Consider : $$ x[n]:=\begin{cases} 1&\text{if $3\leq n\leq 8$}\\ 0&\text{if otherwise} \end{cases} \quad\text{and}\quad h[n]:=\begin{cases} 1&\text{if $4\leq n \leq 15$}\\ 0&\text{if otherwise} \end{cases} $$ I wish to compute the convolution $\displaystyle y[n]=(x*h)[n]=\sum_{k=-\infty}^{\infty}x[k]h[n-k]$. Because I dislike the method of graphing, I wanted to take another approach using unit step function : $$ x[n]=u[n-3]-u[n-8] \quad \text{and} \quad h[n]=u[n-4]-u[n-15] $$ By using distributivity property of the discrete-time convolution, we get : \begin{align*} (x*h)[n]&=(u[n-3]-u[n-8])*(u[n-4]-u[n-15])\\ \\ &=\underbrace{u[n-3]*u[n-4]}_{(y_{1}[n])} -\underbrace{u[n-3]*u[n-15]}_{(y_{2}[n])}-\underbrace{u[n-8]*u[n-4]}_{(y_{3}[n])}+\underbrace{u[n-8]*u[n-15]}_{(y_{4}[n])} \end{align*} Therefore we have four convolutions for which we can see that :
$y_{1}[n]$ = $\begin{cases}\displaystyle\sum_{3}^{n-4}1&\text{if $7\leq n \leq 11$}\\0&\text{if otherwise}\end{cases}=\begin{cases}n-6&\text{if $7\leq n \leq 11$}\\0&\text{if otherwise}\end{cases}$
$y_{2}[n] = \begin{cases}\displaystyle\sum_{3}^{n-15}1&\text{if $18\leq n \leq 33$}\\0&\text{if otherwise}\end{cases}=\begin{cases}n-17&\text{if $18\leq n \leq 33$}\\0&\text{if otherwise}\end{cases}$
$y_{3}[n] =\begin{cases}\displaystyle\sum_{8}^{n-4}1&\text{if $12\leq n \leq 16$}\\0&\text{if otherwise}\end{cases}=\begin{cases}n-11&\text{if $12\leq n \leq 16$}\\0&\text{if otherwise}\end{cases}$
$y_{4}[n]=\begin{cases}\displaystyle\sum_{8}^{n-15}1&\text{if $23\leq n \leq 38$}\\0&\text{if otherwise}\end{cases}=\begin{cases}n-22&\text{if $23\leq n \leq 38$}\\0&\text{if otherwise}\end{cases}$
I must note that I am afraid I don't know if the upper bound in the sum of $(1)$, $(2)$, $(3)$, and $(4)$ are right or wrong. I think what's left to do is compute : $$ y[n]=y_{1}[n]-y_{2}[n]-y_{3}[n]+y_{4}[n] $$ I hope someone can show me how from here we can prove that the answer is : $$ y[n]=\begin{cases} n-6&\text{if $7\leq n\leq 11$}\\ 6&\text{if $12\leq n\leq 18$}\\ 24-n&\text{if $19\leq n \leq 23$}\\ 0&\text{if otherwise} \end{cases} $$