Digital Signal Processing Reference
In-Depth Information
w
(
t
)
x
(
t
)
S
1
S
2
y
(
t
)
(a)
y
1
(
t
)
+
S
1
x
(
t
)
y
(
t
)
∑
S
1
+
−
w
(
t
)
x
(
t
)
∑
y
(
t
)
+
y
2
(
t
)
S
2
S
2
(b)
(c)
(ii)
S
1
:
w
[
k
]
−
w
[
k
−
1]
=
x
[
k
] with
w
[0]
=
0
and
S
2
:
y
[
k
]
−
2
y
[
k
−
1]
=
w
[
k
] with
y
[0]
=
0.
Fig. 2.19. Interconnection of
systems: (a) cascaded
configuration; (b) parallel
configuration; (c) feedback
configuration. Although these
diagrams are for CT systems, the
DT systems can be
interconnected to form the three
configurations in exactly the
same manner.
Solution
(i) Differentiating both sides of the differential equation modeling system
S
2
with respect to
t
yields
d
2
y
d
t
2
+
3
d
y
d
t
=
d
w
d
t
S
2
:
.
Multiplying the differential equation modeling system
S
2
by 2 and adding the
result to the above equation yields
d
2
y
d
t
2
+
5
d
y
d
t
+
6
y
(
t
)
=
d
w
d
t
+
2
w
(
t
)
.
x
(
t
)
Based on the differential equation modeling system
S
1
, the right-hand side of
the equation equals
x
(
t
). The overall relationship of the cascaded system is,
therefore, given by
d
2
y
d
t
2
+
5
d
y
d
t
+
6
y
(
t
)
=
x
(
t
)
.
=
p
−
1 in the difference equation modeling system
S
2
(ii) Substituting
k
yields
S
2
:
y
[
p
−
1]
−
2
y
[
p
−
2]
=
w
[
p
−
1]
,
or, in terms of time index
k
,
S
2
:
y
[
k
−
1]
−
2
y
[
k
−
2]
=
w
[
k
−
1]
.
Subtracting the above equation from the original difference equation modeling
system
S
2
yields
S
2
:
y
[
k
]
−
3
y
[
k
−
1]
+
2
y
[
k
−
2]
=
w
[
k
]
−
w
[
k
−
1]
.
x
[
k
]
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