Example of DT convolution

$ y[n]=x[n]*h[n]=\sum_{k=-\infty}^{\infty} x[k]h[n-k] $

$ x[n] = 2^nu[-n] $ and $ h[n]=u[-n] $

$ y[n] = \sum_{k=-\infty}^{\infty} 2^ku[-k]u[-(n-k)] $

$ y[n] = \sum_{k=-\infty}^0 2^ku[-(n-k)] $

$ y[n] = \sum_{k=-\infty}^0 2^ku[-n+k] $

Let $ r=-k $

$ y[n] = \sum_{r=0}^{\infty} (1/2)^ru[-n-r] $

$ -n-r \ge 0 \Rightarrow -n \ge r $

$ y[n]= \left\{ \begin{array}{ll} \sum_{r=0}^{-n}(1/2)^n &, \text{ if } -n\ge 0\\ 0 &, \text{ else}\end{array}\right. $

$ y[n]= \left\{ \begin{array}{ll} \frac{1-(\frac{1}{2})^{-n+1}}{1-\frac{1}{2}} &, \text{ if } -n\ge 0\\ 0 &, \text{ else}\end{array}\right. $