# Fast Fourier transform of the Gaussian function

As I know, the Fourier transform of the Gaussian function is also Gaussian.

The simple verification for that is shown below. http://mathworld.wolfram.com/FourierTransformGaussian.html

But fast Fourier transform(in Matlab) of the Gaussian function gives me some distribution, which oscillates its sign. However, its power spectrum looks like Gaussian. The results I got from Matlab are shown below.

(1) The Gaussian function in time domain

(2) fft of The Gaussian function in frequency domain

Why does the sign of fast Fourier transform of the Gaussian function oscillate? I appreciate your attention and answers in advance.

• A shift in the time domain is a modulation in the frequency domain. Had your Gaussian been centered at 0 in the time domain, you wouldn't have the oscillation in the frequency domain. – Andy Walls Feb 19 at 13:49
• @AndyWalls But if I just put Gaussian centered at 0, it would have half of the Gaussian shape. Is it okay? – 이동현 Feb 19 at 14:47
• Just as the frequency domain is periodic/cyclic with the DFT, the time domain of the DFT is periodic/cyclic. The right half of your Gaussian would be at zero; the left half of your Gaussian would be at the end of your time domain samples block. – Andy Walls Feb 19 at 16:08
• Also in the frequency domain, you may wish to consider plotting magnitude on one plot and phase on another plot. Plotting the real part on one plot and the imaginary part on another plot isn't very helpful for analysis of engineering problems. – Andy Walls Feb 19 at 16:11