z transform part 1.pdf

8/10/2019 Z Transform part 1.pdf

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The z-transform

Part 1

Dr. Ali Hussein Muqaibel

http://faculty.kfupm.edu.sa/ee/muqaibel/

Dr. Ali Muqaibel


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The material to be covered in this lecture is

as follows:

Dr. Ali Muqaibel

Introduction to the z-transform

Definition of the z-transform

Derivation of the z-transform

Region of convergence for the transform

Examples.


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After finishing this lecture you should be

able to:

Dr. Ali Muqaibel

Find the z-transform for a given signal utilizing the z-

transform definition

Calculate the region of convergence for the transform


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Derivation of the z-Transform

Dr. Ali Muqaibel

The z-transform is the basic tool for the analysis and synthesisof discrete-time systems.

The z-transform is defined as follows:

The coefficient

denote the sample value and

denotes that the sample occurs sample periods afterthe 0 reference. Note that the lower limit of the summation can start from

zero if the signal is causal (Unilateral z-transform)

Rather than starting form the given definition for the z-transform, we may start from the continuous-time functionand derive the z-transform. This is done in the next slide.

nn

z x nT zX


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Derivation of the z-transform

Dr. Ali Muqaibel

The sampled signal may be written as

Since

0for all t except

at , can be replaced by. And Assuming 0 for 0. Then,

Taking Laplace transform yields Rearranging

By sifting property of the delta

function

sn

x t x t t nT

0

s s

n

sx t x tnT nT

0

0

t

n

ss

X s x nt t n eT dt

0

0

st

s

n

X s x nT t nT e dt


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Continue Derivation

Dr. Ali Muqaibel

Defining the complex variable z as the Laplace time-shift

operator becomes, We could have started from the last expression but it is good

to relate to the s-domain

In the -domain the left-half plane corresponds to 0 ismapped to || 1 in the z-plane which is the region insidethe unit circle.

j

1 Re[z]

Im[z]

S-planez-plane

Left-half

Unit

circle


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Region of Convergence (ROC) is converged for 0 (left-half of s-plane). This corresponds

to 1. This is the region inside the unit circle. is NOT converged for 0 (right-half of s-plane). This

corresponds to 1 which is the region outside the unit circle

The mapping of the Laplace variable into the z-plane through is illustrated in the figure.

j

1Re[z]

Im[z]

S-planez-plane

Left-half

Unit

circle


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The Z-Transform in Summary

Dr. Ali Muqaibel

The coefficient denotes the sampled value The square bracket is used to indicate discrete times.

denotes that the sample occurs

sample periods after

the 0 reference. is simply the -second time shift The parameter is simply shorthand notation for the Laplace

time shift operator

For instance, 30 denotes a sample, having value 30, whichoccurs 40 sample periods after the t=0 reference

Matlab has special tools for z-transform: ztrans, iztrans , pretty

Where

and

0


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Example 1:

Dr. Ali Muqaibel

Determine the z-transform for the

following signal

Solution:

We know that

hence

1

2

,

,

1

0

1

2

1

0,

1

,

,

n

n

x n n

n

otherwise

X zn

nx nz

1 0 1 2

1

1

2

2

1 0X 1 2

2

z n x x xx n z z z z z

X z z z z

x


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Example 2: Sampled Step Function

(Important Functions)

Dr. Ali Muqaibel

Consider a unit step sample sequence

defined by

Find the z-transform.

Solution

The sum converges absolutely to

outside the unit circle

t

u(t)

T 2 T 3 T 4 T 5 T

1

0

1 2 3 10

1( ) 1 ........... , 1

1

n

n

U z X z z z z z zz

0

n

n

X z z


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Sampled Dirac Delta Function (an other

important function)

Dr. Ali Muqaibel

The Dirac Delta Function is defined to be

For a delayed version of delta is defined as

Applying the definition of the z-transform

1 0[ ]

0 0

nn

n

1[ ]0

n kn kn k

Dirac

function( )t

0( ) ( ) (0) 1s T

k

X z t z

( ) 1X z


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The Unit Exponential Sequence

Dr. Ali Muqaibel

The unit exponential sequence is defined to be

Apply z-transform definition

,we get

, 0[ ]0 0

k

e kx kk

k

1

1

1[ ] 1

zX k e z z e

1

0 0

1

( ) ( )

1

( ) 1

k k k

k k

X z e z e z

z

X z e z z e

| |where z e

if k e then 11

( )1

zX z

kz z k


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Example 3 with Poles and Zeros

Dr. Ali Muqaibel

Determine thez-transform of the signal

0.5 Depict the ROC and the locations of poles and zeros of X(z) in the z-plane

Solution:

Substituting is the definition of the z-transform

0.5 0.5 .

This is a geometric series of infinite length in the ratio 0.5/z; the sumconverges, provided that . 1 or 0.5. Hence the z-transform is

Pole at , zero at . , ROC is the light blue region

10

0.5 1, 0.5

1 0.5

, 0.50.5

n

n

X z zz z

z zz

Im[z]

0.50

0 1 2

0 1 2

2

1 1 1...

2 2 2

1 11 ...

2 4

X z z z z

X zz z


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Self Test 1:

Dr. Ali Muqaibel

Find the z- transform of the following signal:

Hint :

Answer:

2 2

1

( ) 1X z z a

a z


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Self Test 2:

Dr. Ali Muqaibel

Determine the z-transform of the signal

1 0.5 Depict the ROC and the locations of poles and zeros of in the z-

plane Answer:

the sum converges, provided that 0.5 and 1.

1

0

0 0

0.5

0.51

n

n

n n

n

k

n k

X z zz

zz

1

1 11 , 0.5 1

1 0.5 1

2 1.5, 0.5 1

0.5 1

X z zz z

z zz

z z

Poles at z=0.5, 1, zeros at z=0, 0.75.ROC is the region in between


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Dr. Ali Muqaibel

1

1 11 , 0.5 1

1 0.5 1

2 1.5 , 0.5 10.5 1

X z zz z

z zz

z z

Poles at z=0.5, 1, zeros at z=0, 0.75. ROC is the region in between

Continue Self-test

z transform part 1.pdf

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