z transform
DESCRIPTION
TRANSCRIPT
Waqas Ahmed
Introduction z-Transform Zeros and Poles Region of Convergence z-Transform Properties
Introduction
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
z-Transform
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
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.
Zeros and Poles
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.
A stable system requires that its Fourier transform is uniformly convergent.
Re
Im
1
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.
)()( 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
|||| ,)( azaz
zzX
|||| ,)( az
az
zzX
Re
Im
1aa
Re
Im
1
Which one is stable?Which one is stable?
)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
|||| ,)( azaz
zzX
|||| ,)( az
az
zzX
Re
Im
1aa
Re
Im
1
Which one is stable?Which one is stable?
Region of Convergence
)(
)()(
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) =
)()( nuanx n |||| ,)( azaz
zzX
Re
Im
a
ROC is bounded by the pole and is the exterior of a circle.
)1()( nuanx n|||| ,)( az
az
zzX
Re
Im
a
ROC is bounded by the pole and is the interior of a circle.
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.
Re
Im
a b c
Consider the z-transform with the pole pattern:
Case 1: A right sided Sequence.
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 2: A left sided Sequence.
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 3: A two sided Sequence.
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 4: Another two sided Sequence.
Importantz-Transform Pairs
Sequence z-Transform ROC
)(n 1 All z
)( mn mz All z except 0 (if m>0)or (if m<0)
)(nu 11
1 z
1|| z
)1( nu 11
1 z
1|| z
)(nuan 11
1 az
|||| az
)1( nuan 11
1 az
|||| az
z-Transform Theorems and Properties
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzbYzaXnbynax ),()()]()([Z
Overlay of the above two
ROC’s
xRzzXnx ),()]([Z
xn RzzXznnx )()]([ 0
0Z
xx- RzRzXnx || ),()]([Z
xn RazzaXnxa || )()]([ 1Z
xRzzXnx ),()]([Z
xRzdz
zdXznnx
)()]([Z
xRzzXnx ),()]([Z
xRzzXnx /1 )()]([ 1 Z
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzYzXnynx )()()](*)([Z
33
both. of econvergenc of regions the inside of values for)()()(
)()()(
let
)()(
)()()(
Then
)()()(
zzYzXzW
zzmykxzW
knm
zknykx
zknykxzW
knykxnw
k
k m
m
n
k n
n
n k
k