FFT

Hi, I have a complex time signal x(t)=X1*exp(i*theta1)+X2*exp(i*2*theta). On converting to frequency domain, i expect frequency components at omega1 and twice of omega1 (when theta=omega*t). FFT gives the desired frequencies, but the amplitudes donot correspond to X1 and X2. I understand that this is due to the phase difference between the signals X1*exp(i*theta1) and X2*exp(i*2*theta). My question is how should i get back the amplitudes X1,X2,... when multiple frequency components exist in the signal.

P.S: when only x(t)=X1*exp(i*theta1);am able to get back X1 in the fft.

Thanks in advance, Sumit.

Answer 1

[Wayne King]

Hi Sumit:

   Fs = 1e3;
   t = 0:0.001:1-0.001;
   x = 2*exp(1j*(2*pi*100*t-pi/4))+1.5*exp(1j*(2*pi*200*t+pi/8));
   xdft = fft(x)/length(x);
   abs(xdft(101))  %this is the 100-Hz component
   abs(xdft(201))  % this is the 200-Hz component

Perhaps you forgot to divide the output of fft() by the length of the signal? The DFT bins corresponding to your complex exponentials contain the complex amplitude (amplitude and phase), but if you take the modulus you should recover your X1 and X2.