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Sec. 5–6 Phase Modulation and Frequency Modulation 345
where V p is the peak voltage of the triangular waveform. Substituting this equation into
Eq. (5–66), we obtain
(f) = pA c 2
2D f
1
, 2
2p
Cc 2V (f - fc) 2 6 V p D p
f
s
0, f otherwise
1
+ c , 2
2p
2V
(f - fc) 2 6 V
p D p
f
sS
0, f otherwise
The PSD of the WBFM signal becomes
(f) = c
A 2 c
8¢F , (f c - ¢F) 6 f 6 (f c + ¢F)
s
0, f otherwise
+ c
A 2 c
8¢F , (- f c - ¢F) 6 f 6 (- f c + ¢F)
s
0, f otherwise
(5–68)
where the peak frequency deviation is
¢F = D fV p
2p
(5–69)
This PSD is plotted in Fig. 5–14c. It is recognized that this result is an approximation to the
actual PSD. In this example, the modulation is periodic with period Tm, so that the actual
spectrum is a line spectrum with the delta functions spaced f m = 1Tm hertz apart from each
other (like the spacing shown in Fig. 5–11 for the case of sinusoidal modulation). This
approximation gives us the approximate envelope of the line spectrum. (If the modulation
had been nonperiodic, the exact spectrum would be continuous and thus would not be a line
spectrum.)
In the other example, which uses sinusoidal (periodic) modulation, the exact PSD is
known to contain line components with weights [A c J n (b)] 2 2 located at frequencies f = f c +
nf m . (See Fig. 5–11.) For the case of a high index, the envelope of these weights approximates
the PDF of a sinusoid as given in Appendix B.
The approximate spectra for wideband digital PM signals can also be evaluated by
Eq. (5–66). For digital modulation with rectangular pulse shapes, the PDF of the modulation
voltage, m(t), consists of delta functions located at the discrete voltage levels. Consequently,
the PSD for wideband digital FM is approximated by delta functions. This approximation is
illustrated in Fig. 5–15 for wideband binary frequency shift keying (FSK).