Digital Signal Processing Reference
In-Depth Information
damper system and the modeling of the human immune system are presented
in Chapter 8.
6.10 Block diagram
representations
In the preceding discussion, we considered relatively elementary LTIC systems
described by linear, constant-coefficient differential equations. Most practi-
cal structures are more complex, consisting of a combination of several LTIC
systems. In this section, we analyze the cascaded, parallel, and feedback con-
figurations used to synthesize larger systems.
6.10.1 Cascaded configuration
A series or cascaded configuration between two systems is illustrated in
Fig. 6.12(a). The output of the first system
H
1
(
s
) is applied as input to the
second system
H
2
(
s
). Assuming that the Laplace transform of the input
x
(
t
),
applied to the first system, is given by
X
(
s
), the Laplace transform
W
(
s
)ofthe
output
w
(
t
) of the first system is given by
←→
w
(
t
)
=
x
(
t
)
∗
h
1
(
t
)
W
(
s
)
=
X
(
s
)
H
1
(
s
)
.
(6.52)
The resulting signal
w
(
t
) is applied as input to the second system
H
2
(
s
), which
leads to the following overall output:
←→
y
(
t
)
=
w
(
t
)
∗
h
2
(
t
)
Y
(
s
)
=
W
(
s
)
H
2
(
s
)
.
(6.53)
Substituting the value of
w
(
t
) from Eq. (6.52), Eq. (6.53) reduces to
←→
y
(
t
)
=
x
(
t
)
∗
h
1
(
t
)
∗
h
2
(
t
)
Y
(
s
)
=
W
(
s
)
H
1
(
s
)
H
2
(
s
)
.
(6.54)
In other words, the cascaded configuration is equivalent to a single LTIC system
with transfer function
Fig. 6.12. Cascaded
configuration for connecting
LTIC systems: (a) cascaded
connection; (b) its equivalent
single system.
←→
h
(
t
)
=
h
1
(
t
)
∗
h
2
(
t
)
H
(
s
)
=
H
1
(
s
)
H
2
(
s
)
.
(6.55)
The system
H
(
s
) equivalent to the cascaded configuration is shown in
Fig. 6.12(b).
W
(
s
)
Y
(
s
)
H
(
s
)
= H
1
(
s
)
H
2
(
s
)
X
(
s
)
H
1
(
s
)
H
2
(
s
)
X
(
s
)
Y
(
s
)
(a)
(b)
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