analog properties and z-transform

GANDHINAGAR INSTITUTE OF TECHNOLOGY


TOPIC:- ANALOG PROPERTIES AND Z-TRANSFORMSUBJECT:- SIGNALS & SYSTEMSPrepared by:-

Name of the students

ISHITA AMBANI

ANKITA BADORIA


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

Linear vs. Nonlinear

Time Invariant vs. Time Varying

Causal vs. Non-causal

Stable vs. Unstable


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:

Z-Transforms

n

nznxzX )()(


z-Plane

Re

Im

z = ej

n

nznxzX )()(

( ) ( )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 |||| ,)( azaz

zzX

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

zzX

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 mz All z except 0 (if m>0)or (if m<0)

)(nu 11

1 z 1|| z

)1( nu 11

1 z 1|| z

)(nuan 11

1 az |||| az

)1( nuan 11

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

Engineering