2.5 Z-transform examples

Consider the $z$ -transform given by $H\left(z\right)=z$ , as illustrated below.

The corresponding DTFT has magnitude and phase given below.

What could the system $H$ be doing? It is a perfect all-pass, linear-phase system. But what does this mean?

Suppose $h\left[n\right]=\delta [n-n_0]$ . Then

Thus, $H\left(z\right)=z^{-n_0}$ is the $z$ -transform of a system that simply delays the input by $n_0$ . $H\left(z\right)=z^{-1}$ is the $z$ -transform of a unit-delay.

Now consider $x\left[n\right]={\alpha }^{n}u\left[n\right]$

What if $\left|\frac{a}{z}\right|\ge 1?$ Then $\stackrel{\infty }{\sum _{n=0}}{\left(\frac{\alpha }{n}\right)}^n$ does not converge! Therefore, whenever we compute a $z$ -transform, we must also specify the set of $z$ 's for which the $z$ -transform exists. This is called the region of convergence (ROC). In the above example, the ROC= $\{z:\left|z\right|>\left|\alpha \right|\}$ .

What about the “evil twin” $x\left[n\right]=-{\alpha }^{n}u[-1-n]$ ?

We get the exact same result but with ROC= $\{z:\left|z\right|<\left|\alpha \right|\}$ .