10.0 Z-Transform
10.1 General Principles of
Z-Transform
Z-Transform

Eigenfunction Property
x[n] = zn
y[n] = H(z)zn
h[n]
linear, time-invariant
H z  

n


h
n
z

n  
Chapters 3, 4, 5, 9, 10 (p.2 of 9.0)
jω

Laplace Transform (p.4 of 9.0)

A Generalization of Fourier Transform
from s  j to s    j
X   j  

xt  e  j t dt

  xt  e   e


- t
 jt

dt

t
Fourier transform of x t e
X( + jω)
jω

Laplace Transform (p.5 of 9.0)
Z-Transform

Eigenfunction Property
– applies for all complex variables z
z  e j
   hn e

H e j 
 j n
Fourier Transform
n  
z  re j

n


h
n
z

H z  
n  

Z-Transform
X z  

n


x
n
z

n  
Z
xn
 X z 
Z-Transform
Z-Transform
Z-Transform

A Generalization of Fourier Transform
from z  e j to z  re j
X re    xnre     xn r e
Fourier Transform of xn r
j


j  n
n  
n
n  
n
Im
z = rejω
unit circle
z = ejω
r
ω
1
Re
 jn
Z-Transform
Z-Transform
Z-Transform
Z-Transform
2π
0
Z-Transform
Z-Transform
Z-Transform

A Generalization of Fourier Transform
X z 
z e
j
 
 X e j
reduces to Fourier Transform
– X(z) may not be well defined (or converged) for all z
– X(z) may converge at some region of z-plane, while
x[n] doesn’t have Fourier Transform
– covering broader class of signals, performing more
analysis for signals/systems
Z-Transform

Rational Expressions and Poles/Zeros
X z  
N z 
D z 
roots
zeros
roots
poles
In terms of z, not z-1
– Pole-Zero Plots
specifying X(z) except for a scale factor
Z-Transform
Z-Transform

Rational Expressions and Poles/Zeros
– Geometric evaluation of Fourier/Z-Transform from
pole-zero plots
X z   M
 i z   i 

 j z  j

each term (z-βi) or (z-αj) represented by a vector with
magnitude/phase
Poles & Zeros
Poles & Zeros (p.9 of 9.0)
Z-Transform (p.12 of 10.0)
Z-Transform

Rational Expressions and Poles/Zeros
– Geometric evaluation of Fourier/Z-transform from
pole-zero plots

Example : 1st-order
hn   a nu n
H z  
1

z
, z  a
1  az1 z  a
pole: z  a, zero : z  0
See Example 10.1, p.743~744 of text
 
H e j 
1
1  ae j
See Fig. 10.13, p.764 of text
Z-Transform

Rational Expressions and Poles/Zeros
– Geometric evaluation of Fourier/Z-transform from
pole-zero plots

Example : 2nd-order
sin n  1
n
hn  r
un
sin 
H z  
1
1  2r cos z 1  r 2 z  2
pole: z1  re j , z 2  re  j
double zero : z  0
See Fig. 10.14, p.766 of text
Z-Transform

Rational Expressions and Poles/Zeros
– Specification of Z-Transform includes the region of
convergence (ROC)

Example :
x1 n   a n u n 
X 1 z  
1
1  az
1

z
za
, z  a
x2 n    a n u  n  1
X 2 z  
1
1  az
1

z
za
, z  a
pole: z  a, zero : z  0 in bot h cases
See Example 10.1, 10.2, p.743~745 of text
Z-Transform

Rational Expressions and Poles/Zeros
– x[n] = 0, n < 0
X(z) involes only negative powers of z initially
x[n] = 0, n > 0
X(z) involes only positive powers of z initially
– poles at infinity if
degree of N(z) > degree of D(z)
zeros at infinity if
degree of D(z) > degree of N(z)
Z-Transform (p.6 of 10.0)
Region of Convergence (ROC)

Property 1 : The ROC of X(z) consists of a ring in
the z-plane centered at the origin
– for the Fourier Transform of x[n]r-n
to converge


n  
xn r n   , depending on r
only, not on ω
– the inner boundary may extend to
include the origin, and the outer
boundary may extend to infinity

Property 2 : The ROC of X(z) doesn’t include any
poles
Property 1
Region of Convergence (ROC)

Property 3 : If x[n] is of finite duration, the ROC is
the entire z-plane, except possibly for z
= 0 and/or z = ∞
xn  0, n  N1 , n  N 2
X z  
 xn z
N2
n
n  N1
– If N1  0, N 2  0
z  0  ROC, z    ROC
If N1  0, z    ROC
If N 2  0, z  0  ROC
Property 3, 4, 5
Region of Convergence (ROC)

Property 4 : If x[n] is right-sided, and {z | |z| =
r0 }ROC, then {z | ∞ > |z| > r0}ROC

 xn
n  N1
r0
n

If N1  0, z    ROC
If N1  0, z    ROC
Region of Convergence (ROC)

Property 5 : If x[n] is left-sided and {z | |z| =
r0}ROC, then {z | 0 < |z| < r0}ROC
 xn r
N2
n  
0
n

If N 2  0, z  0  ROC
If N 2  0, z  0  ROC
Region of Convergence (ROC)

Property 6 : If x[n] is two-sided, and {z | |z| =
r0}ROC, then ROC consists of a ring
that includes {z | |z| = r0}
xn  xR n xL n
– a two-sided x[n] may not have ROC
Region of Convergence (ROC)

Property 7 : If X(z) is rational, then ROC is
bounded by poles or extends to zero or
infinity

Property 8 : If X(z) is rational, and x[n] is rightsided, then ROC is the region outside
the outermost pole, possibly includes z =
∞. If in addition x[n] = 0, n < 0, ROC
also includes z = ∞
Region of Convergence (ROC)

Property 9 : If X(z) is rational, and x[n] is left-sided,
the ROC is the region inside the
innermost pole, possibly includes z = 0.
If in addition x[n] = 0, n > 0, ROC also
includes z = 0
See Example 10.8, p.756~757 of text
Fig. 10.12, p.757 of text
Inverse Z-Transform
xn r
n
1
 
 F 1 X r1e j

xn  1
2

xn  1
2j
 X z  z
2
X r1e
j
  21 
2
r e 
j
1
n 1
dz
n
d ,


X r1e j e jn d
z  r1e
j
j
dz  jr1e d
 jzd
– integration along a circle counterclockwise,
{z | |z| = r1}ROC, for a fixed r1
Laplace Transform (p.28 of 9.0)
Z-Transform
Z-Transform (p.13 of 10.0)
Inverse Z-Transform

Partial-fraction expansion practically useful:
X z   
m
Ai
i 1
1  ai z 1
for each t erm
Ai
1  ai z 1
– ROC outside the pole at z  ai  Ai ai un
n
ROC inside the pole at z  ai   Ai ai u n  1
n
Inverse Z-Transform

Partial-fraction expansion practically useful:
– Example:
3
5
z 1
1
2
6
X z  



1 1 
1 1  
1 1  
1 1 
1  z 1  z  1  z  1  z 


 
 

4
3
4
3


 
 

  un
 
n


1
1
ROC   z z 
, xn  
u n   2 1

3
4
3

 
n
  u n  1
n


1
1
1
ROC   z
 z 
, xn  
u n   2 1

3
4
4
3

 
n
  u n  1
n


1
1
ROC   z z 
, xn   
u  n  1  2 1

4
4
3

n
See Example 10.9, 10.10, 10.11, p.758~760 of text
Example
Inverse Z-Transform

Power-series expansion practically useful:
X z  

n


x
n
z

n  
– right-sided or left-sided based on ROC
Inverse Z-Transform

Power-series expansion practically useful:
– Example:
X z  
1
1  az1


ROC  z z  a ,
xn   a n u n 


ROC  z z  a ,
1
1  az1
1
1  az1
xn    a n u  n  1
 1  az1  a 2 z  2  ......
  a 1 z  a  2 z 2  ......
See Example 10.12, 10.13, 10.14, p.761~763 of text

Known pairs/properties practically helpful
10.2 Properties of Z-Transform
xn 
 X z , ROC  R
Z
x1 n  X 1 z , ROC  R1
Z
x2 n 
 X 2 z , ROC  R2
Z

Linearity
ax1n bx2 n
 aX1 z   bX2 z , ROC  R1  R2 
Z
– ROC = R1∩ R2 if no pole-zero cancellation
Linearity & Time Shift
Linearity
Time Shift

Time Shift
xn  n0 
 z
Z
 n0
X z ,
ROC = R, except for possible addition or deletion of the
origin or infinity
– n0 > 0, poles introduced at z = 0 may cancel zeros at
z=0
– n0 < 0, zeros introduced at z = 0 may cancel poles at
z=0
– Similarly for z = ∞

Scaling in z-domain
 z 
Z
n
z0 xn
 X  , ROC  z0 z z  R
z 
 0


– Pole/zero at z = a  shifted to z = z0a
– z0  e j0

Z
e j0n xn

 X e j0 z

rotation in z-plane by ω0
See Fig. 10.15, p.769 of text
– z0  r0e j0
pole/zero rotation by ω0 and scaled by r0
Scaling in Z-domain
Shift in s-plane (p.33 of 9.0)

Time Reversal
1


1




x n
 X  , ROC  
z  R


z
 z

Z

Time Expansion
 xn k , n is a multipleof k
xk  n  
, else
 0
 


xk  n
 X z k , ROC  z1 k z  R
Z
– pole/zero at z = a  shifted to z = a1/k
Time Reversal
Time Expansion
(p.41 of 5.0)
Time Expansion (p.42 of 5.0)

Conjugation
 
x n
 X  z , ROC  R

Z
– if x[n] is real
 
X z   X  z 
a pole/zero at z = z0  a pole/zero at z  z0

*
Convolution
x n x n
 X z X z , ROC  R  R 
Z
1
2
1
2
1
2
ROC may be larger if pole/zero cancellation occurs
– power series expansion interpretation
Conjugation

Multiplication (p.33 of 3.0)
xt a k , y t  bk
FS
FS
FS
xt  yt 
dk 


a b
j  
Conjugation
x t  a

FS

k
a k  ak , if xt  real
j k j
ak  bk
Multiplication (p.34 of 3.0)
Convolution
y3  
k

First Difference/Accumulation

xn xn 1
 1  z
Z
1
X z
ROC = R with possible deletion of z = 0
and/or addition of z = 1
 xk  1  z
n
k  
Z
1
1
X z 

ROC  R  z

z 1
First Difference/Accumulation

Differentiation in z-domain
nxn
  z
Z

dX z 
, ROC  R
dz
Initial-value Theorem
if xn   0, n  0
x0  lim X z 
z 

X z    xn  z  n
k 0

Summary of Properties/Known Pairs
See Tables 10.1, 10.2, p.775, 776 of text
10.3 System Characterization with
Z-Transform
y[n]=x[n]＊h[n]
x[n]
h[n]
X(z)
H(z)
Y(z)=X(z)H(z)
system function, transfer function

Causality
– A system is causal if and only if the ROC of H(z) is
the exterior of a circle including infinity (may
extend to include the origin in some cases)

H z    hn z n , right - sided
n 0
if hn0   0, n0  0
H z  includes xn0  z n0
– A system with rational H(z) is causal if and only
if
(1)ROC is the exterior of a circle outside the
outermost pole including infinity
and (2)order of N(z) ≤ order of D(z)
H(z) finite for z  ∞

Causality (p.44 of 9.0)
– A causal system has an H(s) whose ROC is a righthalf plane
h(t) is right-sided
– For a system with a rational H(s), causality is
equivalent to its ROC being the right-half plane to
the right of the rightmost pole
– Anticausality
a system is anticausal if h(t) = 0, t > 0
an anticausal system has an H(s) whose ROC is a
left-half plane, etc.
Causality (p.45 of 9.0)
Z-Transform (p.6 of 10.0)

Stability
– A system is stable if and only if ROC of H(z)
includes the unit circle
Fourier Transform converges, or absolutely
summable
– A causal system with a rational H(z) is stable if and
only if all poles lie inside the unit circle
ROC is outside the outermost pole

Systems Characterized by Linear Difference
Equations
 a yn  k    b xn  k 
N
k 0
M
k
k 0
k
Y z  ak z  k  X z  bk z  k
N
M
k 0
k 0
H z  
Y z 
X z 
M

k
b
z
 k
k 0
N
a
k 0
k
z
k
zeros
poles
– difference equation doesn’t specify ROC
stability/causality helps to specify ROC

Interconnections of Systems
– Parallel
H1(z)
H(z)=H1(z)+H2(z)
H2(z)
– Cascade
H1(z)
H2(z)
H(z)=H1(z)H2(z)

Interconnections of Systems
– Feedback
+
－
＋
H1(z)
H2(z)
H z  
H1 z 
1 H1 z H 2 z 

Block Diagram Representation
– Example:
yn
1
4
yn  1
H z  
8
yn  2  xn
1
1
1
4

1
2
1
z 
1
z 2
8






1
1




1 1 
1 1 
 1  z  1  z 
4
2



1
3
3 
1 1
1 1
1 z
1 z
4
2
– direct form, cascade form, parallel form
See Fig. 10.20, p.787 of text
10.4 Unilateral Z-Transform

X z u   xn z  n
unilat eralz - t ransform
k 0
X z  

n


x
n
z

bilat eralz - t ransform
k  
– Z{x[n]u[n]} = Zu{x[n]}
for x[n] = 0, n < 0, X(z)u = X(z)
ROC of X(z)u is always the exterior of a circle
including z = ∞
degree of N(z) ≤ degree of D(z) (converged for z = ∞)
10.4 Unilateral Z-Transform
– Time Delay Property (different from bilateral case)
xn  1 x 1  z 1 X z u
Zu
u
xn  2
x 2  x 1 z 1  z  2 X z u
Z


n 0
n 1
n
n






x
n

1
z

x

1

x
n

1
z


– Time Advance Property (different from bilateral
case)
xn 1 zX z 
Zu
u
 zx0
Time Delay Property/Time Advance Property
10.4 Unilateral Z-Transform
– Convolution Property
if x1 n  x2 n  0, n  0
x1 n x2 n X 1 z u X 2 z u
Zu
this is not true if x1[n] , x2[n] has nonzero values for
n<0
Convolution Property
Examples
• Example 10.4, p.747 of text
n
xn   1  sin   n un
4 
 3
n
n
 1  1 e  un  1  1 e  un

2 j  3  
2 j  3 
3 12 z
1
X z  
,
z



1 j4
1 j4
3
( z  3 e )(z  3 e )
j 4
 j 4
Examples
• Example 10.6, p.752 of text
a n , 0  n  N - 1, a  0

xn   0 , else
1  (az1 ) N  1  z N  a N 
X z  
  N 1 

1
1  az
 z  z  a 
( N  1)st order pole at orgin
j  2Nπk 
N - 1 zerosat zk  ae
, k  1,2 ,...N-1
pot ent ialpole/zerocanceledat z  a
ROC   z  0
Examples
• Example 10.6, p.752 of text
Examples
• Example 10.17, p.772 of text


X  z   log 1  az1 , z  a
1


z
dX
az
Z
, z  a

 z

nxn 
1
dz
1  az
a
Z
, z  a


 a(a) n un 
1
1  az
1
n 1
az
Z
, z  a , t imeshift propert y

a(a) un  1 
1
1  az
 (a) n
un  1
xn 
n
Examples
• Example 10.31, p.788 of text (Problem 10.38,
P.805 of text)
1  74 z 1  12 z 2
H z  
1  14 z 1  18 z  2


1
7 1
1 2

 
1

z

4
2 z
1 1
1 2 
1  4 z  8 z 

Direct formrepresentation of thesystem

Problem 10.12, p.799 of text
Whichof thefollowingsystemis approximately
lowpass, highpass,or brandpass?
1
z
(a) H z  
1  89 z 1
, z 
1  89 z 1
(b) H z  
, z 
16 1
64  2
1  9 z  81 z
1
(c)H z  
, z 
64  2
1  81 z
8
9
8
9
8
9
Problem 10.12, p.799 of text
Problem 10.44, p.808 of text
(c)
x1 n   x 2n 
g n   1 x n   ( 1) n x n
2
G  z   1  X  z   X  z 
2

X 1 z  




n


x
n
z
 1
n  

n


g
2n
z

n  

 g n z
n  
 n2
 
Gz
n: even
1
2
    
1
1
1
2

X z  X - z2
2
Problem 10.46, p.808 of text
y n  x n  e8 xn  8 ,
H z   1  e z
8
8
e  1
8
8
z

e

,
8
z
z 0
8-th order pole at z=0 and 8 zeros
causal and stable
G z  
 
8
1


G
z
1  e8 z 8
8-th order zero at z=0 and 8 poles
causal and stable
G z  
1
1  e8 z 1
g n  e8nun
g n  g (8) n 
e
8 8n
 en , n  0,8,16,....
0 , else
Problem 10.46, p.808 of text
x[n]
y[n]
+
z
_
8
e
H(z)
8
……
y[n]
x[n]
e
8
z
8
G(z)