# 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

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### 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 , 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') 

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### Algorithms

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

$w\left(n\right)={e}^{-\frac{1}{2}{\left(\alpha \frac{n}{\left(N-1\right)/2}\right)}^{2}}={e}^{-{n}^{2}/2{\sigma }^{2}},$

where –(N – 1)/2 ≤ n ≤ (N – 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 = (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.