Signals and Systems - Electrical Engineering

9.4 One-Sided Z-Transform 523

The Z-transform is a linear transformation, meaning that

Z(ax[n]+by[n])=aZ(x[n])+bZ(y[n]) (9.10)

for signalsx[n]andy[n]and constantsaandb.

To illustrate the linearity property as well as the connection between the signal and the poles of its
Z-transform, consider the signalx[n]=αnu[n] for real or complex valuesα. Its Z-transform will be
used to compute the Z-transform of the following signals:

n x[n]=cos(ω 0 n+θ)u[n] for frequency 0≤ω 0 ≤πand phaseθ.
n x[n]=αncos(ω 0 n+θ)u[n] for frequency 0≤ω 0 ≤πand phaseθ.

Show how the poles of the corresponding Z-transform connect with the signals.

The Z-transform of the causal signalx[n]=αnu[n] is

X(z)=

∑∞

n= 0

αnz−n=

∑∞

n= 0

(αz−^1 )n=

1

1 −αz−^1

=

z

z−α

ROC:|z|>|α| (9.11)

Using the last expression in Equation (9.11) the zero ofX(z)isz=0 and its pole isz=α, since the
first value makesX( 0 )=0 and the second makesX(α)→∞. Forαreal, be it positive or negative,
the region of convergence is the same, but the poles are located in different places. See Figure 9.2 for
α <0.

Ifα=1 the signalx[n]=u[n] is constant forn≥0 and the pole ofX(z)is atz= 1 ej^0 (the radius
isr=1 and the lowest discrete frequencyω=0 rad). On the other hand, whenα=−1 the signal
isx[n]=(− 1 )nu[n], which varies from sample to sample forn≥0; its Z-transform has a pole at
z=− 1 = 1 ejπ(a radiusr=1 and the highest discrete frequencyω=πrad). As we move the pole
toward the center of thez-plane (i.e.,|α|→0), the corresponding signal decays exponentially for 0<
α <1, and is a modulated exponential of|α|n(− 1 )nu[n]=|α|ncos(πn)u[n] for− 1 < α <0. When
|α|>1 the signal becomes either a growing exponential(α > 1 )or a growing modulated exponential
(α <− 1 ).

FIGURE 9.2
Region of convergence (shaded area) ofX(z)with a pole at
z=α,α < 0 (same ROC if pole is atz=−α).

− 1 α× 1

z-plane