Design of Variable Fractional Delay Filter Using Infinite

Proceedings of the Second APSIPA Annual Summit and Conference, pages 335–340,Biopolis, Singapore, 14-17 December 2010.

Design of Variable Fractional Delay Filter Using Infinite Product Expansion

Chien-Cheng Tseng* and Su-Ling Lee† *National Kaohsiung First University of Science and Technology, Kaohsiung Taiwan

E-mail: [email protected] †Chung-Jung Christian University, Tainan Taiwan

E-mail: [email protected]

Abstract—In this paper, the design of variable fractional delay (VFD) filter using the infinite product expansion is investigated. First, the infinite products of trigonometric functions are applied to transform the VFD filter design into the designs of first and second order digital differentiators. Next, conventional digital differentiators are used to implement the proposed VFD filters efficiently. The main advantage of the proposed VFD filter is that it needs less storage requirement of filter coefficients than the traditional power series expansion design approach. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design methods.

I. INTRODUCTION

Recently, fractional delay filter has become an important device in the applications of modeling of music instruments, beam steering of antenna array, speech coding and synthesis, time adjustment in digital receivers, comb filter design and analog digital conversion etc. [1]-[5]. A comprehensive study of the fractional delay filter design is available in tutorial paper [1]. The ideal frequency response of fractional delay filter is given by

pjeD ωω −=)( (1)

where p is a fractional number in the range ]5.0,5.0[− . Thus, the design problem is how to find a digital filter such that its frequency response fits the ideal response )(ωD as well as possible. If the delay p is adjustable, it is called the variable fractional delay (VFD) design. When delay p is fixed, it is called the fixed fractional delay (FFD) design. So far, several methods have been proposed to solve the VFD design problem such as differentiator-bank method [2], Lagrange or maximally flat method [3][4], and weighted least squares (WLS) method [5]. Each method has its unique features.

In the conventional designs, the power series expansion has been applied to divide the variable fractional delay filter design into some fixed sub-filters designs. Then, the least squares method or differentiator bank method is used to design sub-filters. One of the main advantages of this approach is that the designed variable fractional delay filter can be implemented efficiently by using Farrow structure [2].

In the literature, there exist infinite product expansion and infinite partial fraction expansion of elementary functions except power series expansion [6]. Thus, it is interesting to use these expansion methods to design variable fractional delay filters. The purpose of this paper is to study the infinite product expansion design method and to compare this method with conventional power series method. This paper is organized as follows. In section II, the conventional design of variable fractional delay filter using power series expansion is first described briefly. In section III, the infinite product expansion of trigonometric functions are applied to transform the VFD filter design into the designs of first and second order digital differentiators such that the conventional FIR digital differentiators can be used to implement the proposed VFD filters efficiently. The main advantage of the proposed VFD filter is that it needs less storage requirement of filter coefficients than the power series expansion design approach. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design method and conclusions are made.

II. POWER SERIES EXPANSION METHOD

In this section, the conventional designs of variable fractional delay filter using power series expansion are described briefly. It is well-known that the power series expansion of exponential function is given by

∑∞

=

=0 !k

kx

kxe (2)

Using the variable substitution pjx ω−= , it yields the result

)(!

)(!

)(

1

0

0

+

=

∞

=

−

+−

=

−=

∑

∑

MkM

k

kk

kpj

pOpkj

kpje

ω

ωω

(3)

where )(xO denotes a term which goes to zero at least as x when x approaches zeros and M is a prescribed integer order. Because p is a fractional number in the range

]5.0,5.0[− , the above series expansion can be truncated into the form

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10-0103350340©2010 APSIPA. All rights reserved.

kM

k

kpj p

kje ∑

=

− −≈

0 !)( ωω (4)

Based on this approximation, if the frequency responses of filters )( ωj

k eG are designed to fit the ideal responses k

k jk

)(!)1( ω− well, then the frequency response of the

following filter

kjM

kk

j peGpe )(),(0

1ωω ∑

=

=Φ (5)

will approximate the ideal response )(ωD well. Moreover, the above filter can be implemented efficiently by using the structure I shown in Fig.1 which is called Farrow structure in the literature. Clearly, there are 1+M sub-filters )( ωj

k eG needed to be implemented in the Farrow structure of variable fractional delay filter. The typical least squares method to design sub-filters )( ωj

k eG is described below. The transfer functions of sub-filters are chosen as

n

N

Nnkk zngzG −

−=∑=

1

1

)()( (6)

Then, filter coefficients are determined by minimizing the following cost function:

ωωλπ ω dkjeGJ

kj

k∫−

−=0 !

)()()( kg (7)

where vector Tkkkk NgNgNg ])()1()([ 111 L+−−=g .

Because this is the standard least squares FIR filter design problem, its optimal solution can be obtained easily by using the method in the digital signal processing textbook [7].

III. INFINITE PRODUCT EXPANSION METHOD

In this section, the infinite product expansions of trigonometric functions are first reviewed. Then, the infinite product expansions are used to design variable fractional delay filter. An implementation structure is also developed which only contain first and second order differentiators.

A. Infinite Product Expansion From the book [6], we have the infinite product expansion

of cosine and sine functions:

∏∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=0

22

2

)12(41cos

k kxx

π (8a)

∏∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

122

2

1sink k

xxxπ

(8b)

The above expansions are available in mathematics for as long as the last two hundred and fifty plus years since Leonhard Euler found them [8]. To give a clear view of the approximations of both representations, let us display graphs of their L th partial products:

∏=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=L

k

L

kxxEc

022

2)(

)12(41)(

π (9a)

∏=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

L

k

L

kxxxEs

122

2)( 1)(

π (9b)

It can be seen that )()( LxEc is an even function and )()( LxEs is an odd function. The partial product expansion

approximation in Eq.(9) of 10=L is shown in Fig.2. The solid lines are the ideal curves of )cos( x and )sin( x . The

dashed lines are the curves of )10()(xEc and )10()(xEs . It is clear that the approximation is well for x around zero. Recently, Melnikov proposed a new infinite product expansion of trigonometric and hyperbolic functions based on Green's function [8]. One of these new results is given by

∏∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+

−=

122 )41(

)(412cosk k

xxxxπ

ππ

π (10a)

∏∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+=

122

22

)41(412sin

k kxxx

ππ

π (10b)

To show the approximations of these new representations, let us display graphs of their L th partial products:

∏=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+

−=

L

k

L

kxxxxMc

122

)(

)41()(412)(

ππ

ππ (11a)

∏=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+=

L

k

L

kxxxMs

122

22)(

)41(412)(

ππ

π (11b)

It can be seen that )()( LxMs is an odd function, but )()( LxMc is not an even function. The partial product expansion approximation in Eq.(11) of 10=L is shown in Fig.3. The solid lines are the ideal curves of )cos( x and )sin( x . The dashed lines are the curves of )10()(xMc and )10()(xMs . It is clear that the approximation is well for x around zero. However, compared the results in Fig.2(b) and Fig.3(b), it can be seen that the function )10()(xEc has better approximation than )10()(xMc . So, in next subsection, only the infinite product expansion in Eq.(9) is used to design variable fractional delay filter.

B. Proposed Design Method It is well-known that the Euler formula is given by

)sin()cos( xjxe jx += (12) Substituting Eq.(8) into Eq.(12), we have

∏∏∞

=

∞

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−=1

22

2

022

2

1)12(

41kk

jx

kxjx

kxe

ππ (13)

Using the variable substitution px ω−= , it yields the result:

∏∏∞

=

∞

=

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−=1

22

22

022

22

1)12(

41kk

pj

kppj

kpe

πωω

πωω (14)

336

Because p is a fractional number, the above infinite product can be truncated. So, we have the following form

∏∏==

−⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−≈L

k

L

k

pj

kppj

kpe

122

22

022

22

1)12(

41π

ωωπ

ωω (15)

where L is called the truncation order. If two sub-filters )( ωjeS and )( ωjeF are designed to satisfy the following

approximation condition: 22)()( ωωω −=≈ jeS j (16a)

ωω jeF j ≈)( (16b) , then the following filter

∏

∏

=

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+−

⎟⎟⎠

⎞⎜⎜⎝

⎛+

+=Φ

L

k

jj

L

k

jj

kpeSpeF

kpeSpe

122

2

022

2

2

)2()(41)(

)12()(41),(

π

πω

ω

ωω

(17)

will approximate the ideal response )(ωD well. Moreover, the above filter can be implemented efficiently by using the structure II in Fig.4 where 2=L is chosen. Compared the structures in Fig.1 and Fig.4, three observations are made below: (1) The right side of Eq.(15) can be expressed as a polynomial of p with degree 22 +L . That is,

∑

∏∏+

=

==

−

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−⎟⎟

⎠

⎞⎜⎜⎝

⎛+

−≈

22

0

122

22

022

22

)(

1)12(

41

L

k

kk

L

k

L

k

pj

pa

kppj

kpe

ω

πωω

πωω

(18)

where )(ωka can be obtained by expansion and combination. Compared Eq.(4) with Eq.(18), we see that if 22 += LM , the structure I and II have the same degree of polynomial of p. (2) There are 1+M sub-filters )( zG k needed to be implemented in Farrow structure, but there are only two sub-filters )( zS and )( zF needed to be realized in structure II. So, the memory storage of filter coefficients in structure II is less than that of Farrow structure. (3) From Fig.1, the Farrow structure needs to implement

1+M sub-filters, and the structure II also needs to implement 22 +L sub-filters. So, the implementation complexity of structure II is same as that of Farrow structure if the condition 12 += LM holds. Until now, the design of variable fractional delay filter using infinite product expansion has been described. The remaining problem is how to design sub-filters )( zS and

)( zF used in structure II. In next subsection, this design problem will be studied.

C. Designs of sub-filters Now, let us study the designs of the sub-filters )( zS and

)( zF . First, let us design the filter )( zS . From Eq.(16a), it is obvious that the ideal frequency response of filter )( zS is

2)( ωj which is the ideal response of the second order differentiator. In the following, the least squares method is described below: In this paper, the transfer function of filter

)( zS is chosen as an FIR filter below:

∑−=

−=2

2

)()(N

Nk

kzkszS (19)

where filter coefficients )(ks satisfy the even symmetric condition, that is,

2,,2,1)()( Nkksks L==− (20) Then, the frequency response of )( zS can be written as

)(

)]cos()(2[)0(

))(()0()(

2

2

1

1

ω

ω

ωωω

rs T=

+=

++=

∑

∑

=

=

−

kkss

eeksseS

N

k

N

k

kjkjj

(21)

where vectors are TNsss )]()1()0([ 2L=s (22a)

TN )]cos(2)cos(21[)( 2ωωω L=r (22b) To let frequency response )( ωjeS approximate the response

2)( ωj well, the filter coefficients s are determined by minimizing the following objective error function

c

deTT

T

+−=

+= ∫suUss

rss

2

)()(2

0

21 ωωω

λπ

(23)

where

∫=λπ

ωωω0

)()( dTrrU (24a)

∫−=λπ

ωωω0

2 )( dru (24b)

∫=λπ

ωω0

4 dc (24c)

The function )(1 se is a quadratic function of coefficients s , so the optimal solution is given by

uUsopt1−= (25)

Next, let us design filter )( zF . From Eq.(16b), it is clear that the ideal frequency response of filter )( zF is ωj which is the ideal response of the first order differentiator. So, the design methods of differentiator in the literature can be used to solve this design problem easily. Now, the simple least squares method is described below: In this paper, the transfer function of filter )( zF is chosen as an FIR filter below:

∑−=

−=3

3

)()(N

Nk

kzkfzF (26)

where filter coefficients )(kf satisfy the odd symmetric condition, that is, 3,,2,1)()( Nkkfkf L=−=− (27a)

337

0)0( =f (27b) Then, the frequency response of )( zF can be written as

)(

)]sin()(2[

))(()(

3

3

1

1

ω

ω

ωωω

bf Tj

kkfj

eekfeF

N

k

N

k

kjkjj

=

−=

−=

∑

∑

=

=

−

(28)

where vectors are TNfff )]()2()1([ 3L=f (29a)

TN )]sin()2sin()sin([2)( 3ωωωω L−=b (29b)

To let frequency response )( ωjeF approximate the response ωj well, the filter coefficients f are determined by

minimizing the following objective error function

d

deTT

T

+−=

−= ∫fqQff

bff

2

)()(2

02 ωωωλπ

(30)

where

∫=λπ

ωωω0

)()( dTbbQ (31a)

∫=λπ

ωωω0

)( dbq (31b)

∫=λπ

ωω0

2 dd (31c)

Because the function )(2 fe is a quadratic function of coefficients f , the optimal solution is given by

qQf opt1−= (32)

So far, the designs of sub-filters )( zS and )( zF used in structures II have been presented. In next section, some numerical examples are shown to evaluate the performance of the proposed design approach.

IV. NUMERICAL EXAMPLES

In the following, the numerical examples performed with the MATLAB language on an personal computer are now presented to evaluate the performances of the conventional Farrow structure I and the proposed structure II. Example 1: In this example, the performance of Farrow structure I in Fig.1 is studied. The design parameters are chosen as 401 =N , 7=M and 9.0=λ . Fig.5 shows the magnitudes responses of the sub-filters |)(| ωj

k eG , (k=0,1,2,...,5) which are designed by the least squares method in Eq.(7). And, Fig.6(a)-(d) show the magnitude response, magnitude error, group delay, and group delay error of the designed variable fractional delay filter. In this design, the number of filter coefficients needed to store in memory is

648)1)(12( 1 =++ MN . Moreover, because the sub-filters are not linear phase filters, the number of multiplications to implement the designed Farrow structure in Fig.1 is 648.

Example 2: In this example, the performance of structure II in Fig.4 is investigated. The parameters are chosen as

402 =N , 203 =N , 10=L and 9.0=λ . Fig.7(a)(b) show the magnitude responses of the designed sub-filters

)( zS and )( zF . Fig.8(a)-(d) show the magnitude response, magnitude error, group delay, and group delay error of the designed variable fractional delay filter. Due to the symmetry, the number of filter coefficients needed to store in memory is

61)1( 32 =++ NN . Moreover, because sub-filters )( zS and )( zF are both linear phase filters, so the number of multiplications to implement the designed structure II in Fig.4 is 481)12)(1( 32 =+++ LNN .

Based on the simulation results of the above two numerical examples, we have the following observations: (a) The number of filter coefficients needed to store in memory in the structure II is much smaller than that of the conventional Farrow structure I. (b) Because sub-filters )( zS and )( zF are both linear phase filters and )( zG k are nonlinear phase filters, the number of multiplications to implement the structure II is smaller than that of the conventional Farrow structure I. (c) Compared with Fig.6(d) with Fig.8(d), it can be seen that the group delay errors of both structures are comparable. (d) Compared with Fig.6(b) with Fig.8(b), it is clear that the magnitude error of structures II is larger than the error of the structure I. Thus, it is an interesting topic to study how to reduce the magnitude error of structure II.

V. CONCLUSIONS

In this paper, the design of variable fractional delay (VFD) filter using the infinite product expansion is investigated. First, the infinite products of trigonometric functions are applied to transform the VFD filter design into the designs of digital differentiators. Next, conventional digital differentiators are used to implement the proposed VFD filters efficiently. The main advantage of the proposed VFD filter is that it needs less storage requirement of filter coefficients than the traditional power series expansion design approach. Finally, several numerical examples are demonstrated to show the effectiveness of the proposed design methods. However, only one-dimensional VFD filter is studied in this paper. Thus, it is interesting to extend the proposed infinite product expansion method to design multi-dimensional VFD filter in the future.

REFERENCES

[1] T. I. Laakso, V. Valimaki, M. Karjalainen and U.K. Laine, "Splitting the unit delay: tool for fractional delay filter design," IEEE Signal Processing Magazine, pp.30-60, Jan. 1996.

[2] C.C. Tseng, “Design of variable fractional delay FIR filter using differentiator bank,” Int. Symp. Circuits and Systems, vol.4, pp. 421-424, Apr. 2002.

[3] T.B. Deng, "Coefficient-symmetries for implementing arbitrary-order Lagrange-type variable fractional-delay digital filters," IEEE Trans. on Signal Processing, vol.55, pp.4078-4090, Aug. 2007.

338

[4] T.B. Deng, "Symmetric structures for odd-order maximally flat and weighted-least-squares variable fractional-delay filters," IEEE Trans. on Circuits and Systems-I: Regular Papers, vol.54, pp.2718-2732, Dec. 2007.

[5] Y.D. Huang, S.C. Pei and J.J. Shyu, “WLS design of variable fractional-delay FIR filters using coefficient relationship,” IEEE Trans. on Circuits and Systems-II: Expressed Briefs, vol.56, pp.220-224, Mar. 2009.

[6] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products. Seventh Edition, Academic Press, 2007.

[7] P.S.R. Diniz, E.A.B. da Silva and S.L. Netto, Digital Signal Processing: System Analysis and Design, Cambridge University Press, 2002.

[8] Y.A. Melnikov, “A new approach to representation of trigonometric and hyperbolic functions by infinite product,” J. Math. Anal. Appl. vol.344, pp. 521-534, 2008.

Fig. 1 The Farrow structure I for implementing variable fractional delay filter with M=4, where delay p is adjustable.

-6 -4 -2 0 2 4 6-2

-1

0

1

2

x

sin(

x)

(a)

-6 -4 -2 0 2 4 6-2

-1

0

1

2

x

cos(

x)

(b)

Fig.2 The 10th partial product expansion approximations of Eq.(9) for trigonometric functions. (a) Sine case. (b) Cosine case. The solid lines are true curves of )cos( x and )sin( x . The dashed lines are

the curves of )10()(xEc and )10()(xEs .

-6 -4 -2 0 2 4 6-2

-1

0

1

2

x

sin(

x)

(a)

-6 -4 -2 0 2 4 6-2

-1

0

1

2

x

cos(

x)

(b)

Fig.3 The 10th partial product expansion approximations of Eq.(11) for trigonometric functions. (a) Sine case. (b) Cosine case. The solid lines are true curves of )cos( x and )sin( x . The dashed lines are

the curves of )10()(xMc and )10()(xMs . Fig. 4 The proposed structure II for implementing variable fractional delay filter with L=2, where delay p is adjustable.

p2

out

in

-p

F(z)

S(z)4/π2

S(z)4/(32π2)

S(z)4/(52π2)

S(z) 4/(22π2)

S(z) 4/(42π2)

p

out in

G2(z)

G1(z)

G0(z)

G3(z)

G4(z)

339

0 0.5 10

0.5

1

1.5

normalized frequency ω/π

|G0( ω

)|

0 0.5 10

1

2

3

4

normalized frequency ω/π|G

1( ω)|

0 0.5 10

1

2

3

4

5


|G2( ω

)|

0 0.5 10

1

2

3

4

5


|G3( ω

)|

0 0.5 10

1

2

3

4


|G4( ω

)|

0 0.5 10

0.5

1

1.5

2


|G5( ω

)|

Fig.5 The magnitudes responses of the sub-filters |)(| ωj

k eG , (k=0,1,2,...,5) which are designed by the least squares method.

00.2

0.40.6

0.8

-0.5

0

0.50

0.5

1

1.5

ω/πp

mag

nitu

de re

pons

e

0

0.20.4

0.60.8

-0.5

0

0.50

1

2

x 10-4

ω/πp

mag

nitu

de e

rror

(a) (b)

00.2

0.40.6

0.8

-0.5

0

0.5

-0.4

-0.2

0

0.2

0.4

0.6

ω/πp

grou

p de

lay

0

0.20.4

0.60.8

-0.5

0

0.50

0.5

1

1.5

x 10-3

ω/πp

dela

y er

ror

(c) (d) Fig.6 The designed results of the variable fractional delay filter in the Farrow structure I. (a) Magnitude response (b) Magnitude error (c) Group delay response (d) Group delay error.

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Normalized frequency ω/π

Mag

nitu

de |F

(ej ω

)|

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

Normalized frequency ω/π

Mag

nitu

de |S

(ej ω

)|

Fig.7 The magnitude responses of the designed two sub-filters

)( zS and )( zF .

00.2

0.40.6

0.8

-0.5

0

0.50

0.5

1

1.5

ω/πp

mag

nitu

de re

pons

e

00.2

0.40.6

0.8

-0.5

0

0.50

0.01

0.02

0.03

0.04

ω/pipm

agni

tude

erro

r

(a) (b)

00.2

0.40.6

0.8

-0.5

0

0.5

-0.4

-0.2

0

0.2

0.4

0.6

ω/pip

grou

p de

lay

00.2

0.40.6

0.8

-0.5

0

0.50

0.5

1

1.5

x 10-3

ω/πp

dela

y er

ror

(c) (d) Fig.8 The designed results of the variable fractional delay filter in the proposed structure II. (a) Magnitude response (b) Magnitude error (c) Group delay response (d) Group delay error.

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Design of Variable Fractional Delay Filter Using Infinite