563489578934

50
Signals and Spectra Chap. 2
Interchanging the order of integration on f and t † gives
L
Using Eq. (2–26) produces Eq. (2–40). Parseval’s theorem gives an alternative method for
evaluating the energy by using the frequency domain description instead of the time domain
definition. This leads to the concept of the energy spectral density function.
DEFINITION. The energy spectral density (ESD) is defined for energy waveforms by
(f) = |W(f)| 2
(2–42)
where w(t) 4 W(f). (f) has units of joules per hertz.
Using Eq. (2–41), we see that the total normalized energy is given by the area under the ESD
function:
(2–43)
For power waveforms, a similar function called the power spectral density (PSD) can be
defined. This is developed in Sec. 2–3 and in Chapter 6.
There are many other Fourier transform theorems in addition to Parseval’s theorem.
Some are summarized in Table 2–1. These theorems can be proved by substituting the corresponding
time function into the definition of the Fourier transform and reducing the result to
that given in the rightmost column of the table. For example, the scale-change theorem is
proved by substituting w(at) into Eq. (2–26). We get
q
[w(at)] = w(at)e -j2pft dt
L
Letting t 1 = at, and assuming that a 7 0, we obtain
[w(at)] =
L
q
For a 6 0, this equation becomes
q
-q
q
q
w 1 (t)w * 2 (t) dt = W 1 (f)c w 2 (t)e -j2pft *
dt d df
L L
[w(at)] =
L
q
-q
-q
-q
q
E = (f) df
L
-q
-q
-q
1
a w(t 1)e -j2p(f/a)t 1
dt 1 = 1 a Wa f a b
-1
a w(t 1)e -j2p(f/a)t 1
dt 1 = 1
|a| Wa f a b
† Fubini’s theorem states that the order of integration may be exchanged if all of the integrals are absolutely
convergent. That is, these integrals are finite valued when the integrands are replaced by their absolute values. We
assume that this condition is satisfied.