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The Kaiser window is an approximation to the prolate spheroidal window, for which the ratio of
the mainlobe energy to the sidelobe energy is maximized. For a Kaiser window of a
particular length, the parameter *β* controls the relative
sidelobe attenuation. For a given *β*, the relative sidelobe
attenuation is fixed with respect to window length. The statement `kaiser`

`(n,beta)`

computes a
length `n`

Kaiser window with parameter
`beta`

.

As *β* increases, the relative sidelobe attenuation decreases and the
mainlobe width increases. This screen shot shows how the relative sidelobe attenuation
stays approximately the same for a fixed *β* parameter as the length is
varied.

Examples of Kaiser windows with length 50
and *β* parameters of 1, 4, and 9 are shown
in this example.

To create these Kaiser windows using the MATLAB^{®} command
line, type the following:

n = 50; w1 = kaiser(n,1); w2 = kaiser(n,4); w3 = kaiser(n,9); [W1,f] = freqz(w1/sum(w1),1,512,2); [W2,f] = freqz(w2/sum(w2),1,512,2); [W3,f] = freqz(w3/sum(w3),1,512,2); plot(f,20*log10(abs([W1 W2 W3]))) grid legend('\beta = 1','\beta = 4','\beta = 9')

To create these Kaiser windows using the MATLAB command line, type the following:

w1 = kaiser(50,4); w2 = kaiser(20,4); w3 = kaiser(101,4); [W1,f] = freqz(w1/sum(w1),1,512,2); [W2,f] = freqz(w2/sum(w2),1,512,2); [W3,f] = freqz(w3/sum(w3),1,512,2); plot(f,20*log10(abs([W1 W2 W3]))) grid legend('length = 50','length = 20','length = 101')

There are two design formulas that can help you design FIR filters to meet a set of filter specifications using a Kaiser window. To achieve a relative sidelobe attenuation of –*α* dB, the *β* (`beta`

) parameter is

$$\beta =\{\begin{array}{cc}0.1102\left(\alpha -8.7\right),& \alpha >50,\\ 0.5842{\left(\alpha -21\right)}^{0.4}+0.7886\left(\alpha -21\right),& 50\ge \alpha \ge 21,\\ 0,& \alpha <21.\end{array}$$

For a transition width of $\Delta \omega $ rad/sample, use the length

$$n=\frac{\alpha -8}{2.285\Delta \omega}+1$$.

Filters designed using these heuristics will meet the specifications approximately, but you should verify this. To design a lowpass filter with cutoff frequency 0.5$\pi $ rad/sample, transition width 0.2$\pi $ rad/sample, and 40 dB of attenuation in the stopband, try

```
[n,wn,beta] = kaiserord([0.4 0.6]*pi,[1 0],[0.01 0.01],2*pi);
h = fir1(n,wn,kaiser(n+1,beta),'noscale');
```

The `kaiserord`

function estimates the filter order, cutoff frequency, and Kaiser window beta parameter needed to meet a given set of frequency domain specifications.

The ripple in the passband is roughly the same as the ripple in the stopband. As you can see from the frequency response, this filter nearly meets the specifications:

fvtool(h,1)