Dolph-Chebyshev Window Definition

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Let denote the desired window length. Then the zero-phase Dolph-Chebyshev window is defined in the frequency domain by [146]

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

where is defined by the desired ripple specification:

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

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

$\displaystyle \omega_c \isdef 2\cos^{-1}\left[\frac{1}{x_0}\right].$

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

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

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

Previous: Chebyshev Polynomials
Next: Dolph-Chebyshev Window Main-Lobe Width

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.

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