Solve convolution using graph

Question

I have this two equations :

$u(t) = \widetilde \delta(t)$

and this

$f(t) = 5sinc(5t)$

Fourier transform are this:

$U(t) = 1$

$F(t) = \Pi(f/5)$

Right? How to draw this ? $U(t) * F(t)$

Answer 1

Well, the convolution of any function with a constant function is a constant function. It's easy to see if you write the formula of the convolution.

But since you seem to ask for a graph-based proof or intuition, an interpretation of that convolution is that it's obtained by integrating $U=1$ against the moving square window $F$. Because $U$ is constant, that integration always yields the same result (5 in this case).

Another way to see it is that the Fourier transform of the convolution $U \circledast F$ is the the product of Fourier transforms $u \cdot f$. And we have $$u(t) \cdot f(t) = \delta(t)\cdot 5 sinc(t) = \delta(t)\cdot 5 sinc(0) = 5 \delta(t)$$ Therefore, $$U \circledast F (t) =5$$