inverse z-transform ppt

SIGNAL AND SYSTEM

ELECTRONICS AND COMMUNICATION

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SARDAR VALLABHBHAI PATEL INSTITUTE OF TECHNOLOGY

2ELECTRONICS AND COMMUNICATION

INVERSE Z-TRANSORM

MADE BY:

VISHAL HASRAJANI 130410111033

RAJSI JADHAV 130410111035

MIHIR JAIN 130410111036


DEFINITION z-TRANSFORM

• z-transform provides a valuable technique for analysis and design of discrete time signals and discrete time LTI system.


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The z-TransformDefinition

• The z-transform of sequence x(n) is defined by

n

nznxzX )()(

Let z = ej.

( ) ( )j j n

n

X e x n e

Fourier Transform


z-Plane

Re

Im

z = ej

n

nznxzX )()(

( ) ( )j j n

n

X e x n e

Fourier Transform is to evaluate z-transform on a unit circle.

Fourier Transform is to evaluate z-transform on a unit circle. 6ELECTRONICS AND

COMMUNICATION

Advantages of z-transform

• Stability of LTI system can be determined using z-transform.

• By calculating z-transform of a given signal, DFT and FT can be determined.

• The solution of differential equations can be simplified using z-transform.


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Inverse z-transform

Where C is a counterclockwise closed path encircling the origin and is entirely in the ROC. Contour C must encircle all the poles of X(z).


c

nn dzzzX

jx 1)(

2

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The Inverse Z-Transform

• There are generally three types of inverse z-transform:– Synthetic division method– Partial fraction expansion– Power series expansion


The Inverse z TransformSynthetic Division


Synthetic Division

The Inverse z Transform


Partial Fraction Expansion


Inverse z Transform Example


Inverse Z-Transform by Power Series Expansion

• The z-transform is power series

• In expanded form

• Z-transforms of this form can generally be inversed easily• Especially useful for finite-length series• Example

n

nz nxzX

2112 z 2xz 1x 0xz 1xz 2xzX

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1112

z21

1z21

z

z1z1z21

1z zX

1n21

n1n21

2nnx

2n0

1n21

0n1

1n21

2n1

nx



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inverse z-transform ppt

Engineering