I want to sample (evaluate at regular intervals) the integral of the Kaiser window.
Is there a nice closed-form solution to this, so I don't have to do it numerically?
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1$\begingroup$ sample: I assume that means "calculating the value of the integral at equidistant points in time domain", right? Would be easier in frequency domain... $\endgroup$– Marcus MüllerCommented Jun 4, 2021 at 17:42
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$\begingroup$ Can you, by the way, constrain the points at which you plan to evaluate the integral? Pretty sure the integral of Bessel functions include a Gamma function, and that's "pretty" for natural number arguments. $\endgroup$– Marcus MüllerCommented Jun 4, 2021 at 17:50
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1$\begingroup$ @marcus I've added some clarification to the question. Yes, evaluate at equidistant points, but not aligned in any useful way to the boundaries or scale parameter or anything. $\endgroup$– cloudfeetCommented Jun 4, 2021 at 19:06
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1 Answer
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The Kaiser window of length $L$ is:
$$ w(t) = \begin{cases} \frac{1}{I_0(\beta)} \, I_0\left(\beta \sqrt{1 - \left(\frac{t}{L/2}\right)^2 } \right) \qquad & |t| \le L/2 \\ 0 & |t| > L/2 \\ \end{cases}$$
where
$$ I_0(u) \triangleq \sum\limits_{k=0}^{\infty} \frac{(-1)^k }{(k!)^2}\left( \frac{u}{2} \right)^{2k} $$
is the zeroth-order Bessel function of the first kind.
Now it ain't closed form but you can integrate each term, with respect to $t$, individually and add them.
answered Jun 5, 2021 at 5:17