How can I do Fourier transform of a function using FFT , keeping information of both the amplitude and phase ??

In general when FFT is used to evaluate the fourier transform, only the absolute value is plotted. The phase after performing the FFT turns out to be oscillatory. I am looking for a way to use FFT on various functions to evaluate the fourier transform.

E.g: fourier transform of the function f(x)=(x-1)^2 Exp(-x^2) for simplicity.

Answer 1

[Youssef Khmou]

Hi, Avinash, you can use this function :

 function z=Fast_Fourier_Transform(x,nfft)

 N=length(x);
 z=zeros(1,nfft);
 Sum=0;
 for k=1:nfft
     for jj=1:N
         Sum=Sum+x(jj)*exp(-2*pi*j*(jj-1)*(k-1)/nfft);
     end
 z(k)=Sum;
 Sum=0;% Reset
 end

But the function is slow whith long signals ,so you can try my submission : http://www.mathworks.com/matlabcentral/fileexchange/40002-psd-power-spectral-density-and-amplitude-spectrum-with-adjusted-fft

[Avinash Rustagi]

I was thinking more along the lines of getting to use the built-in functions because ultimately I need to evaluate the transforms of two variable functions. Using "for" loops won't be a very good idea for a large number of data points.

I need to get the fourier transform of f(x,y) and the sampling is done at very small spacings to give me a large number of data points.

I have tried something which works but cannot justify why ?

If "f" is the function and "g" is the fourier transform, I get very good results if I use

g=fftshift(fft(fftshift(f))); Note: the extra "fftshift" before taking the "fft". Using this chain of command , I do not have to worry about the phase. Everything fits in.

[Youssef Khmou]

in the submission, the fft built-in is the core, but the functions computes nfft and give exact frequencies by adjusting frequency axis and amplitudes .

Ok anyway we wait for better answer .