563489578934

Sec. 5–6 Phase Modulation and Frequency Modulation 343
for wide frequency deviation (∆F large), the stability of the carrier frequency f c = f 0 is not
very good, so the VCO is incorporated into a PLL arrangement wherein the PLL is locked to
a stable frequency source, such as a crystal oscillator. (See Fig. 5–13.) The frequency divider
is needed to lower the modulation index of the WBFM signal to produce an NBFM signal
(b ≈ 0.2) so that a large discrete carrier term will always be present at frequency f c N to beat
with the crystal oscillator signal and produce the DC control voltage. [See Fig. 5–11a and
Eqs. (5–63) and (5–64).] This DC control voltage holds the VCO on the assigned frequency
with a tolerance determined by the crystal oscillator circuit.
The power spectral density (PSD) of a WBFM signal may be approximated by using
the probability density function (PDF) of the modulating signal. This is reasonable from an
intuitive viewpoint, since the instantaneous frequency varies directly with the modulating
signal voltage for the case of FM [D f (2 p) being the proportionality constant]. If the modulating
signal spends more time at one voltage value than another, the instantaneous
frequency will dwell at the corresponding frequency, and the power spectrum will have a
peak at this frequency. A discussion of the approximation involved—called the quasi-static
approximation—is well documented [Rowe, 1965]. This result is stated in the following
theorem.
THEOREM.
For WBFM signaling, where
1
s(t) = A c cos cv c t + D f m(s)ds d
L
-q
b f = D f max [m(t)]
2pB
7 1
and B is the bandwidth of m(t), the normalized PSD of the WBFM signal is approximated
by
m(t)
Modulating
signal in
Crystal
oscillator
f osc =f c /N
LPF
Figure 5–13
Frequency
divider
N
VCO
Direct method of generating WBFM.
WBFM
signal
out