signal and system

1. Active Learning Assignment
Semester : 4th
Branch : Electrical Engineering
Subject : Signals & Systems (2141005)
Topic : “Inverse Z Transform (IZT)”
Prepared by : Shah Urvish (160123109016)
Guided by : Prof. Chintan Patel
1Gandhinagar Institute of Technology

2. Contents
•What is z transform (introduction)
•Methods for obtaining IZT
I. power series expansion
II.Partial fraction expansion
III.Residue method
IV.Inspection method
•Examples of all methods
•Applications
•Conclusion
•References
2
Gandhinagar Institute of Technology

3. Introduction
• Evaluation of this integral is beyond the scope of this course. Instead,
as with the Laplace transform, we will restrict our interest in the
inverse transform to rational forms (ratio of polynomials). We will see
shortly that this is convenient since linear constant-coefficient
difference equations can be converted to polynomials using the z-
transform.
3
Gandhinagar Institute of Technology

4. Methods for obtaining izt
4
Gandhinagar Institute of Technology

5. 5
Gandhinagar Institute of Technology

6. Rules for power series method
6
Gandhinagar Institute of Technology

7. 7
Gandhinagar Institute of Technology

8. Partial fraction expansion method(P.F.E)
• This is possible for the Z transform which are rational in nature, that
are expression as the ratio of two polynomials.
•Rational transforms can be factored using the same partial
fractions approach we used for the Laplace transforms.
•The partial fractions approach is preferred if we want a
closed-form solution rather than the numerical solution long
division provides.
 Example:
• In this example, the order of the numerator and
denominator are the same. For this case, we can use a trick of
factoring X(z)/z:
8
2
1
)( 23
3
−−−
+
=
zzz
z
zX
Gandhinagar Institute of Technology

9. 9
( )
( ) 643.02
)(
0825.0429.0866.05.0
)(
5.0
2
1
)(
)(
2866.05.0866.05.0
)(
)866.05.0)(866.05.0)(2(2)(
2
3
866.05.0
1
0
0
3110
23
=



−=
+=





++=
−=
−
=



=
−
+
−+
+
++
+=
−+++−=−−−=
=
−−=
=
z
jz
z
z
z
zX
c
jjz
z
zX
c
z
z
zX
c
z
c
jz
c
jz
c
z
c
z
zX
jzjzzzzzzA
Gandhinagar Institute of Technology

10. • We can compute the inverse using our table of common transforms:
• The exponential terms can be converted to a single cosine using a
magnitude/phase conversion:
10
][23][)866.05.0(1][)866.05.0(1][0][
121
3
1866.05.01
1
1866.05.01
1
0
2
3
866.05.0
1
866.05.0
1
0)(
nu
n
cnu
n
jcnu
n
jcncnx
z
c
zj
c
zj
c
c
z
zc
jz
zc
jz
zc
czX
++−+−−+=
−−
+
−−+
+
−++
+=
−
+
−+
+
++
+=
δ
Gandhinagar Institute of Technology

11. IZT using cauchy’s residual theorem:
11
Gandhinagar Institute of Technology

12. Example
12Gandhinagar Institute of Technology

13. Inspection Method
Make use of known z-transform pairs such as
Example: The inverse z-transform of
13
[ ] az
az1
1
nua 1
Zn
>
−
 →← −
( ) [ ] [ ]nu
2
1
nx
2
1
z
z
2
1
1
1
zX
n
1






=→>
−
=
−
Gandhinagar Institute of Technology

14. Applications
• Use to analysis of digital filters.
• Used to simulate the continuous system.
• Analyse the linear discrete system.
• Used to finding frequency response.
• Analysis to discrete signal.
• For automatic controls in telecommunication.
14Gandhinagar Institute of Technology

15. Conclusion
• After studding this topic I understand how to obtain the z-
transform and how to obtain the Inverse z-transform is by
using in parcel fraction and residual method and many more
applications of mathematics and signal processing and I
under where used z-transform
15
Gandhinagar Institute of Technology

16. References
• Signal and system by R.A. barapate,
• Signal and systems by Alan v. Oppenheim and Alan s.
willsky
16Gandhinagar Institute of Technology

17. Thank you for your Time
Any Queries?
Gandhinagar Institute of Technology 17