Optimal Chebyshev Filters

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Problems with Remez Exchange

Convergence unlikely for FIR filters longer than a few hundred taps or so
Fundamentally real polynomial approximation on the unit circle
- Applies only to linear phase filters
- Cannot design individual (non-minimum phase) ``sub-filters'' in polyphase filter bank form
Really need optimal weighted complex approximation
e.g., minimum phase FIR filter design

Example: Chebyshev window and its transform:

**Figure 11.28:**
$\includegraphics[width=4in]{eps/cheb}$

**Figure B.34:**
$\includegraphics[width=4in]{eps/chebFT}$

written by 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.

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