Digital Signal Processing Reference
In-Depth Information
For an invertible system, the system input can be determined uniquely from its out-
put. As an example, consider the squaring circuit mentioned earlier, which is de-
scribed by
y(t) = x
2
(t) Q x(t) =;
2
y(t).
(2.62)
Suppose that the output of this circuit is constant at 4 V. The input could be either
or Hence, this system is not invertible. An example of an invertible sys-
tem is an ideal amplifier of gain
K
:
+2 V
-2 V.
1
K
y(t).
y(t) = Kx(t) Q x(t) =
(2.63)
A definition related to invertibility is the
inverse
of a system. Before giving
this definition, we define the
identity system
to be that system for which the output
is equal to its input. An example of an identity system is an ideal amplifier with a
gain of unity. We now define the inverse of a system.
Inverse of a System
The inverse of a system (denoted by
T
) is a second system (denoted by
T
i
) that, when
cascaded with the system
T
, yields the identity system.
The notation for an inverse transformation is then
y(t) = T[x(t)] Q x(t) = T
i
[y(t)].
(2.64)
T
i
[
#
]
Hence, denotes the inverse transformation. If a system is invertible, we can
find the unique
x
(
t
) for each
y
(
t
) in (2.64). We illustrate an invertible system in
Figure 2.39. In this figure,
z(t) = T
2
[y(t)] = T
i1
(T
1
[x(t)]) = x(t),
(2.65)
T
2
(
#
) = T
i1
(
#
),
T
1
(
#
).
where the inverse of system
A simple example of the inverse of a system is an ideal amplifier with gain 5.
Note that we can obtain the inverse system by solving for
x
(
t
) in terms of
y
(
t
):
y(t) = T[x(t)] = 5x(t) Q x(t) = T
i
[y(t)] = 0.2y(t).
(2.66)
The inverse system is an ideal amplifier with gain 0.2.
A transducer is a physical device used in the measurement of physical vari-
ables. For example, a
thermistor
(a temperature-sensitive resistor) is one device
used to measure temperature. To determine the temperature, we measure the resis-
tance of a thermistor and use the known temperature-resistance characteristic of
that thermistor to determine the temperature.
Inverse
of
system 1
y
(
t
)
x
(
t
)
z
(
t
)
x
(
t
)
System
1
Figure 2.39
Inverse system.
