FIR Filters with Punctured Radix-8 Symmetric Coefficients: Design and Multiplier-Free Realizations

Abstract

In recent years, several approaches have been investigated to obtain multiplier-free realizations of digital filters. One approach makes use of periodically time-varying (PTV) structures. The idea is to distribute the computation in time and in space. Distribution in time provides reuse of the same hardware by means of PTV coefficients. Distribution in space increases the number of coefficients but simplifies the values of the coefficients. Computation distribution is based on radix-r number representation, and it can be carried out to the extent that each computation involves a simple coefficient that can be realized using only addition and shift (no hardware multiplier). Previous PTV realizations could not exploit the coefficient symmetry of finite impulse response (FIR) filters to reduce the number of coefficients. This paper proposes design and realizations of FIR filters with punctured radix-8 coefficients belonging to the septuple set {0, ±1, ±2, ±4), which can be implemented using only a shift operation without requiring any hardware multiplier. The realizations exploit the coefficient symmetry to reduce the hardware by about one-half. Due to a non-uniform grid of representation, we apply a modified Karmarkar's linear programming algorithm to find the optimum set of discrete coefficients that minimizes the weighted peak ripple error. Comparison with a conventional FIR filter with sum-of-powers-of-two (SOPOT) coefficients shows that the proposed filter is faster and uses less hardware than one with SOPOT coefficients. However, the punctured radix-8 system has a limit on the achievable ripple.