Huge difference between the result of fft function Matlab and analytical Fourier transform of the same function
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I am trying to find a way to obtain the numerical fourier transform of a function (it is not a signal and I only want to obtain the numerical fourier transform of a function). For a test code, I tried to see what is the result of fft matlab for a Gaussian function and compared it with the analytical fourier transform of this Gaussian. I have attached the plot of both results. Why the amplitude of fft result is that huge compared to analytical result? I can use fftshift to shift the result of fft to center but still the issue of amplitudes are there. I am wondering isn't it because it is a Gaussian function and so not periodic over time? Does fft only works if the function we have is spanned from -infinity to +infinity (basically a signal)? or we can use it if we want ot calculate the fourier transform of a function which spans from t-1 to t-2?
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David Goodmanson il 11 Set 2022
Modificato: David Goodmanson il 11 Set 2022
Your frequency grid runs from -5 to 5, which for an fft is -fs/2 to fs/2 (fs being the sampling frequency), so fs = 10. The sampling interval delta_t = 1/fs = 1/10. It appears that you are trying to approximate a continuous Riemann integral
Int g(t) e^(-2pi i f t) dt
by using the fft. That approximation is a sum over indices (done by the fft) times the width of the intervals.
Sum (stuff) * delta_t
So if you multiply by 1/10, that takes the fft result down by exactly the right amount.