7 z transform

7/28/2019 7 Z Transform

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Z Transform (1)

Hany FerdinandoDept. of Electrical Eng.

Petra Christian University


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Z Transform (1) - Hany Ferdinando 2

Overview

Introduction

Basic calculation

RoC

Inverse Z Transform

Properties of Z transform Exercise
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Introduction

For discrete-time, we have not only

Fourier analysis, but also Z transform

This is special for discrete-time only

The main idea is to transform

signal/system from time-domain to z-

domain it means there is no timevariable in the z-domain


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Introduction

One important consequence of

transform-domain description of LTI

system is that the convolutionoperation in the time domain is

converted to a multiplication

operation in the transform-domain


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Introduction

It simplifies the study of LTI system by:

Providing intuition that is not evident in

the time-domain solution Including initial conditions in the solution

process automatically

Reducing the solution process of many

problems to a simple table look up, muchas one did for logarithm before theadvent of hand calculators


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Basic Calculation

They are general formula: Index k or n refer to time variable

If k > 0 then k is from 1 to infinity

Solve those equation with the geometricsseries

k

k

kzxX(z)

n

nx(n)zX(z)or
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Basic Calculation

0k,2

0k0,

k

kx 0k0,

0k,2- k

kx

Calculate:

2zzX(z)

2z

zX(z)
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Basic Calculation

Different signals can have the same

transform in the z-domain strange

The problem is when we got therepresentation in z-domain, how we

can know the original signal in the time

domain


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Region of Convergence (RoC)

Geometrics series for infinite sum has

special rule in order to solve it

This is the ratio between adjacentvalues

For those who forget this rule, please

refer to geometrics series


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RoC Properties

RoC of X(z) consists of a ring in the z-

plane centered about the origin

RoC does not contain any poles

If x(n) is of finite duration then the RoC

is the entire z-plane except possibly z

= 0 and/or z =


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RoC Properties

If x(n) is right-sided sequence and if |z|= ro is in the RoC, then all finite values

of z for which |z| > ro will also be in theRoC

If x(n) is left-sided sequence and if |z|= ro is in the RoC, then all values forwhich 0 < |z| < ro will also be in theRoC


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RoC Properties

If x(n) is two-sided and if |z| = ro is in

the RoC, then the RoC will consists of

a ring in the z-plane which includes the|z| = ro


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Inverse Z Transform

Direct division

Partial expansion

Alternative partial expansion

Use RoC

information


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Direct Division

If the RoC is less thana, thenexpand it to positive power of z

a is divided by (a+z)

If the RoC is greater thana, thenexpand it to negative power of z

a is divided by (z-a)

az

aX(z)
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Partial Expansion

If the z is in the power of two or more,

then use partial expansion to reduce

its order

Then solve them with direct division

n

n

2

2

1

1

n21

m

2m

2

1m

1

m

0

pz

A...

pz

A

pz

A

)p)...(zp)(zp(z

a...zazaza
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Properties of Z Transform

General term and condition:

For every x(n) in time domain, there is

X(z) in z domain with R as RoC

n is always from to


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Linearity

a x1(n) + b x2(n) a X1(z) + b X2(z)

RoC is R1R2

If a X1(z) + b X2(z) consist of all poles of X1(z)and X2(z) (there is no pole-zero cancellation),

the RoC is exactly equal to the overlap of the

individual RoC. Otherwise, it will be larger

anu(n) and anu(n-1) has the same RoC, i.e.|z|>|a|, but the RoC of [anu(n) anu(n-1)] or

d(n) is the entire z-plane


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Time Shifting

x(n-m) z-mX(z)

RoC of z-mX(z) is R, except for the

possible addition or deletion of the originof infinity

For m>0, it introduces pole at z = 0 and

the RoC may not include the origin For m


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Frequency Shifting

ej(Wo)nx(n) X(ej(Wo)z)

RoC is R

The poles and zeros is rotated by the

angle ofWo, therefore if X(z) has

complex conjugate poles/zeros, they

will have no symmetry at all


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Time Reversal

x(-n) X(1/z)

RoC is 1/R


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Convolution Property

x1(n)*x2(n) X1(z)X2(z)

RoC is R1R2

The behavior of RoC is similar to thelinearity property

It says that when two polynomial or powerseries of X1(z) and X2(z) are multiplied, the

coefficient of representing the product areconvolution of the coefficient of X1(z) andX2(z)


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Differentiation

RoC is R

One can use this property as a tool to

simplify the problem, but the whole

concept of z transform must be

understood first

dz

dX(z)znx(n)
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Next

Signals and Systems by A. V. Oppeneim ch10, or

Signals and Linear Systems by Robert A.

Gabel ch 4, or

Sinyal & Sistem (terj) ch 10

For the next class, students have to read Z

transform:

7 z transform

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