Signals and Systems - Electrical Engineering

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

localization of poles and zeros determines the type of filter. However, in the discrete domain there is

a greater variety of filters than in the analog domain.

9.2 Laplace Transform of Sampled Signals...................................................

The Laplace transform of a sampled signal

x(t)=

∑

n

x(nTs)δ(t−nTs) (9.1)

is given by

X(s)=

∑

n

x(nTs)L[δ(t−nTs)]

=

∑

n

x(nTs)e−nsTs (9.2)

By lettingz=esTs, we can rewrite Equation (9.2) as

Z[x(nTs)]=L[xs(t)]

∣∣

z=esTs

=

∑

n

x(nTs)z−n (9.3)

which is called the Z-transform of the sampled signal.

RemarksThe function X(s)in Equation (9.2) is different from the Laplace transforms we considered

before:

n Letting s=j, X()is periodic of period 2 π/Ts(i.e., X(+ 2 π/Ts)=X()for an integer k). Indeed,

X(+ 2 π/Ts)=

∑

n

x(nTs)e−jn(+^2 π/Ts)Ts=

∑

n

x(nTs)e−jn(Ts+^2 π)=X()

n X(s)may have an infinite number of poles or zeros—complicating the partial fraction expansion when

finding its inverse. Fortunately, the presence of the{e−nsTs}terms suggests that the inverse should be done

using the time-shift property of the Laplace transform instead, giving Equation (9.1).

nExample 9.1

To see the possibility of an infinite number of poles and zeros in the Laplace transform of a sam-

pled signal, consider a pulsex(t)=u(t)−u(t−T 0 )sampled with a sampling periodTs=T 0 /N.

Find the Laplace transform of the sampled signal and determine its poles and zeros.

Solution

The sampled signal is

x(nTs)=

{

1 0≤nTs≤T 0 or 0≤n≤N

0 otherwise