# Can convolution of one signal with different signals give the same answer?

Let us consider $x_1(t)$, $x_2(t)$, $x_3(t)$, all the same within some some duration 0 to $T$ but all different outside this interval. Now let us multiply each of these signals with $w(t)$, a window function - nonzero from 0 to $T$ but 0 outside this interval. So the multiplication of this $w(t)$ with each of $x_i(t)$ will give the same signal. This should be same as Inverse Fourier Transform of $W(\omega)$ convolved with $X_i(\omega)$. Does this mean convolution of the same signal with different signals can give the same result? Any comments? Is there a pitfall in my interpretation?

Also I see mathematically equations neatly entered in questions and answers on this site? Where should I start to learn on how to write the equations?

• To enter equations, use latex notation between dollar signs. See the edits I made to your question to get started. – MBaz Apr 30 '15 at 23:52
• The answer to the question in the title is Yes. The signals $x_i(t) = \operatorname{sinc}(t/T_i)$ are the impulse responses of different ideal lowpass filters of different bandwidths. Assume that $T_1 > T_2 > T_3$. Then, $$x_1(t)\star x_2(t) = x_1(t)\star x_3(t) = x_1(t).$$ Don't try to verify this statement in the time domain via convolution: just use $$X_1(f)X_2(f) = X_1(f)X_3(f) = X_1(f)$$ based on the properties of ideal LPFs. (Drawing a sketch of the LPF transfer functions might help....) – Dilip Sarwate May 1 '15 at 1:15
• Thanks a lot. I do not know Latex. How do I get onboard Latex quickly. Is there a quick reference so I can use it directly on this site? – Seetha Rama Raju Sanapala May 1 '15 at 4:20