Weighted-Least-Squares Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry

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: [email protected].

jp).Y. Lian is with the Department of Electrical and Computer Engineering, Na-

tional University of Singapore, Singapore (e-mail: [email protected]).

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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3024 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 54, NO. 8, AUGUST 2006

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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DENG AND LIAN: WLS DESIGN OF VFD FIR FILTERS USING COEFFICIENT SYMMETRY 3025

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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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

......

......

...


13/16


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

......

......


14/16


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)


15/16


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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Reg. Papers, vol. 52, no. 1, pp. 148167, Jan. 2005.

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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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.

Documents

Weighted-Least-Squares Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry