Optimal Design of Complex FIR Filters With
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Wei Rong Lee, Lou Caccetta, Kok Lay Teo, Senior Member, IEEE, and Volker Rehbock
Abstract—This paper presents a method for the frequency-domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses. The method can deal with both the equiripple design problem and the peak constrained least squares (PCLS) design problem. Consequently, the method can also be applied to the equiripple passbands and PCLS stopbands design problem as a special case of the PCLS design. Both the equiripple and the PCLS design problems are converted into weighted least squares optimization problems. They are then solved iteratively with appropriately updated error weighting functions. A novel scheme for updating the error weighting function is developed to incorporate the design requirements. Design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods.
K. L. Teo was with the Department of Applied Mathematics and the Center for Multimedia Signal Processing, Department of Electronics and Information Technology, The Hong Kong Polytechnic University, Kowloon, Hong Kong. He is currently with the Department of Mathematics and Statistics, Curtin University of Technology, Perth 6845, Australia.
W. R. Lee, L. Caccetta, and V. Rehbock are with the Western Australian Centre of Excellence in Industrial Optimization, Department of Mathematics and Statistics, Curtin University of Technology, Perth 6845, Australia.
IEEE TRANSACTIONS ONຫໍສະໝຸດ BaiduSIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
1617
Optimal Design of Complex FIR Filters With Arbitrary Magnitude and Group Delay Responses
Index Terms—Complex finite impulse response (FIR) filters, optimal design, weighted least squares.
I. INTRODUCTION
L INEAR-phase finite impulse response (FIR) digital filters are frequently used in signal processing applications because of their guaranteed stability and freedom from phase distortion. However, linear-phase filters introduce large signal delays when long filters are required. Moreover, the linear-phase restriction is not needed in the stopbands. Imposing the linear-phase requirements only in the passbands results in complex FIR filter designs. Examples include the equiripple design, the peak constrained least squares (PCLS) design [2], and the equiripple passbands and PCLS stopbands (EPPCLSS) design [3]. The need for such designs arises in various applications. They have been extensively studied [3]–[5]. Several methods have been proposed for the solution of these optimization problems. These include linear programming [6]–[9], the multiple exchange algorithm [4], [10]–[13], Lawson’s algorithm [14]–[17], quadratic programming [18], [19], the implicit multiple exchange algorithms [20]–[22], semidefinite programming [3], and a method based on alternate iterations in the time and frequency domains [23]. While these
Digital Object Identifier 10.1109/TSP.2006.872542
methods have generated good solutions, the design criteria used in these paper are not the most appropriate in terms of minimizing the weighted peak ripple (WPR) of the magnitude and the maximum phase deviation from the linear phase in the passbands. This is because the minimum Chebyshev error is not equivalent to the minimum WPR with the maximum phase deviation from the linear phase being less than or equal to the peak ripple of the magnitude in each individual passband. Here, we just compare the values of the phase deviation and the peak ripple of the magnitude where the phase is measured in radians.
In this paper, we first reformulate the equiripple design problem. The error criteria used are the WPR of the magnitude and the maximum phase deviation from the linear phase in the passbands. This reformulation leads to a better equiripple design formulation. A new computational method is then developed for solving these complex FIR filter design problems. The method is based on the ripple-weighted approach introduced in [1] and [24]. We show that this method, with slight modifications, can be readily used to solve the PCLS design problem. Consequently, it can also be used to solve the EPPCLSS design problem. All of these problems are effectively converted into corresponding weighted least squares (WLS) problems. They are then solved iteratively. The success of the WLS approach depends on the appropriate error weighting function. In [26], Algazi et al. pointed out a formal relation between WLS design and equiripple design through Lawson’s algorithm [25]. Lim et al. in [27] presented a novel scheme to derive the error weighting function for an equiripple design. Diniz and Netto presented a modified scheme in [28] for solving WLS-Chebyshev FIR digital filters. However, the schemes proposed in these papers only deal with the magnitude of the filter. For the problems addressed in this paper, both the magnitude and the phase deviation from the linear phase should be dealt with. We thus propose a novel scheme to update the error weighting functions for the new formulation and the design requirements for both the equiripple design and the PCLS design. At each iteration, the error weighting function is updated according to the design requirements and the resulting LS problem is solved. It is well known that the LS solution can be obtained analytically. It is also well known that the large number of frequency points required for accurate design of a long filter does not increase the size of the linear system to be solved at each iteration. Consequently, one does not increase the complexity of the problem to be solved. The only computational tasks involved in our method are to update a sequence of error weighting functions and to evaluate a sequence of analytic solutions of LS problems. Furthermore, the error weighting function for the new scheme does not change for most frequencies at each
Manuscript received November 22, 2004; revised June 15, 2005. The work of K. L. Teo was supported by the Research Grant Council of Hong Kong under Research Grant Ployu5247/04E. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Mariane R. Petraglia
Abstract—This paper presents a method for the frequency-domain design of digital finite impulse response filters with arbitrary magnitude and group delay responses. The method can deal with both the equiripple design problem and the peak constrained least squares (PCLS) design problem. Consequently, the method can also be applied to the equiripple passbands and PCLS stopbands design problem as a special case of the PCLS design. Both the equiripple and the PCLS design problems are converted into weighted least squares optimization problems. They are then solved iteratively with appropriately updated error weighting functions. A novel scheme for updating the error weighting function is developed to incorporate the design requirements. Design examples are included in order to compare the performance of the filters designed using the proposed scheme and several other existing methods.
K. L. Teo was with the Department of Applied Mathematics and the Center for Multimedia Signal Processing, Department of Electronics and Information Technology, The Hong Kong Polytechnic University, Kowloon, Hong Kong. He is currently with the Department of Mathematics and Statistics, Curtin University of Technology, Perth 6845, Australia.
W. R. Lee, L. Caccetta, and V. Rehbock are with the Western Australian Centre of Excellence in Industrial Optimization, Department of Mathematics and Statistics, Curtin University of Technology, Perth 6845, Australia.
IEEE TRANSACTIONS ONຫໍສະໝຸດ BaiduSIGNAL PROCESSING, VOL. 54, NO. 5, MAY 2006
1617
Optimal Design of Complex FIR Filters With Arbitrary Magnitude and Group Delay Responses
Index Terms—Complex finite impulse response (FIR) filters, optimal design, weighted least squares.
I. INTRODUCTION
L INEAR-phase finite impulse response (FIR) digital filters are frequently used in signal processing applications because of their guaranteed stability and freedom from phase distortion. However, linear-phase filters introduce large signal delays when long filters are required. Moreover, the linear-phase restriction is not needed in the stopbands. Imposing the linear-phase requirements only in the passbands results in complex FIR filter designs. Examples include the equiripple design, the peak constrained least squares (PCLS) design [2], and the equiripple passbands and PCLS stopbands (EPPCLSS) design [3]. The need for such designs arises in various applications. They have been extensively studied [3]–[5]. Several methods have been proposed for the solution of these optimization problems. These include linear programming [6]–[9], the multiple exchange algorithm [4], [10]–[13], Lawson’s algorithm [14]–[17], quadratic programming [18], [19], the implicit multiple exchange algorithms [20]–[22], semidefinite programming [3], and a method based on alternate iterations in the time and frequency domains [23]. While these
Digital Object Identifier 10.1109/TSP.2006.872542
methods have generated good solutions, the design criteria used in these paper are not the most appropriate in terms of minimizing the weighted peak ripple (WPR) of the magnitude and the maximum phase deviation from the linear phase in the passbands. This is because the minimum Chebyshev error is not equivalent to the minimum WPR with the maximum phase deviation from the linear phase being less than or equal to the peak ripple of the magnitude in each individual passband. Here, we just compare the values of the phase deviation and the peak ripple of the magnitude where the phase is measured in radians.
In this paper, we first reformulate the equiripple design problem. The error criteria used are the WPR of the magnitude and the maximum phase deviation from the linear phase in the passbands. This reformulation leads to a better equiripple design formulation. A new computational method is then developed for solving these complex FIR filter design problems. The method is based on the ripple-weighted approach introduced in [1] and [24]. We show that this method, with slight modifications, can be readily used to solve the PCLS design problem. Consequently, it can also be used to solve the EPPCLSS design problem. All of these problems are effectively converted into corresponding weighted least squares (WLS) problems. They are then solved iteratively. The success of the WLS approach depends on the appropriate error weighting function. In [26], Algazi et al. pointed out a formal relation between WLS design and equiripple design through Lawson’s algorithm [25]. Lim et al. in [27] presented a novel scheme to derive the error weighting function for an equiripple design. Diniz and Netto presented a modified scheme in [28] for solving WLS-Chebyshev FIR digital filters. However, the schemes proposed in these papers only deal with the magnitude of the filter. For the problems addressed in this paper, both the magnitude and the phase deviation from the linear phase should be dealt with. We thus propose a novel scheme to update the error weighting functions for the new formulation and the design requirements for both the equiripple design and the PCLS design. At each iteration, the error weighting function is updated according to the design requirements and the resulting LS problem is solved. It is well known that the LS solution can be obtained analytically. It is also well known that the large number of frequency points required for accurate design of a long filter does not increase the size of the linear system to be solved at each iteration. Consequently, one does not increase the complexity of the problem to be solved. The only computational tasks involved in our method are to update a sequence of error weighting functions and to evaluate a sequence of analytic solutions of LS problems. Furthermore, the error weighting function for the new scheme does not change for most frequencies at each
Manuscript received November 22, 2004; revised June 15, 2005. The work of K. L. Teo was supported by the Research Grant Council of Hong Kong under Research Grant Ployu5247/04E. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Mariane R. Petraglia