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66
Signals and Spectra Chap. 2
where w 1f2 = [R w (t)]. This is called the Wiener–Khintchine theorem. The theorem,
along with properties for R(τ) and ( f ), are developed in Chapter 6.
In summary, the PSD can be evaluated by either of the following two methods:
1. Direct method, by using the definition, Eq. (2–66). †
2. Indirect method, by first evaluating the autocorrelation function and then taking the
Fourier transform: w 1f2 = [R w (t)].
Furthermore, the total average normalized power for the waveform w(t) can be evaluated by
using any of the four techniques embedded in the following equation:
P = 8w 2 2
(t)9 = W rms
q
= P w (f) df = R w (0)
L
-q
(2–70)
Example 2–11 PSD OF A SINUSOID
Let
The PSD will be evaluated using the indirect method. The autocorrelation is
Using a trigonometric identity, from Appendix A we obtain
R w (t) = A2
2 cos v 0t lim
T: q
which reduces to
R w (t) = 8w(t)w(t + t)9
= lim
T: q
1
T L
1
T L
w(t) = A sin v 0 t
T/2
-T/2
T/2
-T/2
A 2 sin v 0 t sin v 0 (t + t) dt
dt - A2
2
lim
T: q
1
T L
T/2
-T/2
cos(2v 0 t + v 0 t) dt
R w (t) = A2
2 cos v 0t
(2–71)
The PSD is then
w (f) = c A2
2 cos v 0t d = A2
4 [d(f - f 0) + d(f + f 0 )]
(2–72)
as shown in Fig. 2–9. Eq. (2–72) is plotted by running Example2_11.m. The PSD may be compared
to the “voltage” spectrum for a sinusoid found in Example 2–5 and shown in Fig. 2–4.
The average normalized power may be obtained by using Eq. (2–67):
P =
L
q
-q
A 2
4 [d(f - f 0) + d1f + f 0 2] df = A2
2
(2–73)
† The direct method is usually more difficult than the indirect method.