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AM, FM, and Digital Modulated Systems Chap. 5
Spectra of Angle-Modulated Signals
Using Eq. (4–12), we find that the spectrum of an angle modulated signal is given by
S(f) = 1 2 [G(f-f c) + G * (-f - f c )]
(5–49)
where
G(f) = [g(t)] = [A c e ju(t) ]
(5–50)
When the spectra for AM, DSB-SC, and SSB were evaluated, we were able to obtain
relatively simple formulas relating S(f ) to M(f ). For angle modulation signaling, this is not
the case, because g(t) is a nonlinear function of m(t). Thus, a general formula relating G( f) to
M( f) cannot be obtained. This, is unfortunate, but it is a fact of life. That is, to evaluate the
spectrum for an angle-modulated signal, Eq. (5–50) must be evaluated on a case-by-case basis
for the particular modulating waveshape of interest. Furthermore, since g(t) is a nonlinear
function of m(t), superposition does not hold, and the FM spectrum for the sum of two modulating
waveshapes is not the same as summing the FM spectra that were obtained when the
individual waveshapes were used.
One example of the spectra obtained for an angle-modulated signal is given in Chapter 2.
(See Example 2–22.) There, a carrier was phase modulated by a square wave where the peakto-peak
phase deviation was 180. In that example, the spectrum was easy to evaluate because
this was the very special case where the PM signal reduces to a DSB-SC signal. In general,
of course, the evaluation of Eq. (5–50) into a closed form is not easy, and one often has to use
numerical techniques to approximate the Fourier transform integral. An example for the case
of a sinusoidal modulating waveshape will now be worked out.
Example 5–6 SPECTRUM OF A PM OR FM SIGNAL WITH SINUSOIDAL
MODULATION
Assume that the modulation on the PM signal is
m p (t) = A m sin v m t
(5–51)
Then
u(t) = b sin v m t
(5–52)
where b p = D p A m = b is the phase modulation index.
The same phase function u (t), as given by Eq. (5–52), could also be obtained if FM were
used, where
m f (t) = A m cos v m t
(5–53)
and b = b f = D f A m v m . The peak frequency deviation would be ∆F = D f A m 2p.
The complex envelope is
g(t) = A c e ju(t) = A c e jb sin v mt
(5–54)