FIR Filter Design Based on Successive Approximation of Vectors

We present a novel method for the design of finite impulse response (FIR) filters with discrete coefficients that belong in the sum of powers-of-two (POT) space. The importance of this class of filters cannot be overstated, given the ever-increasing number of applications for which a specific hardware implementation is needed. Filters that have coefficients that belong to such a class are also referred to as multiplierless filters, given that the operations performed by the filter can all be implemented by using appropriately designed shifts of the input data, making them a perfect choice whenever implementation simplicity and processing speed are the ultimate goal. To produce such a design, we employ a vector successive approximation technique successfully used in data compression that has a very low computational complexity, the Matching Pursuits Generalized BitPlanes algorithm (MPGBP). We derive optimality conditions for the approximation dictionary. We compare filters obtained with the proposed method with those derived in previous works. Based on this comparative analysis, we show that this new and powerful way of producing the filters' coefficients is also among the simplest available in the literature.

[1]  Andrew G. Dempster,et al.  Multiplierless FIR filter design algorithms , 2005, IEEE Signal Processing Letters.

[2]  Katta G. Murty,et al.  Nonlinear Programming Theory and Algorithms , 2007, Technometrics.

[3]  A. Constantinides,et al.  Finite word length FIR filter design using integer programming over a discrete coefficient space , 1982 .

[4]  C. Ogilvy,et al.  Excursions in Geometry , 1990 .

[5]  Eduardo A. B. da Silva,et al.  On the statistics of matching pursuit angles , 2010, Signal Process..

[6]  Tapio Saramäki,et al.  A systematic algorithm for the design of multiplierless FIR filters , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[7]  D. Ait-Boudaoud,et al.  Genetic approach to design of multiplierless FIR filters , 1993 .

[8]  Khalid Sayood,et al.  Introduction to data compression (2nd ed.) , 2000 .

[9]  Franc Brglez,et al.  Digital filter design with short word-length coefficients , 1978 .

[10]  G. Blelloch.  Introduction to Data Compression * , 2022 .

[11]  Joseph B. Evans.  Efficient FIR filter architectures suitable for FPGA implementation , 1994 .

[12]  Ronald A. DeVore,et al.  Some remarks on greedy algorithms , 1996, Adv. Comput. Math..

[13]  Yong Ching Lim,et al.  Design of Linear Phase FIR Filters in Subexpression Space Using Mixed Integer Linear Programming , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[14]  Tapio Saramäki,et al.  An iterative method for optimizing FIR filters synthesized using the two-stage frequency-response masking technique , 2003, Proceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03..

[15]  Stéphane Mallat,et al.  Matching pursuit of images , 1995, Proceedings., International Conference on Image Processing.

[16]  Y. Lim,et al.  FIR filter design over a discrete powers-of-two coefficient space , 1983 .

[17]  E.A.B. da Silva,et al.  Uniform distribution of points on a hyper-sphere with applications to vector bit-plane encoding , 2001 .

[18]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[19]  Chip-Hong Chang,et al.  Design of Low-Complexity FIR Filters Based on Signed-Powers-of-Two Coefficients With Reusable Common Subexpressions , 2007, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[20]  Dusan M. Kodek.  Performance limit of finite wordlength FIR digital filters , 2005, IEEE Transactions on Signal Processing.

[21]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[22]  H. Samueli,et al.  An improved search algorithm for the design of multiplierless FIR filters with powers-of-two coefficients , 1989 .

[23]  R. Cooke.  Real and Complex Analysis , 2011 .

[24]  Shing-Chow Chan,et al.  Multiplier-less FIR digital filters using programmable sum-of-power-of-two (SOPOT) coefficients , 2002, 2002 IEEE International Conference on Field-Programmable Technology, 2002. (FPT). Proceedings..

[25]  Zhifeng Zhang,et al.  Adaptive time-frequency decompositions , 1994 .

[26]  Pierre Vandergheynst,et al.  On the exponential convergence of matching pursuits in quasi-incoherent dictionaries , 2006, IEEE Transactions on Information Theory.

[27]  Stephen P. Boyd,et al.  Filter Design With Low Complexity Coefficients , 2008, IEEE Transactions on Signal Processing.

[28]  Kai Wang,et al.  Design of High-Order Extrapolated Impulse Response FIR Filters with Signed Powers-of-Two Coefficients , 2011, Circuits Syst. Signal Process..

[29]  Tapio Saramäki,et al.  A Systematic Algorithm for the Design of Lattice Wave Digital Filters With Short-Coefficient Wordlength , 2007, IEEE Transactions on Circuits and Systems I: Regular Papers.

[30]  Eduardo A. B. da Silva,et al.  Digital Signal Processing: System Analysis and Design , 2002 .

[31]  Yong Lian,et al.  A polynomial-time algorithm for designing FIR filters with power-of-two coefficients , 2002, IEEE Trans. Signal Process..

[32]  Francesco Piazza,et al.  Efficient genetic algorithm design for power-of-two FIR filters , 1995, 1995 International Conference on Acoustics, Speech, and Signal Processing.

[33]  K. Steiglitz,et al.  Filter-length word-length tradeoffs in FIR digital filter design , 1980 .

[34]  Yong Ching Lim,et al.  Signed power-of-two term allocation scheme for the design of digital filters , 1999 .

[35]  S. Mallat,et al.  Adaptive greedy approximations , 1997 .

[36]  Alexandre Ciancio,et al.  Video coding using greedy decompositions on generalised bit-planes , 2002 .

[37]  D. Kodek.  Design of optimal finite wordlength FIR digital filters using integer programming techniques , 1980 .