Digital Signal Processing Reference
In-Depth Information
9.6
PROPERTIES OF DISCRETE-TIME SYSTEMS
In Section 9.5, the Euler integrator and the were given as examples of discrete-
time systems. In this section, we present some of the characteristics and properties of
discrete-time systems.
In the following,
x
[
n
] denotes the input of a system and
y
[
n
] denotes the out-
put. We show this relationship symbolically by the notation
a-filter
x[n] : y[n].
(9.63)
As with continuous-time systems, we read this relation as
x
[
n
] produces
y
[
n
]. Rela-
tionship (9.63) has the same meaning as
[eq(9.59)]
y[n] = T(x[n]).
The definitions to be given are similar to those listed in Section 2.7 for continuous-
time systems.
We first define a system that has memory:
Memory
A system has memory if its output at time
n
0
, y[n
0
],
depends on input values other than
x[n
0
].
Otherwise, the system is memoryless.
For a discrete signal
x
[
n
], time is represented by the discrete increment variable
n
.
An example of a simple memoryless discrete-time system is the equation
y[n] = 5x[n].
A memoryless system is also called a
static system
.
A system with memory is also called a
dynamic system
. An example of a sys-
tem with memory is the Euler integrator of (9.5):
y[n] = y[n - 1] + Hx[n - 1].
Recall from Section 9.1 and (9.8) that this equation can also be expressed as
n- 1
y[n] = H
a
x[k],
(9.64)
k=-
q
and we see that the output depends on all past values of the input.
A second example of a discrete system with memory is one whose output is
the average of the last two values of the input. The difference equation describing
this system is
1
2
[x[n] + x[n - 1]].
y[n] =
(9.65)
