### Digital Filter Design Overview

This section (adapted from [428]), summarizes some of the more commonly used methods for digital filter design aimed at matching a*nonparametric*frequency response, such as typically obtained from input/output measurements. This problem should be distinguished from more classical problems with their own specialized methods, such as designing lowpass, highpass, and bandpass filters [343,362], or peak/shelf equalizers [559,449], and other utility filters designed from

*a priori*mathematical specifications.

The problem of fitting a digital filter to a prescribed frequency response may be formulated as follows. To simplify, we set . Given a continuous complex function , corresponding to a causal desired frequency response,

^{9.8}find a stable digital filter of the form where

(9.15) | |||

(9.16) |

with given, such that some

*norm*of the error

is minimum with respect to the filter coefficients

*constrained*to be stable, and since is causal (no positive powers of ), stability implies causality. Consequently, the impulse response of the model is zero for . The filter-design problem is then to find a (strictly) stable -pole, -zero digital filter which minimizes some norm of the error in the frequency-response. This is fundamentally

*rational approximation of a complex function of a real (frequency) variable, with constraints on the poles*. While the filter-design problem has been formulated quite naturally, it is difficult to solve in practice. The strict stability assumption yields a compact space of filter coefficients , leading to the conclusion that a best approximation

*exists*over this domain.

^{9.9}Unfortunately, the norm of the error typically is not a

*convex*

^{9.10}function of the filter coefficients on . This means that algorithms based on gradient descent may fail to find an optimum filter due to their premature termination at a suboptimal local minimum of . Fortunately, there is at least one norm whose global minimization may be accomplished in a straightforward fashion without need for initial guesses or ad hoc modifications of the complex (phase-sensitive) IIR filter-design problem--the

*Hankel norm*[155,428,177,36]. Hankel norm methods for digital filter design deliver a spontaneously

*stable*filter of any desired order without imposing coefficient constraints in the algorithm. An alternative to Hankel-norm approximation is to reformulate the problem, replacing Eq.(8.17) with a modified error criterion so that the resulting problem can be solved by

*linear least-squares*or

*convex optimization*techniques. Examples include

*Pseudo-norm*minimization: (Pseudo-norms can be zero for nonzero functions.) For example, Padé approximation falls in this category. In Padé approximation, the first samples of the impulse-response of are matched exactly, and the error in the remaining impulse-response samples is ignored.*Ratio Error*: Minimize subject to . Minimizing the norm of the ratio error yields the class of methods known as*linear prediction*techniques [20,296,297]. Since, by the definition of a norm, we have , it follows that ; therefore,*ratio error methods ignore the phase of the approximation*. It is also evident that ratio error is minimized by making larger than .^{9.11}For this reason, ratio-error methods are considered most appropriate for modeling the*spectral envelope*of . It is well known that these methods are fast and exceedingly robust in practice, and this explains in part why they are used almost exclusively in some data-intensive applications such as speech modeling and other spectral-envelope applications. In some applications, such as adaptive control or forecasting, the fact that linear prediction error is minimized can justify their choice.*Equation error*: Minimize*linear prediction error*is minimized in the sense, and in certain applications this is ideal. (Equation-error methods thus provide a natural extension of ratio-error methods to include zeros.) Using so-called*Steiglitz-McBride iterations*[287,449,288], the equation-error solution iteratively approaches the norm-minimizing solution of Eq.(8.17) for the L2 norm. Examples of minimizing equation error using the matlab function`invfreqz`are given in §8.6.3 and §8.6.4 below. See [449, Appendix I] (based on [428, pp. 48-50]) for a discussion of equation-error IIR filter design and a derivation of a fast equation-error method based on the Fast Fourier Transform (FFT) (used in`invfreqz`).- Conversion to
*real-valued approximation*: For example,*power spectrum matching*, i.e., minimization of , is possible using the Chebyshev or norm [428]. Similarly,*linear-phase*filter design can be carried out with some guarantees, since again the problem reduces to real-valued approximation on the unit circle. The essence of these methods is that the*phase-response*is eliminated from the error measure, as in the norm of the ratio error, in order to convert a complex approximation problem into a real one. Real rational approximation of a continuous curve appears to be solved in principle only under the norm [373,374]. *Decoupling poles and zeros*: An effective example of this approach is Kopec's method [428] which consists of using ratio error to find the poles, computing the error spectrum , inverting it, and fitting poles again (to ). There is a wide variety of methods which first fit poles and then zeros. None of these methods produce optimum filters, however, in any normal sense.

*log-magnitude*frequency-response error. This is due to the way we hear spectral distortions in many circumstances. A technique which accomplishes this objective to the first order in the norm is described in [428]. Sometimes the most important spectral structure is confined to an interval of the frequency domain. A question arises as to how this structure can be accurately modeled while obtaining a cruder fit elsewhere. The usual technique is a weighting function versus frequency. An alternative, however, is to frequency-warp the problem using a first-order

*conformal map*. It turns out a first-order conformal map can be made to approximate very well frequency-resolution scales of human hearing such as the

*Bark scale*or

*ERB scale*[459]. Frequency-warping is especially valuable for providing an effective weighting function connection for filter-design methods, such as the Hankel-norm method, that are intrinsically do not offer choice of a

*weighted*norm for the frequency-response error. There are several methods which produce instead of directly. A

*fast spectral factorization*technique is useful in conjunction with methods of this category [428]. Roughly speaking, a size polynomial factorization is replaced by an FFT and a size system of linear equations.

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