Digital Signal Processing Reference
In-Depth Information
Table 2.2. Examples of causal and non-causal systems
The CT and DT systems are represented using their input-output relationships. Note that all systems in the table
have memory.
CT systems
DT systems
Causal
Non-causal
Causal
Non-causal
y
(
t
)
=
x
(
t
−
5)
y
(
t
)
=
x
(
t
+
2)
y
[
k
]
=
3
x
[
k
−
1]
+
7
y
[
k
]
=
x
[
k
+
3]
y
(
t
)
=
sin
x
(
t
−
4)
+
3
y
(
t
)
=
sin
x
(
t
+
4)
+
3
y
[
k
]
=
sin(
x
[
k
−
4])
+
3
y
[
k
]
=
sin(
x
[
k
+
4])
+
3
y
(
t
)
=
e
x
(
t
−
2)
y
(
t
)
=
x
(2
t
)
y
[
k
]
=
e
x
[
k
−
2]
y
[
k
]
=
x
[2
k
]
y
(
t
)
=
x
2
(
t
−
2)
y
(
t
)
=
x
(
t
/
2)
y
[
k
]
=
x
2
[
k
−
5]
y
[
k
]
=
x
[
k
/
2]
y
(
t
)
=
x
(
t
−
2)
+
x
(
t
−
5)
y
(
t
)
=
x
(
t
−
2)
+
x
(
t
+
2)
y
[
k
]
=
x
[
k
−
2]
+
x
[
k
−
8]
y
[
k
]
=
x
[
k
+
2]
+
x
[
k
−
8]
y
(
t
)
y
[
k
]
CT
system
inverse
system
DT
system
inverse
system
x
(
t
)
x
(
t
)
x
[
k
]
x
[
k
]
(a)
(b)
Fig. 2.17. Invertible systems.
(a) Inverse of a CT system.
(b) Inverse of a DT system.
Causality is a required condition for the system to be physically realizable. A
non-causal system is a predictive system and cannot be implemented physically.
Table 2.2 presents examples of causal and non-causal systems in CT and DT
domains.
2.2.5 Invertible and non-invertible systems
A CT system is
invertible
if the input signal
x
(
t
) can be uniquely determined
from the output
y
(
t
) produced in response to
x
(
t
) for all time
t
∈
(
−∞, ∞
).
Similarly, a DT system is called
invertible
if, given an arbitrary output response
y
[
k
] of the system for
k
∈
(
−∞, ∞
), the corresponding input signal
x
[
k
] can be
uniquely determined for all time
k
∈
(
−∞, ∞
). To be invertible, two different
inputs cannot produce the same output since, in such cases, the input signal
cannot be uniquely determined from the output signal.
A direct consequence of the invertibility property is the determination of a
second system that restores the original input. A system is said to be invertible
if the input to the system can be recovered by applying the output of the original
system as input to a second system. The second system is called the inverse
of the original system. The relationship between the original system and its
inverse is shown in Fig. 2.17.
Example 2.8
Determine if the following CT systems are invertible.
(i) Incrementally linear system:
y
(
t
)
=
3
x
(
t
)
+
5
.
Search WWH ::
Custom Search