Thinning digital filters: A piecewise-exponential approximation approach

An algorithm is presented for designing finite impulse response (FIR) recursive digital filters that require few multiplies to produce good frequency response, The process of reducing the number of multiplies needed to implement a digital filter is called thinning. This thinning algorithm uses dynamic programming techniques to find the best least-squares piecewise-exponential approximation to a desired impulse response of length P. Because these filters are implemented recursively, the number of arithmetic operations is independent of the model filter length P and is dependent only on the number of pieces or segments S used in the approximation (
S \ll P
). Examples of thinned narrow-band, broad-band, lowpass, and bandpass filters are given. Several of these thinned filters require fewer than one-third the number of multiplications required for the corresponding model filter, while still retaining desirable frequency response characteristics. The effects of coefficient quantization and finite precision arithmetic are also discussed.