# Integral of Kaiser window

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?

• sample: I assume that means "calculating the value of the integral at equidistant points in time domain", right? Would be easier in frequency domain... Commented Jun 4, 2021 at 17:42
• 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. Commented Jun 4, 2021 at 17:50
• @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. Commented Jun 4, 2021 at 19:06

The Kaiser window of length $$L$$ is:
$$w(t) = \begin{cases} \frac{1}{I_0(\beta)} \, I_0\left(\beta \sqrt{1 - \left(\frac{t}{L/2}\right)^2 } \right) \qquad & |t| \le L/2 \\ 0 & |t| > L/2 \\ \end{cases}$$
$$I_0(u) \triangleq \sum\limits_{k=0}^{\infty} \frac{(-1)^k }{(k!)^2}\left( \frac{u}{2} \right)^{2k}$$
Now it ain't closed form but you can integrate each term, with respect to $$t$$, individually and add them.