3.8 Direct frequency domain iir filter design methods  (Page 5/6)

Practical difficulties exist in solving this approximation problem. In some cases, local minima are found rather than theglobal minimum. In other cases, convergence of the minimization algorithm is slow or does not occur at all. Numerical problems canresult from ill-conditioned equations, and there is no guarantee that the designed filter will be stable.

An important factor is the choice of a desired frequency- response function $H_d\left(\omega \right)$ that does not result in the optimum approximation having a large error. This often means not having anabrupt discontinuity between the passband and stopband.

Another factor is the starting of the iterative optimization algorithm with a set of coefficients in $x$ that is close to the optimum. This can be accomplished by using the frequency samplingmethod to give a design that can be used to start a least-squares algorithm. Because the error definedin [link] is in terms of magnitudes, an unstable design can be converted to a stable one by moving the unstable pole at a radius of $r$ in the $z$ -plane to a radius of $1/r$ . This does not change the magnitude frequency response and does stabilize the effect of thatpole [link] .

A generalization of the idea of a squared-error measure is defined by raising the error to the $p$ power where $p$ is a positive integer. This error is defined by

Deczky [link] developed this approach and used the Fletcher-Powell method to minimize [link] . He also applied this method to the approximation of a desired group-delay function. An importantcharacteristic of this formulation is that the solution approaches the Chebyshev or mini-max solution as p becomes large. Initial work shows themethod of iteratively reweighted least squared error (IRLS) as was applied to the FIR filter design in [link] can also be used for $L_p$ and constrained least squared error optimal design of IIR filters [link] .

The chebyshev error criterion for iir filter design

The error measure that often best meets filter design specifications is the maximum error in the frequency responsethat occurs over a band. The filter design problem becomes the problem of minimizing the maximum error (the min-max problem).

Among several approaches to this error minimization, one is by Deczky which minimizes the p-power error of [link] for large p. Generally, $p=10$ or greater approximates a Chebyshev result [link] . Another is by Dolan and Kaiser which uses a penalty-function approach.

Linear programming can be applied to this error measure by linearizing the equations in much the same way asin [link] [link] . In contrast to the FIR case, this can be a practical design method because the order of practical IIRfilters is generally much lower than for FIR filters. A scheme called differential correction has also proven to be effective.

Although the rational approximation problem is nonlinear, an application of the Remes exchange algorithm can be implemented [link] . Since the zeros of the numerator of the transfer function mainly control the stopband characteristics of a filter,and the zeros of the denominator mainly control the passband, the effects of the two are somewhat uncoupled. An application of theRemes exchange algorithm, alternating between the numerator and denominator, gives an effective method for designing IIR filterswith a Chebyshev error criterion. If the order of the numerator and denominator are the same and the desired filter isan ideal lowpass filter, the Remes exchange should give the same result as the elliptic function filter. However,this approach allows any order numerator or denominator to be set and any shape passband to be approximated. There are cases wherea lower-order denominator than numerator results in a filter with fewer required muliplications than an elliptic-function filter [link] .