Z-Plane Analysis

DR. Wajiha Shah

Content

Introduction z-Transform Zeros and Poles Region of Convergence Important z-Transform Pairs Inverse z-Transform z-Transform Theorems and Properties System Function

The z-Transform

Introduction

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

The z-Transform

z-Transform

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

n

nznxzX )()(

Let z = ej.

( ) ( )j j n

n

X e x n e

Fourier Transform

z-Plane

Re

Im

z = ej

n

nznxzX )()(

( ) ( )j j n

n

X e x n e

Fourier Transform is to evaluate z-transform on a unit circle.

Fourier Transform is to evaluate z-transform on a unit circle.

z-Plane

Re

Im

X(z)

Re

Im

z = ej

Periodic Property of FT

Re

Im

X(z)

X(ej)

Can you say why Fourier Transform is a periodic function with period 2?

Can you say why Fourier Transform is a periodic function with period 2?

z-Plane Analysis

Zeros and Poles

Definition

Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<, is called the region of convergence.

n

n

n

n znxznxzX |||)(|)(|)(|

ROC is centered on origin and consists of a set of rings.

ROC is centered on origin and consists of a set of rings.

Example: Region of Convergence

Re

Im

n

n

n

n znxznxzX |||)(|)(|)(|

ROC is an annual ring centered on the origin.

ROC is an annual ring centered on the origin.

xx RzR ||

r

}|{ xx

j RrRrezROC

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.

Example: A right sided Sequence

)()( nuanx n )()( nuanx n

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

n

x(n)

. . .

Example: A right sided Sequence

)()( nuanx n )()( nuanx n

n

n

n znuazX

)()(

0n

nn za

0

1)(n

naz

For convergence of X(z), we require that

0

1 ||n

az 1|| 1 az

|||| az

az

z

azazzX

n

n

10

1

1

1)()(

|||| az

aa

Example: A right sided Sequence ROC for x(n)=anu(n)

|||| ,)( azaz

zzX

|||| ,)( az

az

zzX

Re

Im

1aa

Re

Im

1

Which one is stable?Which one is stable?

Example: A left sided Sequence

)1()( nuanx n )1()( nuanx n

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

x(n)

. . .

Example: A left sided Sequence

)1()( nuanx n )1()( nuanx n

n

n

n znuazX

)1()(

For convergence of X(z), we require that

0

1 ||n

za 1|| 1 za

|||| az

az

z

zazazX

n

n

10

1

1

11)(1)(

|||| az

n

n

n za

1

n

n

n za

1

n

n

n za

0

1

aa

Example: A left sided Sequence ROC for x(n)=anu( n1)

|||| ,)( azaz

zzX

|||| ,)( az

az

zzX

Re

Im

1aa

Re

Im

1

Which one is stable?Which one is stable?

The z-Transform

Region of Convergence

Represent z-transform as a Rational Function

)(

)()(

zQ

zPzX 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) =