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Dolph-Chebyshev Window Length Computation

Given a prescribed side-lobe ripple-magnitude $ r$ and main-lobe width $ 2\omega_c$ , the required window length $ M$ is given by [155]

$\displaystyle M = 1 + \frac{\cosh^{-1}(1/r)}{\cosh^{-1}[\sec(\omega_c/2)]}.$ (4.52)

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

$\displaystyle M \approx 1 + \frac{2}{\omega_c}\cosh^{-1}\left(\frac{1}{r}\right)$ (4.53)

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

$\displaystyle \beta \isdefs (M-1) \omega_c\approx 2\cosh^{-1}\left(\frac{1}{r}\right),$ (4.54)

where $ \beta $ is the parameter often used to design Kaiser windows3.9).


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