2.7 Poles and zeros

Consider

and

Note that the poles and zeros of $X_1\left(z\right)$ and $X_2\left(z\right)$ are identical, but with opposite ROCs. Note also that neither ROC contains the point $\alpha$ .

Consider

We can compute the $z$ -transform of $x_3\left[n\right]$ by simply adding the $z$ -transforms of the two different terms in the sum, which are given by

and

The poles and zeros for these $z$ -transforms are illustrated below.

$X_3\left(z\right)$ is given by

Note that the poles do not change, but the zeros do, as illustrated above.

Now consider the finite-length sequence

The $z$ -transform for this sequence is

We can immediately see that the zeros of $X_4\left(z\right)$ occur when $z^N={\alpha }^N$ . Recalling the “N ^th roots of unity”, we see that the zeros are given by

At first glance, it might appear that there are $N-1$ poles at zero and 1 pole at $\alpha$ , but the pole at $\alpha$ is cancelled by the zero ( $z_0$ ) at $\alpha$ . Thus, $X_4\left(z\right)$ actually has only $N-1$ poles at zero and $N-1$ zeros around a circle of radius $\alpha$ as illustrated below.

So, provided that $\left|\alpha \right|<\infty$ , the ROC is the entire $z$ -plane except for the origin. This actually holds for all finite-length sequences.