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Example Chebyshev Windows and Transforms

Figure 3.31 shows the Dolph-Chebyshev window and its transform as designed by chebwin(31,40) in Matlab, and Fig.3.32 shows the same thing for chebwin(31,200). As can be seen from these examples, higher side-lobe levels are associated with a narrower main lobe and more discontinuous endpoints.

Figure: The length $ 31$ Dolph-Chebyshev window with ripple (side-lobe level) specified to be $ -40$ dB.
\includegraphics[width=\twidth]{eps/cheb31R40}

Figure: The length $ 31$ Dolph-Chebyshev window with ripple (side-lobe level) specified to be $ -200$ dB.
\includegraphics[width=\twidth]{eps/cheb31R200}

Figure: The length $ 101$ Dolph-Chebyshev window with ripple (side-lobe level) specified to be $ -40$ dB.
\includegraphics[width=\twidth]{eps/cheb101R40}

Figure 3.33 shows the Dolph-Chebyshev window and its transform as designed by chebwin(101,40) in Matlab. Note how the endpoints have actually become impulsive for the longer window length. The Hamming window, in contrast, is constrained to be monotonic away from its center in the time domain.

The ``equal ripple'' property in the frequency domain perfectly satisfies worst-case side-lobe specifications. However, it has the potentially unfortunate consequence of introducing ``impulses'' at the window endpoints. Such impulses can be the source of ``pre-echo'' or ``post-echo'' distortion which are time-domain effects not reflected in a simple side-lobe level specification. This is a good lesson in the importance of choosing the right error criterion to minimize. In this case, to avoid impulse endpoints, we might add a continuity or monotonicity constraint in the time domain (see §3.13.2 for examples).


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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
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