2.3 Chebyshev or equal ripple error approximation filters

If one poses the FIR filter design problem by requiring the maximum error over certain bands of frequencies be minimized, we call the resultingfilter a Chebyshev filter or an equal ripple filter. The fact that the minimization of the Chebyshev or $L_{\infty }$ error results in an equal ripple error comes from the alternation theorem. This very powerfultheorem allows one to minimize the Chebyshev error by directly constructing an equal ripple approximation with the proper number ofripples. That is the basis of several very effective algorithms, including the Remez exchange algorithm.

There are several ways one could pose the Chebyshev FIR filter design problem. For a simple length-N linear phase, lowpass filter with a transition band, ifone considers the length N, the passband ripple ${\delta }_p$ , the stopband ripple ${\delta }_s$ , and the transition bandwidth $\Delta ={\omega }_s-{\omega }_p$ , then one can fix or constrain any three of them and minimize the fourth. Or, as Parks and McClellan do, fix the band edges, ${\omega }_p$ and ${\omega }_s$ , and the ratio of ${\delta }_p$ and ${\delta }_s$ and minimize one of them.

The Chebyshev error measure is often used for approximation in digital filter design. This is particularly true when the signals and/or noiseare narrow band or single frequency or when one wants to minimize worst case possibilities. Theoretical justification for its use has been givenby Weisburn, Parks, and Shenoy [link] . For FIR filter design, the Parks-McClellan formulation of the filter design problem andapplication of the Remez exchange algorithm is most commonly used [link] , [link] . It is a particularly interesting and powerful method that should be studied and understood to be fully utilized.

Linear programming was used earlier [link] , [link] , [link] but dropped out of favor when the Parks-McClellan algorithm was introduced.It is now becoming more popular again because of more powerful computers, better algorithms [link] , [link] , and linear programming's ability to allow a variety of constraints [link] .

Still another approach to achieving a Chebyshev approximation is to minimize the $p^{th}$ power of the error using a large value of $p$ or to use an iterative scheme that solves a weighted least squared error with the weights at each stagedetermined by the error of the previous stage [link] . Still another design method that produces an equal ripple error approximation uses aconstrained least squared error criterion [link] , [link] which results in a Chebyshev solution if tight constraints are imposed.

The early work by Herrmann and Schüssler [link] , [link] and the algorithm by Hofstetter, Oppenheim, and Siegel [link] , [link] posed and solved a similar problem but they had only approximate control of ${\omega }_o$ (or ${\omega }_p$ or ${\omega }_s$ ) and always achieved the “extra ripple" design. Given the proper specifications, the Parks-McClellan algorithm coulddesign any filter that the Hofstetter-Oppenheim-Siegel algorithm could, but the opposite is not true. This seems to be one of the reasons theHofstetter-Oppenheim-Siegel algorithm is not commonly used.

The linear phase fir filter chebyshev approximation problem

The Chebyshev error is defined as the maximum difference between the actual and desired response over a band or several bands of frequencies. This is