Design of Cascade Form FIR Filters with Discrete
Valued Coefficients · 1988
When an FIR filter with discrete valued coefficient value is implemented in direct form, the peak ripple of the amplitude re- sponse decreases with increasing filter length up to a certain length. Significant reduction in peak ripple beyond that limit cannot be real- ized easily without increasing the coefficient precision. In this paper, we show that by cascading two direct form FIR filters each with coef- ficients that are sum or difference of two power-of-two terms, it is pos- sible to achieve very small peak ripple. An iterative equalization strat- egy is used in the design of the cascade filter. The success of the method depends on the initial prototype filter being used. Two equally effective methods are presented for selecting the prototype filter which will yield a final design with good roundoff noise property. I. INTRODUCTION N digital filter implementations, especially for high I speed operation, it is often necessary to explore the in- herent savings in the basic arithmetic operations. Within the last decade, a number of approaches have been re- ported to simplify the implementation of one-dimensional FIR filters. Of particular interest are filters whose coef- ficients are restricted to sums of power-of-two terms, thus converting multiplication to simple operations of shift and add 111-(3). The design of these filters can be formulated as an optimization problem in a discrete parameter space. For filters so designed, the peak ripple of the amplitude response decreases with increasing filter length, up to a certain length. Significant reduction in peak ripple beyond that length cannot be realized easily without increasing the coefficient precision. Fig. 1 plots the peak ripple versus filter length for a family of direct form low-pass filter where each filter coef- ficient is a sum or difference of two power-of-two terms (2) and the ratio between the largest and the smallest power-of-two term is 2. The passband and stopband edges of these filters are, respectively, 0.15 and 0.22 times the sampling frequency, and the peak ripples in both bands are equal. These specifications are used in all the exam- ples in this paper. Also shown in the figure is the same plot for filters with infinite precision coefficients. As can be seen, the gap between the peak ripple of the direct form