Comparison to the Optimal Chebyshev FIR Bandpass Filter

To provide some perspective on the results, we will compare the results for the window method to the optimal Chebyshev FIR filter having the same length and design specs as used for the window method above.

An optimal Chebyshev FIR filter is optimal in the minimax sense. That is, the worst-case error (maximum ripple) is minimized. It is also called optimal in the `` '' sense, since the norm of a frequency response error is the maximum magnitude over all frequencies:

The norm is also sometimes 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 effectively found using the Remez multiple exchange algorithm [166,212,65]. The window method and the Remez method together span most practical FIR filter design needs, from ``quick and dirty'' to essentially perfect FIR filters. However, another versatile, effective, and often-used case not covered here is the weighted least squares method, which is implemented in the Matlab function firls (also in the Signal Processing Toolbox). A good reference for further study is [193].

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.