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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, Jn(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 xc(t) = Xccos (wct),
and the modulating signal be
xm(t) =
bsin (wmt)
Then x(t) = Xccos [wct + bsin (wmt)]

Modulation Index

 b = Dw wm = 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/wm < 0.2 (could be as high as 0.5, though)

BW ~ 2wm

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 + wm) = 2*wm (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 (fmax) causes a maximum deviation of 1*fmax in the carrier. From the modulation index formula:

 b = 1 1 = 1

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

 b = 5 1 = 5

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

 b = 25  1 = 25

BESSEL FUNCTIONS

Bessel functions of the first kind are shown in the graph below. In frequency modulation, the carrier and sideband frequencies disappear when the modulation index (
b) is equal to a zero crossing of the function for the nth sideband. For example, the carrier (0th sideband) disappears when the Jn(0,b) plot equals zero. The 1st sideband disappears when the Jn(1,b) plot equals zero.

 Sample of Bessel Function Zero Crossings J0(b) J1(b) J2(b) J3(b) J4(b) J5(b) J6(b) b = 2.40 b = 5.49 b = 8.65 b = 11.8 b = 3.83 b = 7.05 b = 10.2 b = 5.14 b = 8.42 b = 11.6 b = 6.38 b = 8.42 b = 11.6 b = 7.59 b = 11.1 b = 14.4 b = 8.77 b = 12.3 b = 15.7 b = 9.94 b = 13.6 b = 17.0 