convolving signal and noise in frequency domain and inverting back to time domain

Hi,

I'm convolving time series data (a signal and noise) in frequency domain and inverting the convolved time series back to time domain. This seems like a straight forward operation with an fft and ifft. Yet when I do this, the resulting final time series appears to be amplified by some factor. Any explanation about how I can fix this is greatly appreciated.

f = 2;

omega = 2*pi*f;

dt = 0.01;

t = 0:dt:5;

sig = sin(omega*t) * 2;

noise = randi([-1,1],1,length(t));

sig_ns_conv = (ifft(fft(sig) .* fft(noise)));

figure(1)

plot(t,sig,'-r');

hold on;

plot(t,noise,'-b')

plot(t,sig_ns_conv,'-g');

hold off;

Answer 1

[Matt J]

sig_ns_conv = (ifft(fft(sig) .* fft(noise))) * dt;

[Januka Attanayake]

Not sure of this scaling as it seems the resultant signal amplitude is underestimated. Shouldn't the convolution in frequency domain be similar to addition in time domain?

[Matt J]

No, convolution in one domain is equivalent to multiplication in the other domain.

[Januka Attanayake]

Sorry. Yes, that's what I meant. However, the above example doesn't provide similar results. not sure why.

i.e. sig_ns_conv = (ifft(fft(sig) .* fft(noise))); isn't the same as sig .* noise in time domain.

[Matt J]

No, it isn't expected to be. You are multiplying in the frequency domain, so it is equivalent to convolution in the time domain:

f = 2;
omega = 2*pi*f;
dt = 0.01;
t = 0:dt:5;
N=numel(t);
P=2*N-1;

sig = sin(omega*t) * 2;
noise = randi([-1,1],1,N);
 
sig_ns_conv = conv(sig,noise)*dt;

sig_ns_fft = ifft( fft(sig,P) .* fft(noise,P))*dt;

t=0:dt:dt*(P-1);
plot(t, sig_ns_fft, t , sig_ns_conv,'o'); legend('FFT','time domain');

[Matt J]

Januka Attanayake commented:

Many thanks for that clarification Matt.

[Matt J]

You are quite welcome.