Digital Signal Processing Reference
In-Depth Information
In this section, we consider two of the many engineering applications of the Fourier
transform. We define the
frequency response
of a system as the Fourier transform
of the
impulse response
. We also use the term
transfer function
as a pseudonym for
the system
frequency response function
because it describes the relationship be-
tween the input signal and the output signal of a system in the frequency domain.
A more complete discussion of applications of the Fourier transform is provided in
Chapter 6.
5.6
ENERGY AND POWER DENSITY SPECTRA
In this section, we define and show application for the
energy spectral density func-
tion
and the
power spectral density function
. These two functions are used to deter-
mine the energy distribution of an energy signal or the power distribution of a
power signal, in the frequency spectrum. Knowledge of the energy or power distrib-
ution of a signal can be quite valuable in the analysis and design of communication
systems, for example.
An energy signal is defined in Section 5.1 as a waveform,
f(t),
for which
q
-
q
ƒ f(t) ƒ
2
dt 6
q
,
[eq(5.5)]
E =
L
where
E
is the energy associated with the signal. It is noted in that section that en-
ergy signals generally include aperiodic signals that have a finite time duration and
signals that approach zero asymptotically as
t
approaches infinity.
If the signal is written in terms of its Fourier transform,
q
1
2p
L
F(v)e
jvt
dv,
f(t) =
-
q
its energy equation can be rewritten as
q
q
1
2p
L
B
F(v)e
jvt
dv
R
E =
L
f(t)
dt.
-
q
-
q
The order of integration can be rearranged so that
q
q
1
2p
L
B
f(t)e
jvt
dt
R
E =
F(v)
dv.
L
-
q
-
q
