Digital Signal Processing Reference
In-Depth Information
10
DISCRETE-TIME LINEAR
TIME-INVARIANT SYSTEMS
In Chapter 3, we developed important properties of continuous-time linear time-
invariant (LTI) systems; those developments are applied to discrete-time LTI sys-
tems in this chapter. In one sense, discrete-time systems are easier to analyze and
design, since difference equations are easier to solve than are differential equations.
In a different sense, discrete-time systems are more difficult to analyze and design,
since the system characteristics are periodic in frequency. (See Section 9.3.)
In Chapter 9, several properties of discrete-time systems were defined. We
now restate two of these properties.
Consider first
time invariance
. We denote a discrete-time system with input
x
[
n
] and output
y
[
n
] by
x[n] : y[n].
(10.1)
This system is time invariant if the only effect of a time shift of the input signal is the
same time shift of the output signal; that is, in (10.1),
x[n - n
0
] : y[n - n
0
],
(10.2)
where is an arbitrary integer.
Next
linearity
is reviewed. For the system of (10.1), suppose that
n
0
x
1
[n] : y
1
[n],
x
2
[n] : y
2
[n].
(10.3)
This system is linear, provided that the principle of superposition applies:
a
1
x
1
[n] + a
2
x
2
[n] : a
1
y
1
[n] + a
2
y
2
[n].
(10.4)
This property applies for all constants and and for all signals and
In this chapter, we consider only discrete-time systems that are both linear and
time invariant. We refer to these systems as discrete-time LTI systems. We have
several reasons for emphasizing these systems:
a
1
a
2
,
x
1
[n]
x
2
[n].
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