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]
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
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
16.
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)
Red Marked area shows the screen refresh rate which is a discrete function of frequency. The optimal screen refresh rate is determined during starting of computer. However the value can also be changed.