The Kaiser window and its
Fourier transform are given by
: : w_0(x) \triangleq \left\{ \begin{array}{ccl} \tfrac{1}{L}\frac{I_0\left[\pi\alpha \sqrt{1 - \left(2x/L\right)^2}\right]}{I_0[\pi\alpha]},\quad &\left|x\right| \leq L/2\\ 0,\quad &\left|x\right| > L/2 \end{array}\right\} \quad \stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{\sin\bigg(\sqrt{(\pi L f)^2-(\pi \alpha)^2}\bigg)} {I_0(\pi \alpha)\cdot \sqrt{(\pi L f)^2-(\pi \alpha)^2}}, {{efn-ua where \beta \triangleq \pi \alpha,\ \omega \triangleq 2 \pi f,\ M=L. :\frac{\sinh\bigg(\sqrt{(\pi \alpha)^2 - (\pi L f)^2}\bigg)} {I_0(\pi \alpha)\cdot \sqrt{(\pi \alpha)^2 - (\pi L f)^2}}. }} where
: • is the zeroth-order
modified Bessel function of the first kind, • is the window duration, and • is a non-negative
real number that determines the shape of the window. In the frequency domain, it determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design. • Sometimes the Kaiser window is parametrized by , where . For
digital signal processing, the function can be sampled symmetrically as
: :w[n] = L\cdot w_0\left(\tfrac{L}{N} (n-N/2)\right) = \frac{I_0\left[\pi\alpha \sqrt{1 - \left(\frac{2n}{N}-1\right)^2}\right]}{I_0[\pi\alpha]},\quad 0 \leq n \leq N, where the length of the window is N+1, and N can be even or odd. (see
A list of window functions) In the Fourier transform, the first null after the main lobe occurs at f = \tfrac{\sqrt{1+\alpha^2}}{L}, which is just \sqrt{1+\alpha^2} in units of N (
DFT "bins"). As
α increases, the main lobe increases in width, and the side lobes decrease in amplitude. = 0 corresponds to a rectangular window. For large the shape of the Kaiser window (in both time and frequency domain) tends to a
Gaussian curve. The Kaiser window is nearly optimal in the sense of its peak's concentration around frequency 0. ==Kaiser–Bessel-derived (KBD) window==