Assume I have the following two sequences : $$ x[n]=\begin{cases} \alpha&\text{if $a\leq n\leq b$}\\ \\ 0&\text{if otherwise} \end{cases} \qquad \text{and} \qquad h[n]=\begin{cases} \beta&\text{if $c\leq n\leq d$}\\ \\ 0&\text{if otherwise} \end{cases} $$ I was wondering in this case if the convolution : $$ (x*h)[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k] $$ has a specific form applicable for these two piecewise sequences.
I have seen on the internet something like : $$ (x * h)[n]=\sum_{k=\max (a, n-d)}^{\min (b, n-c)} x[k] h[n-k] . $$ However, I can not confirm if this is true. I hope someone can provide a proper proof and thank you very much.
