Analog properties and Z-transform

GANDHINAGAR INSTITUTE OF
TECHNOLOGY
TOPIC:- ANALOG PROPERTIES AND
Z-TRANSFORM
SUBJECT:- SIGNALS & SYSTEMS
Prepared by:-
Name of the students
ISHITA AMBANI
ANKITA BADORIA
GANDHINAGAR INSTITUTE OF TECHNOLOGY

CONTENTS
 INTRODUCTION AND APPLICATIONS
 SYSTEMS AND ITS CLASSIFICATION
 EXAMPLES
 Z-TRANSFORM
 Z-PLANE
 REGION OF CONVERGENCE
 EXAMPLES
 Z-TRANSFORM PAIRS
 APPLICATIONS
 REFERENCES

ANALOG SIGNAL
o An analog signal is a continuous signal that
contains time-varying quantities.
o The illustration in the above figure shows an
analog pattern along side with digital pattern.

APPLICATIONS
 To measure changes in some physical phenomena
such as light, sound, pressure, or temperature.
 For instance, an analog microphone can convert
sound waves into an analog signal.
 Even in digital devices, there is typically some
analog component that is used to take in information
from the external world, which will then get
translated into digital form (using an analog-to-
digital converter.

System
 A System, is any physical set of components that
takes a signal, and produces a signal. In terms of
engineering, the input is generally some electrical
signal X, and the output is another electrical signal
(response) Y.

Classification of system
Continuous vs.
Discrete
Time Invariant vs.
Time Varying
Causal vs. Non-
causal
Stable vs.
Unstable
Linear vs.
Nonlinear

o A system in which the
input signal and output
signal both have
continuous domains is said
to be a continuous system.
o One in which the input
signal and output signal
both have discrete domains
is said to be a discrete
system.
CONTINUOUS DISCRETE

 A linear system is any
system that obeys the
properties of scaling and
superposition
(additivity).
 A nonlinear system is
any system that does not
have at least one of these
properties.
LINEAR NON-LINEAR

TIME VARIANT and TIME-INVARIANT

o A causal system is one in
which the output depends
only on current or past
input, but not future inputs.
o Non-causal is the one in
which output depends on
both past and future
inputs.
CAUSAL NON-CAUSAL

STABLE & UN-STABLE

 It is a DT system in
which output at any
instant of time depends
upon input sample at the
same time.
 Examples:
I. y(n)=5x(n)
II. Y(n)=x^2(n)+5x(n)+10
 It is a system in which
output at any instant of
time depends on input
sample at the same
time as well as at other
instants of time.
 Examples:
I. y(n)=x(n)+5x(n-1)
II. y(n)=3x(n+2)+x(n)
STATIC DT SIGNALS
DYNAMIC DT
SIGNALS

INVERTIBILITY

Just like Laplace transforms are used for evaluation of
continuous functions, Z-transforms can be used for
evaluating discrete functions.
Z-Transforms are highly expedient in discrete
analysis,Which form the basis of communication technology.
Definition:





n
n
znxzX )()(

z-Plane
Re
Im
z = ej






n
n
znxzX )()(
( ) ( )j j n
n
X e x n e 



 

 Give a sequence, the set of values of z for which
the z-transform converges, i.e., |X (z)|<, is called
the region of convergence.
Definition
 






n
n
n
n
znxznxzX |||)(|)(|)(|

Example: A right sided
Sequence
)()( nuanx n
 ||||,)( az
az
z
zX 


Re
Im
a
ROC is bounded by
the pole and is the
exterior of a circle.

Example: A left sided Sequence
)1()(  nuanx n
||||,)( az
az
z
zX 


Re
Im
a
ROC is bounded by
the pole and is the
interior of a circle.

 A ring or disk in the z-plane centered at the origin.
 The Fourier Transform of x(n) is converge absolutely iff the
ROC includes the unit circle.
 The ROC cannot include any poles
 Finite Duration Sequences: The ROC is the entire z-plane
except possibly z=0 or z=.
 Right sided sequences: The ROC extends outward from the
outermost finite pole in X(z) to z=.
 Left sided sequences: The ROC extends inward from the
innermost nonzero pole in X(z) to z=0.
Properties of ROC

Z-Transform Pairs
SEQUENCE Z-TRANSFORM ROC
)(n 1 All z
)( mn m
z All z except 0 (if m>0)
or  (if m<0)
)(nu 1
1
1

 z
1|| z
)1(  nu 1
1
1

 z
1|| z
)(nuan 1
1
1

 az
|||| az 
)1(  nuan 1
1
1

 az
|||| az 

APPLICATIONS OF Z-
TRANSFORMS
 The field of signal processing is essentially a field of signal
analysis in which they are reduced to their mathematical
components and evaluated. One important concept in signal
processing is that of the Z-Transform, which converts
unwieldy sequences into forms that can be easily dealt with
Z-Transforms are used in many signal processing systems.
 Z-transforms can be used to solve differential equations with
constant coefficients.

Analog properties and Z-transform

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