where , 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
The value of α defaults to 2.5.
w=gausswin(N,Alpha) returns an N-point Gaussian window where Alpha is proportional to 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.
Create a 64-point Gaussian window and display the result in WVTool:
This example demonstrates 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 α=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; y = exp(-1/2*(alpha*n/(N/2)).^2); w = gausswin(N,alpha); plot(n,w,'r') hold on; plot(n,y,'k') title('Gaussian Window N = 64');
Obtain the Fourier transform of the Gaussian window and use fftshift to center the Fourier transform at zero frequency (DC).
figure wdft = fftshift(fft(w)); freq = linspace(-pi,pi,length(wdft)); plot(freq,abs(wdft),'linewidth',2) xlabel('Radians/Sample'); title('Fourier Transform of Gaussian Window');
The Fourier transform of the Gaussian window is also Gaussian with a standard deviation that is the reciprocal of the time-domain standard deviation.
 Harris, F.J. "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform." Proceedings of the IEEE®. Vol. 66, No. 1 (January 1978).
 Roberts, Richard A., and C.T. Mullis. Digital Signal Processing. Reading, MA: Addison-Wesley, 1987, pp. 135-136.