This section (adapted from ), 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 [346,365], or peak/shelf equalizers [563,452], 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
with given, such that some norm of the error
The filter coefficients are constrained to lie in some subset , where . The filter coefficients may also be complex, in which case .
The approximate filter is typically 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.9.9Unfortunately, the norm of the error typically is not a convex9.10function 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 [156,432,178,36]. 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
When the norm of equation-error is minimized, the problem becomes solving a set of linear equations.
The above expression makes it clear that equation-error can be seen as a frequency-response error weighted by . Thus, relatively large errors can be expected where the poles of the optimum approximation (roots of ) approach the unit circle . While this may make the frequency-domain formulation seem ill-posed, in the time-domain, 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 [289,452,290], the equation-error solution iteratively approaches the norm-minimizing solution of Eq.(8.17) for the norm.
Examples of minimizing equation error using the matlab function invfreqz are given in §8.6.3 and §8.6.4 below. See [452, Appendix I] (based on [432, 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).
In addition to these modifications, sometimes it is necessary to reformulate the problem in order to achieve a different goal. For example, in some audio applications, it is desirable to minimize the 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 .
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 . Frequency-warping is especially valuable for providing an effective weighting-function for filter-design methods, such as the Hankel-norm method, which have no mechanism for imposing an explicit weighting-function on 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 . Roughly speaking, a size polynomial factorization is replaced by an FFT and a size system of linear equations.