Signals and Systems - Electrical Engineering

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

Letting r=eσTsandω=Ts, we have that

z=rejω

which is a complex variable in polar form, with radius 0 ≤r<∞and angleωin radians. The variable

r is a damping factor andωis the discrete frequency in radians, so the z-plane corresponds to circles of

radius r and angles−π≤ω < π.

n Let us see how the relation z=esTs maps the s-plane into the z-plane. Consider the strip of width

2 π/Tsacross the s-plane shown in Figure 9.1. The width of this strip is related to the Nyquist

condition establishing that the maximum frequency of the analog signals we are considering is

M=s/ 2 =π/Tswheresis the sampling frequency and Tsis the sampling period. If Ts→ 0 ,

we would be considering the class of signals with maximum frequency approaching∞—that is, all signals.

The relation z=esTsmaps the real part of s=σ+j,Re(s)=σ, into the radius r=eσTs≥ 0 , and

the analog frequencies−π/Ts≤≤π/Tsinto−π≤ω < π, according to the frequency connection

ω=Ts. Thus, the mapping of the jaxis in the s-plane, corresponding toσ= 0 , gives a circle of radius

r= 1 or the unit circle.

The right-hand s-plane,σ > 0 , maps into circles with radius r> 1 , and the left-hand s-plane,σ < 0 ,

maps into circles of radius r< 1. Points A, B, and C in the strip are mapped into corresponding points

in the z-plane as shown in Figure 9.1. So the given strip in the s-plane maps into the whole z-plane—

similarly for other strips of the same width. Thus, the s-plane, as a union of these strips, is mapped onto

the same z-plane.

n The mapping z=esTscan be used to illustrate the sampling process. Consider a band-limited signal x(t)

with maximum frequencyπ/Tswith a spectrum in the band[−π/Ts π/Ts]. According to the relation

z=esTsthe spectrum of x(t)in[−π/Tsπ/Ts]is mapped into the unit circle of the z-plane from[−π,π)

on the unit circle. Going around the unit circle in the z-plane, the mapped frequency response repeats

periodically just like the spectrum of the sampled signal.

z=esTs

jΩ

A σ

B

ω

r

A

B

C

C

s-plane z-plane

−jTπ

s

jTπ

s

FIGURE 9.1

Mapping of the Laplace plane into thez-plane. Slabs of width 2 π/Tsin the left-hands-plane are mapped into the

inside of a unit circle in thez-plane. The right-hand side of the slab is mapped outside the unit circle. Thej-axis

in thes-plane is mapped into the unit-circle in thez-plane. The wholes-plane as a union of these slabs is

mapped onto the samez-plane.