New techniques for digital filter design
Ivan Selesnick · Rice Research Repository (Rice University) · 1996
Several new techniques for open problems in the frequency domain design of FIR and IIR digital filters, both iterative and analytic, are presented. This thesis begins by putting forth the notion that explicitly specified transition bands have been introduced in the literature in part as an indirect approach for dealing with discontinuities in the desired frequency response. To overcome this, a rapidly converging algorithm is presented for the design of symmetric FIR filters according to a square error criterion that does not require specified transition bands. 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. Also presented are algorithms for symmetric FIR filter design that modify the Parks-McClellan algorithm and a variation due to Vaidyanathan, to give a fairly complete set of design techniques for the design equiripple symmetric FIR filters. Two types of filters, which are particularly well suited for both the approximation and implementation problems, are frequently overlooked because their design is substantially more difficult than the design of symmetric FIR and classical IIR filters. These two filter types are (1) non-symmetric FIR filters, and (2) IIR filters for which the numerator and denominator degrees need not be equal. For maximally-flat lowpass responses, analytic techniques for these two filter types are presented. For non-symmetric FIR filter design, in which the magnitude and group delay are regarded separately (a classic problem in the design of both digital and analogue filters), it is shown that the delay can be reduced significantly while maintaining a very constant passband group delay, with no degradation in the magnitude response. For IIR filters for which the numerator and denominator degrees are unequal, techniques for the design of generalized Butterworth and Chebyshev filters are presented. This thesis also presents a technique to make more practical the rational Remez exchange algorithm. Lastly, a problem examined by Souto is revisited, and the use of Grobner bases from computational algebraic geometry for this problem is described.