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Frequency Modulation

Bessel function chart - RF CafeFrequency modulation uses the instantaneous frequency of a modulating signal (voice, music, data, etc.) to directly vary the frequency of a carrier signal. Modulation index, β, is used to describe the ratio of maximum frequency deviation of the carrier to the maximum frequency deviation of the modulating signal. The concept was pioneered by Edwin H. Armstrong in the late 1920s and patented in the early 1930s.

Depending on the modulation index chosen, the carrier and certain sideband frequencies may actually be suppressed. Zero crossings of the Bessel functions, Jn(β), occur where the corresponding sideband, n, disappears for a given modulation index, β. 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) = Xc·cos (Ωct),
and the modulating signal be xm(t) = β·sin (Ωmt)

Then x(t) = Xc·cos [Ωct + β·sin (Ωmt)]

Modulation Index

FM Modulation Index formula - RF Cafe

Narrowband FM (NBFM)

Narrowband FM is defined as the condition where β is small enough to make all terms after the first two in the series expansion of the FM equation negligible.

Narrowband Approximation: β = Δω/Ωm < 0.2 (could be as high as 0.5, though)

BW ~ 2ωm

Wideband FM (WBFM)

Wideband FM is defined as when a significant number of sidebands have significant amplitudes.

BW ~ 2Δω

Carson's Rule

J.R. Carson showed in the 1920's that a good approximation that for both very small and very large β,

BW ~ 2 (Δω + Ωm)) = 2*Ωm (1 + β)

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 (β) = 1

FM modulation index = 1 graph - RF Cafe

Here, the maximum frequency (fmax) causes a maximum deviation of 1*fmax in the carrier. From the modulation index formula:

 - RF Cafe

Modulation Index (β) = 5

FM modulation index = 5 graph - RF Cafe

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

 - RF Cafe

Modulation Index (β) = 25

FM modulation index = 25 graph - RF Cafe

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

 - RF Cafe

Note: FM waveforms created with MathCAD 4.0 software.

Related Pages on RF Cafe
- Spectral (Frequency) Inversion
- RF System Cascade Budget Calculation
- Example of System Cascade
- Frequency Modulation
- Frequency Modulation Fundamentals
- Mixer & Frequency Discriminators EW&RH
- Mixer Image Frequency
- Frequency Conversion Calculator
- A Graphical Approach to Mixer Spurious Analysis
- Frequency Mixer, Converter, Multiplier, Modulator Vendors
RF Cascade Workbook 2018 by RF Cafe Rohde & Schwarz FPC1500 Spectrum Analyzer - RF Cafe
QuinStar - RF Cafe Axiom Test Equipment - RF Cafe
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Webmaster:
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    BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The Internet was still largely an unknown entity at the time and not much was available in the form of WYSIWYG ...

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