Window transform:

The Fourier transform of the Kaiser window $w_K(t)$ (where $t$ is treated as continuous) is given by

$$W(\omega) = \frac{M}{I_0(\beta)} \frac{\sinh\left(\sqrt{\beta^2... ...\right)^2}-\beta^2\right)} {\sqrt{\left(\frac{M\omega}{2}\right)^2 - \beta^2}}$$

where $I_0$ is the zero-order modified Bessel function of the first kind:^4.6

$$I_0(x) \isdef \sum_{k=0}^{\infty} \left[ \frac{\left(\frac{x}{2}\right)^k}{k!} \right]^2$$

Notes:

• Reduces to rectangular window for $\beta=0$
• Asymptotic roll-off is 6 dB/octave
• First null in window transform is at $\omega_0=2\beta/M$
• Time-bandwidth product $\omega_0 (M/2) = \beta$ radians if bandwidths are measured from 0 to positive band-limit
• Full time-bandwidth product $(2\omega_0) M = 4\beta$ radians when frequency bandwidth is defined as main-lobe width out to first null
• Sometimes the Kaiser window is parameterized by $\alpha$ , where
  $$\beta\isdef \pi\alpha$$