M. Lang et Ti. Laakso, SIMPLE AND ROBUST METHOD FOR THE DESIGN OF ALLPASS FILTERS USING LEAST-SQUARES PHASE ERROR CRITERION, IEEE transactions on circuits and systems. 2, Analog and digital signal processing, 41(1), 1994, pp. 40-48
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.