An Improved Method for the Design of Variable Fractional-Delay IIR Digital Filters Soo-Chang Pei1, Jong-Jy Shyu2, Cheng-Han Chan3, Yun-Da Huang1 1
Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan, ROC Department of Electrical Engineering, National University of Kaohsiung, Kaohsiung, Taiwan, ROC 3 Department of Computer and Communication Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan, ROC e-mail:
[email protected], jshyu@ nuk.edu.tw,
[email protected],
[email protected] 2
Abstract—In this paper, an improved method is proposed for the design of variable fractional-delay (VFD) IIR digital filters. The objective error function is constituted directly by the root-mean-squared error function of variable frequency response and the stability constrained function, so that the root-mean-squared error of variable frequency response can be minimized while the designed VFD IIR filter is stable. Several design examples will be presented to demonstrate the effectiveness of the proposed method.
where I is a prescribed integer delay, 0 < ω p < π and the adjustable parameter p ∈ [ −0.5, 0.5] . To approximate (1), the transfer function of the IIR digital filter is characterized by
∑ = H ( z, p ) = A ( z, p ) 1 + ∑
(3a)
bn ( p ) = ∑ m = 0 b ( n, m ) p m , 0 ≤ n ≤ Nb .
(3b)
Hence, (2) becomes H ( z, p ) =
∑ ∑ b ( n, m ) p z 1 + ∑ ∑ a ( n, m ) p z M
Nb
m=0 M
n =0 Na
m=0
−n
m
m
−n
(4)
.
n =1
The frequency response of the designed VFD IIR filter can be represented by
∑ ∑ b ( n, m ) p e ω H (e , p) = 1 + ∑ ∑ a ( n, m ) p e ω jω
M
Nb
m =0 M
n =0 Na
m=0
=
m − jn
m − jn
n =1
bT cb (ω , p ) − jbT sb (ω , p )
(5)
1 + aT c a (ω , p ) − jaT s a (ω , p )
where the superscript T denotes a transpose operator,
a = ⎡⎣ a (1, 0 ) ," , a ( N a , 0 ) ," , a (1, M ) ," , a ( N a , M ) ⎤⎦
T
ca (ω , p ) = ⎡⎣ cos (ω ) ," , cos ( N aω ) ," , p M cos ( ω ) ," , p M cos ( N aω ) ⎤⎦
T
s a (ω , p ) = ⎡⎣sin (ω ) ," ,sin ( N aω ) ,"
II. PROBLEM FORMULATION For designing a VFD filter, the desired response can be represented by
978-1-4244-9472-9/11/$26.00 ©2011 IEEE
an ( p ) = ∑ m = 0 a ( n, m ) p m , 1 ≤ n ≤ N a M
This paper is organized as follows. In Section II, the problem is formulated. To design a stable VFD IIR filter, a constrained function has been incorporated into the objective error function. Then, several examples have been presented in Section III to demonstrate the effectiveness of the proposed method. Finally, the conclusions are given in Section IV.
, ω ∈ ⎡⎣0, ω p ⎤⎦
(2)
M
In this paper, we will focus on the design of VFD filters with the hybrid of IIR structure and Farrow structure, and an improved method of [22] will be proposed. The applied objective error function is formulated by combining directly the root-mean-squared error of variable frequency response and the stability constrained function, so that the root-mean-squared error of variable frequency response can be minimized while the stability problem for VFD IIR filter design can be overcome.
− j ( I + p )ω
b
n =0 n Na
in which the filter coefficients an ( p ) and bn ( p ) are expressed as the polynomials of p
I. INTRODUCTION For the past decade, the design of variable fractional-delay (VFD) digital filters became an important branch of digital signal processing due to their wide applications in signal processing and communication systems, such as comb filter design, tunable modulator and sample-rate converter [1]-[5]. In 1988, Farrow proposed an effective structure [6] which is applied to implement VFD digital filters, and the related design has made a great progress since then. First, an excellent tutorial paper by Laakso etc. [7] is proposed, then several works are presented including FIR-based design [8]-[12], allpass-based design [12][13]-[16], and IIR-based design [12][17]-[24] with their respective features.
H d (ω , p) = e
( p ) z−n a ( p ) z−n n =1 n
Nb
B ( z, p )
and
, p M sin ( ω ) ," , p M sin ( N aω ) ⎤⎦
cb (ω , p ) = ⎡⎣1,",cos ( Nbω ) ,", p M ,", p M cos ( Nbω )⎤⎦
394
(6b)
(6c)
T
b = ⎡⎣b ( 0, 0 ) ," , b ( Nb ,0 ) ," , b ( 0, M ) ," , b ( Nb , M ) ⎤⎦
(1)
(6a)
T
T
(6d) (6e)
system. It is found that ε m 2 attains its minimum value when α = 3.16228 × 10−9 which is used in this example. Fig. 1(b) presents the obtained magnitude response of variable frequency response, while the delay response and the absolute error of variable group-delay response are shown in Fig. 1(c) and (d), respectively. The errors for both the proposed method and Zhao-Kwan’s method [22] are the following: Proposed method:
Zhao-K wan's method[22]:
C PU time: 16.406 seconds ε m 2 = 0.0004676%
C PU time: 2.453 seconds
ε m = − 90.36dB ε τ = 0.0043
[3] [4]
[5] [6] [7]
ε m 2 = 0.006937% ε m = − 65.51dB ε τ = 0.0181
[8] [9]
To illustrate the stability of the designed filter, Fig. 1(e) presents the maximum pole radii for p ∈ [ −0.5,0.5] as well as that of [22]. It can be observed that all poles for p ∈ [ −0.5,0.5] are inside the unit circle so the designed VFD IIR filter is stable, and the proposed method achieves an improved design at a cost of increasing the maximum pole radii from around 0.4 in [22] to greater than 0.9 throughout all values of p.
[10]
Example 2: In this example, an allpass-like VFD filter is designed with N a = N b = I = 35 , M=5, ω p = 0.9π and
[13]
[11] [12]
W ( ω ) = 1 . Also as in [22], the designed filter is stable even when α = 0 . The magnitude response, the absolute error of variable frequency response, the group-delay response, the absolute error of variable group-delay response and the maximum pole radii are shown in Fig. 2, and the errors defined in (21) are the following: Proposed method: CPU time: 19.594 seconds
[14] [15] [16]
Zhao-Kwan's method [22]: CPU time: 2.328 seconds
ε m 2 = 0.0005211% ε m = −82.90dB ετ = 0.0219
[17]
ε m 2 = 0.000623% ε m = −76.49dB ετ = 0.0288
[18]
[19]
IV. CONCLUSIONS In this paper, an improved method has been successfully used to design VFD IIR filters. By combining the root-mean-squared error function of variable frequency response and the proposed constrained error function, the presented experiments show that the performance of the designed filters is satisfactory as well as the stability problem can be effectively overcome. Obviously, the proposed method can also be applied to design other types of variable IIR digital filters.
[20] [21] [22] [23] [24]
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