Signals and Systems - Electrical Engineering

516 C H A P T E R 9: The Z-Transform

causal. To express the two-sided Z-transform in terms of the one-sided Z-transform we separate the

sum into two and make each into a causal sum:

X(z)=

∑∞

n=−∞

x[n]z−n=

∑∞

n= 0

x[n]u[n]z−n+

∑^0

n=−∞

x[n]u[−n]z−n−x[0]

=Z

(

x[n]u[n]

)

+

∑∞

m= 0

x[−m]u[m]zm−x[0]

=Z

(

x[n]u[n]

)

+Z

(

x[−n]u[n]

)

|z−x[0]

where the inclusion of the additional termx[0] in the sum from−∞to 0 is compensated by subtract-

ing it, and in the same sum a change of variable(m=−n)gives a one-sided Z-transform in terms of

positive powers ofz, as indicated by the notationZ

(

x[−n]u[n]

)

|z.

9.3.1 Region of Convergence

The infinite summation of the two-sided Z-transform must converge for some values ofz. ForX(z)to

converge it is necessary that

|X(z)|=

∣

∣

∣

∣

∣

∑

n

x[n]z−n

∣

∣

∣

∣

∣

≤

∑

n

|x[n]||r−nejωn|=

∑

n

|x[n]||r−n|<∞

Thus, the convergence ofX(z)depends onr. The region in thez-plane whereX(z)converges, its ROC,

connects the signal and its Z-transform uniquely. As with the Laplace transform, the poles ofX(z)are

connected with its region of convergence.

The poles of a Z-transformX(z)are complex values{pk}such that

X(pk)→∞

while the zeros ofX(z)are the complex values{zk}that make

X(zk)= 0

nExample 9.2

Find the poles and the zeros of the following Z-transforms:

(a)X 1 (z)= 1 + 2 z−^1 + 3 z−^2 + 4 z−^3

(b)X 2 (z)=

(z−^1 − 1 )(z−^1 + 2 )^2

z−^1 (z−^2 +

√

2 z−^1 + 1 )