Z transform and its Properties Z transform of basic signal

1. Chapter # 3
Z-Transform
Digital Signal Processing

2. Contents:
• Definition of Z-Transform.
• One sided and Two sided Z-Transform.
• Z-Transform of Basic Signals.
• Transformation from S-Plane to Z-Plane.
• Zero Pole Representation.
• Properties of Z-Transform.
• Inverse Z-Transform.
• Difference Equations using Z-Transforms.
Digital Signal Processing 2

3. 3
The z-Transform
• Counterpart of the Laplace transform for discrete-time signals
• The z-Transform is often time more convenient to use
• Definition:
• z is a complex variable that can be represented as z=r ej
• The inverse procedure [i.e., obtaining x (n) from X(z)] is called
the inverse z-transform.
 For convenience, the z-transform of a signal x(n) is denoted by
X(z) = Z{x(n)}
 whereas the relationship between x(n) and X(z) is indicated by
z
x(n) X(z)
   




n
n
znxzX

4. One Sided And Two Sided Z-Transform
4
One Sided Z-Transform
The one sided Z-Transform can be used to solve Differential
Equations with initial conditions.
This Property makes it useful for use in Practical systems
It does not contain information about the signal x(n) for
negative values of time.
It is represented by the equation :

X
(z)x(n)zn
n0

5. Two Sided Z-Transform
• Two sided Z-transform requires that the signal be specified for the entire
range :
 
• It represented by the equation :

5
X ( z )  
n  
x ( n ) z n

6. The region of convergence (ROC)
• Since the z-transform is an infinite power series, it exists only
for those values of z for which this series converges. The
region of convergence (ROC) of X(z) is the set of all values of
z for which X(z) attains a finite value. Thus any time we cite a
z-transform we should also indicate its ROC.
Digital Signal Processing 6

7. Solution
Digital Signal Processing 7

8. Example:
8

9. Z-Transform of Basic Signals
9
• Unit Impulse Function
x(n)   (n)
 n   n   n
n  0
X(z) 1


10. • Unit step Function
10

11. • Exponential Function
11
Unit Step function

12. • Unit Ramp Function
12

13. Bilateral Z-Transform
13

14. Bilateral Z-Transform
• Case 1:
• |b| < |a|: In this case the two ROC above do not overlap, as
shown in Fig. 3.1.4(a).
Consequently, we cannot find values of z for which both power
series converge simultaneously. Clearly, in this case, X(z) does
not exist.
Case 2:
• |b| > |a|: In this case there is a ring in the z-plane where
both power series converge
simultaneously, as shown in Fig. 3.1.4(b). Then we obtain The
ROC of X(z) is lal < |zl < |bl
14

15. 15
Two-Sided Exponential Sequence Example
     1-n-u
2
1
-nu
3
1
nx
nn













11
1
0
1
0n
n
1
z
3
1
1
1
z
3
1
1
z
3
1
z
3
1
z
3
1





























11
0
11
1
n
n
1
z
2
1
1
1
z
2
1
1
z
2
1
z
2
1
z
2
1






























z
3
1
1z
3
1
:ROC 1

 
z
2
1
1z
2
1
:ROC 1


 


























2
1
z
3
1
z
12
1
zz2
z
2
1
1
1
z
3
1
1
1
zX
11
Re
Im
2
1
oo
12
1
xx3
1


16. TABLE 3.1 Characteristic Families of Signals with Their Corresponding
ROCs
• We have seen that the ROC of a signal depends both on its duration (finite or
infinite) and on whether it is causal, anticausal, or two-sided. These facts are
summarized in Table 3.1.
16

17. Digital Signal Processing 17
The z-transform
• The z-transform is a function of the complex z variable
• Convenient to describe on the complex z-plane
• If we plot z=ej for =0 to 2 we get the unit circle
Re
Im
Unit Circle

r=1
0
2

18. 18
Region of Convergence
• The set of values of z for which the z-transform converges
• Each value of r represents a circle of radius r
• The region of convergence is made of circles
• Example: z-transform converges for
values of 0.5<r<2
– ROC is shown on the left
– In this example the ROC includes
the unit circle, so DTFT exists
• Not all sequence have a z-transform
Re
Im

19. 19
Right-Sided Exponential Sequence Pole-Zero Example
• For Convergence we require
• Hence the ROC is defined as
• Inside the ROC series converges
• Geometric series formula
         







0n
n1
n
nnn
azznuazXnuanx
az1az
n
1

    az
z
az1
1
azzX
0n
1
n1



 




Re
Im
a 1
o x
• Region outside the circle of
radius a is the ROC
• Right-sided sequence ROCs
extend outside a circle





2
1
21N
Nn
1NN
n
1

20. 20
Finite Length Sequence
 


 

otherwise0
1Nn0a
nx
n
     
az
az
z
1
az1
az1
azzazX
NN
1N1
N11N
0n
n1
1N
0n
nn





 








21. 21
Properties of The ROC of Z-Transform
• The ROC is a ring or disk centered at the origin
• DTFT exists if and only if the ROC includes the unit circle
• The ROC cannot contain any poles
• The ROC for finite-length sequence is the entire z-plane
– except possibly z=0 and z=
• The ROC for a right-handed sequence extends outward from
the outermost pole possibly including z= 
• The ROC for a left-handed sequence extends inward from the
innermost pole possibly including z=0
• The ROC of a two-sided sequence is a ring bounded by poles
• The ROC must be a connected region
• A z-transform does not uniquely determine a sequence without
specifying the ROC

22. Properties of the z-Transform
•
22
 Linearity Property :

23. Cont.…
23

24. Cont.…
24
 Time Shifting Property :
Scaling Property :

25. Cont.…
25

26. Cont.…
26
Time Reversal Property :
x[n]  X (z)
x(n)  X(Z1
)

27. Cont.…
27
 Differentiation Property :

28. Cont.…
28

29. 29
 Convolution property :

30. TABLE 3.2 Properties of the z-Transform
30

31. 31
Stability, Causality, and the ROC
• Consider a system with impulse response h[n]
• The z-transform H(z) and the pole-zero plot shown below
• Without any other information h[n] is not uniquely determined
– |z|>2 or |z|<½ or ½<|z|<2
• If system stable ROC must include unit-circle: ½<|z|<2
• If system is causal must be right sided: |z|>2