
When the process (4.4.1) comes at the MF input, the MF output y (m) (i) is equal to
y
I −1
( m) ( m)
( i) = ∑ x ( j) s( I − i + j)
. (4.1.2)
j=
0

Let us consider two-alternative signal detection problem by using the following two hypotheses
H1 (signal is received) and H0 (no signal at the MF input) as
( m)
( m)
( m)
Н1: x ( i)
= s(
i −τ
) + n ( i)
, (4.1.3а)
( m)
( m)
Н0: x ( i)
= n ( i)
. (4.1.3b)
For solving this problem, we propose to use a new test detection statistic given by third-order
autocorrelation function (TOAF) calculated at the MF output in the point of origin of coordinates.
According to hypothesis H0 (no signal), the response at the MF output is equal to
y
( m)
n
( i)
I 1
= ∑ −
j = 0
n
( m)
130
( j)
s(
I − i + j)
, (4.1.4а)
and according to hypothesis H1 (desirable signal is received), the MF output can be written as
( m) s
I −1
j=
0
( m) τ
( m)
y ( i) = ∑ {[ s( j − ) + n ( j)] s( I − i + j)}
. (4.1.4b)
Sampled, i.e. estimated for an arbitrary m-th realization (4.1.1), detection test statistics in the
form of TOAF computed in the point of its origin of coordinates (0, 0) and corresponding to the hypo-
ˆ )
yn ˆ )
ys (m
(m
theses H0 – R ( 0,
0)
and H1 – R ( 0,
0)
can be defined as
ˆ ˆ 1 1
R R y i n j s I i j
I −1 I −1 I −1
( m) ( m) ( m) 3 ( m)
3
y (0,0) = [ ( )] [ ( ) ( )]
n y =
n
n = − +
I i= 0 I i= 0 j=
0
∑ ∑ ∑ , (4.1.5а)
ˆ ˆ 1 1
R R y i s j n j s I i j
I −1 I −1 I −1
( m) ( m) ( m) 3 ( m) ( m)
3
y (0,0) = [ ( ))] { [ ( ) ( )] ( )}
s y = = − τ + − +
s I i= 0 I i= 0 j=
0
∑ ∑ ∑ . (4.1.5b)
Assume that the proposed test third-order statistic is a random variable with statistical distribu-
tion that asymptotically obeys Gaussian law for М→∞. Hence, conditional probability densities cor-
responding to the signal presence p(
Rˆ
y H1)
and absence p(
Rˆ
y H0
) can be written as
p(
Rˆ
p(
Rˆ
ys
yn
s
H ) =
1
H ) =
0
1
2πσ
( Rˆ
ys
1
2πσ
( Rˆ
yn
e
)
e
)
n
( )
2
⎛ ˆ m ⎞
1 ⎜
R −R
ys
ys
⎟
−
2 ⎜
σ ( Rˆ
) ⎟
y ⎝ s ⎠
( )
2
⎛ ˆ m ⎞
1 ⎜
R −R
yn
yn
⎟
−
2 ⎜
σ ( Rˆ
) ⎟
y ⎝ n ⎠
, (4.1.6)
, (4.1.7)

where
1 1
R R y i s j s I i j
I −1 I −1 I −1
3 3
y = (0,0) [ ( ))] [ ( ) ( )]
s y =
s
s = − +
I i= 0 I i= 0 j=
0
∑ ∑ ∑ is the TOAF at the MF output
computed in the origin of coordinates (0, 0) in absence of noise at the MF input;
1 1 1 1
R R y i n j s I i j
M I −1 M I −1 I −1
( m) 3 ( m)
3
y = (0,0) [ ( )] [ ( ) ( )]
n y =
n
n = − +
M m= 1 I i= 0 M m= 1 I i= 0 j=
0
∑ ∑ ∑ ∑ ∑ is the MF output aver-
aged by M realizations and computed when the signal is absent at the MF input;
σ
1
= − =
M
2 ˆ ˆ ( m)
2
( Ry ) [ (0,0) (0,0)]
s ∑ Ry R
s ys
M m=
1
1 1 1
= ∑{ ∑[ ∑( s( j − τ ) + n ( j)) s( I − i + j)] − ∑[ ∑ s( j) s( I − i + j)]
}
M I I
M I −1 I −1 I −1 I −1
( m) ( m)
3 3 2
m= 1 i= 0 j= 0 i= 0 j=
0
131
is the variance of
the third-order test statistic estimate computed under condition of receiving the additive mixture of
desirable signal and noise at the MF input;
σ
1
= − =
M
2 ˆ ˆ ( m)
2
( Ry ) [ (0,0) (0,0)]
n ∑ Ry R
n yn
M m=
1
is the variance of the
M I −1 I −1 M I −1 I −1
1 1 ( m) 3 1 1
( m)
3 2
= ∑{ ∑[ ∑ n ( j) s( I − i + j)] − ∑ ∑[ ∑ n ( j) s( I − i + j)]
}
M I M I
m= 1 i= 0 j= 0 m= 1 i= 0 j=
0
third-order test statistic estimate computed under condition of receiving only noise (signal is ab-
sent).
Likelihood ratio for considered two-alternative detection problem is defined as
where L is the threshold (commonly L < 1).
p( Rˆ
H H
ys
1)
> 0
L
p(
Rˆ
< , (4.1.8)
H ) H1
yn
0
After substitution the conditional probabilities (4.1.6) and (4.1.7) into formula (4.1.8) and after
simple transformations, we obtain the following condition
( Rˆ − R ) ( Rˆ ) − R ) σ ( Rˆ
)
− ln[ ] . (4.1.9)
σ ( ) σ ( ) σ ( )
( m) 2 ( m)
2 2
y 0
n yn ys ys >
H
ys
2 ˆ 2 ˆ < L 2
R 1 ˆ
y R H
n y R
s yn
Condition (4.1.9) allows us to calculate signal detection probabilities in Gaussian noise with us-
ing set of realizations m=1, 2,…, M observed at the MF input.
For comparison of signal detection performance obtained with the proposed third-order statis-
tics, i.e. TOAF estimate, and the conventional second-order statistics, i.e. correlation integral esti-
mates, below we consider the conventional autocorrelation functions calculated at the MF output.

In contrast to the previously mentioned third-order statistic (see formulas (4.1.5a, b)), we will
denote the second-order statistics by symbol r.
Conditional probability densities corresponding to the signal presence ( ˆ 1)
H r p y and signal ab-
sence ( ˆ 0)
H r p y at the MF input can be represented in the following forms
where
n
p(
rˆ
y
s
H ) =
1
p(
rˆ
y H0
) =
1
e
2π
σ ( rˆ
y )
1
e
2π
σ ( rˆ
y )
1
rˆ rˆ (0) { [ s( j ) n ( j)] s( I i j)}
I −1 I −1
( m) ( m) ( m) ( m)
2
y =
s y = − τ + − +
s I i= 0 j=
0
132
n
s
2
⎛ ( m)
ˆ ⎞
1⎜
r −r
ys
ys
⎟
−
2
⎜
( ˆ )
⎟
⎜ σ ry
⎟
⎝ s ⎠
2
⎛ ( m)
ˆ ⎞
1⎜
r −r
yn
yn
⎟
−
2
⎜
( ˆ )
⎟
⎜ σ ry
⎟
⎝ n ⎠
s
, (4.1.10)
, (4.1.11)
∑ ∑ is the sampled autocorrelation function
estimate calculated in the origin of coordinates for an arbitrary realization (4.1.1);
r
1 I −1
I −1
2
= ry
( 0)
= ∑{ ∑ s(
j)
s(
I − i + j)}
is the signal autocorrelation function computed in the origin
I i=
0 j=
0
ys s
of coordinates with the absence of AWGN at the MF input;
2
1 1
( ) ( ) 1 I − I −
m m
( m)
rˆ
= rˆ
( 0)
= ∑[ ∑ n ( j)
s(
I − i + j)]
yn
yn
I i=
0 j=
0
is the noise autocorrelation function computed in the
origin of coordinates in the case of the signal absence (only noise n (m) (i) arrives at the MF input) at
the MF input;
ryn 1
M I −1
1
i−1
M m=
1I
i=
0 J = 0
( m)
2
( 0)
= ∑ ∑[ ∑ n ( j)
s(
I − i + j)]
is the test statistic averaged by M realizations with the signal
absence at the MF input;
2 1 M 1 I−1 I−1 1 1
( ) ( ) 2 1 I− I−
m m
σ ( rˆ y ) = ∑ { ∑ { ∑ [ s( j − τ + n ( j)] s( I − i + j)} − ∑ [ ∑ s( j) s( I − i j
s M m= 1 I i= 0 j= 0 I i= 0 j=
0
2 2
+ )] } is
the variance of the second-order statistic computed for the case of signal and noise mixture ob-
served at the MF input;

2
2 1 M 1 I −1
I −1
( m)
σ ( rˆ y ) = ∑ { ∑[ ∑ n ( j)
s(
I − i + j)]
−
n M m=
1 I i=
0 j=
0
is the variance of second-order statistic estimate com-
2
1 M 1 I −1
I −1
( m)
2
− ∑ ∑[ ∑ n ( j)
s(
I − i + j)]
}
M m=
1 I i=
0 j=
0
puted for the case of signal absence (only noise is observed).
Likelihood ratio for the second-order statistic is defined as
where L is the threshold which value is selected as L < 1.
p( rˆ
H H
y 1)
> 0
< L , (4.1.12)
p(
rˆ
H ) H1
y
0
After substitution the expressions (4.1.10) and (4.1.11) into (4.1.12) and after simple transfor-
mation, we obtain the following form of the likelihood ratio
( rˆ − r ) ( rˆ − r ) σ ( rˆ
)
− ln[ ] . (4.1.13)
( ) ( ) ( )
( m) 2 ( m)
2 2
y 0
n yn ys ys >
H
ys
2 2 < L 2
σ rˆ ˆ H1
ˆ
y σ r
n y σ r
s yn
The condition (4.1.13) permits to calculate signal detection probability by using second-order
test statistics in the form of autocorrelation function estimated at the MF output.
Fig. 4.1.1):
Three following typical in practice signal models have been used in computer simulations (see
- single pulse of rectangular shape;
- two pulses of triangular shape with different amplitudes;
- linear frequency modulated (LFM) signal s(i) = Аcos[2π(fmin + βi)i], β=( fmax – fmin )/I.
Observation interval is selected the same value for all three signals and set of I = 256 samples.
a b c
Fig. 4.1.1. Signal studied in computer simulations: a) single pulse of rectangular shape;
133

) two pulses of triangular shape; c) LFM signal.
For assessing the behavior of probability densities of third-order statistics ( ˆ 1)
H R p y given in
(4.1.6) and p(
Rˆ
y H0
) in (4.1.7), as well as second-order statistics p(
rˆ
y H1)
given by (4.1.10) and
n
( ˆ 0)
H r p y by (4.1.11), the corresponding histograms have been calculated. The graphs of the histo-
n
grams obtained for LFM signal with fmin = 10 Hz and fmax = 1000 Hz and for М = 10000 realizations
are plotted in Figures 4.1.2 and 4.1.3 for the second- and third-order statistics.
Fig. 4.1.2. The histograms of conditional probability distributions plotted for the third-order sta-
tistics ( ˆ 1)
H R p y (continuous line) and ( ˆ 0)
H r p y (dashed line).
s
Fig. 4.1.3. The histograms of conditional probability distributions plotted for the second-order
statistics p(
rˆ
y H1)
(continuous line) and ( ˆ 0)
H r p
s
134
n
y (dashed line).
Analysis the curves plotted in Figures 4.1.2 and 4.1.3 confirms correctness of the above men-
tioned assumption about Gaussianity of conditional probability densities given by formulas (4.1.6)
n
s
s

and (4.1.7) for the third-order statistics and formulas (4.1.10) and (4.1.11) for the second-order statis-
tic, respectively. One should also pay attention to better separation of the maximums of Gaussian
functions in Figures 4.1.2 and 4.1.3. The latter peculiarity seems promising for enhancement of signal
detection by using the third-order statistics.
It should be stressed, that low input SNR values are of prime interest for computer simulations,
because just under these conditions noise leaks to the MF output and it causes decreasing of correct
detection probability.
Several examples of noisy MF outputs are demonstrated in Fig. 4.1.4 for the case of receiving
the single pulse of rectangular waveform in AWGN for the input SNR equal to 4 dB.
Fig. 4.1.4. Examples of MF outputs (4.1.2) plotted for receiving three arbitrary realizations
(4.1.1) of the single rectangular pulse embedded in AWGN.
Obvious noise leakage at the MF output is clearly seen in Fig. 4.1.4.
The plots of signal detection probabilities as functions of SNR at the MF input are represented
in Figures 4.1.5 – 4.1.9 for three above considered test signals (see Fig. 4.1.1). The threshold L has
been selected as L = 0.01 and the number of realizations participating in statistical computer simula-
tions M = 1000.
Continuous curves in Figures 4.1.5 – 4.1.9 correspond to the third-order statistics and the dashed
curves correspond to the second-order statistics.
SNR at the MF input is calculated as
where
I −1
1
P s = ∑[ s(
i)
− ms
] ; ∑ I
−1
1
= ( )
I
m s s i ;
I
i=
0
2
i=
0
P
SNR = 10lg( ) , (4.1.14)
σ
135
s
2
n
2
σ n is the AWGN variance.

Detection probability
Fig. 4.1.5. Signal detection probability (single pulse of rectangular shape with the width of t1 = 3
samples and the amplitude A1 = 3) as a function of SNR at the MF input (deviation of signal random
Detection probability
shift is equal to τ (m) = ± 20 samples).
Fig. 4.1.6. Signal detection probability (single pulse of rectangular shape with the width of t1 = 3
samples and the amplitude A1 = 3) as a function of SNR at the MF input (no random shift, τ (m) = 0).
Detection probability
Fig. 4.1.7. Signal detection probability (two pulses of triangular shape with the same widths of t1 = t2
= 3 samples and different amplitudes of A1 = 3 and A2 = 7) as a function of SNR at the MF input
(deviation of signal random shift is equal to τ (m) = ± 20 samples).
136
SNR, dB
SNR, dB
SNR, dB

Fig. 4.1.8. Signal detection probability (two pulses of triangular shape with the same widths of t1 = t2
= 3 samples and different amplitudes of A1 = 3 and A2 = 7) as a function of SNR at the MF input (no
ing.
deviation of signal random shift, τ (m) = 0).
Fig. 4.1.9. Signal detection probability (the LFM signal with the amplitude A = 2,
fmin = 10 Hz and fmax = 1000 Hz) as a function of SNR at the MF input.
Analysis of the simulation results demonstrated in Figures 4.1.5 – 9 permits to note the follow-
- The use of third-order statistics provides better performance for detection of signals embedded
in AWGN.
Detection probability
Detection probability
- Correct detection probability tends to unity for MF input SNR equal approximately to 4 dB
(and, sometimes, even less) for third-order test statistic (see Figures 4.1.5 – 9).
- Correct detection probability value tends to unity for MF input SNR equal approximately to 8
dB for second-order test statistic (see Figures 4.1.7 and 4.1.9);
- Correct detection probability depends upon signal random shift deviation and third-order sta-
tistic is less sensitive to these deviations comparing to second-order statistic.
Thus, comparative analysis performed for signal detection performance assessment with the
second- and third-order statistics permits to make up the conclusion about promising perspective of
the suggested approach in telecommunications and radar systems. Application of this approach to ra-
137
SNR, dB
SNR, dB

dio telecommunications systems will be considered in the next subsection and for radar system – in
the next Chapter.
4.2. Application of triple correlation and bispectrum for interference immunity improvement in
digital telecommunications systems
A new noise immunity encoding/decoding technique that exploits advantages of triple correla-
tion and bispectrum has been recently proposed by us in [85]. Below, in this subsection we consider
approach suggested in [85] more in detail.
In many real-life situations, mobile communication multi-path radio channels exhibit influence
of fading and power electrical interferences. The latter may provoke intolerable errors in digital radio
communication systems operating under low and time-varying SNRs. Hence, to provide desirable
communication reliability it is necessary to select an adequate noise immunity encoding/decoding
technique. However, in the case when the interference varies within a wide dynamic range during a
communication session, it is problematic to find a reliable encoding technique for providing required
noise immunity and satisfactory message errors.
Bit error probability pe assessed at the output of correlation detection scheme according to the
maximum-likelihood decoding rule is given by the following well-known condition [82]
p
M E ( 1 − ρ)
M P ( 1 − ρ)
w
w
≤ Q(
) = Q(
B)
, (4.2.1)
2
2 N 2
e σ
0
where Q(x) is the Gaussian error integral; М = 2 k is the coder alphabet capacity; к is the number of
information bits in a code word; Ew = kEb, Pw = kРb; Eb and Рb are the binary signal energy and power,
respectively; N0 and σ 2 are the AWGN spectral density and variance, respectively; В = FkTb is the
signal base; F is the signal spectrum bandwidth; Tb is the time interval fixed for transmission of binary
symbol; ρ is the cross correlation coefficient of the binary signals s1(t) and s2(t) that is equal to
T 1 b
ρ= ∫ s ( t)
s ( t)
dt .
Eb 0
1
2
According to formula (4.2.1), required noise immunity can be achieved in different ways. One
of them employs noise-like signals having large signal base B [86]. However, the use of wideband
signals inevitably requires widening a telecommunication system bandwidth. Unfortunately, the latter
requirement is difficult or impossible to provide in practice due to known limitations imposed on the
channel bandwidth. Moreover, the signal duration kTb must be as short as possible for obtaining desir-
able transmission rate that is usually limited.
One more noise immunity technique widely used in digital radio telecommunication systems is
138

the block coding [82] based on inserting redundancy in information bit stream. According to the block
coding approach, a bit stream is grouped in a sequence of words (blocks). The length of each block is
assumed to be equal to k (М = 2 k ). As a result, the bit error probability pe can be decreased without
increasing the binary signal power Рb in (4.2.1).
However, the block coding technique requires widening the bandwidth for preserving desirable
transmission rate. In case when increasing of bandwidth is impossible, time delay must be increased
for providing required pe.
It follows from the linear block coding theory and maximum-likelihood decoding [82] that cor-
rection of e errors is possible in the case when the Hamming distance dH between two arbitrary words
from the selected code word set satisfies the following inequality
dH ≥ 2e+1. (4.2.2)
Note that condition (4.2.2) requires the memorizing and storing the total number of code words
equal to 2 k . It is well-known that noise immunity increases parallel to increasing the block length.
However, the procedure of multiple correlation comparison of a code word received with a large total
number of the code words requires sufficient processing time and its efficiency decreases with dimin-
ishing SNR at the communication system input.
Thus, noise immunity of digital telecommunication systems can be achieved by:
- using the binary orthogonal signals;
- increasing the signal base;
- inserting the redundancy for error detection and correction.
Searching for improving the digital radio telecommunication system performance in the sense of
enhancement of its noise immunity has run us into an idea of development a new redundant encod-
ing/decoding approach by using the interference protection properties of triple correlation and bispec-
trum. Our strategy is based on the application of the properties of triple correlation and bispectrum
that have been thoroughly considered in Chapter 1 and the results obtained and discussed in the sub-
section 4.1. According to the approach suggested in [85], a new noise immunity encoding/decoding
technique is based on using the code words constructed by quantized samples of triple correlation for
two a priori inserted subsidiary digital sequences. To recover a binary symbol we propose employing
the maximum of bispectrum magnitude (bimagnitude) of the above mentioned subsidiary digital se-
quences as a test statistic. Bimagnitude is a robust estimate with respect to additive interference and it
is non-sensitive to translations of the binary code sequence received.
A distinctive property of the approach suggested is a two-stage encoding procedure. First, two
139

orthogonal subsidiary code sequences are generated. Second, two redundant code sequences are con-
structed in the form of the samples of triple correlation functions (TCFs) of these subsidiary code se-
quences. Below a set of the encoding procedures proposed is listed.
1. Two arbitrary selected orthogonal subsidiary code sequences, for example, a = (00000000)
and b = (22222221) of the length of 8 elements in each one is generated. The sequence a is assumed to
correspond to the logic “zero” and the sequence b corresponds to the logic “unit” in the original binary
message (original information bit stream).
Note that with increasing the mentioned subsidiary code length (more than 8 elements consi-
dered in our illustrative example), noise immunity increases. However, at the same time, the message
time delay becomes larger as well.
(а) (b)
Fig. 4.2.1. The TCFs Ra(l,m) and Rb(l,m) computed by (4.2.3a) and (4.2.3b) for the considered
sequences a (а) and b (b), respectively, and the sample enumeration introduced in the TCFs
(c).
2. Computation of the TCFs Ra(l,m) and Rb(l,m) for the code sequences a = (00000000) and b =
(22222221) as
8
n=
1
140
(c)
R
a
( l, m) = ∑ a( n) a( n + l − 1) a( n + m − 1) , (4.2.3а)

8
R
b
( l, m) = ∑ b( n) b( n + l − 1) b( n + m − 1) , (4.2.3b)
n=
1
where l = 1, 2,…, 8 and m = 1, 2,…, 8 are the translation indices. Note that the indices l, m, and n in
the formulas (4.2.3a) and (4.2.3b) are only positive-valued. The distributions of the TCF samples
computed for the inserted sequences a and b are shown in Fig. 4.2.1.
Analysis of the data represented in Fig. 4.2.1 allows noting the following well-known peculiari-
ties of the TCFs (see subsection 1.2). The TCF samples are distributed in the limits of the typical
hexagon in the (l, m) plane. Since the considered number of translation indices l and m are equal to
even values of l = m = 8, the symmetry of the hexagon with respect to the origin sample R(1,1) in Fig.
4.2.1 is illustrated by using additional 9-th row and column due to the known TCF symmetry property
(see 1.2.6). This TCF symmetry feature permits to compute the TCFs only in the limits of the main
triangular domain shown in Fig. 4.2.1c. Note that such twelve triangular domains form the hexagon in
Fig. 4.2.1. The latter peculiarity will be employed below for constructing a redundant code.
3. Transformation of the decimal Ra(l,m) and Rb(l,m) samples (4.2.3a) and (4.2.3b) to the binary
r-bit values (r = 6 for the illustrative example considered). As a result, redundant binary code arrays
С(l,m) and D(l,m) are obtained and prepared for transmitting a binary message.
Let us represent both code arrays С(l,m) and D(l,m) by the code sequence containing 8 rows of
the length of 8 elements each according to their locations in Fig. 4.2.1. We define the ordering in the
code sequence in the following way. The first row in our code construction includes the samples be-
longing to the central horizontal in Fig. 4.2.1. Note that the first code word in the mentioned code ar-
rays corresponds to the maximum value of TCF. As a result, at the coder output we obtain the follow-
ing code arrays prepared for transmission via a radio channel
⎛ R a (1,1),..., R a (8,1) ⎞
⎜ ⎟
R a (1, 2),..., R a (8, 2)
C(l, m) = ⎜ ⎟,
(4.2.4а)
⎜............................ ⎟
⎜ ⎟
⎝ R a (1,8),..., R a (8,8) ⎠
⎛ R (1,1),..., R (8,1) ⎞
b b
⎜ ⎟
R (1, 2),..., R (8, 2)
D(l, m) = ⎜ b b ⎟ . (4.2.4b)
⎜............................ ⎟
⎜ ⎟
R (1,8),..., R (8,8)
⎝ b b ⎠
Note that the maximum number of the words in the redundant codes (4.2.4a) and (4.2.4b) is
141

equal to 64 elements.
4.The TCF symmetry property
R(l,m) = R(m,l)= R(l–m,–m)= R(m–l,–l)= R(–l,l–m), (4.2.5)
permits to note that TCF (4.2.3b) exhibits proper redundancy, i.e. only three dissimilar values, name-
ly, 57, 54 and 52 belong to the function Rb(l,m) (4.2.3b) in the main triangular region (see Fig. 4.2.1c).
These values repeat several times in the hexagonal domain (l, m) due to the above mentioned TCF
symmetry property. It should be noted, that for the considered case of positive indices, we use only the
first equality in formula (4.2.5).
Note that these three values unambiguously define TCF Rb(l,m). Therefore, the number of code
words can be sufficiently reduced in the truncated code stream without any loss of information trans-
mitted.
Taking into consideration the symmetry property (4.2.5), the new short (truncated) code se-
quences СT(l,m) and DT(l,m) can be constructed in the form of a set containing only the three following
6-bit code words as
СT(l,m) = Ra(1,1); Ra(1,2); Ra(2,3), (4.2.6a)
DT(l,m) = Rb(1,1); Rb(1,2); Rb(2,3). (4.2.6b)
Since the number of code words in (4.2.6a) and (4.2.6b) is sufficiently smaller in comparison to
(4.2.4a) and (4.2.4b), the transmission and decoding performed for the truncated codes takes suffi-
ciently shorter time intervals.
As a real-life example, we consider a digital radio telecommunication system with binary fre-
quency shift keying (FSK) manipulation. Although many other manipulation systems are convention-
ally in use, the FSK manipulation is a common one.
Orthogonal FSK signals s1(t) (logical «zero») and s2(t) (logical «one») transmitted during the
limited bit time interval [0, Tb] can be written as
( 1
1
s
s
t)
= A cos( 2πf
t)
, Tb ≥ t ≥ 0, (4.2.7а)
s
( 2
2
t)
= A cos( 2πf
t)
, Tb ≥ t ≥ 0. (4.2.7b)
s
Let us assume that the received waveform r(t) obtained at the FSK demodulator output is cor-
rupted by additive interference as
r(t) = si(t) + nG(t)+ np(t), i =1, 2, (4.2.8)
where nG(t) is the AWGN whose spectral power density and variance are equal to N0 and σ 2 , respec-
142

tively; np(t) is impulsive noise. Note that the models of non-Gaussian noise are nowadays widely used
in simulations of wireless communication system operation since they better correspond to reality [87
– 89].
A distinctive peculiarity of the approach proposed is two-stage recovery of binary symbols in a
decoding device.
First, standard correlation detection procedure of the binary FSK signals corrupted by additive
noise (4.2.8) is performed with the help of maximum likelihood method. The noise-free signals
(4.2.7a) and (4.2.7b) serve as the references in a coherent correlation demodulator.
As a result, the estimates of binary 6-bit code words C ˆ ( l,
m),
Dˆ
( l,
m)
(or truncated ones
Cˆ
( l,
m),
Dˆ
( l,
m)
) are recovered at the demodulator output. Note that the part of the symbols in the
T
T
information bit stream received may be changed to opposites due to interference.
It should be stressed, that the same twelve triangular domains exist in the total TCF hexagon
domain due to the above mentioned symmetry property (4.2.5) (see Fig. 4.2.1). This important pecu-
liarity allows averaging the TCF samples C ˆ ( l,
m),
Dˆ
( l,
m)
belonging to the corresponding twelve trian-
gular domains. Therefore, noise immunity for codes C ˆ ( l,
m),
Dˆ
( l,
m)
can be improved by using this av-
eraging procedure when compared to the truncated codes Cˆ
( l,
m),
Dˆ
( l,
m)
.
Second, let us obtain the following bispectrum estimates Bˆ a ( p,
q)
and Bˆ b ( p,
q)
performed in the
decoding device as
143
T
T
ˆ ( p,
q)
= FFT[
Rˆ
( l,
m)]
, (4.2.9а)
a
B a
ˆ ( p,
q)
= FFT[
Rˆ
( l,
m)]
, (4.2.9b)
b
B b
where p = 1, 2,…,8 and q = 1, 2,…,8 are the indices of the independent frequency samples; FFT
denotes fast Fourier transform; Rˆ ( l,
m)
and Rˆ ( l,
m)
are the TCF estimates recovered from the code
a b
sequences C ˆ ( l,
m),
Dˆ
( l,
m)
(or truncated code sequences Cˆ
( l,
m),
Dˆ
( l,
m)
).
The absolute values of the bispectrum estimates (4.2.9a) and (4.2.9b) called “bimagnitudes”
serve as test statistics for recovery of the subsidiary code sequences a and b generated in the coder.
A maximum likelihood detector provides accepting a two-alternative decision according to the
following two hypotheses На and Нb as
T
T

⎪⎧
> γ ⇒ H
b
max{ B ˆ(
p,
q)
} = ⎨ , (4.2.10)
⎪⎩
< γ ⇒ H
a
where γ is a threshold value given a priori in the decision device. Note that proper selection of the
subsidiary code sequences a and b will permits to obtain a robust noise protected bimagnitude statistic
max{ B ˆ(
p,
q)
} for the considered noise environment.
After making a decision according to the criterion (4.2.10), the received message is recovered at
the decoder output.
Performance of the proposed encoding/decoding technique is studied, discussed and compared
with a conventional linear repetition block code (6, 1) [82]. This repetition block code has been se-
lected due to the following two reasons. First, this repetition block coding technique in practice exhib-
its perfect effectiveness in mobile radio communication systems where the mobile unit travels at rap-
idly changing speeds and, hence, SNR may vary within wide limits. Second, for adequate comparative
analysis, we have to select the same code parameters, namely, the dimension of the repetition code
and the length of the code word in the coding technique proposed.
As a typical practical example, we consider the original message given in computer ANSI al-
phabet. Binary FSK signals (4.2.7a) and (4.2.7b) with the tone values of f1 = 1200 Hz and f2 = 2200
Hz as well as binary signal duration of Tb = 1/(f2 – f1) have been used in our simulations.
Estimation of noise immunity of the technique proposed has been performed by analysis of the
errorless decoding probability. This probability has been computed in the form of an averaged number
of errorless symbols decoded normalized by their total number belonging to the original message.
Radio communication channel has been modeled as a memoryless channel. Errorless decoding
probability was computed depending on the binary signal amplitude As given in (4.2.7а) and (4.2.7b)
and for fixed AWGN variance. The signal amplitude As has been varied within the limits of (3, 35)
mV. This allows varying input SNR in a wide range.
Note that the considered limits of As correspond to changing the SNR = Рb/σ 2 at the FSK demo-
dulator input from 0.0045 up to 0.61 for the fixed AWGN variance equal to σ 2 = 1000 mW. It should
be stressed that such SNR variation can be caused in practice by multi-path fading channel in mobile
telecommunication systems. Bimagnitude threshold in decision rule (4.2.10) has been selected to be
equal to γ = 1529.
According to the above mentioned symmetry property of TCF, the bit sequences generated by
only nine code words, namely, by one word Ra(1,1) (or Rb(1,1)) corresponding to the maximum TCF
value and four times repeated code words Ra(1,2) and Ra(2,3) (or words Rb(1,2) and Rb(2,3)) were
144

used for transmission of the test message for the truncated codes (4.2.6а) and (4.2.6b).
The segment realizations of the FSK signals corrupted by AWGN and mixture of AWGN and
impulsive noise are shown in Figures 4.2.2 and 4.2.3, respectively.
Fig. 4.2.2. Segment of FSK signal of amplitude
As =10 mV corrupted by AWGN with variance
σ 2 = 1000 mW (the thick curve corresponds
to the original binary FSK signals).
Fig. 4.2.3. Segment of FSK signal of amplitude
As =10 mV corrupted by mixture of
AWGN with variance σ 2 = 100 mW and impulsive
noise with spike amplitude of Ap = 200
mV and probability of negative and positive
impulses equal to p = 5%.
The graphs of errorless decoding probabilities computed for the code sequences (4.2.4a, b) and
(4.2.6a, b) for a test message “xai504” (it means the name of authors department in Kharkov Aviation
Institute, see http://k504.xai.edu.ua/eng/main_eng.php) transmitted over a radio channel corrupted by
AWGN with fixed noise variance of σ 2 = 1000 mW are shown in Figures 4.2.4 and 4.2.5, respectively.
Fig. 4.2.4. Errorless decoding probability as
a function of binary FSK signal amplitude
obtained for the code sequences (4.2.4a, b)
in a radio channel corrupted by AWGN.
Fig. 4.2.5. Errorless decoding probability as
a function of binary signal amplitude obtained
for truncated code sequences (4.2.6a,
b) in a radio channel corrupted by AWGN.
The graphs of the errorless decoding probabilities obtained for the code sequences (4.2.4a, b)
and (4.2.6a, b) for a test message “xai504” in a radio channel corrupted by the mixture of AWGN
145

(σ 2 = 100 mW) and impulsive noise (the spike amplitude is equal to Ap = 200 mV and probability of
negative and positive impulses equals to p = 5%) are shown in Figures 4.2.6 and 4.2.7, respectively.
Fig. 4.2.6. Errorless decoding probability as
a function of binary signal amplitude obtained
for the code sequences (4.2.4a, b) in
a radio channel corrupted by mixture of
AWGN and impulsive noise.
Fig. 4.2.7. Errorless decoding probability as
a function of binary signal amplitude obtained
for code sequences (4.2.6a, b) in a
radio channel corrupted by mixture of
AWGN and impulsive noise.
The results demonstrated in Figures 4.2.4 – 4.2.7 were computed for 30 Monte Carlo runs, i.e.
by using 30 independent radio telecommunication sessions for the same test message in the noisy
memoryless channel. The curves in the graphs obtained for the code (6, 1) are marked by dashed lines
and the results obtained for the code proposed are represented by solid curves.
It is clearly seen from Figures 4.2.4 and 4.2.5 that the binary signal threshold amplitude provid-
ing errorless decoding for block coding (6, 1) is equal to АT = 18 mV (the corresponding threshold
SNR is equal approximately to (Рb/σ 2 )Т = 0.16). The corresponding binary signal threshold amplitudes
obtained for the proposed codes (4.2.4a, b) and (4.2.6a, b) are equal to АT = 7 mV ((Рb/σ 2 )T = 0.0245)
and АT = 12 mV ((Рb/σ 2 )T = 0.072), respectively. Therefore, the benefit achieved for the proposed code
in comparison to the block code (6, 1) (for the corresponding (Рb/σ 2 )Т ratios) is equal to 6.5 times (8.1
dB) for the code (4a, b) and 2.2 times (3.4 dB) for the code (6a, b), respectively.
Comparative analysis of the noise immunity of the TCF-coding for the proposed and conven-
tional code (6, 1) with respect to the practically important case of influence of impulsive noise (see
Figures 4.2.6 and 4.2.7) also demonstrates good robustness of the codes suggested. It follows from the
plots that the conventional code (6, 1) provides errorless decoding with a binary signal threshold am-
plitude equal to АT = 22 mV. The proposed codes (4.2.4a, b) and (4.2.6a, b) provide errorless decoding
in a mixed noise environment for АT = 11 mV and АT = 15 mV, respectively. The better performance
of the proposed TCF-coding technique is clearly seen for the curves plotted in Figures 4.2.6 and 4.2.7.
146

The proposed technique allows improving reliability of digital radio communication systems
operating under low (less than unity) and varying SNR. Unfortunately, message time delay in the cod-
ing technique suggested increases due to insertion of a redundant code. However, we suppose that our
coding technique may be useful in situations when errorless transmission is of great importance, and
time delay is a less important parameter.
The results of this subsection can be useful for digital mobile communication systems operating
under low and variable SNR as well as with messages transmitted over the radio channels corrupted
by a mixture of AWGN and impulsive noise.
147

148

5. Bispectrum estimation in radar applications
5.1. Radar range profile estimation of naval objects
Automatic target recognition (ATR) techniques using radar range profile (RP) estimations are of
particular interest [90 – 94]. There is a variety of approaches to solving ATR problem such as optimal
statistical pattern recognition using wide band and multi-frequency radiation [90], the use of neural
network-based classifiers [94], the target shape reconstruction by bispectrum-based techniques [91 –
93], etc. Such variability of approaches deals with the following. First, there are many different factors
affecting the quality of the obtained RPs and the performance of ATR methods. These factors are tar-
get rotation and translation during data acquisition, noise and/or clutter influence, heterogeneity of
propagation channel, etc. Because of this, the ATR is a very complicated task. Second, the perfor-
mance of ATR systems depends upon the radar characteristics and operation principles, the effective-
ness of signal processing techniques, the robustness of RP characteristics (information features) and
the effectiveness of a classifier.
Considerable attention is paid recently on improving ATR system performance at different stag-
es. For example, wideband radars are used to obtain information about target length and shape by RPs
and improve scatterer distribution estimations [90]. Bispectrum-based methods possess good immuni-
ty to RP translation and permit to quasi-coherently store several observations [91 – 93].
Although naval targets have relatively low velocities in comparison to aerial targets, the naval
object RPs obtained by conventional radar signal processing methods still depend on target aspect an-
gle and they are translation variant. Another peculiar feature that restricts naval object identification in
maritime environment is the presence of sea clutter and radar echo interference caused by backscatter-
ing of electromagnetic waves from the surface of the sea. There are many scientific papers dealing
with sea clutter study (see, for example, [95]) that demonstrate the complicated and non-stationary
nature of the statistical characteristics of the backscattered signal which depend on random dynamic of
sea clutter, grazing angle of observations, radar signal frequency and polarization. Because of this, it
is usually rather difficult to suppress sea clutter to separate naval object RP from sea interference.
In this subsection, we consider the bispectrum-based approach to naval object RP reconstruction
using experimental range naval object portraits obtained by X-band polarimetric radar for the cases of
non-Gaussian sea clutter interference [28]. First, we consider conventional approach to RP estimation
by using averaging the received echo signal envelopes. Let us describe a spatial-temporal model of
received signals that correspond to reflections from a naval object and interference that is caused by
sea surface backscattering. We assume that the dimensions of the object are significantly larger than
149

wavelength. The relative location of transmitting-receiving on-land antenna, the backscattering area Ω
that is determined by antenna beamwidth, the naval object and backscattering sea surface are shown in
Fig. 5.1.1.
yo
On-land
radar
Fig. 5.1.1. Geometric relationship of coastal radar and naval object.
The plane of Cartesian coordinate system XOY is consisted with antenna aperture and its origin
is positioned onto aperture center. Total area Ω can be divided into a set of independent elementary
surfaces ∆θ with reflection coefficients that are different for the metallic object and the sea surface.
Assume that the antenna is fixed and its pattern maximum is oriented to the object effective reflection
center θ0. Define the coordinates of elements ∆θ of the area Ω by vector θ =(θx, θy) with directive co-
sines θx, and θy.
Suppose that a polarimetric radar antenna irradiates narrow-band signal in the form of M RF im-
pulse chain with rectangular shape and without any carrier modulation. Let us present he received sig-
nal ṡ
ik
(t)
as
ṡ ik
( t)
= Ṡ
ik
( t)
exp( j2πf
0
t)
, (5.1.1)
where t ∈ [ −T
/ 2,
T / 2]
denotes time; i,k={H,V} are the indices that correspond to horizontal and vertic-
al polarization;
y
θx
θy θθθθ
θθθθo
x
Sea
⎧1,
t − mT ≤ τ
⎪ r p / 2
Ṡ
ik
( t − mτ
p ) = ⎨
is the complex envelope; τp denotes the pulse length;
⎪0,
t − mT > τ
⎩ r p / 2
150
Ω
R
Target
∆∆∆∆θθθθ

T=TrM is the total radar transmitting/processing time, Tr is the pulse repetition period; m=1,2,3,…,M
defines the pulse index; f0 is the central frequency, f ∈ [ f − ∆F
/ 2,
f + ∆F
/ 2]
where the bandwidth
0 0
∆F

- antenna-sea surface-antenna;
- antenna-sea surface-target-antenna;
- antenna-target-sea surface-antenna;
- antenna-sea surface-target-sea surface-antenna.
Conventional RP estimate is obtained using the averaging for N realizations. Due to statistical
independence of signal and interference, one obtains
ˆ ( n) ( n) ( n) ( n) ( n)
S
ik
( l∆ t) = S
ik
( l∆ t) = S̃ T
[ l∆t − τT ] + S̃ S
[ l∆t − τ
S
] , (5.1.4)
~ ( n)
~ ( n)
where S
T
(...) and S
S
(...) are the n-th echo envelopes smoothed by antenna pattern and correspond-
(n)
ing to the target reflections and to the sea backscattering, respectively; τ
T denotes the time lag inte-
)
grated for all target backscattering centers during the n-th scan; τ
(n
is the time lag integrated for all
sea backscattering elements during the n-th scan; denotes the averaging for N realizations (for N
)
observed scans). Both τ
(n
)
and τ
(n
are random values that alternate from one realization to another.
T
S
The analysis of the relationships (5.1.2 – 4) reveals that:
1) The antenna integrates the multiple reflections and interference into common temporal
signal due to influence of F(θ ) ̇ cut-off property. As the result of random target motions on the sea sur-
face, the object response is a fluctuation process that changes from one scan to another and it pos-
sesses specific PDF.
)
2) The presence of different propagation paths leads to different time lags τ
(n
(
and τ
n
in received signals (5.1.4), and these lags vary randomly from one scan to another.
The target and sea responses are independent processes for which the corresponding PDFs and
correlation intervals differ from each other. The correlation intervals for the processes that correspond
to reflections from sea surface elements are comparable to one scan processing time. Therefore, the
response from the sea is a rapidly fluctuating process that changes from one scan to another. Thus, the
time varying sea echo responses that usually overlap with the object response can destroy the estimate
(5.1.4).
Recall that the bispectrum-based RP estimation techniques have the following two advantages:
1) possibility to preserve Fourier spectrum phase information; 2) insensitivity to processed signal time
shifts. Bispectrum estimate B
ˆ
( p,
q)
̇ of observations (5.1.2) can be derived by direct technique and
written as
152
S
T
S
)

ˆ ˆ j ˆ( p, q) ( n) ( n) ( n)*
Ḃ γ
( p, q) = Ḃ ( p, q) e = Ẋ ( p) Ẋ ( q) Ẋ ( p+ q)
, (5.1.5)
where B
ˆ
( p,
q)
̇ and ˆ γ ( p,
q)
are the magnitude and phase bispectrum estimates, respectively; p=1,2,…,L
(n)
and q=1,2,…,L are the independent frequency indices; X (...) ̇ is the direct Fourier transform of
(5.1.2):
( n)
( n)
( n)
( n)
j2πτ
p
n
j2
p
X p S p e T ( )
πτ
̇ ( ) = ̇ ( )
+ Ṡ
( p)
e S ;
T
S
( n)
( p)
T
~ ( n)
S
T
~ ( n)
and sea S
(...)
forms of the object (...)
S responses, respectively.
153
Ṡ ( n)
and S ( p)
S
̇ are the Fourier trans-
The expression (5.1.5) can be rewritten on basis of (5.1.4) with regard to statistical indepen-
dence of the object and sea radar responses. Then, due to the signal shift invariance property of bis-
pectrum, one gets
( n)
( n)
2 ( )
j2 ( p q)
ˆ ˆ ( n) ( n) − j πτ p+ q
πτ +
( n)* Ḃ ( p, q) = Ḃ ( p, q) + Ṡ ( p) Ṡ ( q) e T Ṡ ( p+ q) e S +
T T T S
( n) ( n) ( n)
( n)
( n) ( n)* j2πτ q
( n) −j2πτ q
( n) ( n)* j2πτ q
( n)
−j2πτ
p
Ṡ ( p) Ṡ ( p+ q) e T Ṡ ( q) e S + Ṡ ( q) Ṡ ( p+ q) e T Ṡ ( p) e S +
T T S T T S
( n) ( n) ( n)
( n)
( n) −j2πτ
p 2 2
2
Ṡ ( p) e T ( n) ( n)* j πτ p
( n) −j
πτ q
( n) ( n)*
j πτ q
Ṡ ( q) Ṡ ( p+ q) e S + Ṡ ( q) e T Ṡ ( p) Ṡ ( p+ q) e S +
T
S S T S S
( n)
( n)
2 ( )
j2 ( p q)
( n)* j πτ p+ q
− πτ +
( n) ( n) ˆ
Ṡ ( p+ q) e T Ṡ ( p) Ṡ ( q) e S + Ḃ ( p, q)
T S S
S
.(5.1.6)
The first term in formula (5.1.6)
ˆ
( n)
( n)
( n)
*
B ̇
T
( p,
q)
= Ṡ
( p)
Ṡ
T T
( q)
Ṡ
T
( p + q)
is the original RP target
bispectrum. The other terms in (5.1.6) are the interference terms that degrade the original RP target
bispectrum estimate. Theoretically, sufficiently good accuracy of B
ˆ
T
( p,
q)
̇ can be obtained under tra-
ditional assumptions that interference is zero-mean, and its PDF is close to symmetric [43]. The case
of sea clutter with non-zero mean and with long tail PDFs needs to be studied and this problem is one
of the subjects of our experimental investigation in this subchapter.
form (IFT)
Bispectrum-based object RP estimate can be represented as the following inverse Fourier trans-
⎧ j ˆ ϕ ( ) ⎫
ˆ
ˆ bisp
r
S ( l)
= IFT ⎨ Ṡ
range
bisp
( r)
e ⎬ , r = 1, 2,…, L, (5.1.7)
⎩
⎭

where the magnitude
ˆ
S
bisp
( r)
̇ and phase ˆ ϕbisp ( r)
object RP Fourier spectrum estimations can be
recovered from (5.1.6) by recursive algorithms [10].
Experimental investigations were carried out in summer period using on-land X-band polarime-
tric radar with f0=9.370 MHz. The fixed antenna was located at the height of y0=8 m over sea level
(see Fig. 5.1.1) on the Black Sea shore in Crimea.
The radar basic characteristics are the following:
- the antenna beam width is 3.0 0 in both azimuth and elevation;
- the transmitted peak power is 10 kW;
- the pulse width is 3 µs;
- the pulse repetition frequency is 400 Hz for the total set of polarization combinations, i.e. for
HH, HV, VH and VV polarizations (the first letter corresponds to the transmitted wave polari-
zation while the second one denotes the received signal polarization), thus, it is 100 Hz for
each polarization component combination;
- two synchronous receiving channels for H and V polarization were employed; the separation
of the channels is over 30 dB;
- the dynamic range of the receiver is not less than 120 dB;
- the ADC capacity is 10 bits;
- the access time is 50 ns;
- the sample pulse step is equal to 250 ns that corresponds to the range bin of 75m.
After passing through analog intermediate frequency amplifier and amplitude detector, the re-
ceived signals were digitized and recorded in the memory block. The intermediate frequency amplifier
and detector characteristics were close to linear. The sampled data were recorded in the form of scans
(realizations) with L=32 samples for each HH, HV, VH and VV polarization. The scan duration for
each polarization was equal to 320 ms. The number of recorded scans for each polarization is equal to
N=256.
The antenna was pointed to the object with the aid of video camera fixed to the antenna. The
anchored metallic buoy served as the naval target. Its size was considerably smaller than the range bin.
The object radar echoes were recorded under small grazing angles of about 0.4 0 .
Fig. 5.1.2 illustrates consecutive scans for HH polarization, i.e. normalized range profiles
(NRPs) as the function of range sample index l. The slant range to buoy was R=1500 m and the buoy
range position approximately corresponds to the range sample index 26.
154

NRP
NRP
NRP
l
NRP
a b
l
NRP
c d
l
NRP
e f
Fig. 5.1.2. Consecutive scans: a) scan#1; b) scan#2; c) scan#3; d) scan#4; e) scan#5; f) scan#6.
The total duration of 6 observed scans equals to 6x320ms=1920ms. Note that random buoy mo-
tion on sea waves as well as random changes of the electromagnetic wave reflection angles can be ex-
155
l
l
l

pected to appear during this sufficiently long observation time interval. Hence, the behavior of the ob-
( n)
( n)
ject ε̇
T ik
( θ , ∆t)
and sea
S ik
( θ , l∆t)
ing every observed scan.
ε̇ reflection components in (5.1.3) can be expected as random dur-
As seen from Fig. 5.1.2, the scan fragments corresponding to buoy location and to sea clutter
have random nature and their appearance considerably changes from one scan to another. For some
scans the buoy is not visible at all (e.g., for the scan #2). Though the target RP should appear itself as
single peak because its size is much smaller than range bin, the object response shape is considerably
distorted. Due to interference, several peaks are observed in the neighborhood of the index 26 – see
the scans ##3, 4, 5, 6. Such RP distortions prevent object classification.
To improve estimation performance, one can simply average the scans for N realizations (see
(5.1.4)). The averaged NRPs obtained for different polarizations are shown in Figures 5.1.3 – 5.1.8.
The Figures 5.1.3 – 5.1.5 correspond to irregular sea waves and absence of wind. The Figures 5.1.6 –
5.1.8 correspond to sea state of 2…2.5 and the wind speed of 7…10 m/s.
As seen from Figures 5.1.3 – 5.1.5, when the sea backscattering is low, the NRPs for different
polarizations are considerably distorted. The averaged object NRPs are spread and their width at the
half-amplitude level varies from approximately 300 m (see the Figures 5.1.4 and 5.1.5) to 450 m (see
Fig. 5.1.3). These values are sufficiently larger than the range bin 75 m that can be expected as the
target response width.
NRP
Fig. 5.1.3. Averaged NRP
(HH, no wind).
l
NRP
Fig. 5.1.4. Averaged RP (HV,
no wind).
l
156
NRP
Fig. 5.1.5. Averaged RP
(VV, no wind).
l

NRP
Fig. 5.1.6. Averaged RP (HH,
wind speed 7…10 m/s; sea
state 2…2.5).
l
NRP
Fig. 5.1.7. Averaged RP
(HV, wind speed 7…10 m/s;
sea state 2…2.5).
l
157
NRP
Fig. 5.1.8. Averaged RP
(VV, wind speed 7…10 m/s;
sea state 2…2.5).
This effect appears due to sea clutter and above mentioned influence of interference from sever-
al electromagnetic wave propagation paths. The effect can be also partly induced by the random mo-
tion of the naval object and averaging of the randomly shifted object responses according to (5.1.4).
The sea clutter level observed in Figures 5.1.3 – 5.1.5 depends on polarization and its maximum
varies from approximately – 6 dB (see Figures 5.1.4 and 5.1.5) to – 13 dB (see Fig. 5.1.3).
With wind present and the sea state 2…2.5 (see Figures 5.1.6 – 5.1.8), the object NRP width at
the half-amplitude level varies approximately between 900 m and 300 m. In this case, sea clutter level
varies approximately from –13 dB (see Figures 5.1.6 and 5.1.8) to –8 dB (see Fig. 5.1.7).
Thus, analysis of the NRPs shown in the Figures 5.1.3 – 5.1.8 permits to note that:
- each observed object response shape is distorted due to the averaging of the set of randomly
shifted received object signal envelopes (see Fig. 5.1.2);
- sea clutter level is sufficiently high that can mask the object response in some cases (see, for ex-
ample, Fig. 5.1.6).
Consequently, conventional RP estimation technique based on the averaging received signal enve-
lopes has low range resolution and low robustness to sea clutter.
The bispectrum-based NRP estimates are shown in Figures 5.1.9 – 5.1.14. The Figures 5.1.9 –
5.1.11 correspond to irregular sea waves and absence of wind. The Figures 5.1.12 – 5.1.14 correspond
to sea state of 2…2.5 and the wind speed of 7…10 m/s. Note, that due to bispectrum shift invariance
these NRPs are centered with respect to the NRP center of gravity (index 16 corresponds to the object
location in Figures 5.1.9 – 5.1.14).
l

NRP
Fig. 5.1.9. Bispectrum-based
NRP (HH, no wind).
NRP
l
Fig. 5.1.12. Bispectrumbased
NRP (HH, wind speed
7…10 m/s; sea state 2…2.5).
l
NRP
Fig. 5.1.10. Bispectrum-based
NRP (HV, no wind).
NRP
Fig. 5.1.13. Bispectrum-based
NRP (HV, wind speed 7…10
m/s; sea state 2…2.5).
l
l
158
NRP
Fig. 5.1.11. RP Bispectrumbased
NRP (VV, no wind).
NRP
Fig. 5.1.14. Bispectrum-based
NRP (VV, wind speed 7…10
m/s; sea state 2…2.5).
With no wind, the NRP width at the half-amplitude level equals approximately to 150 m and the
sea clutter levels have the values –20 dB (Fig. 5.1.10), –24 dB (Fig. 5.1.9) and –26 dB (Fig. 5.1.11).
The bispectrum-based technique performance slightly worsens in the case of the sea state 2…2.5 and
wind speed of 7…10 m/s (see Figures 5.1.12 – 5.1.14). The NRP width is between 150 m (see Figures
5.1.13 and 5.1.14) and 450 m (Fig. 5.1.12) and the sea clutter level becomes equal to –17 dB (Fig.
5.1.12), –28 dB (Fig. 5.1.13), and –26 dB (Fig. 5.1.14).
In Tables 5.1.1 and 5.1.2, the key parameters of the RP estimates (range resolution and sea clut-
ter level) for two considered techniques are summarized.
l
l

Table 5.1.1. Results obtained for averaging received signal envelopes corresponding formula (5.1.4).
Absence of wind Wind speed of 7…10
m/s
HH HV VV HH HV VV
NRP width at the half-amplitude level, m
450 300 300 900 300 300
Sea clutter level, dB
-13 -6 -6 -13 -8 -13
Table 5.1.2. Results obtained by proposed signal processing corresponding formulas (5.1.5 – 5.1.7).
Absence of wind Wind speed of 7…10
m/s
HH HV VV HH HV VV
NRP width at the half-amplitude level, m
150 150 150 450 150 150
Sea clutter level, dB
-24 -20 -26 -17 -28 -26
As seen from Table 5.1.1, the range resolution is worse by from 2 to 12 times in comparison to
the theoretical radar range bin of 75 m.
The results represented in Table 5.1.2 demonstrate the stability of the range resolution that is
worse only by 2 times in comparison to the above mentioned theoretical radar range bin except the
case of wind speed of 7…10 m/s and HH polarization.
The values of the sea clutter levels given in Tables 5.1.1 and 5.1.2 depend upon polarization
type and sea state. As clearly seen from the comparison the data in Tables 5.1.1 and 5.1.2, bispectrum-
based technique allows reducing sea clutter level from 11 dB (HH polarization) to 20 dB (VV polari-
zation) in the case of absence of wind and from 4 dB (HH polarization) to 20 dB (HV polarization) in
the case of wind speed of 7…10 m/s.
Thus, with the bispectrum-based technique the resolution of the naval object on the background
of sea clutter increases markedly. Experimental simulation results demonstrate the promising possi-
bility of improving the range resolution of a priori unknown naval object on the background of sea
clutter by bispectrum-based reconstruction of radar RPs. This bispectrum-based approach does not
require a priori knowledge about object and interference characteristics.
Experimental results have demonstrated sea clutter suppression and good naval object resolu-
tion. These results have been achieved for low resolution, polarimetric X-band radar for different sea
states.
159

5.2. Reduction of aspect dependent speckle distortions in aerial HRRP
The problem of classification and identification of aerial targets of different types by using their
radar high resolution range profiles (HRRPs) is of particular interest for ATR systems [96 – 100].
Aerial HRRPs are usually modeled by using a large number of local discrete point reflectors
spatially distributed on the target surface [96]. The difficulty of using HRRPs for ATR purposes is in
their variability caused by minor target changes in azimuth, elevation and time.
Bispectrum-based approach for solving the problem of designing a preprocessing stage that can
lead to improving the performance of ATR classifiers has been considered in [101]. The main objec-
tive of investigations performed in [101] is to obtain robust features of the target and reduce the sensi-
tivity with respect to its spatial orientation. The proposed approach exploits the properties of the bis-
pectrum to reduce the aspect dependent speckle influence observed in HRRPs.
In practice, as a result of target motion, its HRRP can fluctuate considerably owing to small
(about tenths of one angle degree) aspect variations as well as with due to small target spatial shifts
and rotations. The efficiency of the classifier learning procedure performed in ATR systems largely
depends on the robustness of the information feature vector applied at its input. Unfortunately, HRRPs
variability often provokes many difficulties in obtaining training data.
In the real-life echo radar signal processing, the value of the vector sum in the backscattered mi-
crowave electromagnetic field is determined by the contribution of multiple target “bright points”
measured in a time-fixed range bin. In other words, the echo signal at the output of the matched filter
largely depends on the current target aspect and elevation angles [96]. The following three main
sources of HRRPs variability are of practical interest for those who process them: the rotational range
migration of the intensity peaks associated to separate local point scatterers (“bright points”), the
translation range migration, and the speckle phenomenon. Rotation and translation motion of a target
provoke the displacement of the peaks in HRRPs from one range bin to another according to target
travel. But, the relative distance between the information peaks in target intensity distribution does not
vary. Hence, a separate-measured HRRP undergoes the spatial shifts due to the aircraft translation and
rotation, and a procedure of HRRPs alignment is required before conventional averaging of the meas-
ured HRRPs to provide the “matched” (coherent) accumulation of the set of HRRPs and essential val-
ue of output SNR. However, the alignment of the HRRPs is a separate independent problem that re-
quires additional a priori data about parameters of an aerial target travel. The latter data, unfortunately,
are not always available in practice. Otherwise, the averaged HRRP becomes smeared. The latter pe-
culiarity usually complicates solving the recognition problem in radar ATR systems.
Bispectrum signal shift invariance property (1.2.12) permits to automatically perform “matched”
160

accumulation of the HRRPs by averaging carried out in the bispectral domain. In this case, the proce-
dure of range alignment is, in fact, skipped [99 – 100]. The latter property is one of the major benefits
in the bispectrum-based signal processing comparing to the simple direct “non-matched” (non-
coherent) averaging of the HRRPs ordinary proposed by number of the authors (see, for example, the
reference [96 – 98]). The noted bispectral estimation feature in aggregate with high noise immunity of
bispectrum to AWGN possesses certain benefits comparing to the conventional direct averaging of the
HRRPs measured within a large target observation azimuth and evaluation angle sectors [96].
The speckle phenomenon provokes fluctuations in range profile for much smaller changes of
target aspect angles corresponding to considerably shorter dwell time comparing to the aircraft trans-
lation and rotation. If the electromagnetic waves backscattered by, at least, two separate local point
scatterers are interacted in a single range bin, HRRP considerably varies due to the influence of
speckle. Hence, just a small rotation of an aircraft can cause changing the differential wave propaga-
tion length close to the half of radar wavelength. It can provoke partial mutual suppression of the elec-
tromagnetic waves interacted in the radar range bin. Therefore, even a small variation in aircraft as-
pect angle can cause pronounced changes in HRRP shape and produce the troubles for aerial target
recognition and identification.
Unlike the problem of suppression of rotational and translation range migrations (see the refer-
ences [99 –100]), the problem of the reduction of HRRP sensitivity to speckle caused by small
changes of an aerial target aspect angle (small dwell time interval) has not been considered yet in the
literature. It should be stressed, that in many practical situations just speckle distortions are of relevant
value.
Let us consider a model of the radar echo signal for an aerial target that turns uniformly within a
certain aspect angle sector. A pulse LFM radar operating at the X-band can provide the HRRPs. Tak-
ing into consideration the multiple local point backscatterer model, the received discrete complex-
valued signal yk(i) measured at the output of the matched range-finder filter for an arbitrary current k-
th aspect corresponding to an arbitrary k-th scan can be defined as
k
M
= m
m=
1
y ( i)
∑ a ( i)
exp[ jΦ
( i)]
, (5.2.1)
where am(i) and Φkm(i) are the scatterer amplitude and phase spatial distributions, respectively; i =
1,2,…I is the sample index for the range axis; k=1,2,…,K is the sample index of the aspect angle; the
index m denotes the local m-th target point scatterer.
The expression for k-th HRRP following from (5.2.1) can be written as
161
km

M M m
2 2 2
( ) = {Re[ ( )]} + {Im[ ( )]} ( ) 2 ( ) ( ) cos[ ( ) ( )]
k k k
= ∑ + ∑ ∑
Φ − Φ . (5.2.2)
m m n
km kn
m= 1 m= 2 n=
1
z i y i y i a i a i a i i i
It can be seen from (5.2.2) that spatial intensity distribution in HRRP includes two terms ob-
served for each k-th arbitrary aspect angle. The first term is the sum of intensities of point scatterers
that does not depend on aircraft rotation. The second cross-term depends on the aspect angle and it
depends on each change of the k-th aspect.
According to the results obtained in [99], for a great number of scatterers, the cross-term in
(5.2.2) tends to a random process with zero-mean value, and the correlation coefficient assessed be-
tween two arbitrary consequent HRRPs can be of very small value with aspect changes. The correla-
tion coefficient is approximately equal to unity only in the case when aerial target aspect variations are
not more than one hundredth of a degree. This second, aspect dependent random-valued cross-term in
(5.2.2) causes considerable fluctuations in aircraft HRRP just within a small target turn angle. It
should be also noted that the sensitivity of HRRP to aspect variations increases with radar wavelength
decreasing and aircraft size increasing.
To suppress fluctuation distortions caused by the above mentioned speckle, one needs to per-
form the averaging procedure and it will require employing a certain set of multiple HRRPs regis-
tered.
Let us define the short-time bispectrum estimate as a triple product of the Fourier transforms of
the k-th arbitrary HRRP (5.2.2) registered for a k-th scan. Then, the aspect-varying complex-valued
bispectrum estimate can be written as
*
B ( p,
q)
= Z ( p)
Z ( q)
Z ( p + q)
, (5.2.3)
k
k
k
where Zk(p) is the discrete direct Fourier transform of the k-th HRRP; p=1, 2,…, I and q = 1, 2,…, I
are the independent frequency indices. Note that prior to computations of the bispectrum estimates
(5.2.3), the mean values in the HRRP (5.2.2) must be removed for obtaining unbiased bispectrum es-
timate.
k
As it will be demonstrated below, the ensemble averaging of the short-time aspect-dependent
bispectrum estimates (5.2.3) performed within a proper aspect angle sector as
~
B ( p,
q)
=
N
N
1
∑
N
k = 1
B ( p,
q)
, N

averaged bispectrum estimates (5.2.4) by using the well-known recursive algorithm BLW [10].
It should be noted, that changes in N allow assessing the interesting evolutionary HRRP varia-
bility. The efficient selection of N gives us an opportunity to obtain the optimal (the most robust) in-
formation feature regarding to aspect angle variations and, hence, to improve ATR performance.
HRRP aspect sensitivity has been studied by means of the set of HRRP models computed with
the help of the software developed by the authors of the book [96]. We have investigated the HRRPs
of the following aerial targets: Tu-16 and B1-B bomber aircrafts; MiG-21 tactical fighter aircraft; and
GLCM cruise missile.
The total number of K = 2000 HRRPs where each contains 160 real part and 160 imaginary part
samples registered at the I/Q output of the matched filter were analyzed for above-mentioned aerial
targets. The following initial data were used in our computer simulations:
- LFM signal of Gaussian shape with pulse duration and repetition of 10 µs and 1000 Hz, re-
spectively;
- central wavelength of 3 cm and frequency bandwidth of 80 MHz;
- range resolution equal to ∆r = c/2∆f = 1.875m;
- horizontal polarization;
- length of the window (range cell) for target tracking in range of 80 m and range profile sam-
ple step of 0.5 m;
- constant aerial target angle with the step 2 o and constant angle of pitch of 3 o .
The uniform travel of the aerial target has been investigated and complex-valued HRRP samples
of the form of (5.2.1) have been zero-padded to I = 256 samples before computation the bispectrum
estimates (5.2.3) and (5.2.4).
Variability of the HRRPs has been studied within the total aspect sector of 10 o . Angle-of-
sighting has varied from 180 o (aerial target nose-on aspect) to 170 o for each aerial target.
Quantitative assessing of HRRP variability has been computed by using the formula
⌢
where z ( i)
and z ( i)
0
⌢
N
163
256
∑
zˆ
( i − t)
− zˆ
( i)
0
N
i = 1
v = min
, (5.2.5)
N
256
t
∑
i = 1
zˆ
( i)
0
are the HRRPs corresponding to the nose-on aspect averaging sector and the
N-th aspect averaging sector recovered by bispectrum estimate (5.2.4), respectively; t is the integer-
valued shift (t = 1, 2,…, 256) that is introduced in order to take into account translation invariance

property of bispectrum.
We have tested by computer simulations the following two kinds of averaging procedures to
study and compare speckle reduction in HRRP caused by the aircraft turn. The first non-coherent pro-
cedure deals with the sequences of K/N1 = 40 and K/N2 = 20 HRRPs. Direct “non-matched” averaging
has been executed within the segment length of N1 = 50 and N2 = 100 HRRPs (5.2.2), respectively.
The second coherent procedure suggested was performed in bispectrum domain by averaging the K/N1
= 40 and K/N2 = 20 bispectrum estimates (5.2.3) and recovery the HRRPs from the ensemble averaged
bispectrum estimates (5.2.4) within the segment length of N1 = 50 and N2 = 100, respectively.
As a typical example, the speckle-dependent evolutionary variability of the HRRP computed for
the TU-16 bomber aircraft is illustrated in Fig. 5.2.1. The sequence containing K/N = 20 HRRPs is
represented. These HRRPs have been computed by “non-matched” ensemble averaging of the N = 100
HRRPs performed according to the ordinary procedure (5.2.2). The evolutionary HRRP variability
observed within the aspect angle limits of 180…170 o is clearly shown in Fig. 5.2.1.
Fig. 5.2.1. The evolutionary sequence of the HRRPs computed for the TU-16 bomber aircraft.
The plots of the variability parameter (5.2.5) as a function of the segment number are
represented in Figures 5.2.2 – 9. The segment number K/N1 = 40 in Figures 5.2.2 – 5.2.5 corresponds
to the discrete sequence of the target aspects computed with the step equal to 0.25 o . The segment
number K/N2 = 20 in Figures 5.2.6 – 5.2.9 corresponds to the discrete sequence of target aspects com-
puted with the step equal to 0.5 o .
164

HRRP variability
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Segment number
Non-coherent averaging
Bispectrum estimation
Fig. 5.2.2. HRRP variability as a function of current segment number computed for the B1-B
HRRP variability
0,009
0,008
0,007
0,006
0,005
0,004
0,003
0,002
0,001
0
bomber aircraft.
1 4 7 10 13 16 19 22 25 28 31 34 37
165
Segment number
Non-coherent averaging Bispectrum estimation
Fig. 5.2.3. HRRP variability as a function of current segment number computed for the
GLCM cruise missile.

HRRP variability
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
166
Segment number
Non-coherent averaging
Bispectrum estimation
Fig. 5.2.4. HRRP variability as a function of current segment number computed for the MiG-
HRRP variability
160
140
120
100
80
60
40
20
0
21 tactical fighter aircraft.
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Segment number
Non-coherent averaging
Bispectrum estimation
Fig. 5.2.5. HRRP variability as a function of current segment number computed for the Tu-16
bomber aircraft.

Fig. 5.2.6. HRRP variability as a function of current segment number computed for the B1-B
bomber aircraft.
Fig. 5.2.7. HRRP variability as a function of current segment number computed for the GLCM
cruise missile.
167

HRRP variability
0,3
0,25
0,2
0,15
0,1
0,05
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Segment number
Non-coherent averaging
Bispectrum estimation
Fig. 5.2.8. HRRP variability as a function of current segment number computed for the MiG-
21 tactical fighter aircraft.
Fig. 5.2.9. HRRP variability as a function of current segment number computed for the Tu-16
bomber aircraft.
Analysis of computer simulation results presented in Figures 5.2.2 – 5.2.9 demonstrates the fol-
lowing peculiarities.
- First, HRRP variability largely depends on the aerial aircraft size and the shape of the back-
scattering surface. Speckle fluctuations for relatively smooth backscattering surfaces are of
minimum values: see the plots in Figs. 5.2.3 and 5.2.7 obtained for GLCM cruise missile and
compare them to the bomber and tactical fighter aircrafts.
- HRRP speckle fluctuations increase when aerial target size and the number of backscattering
168

surface irregularities increase: see for comparison the plots computed for MiG-21 in Figures.
5.2.4 and 5.2.8 and Tu-16 in Figures 5.2.5 and 5.2.9.
- Second, HRRP variability value (5.2.5) fluctuates due to speckle influence, and HRRP fluc-
tuations decrease when the number of averaged profiles increases. However, at the same
time, the number of averaged HRRP segments is limited in practice due to appearance of ro-
tation and translation range migrations as well as the effect of the radar dwell time.
- Third, HRRP fluctuations caused by speckle are significantly smaller for the proposed bis-
pectrum-based processing technique in comparison to the conventional direct non-coherent
averaging technique.
The root-mean square deviations of the HRRP variability were computed for quantitative as-
sessment of speckle fluctuations reduction. The benefits obtained in reduction of HRRP speckle fluc-
tuations by the technique suggested compared to the conventional non-coherent averaging can be seen
clearly from the results represented in Table 5.2.1.
Table 5.2.1. Quantitative comparison of two techniques investigated.
Aerial target Root-mean square deviation of HRRP variability
Direct averaging of HRRPs Bispectrum-based processing
50 profiles averaged 100 profiles averaged 50 profiles averaged 100 profiles averaged
B1-B 0.199 0.166 0.036 0.03
GLCM 1.7E-03 1.5E-03 9.8E-04 9.1E-04
MiG-21 0.202 0.075 0.045 0.057
Tu-16 37.23 25.58 15.23 5.67
Thus, the key distinction of our approach [101] from the known approach is in recovery of a set
of HRRPs from the corresponding sequence of bispectrum estimates ensemble averaged by the num-
ber of aspect segments. It makes it possible to reduce radar range profile aspect speckle sensitivity.
Our approach has been tested with computer simulations performed for radar aerial target models of
different types, sizes and backscattering surface irregularities. The use of HRRPs recovered by aver-
aged bispectrum estimates is seen to provide considerably more robust features for aerial target classi-
fication, which can be of interest in radar ATR systems.
5.3. Time-frequency analysis of backscattered signals in ground surveillance Doppler radar
One more important problem in radar ATR systems is the search and obtaining the robust in-
formation features of objects. The Doppler velocity spectra are important characteristics that are wide-
169

ly used for a moving target detection, classification and recognition. The classical energy density
spectrum or magnitude squared Fourier transform estimated during a rather long time interval is in-
adequate for analyzing the spectral content of non-stationary signals observed in ground surveillance
radar systems because the notion of frequency becomes meaningless. An alternative assumption is that
a signal is stationary during a short time interval, and instantaneous frequency (IF) should be used to
understand how the spectral content is changing in time [102]. The IF is an important time-varying
signal parameter which allows one to analyze the behavior of the spectral content simultaneously in
time and in frequency. In recent years, time-frequency (TF) distributions have been extensively em-
ployed in the analysis of time-varying spectra and the estimation of parameters of non-stationary
ground radar backscattered signals, as well as in underwater acoustics [102 – 108].
Note that one approach frequently used in short time-varying spectrum analysis is estimation of
energy per unit time per unit frequency. Spectrogram is a typical representative of these energy-based
estimators for detection and recognition of ground moving targets (for example, see [109]). Since the
possibly existing phase relationships between harmonics are lost in energy and power spectra, it is im-
possible to extract phase coupled frequencies by means of time-varying energy or power spectra. In
other words, energy and power spectra are unable to separate phase coupled instantaneous frequencies
(PCIFs) contained in time sequences processed. This peculiarity is the most serious shortcoming of
energetic versions of evolutionary spectral estimators.
Starting from the classical Wigner-Ville distribution [110] and its modifications with different
kernels [111], the optimal estimates for the analysis of linearly frequency modulated (LFM) signals
have been obtained. However, one of the most important drawbacks of these distributions is the ap-
pearance of a large number of spurious cross-terms when processing nonlinearly FM signals or multi
component FM signals. These cross-terms often mask the true features of the signal analyzed.
The third-order Wigner distribution or Wigner bispectrum has been originally proposed by N. L.
Gerr in [112] as an extension of the Wigner-Ville distribution for high-order spectral analysis. This
distribution permits to analyze the evolutionary behavior of the third-order moment functions of a
non-Gaussian process in the time-bifrequency hyperplane. Other authors have extended this definition
[113] and studied theoretically the properties of third-order Wigner distributions [114].
The attractive benefits of the Wigner bispectrum follow from the conventional properties of the
bispectrum analysis and are described in detail in Chapter 1. They are as follows: 1) low sensitivity to
any additive symmetrically distributed noise; this property allows to reduce the noise influence; 2) in-
sensitivity to target translation and rotation; this property permits to remove the artifacts caused by
target translations and rotations; 3) robust detection phase coupling and discriminating phase coupled
170

Fourier spectrum components from those that are not; this can be a useful tool for extracting relevant
features of an unknown object.
However, there are serious shortcomings of higher-order Wigner distributions. These are huge
computational complexity and low processing rate caused by the necessity of using 2-D FFT, as well
as the impossibility of visual representation of evolution of higher-order spectrum estimates in the
time-bifrequency hyperplane for analysis. Due to these, higher-order Wigner distributions are not
widely employed in real-life radar signal processing.
An approach based on extraction of phase coupled Doppler frequency pairs from time-varying
bispectrum estimates of transient sample sequences separated from the ground surveillance radar
backscattered signal by a sliding window has been proposed in our papers [115 – 117].
First, we focus on the Doppler ground surveillance radar echo signal model for a walking hu-
man. A walking person can be considered as a complex physical phenomenon of simultaneous motion
of different body parts: the torso, the legs, and the arms. The parameters of the radar echo depend on
several features of scattering by moving and spatially distributed surfaces. The size of the scattering
surface (usually compared with respect to operation wavelength in radar applications), the electrical
characteristics of the human body (admittance and radiation efficiency of the human body parts), the
scattering surface velocity, the grazing angle and reflecting power may contribute significantly to the
backscattered signal. Hence, for a human body the backscattered field is a sum of a number of contri-
butions.
The continuous-wave radar operating at the microwave band can provide the recording of the
radar signatures corresponding to a walking human with high-resolution Doppler frequency shifts.
The echo signals include the combined reflections from the three main body parts involved in human
walking. They are the swings of torso, legs and arms having different areas of scattering surfaces and
moving with different (and possibly, time-varying) velocities.
It should be stressed that the time-varying phase of the backscattered field is of paramount im-
portance. Since the field backscattered by a moving human is a sum of a number of contributions of
body parts, the phase of the total backscattered field does not vary with the same rate as that of the
different components. Hence, in the assumption of a linear approximation, these swinging body parts
provoke a sum of different time-varying Doppler frequency shifts in the backscattered signal and these
frequencies may be phase coupled due to the conduction currents flowing in the human skin.
It should be especially noted, that backscattered radar fields obtained by using vertical and hori-
zontal polarizations differ one from another just by different spatial phase distribution of the reflection
coefficients for vertical and horizontal polarizations. In other words, a human body can be considered
171

as radiating antenna having different and time-varying spatial amplitude and phase distributions for
vertical and horizontal polarizations. This important peculiarity can serve for retrieving new features
in object classification.
Taking into consideration these peculiarities, one can assume that a received radar return is a
nonstationary and multi component signal. For a multipoint scattering surface (containing a great
number of scattering centers), the observed discrete-time received multi component and nonlinearly
FM returns for vertical y V (i) and horizontal y H (i) polarizations can be defined, respectively, as
M
V V V
y ( i) = ∑ A ( i) cos[ Φ ( i)]
=
m=
1
m m
M
V V 2 4π
= ∑ a ( i) cos[ ϕ ( i)] F ( θ ) cos{ [ r ( i) −r
( i)]}
λ
0
y
m=
1
m m m m
0
H
M
H
( i)
= ∑ A ( i)
cos[ Φ ( i)]
=
m=
1 m
172
H
m
M
H
H 2 4π
= ∑ a ( i)
cos[ ϕ ( i)]
F ( θ ) cos{ [ r ( i)
− r ( i)]}
m=
1 m
m
m λ m 0
0
, (5.3.1a)
, (5.3.1b)
where a (i)
V
m and a (i)
H
m are the time-varying magnitudes of the local reflection coefficients corres-
ponding to the m-th object scattering center for vertical and horizontal polarizations, respectively;
V
H
ϕ (i)
and ϕ (i)
are the local time-varying phases for vertical and horizontal polarizations, respective-
m
m
ly; i = 1, 2,…I is the temporal sample index; F(θ) is the radar antenna amplitude directional pattern (it
is supposed that the same antenna is used for transmission and reception, the pattern shape is the same
in the H and E planes); θm is the angle location of the m-th object scattering center; rm(i) and r0(i) are
the time-varying distances between the antenna phase center and the arbitrary m-th moving object
scattering center and the object phase center, respectively; λ0 is the radiated wavelength; Am(i) and
Φm(i) are the total time-varying radar echo signal magnitude and phase corresponding to the m-th ob-
ject scattering center.
According to (5.3.1a, 5.3.1.b), the backscattered radar signal is a result of collecting M contribu-
tions caused by a large number of scattering centers. The sum of all backscattering amounts to the to-
tal radar return obtained from a walking human.
Two motion components are involved in the model (5.3.1a, b): the translation motion of the hu-
man torso (this one corresponds to an approximately constant low Doppler frequency shift due to the
relatively constant and low speed of the human body) and the swinging motion of legs and arms (this
one corresponds to the sum of different time-varying Doppler frequency shifts caused by different
non-uniform time-varying velocities of swinging body parts).
The time-varying phase Φm(i) in (5.3.1a, b) can be represented in the form of the following po-

lynomial of a priori unknown order R depending on the characteristic features of a moving human
Φ
m
R
( i)
=
r
∑b
i , (5.3.2)
mr
r = 0
where bmr are the unknown constant coefficients controlling the weighting of polynomial phase val-
ues.
The discrete-time IF can be expressed as
1 dΦ
R
m ( i)
1
r −1
fm
( i)
= = ∑ rbmri
.
2π
di 2π
r = 0
(5.3.3)
Depending on the physical properties of the moving object, the polynomial (5.3.3) can describe,
for example, the quadratic FM law (for r = 3). In general, the radar echo signal model (5.3.1 –3) de-
scribes a nonstationary signal of multi component structure as a weighted sum of mono component
signals, each one with its own IF fm(i). This frequency is proportional to the radial velocity of the m-th
object scattering center. In other words, an unknown set of IFs might be contained in the observed
signal.
It should be especially noted that the phases of some IFs may be related, i.e. some of the IFs
might be phase coupled.
The latter assumption is the key hypothesis in the approach suggested in [115 – 117]. It is capa-
ble of providing new data in object identification, classification and recognition in such circums-
tances.
It should be especially stressed that the authors commonly pay attention only to the quadratic
phase coupling phenomenon caused by passing a linear non-Gaussian process through non-linear de-
vice (see [2]). We are the first who have found and investigated experimentally existence of phase
coupled spectral components in multi component and nonstationary radar signals backscattered by a
moving human [115].
We explain our approach physically by the fact that the swinging torso, legs, and arms are not
independent sources of Doppler frequency shifts, but are related via the “carrier” (this “carrier” is the
human torso) that can be considered as the “common basis” for the swinging legs and arms. Further-
more, from the radio physical point of view, the microwave conductivity currents flowing in the hu-
man skin (actually, they are the source of backscattered microwave field) also may cause certain phase
relationships in multi component radar echo signal. Therefore, one can expect the presence of phase
coupled harmonics in radar return (5.3.1a, b).
According this hypothesis, our goal is to detect and verify the presence of phase coupling as
well as to estimate evolutionary behavior of the PCIFs in the nonstationary and possibly multi compo-
nent radar echoes collected from a moving object.
173

The main idea of our approach utilizes one of the properties of the bispectrum. It is well-known
[2] that the bispectrum allows measuring the magnitude and phase of the autocorrelation of a signal at
different Fourier frequencies. When a phase relationships exist, the phase coupled components contri-
bute to the bispectrum estimate. In other words, the bispectrum magnitude (bimagnitude) is nonzero
only if the responses at certain frequencies are correlated. Therefore, the quantitative degree of coupl-
ing between the harmonics may be measured by the bispectrum analysis. On the other hand, for a sta-
tionary zero-mean Gaussian process, the bispectrum is identically zero. It means that there are no
phase coupled frequencies in a linear Gaussian process. Therefore, unlike the energy spectrum, the
bimagnitude distribution contains peaks caused by the coherence between the bifrequency compo-
nents.
Let us define the short-time bispectrum estimate as a triple product of the Fourier transforms of
a transient signal y(i,n) whose time duration is significantly shorter than the total observation interval
of I samples recorded in the observation (5.3.1a, b). Note that the transient signal y(i,n) can be sepa-
rated from the total observation (5.3.1a, b) by a sliding window that step by step moves along the
process recorded and takes n = 1, 2, …, N non-overlapping positions.
Then, the time-varying bispectrum estimate can be calculated as the following triple product
*
B(
p,
q;
n)
= Y ( p,
n)
Y ( q,
n)
Y ( p + q,
n)
=
, (5.3.4)
= B(
p,
q;
n)
exp[ jβ
( p,
q;
n)]
where Y ( p,
n)
= Y ( p,
n)
exp[ jϕ(
p,
n)]
is the discrete time-varying Fourier transform of the transient sig-
nal y(i,n); B ( p,
q;
n)
and β(p,q;n) are the time-varying bimagnitude and biphase, respectively;
p=1,…,P and q=1,…,P ( P

adar signal of the total length of 200000 samples was recorded as a continuous radar echo return ob-
tained from a metallic sphere (a pendulum) swinging in coherent continuous-wave radiation with the
operating wavelength of λ0=2 cm. A typical echo signal segment of the length of approximately 4000
samples is shown in Fig. 5.3.1.
Because the radar radiation was backscattered by a spatially distributed object (the diameter of
the sphere significantly exceeded the radar wavelength λ0) the radar echo is not a pure LFM signal in
the strict sense as in the case of a point scatterer. As clearly seen from Fig. 5.3.1, the process is of
chirp-like nature and demonstrates nonstationary behavior.
Fig.5.3.1. The chirp-like radar return recorded from a swinging sphere:
the signal magnitude as a function of a sample number.
The plot of the bimagnitude estimate of the transient signal truncated by the window of simple rectangular
shape with duration of P=256 samples and an arbitrary location ( i ∈ [ 514,
770]
) is shown in
Fig. 5.3.2.
Fig. 5.3.2. Bimagnitude as a function of frequency samples p and q measured for an arbitrary radar
transient echo signal segment for a swinging sphere.
The horizontal plane in this graph is the bifrequency plane (pxq=256x256 frequency samples)
and the bimagnitude values are on the vertical axis.
Several pronounced sharp bimagnitude peaks in the bifrequency plane are clearly seen in Fig.
5.3.2. This indicates the presence of phase coupled frequencies in the radar echo returns obtained from
175

the swinging metallic sphere.
The analysis of the bimagnitude function shown in Fig. 5.3.2 has suggested the idea to project
the 3-D PCIF bimagnitude distribution to 2-D time-frequency (TF) plane convenient for analysis [115
–117].
Experimental verification of the suggested approach has been performed by a coherent, homo-
dyne, continuous-wave radar operating at the wavelengths of λ0 = 2 cm, 3 cm and 8 mm with the ver-
tical polarization and a polarimetric radar operating at the wavelength of λ0=8.8 mm with the vertical
and horizontal polarizations have been designed at the Institute of Radiophysics and Electronics of the
National Academy of Sciences (Kharkov, Ukraine) [118]. The output of the 10 bits ADC allows col-
lecting low frequency radar returns containing Doppler frequency shifts. Experimental investigations
were carried out during the spring and summer periods at the site of the Institute. Fixed transmitting
and receiving horn antennas were mounted at the height of 150 cm above the ground.
A block diagram and photograph of the radar system are shown in Figures 5.3.3 and 5.3.4.
Channel 2
IMPATT
Power
Supply
PC
Ferrite
Circulator
176
Channel 1 LF
Amplifier
Magic-T
Mixer
FC-1
FC-2
Hor
Twist Ver
Mixer
LF
Amplifier
Fig. 5.3.3. Block diagram of the radar.

Fig. 5.3.4. The radar system (f =15.2 GHz; power radiated =85 mW; antenna beam width =6 0 in
both E and H planes; sensitivity of receiver =10 -15 W/Hz).
First, we analyze the radar echo signals collected by the radar operating with vertical polariza-
tion. The radar returns from a swinging metallic sphere (a calibration test object) and a walking person
were recorded during the radar data collection time of 63 s that corresponds to 200000 digitized signal
samples. This record of the mentioned total length was divided into N segments by a rectangular
shaped window with the width of 256 samples which was sliding along the signal. The dividing pro-
cedure was non-overlapping. As a result, maximum number of N = 780 short segments (N transient
partial records) can be registered.
The window width of 256 samples provides a frequency resolution of approximately 12.4 Hz.
The sequence of time-varying bispectrum estimates of 256x256 samples has been calculated in the
form of triple products of time-varying Fourier transforms (5.3.4) for each n-th selected signal seg-
ment. The peaks of bimagnitude B ( p,
q;
n)
have been analyzed and their values exceeding some thre-
shold level were selected for analysis. The bimagnitude values less than one tenth of maximum bi-
magnitude value have not been taken into account because these values correspond to such PCIF co-
herence coefficients that can be neglected.
Note that self-phase coupling is an artifact of the instantaneous bimagnitude estimates. It occurs
when 2p=q or 2q=p. Therefore, the corresponding bimagnitude slices located at (p, p) were eliminated
from further consideration. Finally, the values of the selected bimagnitude peaks were stored.
It should be stressed, that in the analysis of evolutional behavior of the PCIF, the 3-D time-
varying bispectrum estimates are difficult to visualize. Thus, we propose to map the phase coupling
177

frequency values (which correspond to the peak of bimagnitude in the bifrequency plane) onto the TF
plane as two separate values on the frequency axis in the coordinates p and q. The graphs of TF distri-
butions plane were plotted on the basis of stored time-varying bimagnitude data [117].
We first verified our approach with radar echoes generated by the multipoint scattering surface
of the oscillating metallic sphere. The metallic sphere was hung up with a thin line and oscillated as a
pendulum in the far field of the radar. The radar echo signal received was analyzed during the time
interval of 1200 ms (N=15) that approximately corresponded to one total pendulum oscillation period.
Fig. 5.3.5. PCIF TF distribution for an oscillating sphere.
The behavior of PCIF in the TF plane (PCIF TF distribution) is illustrated in Fig. 5.3.5.
The time-frequency analysis of the graph in Fig. 5.3.5 reveals the following interesting proper-
ties of the swinging sphere. First, there is the horizontal (parallel to the time axis) Doppler frequency
shift response. This component of the phase coupling IF distribution in TF domain corresponds to the
approximately constant Doppler frequency shift due to the presence of the zero-order term in the po-
lynomial expansion (5.3.3). Second, the quadratic law can be clearly seen for the PCIF behavior in the
TF domain of the graph.
It should be stressed that the phase coupling phenomenon differs from a walking human. The
phase coupling in the backscattered radar return obtained from an oscillating sphere can be explained
by the spatial phase distribution of microwave conductivity currents flowing on a spherical surface.
These currents are phase coupled, i.e. the spatial phase values of the microwave currents in different
local backscattering centers distributed in the spherical surface are coupled by a certain law. In the
course of time, the pendulum-like oscillations of the 3-D spherical shape cause simultaneous dis-
placements of the all local backscattering centers. It is experimentally confirmed that the 3-D micro-
wave current spatial distribution turns into the PCIF distribution in Fig. 5.3.5 owing to behavior of the
oscillations of the sphere.
The same results were obtained for the wavelengths of λ0 = 3 cm and 8 mm. Therefore, the PCIF
178

distribution in the TF domain obtained on basis of the time-varying bispectrum estimates of radar re-
turn signal can be considered as a promising information feature for object classification.
The other radar target in our experimental investigations was a walking human. Unlike a swing-
ing metallic sphere, a human body is a much more complicated backscattering radar object.
An example of the received signal (duration of recording is 960 ms) backscattered by a walking
human is displayed in Fig. 5.3.6.
Fig. 5.3.6. The radar echo sample recorded from a walking human: the signal magnitude as a
function of a sample number.
As compared to the signal in Fig. 5.3.1, more magnitude and chirp-like fluctuations are observed
in this arbitrary signal segment.
It is reasonable to suppose that the signal in Fig. 5.3.6 is a nonstationary multi component one.
This assumption has been verified by the histogram of the PCIF distribution computed for a walking
human and plotted below in Fig. 5.3.7. The presence of a large number of PCIFs in the signal
represented in Fig. 5.3.6 is confirmed by the histogram in Fig. 5.3.7.
Fig. 5.3.7. The histogram of the PCIF distribution
obtained for a walking human.
The results of extraction of the PCIFs from the time-varying bispectrum estimates of the signal
in Fig. 5.3.6 are illustrated by the PCIF distribution plotted in Fig. 5.3.8. A sliding window of the size
of 256 samples covered N=12 positions without overlapping and the threshold corresponding to one
tenth of maximum bimagnitude value have been used.
179

Fig. 5.3.8. PCIF distribution for a walking human (vertical polarization, λ0 = 2 cm).
The analysis of the PCIF distribution in Fig. 5.3.8 demonstrates the presence of several frequen-
cy peaks in the TF domain which confirms the above stated assumption on the multi component struc-
ture of the radar echo signal backscattered by a walking human. We would like to stress the presence
of the main- (the swinging human torso) and micro-Doppler (swinging arms and legs) components.
The swinging torso provokes the appearance of peaks in the low-frequency range. These peaks
can be seen for the frequencies of about 30 Hz. The peaks corresponding to the swinging arms and
legs are observed in the high frequency area from approximately 120 Hz to 240 Hz.
Polarimetric measurements allow obtaining additionally new promising information features
like PCIFs extracted from short-time cross-bispectrum estimates by using radar returns recorded for
the vertical (V) and horizontal (H) polarization for an object moving in a growing vegetation like trees
and bushes.
For this reason we pay attention in this subsection to analysis of the PCIF distributions obtained
by the polarimetric radar [116, 117]. The question is to know whether or not there is a difference be-
tween the radar Doppler TF signatures obtained with the vertical and horizontal polarizations and how
much they differ one from another? Most radar specialists without a doubt answer that for metallic
radar objects there is no difference (see, for example, [119]). Even if this answer is usually correct for
metallic objects, this is not always correct for the fields backscattered by a moving human body.
First, we illustrate the difference between the PCIF distributions obtained for V and H polariza-
tions. In fact, according to the radar echo signal model (5.3.1a, b), the last terms (electrical distances)
in the quadruple products in the sums are the same for both vertical and horizontal polarizations. The
V
distinction therefore must arise due to different behavior of the local time varying phases ϕ (i)
and
H
m
ϕ (i)
. The latter peculiarity can be explained by the difference in phase spatial distributions in a hu-
man skin for vertical and horizontal polarizations.
180
m

For experimental verification of our hypothesis, let us analyze the difference between the bi-
magnitude estimates obtained for the transient radar echo signals for V and H polarizations for a hu-
man walking towards the radar. These signals were truncated by a window of simple rectangular
shape with a duration of P = 256 samples and an arbitrary location. The frequency resolution for the
polarimetric radar was equal to 31 Hz.
The plots of the bimagnitude estimates calculated for vertical and horizontal polarizations are
shown in Figures 5.3.9 and 5.3.10, respectively. The difference between these bimagnitude estimates
is clearly seen because the number of bimagnitude peaks, their levels and PCIF values differ consider-
ably one from another for V (Fig. 5.3.9) and H (Fig. 5.3.10) polarizations. It confirms our assumption
about the difference between the spatial phase distributions in the human body for the vertical and ho-
rizontal polarizations.
Let us consider now experimental example that demonstrates the difference between the PCIF
distributions obtained with V and H polarizations. The PCIF distributions computed for a human
standing still but with swinging arms are represented in Figures 5.3.11 and 5.3.12 for V and H polari-
zations, respectively. As can be seen from these Figures, a human swinging torso causes the appear-
ance of the low-frequency peaks in TF distribution which are concentrated approximately near the
frequency of 60 Hz both for V and H polarizations. The swinging arms cause high-frequency peaks
evolutionary behaviors of which are different for vertical (Fig. 5.3.11) and horizontal (Fig. 5.3.12) po-
larizations. The TF distribution in Fig. 5.3.11 contains more high-frequency peaks in comparison to
the horizontal polarization in Fig. 5.3.12.
Fig. 5.3.9. Bimagnitude as a function of frequencies p and q obtained for a human walking to-
wards the radar (vertical polarization).
181

Fig. 5.3.10. Bimagnitude as a function of frequencies p and q obtained for a human walking towards the
radar (horizontal polarization).
Fig. 5.3.11. PCIF distribution for a human standing still but with swinging arms (vertical polarization).
Fig. 5.3.12. PCIF distribution for a human standing still but with swinging arms (horizontal polarization).
182

Thus, it has been demonstrated in [116, 117] that in opposite to the conventional energy-based
estimators which do not allow detection the difference between short time-varying Doppler spectra
obtained for the vertical and horizontal polarizations, the time-varying bispectrum estimates produce
an opportunity for recognizing the difference between PCIF distributions computed for the V and H
polarizations. The approach suggested can serve as a new classification feature for improving the de-
tection and recognition capability in radar ATR systems operating in vegetation clutter like trees and
bushes.
TF analysis of ground surveillance radar signals by using time-varying cross-bispectrum esti-
mates containing radar echoes for V and H polarizations has been proposed in our paper [116].
Let us define the short-time cross-bispectrum estimate as a triple product of the short-time
Fourier transforms of the transient signals y V (i,n) and y H (i,n) separated from the V polarization
(5.3.1a) and H polarization (5.3.1b) responses by sliding window that step by step shifts along the
process studied and takes n=1, 2,…, N non-overlapping locations. The short-time cross-bispectrum
estimates can be written as
*
B ( p, q; n) = Y ( p, n) Y ( q, n) Y ( p + q, n) = BVH ( p, q; n) exp[ jβ ( p, q; n)]
, (5.3.5)
VH V V H VH
where |BVH (p,q;n)| and βVH (p,q;n) are the time-varying cross-bimagnitude and cross-biphase, respec-
tively.
It should be stressed, that if there exists phase coupling between some two harmonics in back-
scattered signal (5.3.1a) for V and (5.3.1b) for H polarization, a peak in the cross-bimagnitude |BVH
(p,q;n)| must appear at the intersection of two corresponding samples p and q in the bifrequency plane.
Otherwise, for a radar Gaussian process backscattered by growing vegetations like trees and bushes,
the cross-bimagnitude tends to zero due to the fact that the corresponding backscattered field tends to
have Gaussian distribution.
Therefore, we have supposed that the cross-bispectrum estimates (5.3.5) would contain the
peaks caused by coupling certain IFs in non-Gaussian signal backscattered by a moving human. This
important peculiarity can serve for retrieving new detection and recognition features for a radar ob-
jects moving in growing vegetation clutter.
A coherent, homodyne, continuous-wave and polarimetric ground surveillance radar operating
at the millimeter wavelength of λ0=8.8 mm has been designed at the Institute of Radiophysics and
Electronics of the National Academy of Sciences (Kharkov, Ukraine) [118]. The radar parameters are:
the power radiated is equal to 15 mW; the antenna beam width is 6 0 in both E and H planes; the value
183

of receiver noise figure is of 20.2 dB; the level of side lobes is of - 24 dB; the cross polarization level
is less than -30 dB; the 16 bit two-channel ADC has been used. Transmitting-receiving horn antennas
were mounted at the height of 150 cm above the ground. Experimental investigations were carried out
during the summer period at the site of the Institute.
The radar returns obtained at V and H polarization from a walking human and vegetation like
trees and bushes at light breeze were recorded in PC memory during the data collection time of ap-
proximately one minute. The sequence of the backscattered signal samples has been truncated by a
rectangular window having the width of P=256 samples and sliding along the signal. As a result of
truncation, N short signal segments have been obtained. The set of N cross-bispectrum estimates
(5.3.5) of 256x256 samples each has been computed. The peaks of cross-bimagnitude B ( p,
q;
n)
VH
have been analyzed and their values were projected in time-frequency domain. The plots of time-PCIF
distributions measured for a human walking in vegetation clutter and vegetation clutter are shown in
Figures 5.3.13 and 5.3.14, respectively. Several pronounced cross-bimagnitude peaks are observed in
Fig. 5.3.13: the swinging torso provokes the peaks in the low-frequency range and the swinging arms
and legs replicas are observed in a high frequency range. Time-PCIF distribution measured for vegeta-
tion clutter represented in Fig. 5.3.14 essentially differs from a walking human: the cross-bimagnitude
peaks are observed only in low-frequency range and their values are sufficiently smaller in compari-
son to a walking human. It permits to divide and recognize a human walking in vegetation clutter by
the proposed time-PCIF signature.
Fig. 5.313. Time-PCIF distribution for a human
walking in vegetation clutter.
Fig. 5.3.14. Time-PCIF distribution for vegetation
clutter.
Experimental results demonstrated for the cross-bimagnitude estimates can find application in
ground radar surveillance automatic target recognition systems for security purposes.
184

Relative Amplitude (dB)
One of the most important problems usually arising in ground surveillance Doppler radars be-
fore object recognition is detection of a moving target observed for low input SNR. Because of this,
paramount interest consists in studying the detection performance and robustness to AWGN for the
bispectrum-based approach considered in papers [115 – 118]. This important feature has been investi-
gated experimentally in our paper [120]. Ground surveillance radar having the characteristics de-
scribed above served for experimental investigations of noisy backscattered signals recorded with a
human walking and running towards or away the radar in vegetation and precipitation.
The amplitudes of phase coupled Doppler harmonics extracted using bispectrum estimates cal-
culated in the form of (5.3.4) have been measured.
Relative information peak phase coupled amplitudes mapped onto the plane relative amplitude-
frequency are computed as
B( p, q)
Relative Amplitude ( dB)
= 20log
, (5.3.6)
B( p, q)
where |B(p,q)|max is the maximum bimagnitude value.
Sliding window width equal to 1024 samples has taken N = 24 locations with 50 percent over-
lapping. The results of these bispectrum-based measurements are illustrated by graphs plotted in Fig-
ures 5.3.15 – 5.3.19.
Fig. 5.3.15. Human walking away from the radar
(bispectrum estimation).
185
max
Fig. 5.3.16. Human walking away from the radar
(SPECLAB estimation).
The corresponding conventional amplitude Doppler spectra computed by using SPECLAB
software with FFT size of 32768 samples are demonstrated in Figures 5.3.16, 5.3.18, and 5.3.20 for
comparative study and analyzing the benefits of bispectrum-based approach with respect to Doppler
Fourier spectrum analysis. The total measurement time interval is equal to approximately 3s for analy-
sis of Doppler spectrum content.
Frequency (Hz)

Relative Amplitude (dB)
Fig. 5.3.17. Human walking towards the radar
(bispectrum estimation).
Relative Amplitude (dB)
Fig. 5.3.18. Human walking towards the radar
(SPECLAB estimation).
Fig. 5.3.19. Human running away from the ra- Fig. 5.3.20. Human running away from the radar
(bispectrum estimation).
dar (SPECLAB estimation).
Comparison of the graphs shown in Figs 5.3.15 and 5.3.16 permits to note that the differences
between information Doppler peak at 400 Hz (this Doppler frequency value has been a priori calcu-
lated) and interference levels estimated at the frequency of 2 KHz are equal to 68dB in Fig. 5.3.15 and
45 dB in Fig. 5.3.16, respectively. Therefore, the benefit obtained by the bispectrum-based approach
in signal-to-interference ratio is equal to 23 dB. Comparative analysis of results represented in Figures
5.3.17 and 5.3. 18 as well as in Figures 5.3.19 and 5.3.20 allows concluding that the benefits provided
by the bispectrum-based approach in SNR terms are equal approximately to 15 dB.
One can conclude that the improvement of signal-to-interference ratio is achieved comparing to
the conventional Doppler Fourier spectrum estimation due to the coherent accumulation of the bimag-
nitude peaks caused by phase coupling of Doppler frequencies and suppressing the non-coherent inter-
ference frequency contributions.
Frequency (Hz)
Frequency (Hz)
186

5.4. Parametrical short-time bispectral estimation for time-frequency analysis of multi-
frequency signals
A new approach to time-frequency analysis of multi component chirp-like signals by using the
extraction of PCIF pairs from short-time parametrical bispectral density estimation is presented in our
paper [121]. We describe in this subsection the suggested approach in detail.
Logical question can arise – why parametrical methods of bispectrum estimation are interesting
for radar applications considered in this Chapter? The answer is: because, first, they possess high
spectral resolution which is not dependent on the observed signal duration and, second, spectral side
lobes typical for conventional Fourier spectral analysis are absent in the estimates obtained by using
parametrical technique. Moreover, the most conventional approaches for the correlation functions and
power spectra estimation deal with the basic assumption of a process stationarity and ergodicity. To
obtain robust estimates the time averaging is usually used. The ensemble averaging procedure allows
eliminating random time variations of signal parameters to be estimated. However, in many practical
applications, the signals under investigations are the non-stationary processes having essentially time-
varying parameters. In this case, the ordinary time averaging procedure does not permit assess real
evolutionary changes of the estimated signal parameters. The typical example of such non-stationary
process considered in this Section is the sum of Doppler radar chirp-like signals backscattered by
ground moving objects.
To obtain a required high spectral resolution in ground surveillance which does not depend on
observation interval, we will use the parametric bispectral estimation approach [23].
First, the observed radar digital signal record must be divided into N short segments for obtain-
ing the sequence of N short quasi-stationary processes. The time window of the width of I samples
that step by step slides along the process analyzed and takes m = 1, 2,…, M positions with 50 % over-
lapping is used.
Autoregressive (AR) quasi-stationary signal model observed within an arbitrary n-th segment (n
( )
= 1,2, …,N) in the form of real-valued samples x ( i)
m
, i = 1,2,…,I can be written as the following dif-
ferential equations
x
( m)
n
( i)
n
p
= −∑
k = 1
p
+∑
k = 1
187
a
k n
x
( m)
n
( i − k)
+ n(
i)
, (5.4.1а)
( m)
( m)
or x ( i)
a x ( i − k)
= n(
i)
, (5.4.1b)
n
where n(i) is the stationary non-Gaussian i.i.d. excitation p-order process with Е{n(i)}=0, Е{ n 2 (i)} =
k n
n

σ 2 ; Е{n 3 (i)} = β ≠ 0 and Е{…} denotes expectation procedure. It is also assumed that the processes n(i)
( )
and x ( i)
m
n are independent.
(5.4.1) as
TAF estimate formed in an arbitrary n-th segment can be defined according to the AR (p) model
Rˆ
1
n ( l,
l)
= ( l,
l)
, (5.4.2)
M
188
M
( m)
∑ rˆ
n
m=
1
I
( m)
1 ( m)
( m)
2
where rˆ
n ( l,
l)
= ∑ xn
( i)
w(
i)[
xn
( i + l)
w(
i + l)]
, l = – p, –p+1, –p+2,…,0,…, p–1, p; w(i) is the
I i=
1
weighting window. The TAF estimate (5.4.2) can be represented in the form of the following Toeplitz
matrix
⎡Rˆ
( 0,0)
Rˆ
( 1,1)
.... Rˆ
( p,
p)
⎤
⎢
⎥
⎢Rˆ
( −1,
−1)
Rˆ
( 0,0)
... Rˆ
( p −1,
p −1)
⎥
⎢
⎥
= ⎢.
R ˆ
⎥
n
. (5.4.3)
⎢.
⎥
⎢
⎥
⎢.
⎥
⎢ ˆ
⎥
⎣R(
−p,
−p)
Rˆ
( −p
+ 1,
−p
+ 1)
... Rˆ
( 0,0)
⎦
To find the p-th order AR(p) model coefficients akn in (5.4.1), it is necessary to solve the corre-
sponding matrix equation with respect to the vector an
ˆ = bˆ
, (5.4.4)
R na
n
where an = [1, an1, an2, an3,…, anp] T is the matrix-vector of the coefficients computed for n-th segment in
the process under analysis (n =1,2,…,N); n bˆ = [b0, 0, 0,…,0] T ; b0 is the constant that is introduced in
the initial conditions and its value does not influence the Doppler PCIF value; T denotes the transposi-
tion procedure.
Parametrical bispectrum estimate ̇ ˆ
( q,
s)
derived for an arbitrary n-th segment can be repre-
B n
sented by using the triple product of the transfer function of an equivalent AR(p) model filter as
where
* j ( q, s)
n 0 n n n n
n
B
ˆ
( q, s) b H
ˆ
( q) H
ˆ
( s) H
ˆ
( s q) B
ˆ
( q, s) e γ
̇ = ̇ ̇ ̇ + = ̇ , (5.4.5)
ˆ
1
Ḣ
n ( q)
=
is the equivalent filter transfer function; s = 1,2,…,I and q =
p
2π
[ 1 + ani
exp( − j iq)
]
I
∑
i=
1
1,2,…,I are the frequency indices in bispectrum plane.
If there exists phase coupling between two frequencies (two frequency samples q and s) for an

arbitrary n-th segment, a peak in the bimagnitude estimate B
ˆ
( q,
s)
n
̇ (5.4.5) appears at the intersection
of the two corresponding samples q and s in the bifrequency plane. This peak indicates the presence of
phase coupling. Otherwise, in the corresponding power spectrum estimate, several peaks appear re-
gardless whether phase coupling exists or not.
Note that prior to obtaining the time-varying parametric bispectrum estimates (5.4.5), the mean
values in the input signal must be removed in order to eliminate a large amplitude spurious peak in the
bifrequency plane at its origin (p=q=0) that might mask information peaks corresponding to the
PCIFs.
Below we consider several examples demonstrating time-frequency distributions of the PCIFs
recovered from parametrical bispectral estimates by using AR models.
Example 1. We consider the following test multi-component process the model of which is giv-
en by the sum of six chirp signals having different modulation indices βk and two pairs of PCIFs as
x1(i)=cos[2π(f01 + β1 i)i+φ1]+cos[2π(f02+β2 i)i +φ2]+
+cos[2π(f03+β3 i)i+φ3]+cos[2π(f04+β4 i)i+φ4]+ (5.4.6)
+cos[2π(f05+β5 i)i+φ5]+cos[2π(f06+β6 i)i+φ6],
where βk = (fTk – f0k)/T, k = 1,2,…,6; f03 = f01 + f02 and fT3 = fT1 + fT2; f06 = f04 + f05 and fT6 = fT4
+ fT5; f01=1500 Hz, f02=875 Hz, f04=240 Hz, f05=2000 Hz; fT1= 2000 Hz , fT2= 1440 Hz , fT4= 800 Hz,
fT5= 2400 Hz; the initial phases φ1,.., φ6 are independent values uniformly distributed within the inter-
val of [0, 2π] and φ3= φ1 + φ2, φ6 = φ4 + φ5; T is the total observation time interval corresponding in
our example to 10 4 time samples. The sampling rate in the examples considered is equal to 8000 Hz.
Example 2: The second example is the multi-component process containing the following
weighted sum of five chirp signals having different modulation indices and two PCIFs
x2(i)=0,5cos[2π(f01+β1 i)i+φ1]+0,15cos[2π(f02+β2 i)i+φ2]+
+ 0,15 cos[2π(f03+β3 i)i+φ3]+0,1 cos[2π(f04 +β4 i)i+φ4]+ (5.4.7)
+ 0,1 cos[2π(f05 + β5 i)i + φ5],
where the relationships between the separate PCIFs are defined as (f05 + β5 i) = (f01 + β1 i) + (f03 +
β3 i); βk = (fTk – f0k)/T, k = 1,2,…,5; f01=233 Hz, f02=23 Hz, f03=854 Hz, f04=171 Hz, f05=1087 Hz; fT1=
750 Hz , fT2= 54 Hz, fT3= 1118 Hz fT4= 298 Hz, fT5= 1868 Hz; the initial phases φ1,.., φ5 are indepen-
dent values uniformly distributed within the interval [0, 2π] and φ5= φ1 + φ3.
Example 3: The third example is the radar signal backscattered by a walking human and re-
corded by the coherent homodyne continuous-wave radar operating in K-band [117]. This radar signal
was registered for a human walking towards the Doppler ground surveillance radar.
189

Fig. 5.4.1. Example1. TF PCIF distribution computed for the data:
AR order p=20; I=64; M=10; N=10; Dirichlet window; normalized bimagnitude threshold is equal
to 0.3.
Fig. 5.4.2. Example 2. TF PCIF distribution computed for the data:
AR order p=20; I=64; M=10; N=10; Hamming window; normalized bimagnitude threshold is equal
to 0.5.
190

Fig. 5.4.3. Example 3. PCIF TF distribution computed for the data:
AR order p=50; I=64; M=10; N=20; Blackman window; normalized bimagnitude threshold is equal
to 0.3
Three TF distributions demonstrated in Figures 5.4.1–5.4.3 are plotted by mapping the pairs of
the PCIFs onto the time-frequency plane. The locations of the normalized bimagnitude peaks corres-
pond to the PCIF intersections.
It is clearly seen from the TF distributions in Figures 5.4.1 and 5.4.2 that the behavior of f01, f02,
and f04, f05 and f01 and f03 PCIF’s pairs inclines evolutionary change according to the laws given in the
multi component signals (5.4.6) and (5.4.7), respectively. At the same time, the frequencies that do not
have phase relationships (see the frequencies f02 and f04 in Fig. 5.4.2) do not provoke the peaks in the
TF distribution.
The radar time-frequency signature represented in Fig. 5.4.3 demonstrates the TF PCIF distribu-
tion measured for a human walking towards the radar. The analysis of this signature shows the pres-
ence of several PCIF peaks in low, middle and high frequencies corresponding to the above-
mentioned assumption about phase coupled Doppler frequencies provoked by human swinging torso,
arms and legs.
It should be stressed, that good frequency resolution is achieved in low frequency sub-band that
can give a promising possibility to extract information feature in case of small radial velocities of the
objects to be recognized by ground surveillance radars.
191

Thus, the approach based on parametrical bispectral estimation can serve as the promising tool
for obtaining additional information features for object identification and classification in ground sur-
veillance systems for security applications.
192

6. Conclusions
Properties and benefits of cumulant and moment functions, i.e. higher-order correlation func-
tions and spectra, were considered from the particular point of view of real-life digital signal
processing in AWGN and non-Gaussian noise environment. The main attention in the work has been
paid to non-parametric techniques for signal recovery by using triple correlation function and bispec-
trum estimates, although a parametrical estimation technique has been also studied in application to
radar signal processing.
Statistical features of noisy bispectrum estimates have been analyzed and investigated by com-
puter simulations performed for different types of signals embedded in AWGN. It was demonstrated
that the noise leaked in bispectrum estimate possesses non-Gaussian behavior both in real and imagi-
nary parts in bispectrum estimates. On the basis of these statistical investigations, novel bispectrum-
filtering techniques have been developed and studied. Suggested bispectrum-filtering techniques are
based on linear and non-linear, as well as non-adaptive and adaptive filtering the bimagnitude and bi-
phase, as well as filtering real and imaginary parts of bispectrum estimates. Novel modifications of
adaptive DCT based filters have been designed and shown to perform well for a wide range of input
SNR and signal waveforms. The benefits of the combined bispectrum-filtering approach are demon-
strated by numerous examples of computer simulation. Recommendations on selection the appropriate
filter type and parameters have been provided.
Instead of traditional calculation of phase bispectrum usually provoking phase wrapping, a new
approach without addressing to direct standard phase calculation is proposed. The suggested recursive
signal waveform reconstruction procedure is represented in the form of computation the quadrature
components of complex-valued normalized bispectrum and signal Fourier spectrum. It provides an
essential reduction of distortions caused by phase wrapping in reconstructed signal waveforms.
Novel bispectrum-based techniques for reconstruction jittered and noisy digital images are pro-
posed by using additive and multiplicative pre-distortion functions inserted in image rows. The optim-
al pre-distortion function parameters have been defined for several standard digital images.
The problem of detection deterministic signals embedded in AWGN by using new proposed sta-
tistic like third-order autocorrelation function formed at the matched filter output has been investi-
gated. By computer simulations, the merits of the proposed test statistic comparing to second-order
statistic formed as conventional correlation integral have been demonstrated. In particular, it resulted
in higher signal detection probability.
193

A new noise immunity encoding/decoding technique operating in noisy radio telecommunica-
tion channel is suggested and studied by statistical computer simulations. This approach is based on
inserting new code words constructed by quantized samples of triple correlation for two a priori in-
serted subsidiary digital sequences. Performance of the proposed encoding/decoding technique is stu-
died by computing the errorless decoding probability as a function of binary signal amplitude in radio
communication channel corrupted by AWGN as well as mixture of AWGN and impulsive noise. Bet-
ter performance of the suggested approach compared to the linear repetition code for low SNR value
in communication channel has been demonstrated.
The benefits of the proposed bispectrum-based approach were demonstrated by experimental
study of radar target detection, classification and identification for naval, aerial and ground moving
objects. Experimental results represented in this work demonstrate sea clutter suppression and good
naval object range resolution provided by polarimetric X-band radar. Reduction of aspect angle de-
pendent speckle distortions in aerial high resolution range profiles were illustrated by computer simu-
lation results performed for several aerial targets of different types, sizes and backscattering surface
irregularities like bomber aircraft, cruise missile and fighter. Essential reduction of speckle distortions
in high resolution range profiles obtained by the technique proposed was shown.
Experimental results of time-frequency analysis of backscattered signals recorded by ground
surveillance Doppler radar were represented and discussed. The new approach proposed permitted to
extract the phase coupled instantaneous frequency pairs from non-stationary and multi component
chirp-like radar signals. This makes it possible to obtain new information features for better target
recognition and classification. The approach suggested can serve for improving the detection and rec-
ognition performances in radar ATR systems operating in vegetation clutter.
194

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