Z-transforms can be used to analyze systems described by difference equations. The Z-transform relates the terms of a discrete-time sequence to a complex function of a complex variable z. Some key applications of Z-transforms include analyzing Fibonacci series, Newton's interpolation formula, and compound interest problems. Important results for Z-transforms include theorems regarding shifting, constants, initial values, final values, and convolution. Z-transforms are also connected to other transforms like the Fourier transform and Laplace transform through relationships like the discrete-time Fourier transform and bilinear transform.

1. By
Ram Kumar K R
Ganesh Arkalgud
Anshul Bansal
Z-Transforms and their applications

2. Introduction:
 Definition: The Z-transform of a sequence un
defined for discreet values n=0,1,2,3….(and un=0
for n<0).This is denoted by Z(un) and is defined as
where u is a function of Z.
 For every application of Laplace Transform there
is a corresponding application for the Z-Transform
)()(
0
zUzuuZ
n
n
nn  




3. Difference Equations
 Definition : Equations relating various terms of a
sequence .
 Example : String with finite number of beads at
equally spaced points

4. Applications of Difference Equation:
 Fibonacci series
 Newton’s Interpolation formula
 Compound Interest, etc.,
....
0
2
!2
)1(
0!10
)(
21







y
pp
y
p
yxf
n
a
n
a
n
a

5. Important results on Z-Transforms
 an
 Recurrence
 Linearity
 Damping Rule 











a
z
UuaZ
vbZuaZbvauZ
nZ
dz
d
znZ
az
z
aZ
n
n
nnnn
pp
n
)(
)()()(
)]([)(
)(
1

6. Important results on Z-Transforms
 Shifting Theorem
Z(a yk-1) = z-1 Y[z]
 Multiplication by a constant
Z(a yk) = a Y[z]

7. Some Basic Theorems
 Initial Value Theorem
If
then
 Final Value Theorem

8. Some Basic Theorems
 Convolution Theorem

9. Connecting Z-Transforms with
other Transforms

10. Fourier Transform:
• The Fourier transform ƒ̂ of an integrable function ƒ:
R → C
for every real number ξ.

11. Connecting Z and Fourier
transforms
 The DTFT is a special case of the Z-transform.
The bilateral Z-transform is defined as:
 special case is 
 Since ,
 it is the evaluation of the Z-transform around
the unit circle in the complex plane.
1||
][)(


 





i
i
n
n
e
ez
znxzX

12. Connecting Z and Laplace
Transforms
 Bilinear transform
The relation between Laplace and Z transform is
given by
The relation between Z and Laplace transforms is
given by

13. Applications of Z-transforms

14. Sampled systems
 inputs and outputs are related by difference
equations and Z-transform techniques are used
to solve those difference equations.
 VOICE TRANSMISSION : To band-limit the signal
and filter noise from the signal.
 Calculation of a signal to control a system

15. Data Reading from a CD
 Digital Signals
 Transmission systems - Telephone

17. References
 Higher Engineering Mathematics by B.S. Grewal
 Higher Engineering Mathematics by B.V. Ramana
 Z-Transforms and applications by E. J.
Mastascusa. (http://www.facstaff.bucknell.edu/mastascu/econtrolhtml/Sa
mpled/Sampled1.html)

18. Thank You