gausswin

Gaussian window

Syntax

w = gausswin(N)
w = gausswin(N,Alpha)

Description

w = gausswin(N) returns an N-point Gaussian window in a column vector, w. N is a positive integer.

w = gausswin(N,Alpha) returns an N-point Gaussian window with Alpha proportional to the reciprocal of the standard deviation. The width of the window is inversely related to the value of α. A larger value of α produces a more narrow window. The value of α defaults to 2.5.

    Note   If the window appears to be clipped, increase N, the number of points.

Examples

expand all

Gaussian Window

Create a 64-point Gaussian window. Display the result in wvtool.

L = 64;
wvtool(gausswin(L))

Gaussian Window and the Fourier Transform

This example shows that the Fourier transform of the Gaussian window is also Gaussian with a reciprocal standard deviation. This is an illustration of the time-frequency uncertainty principle.

Create a Gaussian window of length 64 by using gausswin and the defining equation. Set $\alpha = 8$, which results in a standard deviation of 64/16 = 4. Accordingly, you expect that the Gaussian is essentially limited to the mean plus or minus 3 standard deviations, or an approximate support of [-12, 12].

N = 64;
n = -(N-1)/2:(N-1)/2;
alpha = 8;

w = gausswin(N,alpha);

stdev = (N-1)/(2*alpha);
y = exp(-1/2*(n/stdev).^2);

plot(n,w)
hold on
plot(n,y,'.')
hold off

xlabel('Samples')
title('Gaussian Window, N = 64')

Obtain the Fourier transform of the Gaussian window at 256 points. Use fftshift to center the Fourier transform at zero frequency (DC).

nfft = 4*N;
freq = -pi:2*pi/nfft:pi-pi/nfft;

wdft = fftshift(fft(w,nfft));

The Fourier transform of the Gaussian window is also Gaussian with a standard deviation that is the reciprocal of the time-domain standard deviation. Include the Gaussian normalization factor in your computation.

ydft = exp(-1/2*(freq/(1/stdev)).^2)*(stdev*sqrt(2*pi));

plot(freq/pi,abs(wdft))
hold on
plot(freq/pi,abs(ydft),'.')
hold off

xlabel('Normalized frequency (\times\pi rad/sample)')
title('Fourier Transform of Gaussian Window')

More About

expand all

Algorithms

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

w(n)=e12(αn(N1)/2)2,

where –(N – 1)/2 ≤ n ≤ (N – 1)/2 and α is inversely proportional to the standard deviation of a Gaussian random variable. The exact correspondence with the standard deviation, σ, of a Gaussian probability density function is σ = (N – 1)/(2α).

References

[1] Harris, Fredric J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proceedings of the IEEE®. Vol. 66, January 1978, pp. 51–83.

[2] Roberts, Richard A., and C. T. Mullis. Digital Signal Processing. Reading, MA: Addison-Wesley, 1987, pp. 135–136.

Was this topic helpful?