Define a DT LTI System
$ \,\ x[n] = 5*u[n-5] + 6*u[n+6] $
h[n] and H(z)
We obtain $ h[n] $ by finding the response of $ x[n] $ to the unit impulse response ($ \delta[n] $).
$ \,\ h[n] = 5*\delta[n-5] + 6*\delta[n+6] $
$ \,\ H[z] = \sum_{m=-\infty}^\infty h[m] * Z $($ -m $)
$ \,\ H[z] = \sum_{m=-\infty}^{\infty} (5*\delta[n-5] + 6*\delta[n+6]) * Z $($ -m $)
By the sifting property, this sum equals:
$ \,\ H[z] = 5*Z $-5$ \,\ + 6*Z $6
