Convolution of $h(t) = u(t+2) - u(t-2)$ and $ f(t) = tu(t) - tu(t-2)$.

Question

Could someone please explain how I perform the convolution? My professor only taught me how to use the table, but I have been teaching myself from the book. I know that convolution is associative and has a time shift property but I am confused on how I apply it for f(t).

Answer 1

I think your best bet is to compute the convolution graphically. Notice that $h(t)$ is a rectangular pulse centred on the $y$-axis, and that $f(t)$ is one cycle of a sawtooth wave. Choose one, reflect it about the $y$-axis, and compute the overlap between the two graphs (scaled by the product of their magnitudes) as a function of how much your reflected graph is translated left or right.

Alternatively, you could use the convolution property of the Laplace transform:

$$h(t) \star f(t) = \mathcal{L}^{-1}(H(s)F(s)) $$

where $H(s) = \mathcal{L}(h(t)), F(s) = \mathcal{L}(f(t)).$