# 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

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

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

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### 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.