SIMPLE AND ROBUST METHOD FOR THE DESIGN OF ALLPASS FILTERS USING LEAST-SQUARES PHASE ERROR CRITERION

In this paper we consider a simple scheme for the design of allpass fi lters for approximation (or equalization) of a given phase function us ing least-squares error criterion. Assuming that the desired phase res ponse is prescribed at a discrete set of frequency points, we formulat e a general least-squares equation-error solution with a possible weig ht function. Based on the general formulation and detailed analysis of the introduced error, we construct a new algorithm for phase approxim ation. In addition to iterative weighting of the equation error, the n ominal value of the desired group delay is also adjusted iteratively t o minimize the total phase error measure in equalizer applications. Th is new feature essentially eliminates the difficult choice of the nomi nal group delay which is known to have a profound effect on the stabil ity of the designed allpass filter. The proposed method can be used fo r highpass and bandpass equalization as well, where the total phase er ror can be further reduced by introducing an adjustable-phase offset i n the optimization. The performance of the algorithm is analyzed in de tail with examples. First we examine the approximation of a given phas e function. Then we study the equalization of the nonlinear phase of v arious lowpass filters. Also, a bandpass example is included. Finally we demonstrate the use of the algorithm for the design of approximatel y linear-phase recursive filters as a parallel connection of a delay l ine and an allpass filter. The examples show that the proposed simple and easy-to-use algorithm yields excellent allpass filters for various applications in practice.