z plane

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Z-Plane Analysis

DR. Wajiha Shah

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Content Introduction

z -Transform

Zeros and Poles

Region of Convergence

Important z -Transform Pairs Inverse z -Transform

z -Transform Theorems and Properties

System Function

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The z-Transform

Introduction

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Why z-Transform? A generalization of Fourier transform

Why generalize it? – FT does not converge on all sequence

– Notation good for analysis

– Bring the power of complex variable theory deal with

the discrete-time signals and systems

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The z-Transform

z-Transform

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

∑∞

−∞=

−=n

n z n x z X )()(

Let z = e− jω .

( ) ( ) j j n

n

X e x n eω ω

∞−

=−∞

= ∑

Fourier

Transform

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z -Plane

Re

Im

z = e− jω

ω∑

∞

−∞=

−=n

n z n x z X )()(

( ) ( )

j j n

n X e x n e

ω ω

∞−

=−∞= ∑Fourier Transform is to evaluate z-transform

on a unit circle.

Fourier Transform is to evaluate z-transform

on a unit circle.

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z -Plane

Re

Im

X ( z )

Re

Im

z = e− jω

ω

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Periodic Property of FT

Re

Im

X ( z )

π−π ω

X (e jω)

Can you say why Fourier Transform is

a periodic function with period 2π?

Can you say why Fourier Transform is

a periodic function with period 2π?

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z-Plane Analysis

Zeros and Poles

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DefinitionGive a sequence, the set of values of z for which the

z -transform converges, i.e., | X ( z )|<∞, is called theregion of convergence.

∞<== ∑∑

∞

−∞=

−∞

−∞=

−

n

n

n

n z n x z n x z X |||)(|)(|)(|

ROC is centered on origin and

consists of a set of rings.

ROC is centered on origin and

consists of a set of rings.

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Example: Region of Convergence

Re

Im

∞<== ∑∑∞

−∞=

−∞

−∞=

−

n

n

n

n z n x z n x z X |||)(|)(|)(|

ROC is an annual ring centered

on the origin.

ROC is an annual ring centered

on the origin.

+− << x x R z R ||r

}|{ +−

ω <<== x x

j Rr Rre z ROC

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Stable Systems

Re

Im

1

A stable system requires that its Fourier transform is

uniformly convergent. Fact: Fourier transform is to

evaluate z -transform on a unit

circle.

A stable system requires the

ROC of z -transform to include

the unit circle.

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Example: A right sided Sequence

)()( nuan x n=

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

x(n)

. . .

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)()( nuan x n=

n

n

n z nua z X −∞

−∞=∑= )()(

∑∞

=

−=0n

nn z a

∑∞

=

−=0

1)(

n

naz

For convergence of X ( z ), we

require that

∞<∑∞

=

−

0

1 ||n

az 1|| 1 <−az

|||| a z >

a z

z

az az z X

n

n

−=

−== −

∞

=

−∑ 10

1

1

1)()(

|||| a z >

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a− a


ROC for x(n)=anu(n)

|||| ,)( a z a z

z z X >

−

=

Re

Im

1 a− a

Re

Im

1

Which one is stable?Which one is stable?

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Example: A left sided Sequence

)1()( −−−= nuan x n

1 2 3 4 5 6 7 8 9 10-1-2-3-4-5-6-7-8

n

x(n)

. . .

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)1()( −−−= nuan x n

n

n

n z nua z X −∞

−∞=

∑ −−−= )1()(

For convergence of X ( z ), we

require that

∞<∑∞

=

−

0

1 ||n

z a 1|| 1 <− z a

|||| a z <

a z

z

z a z a z X

n

n

−=

−−=−= −

∞

=

−∑ 10

1

1

11)(1)(

|||| a z <

n

n

n z a −−

−∞=∑−=

1

n

n

n z a∑∞

=

−−=1

n

n

n z a∑∞

=

−−=0

1

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a− a


ROC for x(n)=−anu(− n−1)

|||| ,)( a z a z

z z X <

−

=

Re

Im

1 a− a

Re

Im

1

Which one is stable?Which one is stable?

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The z-Transform

Region of

Convergence

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Represent z -transform as a

Rational Function

)(

)(

)( z Q

z P

z X =where P ( z ) and Q( z ) are

polynomials in z .

Zeros: The values of z ’s such that X ( z ) = 0

Poles: The values of z ’s such that X ( z ) = ∞

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)()( nuan x n= |||| ,)( a z

a z

z z X >

−

=

Re

Im

a

ROC is bounded by the

pole and is the exterior of a circle.

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)1()( −−−= nuan x n|||| ,)( a z

a z

z z X <

−

=

Re

Im

a

ROC is bounded by the

pole and is the interior of a circle.

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Example: Sum of Two Right Sided Sequences

)()()()()(31

21 nunun x nn −+=

31

21

)(+

+−

= z

z

z

z z X

Re

Im

1/2

))((

)(2

31

21

121

+−−= z z

z z

−1/3

1/12

ROC is bounded by poles

and is the exterior of a circle.

ROC does not include any pole.

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Example: A Two Sided Sequence

)1()()()()(21

31 −−−−= nunun x nn

21

31

)(−

++

= z

z

z

z z X

Re

Im

1/2

))((

)(2

21

31

121

−+−= z z

z z

−1/3

1/12

ROC is bounded by poles and is a ring.


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Example: A Finite Sequence

10 ,)( −≤≤= N nan x n

n N

n

n N

n

n z a z a z X )()( 11

0

1

0

−−

=

−−

=∑∑ ==

Re

Im

ROC: 0 < z < ∞


1

1

1

)(1−

−

−

−=

az

az N

a z

a z

z

N N

N −−= −1

1

N -1 poles

N -1 zeros

Always StableAlways Stable

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Properties of ROC

A ring or disk in the z-plane centered at the origin.

The Fourier Transform of x(n) is converge absolutely iff the ROC

includes the unit circle. The ROC cannot include any poles

Finite Duration Sequences: The ROC is the entire z -plane except

possibly z =0 or z =∞.

Right sided sequences: The ROC extends outward from the outermost

finite pole in X ( z ) to z =∞.

Left sided sequences: The ROC extends inward from the innermost

nonzero pole in X ( z ) to z =0.

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More on Rational z -Transform

Re

Im

a b c

Consider the rational z -transform

with the pole pattern:

Find the possibleROC’sFind the possibleROC’s

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Re

Im

a b c



Case 1: A right sided Sequence.

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Re

Im

a b c



Case 2: A left sided Sequence.

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Re

Im

a b c



Case 3: A two sided Sequence.

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Re

Im

a b c



Case 4: Another two sided Sequence.

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The z-Transform

Important

z -Transform Pairs

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Z-Transform Pairs

Sequence z -Transform ROC

)(nδ 1 All z

)( mn −δ m z − All z except 0 (if m>0)

or ∞ (if m<0)

)(nu 11

1−− z

1|| > z

)1( −−− nu1

1

1−

− z 1|| < z

)(nua n 11

1−− az

|||| a z >

)1( −−− nua

n 1

1

1−

− az

|||| a z <

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Z-Transform Pairs

Sequence z -Transform ROC

)(][cos 0 nunω 21

0

1

0

]cos2[1

][cos1−−

−

+ω−

ω−

z z

z 1|| > z

)(][sin 0 nunω 21

0

1

0

]cos2[1

][sin−−

−

+ω−ω

z z

z 1|| > z

)(]cos[ 0 nunr n ω 221

0

1

0

]cos2[1

]cos[1−−

−

+ω−

ω−

z r z r

z r r z >||

)(]sin[ 0 nunr n ω 221

0

1

0

]cos2[1

]sin[−−

−

+ω−ω

z r z r

z r r z >||

−≤≤

otherwise0

10 N nan

11

1−

−

−−

az

z a N N

0|| > z

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The z-Transform

Inverse z -Transform

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The z-Transform

z -Transform Theorems

and Properties

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Linearity x R z z X n x ∈= ),()]([Z

y R z z Y n y ∈= ),()]([Z

y x R R z z bY z aX nbynax ∩∈+=+ ),()()]()([Z

Overlay of

the above two

ROC’s

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Shift x R z z X n x ∈= ),()]([Z

x

n R z z X z nn x ∈=+ )()]([ 0

0Z

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Multiplication by an Exponential Sequence

+<<= x x- R z R z X n x || ),()]([Z

x

n Ra z z a X n xa ⋅∈= − || )()]([ 1Z

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Differentiation of X (z )

x R z z X n x ∈= ),()]([Z

x R z dz

z dX z nnx ∈−=

)()]([Z

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Conjugation x R z z X n x ∈= ),()]([Z

x R z z X n x ∈= *)(*)](*[Z

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Reversal x R z z X n x ∈= ),()]([Z

x R z z X n x /1 )()]([ 1 ∈=− −Z

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Real and Imaginary Parts

x R z z X n x ∈= ),()]([Z

x R z z X z X n xe ∈+= *)](*)([)]([21R

x jR z z X z X n x ∈−= *)](*)([)]([

21Im

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Initial Value Theorem

0for ,0)( <= nn x

)(lim)0( z X x z ∞→

=

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Convolution of Sequences

x R z z X n x ∈= ),()]([Z

y R z z Y n y ∈= ),()]([Z

y x R R z z Y z X n yn x ∩∈= )()()](*)([Z

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Convolution of Sequences

∑∞

−∞=

−=k

k n yk xn yn x )()()(*)(

∑ ∑∞

−∞=

−∞

−∞=

−=

n

n

k

z k n yk xn yn x )()()](*)([Z

∑ ∑∞

−∞=

−∞

−∞=

−=k

n

n

z k n yk x )()( ∑ ∑∞

−∞=

−∞

−∞=

−=k

n

n

k z n y z k x )()(

)()( z Y z X =

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The z-Transform

System Function

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Shift-Invariant System

h(n)h(n)

x(n) y(n)= x(n)*h(n)

X ( z ) Y ( z )= X ( z ) H ( z ) H ( z )

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Shift-Invariant System

H ( z ) H ( z )

X ( z ) Y ( z )

)(

)()(

zX

z Y z H =

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N

th

-Order Difference Equation

∑∑==

−=− M

r

r

N

k

k r n xbk n ya00

)()(

∑∑=

−

=

− = M

r

r

r

N

k

k

k z b z X z a z Y 00

)()(

∑∑==

−

=

−N

k

k

k

M

r

r

r z a z b z H 00

)(

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Representation in Factored Form

∏

∏

=

−

=

−

−

−=

N

k

r

M

r

r

z d

z c A

z H

1

1

1

1

)1(

)1(

)(

Contributes poles at 0 and zeros at cr

Contributes zeros at 0 and poles at d r

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Stable and Causal Systems

∏

∏

=

−

=

−

−

−=

N

k

r

M

r

r

z d

z c A

z H

1

1

1

1

)1(

)1(

)(Re

Im

Causal Systems : ROC extends outward from the outermost pole.

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Stable and Causal Systems

∏

∏

=

−

=

−

−

−=

N

k

r

M

r

r

z d

z c A

z H

1

1

1

1

)1(

)1(

)(Re

Im

Stable Systems : ROC includes the unit circle.

1

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ExampleConsider the causal system characterized by

)()1()( n xnayn y +−=

1

1

1)( −

−

=az

z H

Re

Im

1

a

)()( nuanh n=

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Determination of Frequency Response

from pole-zero pattern

A LTI system is completely characterized by its

pole-zero pattern.

))((

)(21

1

p z p z

z z z H

−−

−=

Example:

))(()(

21

1

00

0

0

pe pe

z ee H

j j

j j

−−−

= ωω

ωω

0ω je

Re

Im

z 1

p1

p2

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pole-zero pattern.

))((

)(21

1

p z p z

z z z H

−−

−=

Example:

))(()(

21

1

00

0

0

pe pe

z ee H

j j

j j

−−−

= ωω

ωω

0ω je

Re

Im

z 1

p1

p2

|H (e jω )|=?|H (e jω )|=? ∠ H (e jω )=?∠ H (e jω )=?

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Example1

1

1)( −

−

=az

z H

Re

Im

a

0 2 4 6 8

- 1 0

0

1 0

2 0

0 2 4 6 8

- 2

- 1

0

1

2

d B

z plane

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