082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

19.8 Bessel’s equation and Bessel functions 599
order to explain how FM works it is helpful to start with the electronic process by
which the FM signal is generated and to then derive the properties of the modulated
signal mathematically.
The FM signal is usually produced by a device called a voltage controlled oscillator
(VCO). The VCO generates a sinusoidal signal on its output, the frequency
of which is controlled by a voltage on its signal input. If we were to connect a d.c.
voltage to the input then we would observe a simple sine wave on the output. The
frequencyofthesinewavecouldbechangedbychangingthed.c.voltage.Inananalogueradiobroadcastsystemhoweverwewishtotransmitatime-varyingsignaland
this, rather thanad.c. voltage, isused as inputtothe VCO.
Consider the case of a sinusoidal modulating signal, sin(2πf m
t), that we wish
totransmitsuchasthatshowninFigure19.20.Here f m
isthefrequencyofthismodulating
signal. Another sinusoidal signal, termed the carrier, is used to carry the
modulatedsignal.Thefrequencyofthecarrierisdenoted f c
andisthefrequencyyou
wouldtunetoonaradiodialtoreceivetheradiostation.Letussuppose,forexample,
that the carrier isAcos(2πf c
t) as illustratedinFigure 19.21.
VCO input voltage
1
–1
Figure19.20
A sinusoidalmodulating signal: the signal we wishto transmit.
Time, t
Carrier signal
A
–A
Time, t
Figure19.21
A sinusoidalcarrier signal.
Itis usefultodefine a term, β, which isknown as themodulation index:
β = f
f m
where f isthemaximumchangeinthefrequencyoftheoutputfromtheVCOwhen
the modulating signal is applied. In other words the largest instantaneous frequency
of the carrier is f c
+ f and the smallest is f c
− f. It may seem intuitive that the
frequencieswithintheoutputsignalwouldbebetween f c
+f and f c
−f,butthe
situation is actually more complex. The changing signal produces a different set of
output frequencies. Consider the following analysis.
We can describe this modulation process mathematically as
v(t) =Acos [ 2πf c
t + βsin(2πf m
t) ]
➔