Digital Signal Processing Reference
In-Depth Information
2
CONTINUOUS-TIME
SIGNALS AND SYSTEMS
As stated in Chapter 1, engineers must model two distinct physical phenomena.
The first is physical
systems,
which can be modeled by mathematical
equations
. For
example, continuous-time, or analog, systems (systems that contain no sampling)
can be modeled by ordinary differential equations with constant coefficients. An ex-
ample of such a system is a linear electrical circuit. Of course, it is important to re-
member that the accuracy of these mathematical models may vary.
A second physical phenomenon to be modeled is called a
signal
.
Physical sig-
nals
are modeled by
mathematical functions
. One example of a physical signal is the
voltage that is applied to the speaker in a radio. Another example is the tempera-
ture at a designated point in a particular room. This signal is a function of time,
since the temperature varies with time. We can express this temperature as
temperature at a point = u(t),
(2.1)
where has the units of, for example, degrees Celsius. To be more precise in this ex-
ample, the temperature in a room is a function of time and of space. We may designate
a point in a room in the rectangular coordinates
x
,
y
, and
z
. Equation (2.1) then becomes
u(t)
temperature in a room = u(x, y, z, t),
(2.2)
where the point in a room is identified by the three space coordinates
x
,
y
, and
z
.
The signal in (2.1) is a function of one independent variable, whereas the signal in
(2.2) is a function of four independent variables.
In this topic, we limit signals to having one independent variable (except in
Section 12.7, where we briefly discuss images that are functions of two dimensions).
In general, this independent variable will be time
t
. Signals are divided into two nat-
ural categories. The first category to be considered is
continuous-time signals
, or
simply,
continuous signals
. A signal of this type is defined for all values of time. A
continuous-time signal is also called an
analog signal
. A
continuous-time system
is a
system in which only continuous-time signals appear.
There are two types of continuous time signals. A continuous-time signal
x
(
t
) can be a
continuous-amplitude signal,
for which the time-varying amplitude
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