Minimax Design of Variable Fractional-Delay FIR Digital Filters by
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
693
Minimax Design of Variable Fractional-Delay FIR Digital Filters by Iterative Weighted Least-Squares Approach Jong-Jy Shyu, Member, IEEE, Soo-Chang Pei, Fellow, IEEE, Cheng-Han Chan, and Yun-Da Huang
Abstract—In this letter, an iterative weighted least-squares (LS) approach is proposed for the design of variable fractional-delay (VFD) finite impulse response (FIR) digital filters in minimax sense. For deriving the elements of relevant matrices, there is no need for numerical integration or closed-form formulation and they obey given specifications. Comparing with the existing method, the proposed method does not lead to a misleading problem, and the performance of convergence is also better. Especially, it is found that the used LS weighting function is only dependent on the frequency, not on the tunable parameter, which makes it easier for the minimax design of VFD FIR digital filters. Index Terms—Finite impulse response (FIR) digital filter, iterative weighted least-squares approach, variable filter, variable fractional-delay filter.
that the proposed method can be applied to design VFD FIR digital filters in weighted LS sense. Moreover, the minimax design can also be achieved by applying a proper iterative process [18]. The proposed method is based on the Farrow structure, which is different from [12] where two-stage structure is used. Comparing with the existing method [13], there is no misleading problem and the performance of convergence is also better for the proposed method. To demonstrate the effectiveness of the proposed method, a design example is presented and it is found that the used weighting function depends only on the frequency , not on the tunable parameter for the minimax design of VFD FIR digital filters. II. PROBLEM FORMULATION AND NUMERICAL EXAMPLE
I. INTRODUCTION ECENTLY, the design of variable fractional-delay (VFD) digital filters has received considerable attention due to their wide applications in signal processing and communication systems [1]–[13]. The VFD digital filter is a branch of variable digital filters which are applied to where the frequency characteristics need to be adjustable. Generally, the system is implemented by the Farrow structure [1], [5] where a parameter is used to change continuously the delay online without redesigning a new filter. Among the existing literature, [12] and [13] deal with the design of VFD finite impulse response (FIR) digital filters in minimax sense where the maximum absolute error of variable frequency response is minimized as much as possible. In the mean time, some techniques are also proposed for the minimax design of other variable digital filters such as adjustable-bandwidth FIR filters [14], [15]. In this letter, the minimax design of VFD FIR digital filters is investigated again. The proposed method is originally used to design quadrantally symmetric 2-D FIR filters and higher-dimensional FIR filters in LS sense [16], [17]. However, we do not derive the closed- form formulations as in [16] and [17], and the weighting function is incorporated in the modified algorithm so
R
For designing a VFD FIR filter, the desired response is given by
(1) where and are the order and the variable fractional-delay of the designed FIR digital filter, respectively. To approximate the desired response in (1), the used transfer function is characterized by (2) where the coefficients of
are expressed as the polynomials
(3) hence (2) becomes
Manuscript received May 15, 2008; revised July 14, 2008. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Per Lowenborg. J.-J. Shyu and Y.-D. Huang are with the Department of Electrical Engineering, National University of Kaohsiung, Kaohsiung, Taiwan (e-mail: [email protected]; [email protected]). S.-C. Pei is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). C.-H. Chan is with the Department of Computer and Communication Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/LSP.2008.2005049
(4) which can be implemented by the Farrow structure [1]. In this letter, only even is considered, and the case for odd can be developed in a similar manner. By the symmetric/antisymmetric
1070-9908/$25.00 © 2008 IEEE
694
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
properties of (1) with respect to , (4) can be divided into two subfunctions as follows:
Let and defined by
be
and
matrices (12a)
and (5)
(12b) respectively, the following objective error function is used in this letter:
where for even for odd .
(6)
Also, according to the frequency characteristics of (1), the frequency response of the designed system can be formulated into
(13) where
(7) where (14a)
(8a) (14b) a
(8b) Furthermore, substituting
into (1) and (7)
grid is chosen for the error evaluation and is a positive weighting function. In this letter, and are used. It is especially noted that the weighting function depends only on the transform-domain parameter , not on the tunable variable for the minimax design, which will be shown later. Equation (14) can be expressed in matrix forms as
(9a) and (9b) (15a) which lead to
and (10)
Hence, (7) can be represented by [8]
(15b) (11)
where denotes a trace operator, the superscript a transpose operator
denotes
SHYU et al.: MINIMAX DESIGN OF VARIABLE FRACTIONAL-DELAY FIR DIGITAL FILTERS
695
for the first iteration only, and search for , , and for all iterations. is nearly Step 4) Check whether the error equiripple (such that the peak absolute error of variable frequency response can be minimized) by
Step 3) Find
(16a)
(16b)
(16c)
where is a preassigned very small positive constant. If the condition is satisfied, stop the process; otherwise, go to the next step. Step 5) Compute the unnormalized weighting function
(16d) where is the number of ripples in its maximum value
(16e)
, and find
and (16f) Then update the weighting function by
For the given objective error function (13), the solution can be obtained by differentiating with respect to and setting the result to zero, which yields (17) Similarly (18) To design a VFD FIR filter in minimax sense, the iterative method in [18] is modified and applied. Before describing the proposed algorithm, some notations are defined as follows: the absolute error of variable frequency response defined by
the variable where the maximum of occurs for the first iteration; the th absolute error ripple of with (except that the first ripple interval ); ripple interval is ; ; the relative peak error in variable
defined by
The proposed iterative method is described as follows. Step 1) Initiate the weighting function
Step 2) Find the coefficient matrices (18), respectively.
and
by (17) and
and go to Step 2. To evaluate the performance, the maximum absolute error of variable frequency response and the maximum delay error are defined by (19a) and
(19b) is the actual group-delay of respectively, where . For the computation of (19), the frequency and the variable are uniformly sampled at step sizes and 0.5/200, respectively, where is the order of the designed filter. Example: We consider the design of a VFD FIR filter , , in the example. with , During the first iteration, it is found that , , and the . If relative peak error is used, the design process stops at the fifth iteration , , and when . Fig. 1(a) and (b) presents the final magnitude and group-delay responses, while the absolute errors of variable frequency response for the first iteration and the fifth iteration are shown in Fig. 1(c) and (d), respectively. From Fig. 1(d), it is found that the absolute errors of frequency response for all variable are almost equiripple, which can be , 0.3, 0.2, and further judged by the error curves for 0.1 shown in Fig. 2(a)–(d), respectively. Also, the curve for for is illustrated in the relative peak error Fig. 2(e) which shows the excellent equiripple characteristic of the obtained result. For showing the good convergence of the iterative method, Fig. 2(f) presents the trace of the relative peak
696
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
, is comparable to that imum absolute error, , there is no misleading problem in [13] where for the proposed method, and the convergence is also better than in the method in [13] where eight iterations are needed. III. CONCLUSIONS In this letter, an iterative weighted LS approach has been successfully applied to the minimax design of VFD FIR digital filters. Conventionally, a two-dimensional weighting function was generally used for the VFD FIR filter design by weighted leastsquares approach. However, in fact, it is found that the weighting function is one-dimensional function of the frequency for the minimax design of VFD FIR digital filters, which makes the design easier. A design example has been presented to demonstrate the effectiveness and good convergence of the proposed method.
N = 60 M = 9
Fig. 1. Minimax design of a VFD FIR digital filter with , , and . (a) Magnitude response. (b) Variable fractional-delay response. (c) Absolute error of variable frequency response for the first iteration. (d) Absolute error of variable frequency response for the fifth iteration.
! = 0:9
N = 60, M = 9, and p = 0:5 p = 0:3, (c) p = 0:2, (p) 0 p 0:5. (f)
Fig. 2. Minimax design of a VFD FIR digital filter with . The error curves for (a) , (b) . (e) The relative peak error curve for and (d) Trace of the relative peak error .
! = 0:9 p = 0:1
E(!; p) (p )
error which varies between and after the sixth iteration. For comparing with the existing method [13], the VFD filter , , and the process stops is designed again with is used. Beside the maxat the fourth iteration when
REFERENCES [1] C. W. Farrow, “A continuously variable digital delay element,” in Proc. ISCAS, 1988, pp. 2641–2645. [2] T. I. Laakso, V. Välimäki, M. Karjalainen, and U. K. Laine, “Splitting the unit delay: Tools for fractional delay filter design,” IEEE Signal Process. Mag., vol. 13, no. 1, pp. 30–60, Jan. 1996. [3] W.-S. Lu and T.-B. Deng, “An improved weighted least-squares design for variable fractional delay FIR filters,” IEEE Trans. Circuits Syst. I, Analog Digit. Signal Process., vol. 46, no. 8, pp. 1035–1040, Aug. 1999. [4] T.-B. Deng, “Discretization-free design of variable fractional-delay FIR digital filters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 48, no. 6, pp. 637–644, Jun. 2001. [5] C. K. S. Pun, Y. C. Wu, S. C. Chan, and K. L. Ho, “On the design and efficient implementation of the farrow structure,” IEEE Signal Process. Lett., vol. 10, no. 7, pp. 189–192, Jul. 2003. [6] C.-C. Tseng, “Design of variable fractional delay FIR filter using symmetry,” in Proc. ISCAS, 2004, pp. 477–480. [7] H. Zhao and J. Yu, “A simple and efficient design of variable fractionaldelay FIR filters,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 2, pp. 157–160, Feb. 2006. [8] T.-B. Deng and Y. Lian, “Weighted-least-squares design of variable fractional-delay FIR filters using coefficient symmetry,” IEEE Trans. Signal Process., vol. 54, no. 8, pp. 3023–3038, Aug. 2006. [9] T.-B. Deng, “Symmetric structures for odd-order maximally flat and weighted-least- squares variable fractional-delay filters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 12, pp. 2718–2732, Dec. 2007. [10] J. T. Olkkonen and H. Olkkonen, “Fractional delay filter based on the B-spline transform,” IEEE Signal Process. Lett., vol. 14, no. 2, pp. 97–100, Feb. 2007. [11] J. T. Olkkonen and H. Olkkonen, “Fractional time-shift B-spline filter,” IEEE Signal Process. Lett., vol. 14, no. 10, pp. 688–691, Oct. 2007. [12] E. Hermanowicz and H. Johansson, “On designing minimax adjustable wideband fractional delay FIR filters using two-rate approach,” in Proc. Eur. Conf. Circuit Theory and Design, 2005, pp. III/437–III/440. [13] J.-C. Liu and S.-J. You, “Weighted least squares near-equiripple approximation of variable fractional delay FIR filters,” IET Signal Process., vol. 1, no. 2, pp. 66–72, Jun. 2007. [14] P. Löwenborg and H. Johansson, “Minimax design of adjustablebandwidth linear- phase FIR filters,” IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 53, no. 2, pp. 431–439, Feb. 2006. [15] H. H. Dam, A. Cantoni, K. L. Teo, and S. Norholm, “Variable digital filter with least-square criterion and peak gain constraints,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 1, pp. 24–28, Jan. 2007. [16] M. O. Ahmad and J.-D. Wang, “An analytical least square solution to the design problem of two-dimensional FIR filters with quadrantally symmetric or antisymmetric frequency response,” IEEE Trans. Circuits Syst., vol. 36, no. 7, pp. 968–979, Jul. 1989. [17] S.-C. Pei and J.-J. Shyu, “Efficient fast design of three- and higher-dimensional FIR digital filters by analytical least squares method,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 43, no. 4, pp. 339–344, Apr. 1996. [18] C. Y. Chi and Y. T. Kou, “A new self-initiated optimum WLS approximation method for the design of linear phase FIR digital filters,” in Proc. ISCAS, 1991, pp. 168–217.
693
Minimax Design of Variable Fractional-Delay FIR Digital Filters by Iterative Weighted Least-Squares Approach Jong-Jy Shyu, Member, IEEE, Soo-Chang Pei, Fellow, IEEE, Cheng-Han Chan, and Yun-Da Huang
Abstract—In this letter, an iterative weighted least-squares (LS) approach is proposed for the design of variable fractional-delay (VFD) finite impulse response (FIR) digital filters in minimax sense. For deriving the elements of relevant matrices, there is no need for numerical integration or closed-form formulation and they obey given specifications. Comparing with the existing method, the proposed method does not lead to a misleading problem, and the performance of convergence is also better. Especially, it is found that the used LS weighting function is only dependent on the frequency, not on the tunable parameter, which makes it easier for the minimax design of VFD FIR digital filters. Index Terms—Finite impulse response (FIR) digital filter, iterative weighted least-squares approach, variable filter, variable fractional-delay filter.
that the proposed method can be applied to design VFD FIR digital filters in weighted LS sense. Moreover, the minimax design can also be achieved by applying a proper iterative process [18]. The proposed method is based on the Farrow structure, which is different from [12] where two-stage structure is used. Comparing with the existing method [13], there is no misleading problem and the performance of convergence is also better for the proposed method. To demonstrate the effectiveness of the proposed method, a design example is presented and it is found that the used weighting function depends only on the frequency , not on the tunable parameter for the minimax design of VFD FIR digital filters. II. PROBLEM FORMULATION AND NUMERICAL EXAMPLE
I. INTRODUCTION ECENTLY, the design of variable fractional-delay (VFD) digital filters has received considerable attention due to their wide applications in signal processing and communication systems [1]–[13]. The VFD digital filter is a branch of variable digital filters which are applied to where the frequency characteristics need to be adjustable. Generally, the system is implemented by the Farrow structure [1], [5] where a parameter is used to change continuously the delay online without redesigning a new filter. Among the existing literature, [12] and [13] deal with the design of VFD finite impulse response (FIR) digital filters in minimax sense where the maximum absolute error of variable frequency response is minimized as much as possible. In the mean time, some techniques are also proposed for the minimax design of other variable digital filters such as adjustable-bandwidth FIR filters [14], [15]. In this letter, the minimax design of VFD FIR digital filters is investigated again. The proposed method is originally used to design quadrantally symmetric 2-D FIR filters and higher-dimensional FIR filters in LS sense [16], [17]. However, we do not derive the closed- form formulations as in [16] and [17], and the weighting function is incorporated in the modified algorithm so
R
For designing a VFD FIR filter, the desired response is given by
(1) where and are the order and the variable fractional-delay of the designed FIR digital filter, respectively. To approximate the desired response in (1), the used transfer function is characterized by (2) where the coefficients of
are expressed as the polynomials
(3) hence (2) becomes
Manuscript received May 15, 2008; revised July 14, 2008. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Per Lowenborg. J.-J. Shyu and Y.-D. Huang are with the Department of Electrical Engineering, National University of Kaohsiung, Kaohsiung, Taiwan (e-mail: [email protected]; [email protected]). S.-C. Pei is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). C.-H. Chan is with the Department of Computer and Communication Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/LSP.2008.2005049
(4) which can be implemented by the Farrow structure [1]. In this letter, only even is considered, and the case for odd can be developed in a similar manner. By the symmetric/antisymmetric
1070-9908/$25.00 © 2008 IEEE
694
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
properties of (1) with respect to , (4) can be divided into two subfunctions as follows:
Let and defined by
be
and
matrices (12a)
and (5)
(12b) respectively, the following objective error function is used in this letter:
where for even for odd .
(6)
Also, according to the frequency characteristics of (1), the frequency response of the designed system can be formulated into
(13) where
(7) where (14a)
(8a) (14b) a
(8b) Furthermore, substituting
into (1) and (7)
grid is chosen for the error evaluation and is a positive weighting function. In this letter, and are used. It is especially noted that the weighting function depends only on the transform-domain parameter , not on the tunable variable for the minimax design, which will be shown later. Equation (14) can be expressed in matrix forms as
(9a) and (9b) (15a) which lead to
and (10)
Hence, (7) can be represented by [8]
(15b) (11)
where denotes a trace operator, the superscript a transpose operator
denotes
SHYU et al.: MINIMAX DESIGN OF VARIABLE FRACTIONAL-DELAY FIR DIGITAL FILTERS
695
for the first iteration only, and search for , , and for all iterations. is nearly Step 4) Check whether the error equiripple (such that the peak absolute error of variable frequency response can be minimized) by
Step 3) Find
(16a)
(16b)
(16c)
where is a preassigned very small positive constant. If the condition is satisfied, stop the process; otherwise, go to the next step. Step 5) Compute the unnormalized weighting function
(16d) where is the number of ripples in its maximum value
(16e)
, and find
and (16f) Then update the weighting function by
For the given objective error function (13), the solution can be obtained by differentiating with respect to and setting the result to zero, which yields (17) Similarly (18) To design a VFD FIR filter in minimax sense, the iterative method in [18] is modified and applied. Before describing the proposed algorithm, some notations are defined as follows: the absolute error of variable frequency response defined by
the variable where the maximum of occurs for the first iteration; the th absolute error ripple of with (except that the first ripple interval ); ripple interval is ; ; the relative peak error in variable
defined by
The proposed iterative method is described as follows. Step 1) Initiate the weighting function
Step 2) Find the coefficient matrices (18), respectively.
and
by (17) and
and go to Step 2. To evaluate the performance, the maximum absolute error of variable frequency response and the maximum delay error are defined by (19a) and
(19b) is the actual group-delay of respectively, where . For the computation of (19), the frequency and the variable are uniformly sampled at step sizes and 0.5/200, respectively, where is the order of the designed filter. Example: We consider the design of a VFD FIR filter , , in the example. with , During the first iteration, it is found that , , and the . If relative peak error is used, the design process stops at the fifth iteration , , and when . Fig. 1(a) and (b) presents the final magnitude and group-delay responses, while the absolute errors of variable frequency response for the first iteration and the fifth iteration are shown in Fig. 1(c) and (d), respectively. From Fig. 1(d), it is found that the absolute errors of frequency response for all variable are almost equiripple, which can be , 0.3, 0.2, and further judged by the error curves for 0.1 shown in Fig. 2(a)–(d), respectively. Also, the curve for for is illustrated in the relative peak error Fig. 2(e) which shows the excellent equiripple characteristic of the obtained result. For showing the good convergence of the iterative method, Fig. 2(f) presents the trace of the relative peak
696
IEEE SIGNAL PROCESSING LETTERS, VOL. 15, 2008
, is comparable to that imum absolute error, , there is no misleading problem in [13] where for the proposed method, and the convergence is also better than in the method in [13] where eight iterations are needed. III. CONCLUSIONS In this letter, an iterative weighted LS approach has been successfully applied to the minimax design of VFD FIR digital filters. Conventionally, a two-dimensional weighting function was generally used for the VFD FIR filter design by weighted leastsquares approach. However, in fact, it is found that the weighting function is one-dimensional function of the frequency for the minimax design of VFD FIR digital filters, which makes the design easier. A design example has been presented to demonstrate the effectiveness and good convergence of the proposed method.
N = 60 M = 9
Fig. 1. Minimax design of a VFD FIR digital filter with , , and . (a) Magnitude response. (b) Variable fractional-delay response. (c) Absolute error of variable frequency response for the first iteration. (d) Absolute error of variable frequency response for the fifth iteration.
! = 0:9
N = 60, M = 9, and p = 0:5 p = 0:3, (c) p = 0:2, (p) 0 p 0:5. (f)
Fig. 2. Minimax design of a VFD FIR digital filter with . The error curves for (a) , (b) . (e) The relative peak error curve for and (d) Trace of the relative peak error .
! = 0:9 p = 0:1
E(!; p) (p )
error which varies between and after the sixth iteration. For comparing with the existing method [13], the VFD filter , , and the process stops is designed again with is used. Beside the maxat the fourth iteration when
REFERENCES [1] C. W. Farrow, “A continuously variable digital delay element,” in Proc. ISCAS, 1988, pp. 2641–2645. [2] T. I. Laakso, V. Välimäki, M. Karjalainen, and U. K. Laine, “Splitting the unit delay: Tools for fractional delay filter design,” IEEE Signal Process. Mag., vol. 13, no. 1, pp. 30–60, Jan. 1996. [3] W.-S. Lu and T.-B. Deng, “An improved weighted least-squares design for variable fractional delay FIR filters,” IEEE Trans. Circuits Syst. I, Analog Digit. Signal Process., vol. 46, no. 8, pp. 1035–1040, Aug. 1999. [4] T.-B. Deng, “Discretization-free design of variable fractional-delay FIR digital filters,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 48, no. 6, pp. 637–644, Jun. 2001. [5] C. K. S. Pun, Y. C. Wu, S. C. Chan, and K. L. Ho, “On the design and efficient implementation of the farrow structure,” IEEE Signal Process. Lett., vol. 10, no. 7, pp. 189–192, Jul. 2003. [6] C.-C. Tseng, “Design of variable fractional delay FIR filter using symmetry,” in Proc. ISCAS, 2004, pp. 477–480. [7] H. Zhao and J. Yu, “A simple and efficient design of variable fractionaldelay FIR filters,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 53, no. 2, pp. 157–160, Feb. 2006. [8] T.-B. Deng and Y. Lian, “Weighted-least-squares design of variable fractional-delay FIR filters using coefficient symmetry,” IEEE Trans. Signal Process., vol. 54, no. 8, pp. 3023–3038, Aug. 2006. [9] T.-B. Deng, “Symmetric structures for odd-order maximally flat and weighted-least- squares variable fractional-delay filters,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 54, no. 12, pp. 2718–2732, Dec. 2007. [10] J. T. Olkkonen and H. Olkkonen, “Fractional delay filter based on the B-spline transform,” IEEE Signal Process. Lett., vol. 14, no. 2, pp. 97–100, Feb. 2007. [11] J. T. Olkkonen and H. Olkkonen, “Fractional time-shift B-spline filter,” IEEE Signal Process. Lett., vol. 14, no. 10, pp. 688–691, Oct. 2007. [12] E. Hermanowicz and H. Johansson, “On designing minimax adjustable wideband fractional delay FIR filters using two-rate approach,” in Proc. Eur. Conf. Circuit Theory and Design, 2005, pp. III/437–III/440. [13] J.-C. Liu and S.-J. You, “Weighted least squares near-equiripple approximation of variable fractional delay FIR filters,” IET Signal Process., vol. 1, no. 2, pp. 66–72, Jun. 2007. [14] P. Löwenborg and H. Johansson, “Minimax design of adjustablebandwidth linear- phase FIR filters,” IEEE Trans. Circuits Syst. I, Reg. Paper, vol. 53, no. 2, pp. 431–439, Feb. 2006. [15] H. H. Dam, A. Cantoni, K. L. Teo, and S. Norholm, “Variable digital filter with least-square criterion and peak gain constraints,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 54, no. 1, pp. 24–28, Jan. 2007. [16] M. O. Ahmad and J.-D. Wang, “An analytical least square solution to the design problem of two-dimensional FIR filters with quadrantally symmetric or antisymmetric frequency response,” IEEE Trans. Circuits Syst., vol. 36, no. 7, pp. 968–979, Jul. 1989. [17] S.-C. Pei and J.-J. Shyu, “Efficient fast design of three- and higher-dimensional FIR digital filters by analytical least squares method,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 43, no. 4, pp. 339–344, Apr. 1996. [18] C. Y. Chi and Y. T. Kou, “A new self-initiated optimum WLS approximation method for the design of linear phase FIR digital filters,” in Proc. ISCAS, 1991, pp. 168–217.
