6 z transform
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The z-Transform
Engr. Leah Q. Santos
Faculty, Eng’g Dept.
Introduction
Why z-Transform? A generalization of Fourier transform Why generalize it?
FT does not converge on all sequenceNotation good for analysisBring the power of complex variable theory
deal with the discrete-time signals and systems
z-Transform
Definition
A convenient yet invaluable tool for representing, analyzing,and designing Discrete-time signals and systems.
Mathematically:
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?
Region of Convergence(ROC)
Definition The set of all values of Z for which X(z)
attains a finite values
X(z) = ∑ x(n)z-n limit: -∞<n<∞
X(z) = ∑ x(n)z-n limit: 0<n<∞
for 2-sided z-transform
for 1-sided z-transformROC is centered on origin and consists of a set of rings.
ROC is centered on origin and consists of a set of rings.
Properties of ROC
A ring or disk in the z-plane centered at the origin. The Fourier Transform of x(n) is converge absolutely if 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.
Example:
Soln’s:
Importantz-Transform Pairs
Z-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 Pairs
Sequence z-Transform ROC
)(][cos 0 nun 210
10
]cos2[1
][cos1
zz
z1|| z
)(][sin 0 nun 210
10
]cos2[1
][sin
zz
z1|| z
)(]cos[ 0 nunr n 2210
10
]cos2[1
]cos[1
zrzr
zrrz ||
)(]sin[ 0 nunr n 2210
10
]cos2[1
]sin[
zrzr
zrrz ||
otherwise0
10 Nnan
11
1
az
za NN
0|| z
Z-Transform PairsSequence z-Transform
z-Transform Theorems and Properties
Linearity
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzbYzaXnbynax ),()()]()([Z
Overlay of the above two
ROC’s
ShiftxRzzXnx ),()]([Z
xn RzzXznnx )()]([ 0
0Z
Multiplication by an Exponential Sequence
xx- RzRzXnx || ),()]([Z
xn RazzaXnxa || )()]([ 1Z
Differentiation of X(z)
xRzzXnx ),()]([Z
xRzdz
zdXznnx
)()]([Z
Conjugation
xRzzXnx ),()]([Z
xRzzXnx *)(*)](*[Z
Reversal
xRzzXnx ),()]([Z
xRzzXnx /1 )()]([ 1 Z
Real and Imaginary Parts
xRzzXnx ),()]([Z
xRzzXzXnxe *)](*)([)]([ 21R
xj RzzXzXnx *)](*)([)]([ 21Im
Initial Value Theorem
0for ,0)( nnx
)(lim)0( zXxz
Convolution of Sequences
xRzzXnx ),()]([Z
yRzzYny ),()]([Z
yx RRzzYzXnynx )()()](*)([Z
Convolution of Sequences
k
knykxnynx )()()(*)(
n
n
k
zknykxnynx )()()](*)([Z
k
n
n
zknykx )()(
k
n
n
k znyzkx )()(
)()( zYzX
Causal System
- One which produces an output only when there is an input
- x(n) is zero before time 0.
x(n)=0 n<0
x(n) 0<n<∞
for one sided z-transform
Properties:
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Find the possible ROC’s
Find the possible ROC’s
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 1: A right sided Sequence.
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 2: A left sided Sequence.
More on Rational z-Transform
Re
Im
a b c
Consider the rational z-transform with the pole pattern:
Case 3: A two sided Sequence.
Stable and Causal Systems
Re
Im
Causal Systems : ROC extends outward from the outermost pole.
Stable and Causal Systems
Re
ImStable Systems : ROC includes the unit circle.
1
ROC
4
Examples:
1. ( ) 4 ( )nx n u n
1
( )
1
1 4
na u n
z
: 4ROC z
ROC
5 4
Examples:
2. ( ) [5(4 ) 6(5 )] ( )n nx n u n
1 1
( )
5 6
1 4 1 5
na u n
z z
' : 4; 5
: 5
ROC s z z
ROC z
Example: Sum of Two Right Sided Sequences
)()()()()( 31
21 nununx nn
31
21
)(
z
z
z
zzX
Re
Im
1/2
))((
)(2
31
21
121
zz
zz
1/3
1/12
ROC is bounded by poles and is the exterior of a circle.
ROC does not include any pole.
Example: A Two Sided Sequence
)1()()()()( 21
31 nununx nn
21
31
)(
z
z
z
zzX
Re
Im
1/2
))((
)(2
21
31
121
zz
zz
1/3
1/12
ROC is bounded by poles and is a ring.
ROC does not include any pole.
Example: A Finite Sequence
10 ,)( Nnanx n
nN
n
nN
n
n zazazX )()( 11
0
1
0
Re
Im
ROC: 0 < z <
ROC does not include any pole.
1
1
1
)(1
az
az N
az
az
z
NN
N
1
1
N-1 poles
N-1 zeros
Always StableAlways Stable
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