z transform

Department of Computer Eng.

Sharif University of Technology

Discrete-time signal processing

Chapter 3:Chapter 3:THE Z-TRANSFORMTHE Z-TRANSFORM

Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer and Buck, ©1999-2000 Prentice Hall Inc.

3.1 The Z-Transform

• Counterpart of the Laplace transform for discrete-time signals• Generalization of the Fourier Transform

Fourier Transform does not exist for all signals

• 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

Chapter 3: The Z-Transform 2

n

nznxzX

nj

n

j enxeX

jn

n

z

n

n

rez

znxzX

zXnx

zXznxnx

0

)(

)(

)(

r: فاز : اندازه

یک طرفه zتبدیل

دو طرفه zتبدیل

3.1 The Z-Transform

The z-transform and the DTFT

• Convenient to describe on the complex z-plane• If we plot z=ej for =0 to 2 we get the unit circle


Re

Im

Unit Circle

r=1

0

2 0 2

jeX

Convergence of the z-Transform

• DTFT does not always convergeExample: x[n] = anu[n] for |a|>1 does not have a DTFT

• Complex variable z can be written as r ej so the z-transform

convert to the DTFT of x[n] multiplied with exponential sequence r –n

• For certain choices of r the sum

maybe made finite


n

njn

n

njj enxenxreX r r

nj

n

j enxeX

n

nx r n-

Region of Convergence (ROC)

• ROC: The set of values of z for which the z-transform converges• The region of convergence is made of circles


Re

Im• Example: z-transform converges for

values of 0.5<r<2ROC is shown on the left

In this example the ROC includes the unit circle, so DTFT exists

• Example:Doesn't converge for any r.

DTFT exists.

It has finite energy.

DTFT converges in a mean square sense.

• Example: Doesn't converge for any r.

It doesn’t have even finite energy.

But we define a useful DTFT with impulse function.

nnx ocos

sin cnx nn

Region of Convergence (ROC)

Example 1: Right-Sided Exponential Sequence

• For Convergence we require

• Hence the ROC is defined as

• Inside the ROC series converges to


0n

n1

n

nnn azznuazX nuanx

0n

n1az

az1azn1

az

zaz11

azzX0n

1

n1

Re

Im

a 1o x

• Region outside the circle of radius a is the ROC

• Right-sided sequence ROCs extend outside a circle

) دنباله چپگرا(

az

z

azzazX

azzaza

ROC

zaza

zaznuazX

n

n

n n

nn

n

n

n

n

nn

11

1

0

1

1 0

11

1

1

1

1

11

1

:

1

1

1 nuanx n

a

Example 2: Left-Sided Exponential Sequence

Example 3: Two-Sided Exponential Sequence


1-n-u21

-nu31

nxnn

11

10

1

0

1

3

11

1

3

11

3

1

3

1

3

1

zz

zz

zn

n

11

011

1

n

n1

z21

1

1

z21

1

z21

z21

z21

z31

1 z31

:ROC 1

z21

1 z21

:ROC 1

21

z31

z

121

zz2

z21

1

1

z31

1

1zX

11

Im

21

oo

121

xx31

Example 4: Finite Length Sequence


otherwise0

10 Nnanx

n

N=16Pole-zero plot

Nnunuanx n

0

:

1

1

1)(

1

0

11

1

1

11

0

11

0

zazaz

ROC

az

az

z

az

azazzazX

N

n

n

NN

N

NN

n

nN

n

nn

Some common Z-transform pairs


SEQUENCETRANSFORMROC

1z

0m ifor

0m if 0except z All

1z11

1 z

11

1 z

mz

mn

nu

nu

n

1

1 z ALL


1:

cos21

cos1cos

:1

1

:1

:1

11

:1

1

210

10

0

21

1

21

1

1

1

zROCzz

znun

azROCaz

aznuna

azROCaz

aznuna

azROCaz

nua

azROCaz

nua

Z

Zn

Zn

Zn

Zn


0:1

1

0

10

:cos21

sinsin

1

2210

10

0

zROCaz

za

otherwise

Nna

rzROCzrzr

zrnunr

NNZ

n

Zn

rzROC

zrzr

zrnunr

zROCzz

znun

Zn

Z

:cos21

cos1cos

1:cos21

sinsin

2210

10

0

210

10

0


3.2 Properties of The ROC of Z-Transform

• 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


Stability, Causality, and the ROC

• Consider a system with impulse response h[n]• The z-transform H(z) and the pole-zero plot shown below• Without any other information h[n] is not uniquely determined

|z|>2 or |z|<½ or ½<|z|<2• If system stable ROC must include unit-circle: ½<|z|<2• If system is causal must be right sided: |z|>2


3.4 Z-Transform Properties: Linearity

• Notation

• Linearity

– Note that the ROC of combined sequence may be larger than either ROC– This would happen if some pole/zero cancellation occurs– Example:

•Both sequences are right-sided•Both sequences have a pole z=a•Both have a ROC defined as |z|>|a|•In the combined sequence the pole at z=a cancels with a zero at z=a•The combined ROC is the entire z plane except z=0


xZ RROC zXnx

21 xx21

Z21 RRROC zbXzaXnbxnax

N-nua-nuanx nn

Z-Transform Properties: Time Shifting

• Here no is an integer– If positive the sequence is shifted right– If negative the sequence is shifted left

• The ROC can change– The new term may add or remove poles at z=0 or z=

• Example


xnZ

o RROC zXznnx o

41

z z

41

1

1z zX

1

1

1-nu41

nx1-n

Z-Transform Properties: Multiplication by Exponential

• ROC is scaled by |zo|

• All pole/zero locations are scaled

• If zo is a positive real number: z-plane shrinks or expands

• If zo is a complex number with unit magnitude it rotates

• Example: We know the z-transform pair

• Let’s find the z-transform of


xooZn

o RzROC z/zXnxz

1z:ROC z-1

1nu 1-

Z

nure21

nure21

nuncosrnxnjnj

on oo

rz zre1

2/1zre1

2/1zX

1j1j oo

Z-Transform Properties: Differentiation

• Example: We want the inverse z-transform of

• Let’s differentiate to obtain rational expression

• Making use of z-transform properties and ROC


x

Z RROC dz

zdXznnx

az az1logzX 1

1

11

2

az11

azdz

zdXz

az1az

dzzdX

1nuaannx 1n

1nuna

1nxn

1n

Z-Transform Properties: Conjugation


x**Z* RROC zXnx

n

n

n n

n n

n n

n n

X z x n z

X z x n z x n z

X z x n z x n z Z x n

Z-Transform Properties: Time Reversal

• ROC is inverted• Example:

• Time reversed version of


x

Z

R1

ROC z/1Xnx

nuanx n

nuan

111-

1-1

az za-1

za-az1

1zX

Z-Transform Properties: Convolution

• Convolution in time domain is multiplication in z-domain• Example: Let’s calculate the convolution of

• Multiplications of z-transforms is

• ROC: if |a|<1 ROC is |z|>1 if |a|>1 ROC is |z|>|a|• Partial fractional expansion of Y(z)


2x1x21

Z21 RR:ROC zXzXnxnx

nunx and nuanx 2n

1

az:ROC az11

zX 11

1z:ROC z1

1zX 12

1121 z1az11

zXzXzY

1z :ROC assume 11

1

1

111

az

a

zazY nuanu

a11

ny 1n

Some Z-transform properties


3.3 The Inverse Z-Transform

• Formal inverse z-transform is based on a Cauchy integral• Less formal ways sufficient most of the time

– Inspection method– Partial fraction expansion– Power series expansion

• Inspection Method

Make use of known z-transform pairs such as

Example: The inverse z-transform of


az az11

nua 1Zn

nu21

nx 21

z z

21

1

1zX

n

1

Inverse Z-Transform by Partial Fraction Expansion

• Assume that a given z-transform can be expressed as

• Apply partial fractional expansion

• First term exist only if M>N

– Br is obtained by long division• Second term represents all first order poles• Third term represents an order s pole

– There will be a similar term for every high-order pole • Each term can be inverse transformed by inspection


N

0k

kk

M

0k

kk

za

zbzX

s

1mm1

i

mN

ik,1k1

k

kNM

0r

rr

zd1

Czd1

AzBzX

Inverse Z-Transform by Partial Fraction Expansion

• Coefficients are given as

• Easier to understand with examples


s

1mm1

i

mN

ik,1k1

k

kNM

0r

rr

zd1

Czd1

AzBzX

kdz

1kk zXzd1A

1idw

1sims

ms

msi

m wXwd1dwd

d!ms

1C

Example 5: 2nd Order Z-Transform


21

z :ROC z

21

1z41

1

1zX

11

1

2

1

1

z21

1

A

z41

1

AzX

1

41

21

1

1zXz

41

1A1

41

z

11

2

21

41

1

1zXz

21

1A1

21

z

12

Example 5 Continued

• ROC extends to infinity – Indicates right sided sequence


21

z z

21

1

2

z41

1

1zX

11

nu41

-nu21

2nxnn

Example 6

• Long division to obtain Bo


1z z1z

21

1

z1

z21

z23

1

zz21zX

11

21

21

21

1z5

2z3z

21z2z1z

23

z21

1

12

1212

11

1

z1z21

1

z512zX

1

2

1

1

z1A

z21

1

A2zX

9zXz21

1A

21

z

11

8zXz1A1z

12

Example 5 Continued

• ROC extends to infinity– Indicates right-sided sequence


1z z1

8

z21

1

92zX 1

1

n8u-nu21

9n2nxn

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


n

nznxzX

2112 2 1 0 1 2 zxzxxzxzxzX

12

1112

z21

1z21

z

z1z1z21

1z zX

1n21

n1n21

2nnx

2n0

1n21

0n1

1n21

2n1

nx

Example 6

z transform

Documents

ztransformthe z

roc of z

r ejsubstituting z

connected regiona ztransform

pairssome common z

plotsome common z

complex zplaneif

dtftcomplex variable