An efficient implementation of Lawson's algorithm with application to complex Chebyshev FIR filter design

This paper presents an efficient implementation of Lawson's algorithm and illustrates its application in complex Chebyshev FIR filter design. It is shown that the update terms in Lawson's algorithm can be efficiently achieved by computing a proper subspace projection, provided that the number of points involved in Lawson's algorithm is sufficiently small (not much greater than n where n is the number of variables). An application of this particular implementation form for Lawson's algorithm is then demonstrated by using it to solve the subproblems involved in a multiple exchange algorithm. In particular, this exchange algorithm is based on generalizing Remez second exchange algorithm to the complex case which requires solving a sequence of subproblems where each subproblem is itself a complex Chebyshev approximation problem defined over a finite number of points; the subproblems are systematically defined using a simple exchange procedure. The effectiveness of this multiple exchange algorithm for designing complex FIR filters is illustrated through various design examples, including a long filter with length n=125. >

[1]  H. S. Shapiro,et al.  A Unified Approach to Certain Problems of Approximation and Minimization , 1961 .

[2]  Yong Ching Lim,et al.  A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design , 1992, IEEE Trans. Signal Process..

[3]  P. Tang.  A fast algorithm for linear complex Chebyshev approximations , 1988 .

[4]  G. Lorentz.  Approximation of Functions , 1966 .

[5]  A. K. Cline.  Rate of Convergence of Lawson's Algorithm , 1972 .

[6]  J. Rice,et al.  The Lawson algorithm and extensions , 1968 .

[7]  Thomas W. Parks,et al.  Accelerated design of FIR filters in the complex domain , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[8]  John S. D. Mason,et al.  Complex Chebyshev approximation for FIR digital filters , 1991, IEEE Trans. Signal Process..

[9]  Ching-Yih Tseng.  Further results on complex Chebyshev FIR filter design using a multiple exchange algorithm , 1993, 1993 IEEE International Symposium on Circuits and Systems.

[10]  Soo-Chang Pei,et al.  Design of real FIR filters with arbitrary complex frequency responses by two real Chebyshev approximations , 1992, Signal Process..

[11]  John David Fisher.  DESIGN OF FINITE-IMPULSE RESPONSE DIGITAL FILTERS , 1974 .

[12]  C.-Y. Tseng.  A numerical algorithm for complex Chebyshev FIR filter design , 1992, [Proceedings] 1992 IEEE International Symposium on Circuits and Systems.

[13]  Thomas W. Parks,et al.  Design of FIR filters in the complex domain , 1987, IEEE Trans. Acoust. Speech Signal Process..

[14]  J. McClellan,et al.  Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase , 1972 .

[15]  Matthias Schulist,et al.  Improvements of a complex FIR filter design algorithm , 1990 .

[16]  Ching-Yih Tseng,et al.  A Multiple-Exchange Algorithm for Complex Chebyshev Approximation by Polynomials on the Unit Circle , 1996 .

[17]  Klaus Preuss,et al.  On the design of FIR filters by complex Chebyshev approximation , 1989, IEEE Trans. Acoust. Speech Signal Process..

[18]  C. K. Yuen,et al.  Theory and Application of Digital Signal Processing , 1978, IEEE Transactions on Systems, Man, and Cybernetics.

[19]  E. Cheney.  Introduction to approximation theory , 1966 .

[20]  L. Trefethen.  Near-circularity of the error curve in complex Chebyshev approximation , 1981 .

[21]  Jae S. Lim,et al.  Design of FIR filters by complex Chebyshev approximation , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[22]  Å. Björck,et al.  Solution of Vandermonde Systems of Equations , 1970 .