Determine the z-transform for the unit impulse $\left\{\delta_{k}\right\}=\{1,0,0,0, \ldots\}$
The z-transform of $\left\{\delta_{k}\right\}$ is given by:
$$
Z\left\{\delta_{k}\right\}=F(z)=\sum_{k=0}^{\infty} \frac{\delta_{k}}{z^{k}}=\frac{1}{z^{0}}+\frac{0}{z^{1}}+\frac{0}{z^{2}}+\ldots=1
$$
i.e. $\quad \mathbf{Z}\left\{\boldsymbol{\delta}_{k}\right\}=\mathbf{1}$ valid for all values of $\mathrm{z}$ )