Dolph-Chebyshev Window Length Computation

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Given a prescribed side-lobe ripple-magnitude and main-lobe width $2\omega_c$ , the required window length is given by [146]

$\displaystyle M = 1 + \frac{\cosh^{-1}(1/r)}{\cosh^{-1}[\sec(\omega_c/2)]}.$

For $\omega_c\ll\pi$ (the typical case), the denominator is close to $\omega_c/2$ , and we have

$\displaystyle M \approx 1 + \frac{2}{\omega_c}\cosh^{-1}\left(\frac{1}{r}\right)$

Thus, half the time-bandwidth product in radians is approximately

$\displaystyle \beta \isdef (M-1) \omega_c\approx 2\cosh^{-1}\left(\frac{1}{r}\right),$

where $\beta$ is the parameter often used to design Kaiser windows (§3.9).

Previous: Dolph-Chebyshev Window Main-Lobe Width
Next: Gaussian Window and Transform

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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