Z-Plane Analysis DR. Wajiha Shah. Content Introduction z-Transform Zeros and Poles Region of...
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Transcript of Z-Plane Analysis DR. Wajiha Shah. Content Introduction z-Transform Zeros and Poles Region of...
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) =