Digital Signal Processing Reference
In-Depth Information
Any useful signal
f(t)
that meets the condition
q
2
dt 6
q
E =
L
ƒ f(t) ƒ
(5.5)
-
q
is absolutely integrable. In (5.5),
E
is the energy associated with the signal, which
can be seen if we consider the signal
f(t)
to be the voltage across a
1-Æ
resistor. The
power delivered by
f(t)
is then
2
/R = ƒ f(t) ƒ
2
,
p(t) = ƒ f(t) ƒ
and the integral of power over time is energy.
A signal that meets the condition of containing finite energy is known as an
energy signal
. Energy signals generally include nonperiodic signals that have a finite
time duration (such as the rectangular function, which is considered in several ex-
amples) and signals that approach zero asymptotically so that
f(t)
approaches zero
as
t
approaches infinity.
An example of a mathematical function that does not have a Fourier trans-
form, because it does not meet the Dirichlet condition of absolute integrability, is
However, the frequently encountered signal
f(t) = e
-t
.
f(t) = e
-t
u(t)
does meet
the Dirichlet conditions and does have a Fourier transform.
We have mentioned the use of Fourier transforms of useful signals that do not
meet the Dirichlet conditions. Many signals of interest to electrical engineers are
not energy signals and are, therefore, not absolutely integrable. These include the
unit step function, the signum function, and all periodic functions. It can be shown
that signals that have infinite energy, but contain a finite amount of power, and
meet the other Dirichlet conditions do have valid Fourier transforms [1-3].
A signal that meets the condition
T/2
-T/2
ƒ f(t) ƒ
1
T
L
2
dt 6
q
P =
lim
T:
q
(5.6)
is called a
power signal
.
The power computed by Equation (5.6) is called
normalized average power
. In
electrical signal analysis, normalized power is defined as the power that a signal de-
livers to a
1Æ
load. By the normalized power definition, the signal
f(t)
in (5.6) can
represent either voltage or current as an electrical signal, because
2
2
P = V
rms
= I
rms
when R = 1Æ.
The concept of normalized average power is often used to describe the strength of
communication signals.
The step function, the signum function, and periodic functions that meet the
Dirichlet conditions except for absolute integrability are power signals. We will see
that the Fourier transforms that we derive for power signals contain impulse func-
tions in the frequency domain. This is a general characteristic of power signals and
