Digital Signal Processing Reference
In-Depth Information
Fig. 2.10. Incrementally linear
system expressed as a linear
system with an additive offset.
linear
system
+
x
(
t
)
input
signal
y
(
t
)
output
signal
y
zi
(
t
)
linearity of a system. A CT system
y
(
t
)
=
x
2
(
t
) is clearly a non-linear system,
yet it satisfies the zero-input, zero-output property. For the system to be linear,
it must satisfy Eq. (2.30).
Incrementally linear system
In Example 2.1, we proved that the amplifier
y
(
t
)
=
3
x
(
t
) represents a linear system, while the amplifier with additive bias
y
(
t
)
=
3
x
(
t
)
+
5 represents a non-linear system. System
y
(
t
)
=
3
x
(
t
)
+
5 sat-
isfies a different type of linearity. For two different inputs
x
1
(
t
) and
x
2
(
t
), the
respective outputs of system
y
(
t
)
=
3
x
(
t
)
+
5 are given by
input
x
1
(
t
)
y
1
(
t
)
=
3
x
1
(
t
)
+
5;
input
x
2
(
t
)
y
2
(
t
)
=
3
x
2
(
t
)
+
5
.
Calculating the difference on both sides of the above equations yield
y
2
(
t
)
−
y
1
(
t
)
=
3[
x
2
(
t
)
−
x
1
(
t
)]
or
y
(
t
)
=
3
x
(
t
)
.
In other words, the change in the output of system
y
(
t
)
=
3
x
(
t
)
+
5 is linearly
related to the change in the input. Such systems are called incrementally linear
systems.
An incrementally linear system can be expressed as a combination of a linear
system and an adder that adds an offset
y
zi
(
t
) to the output of the linear sys-
tem. The value of offset
y
zi
(
t
) is the zero-input response of the original system.
System S
1
,
y
(
t
)
=
3
x
(
t
)
+
5, for example, can be expressed as a combination of
a linear system S
2
,
y
(
t
)
=
3
x
(
t
), plus an offset given by the zero-input response
of S
1
, which equals
y
zi
(
t
)
=
5. Figure 2.10 illustrates the block diagram repre-
sentation of an incrementally linear system in terms of a linear system and an
additive offset
y
zi
(
t
).
Example 2.2
Consider two DT systems with the following input-output relationships:
(a) differencing system
y
[
k
]
=
3(
x
[
k
]
−
x
[
k
−
2]);
(2.37)
(b) sinusoidal system
y
[
k
]
=
sin(
x
[
k
])
.
(2.38)
Determine if the DT systems are linear.
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