Faculty Bibliography

This brief describes an exchange algorithm for the frequency domain design of linear-phase FIR equiripple filters where the Chebyshev error in each band is specified. The algorithm is a hybrid of the algorithm of Hofstetter, Oppenheim and Siegel and the Parks-McClellan algorithm. The brief also describes a modification of the Parks-McClellan algorithm where either the passband or the stopband ripple size is specified and the other is minimized.

This brief describes a modification of a technique proposed by Vaidyanathan for the design of filters having Bat passbands and equiripple stopbands. The modification ensures that the passband is monotonic and does so without the use of concavity constraints. Another modification described in this brief adapts the method of Vaidyanathan to the design of low-pass differentiators having a specified degree of tangency at omega = 0.

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 L(2) 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.

We consider the design of 2-D linear phase finite impulse response (FIR) filters according to the least squares (LS) error criterion subject to equality and/or inequality constraints, Since we use a frequency domain formulation, these constraints can be used to explicitly prescribe (frequency-dependent) error tolerances, the maximum, minimum, or fixed values of the frequency response at certain points and/or regions. Our method combines Lagrange multiplier and Kuhn-Tucker theory to solve a much wider class of problems than do standard methods, It allows arbitrary compromises between the LS and the equiripple design.

We describe a set of programs for circular convolution and prime length fast Fourier transforms (FFT's) that are relatively short, possess great structure, share many computational procedures, and cover a large variety of lengths, The programs make clear the structure of the algorithms and clearly enumerate independent computational branches that can be performed in parallel, Moreover, each of these independent operations is made up of a sequence of suboperations that can be implemented as vector/parallel operations, This is in contrast with previously existing programs for prime length FFT's: They consist of straight line code, no code is shared between them, and they cannot be easily adapted for vector/parallel implementations. We have also developed a program that automatically generates these programs for prime length FFT's, This code-generating program requires information only about a set of modules for computing cyclotomic convolutions.