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)

In the following examples, the carrier frequency is eleven time the modulation
frequency. Red (dashed) lines represent the modulation envelope. Blue (solid) lines represent the modulated carrier.
Modulation Index (b) = 1 
Here, the maximum frequency (f_{max}) causes a maximum deviation of 1*f_{max}
in the carrier. From the modulation index formula:

Modulation Index (b) = 5 
Here, the maximum frequency (f_{max}) causes a maximum deviation of 5*f_{max}
in the carrier. From the modulation index formula:

Modulation Index (b) = 25 
Here, the maximum frequency (f_{max}) causes a maximum deviation of 25*f_{max}
in the carrier. From the modulation index formula:

Note: FM waveforms created with MathCAD 4.0 software.

