Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
The z-plane is a complex plane in which the zeros and poles of a z-transform are plotted, which is used to visualise the
properties of a signal or a system. The positions of the poles and zeros on a z-plane determine the frequency properties
and degree of stability. On the other hand, in designing a digital system, the poles and zeros can be chosen to put in
appropriate locations for achieving certain required performance.
z-plane
imaginary
real
Figure 5.3 z-plane
Example 5.3: From the z -transform pair table, we Figure know the 5.3 unit z-plane step pair as
z
u[
n]
↔ z −1
The z-transform of the unit step has one zero at origin as X ( z)
= 0 and one pole z =1 as X z)
→ ∞ , shown
0
in Figure 5.3 in which the pole is represented by a cross and the zero is represented by a circle.
z=
(
z=1
Example 5.4: Find zeros and poles for a z-transform
2
z ( z −1.2)(
z + 1)
X ( z)
=
.
( z − 0.5 + j0.7)(
z − 0.5 − j0.7)(
z − 0.8)
.
Re-write it as
X ( z)
=
( z − 0)( z − 0)( z −1.2)(
z − ( −1))
{ z − (0.5 − j0.7)
}{ z − (0.5 + j0.7)
}(
z − 0.8)
It can be obtained: 4 zeros: 0,0, 1.2, -1; and 3 poles: (0.5-j0.7), (0.5+j0.7), 0.8.
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