Download - Z Transform (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University.

Z Transform (1)

Hany FerdinandoDept. of Electrical Eng.

Petra Christian University

Z Transform (1) - Hany Ferdinando 2

Overview

Introduction Basic calculation RoC Inverse Z Transform Properties of Z transform Exercise


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 time variable in the z-domain


Introduction

One important consequence of transform-domain description of LTI system is that the convolution operation in the time domain is converted to a multiplication operation in the transform-domain


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, much as one did for logarithm before the advent of hand calculators


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 geometrics

series

k

kkzxX(z)

n

nx(n)zX(z)or


Basic Calculation

0k,2

0k0,k

kx

0k0,

0k,2- k

kx

Calculate:

2z

zX(z)

2z

zX(z)


Basic Calculation

Different signals can have the same transform in the z-domain strange

The problem is when we got the representation in z-domain, how we can know the original signal in the time domain…


Region of Convergence (RoC)

Geometrics series for infinite sum has special rule in order to solve it

This is the ratio between adjacent values

For those who forget this rule, please refer to geometrics series


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 = ∞


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 the RoC

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


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


Inverse Z Transform

Direct division Partial expansion Alternative partial expansion

Use RoCinformation


Direct Division

If the RoC is less than ‘a’, then expand it to positive power of z a is divided by (–a+z)

If the RoC is greater than ‘a’, then expand it to negative power of z a is divided by (z-a)

az

aX(z)


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

m2m

21m

1m

0

pz

A...

pz

A

pz

A

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

a...zazaza


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 ∞


Linearity

a x1(n) + b x2(n) ↔ a X1(z) + b X2(z)

RoC is R1∩R2

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 (n) is the entire z-plane


Time Shifting

x(n-m) ↔ z-mX(z) RoC of z-mX(z) is R, except for the

possible addition or deletion of the origin of infinity

For m>0, it introduces pole at z = 0 and the RoC may not include the origin

For m<0, it introduces zero at z = 0 and the RoC may include the origin


Frequency Shifting

ej(o)nx(n) ↔ X(ej(o)z) RoC is R The poles and zeros is rotated by the

angle of o, therefore if X(z) has complex conjugate poles/zeros, they will have no symmetry at all


Time Reversal

x(-n) ↔ X(1/z) RoC is 1/R


Convolution Property

x1(n)*x2(n) ↔ X1(z)X2(z) RoC is R1∩R2 The behavior of RoC is similar to the

linearity property It says that when two polynomial or power

series of X1(z) and X2(z) are multiplied, the coefficient of representing the product are convolution of the coefficient of X1(z) and X2(z)


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)


Next…

Signals and Systems by A. V. Oppeneim ch 10, 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: