Explanation of the result of a solved exercise concerning the unit step response of a system
When they say:

$$y(n)=x(n)\ast h(n)=\sum_{m=-\infty}^{\infty}x(m)\cdot h(n-m)$$

they are correct, but in the next passage they oversimplify a bit.

Let's do it step by step:

\begin{align*}
&amp;\sum_{m=-\infty}^{\infty}x(m)\cdot h(n-m) = \sum_{m=-\infty}^{\infty}u(m)\cdot u(n-m) = \\
&amp;= \sum_{m=0}^{\infty} 1 \cdot u(n-m) = \sum_{m=0}^{\infty} u(n-m) \\
\end{align*}

The function to be summed is \$u(n-m)\$, which is equal to 1 if and only if 
\$n-m&gt;0 \Leftrightarrow m &lt; n\$.

Since the sum is carried out only for \$m&gt;=0\$, if \$n&gt;=0\$ the result is actually \$n+1\$, but if \$n&lt;0\$ the result is 0, therefore they should have written:

\[
 \sum_{m=-\infty}^{\infty}x(m)\cdot h(n-m) = (n+1)u(n)
\]

rightaway.

The following image may help:



the red dots are the function values, the green shaded area highlights the value range that will be summed. The image was drawn with \$n = 3\$. You may infer that if \$n&lt;0\$ the green area will disappear, indicating there is nothing to sum.