Constrained least square design of FIR filters without specified transition bands

This paper puts forth the notion that explicitly specified transition bands have been introduced in the filter design literature in part as an indirect approach for dealing with discontinuities in the desired frequency response. We suggest that the use of explicitly specified 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 filters according to an integral square error criterion that does not require the use of specified 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 filters and gives the best L₂ filter and a continuum of Chebyshev filters as special cases. It is distinct from many other filter design methods because it does not exclude from the integral square error a region around the cut-off frequency, and yet, it overcomes Gibbs' phenomenon without resorting to windowing or 'smoothing out' the discontinuity of the ideal lowpass filter.