Problem
Compute the convolution
$ z[n]=x[n]*y[n] \ $
between
$ x[t] = u[n] - u[n-1] \ $
and
$ y[t] = u[n+2] - u[n-2] \ $.
My Solution
$ \begin{align} z[n] &= x[n]*y[n]\\ &= \sum_{k=-\infty}^{\infty} x[k]y[n-k]\\ &= \sum_{k=-\infty}^{\infty} (u[k] - u[k-1])(u[n-k+2] - u[n-k-2]) \\ &= \sum_{k=0}^{1} (u[n-k+2] - u[n-k-2]) \\ &= \begin{cases} u[n+2] - u[n-2] + u[n+1] - u[n-3] ,& 0 \leq n \leq 1 \\ 0 ,& \text{else} \end{cases}\\ &= \begin{cases} 2 ,& n = 0 \\ 2 ,& n = 1 \\ 0 ,& \text{else} \end{cases} \end{align} $
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