Optimal kernels for time-frequency analysis - Rice University Digital ...

Optimal kernels for time-frequency analysis
Richard G. Baraniuk Douglas L. Jones
Electrical and Computer Engineering
252 Engineering Research Laboratory
University of Illinois
Coordinated Science Laboratory
University of illinois
1 101 W. Springfield Ave.
Urbana, IL 61801 Urbana, IL 61801
(217) 333-0766 (217) 244-6823
ABSTRACT
Current bilinear time-frequency representations apply a fixed kernel to smooth the Wigner distribution. However, the
choice of a fixed kernel limits the class ofsignals that can be analyzed effectively. This paper presents optimality criteria
for the design of signal-dependeni kernels that suppress cross-components while passing as much auto-component
energy as possible, irrespective of the form of the signal. A fast algorithm for the optimal kernel solution makes the
procedure competitive computationaily with fixed kernel methods. Examples demonstrate the superior performance
of the optimal kernel for a frequency modulated signal.
1. INTRODUCTION
Time-Frequency Distributions (TFD's), which indicate the energy content of a signal as a function of both time and
frequency, are a powerful tool for time-varying signal analysis. The Wigner Distribution (WD)
W(t,w) = t: s(t + )s*(t _ )e_Tdr
is of great interest due to a number of attractive properties' . However, it also has spurious cross-components and
high noise sensitivity, both of which obscure the true signal features. Therefore, the WD is often convolved with a
two-dimensional smoothing function that suppresses cross-components at the expense of signal energy concentration.
It is well known that all bilinear TFD's can be represented as smoothed versions of the WD2; that is, if P(t,w) is a
bilinear TFD, then
P(t,w) = W(t,w) * *(t,w) (2)
for some function (" ** " denotes two-dimensional convolution). Equation (2) may be rewritten usingthe twodimensional
inverse Fourier transform as
C(O, r) = A(6, r)'I(9, r), (3)
where C(9, r), the inverse Fourier transform of P(t, w), is known as the characteristic function of the distribution;
A(O, r), the transform of the WD, is called the Ambiguity Function (AF); and (9, r), the transform of the smoothing
function is known as the kernel of the TFD. The AF is also given directly by
A(O, r) =i: s(t + )s*(t _ JOtft (4)
Equation (3) indicates that we can interpret any bilinear TFD as the two-dimensional Fourier transform of a weighted
version of the AF.
The kernel is frequently chosen to weight the AF such that the auto-components of the distribution are passed while
the cross-components and noise are suppressed. In principle, this is possible when the auto-components and crosscomponents
do not overlap. Many kernels have been proposed, but selection of a fixed kernel limits the class of signals
for which the representation will perform well. That is, for any fixed kernel, it is always possible to find signals for
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(1)

which the TFD exhibits poor auto-component concentration or little cross-component suppression. (The same problem
limits the performance of wavelet time-frequency analysis3, since the choice of a fixed analyzing wavelet restricts the
class of signals which can be analyzed effectively.)
The limitations of fixed kernel time-frequency analysis can be illustrated by analyzing a simple signal with several
different TFD's. The WD of the sum of two chirp signals of large effective time envelope is shown in Fig. 1. Although
the auto-components are highly concentrated, there is a large cross-component. The Choi-Williams distribution4,
which has an exponential kernel
cw(O,T)=e (5)
works well for signals whose components have distributions nearly parallel with the time or frequency axis. It performs
poorly, however, for signals with substantial frequency modulation, as seen in Fig. 2, since the kernel severely truncates
the auto-components ofsuch signals. The kernel generating the spectrogram is related to the AF of the analysis window
w(t) by
S(9,T) = i: w(t + )w*(t )ei0tdt.
(6)
Results are excellent for signal components that resemble the window5, but all mismatched components are distorted.
Figure 3 displays a spectrogram computed using a Hamming window of length similar to the effective time width of
the signal. It is poorly concentrated and obscures the true nature of the signal. If the analysis window, and hence
the kernel, is matched to one of the signal components, the matched-filter spectrogram results. See Fig. 4. While
the matched-filter technique can yield excellent results, it works for only one type of signal component and requires a
priori knowledge of the form of the component.
Since the best kernel function depends on the signal to be analyzed, we expect to obtain good performance for a broad
class of signals only by using a signal-dependent kernel. Signal-dependent kernels are proposed by several authors.
The adaptive spectrogram representation for speech signals developed by Glinski6 adapts the window based on a
segmentation (provided by the user) of the signal into pitch periods. Jones and Boashash7 adapt the modulation
rate of a fixed window to match an estimate of the signal's instantaneous frequency. Optimal smoothing kernels are
considered by Andrieux ei al.8 , but only for simple signals of the form s(t) = e2v(t) , and only for the restrictive class
of Gaussian kernels. Kadambe, Boudreaux-Bartels and Duvaut9 utilize an adaptive filtering technique coupled with
AR modeling and clustering to design kernels. Nuttall'° designs a kernel composed of Gaussian components based
on information that the user provides after viewing the WD. Jones and Parks'1 develop a technique using Gaussian
kernels which vary with time and frequency to maximize a local measure of signal-energy concentration.
Each of the methods described above either is ad hoc, excessively restricts the class of allowable kernels, is computationally
expensive or requires human intervention. We propose a new procedure for selecting a signal-dependent
kernel. Given a signal, the method automatically designs a kernel that is optimal with respect to a set of performance
criteria. Since the class of kernels that we consider is large, good performance is expected for a wide range of signals.
The procedure also has a computational complexity that is comparable to fixed-kernel techniques.
2. OPTIMAL KERNEL DESIGN
Rather than choosing an ad hoc method for signal-dependent kernel selection, it seems appropriate to formulate the
procedure as an optimization problem. The problem formulation requires a class of kernels from which the optimal
kernel is chosen, and a performance index that measures the quality of the time-frequency representation with respect
to criteria deemed important by the designer. The kernel that maximizes the value of the performance measure is
selected as the optimal kernel for the signal.
The class of kernels must be large enough to allow for good performance for all signals of interest in a given application.
Likewise, the performance measure must be chosen to yield a tractable optimization problem that can be
solved efficiently. An example of a useful performance index is a measure of the signal-energy concentration of the
distribution". Clearly, the choice of kernel class and performance measure is crucial to the success of the method.
However, once a satisfactory class and measure are found, kernel design for a wide range of signals is reduced to solving
an optimization problem.
The optimal design concept can be generalized to classes of TFD's other than the bilinear by defining a subclass of
182 / SPIE Vol. 1348 Advanced Signal-Processing Algorithms, Architectures, and Implementations (1990)
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allowable TFD's and a performance index. The formulation of optimization problems for TFD design is relatively
simple in the bilinear case, because a bilinear TFD is completely specified by its two-dimensional smoothing kernel.
Thus, we can find the optimal bilinear TFD for a signal simply by solving for the optimal kernel.
2.1 Continuous optimization formulation
This section develops an optimization problem formulation for kernel design that relies on the AF of the signal and
the characteristic function representation of a TFD indicated by (3). We propose optimality criteria based on the AF
for three reasons. First, the multiplicative operation of the kernel on the AF is easier to visualize than convolution of
the WD with the smoothing function, which simplifies the construction of a quality measure. Second, the AF serves
to separate the auto and cross-components5 . Third, the AF may lead to efficient computation of the optimal TFD,
since the TFD is merely the two-dimensional Fourier transform of the product of the optimal kernel and the AF.
We consider an optimal kernel in the continuous case to be one that satisfies the following optimization problem:
p2 p00
max / / (7)
Jo Jo
subject to (O, 0) 1 and
I(ri,/) I(r2,'i/.') V r1 < r2 ,V b, (8)
p2r p00
and subject to I I I'I(r,cb)I2rdrdb < a
Jo Jo
(9)
where A(r, ) is the AF of the signal in polar coordinates, and the kernel (r, b) is assumed to be real and positive.
The performance measure (7) expresses our desire to pass as much auto—component energy as possible into the TFD
for a kernel of fixed volume a, . The second constraint (8) forces the kernel to be radially nonincreasing. Since the
AF auto—components are centered at the origin, this encourages the kernel to preferentially pass auto-components.
The final constraint (9) restricts the size of the kernel so that cross-components are suppressed. An advantage of this
formulation is that the constraints are insensitive to both the time-scale and orientation of the signal in time-frequency.
2.2 Discrete optimization formulation
In practice, TFD's are computed at discrete time and frequency locations, so we reformulate the optimization problem
by discretizing equations (7)—(9). With suitably dense sampling, the discrete formulation converges to the continuous
formulation. Performing the discretization, we define an optimal discrete kernel to be one that satisfies:
fAd(rn, n)'1d(m, n)12, (10)
nx ;:
subject to d(O,O) = 1 and
Jd(m, n) is radially nonincreasing, (11)
and subject to :i Id(m, n)f2 ad (12)
where Ad(m, n) is the N x N discrete AF of the signal to be analyzed, and the kernel d(rn, n) is assumed to bereal
and positive. Note that since the AF is conjugate symmetric through the origin
the optimal kernel can be computed from a half-plane of AF samples.
Ad(rn,fl) = A(—m,_n), (13)
The constraint that the kernel be radially nonincreasing can be implemented exactly only on a polar grid of samples.
However, computing the AF and resulting TFD on a polar grid requires either the computation of a polar Fourier
transform, for which no fast algorithm exists, or a costly interpolation from a rectangular grid. Therefore, we approximate
the polar grid by a set of paths on a rectangular grid. Figure 5 illustrates a tree structure that approximates the
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adial dependencies of the kernel for the upper half-plane of a 64x64 rectangular grid. The nonincreasing constraint
is enforced along each path from the origin to the edge. The branches of the tree are constructed to minimize the
maximum deviation from the branch to the true radial line.
3. OPTIMAL KERNEL SOLUTION
Since the performance measure and constraints are linear in IdI2, the optimal kernel may be found by applying linear
programming'2 to solve for the N2 unknowns 1'd(m, n) (since 1d is assumed to be real and positive, knowing NdI2is
equivalent to knowing d). Moreover, it can be shown that the optimal kernel takes on essentially only the values of
one and zero.
The optimal TFD can thus be determined as follows. First, the discrete AF of the signal to be analyzed is computed.
Next, the linear program (1O)—(12) is solved for the optimal kernel, which is then multiplied by the AF. The twodimensional
Fourier transform of the product is the optimal TFD.
3.1 Fast Algorithm for Solution
A solution for the optimal kernel using standard linear programming methods may be simple, but it is also computationally
expensive. Use of the simplex algorithm would cause the optimal kernel computation to dominate the total
cost of computing the optimal TFD. However, we have found an extremely efficient inductive procedure that computes
the optimal kernel with O[N2] operations to find the optimal kernel of size a. Since this number is small in comparison
to the O[N2 log N] computations required to find the AF or WD, time-frequency analysis with signal-dependent
kernels is competitive computationally with traditional fixed kernel methods.
3.2 Implementation Issues
Although the one-zero kernel is optimal according to the constraints stated in section 2.2, its sharp cutoff may introduce
ringing in the optimal TFD. Thus, some form ofsmoothing may be desired. One simple approach, used in the examples,
tapers the kernel.
Adjustment of the parameter cd controls the tradeoff between cross-component suppression and smearing of the
auto-components. A lower bound on reasonable values for a can be derived from uncertainty principle arguments.
4. EXAMPLES
In order to compare the results of the optimal kernel design procedure with other TFD's, the optimal kernel was
computed for the same signal discussed in the introduction. The AF and kernel were of size 64 x 64, the parameter
ad was set to 30 and tapering was applied. The AF, optimal kernel and resulting TFD are shown in Figs. 6,7 and 8.
The cross-component visible in all of the other TFD's except the matched-filter spectrogram (see Fig. 4) is virtually
eliminated, yet the distribution is still quite concentrated — much more so than the matched-filter spectrogram.
Figure 9 illustrates the WD of the same signal corrupted by additive white Gaussian noise. The SNR of the resulting
signal is 0dB. The optimal kernel was computed using the same parameters as above. The cross-component and noise
suppression of the optimal TFD, shown in Fig. 10, are excellent, indicating that the kernel design procedure is robust
in the presence of significant additive noise.
5. CONCLUSION
An optimization procedure has been presented for the automatic determination of signal-dependent smoothing kernels
for time-frequency analysis. Due to the signal-dependent nature of the kernel, the quality of the resulting timefrequency
representation is insensitive to the time-scale and orientation of the signal. A fast algorithm for the optimal
kernel solution makes the method competitive computationally with traditional fixed kernel methods. The procedure
appears to yield excellent results for a much larger class of signals than any fixed kernel representation. The technique
performs well even in the presence of substantial additive noise.
184 / SPIE Vol. 1348 Advanced Signal-Processing Algorithms, Architectures, and Implementations (1990)
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6. ACKNOWLEDGMENTS
This work was supported by the Music Group of the Computer-based Education Research Laboratory and the Joint
Services Electronics Program, Grant No. N00014-90-J--1270.
7. REFERENCES
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3. P. Flandrin, 0. Rioul, "Affine Smoothing of the Wigner-Ville Distribution," IEEE ICASSP-1990, pp. 2455—2458,
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4. 11.-I. Choi and W.J. Williams, "Improved Time-Frequency Representation of Multicomponent Signals Using Exponential
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5. P.Flandrin, "Some Features of Time-Frequency Representations of Multicomponent Signals," IEEE IGASSP-1 984,
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6. S.C. Glinski, "Diphone Speech Synthesis Based on a Pitch-Adaptive Short-Time Fourier Transform," Ph.D Thesis,
University of Illinois, Urbana, 1981.
7. G. Jones and B. Boashash, "Instantaneous Frequency, Instantaneous Bandwidth and the Analysis of Multicomponent
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8. J.C. Andrieux, M.R. Felix, G. Mourgues, P. Bertrand, B. Izrar and V.T. Nguyen, "Optimum Smoothing of the
Wigner—Ville Distribution," IEEE Transactions on Acoustics, Speech, and Signal Processing ASSP—35(6), pp.
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9. 5. Kadambe, G.F. Boudreaux-Bartels and P. Duvaut, "Window Length Selection for Smoothing the Wigner Distribution
by Applying an Adaptive Filter Technique," IEEE ICASSP-1989, pp. 2226—2229, 1989.
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in Noise," NUSC Technical Report 8225, February 16, 1988.
11. D.L. Jones and T.W. Parks, "A High Resolution Data-Adaptive Time-Frequency Representation," IEEE ICASSP-
87, Dallas TX, pp. 681—684, April 1987.
12. D.G. Luenberger, Introduction to Linear and Nonlinear Programming, Addison-Wesley Co., Reading, MA, 1973.
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Fig.1. Magnitude of the Wigner distribution.
Fig. 3. Magnitude of the spectrogram computed
with a Hamming window of duration equal to
the effective time width of the signal.
Fig. 2. Magnitude of the Choi-Williams disthbution
computed with smoothing parameter a =20.
Fig. 4. Magnitude of the matched-filter spectrogram.
Fig. 5. Approximation of a radial dependency graph'on a rectangular grid.
Architectures, and Implementations (1990)
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Fig. 6. Magnitude of the ambiguity function. Fig. 7. The tapered optimal kernel, a