Q4Advanced Engineering Mathematics I

The Z-transform of a discrete-time signal $x_n$ is defined as: $Z\{x_n\}={\sum }_{n=-\infty }^{\infty }{x}_n{z}^{-n}$

For the unit impulse function, ${\delta }_n=1$ for $n=0$ and ${\delta }_n=0$ for $n\ne 0$.

Substituting this into the definition: $Z\{{\delta }_n\}=\cdots +0\cdot z^1+1\cdot z^0+0\cdot z^{-1}+\dots$ $=z^0=1$.

Therefore, $Z\{{\delta }_n\}=1$.