Digital Signal Processing Reference
In-Depth Information
x
[
n
]
y
[
n
]
x
[
n
]
System
T
i
(
•
)
System
T
(
•
)
Figure 9.27
Identity system.
Invertibility is related to the
inverse
of a system.
Inverse of a System
The inverse of a system
T
is a second system
T
i
that, when cascaded with
T
, yields the
identity system.
The identity system is defined by the equation
y[n] = x[n].
Consider the two sys-
tems of Figure 9.27. System
T
i
is the inverse of system
T
if
y[n] = T
i
[T(x[n])] = x[n].
(9.67)
All physical systems are causal, whether continuous or discrete.
Causal Systems
A system is
causal
if the output at any time is dependent on the input only at the pre-
sent time and in the past.
We have defined the unit delay as a system with an input of
x
[
n
] and an output of
as shown in Figure 9.17. An example of a noncausal system is the unit ad-
vance, which has an input of
x
[
n
] and an output of
x[n - 1],
x[n + 1].
Another example of a
noncausal system is an averaging system, given by
1
3
[x[n - 1] + x[n] + x[n + 1]],
y[n] =
which requires us to know a future value,
x[n + 1],
of the input signal in order to
calculate the current value,
y
[
n
], of the output signal.
We denote the unit advance with the symbol A realizable system that
contains a unit advance is the system of Figure 9.28. We realize it by first delaying a
signal and then advancing it. However, we cannot advance a signal more than it has
been delayed. Although this system may appear to have no application, the proce-
dure is used in filtering signals “off line,” or in nonreal time. If we store a signal in
computer memory, we know “future” values of the signal relative to the value that
D
-1
.
Unit
delay
Unit
delay
Unit
advance
x
[
n
]
x
[
n
1
]
x
[
n
2]
x
[
n
1]
D
1
D
D
Figure 9.28
Realizable system with a unit
advance.
