Optimal FIR filter design
Michael Komodromos · 1993
The design of Finite Impulse Response (FIR) digital filters that considers both phase and magnitude specifications is investigated. This dissertation is divided into two parts. In Part I we present our implementation of an algorithm for the design of minimum phase filters. In Part II we investigate the design of FIR filters in the complex domain and develop a new powerful design method for digital FIR filters with arbitrary specification of magnitude and phase;Part I considers the design of minimum-phase filters. The method presented uses direct factorization of the transfer function of a companion Parks-McClellan linear-phase filter of twice the length of the desired minimum-phase filter. The minimum-phase filter is derived with excision of half the zeros of the companion linear-phase filter. The zeros of the prototype filter are found using Laguerre's method. We will present our implementation of the design method, and describe some practical aspects and problems associated with the design of minimum-phase filters;Part II investigates the design of optimal Chebychev FIR filters in the complex domain. The design of FIR filters with arbitrary specification of magnitude and phase is formulated into a problem of complex approximation. The method developed is capable of designing filters with real or complex coefficients. Complex impulse response designs are an extension of the real coefficient case based on a proper selection of the approximating basis functions;The minimax criterion is used and the complex Chebychev approximation is posed as a minimization problem in linear optimization. The primal problem is converted to its dual and is solved using an efficient, quadratically convergent algorithm developed by Tang (14). The relaxation of the linear-phase constraint results in a reduction of the number of coefficients compared to linear-phase designs. Linear-phase filters are a special case of our filter design approach. We examine the design of frequency selective filters with or without the conjugate symmetry, the design of one-sided, two-sided, narrowband and fullband Hilbert Transformers and differentiators.