Signals and Systems - Electrical Engineering

9.3 Two-Sided Z-Transform 519

an exponentialRn 1 u[n] such that|xc[n]|<MRn 1 forn≥0 for some valueM>0. Then, forX(z)to
converge we need that

|Xc(z)|≤

∑∞

n= 0

|xc[n]||r−n|<M

∑∞

n= 0

∣

∣

∣

∣

R 1

r

∣

∣

∣

∣

n

<∞

or thatR 1 /r<1, which is equivalent to|z|=r>R 1. As indicated, this ROC does not include any
poles ofXc(z)—it is the outside of a circle containing all the poles ofXc(z).

Likewise, for an anti-causal signalxa[n], if we choose a radiusR 2 that is smaller than the radius of all
the poles ofXa(z), the region of convergence is|z|=r<R 2. This ROC does not include any poles of
Xa(z)—it is the inside of a circle that does not contain any of the poles ofXa(z).

If the signalx[n] is noncausal, it can be expressed as

x[n]=xc[n]+xa[n]

where the supports ofxa[n] andxc[n] can be finite or infinite or any possible combination of these
two. The corresponding ROC ofX(z)=Z{x[n]}would then be

0 ≤R 1 <|z|<R 2 <∞

This ROC is a torus surrounded on the inside by the poles of the causal component, and in the
outside by the poles of the anti-causal component. If the signal has finite support, thenR 1 =0 and
R 2 =∞, coinciding with the result for finite-support signals.

For the Z-transformX(z)of an infinite-support signal:

n A causal signalx[n]has a region of convergence|z|>R 1 whereR 1 is the largest radius of the poles of

X(z)—that is, the region of convergence is the outside of a circle of radiusR 1.

n An anti-causal signalx[n]has as region of convergence the inside of the circle defined by the smallest

radiusR 2 of the poles ofX(z), or|z|<R 2.

n A noncausal signalx[n]has as region of convergenceR 1 <|z|<R 2 , or the inside of a torus of inside

radiusR 1 and outside radiusR 2 corresponding to the maximum and minimum radii of the poles ofXc(z)

andXa(z), which are the Z-transforms of the causal and anti-causal components ofx[n].

nExample 9.4

The poles ofX(z)arez=0.5 andz=2. Find all the possible signals that can be associated with it

according to different regions of convergence.

Solution

Possible regions of convergence are:

n {R 1 :|z|> 2 }—the outside of a circle of radius 2, we associateX(z)with a causal signalx 1 [n].