# Documentation

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# kaiser

## Syntax

w = kaiser(L,beta)

## Description

w = kaiser(L,beta) returns an L-point Kaiser window in the column vector w. beta is the Kaiser window parameter that affects the sidelobe attenuation of the Fourier transform of the window. The default value for beta is 0.5.

## Examples

collapse all

Create a 200-point Kaiser window with a beta of 2.5. Display the result using wvtool.

w = kaiser(200,2.5);
wvtool(w)

## Algorithms

The coefficients of a Kaiser window are computed from the following equation:

$w\left(n\right)=\frac{{I}_{0}\left(\beta \sqrt{1-{\left(\frac{n-N/2}{N/2}\right)}^{2}}\right)}{{I}_{0}\left(\beta \right)},\text{ }0\le n\le N,$

where I0 is the zeroth-order modified Bessel function of the first kind. The length L = N + 1. Thus kaiser(L,beta) is equivalent to

besseli(0,beta*sqrt(1-(((0:L-1)-(L-1)/2)/((L-1)/2)).^2))/besseli(0,beta).

To obtain a Kaiser window that designs an FIR filter with sidelobe attenuation of α dB, use the following β.

$\beta =\left\{\begin{array}{ll}0.1102\left(\alpha -8.7\right),\hfill & \alpha >50\hfill \\ 0.5842{\left(\alpha -21\right)}^{0.4}+0.07886\left(\alpha -21\right),\hfill & 50\ge \alpha \ge 21\hfill \\ 0,\hfill & \alpha <21\hfill \end{array}$

Increasing β widens the mainlobe and decreases the amplitude of the sidelobes (i.e., increases the attenuation).

## References

[1] Kaiser, James F. “Nonrecursive Digital Filter Design Using the I0-Sinh Window Function.” Proceedings of the 1974 IEEE® International Symposium on Circuits and Systems. April, 1974, pp. 20–23.

[2] Digital Signal Processing Committee of the IEEE Acoustics, Speech, and Signal Processing Society, eds. Selected Papers in Digital Signal Processing. Vol. II. New York: IEEE Press, 1976.

[3] Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999, p. 474.