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# convolution of Gaussian pulses

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#### purnapragna

gaussian convolution

What is the final result if i convolve two Gaussian pulses i.e.,
$e^{-t^2}*e^{-t^2}=?$
If i convolve for $n$ times what is the result?

thnx

purna!

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convolution gaussian

the result is

1/2*pi^(1/2)*2^(1/2)*exp(-1/2*t^2)

i used matlab for this by convert exp(-t^2) into freq domain the i multiplyed it by itself
then i reconvert it in time domain again
the code is

syms t
a=fourier(exp(-t^2));
a=a^2;
i=ifourier(a)
pretty(i)

for n time convolution you should do this n times

convolution of gaussian

Hey after i have done in Frequency domain i got the Gaussian pulse as the answer
But why like this can anybody explain?

thnx

purna!

convolution of two gaussians

Hi purna,

The result is a gaussian pulse sqrt(2) times wider.
You get this result by directly solving the convolution integral.

Another interesting way to consider this is as follows:
The Fourier transform (FT) of a gaussian pulse (in time) is a gaussian function (in frequency). This if found by solving the integral of the FT.
The product of the gaussian function by itself is a gaussian function, sqrt(2) times narrower. This is easy to show.
Going back to time domain, and using FT properties, you conclude that the convolution of a gaussian pulse by itself is a gaussian pulse sqrt(2) times wider.
Regards

Z

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