i need to evaluate the convolution sum of x[n] * h[n].
x[n] is the step function u[n].
I know how the output should look like but i don't know how i can calculate it.
I think the lower border is 0, cause the step function is 1 for n >= 0.
But the upper border is infinite, it's only approaching to 0.
\begin{aligned}
g[n] & = \sum_{m=0}^{\infty} x[m] h[n-m]\\
x[n] & = u[n] \\
h[n] & = (\frac{1}{2})^n u[n] \\
\end{aligned}
Hope someone can help me…
Best Answer
\begin{equation} g[n] = \sum_{m=0}^{\infty} x[m] h[n-m]\ \end{equation}
i.e., we have \begin{equation} g[n] = \sum_{m=0}^\infty u[m] \left(\frac{1}{2}\right)^{n-m} u[n-m]\ \end{equation}
Since $u[m]>0$ when $m \geq 0$ and $u[n-m]>0$ when $n-m \geq 0$, we have
\begin{equation} g[n]= \sum_{m=0}^{n} \left(\frac{1}{2}\right)^{n-m}\ \end{equation}
This is just a simple geometric series. The answer is $2(1-0.5^{n+1})$.