15.2 ANALOG COMMUNICATION SYSTEMS 693From thenoise performancepoint of view, all the AM systems yield the same performance
for the same messages and transmitted powers. Note, however, that the bandwidth of the DSB
system is twice that of the SSB or VSB systems, so its input noise power is twice as large.
Frequency Modulation
So far we have considered AM of the carrier as a means for transmitting the message signal.
AM methods are also known aslinear modulation methods, although conventional AM is not
linear in the strict sense. Other classes of modulation methods are frequency modulation (FM)
and phase modulation (PM). In FM systems the frequency of the carrierfcis changed by the
message signal, and in PM systems the phase of the carrier is changed according to the variations
of the message signal. FM and PM, which are quitenonlinear, are referred to asangle-modulation
methods. Angle modulation is more complex to implement and much more difficult to analyze
because of its inherent nonlinearity. FM and PM systems generally expand the bandwidth such
that the effective bandwidth of the modulated signal is usually many times the bandwidth of the
message signal. The major benefit of these systems is their high degree of noise immunity. Trading
off bandwidth for high-noise immunity, the FM systems are widely used in high-fidelity music
broadcasting and point-to-point communication systems where the transmitter power is rather
limited.
A sinusoid is said to be frequency-modulated if its instantaneous angular frequencyωFM(t)
is a linear function of the message,
ωFM(t)=ωc+kFMf(t) (15.2.9)wherekFMis a constant with units of radians per second per volt whenf(t) is a message-signal
voltage, andωcis the carrier’s nominal angular frequency. Instantaneous phase being the integral
of instantaneous angular frequency, the FM signal can be expressed as
SFM(t)=Accos[
ωct+φc+kFM∫
f(t)dt]
(15.2.10)whereAcis a constant amplitude andφcis an arbitrary constant phase angle. The maximum
amount of deviation thatωFM(t)of Equation (15.2.9) can have from its nominal value is known
aspeak frequency deviation, given by
ω=kFM|f(t)|max (15.2.11)Because FM involves more than just direct frequency translation, spectral analysis and bandwidth
calculations are difficult in general, except for a few message forms. However, practical experience
indicates that the following relations hold for the FM transmission bandwidth:
WFM∼= 2 ( ω+Wf) (15.2.12)known asCarson’s rulefornarrow-bandFM with ω < Wfor
WFM∼= 2 ( ω+ 2 Wf) (15.2.13)forwide-bandFM with ω >> Wf, whereWfis the spectral extent off(t); i.e., the message
signal has a low-pass bandwidthWf. For example, commercial FM broadcasting utilizesWf=
2 π( 15 × 103 )rad/s (corresponding to 15 kHz) and ω= 5 Wfsuch thatωFM= 14 Wf. Because
the performance of narrow-band FM with noise is roughly equivalent to that of AM systems, only
wide-band FM that exhibits a marked improvement will be considered here.
Figure 15.2.11 illustrates the close relationship between FM and PM. Phase modulating
the integral of a message is equivalent to the frequency modulation of the original message,