Convolution of two heaviside functions

Question

how can i find convolution of two heaviside functions centered at 1/2 and -1/2. I have tried to find the corresponding integral, but i stack due the centers are different.

Answer 1

Let $H(t)$ be the Heaviside function. Note that $H(\tau-1/2)\neq 0$ only when $\tau>1/2$ and that $H(t-\tau+1/2) \neq 0$ only when $t-\tau > -1/2 \implies \tau < t + 1/2$. These only occur simultaneously for $t\geq 0$

So, the convolution will be $0$ when $t<0$. For $t>0$, we compute $$ \begin{align} H(t-1/2) * H(t+1/2) &= \int_{-\infty}^\infty H(\tau-1/2)H(t-\tau+1/2)\,d\tau \\ &= \int_{1/2}^{t+1/2} H(\tau-1/2)H(t-\tau+1/2)\,d\tau \\ &= \int_{1/2}^{t+1/2} 1\,d\tau = t \end{align} $$ That is, we have $$ H(t-1/2) * H(t+1/2) = \begin{cases} t & t \geq 0\\ 0 & t < 0 \end{cases} = t\,H(t) $$