Digital Signal Processing Reference
In-Depth Information
3.
The system is
causal,
because the output does not depend on the input at a future time.
4.
The system is
stable,
the output is bounded for all bounded inputs, because the multipli-
er sin (
2t
) has a maximum value of 1. If
ƒ x(t) ƒ F M, ƒ y(t) ƒ F M
, also.
5.
The system is
time varying
. From (2.73) and (2.74),
y
d
(t) = y(t)
x(t - t
0
)
= sin2tx(t - t
0
)
and
y(t)
t - t
0
= sin2(t - t
0
)x(t - t
0
).
6.
The system is
linear,
since
a
1
x
1
(t) + a
2
x
2
(t) : sin2t[a
1
x
1
(t) + a
2
x
2
(t)]
= a
1
sin 2tx
1
(t) + a
2
sin 2tx
2
(t)
= a
1
y
1
(t) + a
2
y
2
(t).
■
Testing for linearity by using superposition
EXAMPLE 2.20
As a final example, consider the system described by the equation a linear ampli-
fier. This system is easily shown to be linear by the use of superposition. However, the system
an amplifier that adds a dc component, is nonlinear. By superposition,
y(t) = 3x(t),
y(t) = [3x(t) + 1.5],
y(t) = 3[a
1
x
1
(t) + a
2
x
2
(t)] + 1.5 Z a
1
y
1
(t) + a
2
y
2
(t).
This system is not linear, because a part of the output signal is independent of the input
signal.
■
Analysis similar to that of Example 2.20 shows that systems with nonzero ini-
tial conditions are not linear. They can be analyzed as linear systems only if the
nonzero initial conditions are treated as inputs to the system.
In this section, several important properties of continuous-time systems have
been defined; these properties allow us to classify systems. For example, probably
the most important general system properties are linearity and time invariance,
since the analysis and design procedures for LTI systems are simplest and easiest to
apply. We continually refer back to these general system properties for the remain-
der of this topic.
In this chapter, we introduce continuous-time signals and systems, with emphasis
placed on the modeling of signals and the properties of systems.
First, three transformations of the independent variable time are defined: rever-
sal, scaling, and shifting. Next, the same three transformations are defined with respect
to the amplitude of signals. A general procedure is developed for handling all six trans-
formations. These transformations are important with respect to time signals, and as
we will see in Chapter 5, they are equally important as transformations in frequency.
