FREQUENCY
MODULATION
Frequency modulation uses the
instantaneous frequency of a modulating signal (voice, music, data, etc.)
to directly vary the frequency of a carrier signal. Modulation index, b,
is used to describe the ratio of maximum frequency deviation of the
carrier to the maximum frequency deviation of the modulating signal.
Depending
on the modulation index chosen, the carrier and certain sideband
frequencies may actually be suppressed. Zero crossings of the Bessel
functions, J_{n}(b),
occur where the corresponding sideband, n, disappears for a given
modulation index, b.
The composite spectrum for a single tone consists of lines at the carrier
and upper and lower sidebands (of opposite phase), with amplitudes
determined by the Bessel function values at those frequencies.
FM
General Equation

Let the
carrier be x_{c}(t) = X_{c}·cos
(w_{c}t),
and the modulating signal be
x_{m}(t) = b·sin
(w_{m}t) 
Then x(t)
= X_{c}·cos [w_{c}t
+ b·sin
(w_{m}t)] 
Modulation
Index

b
=

Dw
w_{m}

=

maximum
carrier frequency deviation
modulation frequency 

Narrowband
FM (NBFM)

Narrowband
FM is defined as the condition where b
is small enough to make all terms after the first two
in the series expansion of the FM equation negligible.
Narrowband
Approximation: b =
Dw/w_{m}
< 0.2 (could
be as high as 0.5, though)
BW ~ 2w_{m}

Wideband
FM (WBFM)

Wideband
FM is defined as when a significant number of
sidebands have significant amplitudes.
BW ~ 2Dw

Carson's
Rule

J.R.
Carson showed in the 1920's that a good approximation
that for both very small and very large b,
BW ~ 2
(Dw
+ w_{m})
= 2*w_{m}
(1 + b)


