Create a 64-point Gaussian window. Display the result in
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 , 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')
L — Window length
Window length, specified as a positive integer.
alpha — Width factor
2.5 (default) | positive real scalar
Width factor, specified as a positive real scalar.
inversely proportional to the width of the window.
w — Gaussian window
Gaussian window, returned as a column vector.
The coefficients of a Gaussian window are computed from the following equation:
where –(L – 1)/2 ≤ n ≤ (L – 1)/2, and α is inversely proportional to the standard deviation, σ, of a Gaussian random variable. The exact correspondence with the standard deviation of a Gaussian probability density function is σ = (L – 1)/(2α).
 Hansen, Eric W. Fourier Transforms: Principles and Applications. New York: John Wiley & Sons, 2014.
 Oppenheim, Alan V., Ronald W. Schafer, and John R. Buck. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice Hall, 1999.