PRESENTED BY;

ISRA.A.R

ECE DEPARTMENT

PERIYAR MANIAMMAI UNIVERSITY

THANJAVUR

INTRODUCTION Counterpart of the Laplace transform for discrete-time signals

The z-Transform is often time more convenient to use

Definition:

Compare to DTFT definition:

z is a complex variable that can be represented as z=r ej

Substituting z=ej will reduce the z-transform to DTFT

n

nz nxzX

nj

n

j e nxeX

REGION OF CONVERGENCE

Region of Convergence for a discrete time signal x[n] is defined

as a continuous region in z plane where the Z-Transform

converges.

In order to determine RoC, it is convenient to represent the

Z-Transform as

X(z) = P(z) /Q(z)

The roots of the equation P(z) = 0 correspond to the ’zeros’ of X(z)

The roots of the equation Q(z) = 0 correspond to the ’poles’ of X(z)

The RoC of the Z-transform depends on the convergence of the polynomials

P(z) and Q(z),

RIGHT HANDED Z-TRANSFORM

Let x[n] be causal signal given by x[n] = a^n u[n]

The Z - Transform of x[n] is given by

𝑛=−∞

𝑛=∞

x[n]z−n

= z/z-a

The ROC is defined by |az−1| < 1 or |z| > |a|

The RoC for x[n] is the entire region outside the circle z = ae^jω

6

Z-TRANSFORM AND DFDT

Re

Im

Unit Circle

r=1

0

2 0 2

jeX

EXAMPLES OF CONVERGENCE

LEFT HANDED Z-TRANSFORM

Copyright (C) 2005 Güner

Arslan351M Digital Signal Processing 9

• The ROC is a ring or disk centered at the origin

• DTFT exists if and only if the ROC includes the unit circle

• The ROC cannot contain any poles

• The ROC for finite-length sequence is the entire z-plane except possibly z=0 and z=

The ROC for a right-handed sequence extends outward from the outermost pole possibly including z=

The ROC for a left-handed sequence extends inward from the innermost pole possibly including z=0

The ROC of a two-sided sequence is a ring bounded by poles

The ROC must be a connected region

A z-transform does not uniquely determine a sequence without specifying the ROC

APPLICATIONS OF Z-TRANSFORM

Uses to analysis of digital filters.

Used to simulate the continuous systems.

Analyze the linear discrete system.

Used to finding frequency response.

Analysis of discrete signal.

Helps in system design and analysis and also checks the systems

stability.

For automatic controls in telecommunication.

Enhance the electrical and mechanical energy to provide dynamic

nature of the system