Obtaining correct frequency and normalisation using convolution function

I wish to better understand the matlab convolution function, conv. My signal is the product of two sin waves of different frequencies in the time domain and so this gives two more sin waves at the sum and difference frequencies. If I directly take the Fourier transform of my signal I see two peaks in the frequency domain as expected. However when I take the convolution of the two individual sin waves I do not get the same result as expected. I can still see the two peaks expected, but they are much more spaced. Is there some way to shift the convolution so it is at the correct frequency?

Here is a simple code to show what I have been trying

```

time=linspace(0,1,10000);
 
sig=sin(2*pi*10.*time)*100;
sig2=sin(2*pi*20.*time)*100;
 
sigT=sig.*sig2;
 
plot(sigT)

FTsig=fft(sig);
FTsig2=fft(sig2);
 
expFTtot=median(diff(time))*conv(FTsig,FTsig2,'full');
expFTtot2=conv(FTsig2,FTsig,'full');
 
ft=fft(sigT);
 
figure(2)
loglog(abs(ft(1:end/2)),'-.')
hold on
loglog(abs(expFTtot(1:end/2)))
 
loglog(abs(expFTtot2(1:end/2)))

```

Answer 1

[Matt J]

It's usually a good idea to do things in a certain order. First, plan the sampling axes both for time and frequency. Note that because your final spectrum will have peaks at 10 Hz and 30 Hz, we want a frequency sampling interval dF that divides evenly into both of these,

N=10000; 
dF=0.1; %frequency sampling interval
dT=1/N/dF; %time sampling interval

Axis = (0:N-1)-ceil((N-1)/2);
tAxis=Axis*dT; %time axis
fAxis=Axis*dF; %frequency axis

Next, we sample,

sig1=sin(2*pi*10.*tAxis)*100;  %Sample the signals
sig2=sin(2*pi*20.*tAxis)*100;
sigT=sig1.*sig2;

Next, we do the Fourier analysis,

Fourier=@(z) ifftshift(fft( fftshift(z) ))*dT; %Fourier domain
FTsig1=Fourier(sig1);  
FTsig2=Fourier(sig2);
FTsigT=Fourier(sigT);

expFTtot=conv(FTsig1,FTsig2,'same')*dF; %check with convolution

And finally plot. We can see below that there is nice agreement and the peaks are at the appropriate locations.

semilogy(fAxis,abs(FTsigT),'--b', fAxis, abs(expFTtot),'r:') %plot
legend('FFT','Convolution')
xlabel 'Frequency (Hz)'
xlim([-50,50])