Simple and robust method for the design of allpass filters using least-squares phase error criterion

We consider a simple scheme for the design of allpass filters for approximation (or equalization) of a given phase function using a least-squares error criterion. Assuming that the desired phase response is prescribed at a discrete set of frequency points, we formulate a general least-squares equation-error solution with a possible weight function. Based on the general formulation and detailed analysis of the introduced error, we construct a new algorithm for phase approximation. In addition to iterative weighting of the equation error, the nominal value of the desired group delay is also adjusted iteratively to minimize the total phase error measure in equalizer applications. This new feature essentially eliminates the difficult choice of the nominal group delay which is known to have a profound effect on the stability of the designed allpass filter. The proposed method can be used for highpass and bandpass equalization as well, where the total phase error can be further reduced by introducing an adjustable-phase offset in the optimization. The performance of the algorithm is analyzed in detail with examples. First we examine the approximation of a given phase function. Then we study the equalization of the nonlinear phase of various lowpass filters. Also, a bandpass example is included. Finally we demonstrate the use of the algorithm for the design of approximately linear-phase recursive filters as a parallel connection of a delay line and an allpass filter