Digital Signal Processing Reference
In-Depth Information
1.
Many physical systems can be modeled accurately as LTI systems. For ex-
ample, most digital filters are designed to be both linear and time invariant.
2.
We can solve the equations that model LTI systems, for both continuous-
time and discrete-time systems. No general procedures exist for the solu-
tion of the describing equations of non-LTI systems.
3.
Much information is available for both the analysis and the design of LTI
systems. This is especially true for the design of LTI digital filters.
The developments in this chapter are analogous to those of Chapter 3 for
continuous-time systems. Some of the developments are the same as those of
Chapter 3, while others differ significantly. Because almost all signals and systems
of this chapter are discrete time, we often omit this term in the descriptions.
10.1
IMPULSE REPRESENTATION OF DISCRETE-TIME SIGNALS
In this section, a relation is developed that expresses a general signal
x
[
n
] as a func-
tion of impulse functions. This relation is useful in deriving properties of LTI
discrete-time systems.
Recall the definition of the discrete-time impulse function (also called the unit
sample function):
1,
n = n
0
b
d[n - n
0
] =
n Z n
0
.
(10.5)
0,
An impulse function has a value of unity when its argument is zero; otherwise, its
value is zero. From this definition, we see that
x[n]d[n - n
0
] = x[n
0
]d[n - n
0
].
Consider the signal
x
[
n
] in Figure 10.1(a). For simplicity, this signal has only
three nonzero values. We define the following signal, using (10.5):
x[-1],
n =-1
b
x
-1
[n] = x[n]d[n + 1] = x[-1]d[n + 1] =
n Z-1
.
0,
In a like manner, we define the signals
x[0],
n = 0
b
x
0
[n] = x[n]d[n] = x[0]d[n] =
n Z 0
;
0,
x[1],
n = 1
b
x
1
[n] = x[n]d[n - 1] = x[1]d[n - 1] =
n Z 1
.
0,
