Problem
Compute the convolution
$ z[n]=x[n]*y[n] \ $
between
$ x[n] = e^{jwn}u[n] \ $
and
$ y[n] = e^{jwn}u[n-6] \ $.
My Solution
$ \begin{align} z[n] &= x[n]*y[n]\\ &=\sum_{k=-\infty}^{\infty}x[k]y[n-k] \\ &=\sum_{k=-\infty}^{\infty}e^{jwk}u[k]e^{jw(n-k)}u[n-k-6] \\ &=\begin{cases} \sum_{k=0}^{n-6}e^{jwk}e^{jwn}e^{-jwk}, & n \geq 6 \\ 0, & n < 5 \\ \end{cases}\\ &=\begin{cases} e^{jwn}\sum_{k=0}^{n-6}1, & n \geq 6 \\ 0, & n < 5 \\ \end{cases}\\ &=\begin{cases} e^{jwn}(n-5), & n \geq 6 \\ 0, & n < 5 \\ \end{cases}\\ \end{align} $
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