DT LTI System
$ y[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}x[n] \; \; $ (scaled DT integral)
h[n]
$ h[n] = \sum_{n=-\infty}^{\infty}\frac{1}{2}\delta [n] = \frac{1}{2}u[n] $
H(z)
$ H(z) = \sum_{m=-\infty}^{\infty}h[m] e^{-j \omega m} = \sum_{m=-\infty}^{\infty} \frac{1}{2}u[m] e^{-j \omega m} = \sum_{m=0}^{\infty} \frac{1}{2}e^{-j \omega m} = \sum_{m=0}^{\infty} (\frac{1}{2 e^{j \omega}})^m = \frac{1}{1-\frac{1}{2 e^{j \omega}}} $ (geometric series $ r^n $ where $ |r| < 1 $)
