Kaiser Window :: Special Topics (Signal Processing Toolbox)

Signal Processing Toolbox

Kaiser Window

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 beta controls the sidelobe height. For a given beta , the sidelobe height is fixed with respect to window length. The statement kaiser(n,beta) computes a length n Kaiser window with parameter beta.

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

The time and frequency domain plots of the length 50 Kaiser windows with different betas are in this illustration. The window with beta 1 is widest in the time domain, followed by beta 4 and beta 9 as the most narrow. As expected, in the frequency domain, the mainlobe of the beta 9 is widest, followed by the narrower beta 4 and finally the beta 1 with the most narrow mainlobe.

To create these Kaiser windows using the MATLAB command line,

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',3)

As beta increases, the sidelobe height decreases and the mainlobe width increases. This wintool shows how the sidelobe height stays the same for a fixed beta parameter as the length is varied.

To create these Kaiser windows using the MATLAB command line:

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')

Kaiser Windows in FIR Design

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 sidelobe height of - alpha dB, the beta parameter is

For a transition width of capital delta rad/s, use the length

Filters designed using these heuristics will meet the specifications approximately, but you should verify this. To design a lowpass filter with cutoff frequency rad/s, transition width rad/s, 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);
```

Generalized Cosine Windows Chebyshev Window