I'm testing the phase output of an fft of a sin signal and a cos signal.
The script below creates the signals and performs an FFT on them. Bins who's amplitude is below a threshold are zeroed for the phase spectrum because I am only interested in the phase of the signals.
% 10khz 10 second long time interval
t = 0:1 / 10000:10;
%1khz cos
c = cos(2 * pi * 1000 .* t);
%1khz sin
s = sin(2 * pi * 1000 .* t);
%ffts
C = fft(c)/length(c);
S = fft(s)/length(s);
%magnitude and phases of ffts
CA = abs(C); %cos magnitude
SA = abs(S); %sin magnitude
Cthresh = max(CA) * 0.5;
Sthresh = max(SA) * 0.5;
%find all indeces below the threshold
Crange = find(CA < Cthresh);
Srange = find(SA < Sthresh);
%set the indeces below the threshold to 0 - phase will be meaningless for
%noise values
CP = angle(C);
CP(Crange) = 0;
SP = angle(S);
SP(Srange) = 0;
If you plot CP – the phase of the cos – you will get a phase of 0.3142 in the bins corresponding to the frequency of the cos signal and zeros elsewhere. This is pi/10. I'm expecting to get pi. Why is this?
If you plot SP you get values of 1.2566. I'm expecting to get pi/2 or 1.5708. 80% of the expected value. What is causing these errors?
Best Answer
If your input signal is not perfectly periodic in the FFT aperture length (an exact integer number of full periods), the sinusoids will be discontinuous across the ends of the FFT aperture. Thus you will get a phase that is the average of the two different phases at both ends of the FFT input vector.
If you want a more sensible phase, reference the phase of your sinusoids to the center of the FFT input vector, and do an fft shift before the FFT. This will result in a continuous sinusoid at the zero phase reference position, with a single phase instead of a weird average value.
Also note that matlab may reference the phase to the second point in a sampled sinusoid, e.g. vectorElement[i=1], not the first, vectorElement[i=0]. This will have a phase of pi/10 for a sinusoid of period = 20 samples.