Digital Filter Design Using Computer Algebra Systems

Abstract

Digital filter design can be performed very efficiently using modern computer tools. The drawback of the numeric-based tools is that they usually generate a tremendous amount of numeric data, and the user might easily lose insight into the phenomenon being investigated. The computer algebra systems successfully overcome some problems encountered in the traditional numeric-only approach. In this paper, we introduce an original approach to algorithm development and digital filter design using a computer algebra system. The main result of the paper is the development of an algorithm for an infinite impulse response (IIR) filter design that, theoretically, is impossible to be implemented using the traditional approach. We present a step-by-step procedure which includes derivations of closed-form expressions for (1) the transfer functions of the implemented digital filter which contains the algebraic loop; (2) the closed-form expression for computing the number of requested iteration steps; and (3) the error function representing the difference of the output sample values of the new filter and that of the conventional filter. We demonstrate how one can use some already-known multiplierless digital filter as a start-up filter to design a new digital filter whose passband edge frequency can be simply moved by using a single parameter. As a result, we obtain a multiplierless IIR filter, which belongs to the family of low-power digital filters where multipliers are replaced with a small number of adders and shifters.