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Sec. 5–6 Phase Modulation and Frequency Modulation 341
where b is either the phase modulation index or the frequency modulation index and B is the
bandwidth of the modulating signal (which is f m for sinusoidal modulation). † This formula
gives a rule-of-thumb expression for evaluating the transmission bandwidth of PM and FM
signals; it is called Carson’s rule. B T is shown in Fig. 5–11 for various values of b. Carson’s
rule is very important because it gives an easy way to compute the bandwidth of anglemodulated
signals. Computation of the bandwidth using other definitions, such as the 3-dB
bandwidth, can be very difficult, since the spectrum of the FM or PM signal must first be
evaluated. This is a nontrivial task, except for simple cases such as single-tone (sinusoidal)
modulation or unless a digital computer is used to compute the approximate spectrum.
Because the exact spectrum of angle-modulated signals is difficult to evaluate in
general, formulas for the approximation of the spectra are extremely useful. Some relatively
simple approximations may be obtained when the peak phase deviation is small and when the
modulation index is large. These topics are discussed in the sections on narrowband angle
modulation and wideband FM that follow.
Narrowband Angle Modulation
When u (t) is restricted to a small value, say, |u (t)| 6 0.2 rad, the complex envelope g(t) = A c
e ju may be approximated by a Taylor’s series in which only the first two terms are used. Thus,
because e x ≈ 1 + x for |x| 1,
g(t) L A c [1 + ju(t)]
(5–62)
Using this approximation in Eq. (4–9) or Eq. (5–1), we obtain the expression for a narrowband
angle-modulated signal:
s(t) = A c cos v c t - A c u(t) sin v c t
(5–63)
discrete
carrier term
sideband term
This result indicates that a narrowband angle-modulated signal consists of two terms: a
discrete carrier component (which does not change with the modulating signal) and a sideband
term. This signal is similar to an AM-type signal, except that the sideband term is 90
out of phase with the discrete carrier term. The narrowband signal can be generated by
using a balanced modulator (multiplier), as shown in Fig. 5–12a for the case of narrowband
frequency modulation (NBFM). Furthermore, wideband frequency modulation
(WBFM) may be generated from the NBFM signal by using frequency multiplication, as
shown in Fig. 5–12b. Limiter circuits are needed to suppress the incidental AM [which is
21 + u 2 (t) as the result of the approximation of Eq. (5–62)] that is present in the NBFM
signal. This method of generating WBFM is called the Armstrong method or indirect
method.
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† For the case of FM (not PM) with 2 6 B 6 10, Carson’s rule, Eq. (5–61), actually underestimates B T somewhat.
In this case, a better approximation is B T = 2(b + 2)B. Also, if the modulation signal contains discontinuities,
such as square-wave modulation, both formulas may not be too accurate, and the B T should be evaluated by examining
the spectrum of the angle-modulated signal. However, to avoid confusion in computing B T , we will assume that
Eq. (5–61) is approximately correct for all cases.