Digital Signal Processing Reference
In-Depth Information
Therefore, system (ii) is stable.
y
[
k
]
=
2
(iii)
x
[
k
−
m
]
.
(2.56)
m
=−
2
The output is expressed as follows:
y
[
k
]
=
x
[
k
−
2]
+
x
[
k
−
1]
+
x
[
k
]
+
x
[
k
+
1]
+
x
[
k
+
2]
.
If
x
[
k
]
≤
B
x
for all
k
, then
y
[
k
]
≤
5
B
x
for all
k
. Therefore, the system is
stable.
k
(iv)
y
[
k
]
=
x
[
m
]
.
(2.57)
m
=−∞
The output is calculated by summing an infinite number of input signal values.
Hence, there is no guarantee that the output will be bounded even if all the input
values are bounded. System (iv) is, therefore, not a stable system.
2.3 Interconnectio
n of systems
In signal processing, complex structures are formed by interconnecting simple
linear and time-invariant systems. In this section, we describe three widely used
configurations for developing complex systems.
2.3.1 Cascaded configuration
As shown in Fig. 2.19(a), a series or cascaded configuration between two sys-
tems is formed by interconnecting the output of the first system
S
1
to the input
of the second system
S
2
. If the interconnected systems
S
1
and
S
2
are linear, it
is straightforward to show that the overall cascaded system is also linear. Like-
wise, if the two systems
S
1
and
S
2
are time-invariant, then the overall cascaded
system is also time-invariant. Another feature of the cascaded configuration
is that the order of the two systems
S
1
and
S
2
may be interchanged without
changing the output response of the overall system.
Example 2.12
Determine the relationship between the overall output and input signals if
the two cascaded systems in Fig. 2.19(a) are specified by the following
relationships:
d
w
d
t
(i)
S
1
:
+
2
w
(
t
)
=
x
(
t
) with
w
(0)
=
0
and
S
2
:
d
y
d
t
+
3
y
(
t
)
=
w
(
t
) with
y
(0)
=
0;
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