Kaiser Window

Jim Kaiser discovered a simple approximation to the DPSS window based upon Bessel functions [115], generally known as the Kaiser window (or Kaiser-Bessel window).

Definition:

$$w_K(n) \isdefs \left\{ \begin{array}{ll} \frac{ I_0 \left( \beta \sqrt{ 1 - \left( \frac{n}{M/2}\right)^2} \right)}{ I_0(\beta) }, & -\frac{M-1}{2} \leq n \leq \frac{M-1}{2} \\ 0, & \mbox{elsewhere} \\ \end{array} \right.$$ (4.39)

Window transform:

The Fourier transform of the Kaiser window $w_K(t)$ (where $t$ is treated as continuous) is given by^4.11

$$W(\omega) = \frac{M}{I_0(\beta)} \frac{\sinh\left(\sqrt{\beta^2 - \left(\frac{M \omega}{2}\right)^2}\right)} {\sqrt{ \beta^2 - \left(\frac{M\omega}{2}\right)^2}} = \frac{M}{I_0(\beta)} \frac{\sin\left(\sqrt{\left(\frac{M \omega}{2}\right)^2-\beta^2}\right)} {\sqrt{\left(\frac{M\omega}{2}\right)^2 - \beta^2}}$$ (4.40)

where $I_0$ is the zero-order modified Bessel function of the first kind:^4.12

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

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 parametrized by $\alpha$ , where

  $$\beta\isdefs \pi\alpha$$ (4.42)