Adjustable Fractional Delay FIR Filters Design Using Multirate and ...

1
Adjustable Fractional Delay FIR Filters Design
Using Multirate and Frequency Optimization
Techniques
G. Ramirez-Conejo, J. Diaz-Carmona, Member, IEEE, J. Delgado-Frias, Senior, IEEE,
G. Jovanovic-Dolecek, Fellow, IEEE and J. Prado-Olivarez
Abstract— One of the most efficient implementation for
adjustable fractional delay FIR filter is the Farrow structure,
allowing on line fractional delay value update with a fixed
branch filters set. A wideband fractional delay FIR filter
requires high number of branch filters and high branch filters
length, which results in a complex arithmetic implementation.
This paper describes a multirate approach in order to reduce
modified Farrow structure complexity for wideband fractional
delay FIR filters. The structure coefficients are computed
through a global minimax frequency optimization process with a
proposed initial solution. According to the obtained results the
use of this initial proposed solution allows fractional delay filters
design meeting severe specifications errors with on line value
update and a reduction of multipliers up to 48% compared with
design methods results recently reported.
Index Terms—Digital filters, FIR filters, Fractional sample
delay, Multirate.
F
I. INTRODUCTION
RACTIONAL delay (FD) filters are a very important class
of digital filters. FD filter frequency response is
characterized of having a unit magnitude response and a
specified fixed fractional delay response. Hence FD filter
coefficients are obtained meeting both specifications through
a widely type of reported design methods.
In many digital signal processing applications a delay value
equal to a fraction of the sampling period is required. In most
of them on line update of the adjustable fractional value is
needed. Such applications include sample rate conversion in
software-defined radio applications [1], echo cancellation,
phased array antenna systems, and modeling of musical
instruments [2].
There are several FD design methods [2], among them the
use of a polynomial approach allows online desired fractional
delay value update using a Farrow structure [3], or a modified
This work was supported by the Mexican National Counsil of Science and
Technology (CONACyT) under Scholarship Grant number 214724.
G. Ramirez-Conejo, J. Diaz-Carmona and J. Prado-Olivarez are with
Department of Electronics Engineering, Technological Institute of Celaya,
Celaya, GTO, MEXICO (e-mail: jdiaz@itc.mx ).
J. Delgado-Frias is with the School of Electrical Engineering, Washington
State Univeristy, Pullman, WA, USA (e-mail: jdelgado@eecs.wsu.edu).
G. Jovanovic-Dolecek is with the Electronics Department, National
Institute of Physics Optics and Electronics, Puebla, PUE, MEXICO (e-mail:
gordana@inaoep.mx ).
Farrow structure [4]. Both structures are composed of L+1
parallel FIR filters C l (z), each one with length N, where L is
the chosen polynomial order, as it is shown in Fig. 1. In a
modified Farrow structure = 2-1, where is the required
fractional delay value, 0

e
l
N
/ 2
1
l
N
, (2)
l
l
n0 2
2 l!
C 1
n l,
n,
where 0 l L and:
2
LMS optimization. In the same way the combination of this
multirate structure and a frequency optimization technique
were used in subsequent proposals [19]-[20]. In [19] a two
stage FDF jointly optimized technique is applied. In [20] a
complexity reduction is achieved by using an approximately
linear-phase IIR filter instead of a linear-phase FIR filter in
the interpolation process.
This paper describes the use of the multirate structure in a
frequency design approach in order to reduce modified
Farrow structure complexity for wideband FD FIR filters. A
global minimax frequency optimization is used for structure
coefficients computing, where a coefficients set is proposed as
initial solution. This initial coefficient set is obtained as an
individual solution for the branch filters approximations in a
least mean squares sense. According to the obtained results
the use of the proposed frequency optimization approach
allows a fractional delay structure implementation meeting
small passband magnitude and phase delay errors with on line
fractional delay value update requiring small number of
arithmetic operations.
In next section the individual and global frequency error
functions, used in the optimization procedure, are defined for
the modified Farrow structure. The multirate structure is
briefly described in third section. In section four the proposed
design method is presented, which is illustrated through two
design examples in section fifth. Finally conclusions are
presented in last section.
II. FREQUENCY ERROR FUNCTIONS
The frequency design method in [7], is based on two facts:
a) the FIR branch filters C l (z) in the modified Farrow structure
are linear phase, b) it is possible to approximate the input
signal through Taylor series in a modified Farrow structure.
The l order differential approximation to the continuous-time
interpolated input signal is done through the branch filter
C l (z), with a frequency response given as:
C
l
e
j
Fig 1. Farrow structure.
e
N 1
j
2
j
l
2 l!
l
. (1)
The individual error function for each branch filter
approximation is given as:
2 cosn
1/
2
,
l even,
l,
n,
(3)
2 sinn
1/
2
,
l odd.
The global structure errors are defined as:
Complex error:
H
, 0 0 1
e . (4)
c
H id
Magnitude error:
m
H
1,
0 0 1
e . (5)
Phase delay error:
e
p
( D ), 0
p
0 1, (6)
where H() and () are the frequency and phase responses,
respectively, of the designed FD filter. H id () is the frequency
response of the ideal FD filter, p is the passband frequency,
and D is the transport delay.
III. MULTIRATE STRUCTURE
The multirate structure in [15], is proposed for designing
FD filters in time domain. The input signal bandwidth is
reduced by increasing the sampling frequency. In this way
Lagrange interpolation is used in filter coefficients computing
for a FD filter with a wide bandwidth.
This multirate structure, shown in Fig. 2, is composed of
three stages. The first one is an upsampler and a half-band
image suppressor filter H HB (z) for incrementing twice the
input sampling frequency. Second stage is the FD filter
H FD (z), which is designed in time domain through Lagrange
interpolation [4]. Since the signal processing frequency of
filter H FD (z) is twice the input sampling frequency, such filter
can be designed to meet only half of the required bandwidth.
Last stage deals with a downsampler for decreasing the
sampling frequency to its original value.
Such multirate structure can be implemented as the singlesampling-frequency
structure shown in Fig. 3, where filters
H 0 (z) and H 1 (z) are the first and second polyphase components
of the half-band filter H(z), respectively. In the same H FD1 (z)
and H FD0 (z) are the polyphase components of FD filter H FD (z).
The design of the FD filter as a modified Farrow structure
in the multirate structure and using some structure reductions
results in the final FD structure shown in Fig. 4 [17],[18],with
= 4-1, where is the required fractional delay value, and
the filters C l,1 (z) and C l,0 (z) are the first and second polyphase
components of branch filter C l (z), respectively.
p
p

3
X(z)
2
2 H(z) H FD
(z) 2
Fig 2. Multirate structure.
Y(z)
X(z)
H 0
(z)
H 1
(z)
C L,1
(z) C L-1,1
(z) C 1,1
(z) C 0,1
(z)
C L,0
(z) C L-1,0
(z) C 1,0
(z) C 0,0
(z)
Y(z)
2
Fig 4. Resulting FD structure.
X(z) H 0 () z H FD1 () z Yz ()
IV. PROPOSED DESIGN METHOD
The coefficients computing of the resulting FD structure,
shown in Fig. 4, is done through frequency optimization for
the global structure magnitude approximation to the ideal
frequency response in a minimax sense. The objective
function is defined as:
max max
, (7)
0 1,
0
p
m e m
where e m () is the global magnitude error. This objective
function is minimized until the magnitude error specification
m is met. Since approximation must meet both magnitude and
phase errors, then global phase delay error is constrained to
meet next phase delay restriction:
max max , (8)
0
10
p
p e p p
where e p () is the global phase error and p is the phase delay
error specification.
As is well known the initial solution is a very important
factor for a minimax optimization process, [11], the proposed
initial coefficients set is the least mean squares solution of the
individual branch filters approximations to differentiators,
hence the approximation error of the l th branch filter is given
as:
E
l
H 1 () z
p
2
0
e
d
l
H FD0 () z
2
Fig 3. Resulting structure for the FD filter.
2
, (9)
where the upper frequency limit is half the desired FD
bandwidth, this is due to the increment of the sampling
frequency in upsampler stage.
The half-band filter H HB (z) can be designed as a Doph-
Chebyshev window or as an equirriple filter. The final halfband
filter coefficients are obtained from the optimization
process.
V. DESIGN EXAMPLES
Two design examples are described where the proposed
FD filter design results are compared with already reported
ones. The minimax optimization process was performed
through the function fminimax available in the MATLAB
Optimization Toolbox.
Example 1: In this example the specifications are
p m = 0.01 and p =0.001, same design example as
[12]. The given criteria is met with N = 7 and L = 4 and a halfband
filter length of 55. The overall structure requires M = 32
multipliers, A = 47 adders, resulting in a m = 0.0094448 and
p = 0.00096649. The obtained magnitude and phase delay
responses for = 0 to 0.5 with 0.1 delay increment are
depicted in Fig. 5. In Table I are shown the obtained results
with the proposed method and thus reported by other design
methods. Our design requires less multipliers and adders than
[4], [11], the same number of multipliers and nine less adders
Method
TABLE I.
EXAMPLE 1 RESULTS COMPARISON
Arithmetic complexity
N L M A m p
Vesma 97 [4] 26 4 69 91 0.006571 0.0006571
Johansson 03
[11]
28 5 57 72 0.005608 0.0005608
Yli 06 [12] 28 4 32 56 0.009069 0.0009069
Yli 06 [13] 28 4 31 50 0.009742 0.0009742
Yli 07 [14] 28 4 30 - 0.009501 0.0009501
Proposed 7 4 32 47 0.0094448 0.0009664
Magnitude Response
Phase Delay Response
1.01
1
- Not reported
0.99
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Angular Frequency /
a)
14.5
14.4
14.3
14.2
14.1
14
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Angular Frequency /
b)
Fig 5. FD Filter frequency responses for =0.0 to 0.5 in Example 1.
a) Magnitude response, b) Phase delay response.

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than [12], one more multiplier and three less adders than [13],
and two more multipliers than [14].
Example 2: This example shows that the purpose
optimization method can be extended for minimax
approximation of a global complex error. For this proupose
the filter design example described in [11] is used, which
specifications are p and maximum global complex
error of c = 0.0042. Such specifications are met with N = 7
and L = 4 and a half-band filter length of 69. The overall
structure requires M = 35 multipliers with a resulting
maximum complex error c = 0.0036195. The fractional delay
range is between 17.5 and 18.5. In Table II the obtained
results are compared with the reported in existing methods.
The proposed method requires less multipliers than [11], [16]
and case A of [19]. Reported multipliers of [20] and case B of
[19] are less that the obtained with the proposed design
method. It should be pointed out that in [20] an IIR half-band
filter is used and in case B of [19] and [20] a switching
technique between two multirate structures must be
implemented. The complex error magnitude and magnitude
response of the proposed design for = -0.5 to 0 with 0.1
delay increment are shown in Fig. 6.
VI. CONCLUSIONS
The use of a frequency design method for wideband
fractional delay FIR filters using a multirate Farrow structure is
described, where resulting filter coefficients are obtained with
a global structure approximation to the ideal frequency
response of a fractional delay filter through minimax
optimization. The proposed design method is based on starting
the optimization process from an initial structure coefficients
set, which is obtained as a least mean squares solution of
individual branch filters approximations to ideal differentiator
responses. As the presented examples demostrate, the
proposed method can be used in the design of wideband
fractional delay FIR filters meeting magnitude and phase
specifications with online fractional delay value capability, and
TABLE II.
Method
EXAMPLE 2 RESULTS COMPARISON
Arithmetic complexity
N L M
Johannson 03 [11] 39 6 73
Johannson 03 a [11] 31 5 50
Hermanowicz 04 [16] 11 6 60(54)
Hermanowicz 05 [19] 7 5 36
Hermanowicz 05 b [19] 7 3 26
Johannson 06 [20] - 6 32
Johannson 06 b [20] - 3 22
Proposed 7 4 35
a. Minimax design with subfilters jointly optimized.
b. Switching structures. - Not reported
Magnitude Response
4 10-3 3
2
1
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x
Normalized Angular Frequency /
a)
1.005
Complex Error Magnitude
1
0.995
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Normalized Angular Frequency /
b)
Fig 6. Filter frequency responses for =-0.5 to 0 in Example 2. a)
Complex error magnitude, b) Magnitude response.
smaller arithmetic complexity than existing fractional delay
FIR design methods.
VII. ACKNOWLEDGMENT
The authors would like to thank to the School of Electrical
Engineering and Computer Science at Washington State
University for the facilities in the project development, and to
CONACyT for scholarship grant number 214724.
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IX. BIOGRAPHIES
Guillermo Ramírez-Conejo received B. S. and M.
S. degrees from the Technological Institute of
Celaya (ITC), Celaya Gto. México, in 2007 and
2010, respectively, both in Electronic Engineering.
ITC. In first semester of 2009 he was professor at
the Basic Science Department, ITC. In second
semester of 2009 he was a visiting scholar at School
of Electrical Engineering and Computer Science,
Washington State University, Pullman, USA, where
he performed last research stage of his M. S. thesis
project. Currently he is a faculty member at the Mechatronics Engineering
Department in the ITC. His research interest includes reconfigurable
computing and digital filters.
Javier. Diaz-Carmona received a B. S. degree in
Electronics Engineering in 1990 from the
Technological Institute of Celaya (ITC), Celaya Gto,
Mexico, a M. S. degree in 1997 and a Doctor in
Science degree in Electronics in 2003 from the
National Institute of Astrophysics Optics and
Electronics (INAOE), Puebla, Mexico. Since 1991 he
is faculty member at the Electronics Engineering
Department in the ITC, currently as full time professor
and researcher. He is author and co-author of more
than 40 papers. His research interests include Digital Signal Processing,
digital filter design techniques and Microcontoller, DSP/FPGA embedded
systems applications. He is member of IEEE and National Researcher System
(SNI) Mexico.
Binghamton, University of Virginia, and Princeton University. His research
interests include high-performance VLSI systems, reconfigurable
architectures, network routers, and nanotechnology. He has co-authored over
170 journal and conference papers and co-edited five books. He has been
granted twenty-seven patents. Dr. Delgado-Frias has chaired several
international conferences, recently he chaired the 53rd IEEE Midwest
Symposium on Circuits and Systems 2010. He received the SUNY System
Chancellor’s Award for Excellence in Teaching in 1994. He is a senior
member of the IEEE, and a member of the Association for Computer
Machinery (ACM) and American Society for Engineering Education (ASEE).
Gordana Jovanovic-Dolecek received a B. S.
degree from the Department of Electrical
Engineering, University of Sarajevo, an M. S.
degree from University of Belgrade, and a PhD
degree from the Faculty of Electrical Engineering,
University of Sarajevo, where she was professor
until 1993, and 1993-1995 she was with the
Institute Mihailo Pupin, Belgrade. In 1995 she
joined to the National Institute of Astophysics
Optics and Electronics (INAOE), Department for
Electronics, Puebla, Mexico, where she currently works as a professor and
researcher. During 2001-2002, 2006 she was at Department of Electrical &
Computer Engineering, University of California, Santa Barbara, USA, as
visiting researcher. In 2008-2009 she was as visiting researcher at the
Department of Electrical and Computer Engineering, San Diego State
University, USA. She is the author of three books, editor of one book, and
author of more than 200 papers. Her research interests include digital signal
processing and digital communications. She is a Fellow member of IEEE, the
member of Mexican Academy of Science, and the member of National
Researcher System (SNI) Mexico.
Juan Prado-Olivarez. He received the B. S.
degree in Electronics Engineering in 1993 from
the Technological Instiute of Celaya, Celaya Gto,
Mexico, a M. S. degree from School of Electrical,
Mechanical and Electronics Engineering of the
Univeristy of Guanajuato, Salamanca Gto, and a
PhD degree in 2006 from the Laboratoire de
Instrumentation et microélectronique de Nancy,
Nancy, France. Since 2007 he is faculty member
at the Electronics Engineering Department in the
ITC, currently as full time professor and researcher. His topics of interest are
biomedical sensors, spectroscopy impedance and noninvasive measurement of
physiological parameters. He is member of the National Researcher System
(SNI) Mexico.
José G. Delgado-Frias received a B. S. degree from
the National Autonomous University of Mexico,
Mexico City, Mexico, an M.S. degree from the
National Institute for Astrophysics, Optics and
Electronics (INAOE), Puebla, Mexico, and a Ph.D.
degree from Texas A&M University, College
Station, TX, all in electrical engineering. He is
currently professor at the School of Electrical
Engineering and Computer Science, Washington
State University, Pullman, where he holds the
Boeing Centennial Chair in Computer Engineering. He has held academic
positions at Oxford University, State University of New York (SUNY) at