Extension of convolution of Unit Step Function to Square wave

Question

I have performed the convolution of the unit step function $u(t) - u(t-1)$ and the function $e^{-t}$. I have also executed, on a computer, the convolution of square wave with the same exponential decay signal. Is there any way I can generalise the results obtained in the unit step function problem to a square wave? just to validate the computer results.

Answer 1

Let $h(t) = u(t)- u(t-1),\,\,$ (a difference of unit step functions),

and let

$s(t) = \sum_{k=-\infty}^\infty \delta(t-2k)$

A square wave could be written as $w(t) = s(t) \star h(t)$.

You have calculated $g(t) = h(t) \star e^{-t}$ and want

$$w(t) \star e^{-t}= s(t) \star h(t) \star e^{-t} = s(t) \star g(t).$$

Just convolve your result $g(t)$ with a series of delta functions $s(t)$.