frequency domain convolution problem

I have two rectangular signals in time domain, both of them have amptitude of 1, but different width. The signal 1 have width of 300 and siganl 2 have width of 600. If they were mutiply in time domain, the outcome will be only signal 1, so the output spectrum should be exactly the same as signal 1.

Since the multiplication of the time domain should be equal to convolution of frequency domain, so I convolve their spectrum. The output should be the same as signal 1 spectrum, but instead I get rubbish.

What have I done wrong? Thanks!

clear all

fs = 1;

Tm = 150/fs;
T = Tm/2;
f = linspace(1e-4,0.5,1e6);

figure
Signal_1 = Tm*sin(2*pi*T.*f./fs)./(2*pi*T.*f./fs);    % Signal 1 spectrum, sinc function
plot(f, abs(Signal_1))

Tm2 = 300/fs;
T2 = Tm/2;
Signal_2 = Tm2*sin(2*pi*T2.*f./fs)./(2*pi*T2.*f./fs);     % Signal 2 spectrum, also sinc function
plot(f, abs(Signal_2))

Output = conv(Signal_1, Signal_2, "same");      % Convolution of Signal 1 and Signal 2 in frequency domain
plot(f, Output)

Answer 1

[Matt J]

You have only generated half a sinc, rather than a full sinc from -infinity to +infinity.

Keep in mind as well that there will be discretization effects from truncating the sincs to a a finite window and from discrete sampling.

fs = 1;

Tm = 150/fs;
T = Tm/2;
f = linspace(-0.5,0.5,1e4);
Signal_1 = Tm*sin(2*pi*T.*f./fs)./(2*pi*T.*f./fs);    % Signal 1 spectrum, sinc function

Tm2 = 300/fs;
T2 = Tm/2;
Signal_2 = Tm2*sin(2*pi*T2.*f./fs)./(2*pi*T2.*f./fs);     % Signal 2 spectrum, also sinc function

Output = conv(Signal_1, Signal_2, "same")*(f(2)-f(1));      % Convolution of Signal 1 and Signal 2 in frequency domain


plot(f, Signal_2,'-k',f(1:5:end), Output(1:5:end),'x');legend('Signal 2','Output')

xlim([-0.05,+0.05]);