topic review, theory, easy to understand ,exam need ,etc

1. SUBMITED by – SHUBHAM KORDE
Maths-III [TAE]
SUBMITED TO- prof. GUPTA Madam

2. 2
The z-Transform
Introduction
• Counterpart of the Laplace transform for discrete-time signals
• Generalization of the Fourier Transform
– Fourier Transform does not exist for all signals
• The z-Transform is often time more convenient to use
• Definition:
• Compare to DTFT definition:
• z is a complex variable that can be represented as z=r ej
• Substituting z=ej will reduce the z-transform to DTFT
 




n
n
znxzX
   nj
n
j
enxeX 





3. 3
The z-transform and the DTFT
THEORY
• 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 0 2

 j
eX

4. 4
Convergence of the z-Transform
• DTFT does not always converge
– Infinite sum not always finite if x[n] no absolute summable
– Example: x[n] = anu[n] for |a|>1 does not have a DTFT
• Complex variable z can be written as r ej so the z-transform
• DTFT of x[n] multiplied with exponential sequence r -n
– For certain choices of r the sum maybe made finite
    







n
njn
n
njj
ernxernxreX
   nj
n
j
enxeX 




 

n
n-
rnx

5. 5
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 th
e unit circle, so DTFT exists
• Not all sequence have a z-transform
• Example:
– Does not converge for any r
– No ROC, No z-transform
– But DTFT exists?!
– Sequence has finite energy
– DTFT converges in the mean-square
d sense
Re
Im
  ncosnx o

6. 6
Right-Sided Exponential Sequence Example
• For Convergence we require
• Hence the ROC is defined as
• Inside the ROC series converges to
• Geometric series formula
    







0n
n1
n
nnn
azznuazXnuanx





0n
n
1
az
az1az
n
1

   az
z
az1
1
azzX
0n
1
n1













2
1
21N
Nn
1NN
n
a1
aa
a
Re
Im
a 1
o x
• Region outside the circle of r
adius a is the ROC
• Right-sided sequence ROCs e
xtend outside a circle

7. 7
Example
• For the term with infinite exponential to vanish we need
– Determines the ROC (same as the previous approach)
• In the ROC the sum converges to
    







0n
n1
n
nnn
azznuazXnuanx





2
1
21N
Nn
1NN
n
1
     









0n
1
101
n1
az1
azaz
az
za1az1

  






0n
1
n1
az1
1
azzX
|z|>2

8. 8
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


9. 9
Finite Length Sequence



 

otherwise0
1Nn0a
nx
n
  
az
az
z
1
az1
az1
azzazX
NN
1N1
N11N
0n
n1
1N
0n
nn





 








10. 10
Applications 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 th
e outermost pole possibly including z= 
• The ROC for a left-handed sequence extends inward from the i
nnermost 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

11. 11
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

12. CONCLUSION
• Thw purpose of this is to example is to show how to find
z transform of the system output when the system is given as a
differntial equation
• The linearity and the translation property of z transform are
use here
• it is intresting to see another way to relate yhe input and the
of the system
• Actually, z-transform is one way for solving the constant
coefficient difference equation.
12

13. 13