The final class of IIR filter design methods we'll introduce are broadly categorized as optimization methods. These IIR filter design techniques were developed for the situation when the desired IIR filter frequency response was not of the standard low-pass, bandpass, or highpass form. When the desired response has an arbitrary shape, closed-form expressions for the filter's z-transform do not exist, and we have no explicit equations to work with to determine the IIR filter's coefficients. For this general IIR filter design problem, algorithms were developed to solve sets of linear, or nonlinear, equations on a computer. These software routines mandate that the designer describe, in some way, the desired IIR filter frequency response. The algorithms, then, assume a filter transfer function H(z) as a ratio of polynomials in z and start to calculate the filter's frequency response. Based on some error criteria, the algorithm begins iteratively adjusting the filter's coefficients to minimize the error between the desired and the actual filter frequency response. The process ends when the error cannot be further minimized, or a predefined number of iterations has occurred, and the final filter coefficients are presented to the filter designer. Although these optimization algorithms are too mathematically complex to cover in any detail here, descriptions of the most popular optimization schemes are readily available in the literature [14,16,20–25].

The reader may ask, "If we're not going to cover optimization methods in any detail, why introduce the subject here at all?" The answer is that if we spend much time designing IIR filters, we'll end up using optimization techniques in the form of computer software routines most of the time. The vast majority of commercially available digital signal processing software packages include one or more IIR filter design routines that are based on optimization methods. When a computer-aided design technique is available, filter designers are inclined to use it to design the simpler low-pass, bandpass, or highpass forms even though analytical techniques exist. With all due respect to Laplace, Heaviside, and Kaiser, why plow through all the z-transform design equations when the desired frequency response can be applied to a software routine to yield acceptable filter coefficients in a few seconds?

As it turns out, using commercially available optimized IIR filter design routines is very straightforward. Although they come in several flavors, most optimization routines only require the designer to specify a few key amplitude and frequency values, the desired order of the IIR filter (the number of feedback taps), and the software computes the final feed forward and feedback coefficients. In specifying a low-pass, IIR filter for example, a software design routine might require us to specify the values for dp, ds, f1, and f2 shown in Figure 6-35. Some optimization design routines require the user to specify the order of the IIR filter. Those routines then compute the filter coefficients that best approach the required frequency response. Some software routines, on the other hand, don't require the user to specify the filter order. They compute the minimum order of the filter that actually meets the desired frequency response.

Figure 6-35. Example low-pass IIR filter design parameters required for the optimized IIR filter design method.

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