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Optimal Chebyshev FIR Filters

As we've seen above, the defining characteristic of FIR filters optimal in the Chebyshev sense is that they minimize the maximum frequency-response error-magnitude over the frequency axis. In other terms, an optimal Chebyshev FIR filter is optimal in the minimax sense: The filter coefficients are chosen to minimize the worst-case error (maximum weighted error-magnitude ripple) over all frequencies. This also means it is optimal in the $ L-infinity$ sense because, as noted above, the $ L-infinity$ norm of a weighted frequency-response error $ E(\omega) = W(\omega) [ H(\omega) -
D(\omega)]$ is the maximum magnitude over all frequencies:

$\displaystyle \left\Vert\,E\,\right\Vert _\infty \eqsp \max_{-\pi \leq \omega < \pi} \left\vert E(\omega)\right\vert$ (5.32)

Thus, we can say that an optimal Chebyshev filter minimizes the $ L-infinity$ norm of the (possibly weighted) frequency-response error. The $ L-infinity$ norm is also called the uniform norm. While the optimal Chebyshev FIR filter is unique, in principle, there is no guarantee that any particular numerical algorithm can find it.

The optimal Chebyshev FIR filter can often be found effectively using the Remez multiple exchange algorithm (typically called the Parks-McClellan algorithm when applied to FIR filter design) [176,224,66]. This was illustrated in §4.6.4 above. The Parks-McClellan/Remez algorithm also appears to be the most efficient known method for designing optimal Chebyshev FIR filters (as compared with, say linear programming methods using matlab's linprog as in §3.13). This algorithm is available in Matlab's Signal Processing Toolbox as firpm() (remez() in (Linux) Octave).5.13There is also a version of the Remez exchange algorithm for complex FIR filters. See §4.10.7 below for a few details.

The Remez multiple exchange algorithm has its limitations, however. In particular, convergence of the FIR filter coefficients is unlikely for FIR filters longer than a few hundred taps or so.

Optimal Chebyshev FIR filters are normally designed to be linear phase [263] so that the desired frequency response $ D(\omega)$ can be taken to be real (i.e., first a zero-phase FIR filter is designed). The design of linear-phase FIR filters in the frequency domain can therefore be characterized as real polynomial approximation on the unit circle [229,258].

In optimal Chebyshev filter designs, the error exhibits an equiripple characteristic--that is, if the desired response is $ D(\omega)$ and the ripple magnitude is $ \epsilon$ , then the frequency response of the optimal FIR filter (in the unweighted case, i.e., $ W(\omega)=1$ for all $ \omega$ ) will oscillate between $ D(\omega)+\epsilon$ and $ D(\omega)-\epsilon$ as $ \omega$ increases. The powerful alternation theorem characterizes optimal Chebyshev solutions in terms of the alternating error peaks. Essentially, if one finds sufficiently many for the given FIR filter order, then you have found the unique optimal Chebyshev solution [224]. Another remarkable result is that the Remez multiple exchange converges monotonically to the unique optimal Chebyshev solution (in the absence of numerical round-off errors).

Fine online introductions to the theory and practice of Chebyshev-optimal FIR filter design are given in [32,283].

The window method4.5) and Remez-exchange method together span many practical FIR filter design needs, from ``quick and dirty'' to essentially ideal FIR filters (in terms of conventional specifications).

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University