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 [136]

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

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

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

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

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

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