Adding Random phase causes fft anomalies?

Hi

If I add random_phase to my waveforms then the fft goes skewy - basically it breaks down and is not producing what it should be like when random_phase is not present in sinewave generation loop.

%Setup sampling times and frequency range
f=(20.2 : 0.1 : 21.1)*10^9;
Fs = 3*max(f);
Ts = 1/Fs;
end_t = 0.5*10^-6
dt=0: Ts : end_t -Ts

a=0; b=pi; %for random phase calculation

%Below part adds a new sine wave every 0.01*10^9 within frequency range

for a=1:length(f)-1
random_phase = (b-a).*rand(1,length(dt))+a;
y(a,:) = 5*sin(2*pi.*f(a) .* dt + random_phase);  %remove random_phase and scripts works fine. 
end

%combined waveform
waveform = sum(y);

%setup frequency domain for FFT
N=length(waveform);
freq_domain = (0:N-1);
freq_domain = f_domain*Fs/N

ft=2*abs(fft(waveform)/N);

figure(1)
bar(freq_domain, ft);
ax=gca; ax.XAxis.Exponent = 9;
xlim([20 *10^9 21.4*10^9]);

Completely stumped how random phase messes up my fft...

Answer 1

[Star Strider]

The random phase will shift your sine curve randomly in time, essentially destroying the periodicity. You can see this easily if you plot your signal in the time domain:

figure(2)
plot(dt, waveform)
grid
subplot(2,1,2)
plot(dt, waveform)
axis([0  2E-8    ylim])
grid

Also, if you use plot rather than bar, the Fourier transform is easier to see.

Answer 2

[Joel Miller]

If you are trying to randomly shift the phase of the sinusoids, sin(2*pi.*f(a) .* dt), then random_phase should be a scalar. When random_phase is a vector of length dt, you are adding a different phase to each time increment of the sinusoid. That is a reasonable definition of random (phase) noise. You can see this by fourier transforming just random_phase.