Digital Signal Processing Reference
In-Depth Information
which describes the distribution of the energy in an
energy signal
in the frequency
spectrum, and the
power spectral density function
1
T
ƒF
T
(v) ƒ
2
,
[eq(5.53)]
f
(v) K
lim
T:
q
which describes the distribution of the power in a
power signal
in the frequency
spectrum.
The input-output relationship for an
energy signal
transmitted through a lin-
ear system is determined to be
2
[eq(5.59)]
g
(v) = ƒH(v) ƒ
f
(v),
where
f
(v)
is the energy spectral density of the input signal and
g
(v)
is the
energy spectral density of the output signal.
The input-output relationship for a power signal transmitted through a linear
system is shown to be
2
[eq(5.60)]
g
(v) = ƒH(v) ƒ
f
(v),
where
f
(v)
is the power spectral density of the input signal and
g
(v)
is the power
spectral density of the output signal.
In this chapter, we define the Fourier transform (5.1) and the inverse Fourier trans-
form (5.2).
The sufficient conditions for the existence of the integral (5.1) are called the
Dirichlet conditions. In general, the Fourier transform of exists if it is reason-
ably well behaved (if we could draw a picture of it) and if it is absolutely integrable.
These conditions are sufficient, but not necessary. It is shown that many practical
signals that do not fit these conditions do, in fact, have Fourier transforms.
The Fourier transform of a time-domain signal is called the frequency spec-
trum of the signal. The frequency spectrum is often plotted in two parts: is
plotted as the
magnitude spectrum,
and is plotted as the
phase spectrum
.
A third representation of the frequency spectrum is a plot of
f(t)
ƒF(v) ƒ
arg[F(v)]
2
,
ƒF(v) ƒ
which is called
the
energy spectrum
.
Several useful properties of the Fourier transform are introduced and are listed
in Table 5.1. Fourier transforms of several time-domain functions are derived and
are listed in Table 5.2.
The Fourier transform of the
impulse response
of a linear system is shown to
be the system's
frequency response
, which is also the
transfer function
of the system in
the frequency domain. This leads to the important result that if a system with transfer
