Digital Signal Processing Reference
In-Depth Information
Y
1
(
s
)
H
1
(
s
)
+
X
(
s
)
Y
(
s
)
X
(
s
)
H
(
s
)
= H
1
(
s
) +
H
2
(
s
)
Y
(
s
)
∑
+
H
2
(
s
)
(a)
Y
2
(
s
)
(b)
Fig. 6.13. Parallel configuration for connecting LTIC systems: (a) parallel connection; (b) its equivalent
single system.
6.10.2 Parallel configuration
The parallel configuration between two systems is illustrated in Fig. 6.13(a).
A single input
x
(
t
) is applied simultaneously to the two systems. The overall
output
y
(
t
) is obtained by adding the individual outputs
y
1
(
t
) and
y
2
(
t
)ofthe
two systems. The individual outputs of the two systems are given by
←→
system (1)
y
1
(
t
)
=
x
(
t
)
∗
h
1
(
t
)
Y
1
(
s
)
=
X
(
s
)
H
1
(
s
);
(6.56)
←→
system (2)
y
2
(
t
)
=
x
(
t
)
∗
h
2
(
t
)
Y
2
(
s
)
=
X
(
s
)
H
2
(
s
)
.
(6.57)
Combining the two outputs, the overall output
y
(
t
)isgivenby
←→
y
(
t
)
=
y
1
(
t
)
+
y
2
(
t
)
Y
(
s
)
=
Y
1
(
s
)
+
Y
2
(
s
)
.
(6.58)
Substituting Eqs. (6.56) and (6.57) into the above equation yields
←→
y
(
t
)
=
x
(
t
)
∗
[
h
1
(
t
)
+
h
2
(
t
)]
Y
(
s
)
=
X
(
s
)[
H
1
(
s
)
+
H
2
(
s
)]
.
(6.59)
In other words, the parallel configuration is equivalent to a single LTIC system
with transfer function
←→
h
(
t
)
=
h
1
(
t
)
+
h
2
(
t
)
H
(
s
)
=
H
1
(
s
)
+
H
2
(
s
)
.
(6.60)
The parallel configuration and its equivalent single-stage system are shown in
Fig. 6.13.
6.10.3 Feedback configuration
The feedback connection between two systems is shown in Fig. 6.14(a). In a
feedback system, the overall output
y
(
t
) is applied at the input of the second
system
H
2
(
s
). The output
w
(
t
) of the second system is fed back into the input
of the overall system through an adder. In terms of the applied input
x
(
t
) and
w
(
t
), the output of the adder is given by
E
(
s
)
=
X
(
s
)
−
W
(
s
)
.
(6.61)
The outputs of the two LTIC systems are given by
Y
(
s
)
=
system (1)
E
(
s
)
H
1
(
s
);
(6.62)
system (2)
W
(
s
)
=
Y
(
s
)
H
2
(
s
)
.
(6.63)
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