Hi all,
I have a question regarding the computation of the discrete Fourier transfrom of a real valued Gaussian function using the FFT routine in MATLAB. First I define the discrete grids in time and frequency
tn = linspace(-10.0,10.0,128);
fn = tn/(20.0*20.0/128);
gauss = exp(-tn.^2);
The Gaussian function is shown below
The discrete Fourier transform is computed by
fftgauss = fftshift(fft(gauss));
and shown below (red is the real part and blue is the imaginary part)
Now, the Fourier transform of a real and even function is also real and even. Therefore, I'm a bit surprised by the somewhat significant nonzero imaginary part of fftgauss. What is more surprising to me is the oscillations in the real part of fftgauss --- is this due to the discreteness of the transform? The continuous Fourier transform of a real valued Gaussian function is a real valued Gaussian function too...
In order to answer this question, I have written a simple discrete Fourier transform, see below
dftgauss = zeros(128);
for n = 1:128
for m = 1:128
dftgauss(n) = dftgauss(n) + gauss(m)*exp(2.0*pi*i*fn(n)*tn(m));
end
end
and dftgauss is shown below
Clearly, fftgauss and dftgauss are different, though the real part of dftgauss is equal to abs(fftgauss). My discrete Fourier transform actually gives the result that I expected (The continuous Fourier transform of a real valued Gaussian function is a real valued Gaussian function too...).
In short: Why is the real part of fftgauss oscillating?
