Constrained Least Square Design of FIR Filters Without Speci ed Transition Bands

This paper puts forth the notion that explicitly speciied transition bands have been introduced in the lter design literature in part as an indirect approach for dealing with discontinu-ities in the desired frequency response. We suggest that the use of explicitly speciied transition bands is sometimes inappropriate because, to satisfy a meaningful optimality criterion, their use implicitly assumes a possibly unrealistic assumption on the class of input signals. This paper also presents an algorithm for the design of peak constrained lowpass FIR lters according to an integral square error criterion that does not require the use of speciied transition bands. This rapidly converging, robust, simple multiple exchange algorithm uses Lagrange multipliers and the Kuhn-Tucker conditions on each iteration. The algorithm will design linear and minimum phase FIR lters and gives the best L 2 lter and a continuum of Chebyshev lters as special cases. It is distinct from many other lter design methods because it does not exclude from the integral square error a region around the cut-oo frequency, and yet, it overcomes Gibbs' phenomenon without resorting to windowing or`smoothing out' the discontinuity of the ideal lowpass lter. 7 The trade-oo curve for integral square error and peak size error: jjEjj 2 2 versus s. This curve illustrates the trade-oo for lters of length 61 designed with ! o = 0:3, and = p =