Decoupling Minimax Design of Low-Complexity Variable Fractional-Delay FIR Digital Filters

This paper presents a simple linear programming (LP) technique for designing high-accuracy low-complexity finite-impulse-response (FIR) variable fractional-delay (VFD) digital filters in the minimax error sense. The objective of the minimax design is to minimize the maximum absolute error of the variable frequency response (VFR) of an FIR VFD filter, which is a nonlinear problem and difficult to solve. This paper shows that the minimax design can be approximately decomposed into a pair of separate LP subproblems by decoupling the minimization of the real-part VFR error from that of the imaginary-part error. As a result, the original nonlinear minimax design problem can be easily solved by solving the two LP subproblems separately. To reduce the VFD filter complexity, we also propose a one-by-one increase scheme for optimizing the subfilter orders in the Farrow structure such that a given design specification (maximum absolute error of VFR) can be exactly satisfied. Both even-order and odd-order design examples are given to illustrate that the decoupling minimax method is not only simple, but also can achieve excellent high-accuracy low-complexity FIR VFD filters.

[1]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[2]  Che-Ho Wei,et al.  A new variable fractional sample delay filter with nonlinear interpolation , 1992 .

[3]  M. M. Fahmy,et al.  A design technique for variable digital filters , 1989 .

[4]  Tian-Bo Deng,et al.  Design of linear phase variable 2-D digital filters using real-complex decomposition , 1998 .

[5]  Tian-Bo Deng.  Weighted least-squares method for designing arbitrarily variable 1-D FIR digital filters , 2000, Signal Process..

[6]  W.-S. Lu,et al.  Weighted least-squares method for designing variable fractional delay 2-D FIR digital filters , 2000 .

[7]  Tian-Bo Deng.  Robust Structure Transformation for Causal Lagrange-Type Variable Fractional-Delay Filters , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[8]  Kenneth Steiglitz,et al.  Design of FIR digital phase networks , 1981 .

[9]  Soo-Chang Pei,et al.  Design of Variable Two-Dimensional FIR Digital Filters by McClellan Transformation , 2009, IEEE Trans. Circuits Syst. I Regul. Pap..

[10]  Jie-Cherng Liu,et al.  Weighted least squares near-equiripple approximation of variable fractional delay FIR filters , 2007 .

[11]  Graham A. Jullien,et al.  A linear programming approach to recursive digital filter design with linear phase , 1982 .

[12]  Tian-Bo Deng.  Closed-form design and efficient implementation of variable digital filters with simultaneously tunable magnitude and fractional delay , 2004, IEEE Trans. Signal Process..

[13]  Tian-Bo Deng.  Closed-Form Mixed Design of High-Accuracy All-Pass Variable Fractional-Delay Digital Filters , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[14]  Jong-Jy Shyu,et al.  Minimax phase error design of allpass variable fractional-delay digital filters by iterative weighted least-squares method , 2009, Signal Process..

[15]  Tian-Bo Deng.  Discretization-free design of variable fractional-delay FIR digital filters , 2001 .

[16]  J. O. Coleman,et al.  Design of nonlinear-phase FIR filters with second-order cone programming , 1999, 42nd Midwest Symposium on Circuits and Systems (Cat. No.99CH36356).

[17]  Tian-Bo Deng.  Design of recursive 1-D variable filters with guaranteed stability , 1997 .

[18]  Tian-Bo Deng.  Decomposition-based design of linear phase variable digital filters , 1999 .

[19]  Yong Lian,et al.  Weighted-Least-Squares Design of Variable Fractional-Delay FIR Filters Using Coefficient Symmetry , 2006, IEEE Transactions on Signal Processing.

[20]  Tian-Bo Deng.  Design of variable 2-D linear phase recursive digital filters with guaranteed stability , 1998 .

[21]  T. Soma,et al.  Variable digital filter design using the outer product expansion , 1994 .

[22]  Tian-Bo Deng.  Coefficient-Symmetries for Implementing Arbitrary-Order Lagrange-Type Variable Fractional-Delay Digital Filters , 2007, IEEE Transactions on Signal Processing.

[23]  Tian-Bo Deng.  Generalized WLS Method for Designing All-Pass Variable Fractional-Delay Digital Filters , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[24]  Chien-Cheng Tseng,et al.  Improved design of digital fractional-order differentiators using fractional sample delay , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[25]  Tian-Bo Deng,et al.  SVD-based design and new structures for variable fractional-delay digital filters , 2004, IEEE Transactions on Signal Processing.

[26]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[27]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[28]  Tian-Bo Deng,et al.  Symmetric Structures for Odd-Order Maximally Flat and Weighted-Least-Squares Variable Fractional-Delay Filters , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[29]  Chien-Cheng Tseng.  Closed-Form Design of Digital IIR Integrators Using Numerical Integration Rules and Fractional Sample Delays , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[30]  Chien-Cheng Tseng,et al.  Design of 1-D and 2-D variable fractional delay allpass filters using weighted least-squares method , 2002 .

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

[32]  Tian-Bo Deng.  Design of arbitrary-phase variable digital filters using SVD-based vector-array decomposition , 2005, IEEE Trans. Circuits Syst. I Regul. Pap..

[33]  Tian-Bo Deng,et al.  SVD-based design of fractional-delay 2-D digital filters exploiting specification symmetries , 2003, IEEE Trans. Circuits Syst. II Express Briefs.

[34]  Tian-Bo Deng.  Minimax Design of Low-Complexity Allpass Variable Fractional-Delay Digital Filters , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[35]  Tian-Bo Deng,et al.  Hybrid Structures for Low-Complexity Variable Fractional-Delay FIR Filters , 2010, IEEE Transactions on Circuits and Systems I: Regular Papers.

[36]  Tian-Bo Deng.  Noniterative WLS design of allpass variable fractional-delay digital filters , 2006, IEEE Transactions on Circuits and Systems I: Regular Papers.

[37]  Stephen P. Boyd,et al.  FIR filter design via semidefinite programming and spectral factorization , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

[38]  Tian-Bo Deng,et al.  High-Resolution Image Interpolation Using Two-Dimensional Lagrange-Type Variable Fractional-Delay Filter , 2005 .

[39]  Tian-Bo Deng,et al.  Design and parallel implementation of FIR digital filters with simultaneously variable magnitude and non-integer phase-delay , 2003, IEEE Trans. Circuits Syst. II Express Briefs.

[40]  Tian-Bo Deng.  Variable 2-D FIR digital filter design and parallel implementation , 1999 .

[41]  Tian-Bo Deng.  Design of complex-coefficient variable digital filters using successive vector-array decomposition , 2005, IEEE Trans. Circuits Syst. I Regul. Pap..

[42]  C. W. Farrow,et al.  A continuously variable digital delay element , 1988, 1988., IEEE International Symposium on Circuits and Systems.

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

[44]  Soo-Chang Pei,et al.  Two-Dimensional Farrow Structure and the Design of Variable Fractional-Delay 2-D FIR Digital Filters , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[45]  Soo-Chang Pei,et al.  A generalized approach to the design of variable fractional-delay FIR digital filters , 2008, Signal Process..

[46]  Tian-Bo Deng.  Design of Variable FIR Digital Filters with Arbitrary Frequency Responses , 1997 .

[47]  C. Tseng,et al.  Minimax design of stable IIR digital filter with prescribed magnitude and phase responses , 2002 .