563489578934

Sec. 2–5 Fourier Series 81
The PSD is then
q
(f) = [R(t)] = c a |c n |2 e jnv 0 t d
q
= a |c n | 2 [e jnv0t q
] = a |c n | 2 d(f - nf 0 )
-q
-q
(2–128)
Equation (2–126) not only gives a way to evaluate the PSD for periodic waveforms, but also
can be used to evaluate the bandwidths of the waveforms. For example, the frequency interval
in which 90% of the waveform power was concentrated could be found.
-q
Example 2–16 PSD FOR A SQUARE WAVE
The PSD for the periodic square wave shown in Fig. 2–12a will be found. Because the waveform
is periodic, Eq. (2–126) can be used to evaluate the PSD. Consequently this problem becomes one
of evaluating the FS coefficients. Furthermore, the FS coefficients for a square wave are given by
Eq. (2–120). Thus,
q
(f) = a a A 2
(2–129)
2 b a sin(np/2) 2
b d(f - nf 0 )
np/2
-q
This PSD is shown by the solid lines in Fig. 2–13 where the delta functions (which have
infinite amplitudes) are represented with vertical lines that have length equal to the weight
(i.e., area) of the corresponding delta function. Also, see Example2_16.m for a plot of the PSD.
–––
A 2
4
–––
A 2
8
– 4f 0 – 3f 0 – 2f 0 – f 0 0 f 0 2f 0 3f 0 4f 0
Figure 2–13 PSD for a square wave used in Example 2–16.