Part A | 9.1
202 Part A Fundamentals
T s
C
R
+
+
–
–
v i (t)
v o (t)
Fig. 9.7 Switched RC circuit modeled by the ZOH system
In effect,
X.z/j z 1 Dexp.sTs/ D X • .s/ :
(9.23)
Variable z
1 is used for describing a signal, or system
as we will see later on. Its significance lies in the fact
that multiplying by z
1 in the transform domain translates to a time shift by one sampling period, T s , in the
discrete-time domain n. Indeed,
y.n/ D x.n 1/ , Y.z/ D z
1 X.z/ :
(9.24)
Furthermore, here are some more of the most important
properties of the Z-transform:
1. Linearity
X 1 .z/ D Zfx 1 .n/g
X 2 .z/ D Zfx 2 .n/g
k 1 ; k 2 2 R
9
> =
> ;
) k 1 X 1 .z/ C k 2 X 2 .z/
D Zfk 1 x 1 .n/ C k 2 x 2 .n/g :
(9.25)
2. Delay or advance by N discrete-time units
Zfx.n/g D X.z/
)
8
ˆ ˆ <
ˆ ˆ :
Zfx.n N/g D z
N X.z/
Zfx.n C N/g D z
N X.z/
N1 X
kD0
z
Nk x.k/
:
(9.26)
3. Initial or final value theorems
Zfx.n/g D X.z/
)
8
<
:
x.n D 0/ D lim
z!1
.X.z//
lim
n!1
.x.n// D lim
z!1
..1 z
1
/X.z//
:
(9.27)
Note: the final value theorem .n ! 1/ holds only
if X.z/ is an analytic function of complex variable z.
4. Discrete convolution theorem
X 1 .z/ D Zfx 1 .n/g
X 2 .z/ D Zfx 2 .n/g
x.n/ D x 1 .k/ ˝ x 2 .k/ ,
C1 X
kDD1
x 1 .k/ x 2 .n k/
9
> > > > =
> > > > ;
) X.z/ D Zfx.n/g D X 1 .z/ X 2 .z/ :
(9.28)
Note: because x.n/ needs to be causal, i. e., can only
depend on values of signals x 1 and x 2 prior or up to
instant n, the bounds of the convolutional summation are set as follows,
.x 1 .k/ ˝ x 2 .k//.n/ D
n
X
kD0
x 1 .k/ x 2 .n k/ :
It is also pointed out here that the commutative and
associative properties hold for the convolution operator applied on two discrete-time signals.
Finally, it is pointed out that the change of variable,
z D exp.sT s /, on the basis of which the Z-transform is
obtained from the Laplace transform of signal x • .t/,
defines a single-valued mapping of complex frequency
s-plane to the plane of, also complex, delay operator
z. Remember that the stability region on the complex
plane s D a C i!, is the left-hand half-plane defined by
inequality ˛ Ä 0. In effect, the stability region on the
delay operator z-plane is defined by inequality
jzj Ä 1 :
(9.29)
Since
z D e
sTs
D e
aTs e
i!Ts
)
(
jzj D e
aTs
arg.z/ D e
i!Ts
: (9.30)
That is, the stability region is the unit disc. Similarly,
the origin of the s-plane is mapped to point z D A C j0
on the z-plane, while the imaginary axis of the s-plane is
wrapped around the unit circle on the z-plane. Finally,
the real s axis is mapped to the real z semi-axis. The
Z-transform of several elementary discrete-time signals
are shown in Table 9.1. In the interest of completeness
and allowing for comparisons a Laplace transform table
is provided in Table 9.2.
9.1.5 Discrete-Time LTI Systems
Difference Equations
A special class of dynamic systems, widely used in
systems and signal theory, is linear time invariant sys-
202 Part A Fundamentals
T s
C
R
+
+
–
–
v i (t)
v o (t)
Fig. 9.7 Switched RC circuit modeled by the ZOH system
In effect,
X.z/j z 1 Dexp.sTs/ D X • .s/ :
(9.23)
Variable z
1 is used for describing a signal, or system
as we will see later on. Its significance lies in the fact
that multiplying by z
1 in the transform domain translates to a time shift by one sampling period, T s , in the
discrete-time domain n. Indeed,
y.n/ D x.n 1/ , Y.z/ D z
1 X.z/ :
(9.24)
Furthermore, here are some more of the most important
properties of the Z-transform:
1. Linearity
X 1 .z/ D Zfx 1 .n/g
X 2 .z/ D Zfx 2 .n/g
k 1 ; k 2 2 R
9
> =
> ;
) k 1 X 1 .z/ C k 2 X 2 .z/
D Zfk 1 x 1 .n/ C k 2 x 2 .n/g :
(9.25)
2. Delay or advance by N discrete-time units
Zfx.n/g D X.z/
)
8
ˆ ˆ <
ˆ ˆ :
Zfx.n N/g D z
N X.z/
Zfx.n C N/g D z
N X.z/
N1 X
kD0
z
Nk x.k/
:
(9.26)
3. Initial or final value theorems
Zfx.n/g D X.z/
)
8
<
:
x.n D 0/ D lim
z!1
.X.z//
lim
n!1
.x.n// D lim
z!1
..1 z
1
/X.z//
:
(9.27)
Note: the final value theorem .n ! 1/ holds only
if X.z/ is an analytic function of complex variable z.
4. Discrete convolution theorem
X 1 .z/ D Zfx 1 .n/g
X 2 .z/ D Zfx 2 .n/g
x.n/ D x 1 .k/ ˝ x 2 .k/ ,
C1 X
kDD1
x 1 .k/ x 2 .n k/
9
> > > > =
> > > > ;
) X.z/ D Zfx.n/g D X 1 .z/ X 2 .z/ :
(9.28)
Note: because x.n/ needs to be causal, i. e., can only
depend on values of signals x 1 and x 2 prior or up to
instant n, the bounds of the convolutional summation are set as follows,
.x 1 .k/ ˝ x 2 .k//.n/ D
n
X
kD0
x 1 .k/ x 2 .n k/ :
It is also pointed out here that the commutative and
associative properties hold for the convolution operator applied on two discrete-time signals.
Finally, it is pointed out that the change of variable,
z D exp.sT s /, on the basis of which the Z-transform is
obtained from the Laplace transform of signal x • .t/,
defines a single-valued mapping of complex frequency
s-plane to the plane of, also complex, delay operator
z. Remember that the stability region on the complex
plane s D a C i!, is the left-hand half-plane defined by
inequality ˛ Ä 0. In effect, the stability region on the
delay operator z-plane is defined by inequality
jzj Ä 1 :
(9.29)
Since
z D e
sTs
D e
aTs e
i!Ts
)
(
jzj D e
aTs
arg.z/ D e
i!Ts
: (9.30)
That is, the stability region is the unit disc. Similarly,
the origin of the s-plane is mapped to point z D A C j0
on the z-plane, while the imaginary axis of the s-plane is
wrapped around the unit circle on the z-plane. Finally,
the real s axis is mapped to the real z semi-axis. The
Z-transform of several elementary discrete-time signals
are shown in Table 9.1. In the interest of completeness
and allowing for comparisons a Laplace transform table
is provided in Table 9.2.
9.1.5 Discrete-Time LTI Systems
Difference Equations
A special class of dynamic systems, widely used in
systems and signal theory, is linear time invariant sys-