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Dolph-Chebyshev Window Definition

Let $ M$ denote the desired window length. Then the zero-phase Dolph-Chebyshev window is defined in the frequency domain by [155]

$\displaystyle W(\omega) = \frac{T_{M-1}[x_0 \cos(\omega/2)]}{T_{M-1}(x_0)}$ (4.48)

where $ x_0>1$ is defined by the desired ripple specification:

$\displaystyle \vert W(\omega)\vert \le r = \frac{1}{T_{M-1}(x_0)}, \quad \forall\vert\omega\vert\ge\omega_c,$ (4.49)

where $ \omega_c$ is the ``main lobe edge frequency'' defined by

$\displaystyle \omega_c \isdefs 2\cos^{-1}\left[\frac{1}{x_0}\right].$ (4.50)

Expanding $ W(\omega)$ in terms of complex exponentials yields

$\displaystyle W(\omega) = \sum_{n=-M_h}^{M_h} w(n) e^{-j \omega n}$ (4.51)

where $ M_h\isdef (M-1)/2$ . Thus, the coefficients $ w(n)$ give the length $ M$ Dolph-Chebyshev window in zero-phase form.


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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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