ECE
8443
– PatternContinuous
Recognition
EE
3512
– Signals:
and Discrete
LECTURE 31: THE Z-TRANSFORM
AND ITS ROC PROPERTIES
• Objectives:
Relationship to the Laplace Transform
Relationship to the DTFT
Stability and the ROC
ROC Properties
Transform Properties
• Resources:
MIT 6.003: Lecture 22
Wiki: Z-Transform
CNX: Definition of the Z-Transform
CNX: Properties
RW: Properties
MKim: Applications of the Z-Transform
URL:
Definition Based on the Laplace Transform
• The z-Transform is a special case of the Laplace transform and results from
applying the Laplace transform to a discrete-time signal:

X ( s) 
 x(t )e
 st
t  nt
dt  lim

t 0
x(nt ) t 
e 




n  
st  n
x[ n ]
 X(z) 
z

 x[n]z
n
n  
• Let us consider how this transformation maps the s-plane into the z-plane:
 s = j:
z  e st  e jt 
(  j ) t
 et
 s =  + j: z  e
z  e jt  1  j - axis maps to the unit circle
   0 (LHP) maps to the inside of the unit circle
 Recall, if a CT system
is stable, its poles lie
in the left-half plane.
 Hence, a DT system is
stable if its poles are
inside the unit circle.
 The z-Transform behaves
much like the Laplace
transform and can be
applied to difference equations
to produce frequency and time domain responses.
EE 3512: Lecture 31, Slide 1
ROC and the Relationship to the DTFT
• We can derive the DTFT by setting z = rej:
z  re j
X(z) 
 x[n]re 

j  n

n  

 ( x[n]r
n
n  

)e - jn  F x[n]r  n

• The ROC is the region for which:  x[n]r
n


n  
 Depends only on r = |z| just like the ROC in the s-plane for the Laplace
transform depended only on Re{}.
 If the unit circle is in the ROC, then the DTFT, X(ej), exists.
 Example: x[n]  a n u[n] (a right-sided signal)
X(z) 

 a u[n]z
n
n
n  

 a z
n
n
n 0

  (az 1 ) n
n 0
1  (az 1 ) 

?
1
1  az
1
1  az 1
 The ROC is outside a circle of radius a,
and includes the unit circle, which means
its DTFT exists. Note also there is a zero
at z = 0.
 If az 1  1, or, z  a : X(z) 
EE 3512: Lecture 31, Slide 2
Stability and the ROC
• For a > 0: x[n]  a n u[n]  X(z) 
• If the ROC is
outside the unit
circle, the signal
is unstable.
EE 3512: Lecture 31, Slide 3
1
1  az 1
for
x[n]  u[n]
X(z) 
1
1  z 1
for
z  a
• If the ROC
includes the unit
z  1 circle, the signal
is stable.
Stability and the ROC (Cont.)
• For a < 0:
x[n]  a n u[n]  X(z) 
• If the ROC is
outside the unit
circle, the signal
is unstable.
EE 3512: Lecture 31, Slide 4
1
1  az 1
for
x[n]  u[n]
X(z) 
1
1  z 1
for
z  a
• If the ROC
includes the unit
z  1 circle, the signal
is stable.
More on ROC
• Example: x[n]   a n u[ n  1] (left  sided signal)
X ( z) 
  a u[n  1]z

n
n  
n
1

   a 1 z

n
n  
1  (a 1 z ) 
 1
?
1  a 1 z
If: a 1 z  1, or, z  a
1
1  a 1 z
1
X(z)  1 


1  a 1 z 1  a 1 z 1  a 1 z
 a 1 z

1  a 1 z
1

1  az 1
z

za
The z-Transform is the same, but the region of convergence is different.
EE 3512: Lecture 31, Slide 5
Stability and the ROC
• For: x[n]  a n u[n  1]  X(z) 
• If the ROC
includes the unit
circle, the signal
is stable.
EE 3512: Lecture 31, Slide 6
1
1  az 1
for
x[n]  u[n]
X(z) 
1
1  z 1
for
z  a
• If the ROC
includes the unit
z  1 circle, the signal
is unstable.
Properties of the ROC
• The ROC is an annular ring in the z-plane centered
about the origin (which is equivalent to a vertical
strip in the s-plane).
• The ROC does not contain any poles (similar to
the Laplace transform).
• If x[n] is of finite duration, then the ROC is the entire
z-plane except possibly z = 0 and/or z = :
X(z) 

 x[n]z
n
n  
DT :
x[n]   [n]
CT :
 X [ z]  1
ROC all z
x[n]   [n  1]  X [ z ]  z 1 ROC z  0
x(t )   (t )
 X ( s)  1
x(t )   (t  T )  X ( s )  e  sT Re( s )  
x[n]   [n  1]  X [ z ]  z ROC z   x(t )   (t  T )  X ( s )  e sT Re( s )  
• If x[n] is a right-sided sequence, and if |z| = r0 is in the ROC, then all finite
values of z for which |z| > r0 are also in the ROC.
• If x[n] is a left-sided sequence, and if |z| = r0 is in the ROC, then all finite values
of z for which |z| < r0 are also in the ROC.
EE 3512: Lecture 31, Slide 7
Properties of the ROC (Cont.)
• If x[n] is a two-sided sequence, and if |z| = r0 is in the ROC, then the ROC
consists of a ring in the z-plane including |z| = r0.
right-sided
• Example: x[n]  b n
left-sided
b0
x[n]  b n u[n]  b  n u[ n]
1
z b
1
1  bz
1
1
b  n u[ n  1] 
z

b
1  b 1 z 1
1
1
1
X ( z) 

 z b
1  bz 1 1  b 1 z 1 b
b n u[n]

EE 3512: Lecture 31, Slide 8
two-sided
Properties of the Z-Transform
• Linearity: ax1[n]  bx2 [n]  aX 1[ z]  bX 2 [ z]
Proof:
Zax1 [n]  bx 2 [n] 

 (ax [n]  bx [n]) z
1
n  
n
2


 ax [n]z
n  
1
n


 bx [n]z
n  
n
2
n
• Time-shift: x[n  n0 ]  z 0 X [ z ]
Proof:
Zx[n  n0 ] 


 x[n  n
n  

 x[m]z
m  
0
]z
m
n


 x[m]z
 ( m  n0 )
m  
z
 n0
z
 n0

 x[m]z
m
 z  n0 X [ z ]
m  
What was the analog for CT signals and the Laplace transform?
dX [ z ]
• Multiplication by n: nx[n]   z
dz
Proof:
X(z) 

 x[n]z
n
n  


dX ( z )
dX ( z )
 n 1
 n  x[n]z
 z
 n  x[n]z  n  Znx[n]
dz
dz
n  
n  
EE 3512: Lecture 31, Slide 9
 X 1[ z ]  X 2 [ z ]
Summary
• Definition of the z-Transform: X(z) 

 x[n]z
n
n  
• Explanation of the Region of Convergence and its relationship to the
existence of the DTFT and stability.
• Properties of the z-Transform:
 Linearity:
 Time-shift:
 Multiplication by n:
ax1[n]  bx2 [n]  aX 1[ z]  bX 2 [ z]
x[n  n0 ]  z  n0 X [ z ]
dX [ z ]
nx[n]   z
dz
• Basic transforms (see Table 7.1) in the textbook.
EE 3512: Lecture 31, Slide 10