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8/8/2019 Weighted-Least-Squares Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 8, AUGUST 2006 3023
Weighted-Least-Squares Design ofVariable Fractional-Delay FIR
Filters Using Coefficient SymmetryTian-Bo Deng, Senior Member, IEEE, and Yong Lian, Senior Member, IEEE
AbstractOur previous work has shown that the coefficientsymmetry can be efficiently exploited in designing variable fi-nite-impulse-response (FIR) filters with simultaneously tunablemagnitude and fractional-delay responses. This paper presents theoptimal solutions for the weighted-least-squares (WLS) design ofvariable fractional-delay (VFD) FIR filters with same-order anddifferent-order subfilters through utilizing the coefficient sym-metry along with an imposed coefficient constraint. In derivingthe closed-form error functions, since the Taylor series expan-sions of s i n ( ) and c o s ( ) are used, the numerical integralsusing conventional quadrature rules can be completely removed,which speeds up the WLS design and guarantees the optimalityof the final solution. Two design examples are given to illustratethat the proposed WLS methods can achieve better design withsignificantly reduced VFD filter complexity and computationalcost than the existing ones including the WLS-SVD approach.Consequently, the proposed WLS design is the best among all theexisting WLS methods so far.
Index TermsCoefficient constraint, coefficient symmetry,Taylor series expansion, variable digital filter, variable frac-tional-delay (VFD) filter, weighted-least-squares (WLS) design.
I. INTRODUCTION
VARIABLE digital filters can be classified into two main
categories. The first one includes the digital filters with
variable magnitude responses [1][12], such variable digital
filters are useful in implementing variable filter banks for
audio signal processing [10], adaptive noise reduction [11],
and other applications that require quick tuning of magnitude
responses of digital filters during the signal processing process.
The second category includes the digital filters with variable
fractional-delay (VFD) responses [13][29], such VFD filters
are useful in the applications such as discrete-time signal
interpolation [14], timing offset recovery in digital receivers
[15], and image interpolation [16], [17]. The most generalVFD filters have also independently tunable magnitude re-
sponses [21], [22], [29], such VFD filters can be used in the
applications where frequency-selective filtering is also nec-
essary. One of the most important features of variable filters
Manuscript receivedAugust 25, 2004; revised August 16, 2005. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Anamitra Makur.
T.-B. Deng is with the Department of Information Science, Faculty of Sci-ence, Toho University, Chiba 274-8510, Japan (e-mail: deng@is.sci.toho-u.ac.
jp).Y. Lian is with the Department of Electrical and Computer Engineering, Na-
tional University of Singapore, Singapore (e-mail: eleliany@nus.edu.sg).
Digital Object Identifier 10.1109/TSP.2006.875385
is that the frequency-domain characteristics can be quickly
changed without redesigning a new filter, which is flexible
and convenient for online tuning. This paper deals with the
optimal design of finite-impulse-response (FIR) VFD filters
in the weighted-least-squares (WLS) error sense. Among the
existing VFD filter design methods, the Lagrange-type VFD
filter is simple, but its frequency response is unbalanced in the
whole frequency band, i.e., its low-band frequency response
is superior to that of the high-frequency band as demonstratedin [18]. Consequently, it is difficult to achieve a satisfactory
design in the whole frequency band by using the Lagrange-type
VFD filter. To solve this problem, WLS techniques have been
proposed [18][20] for achieving more accurate VFD filters in
the whole frequency band. A general WLS approach has also
been proposed for designing a lowpass FIR filter with inde-
pendently variable magnitude and fractional-delay responses
through using a pair of spectral parameters [21], [22], where
a coefficient symmetry is theoretically proved and efficiently
exploited for reducing filter complexity. If the magnitude
response is fixed, then the design simply reduces to the VFD
filter case [23], while the coefficient symmetry developed in[21] and [22] still holds. Therefore, the VFD filter design can
be performed more efficiently as compared with the design
without using coefficient symmetry [18][20]. The same coef-
ficient symmetry can also be derived from the desired impulse
response of a VFD filter [24]. There are other two different
ways to exploit coefficient symmetry in the VFD filter design:
One assumes that the continuous-time impulse response of
the analog filter is symmetric with respect to its midpoint, and
then it is approximated by using piecewise polynomials [25];
another one uses the Taylor series expansion of the desired vari-
able-frequency response [26], [27]. By truncating the Taylor
series and keeping the first several terms, we can approximate
each term separately through designing a linear-phase fixed-co-efficient FIR filter (subfilter). However, as demonstrated in [28],
since the Taylor series converges very slowly, the complexity
of the resulting VFD filter is much higher than that from the
WLS-SVD approach [28]. At this point, the WLS-SVD method
is the most powerful among all the existing ones in terms of
both filter complexity and design accuracy. This is because
the WLS-SVD generates fast-convergent specifications for
the linear-phase subfilters and one-dimensional polynomials.
The WLS-SVD method can also be generalized for designing
digital filters with variable magnitude and VFD responses [29].
In this paper, we first impose a coefficient constraint on the
VFD filter coefficients, which leads to further reduction of the
1053-587X/$20.00 2006 IEEE
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filter complexity without degrading the final design accuracy as
compared with the WLS methods that exploit coefficient sym-
metry only [23], [24]. Then, two WLS methods are proposed for
designing VFD filters with same-order and different-order sub-
filters through utilizing both the coefficient symmetry and co-
efficient constraint. In deriving the closed-form error functions,
the Taylor series expansions of and and thecorresponding closed-form integrals are used. As a result, the
numerical integrals using conventional quadrature rules such as
adaptive Newton-Cotes 8 panel rule [19], [20], rectangle rule, or
trapezoid rule [24] can be completely removed, which reduces
the computational complexity and enhances the final design ac-
curacy. Therefore, the WLS methods are optimal because the
final solutions are not affected by the numerical integrals.
This paper is organized as follows. Section II first imposes
a coefficient constraint on the VFD filter coefficients and for-
mulates the WLS design using both the coefficient constraint
and coefficient symmetry, then a design example is given for
comparing the new WLS method with the existing ones. In
Section III, we generalize the preceding WLS method for de-signing VFD filters with different-order subfilters and present
an example to show that the generalized WLS method is the
best one among all the existing WLS methods so far. Finally,
Section IV concludes the paper.
II. WLS DESIGN USING COEFFICIENT SYMMETRY AND
COEFFICIENT CONSTRAINT
In this section, we first formulate the WLS design of VFD
filters through exploiting coefficient symmetry along with an
imposed coefficient constraint [23]. Then, closed-form error
functions are derived without using conventional numerical
integrals. Finally, we provide an optimal solution for the WLS
design and use a typical example to show the effectiveness of
the proposed WLS method.
A. Design Formulation and Coefficient Constraint
The objective of designing a VFD filter is to find a variable
transfer function that approximates the desired vari-
able-frequency response
(1)
accurately in the passband
where is the normalized angular frequency, is a fixed
number for specifying the passband, and the parameter repre-
sents the desired fractional group delay within the continuously
variable range
Here, we assume the variable transfer function to be
(2)
whose coefficients are expressed as the polynomials ofthe parameter as
(3)
Substituting (3) into (2) yields
(4)
and its frequency response is
(5)
Our objective here is to find the optimal coefficients
such that the weighted squared error of the variable-frequency
response
(6)
is minimized, where is a nonnegative weighting func-
tion, and
(7)
is the complex-valued error between the actual and desired vari-
able-frequency responses. To obtain a closed-form error func-
tion defined in (6), we make the following assumptions.
1) Weighting function is separable, i.e.,
(8)
2) and are piecewise constant.
3) is even-symmetric with respect to , i.e.,
Although the above assumptions make the WLS design not
general, our computer simulations have shown that the above
weighting function is effective in the practical VFD
filter design. Strictly speaking, there is not an explicit wayto find a more general (non-separable) weighting function
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that can guarantee a better design. Hence, the above
assumptions are reasonable.
In [21], we have proved that the coefficients in (5)
have the following symmetry:
(9)
i.e.,
for even (even-symmetry)
for odd (odd-symmetry).
Obviously, if is odd and equals zero, then
i.e.,
if is odd
Without loss of generality, we consider here the VFD filter de-
sign with odd , say . In this case, the number
of independent VFD filter coefficients to be found for
minimizing (6) is
whereas the existing general WLS designs without exploiting
coefficient symmetry require
filter coefficients [18][20]. Therefore, the total number of the
VFD filter coefficients can be reduced by 50%. Applying the
coefficient symmetry (9) to the variable-frequency response (5)
obtains
(10)
with
for (11)and
To further reduce the number of the VFD filter coefficients, we
substitute into (10) and yield
Since the desired variable-frequency response for is
if we let
for
i.e.,
(12)
then
which implies that the VFD filter (2) causes no filtering error for
. Moreover, exploiting the coefficient constraint (12) can
further reduce the number of the VFD filter coefficients by
. Therefore, the total number of the VFD filter coefficients
becomes
whereas that in [23] and [24] is
Using the coefficient symmetry (9) and the coefficient constraint
(12), we can rearrange the variable transfer function (4) as
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(13)
where
(14)
is a constant, and
(15)
are zero-phase subfilters with even-symmetric coefficients
whereas
(16)
are -phase subfilters with odd-symmetric (antisymmetric)
coefficients
The VFD filter (13) can be implemented as the Farrow structure
[13] or the more efficient structure called evenodd structure
[28].
In [26] and [27], the VFD filters are also constructed through
designing linear-phase subfilters, where the Taylor series ex-
pansion of the desired frequency response is used. However,
it should be noted that the subfilters , , and
developed here are not directly related to the terms of the trun-
cated Taylor series of in (1). As shown in [28], theTaylor series converges very slowly, which implies that more
subfilters must be used in order to achieve comparable design
accuracy to the WLS-SVD method. It has been clearly shown
in [28] that even if one can approximate each term perfectly,
i.e., no design errors occur in the design of subfilters, although
this is impossible in practice, the VFD filter from the Taylor se-
ries method is still much worse than that from the WLS-SVD
approach [28]. We will show later that the new WLS method
exploiting the coefficient symmetry (9) along with the imposed
coefficient constraint (12) can even get much better design than
the WLS-SVD approach. It is also clear that the coefficient sym-
metry (9) and the coefficient-constrtaint (12) are derived from
the correspondence between the desired and actual variable-fre-quency responses, but not the Taylor series expansion.
The frequency response of the VFD filter (13) can be
rewritten as
(17)
If we let
for
for ,
for ,
......
......
......
......
(18)
where the subscripts e and o stand for even and odd,
respectively, then the variable-frequency response (17) can be
rewritten in the matrix form as
and the frequency response error in (7) becomes
with
To find the optimal filter coefficients , we just need to
find the optimal coefficient matrices and by minimizing
the error function
(19)
where only the interval needs to be considered dueto the symmetry.
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B. Closed-Form
To derive a closed-form , we expand the
in (19) as
(20)
Since
(21)
and
we have
(22)
with
(23)
By substituting (22) into (19), we get
(24)
where
(25)
(26)
(27)
(28)
(29)
Substituting (23) into (25)(29) leads to the closed-form error
functions
constant
where the constant matrices can be computed
through using the Taylor series expansions of and
along with their corresponding closed-form integrals
(see Appendixes IIII for the details). Consequently, we obtain
the final closed-form error function (24) as
constant (30)
C. Optimal Solution
To find the optimal coefficient matrices and for mini-
mizing the error function (30), we differentiate with
respect to and , and then set the derivatives to zero as
Since
(31)
and the matrices , , , and are symmetric, we have
and
(32)
i.e.,
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Consequently, the optimal coefficient matrices can be deter-
mined by
(33)
However, the above computations using direct inversions usu-ally cannot guarantee a numerically stable solution because the
condition numbers of , , , and are usually rather
large, which indicates that those matrices are nearly singular,
thus the ill-conditioned problem will seriously affect the final
solution (33). In this paper, we take the following measure to
deal with this ill-conditioned problem so that a numerically sta-
bilized optimal solution can be guaranteed.
Since the matrices , , , and are symmetric, pos-
itive-definite, they can be decomposed by using the Cholesky
factorization as
(34)
where , , , and are upper triangular matrices.
Hence, the inverses of , , , and can be indirectly
determined as
(35)The most important advantage of using Cholesky decomposi-
tion here is that the condition numbers of the triangular ma-
trices , , , and are much smaller than those of ,
, , and ; thus, the inverses (35) can be accurately com-
puted without ill-conditioned problems. Substituting (35) into
(33) yields a numerically stable solution
(36)
D. Design ExampleThis section presents an example to illustrate that the pro-
posed WLS design method can achieve higher design accuracy
with significantly reduced computational cost and VFD filter
complexity than the existing ones that use numerical integrals
[20], [24].
Example I: The variable design specification (1) is approxi-
mated within
(37)
such that the maximum absolute error of the variable-frequency
response is below 100 dB. This is a typical design problemthat has been treated in the literature [18][20], [23], [24].
TABLE I
TRUNCATION ERRORS FOR A AND A
To compare the proposed WLS design with the one in [24],
we use the variable transfer function (4) with
First, let usseehowto determine the valueof in (57) and (61).
Table I lists the normalized root-mean-squared (rms) truncation
errors of and defined by
100
100
as the number increases, where the ideal and in the
denominators are computed from (57) and (61) by using a very
large 20 without weightings, i.e.,
(38)
It is observed from Table I thatif we take 10, the truncation
errors approach zero. Hence, we set 10. If the weighting
functions and are set as (38), then the WLS de-
sign is simply the normal least-squares (LS) design.To evaluate the VFD filter design accuracy, the normalized
RMS error of the variable-frequency response defined by
100 (39)
the maximum absolute error in decibels
(40)
and the maximum group-delay error
(41)
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TABLE II
DESIGN ERRORS AND FILTER COMPLEXITIES
are used, where is the actual fractional group delay that
is the function of the frequency and the desired fractional
group delay .
To c ompute the errors , , and , the f requency in
(37) is uniformly sampled at the step size , and the frac-
tional delay in (37) is uniformly sampled at the step size 1/60.
To compare the proposed LS design with the one using simple
quadrature rule such as rectangle rule (also called the midpointrule) [24], Table II lists the design errors, computational costs in
Flops, and the numbers of VFD filter coefficients. It is observed
that the proposed LS approach can achieve higher design accu-
racy with significantly reduced computational cost (only about
0.01%) than the method [24]. Moreover, the number of the VFD
filter coefficients is further reduced by (34), which is
owing to the exploitation of the imposed coefficient constraint
(12). The same conclusion is also applicable to the general WLS
design. Therefore, the proposed WLS approach is preferred in
terms of higher design accuracy, less computational cost, and
lower filter complexity.
It is also clear from Table II that the LS design does not meet
the requirement that the maximum error must be below
100 dB. Therefore, appropriate weighting functions
and must be found for suppressing the peak errors. Our
computer simulations by trial-and-error method have shown that
if the weighting functions are chosen as
if
if
if
if
if(42)
and the design specification (1) is approximated in the range
where the small numbers
are added for suppressing the peak errors around the edges
and , then the resulting VFD filter (4) with 33
and 7 satisfies the design requirement (maximum error
below 100 dB) as shown in Table II. We have also de-signed a VFD filter satisfying the same design specification by
Fig. 1. Frequency response error from the same-order WLS design.
using the existing WLS method [20] with the same weightingfunctions and same filter order, the results are listed in Table II.
Although the two WLS methods yield comparable design re-
sults, the proposed one requires only about 6% of the computa-
tional cost in Flops required by the existing one [20]. More im-
portant, the new WLS method exploiting coefficient symmetry
and coefficient constraint requires far fewer VFD filter coeffi-
cients (234 coefficients) than the WLS method [20] (536 coeffi-
cients), which significantly reduces the hardware cost for imple-
menting the resulting VFD filter. Fig. 1 illustrates the absolute
error (in decibels) of the variable-frequency response from the
proposed WLS design, whose maximum error is 100.06 dB
(below 100 dB).One may ask how to find the weighting functions (42). Gen-
erally speaking, there is not a systematic way to find the op-
timal weighting functions and . One needs to start
with the pure LS design, and then by observing the distribu-
tion of the variable-frequency response errors, larger weights
are put around the region where peak errors occur. Such a trial-
and-error procedure is repeated until the maximum error is sup-
pressed below the design requirement. Furthermore, it is also
effective to add the small numbers and as above for sup-
pressing the error jumps around the edges and .
Based on the Taylor series expansion of the desired vari-
able-frequency response, a VFD filter can also be designed by
using a set of linear-phase same-order or different-order sub-filters [26], [27], where each subfilter approximates a different
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term of the truncated Taylor series. In [27], a minimax design
technique is proposed for minimizing the peak error of the vari-
able-frequency response of an odd-order VFD filter that requires
240 coefficients to satisfy the above design requirement, but our
WLS design here requires fewer (234) filter coefficients. The
reason why the proposed WLS design can even surpass the min-
imax design is because the Taylor series decays very slowly asshown in [28], which implies that more subfilters must be used
to obtain a satisfactory design. To further reduce the VFD filter
complexity in the minimax design, the resulting subfilters must
be simultaneously optimized [27].
Generally speaking, it is unfair to compare a WLS design
with a minimax design because the two approaches minimize
different error functions. The former minimizes the total energy
of the variable-frequency response errors, while the latter mini-
mizes the peak (maximum) error. Therefore, the total error en-
ergy of the minimax design is usually larger than that of the
WLS design, while its peak error is usually smaller if the same
filter complexity is used. It is also difficult to say one criterion is
better than the other. If one wants to minimize the total error en-ergy, then the LS or WLS design should be chosen. Conversely,
if one intends to minimize the maximum error, then the min-
imax design is preferred.
In [28], a simple and powerful WLS-SVD design technique
has been proposed for designing VFD filters in the WLS error
sense, where both different-order subfilters and coefficient sym-
metry are used. Our computer simulations have verified that
since the subfilters and in (13) are of the same order,
the WLS-SVD technique is still superior to the proposed one. In
the following section, we generalize the above same-order WLS
design technique to the different-order case, and show that the
different-order WLS design can significantly reduce the VFDfilter complexity.
III. GENERALIZED VFD FILTER DESIGN
It is clear from (15) and (16) that the linear-phase subfil-
ters and are of the same order (2N), which is the
main reason why the WLS design technique proposed in the
preceding section is superior to other existing WLS ones ex-
cept the only one [28] (WLS-SVD approach) that utilizes dif-
ferent-order subfilters. In this section, we generalize the above
WLS design method to the case that the subfilters and
may take different orders and show that the generalizedone can surpass all the existing WLS methods, including the
WLS-SVD approach, in design accuracy and filter complexity.
A. Generalized WLS Design
By considering that the th columns of and in (18)
correspond to the coefficients of the subfilters and ,
respectively, if the orders of and are different, then
the coefficients of and cannot be expressed in a
matrix form like (18). Here, we want to design a VFD filter
with different-order subfilters. Without loss of generality, we
formulate the design problem here under the assumption that
is odd, say . Let the number in (15) forbe and that in (16) for be , where the subscript
e denotes that the coefficients of the subfilters are even-
symmetric, and o denotes that the coefficients of the subfilters
are odd-symmetric, then the frequency responsesof
and can be expressed as
and
respectively, where
and the vectors and are related to the coefficients of
and . Thus, the variable-frequency response of the
VFD filter with different-order subfilters can be rewritten as
(43)
with
and
... ...
Our objective here is to find the optimal coefficient vectors
and such that the weighted squared error
(44)
is minimized, where
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is the frequency response error. Let
then the squared frequency response error in (44)becomes
(45)
Substituting (45) into (44) yields
(46)
with
(47)
(48)
Because is the function of only , and is the
function of only , the error function (46) can be minimized
through minimizing and separately. After some
manipulations, the closed-form error functions andcan be obtained as
constant (49)
constant (50)
where and are vectors, and and are symmetric
matrices. (For detailed derivations of (49) and (50), see
Appendix IV and Appendix V.) To minimize in (49),
we differentiate it with respect to and then set the derivative
to zero as
Since
thus
Consequently, the optimal coefficient vector can be deter-
mined by
Furthermore, the symmetric, positive-definite matrix can be
decomposed by using the Cholesky factorization as
where is an upper triangular matrix, so the numerically stable
optimal coefficient vector can be computed as
(51)
Similarly, to minimize in (50), we differentiate
with respect to and then set the derivative to zero as
Since
we get
thus
Moreover, the symmetric, positive-definite matrix can be de-
composed as
through using the Cholesky factorization, where is an upper
triangular matrix, thus the numerically stable optimal coefficient
vector can be determined as
(52)
B. Design Example
This section presents an example to show that the generalized
WLS design method can achieve higher design accuracy withreduced filter complexity than the existing WLS-SVD approach
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TABLE III
TRUNCATION ERRORS FOR u AND v
[28] that is the best one among all the existing WLS design
methods at this point.Example II: The variable design specification is the same as
that given in (37).
Before performing the WLS design using the generalized
WLS approach, the number in (68) and (73) must be fixed.
Table III lists the normalized rms truncation errors of and
defined by
100
100
as the number increases, where the ideal and in the de-
nominators are computed from (68) and (73) by using a very
large 20 , the weighting functions and subfilter orders
are set as
(53)
From Table III, it is observed that if we take 10, the
truncation errors approach zero. Hence, the number is set to10.
To compare the generalized WLS design with the WLS-SVD
approach [28], we use the same error criteria , , and
defined in (39), (40), and (41), respectively. When the gener-
alized WLS method is used to design a VFD filter consisting
of the same-order subfilters in (53), it requires 1640451 Flops,
which indicates that the generalized WLS method requires more
operations than the same-order WLS approach (183958 Flops)
proposed in the preceding section. That is, if a VFD filter with
the same-order subfilters is to be designed, the preceding same-
order WLS method is preferred.
Next, let us see the effectiveness of the generalized WLS
method in reducing the VFD filter complexity. Table II lists thedesign errors, computational cost in Flops, and the total number
of VFD filter coefficients, where the orders of subfilters
are
and those of are
respectively, and the weighting functions are chosen as
if
if
(54)
and the design specification (1) is approximated in the range
where the small numbers
are added for suppressing the peak errors around the edges
and .
As in selecting the weighting functions and ,
the small numbers , , and the subfilter orders are also found
through trial-and-error. To find appropriate subfilter orders, we
first start with a same-order LS design whose peak error is largerthan the design requirement ( 100 dB). Then, the subfilter or-
ders are adjusted (increased or decreased) separately such that
the peak error almost remains the same, but the total number of
VFD filter coefficients is gradually reduced. This adjustment is
repeated until the total number of VFD filter coefficients cannot
be further reduced. Finally, the weighting functions and
are gradually added and adjusted such that the peak error
is suppressed below the design requirement.
Fig. 2 illustrates the absolute error (in decibels) of the vari-
able-frequency response from the generalized WLS design,
whose maximum error is 100.51 dB. The actual variable frac-
tional-delay (VFD) response and its absolute error are depictedin Figs. 3 and 4, respectively, which show that extremely flat
VFD response has been obtained, whose maximum deviation
is 0.000237. Table II also shows the following.
1) The generalized WLS method requires fewer VFD filter
coefficients (150) than the WLS-SVD approach (188).
2) The generalized WLS method yields smaller maximum
frequency response error ( 100.51 dB) as compared with
the WLS-SVD approach ( 98.29 dB).
3) The normalized frequency response error (0.000222%)
from the generalized WLS design is smaller than that from
the WLS-SVD approach (0.000555%).
4) The generalized WLS design requires much less compu-
tational cost (773963 Flops) than the WLS-SVD approach(7839429 Flops).
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Fig. 2. Frequency response error from the generalized WLS design.
Fig. 3. Variable fractional-delay from the generalized WLS design.
Fig. 4. Variable fractional-delay error from the generalized WLS design.
As a result, the generalized WLS method is the best so far amongall the existing WLS methods for designing FIR VFD filters in
terms of higher design accuracy, reduced computational cost,
and less filter complexity.
IV. CONCLUSION
Two closed-form WLS methods for designing FIR VFD fil-
ters have been proposed; one uses the same-order subfilters, and
the other (generalized WLS method) uses different-order subfil-
ters. The former canachieve higher design accuracy with signifi-
cantly reduced filter complexity than other WLS methods except
the WLS-SVD approach [28], while the latter can further reduce
the VFD filter complexity through utilizing different-order sub-
filters. Our design example has shown that the generalized WLS
method is superior to all other existing WLS ones including
the WLS-SVD technique in terms of higher design accuracy,
reduced computational cost, and less filter complexity. As com-
pared with the existing WLS methods, the new WLS designmethods have the following advantages.
1) A coefficient constraint has been imposed on the WLS de-
sign formulation for reducing the VFD filter complexity.
Utilizing the coefficient constraint along with the coeffi-
cient symmetry proven in [21], we can significantly reduce
the filter complexity (the total number of VFD filter co-
efficients), and thus reduce the hardware cost for imple-
menting the resulting VFD filters.
2) Since the closed-form error functions are derived through
using the Taylor series expansions of and
and the corresponding closed-form integrals, the numerical
integrals using conventional adaptive quadrature rules orsimple midpoint rule can be completely removed, which
speeds up the WLS design and guarantees the optimality
of the final solution.
In this paper, we have only exploited the coefficient symmetry
and coefficient constraint in the WLS design of FIR VFD filters
with even-order subfilters. Further research needs to be done to
investigate and exploit coefficient symmetry for the odd-order
case.
APPENDIX I
CLOSED-FORM
Based on the separable weighting function (8), in (25)
can be rewritten as
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3034 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 8, AUGUST 2006
with
(55)
Due to the nonseparable function in (55), the matrix
can be evaluated by using numerical integrals [19][22].
However, such numerical integrals can be avoided by applyingthe Taylor series (Maclaurin series) expansion
where all the terms are separable with respect to and [23].
Therefore
In practice, only the first terms in the above series are usedfor computing , and the remaining terms are truncated, i.e.,
(56)
where a small number , say , usually can achieve
sufficiently satisfactory approximation. Hence, the matrix
can be approximated as
(57)
with
(58)
It should be noted here that the approximation (57) can be made
as accurately as desired by increasing the number , thus one
needs not to worry about the truncation error here. In addition,
the latter integral in (58)
can be computed in a closed-form by using the recurrence
formula
along with
APPENDIX II
CLOSED-FORM AND
The error function in (26) can be obtained as
where
is a Hankel matrix that can be obtained by computing its first
column and last row, and the symmetric matrix
can be obtained by computing only the elements along and
below the main diagonal of the matrix as shown in the equation
at the bottom of the page. Similarly, the error function
in (27) can be evaluated as
......
......
...
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DENG AND LIAN: WLS DESIGN OF VFD FIR FILTERS USING COEFFICIENT SYMMETRY 3035
where
is also a Hankel matrix, and the symmetric matrix
can be obtained by computing only the diagonal elements and
those in the lower triangular matrix shown at the bottom of the
page.
APPENDIX III
CLOSED-FORM
Substituting in (23) into (28) yields
with
(59)
Due to the nonseparable function in (59), the matrix
can be computed by using numerical integrals [19][22].
However, such numerical integrals can be removed by applying
the Taylor series (Maclaurin series) expansion
where all the terms are separable with respect to and . In
practice, only the first terms in the above series are used for
computing , and the remaining ones are truncated as
(60)
Usually, a small number , say 10, makes the trunca-
tion error almost zero. Therefore, the matrix can be approx-
imated as
(61)
with
(62)
It should be noted here that the approximation (61) can be made
as accurately as desired by increasing the number , thus one
needs not to worry about the truncation error here. In addition,
the latter integral
in (62) can be computed in a closed form by using the recurrence
formula
along with
......
......
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3036 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 8, AUGUST 2006
APPENDIX IV
DERIVATION OF
The error function in (47) can be rewritten as
where
(63)
with
(64)
and
(65)
with
(66)
and
constant
(67)
To compute the vector in (64), the Taylor series expansion
(56) is used to get
(68)
with
computed in closed-form integrals. The symmetric matrix in
(66) can also be computed through using closed-form integrals.
Combining (63), (65) and (67) together yields the error function
in (49).
APPENDIX V
DERIVATION OF
The error function in (48) can be rewritten as
where
(69)
with
(70)
and
(71)
with
computed using closed-form integrals. Finally
constant (72)
To compute the vector in (70), the Taylor series expansion
(60) is used to get
(73)
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DENG AND LIAN: WLS DESIGN OF VFD FIR FILTERS USING COEFFICIENT SYMMETRY 3037
with
computed in closed-form integrals. Combining (69), (71), and
(72) together leads to the error function in (50).
ACKNOWLEDGMENT
The authors would like to thank the anonymous reviewers for
their constructive comments on the original manuscript.
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Tian-BoDeng (M92SM99)receivedthe Ph.D. de-gree in electronic engineering from Tohoku Univer-sity, Sendai, Japan, in 1991.
From 1991 to 1992, he was a Research Associatewith the Department of Information and ComputerSciences, Toyohashi University of Technology,Toyohashi, Japan. In 1992, he was selected by theJapanese Government as a Special Researcher forcarrying out the Basic-Science-Program at the Insti-tute of Physical and Chemical Research (RIKEN),Wako, Japan. In 1994, he joined the Department of
Information Science, Faculty of Science, Toho University, Funabashi, Japan,as an Assistant Professor, and has been an Associate Professor since 1998.
From 1998 to 1999, he was also a Visiting Professor with the Department ofElectrical and Computer Engineering, University of Victoria, BC, Canada.His research interests include speech processing, design theory of constantmultidimensional digital filters, and design theory of variable one-dimensional
and variable multidimensional digital filters.
Yong Lian (M90SM99) received the B.Sc. degreefrom the School of Management of Shanghai JiaoTong University, China, in 1984 and the Ph.D. degreefromthe Department of Electrical Engineering of Na-tional University of Singapore, Singapore, in 1994.
From 1984 to 1996, he was with the Institute of
Microcomputer Research of Shanghai Jiao TongUniversity, Brighten Information Technology, Ltd.,
SyQuest Technology International, and Xyplex,Inc. In 1996, he joined the National University ofSingapore, Singapore, where he is currently an As-
sociate Professor with the Department of Electrical and Computer Engineering.
His research interests include digital filter design, VLSI implementation ofhigh-speed digital systems, biomedical instruments, and radio-frequencyintegrated circuits design.
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3038 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 8, AUGUST 2006
Dr. Lian received the 1996 IEEE Circuits and Systems SocietysGuillemin-Cauer Award. He currently serves as Associate Editors of theIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I and of the Journal ofCircuits Systems and Signal Processing. He is the Guest Editor of the SpecialIssue on Biomedical Circuits and Systems in the IEEE T RANSACTIONS ON
CIRCUITS AND SYSTEMS I and of Special Issue on Computationally EfficientDigital Filters: Design Techniques and Applications in the Journal of Circuits,
Systems and Signal Processing. He was an Associate Editor of the IEEE IEEE
TRANSACTIONS ON
CIRCUITS AND
SYSTEMS
PART
II from 2002 to 2003 and theGuest Editor of Special Issue on Frequency-Response Masking Technique andIts Applications in the Journal of Circuits, Systems and Signal Processing in
2003. He is involved in various IEEE activities, including serving as an IEEE
Circuits and Systems (CAS) Society Distinguished Lecturer, Vice Chairmanof the Biomedical Circuits and Systems Technical Committee of CAS Society,Committee Member of Digital Signal Processing Technical Committee ofCAS Society, Chair of Singapore CAS Chapter, General Co-Chair of the FirstIEEE International Workshop on Biomedical Circuits and Systems, TechnicalProgram Co-Chair of the 2006 IEEE International Conference on BiomedicalCircuits and Systems, and Technical Program Co-Chair of the 2006 IEEEAsia Pacific Conference on Circuits and Systems. He has served on technical
program committees, organizing committees, and session chairs for manyinternational conferences.
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