Splitting the Unit Delay - IEEE Signal Processing ... - IEEE Xplore

Another major disadvantage is the possible instability ofrecursive filters. In general, the obtained solution has to bechecked so that all the poles of the filter remain within theunit circle in the z-domain, which makes real-time coefficientupdate difficult.Here we omit the problem of magnitude approximation byconsidering only the design of allpass filters, a special subclassof recursive filters. (For a report on fractional delay IIRfilter design, see [ 1 131.) Allpass filters have unity magnituderesponse in the whole frequency band by definition, whichmeans that one can concentrate on the approximation of thedesired phase (or group delay, or phase delay) characteristics.This reduces the available degrees of freedom but also makesthe design task much easier.The z transfer function of an Nth-order allpass filter is ofthe form!where the numerator polynomial is a mirrored version of the(supposedly stable) denominator D(z). The coefficients areassumed to be real-valued. The direct form I implementationof the allpass filter is shown in Fig. 9. The phase response ofthe allpass filter can be expressed as. Direct form I implementation of an Nth-order allpass filter.wherewheref N 7A=diag[O 1 ... N](69c)Naturally, for the phase delay it holds= arctan{$)where c and s are appropriate cosine and sine vectors asdefined in Eq. 30, and a is the coefficient vectorwith a0 = 1. The group delay of the allpass filter is related tothat of the denominator similarly to Eq. 65, orwhere the delay of the denominator can be expressed asUnfortunately, the phase, phase delay, and group delay areall related to the filter coefficients in a very nonlinear manner,as the above equations show. This means that one cannotexpect as simple design formulas for the allpass filter coefficientsas for FIR filters. Instead, one can almost exclusivelyfind only iterative optimization techniques for minimizationof traditional error criteria.In the following discussion, we shall review in greaterdetail only the simplest allpass design techniques that havesome potential for applications requiring real-time coefficientupdate. Among the above delay measures, the phase isperhaps the most suitable for least squared error design.Consequently, much of the following is based on the recentresults on least squares phase approximation presented in [60,63,53,77]. These schemes are easy to program and can alsobe modified for approximately equiripple phase error solutions,as well as for corresponding phase delay approximations,in contrast to many other methods that may be moredifficult to use.JANUARY 1996 IEEE SIGNAL PROCESSING MAGAZINE 47