A continuous wave modulation method, which varies frequency in proportion to the amplitude of the modulating signal. Invented by Major Edwin Howard Armstrong, also known for two other key innovations, superheterodyning and regeneration, Frequency Modulation (FM) is the basis for most common wireless communications technologies, from radio and television to wireless LAN.

### History of Frequency Modulation

Armstrong pioneered FM in the 1930s, with initial success in 1933 leading to commercially sponsored pilot broadcasting services in New York and New England in 1939. Initially, the Radio Corporation of America were reluctant to license Armstrong's new technology, claiming that listeners weren't interested in higher quality radio, only in cheaper radio sets. More importantly, RCA had invested significant sums in AM transmitters.

After the second world war (during which FM had been extremely useful to the allied forces, especially since Armstrong had waived his royalty fees), the FCC moved the FM band from 44-50 MHz to its current allocation of 88-108 MHz. This obsoleted all the current installations of FM transmitters and receivers, and halted the progress of FM rollout until Armstrong could redesign his systems to work at the new frequencies.

After a long string of expensive and drawn-out lawsuits brought by Armstrong against RCA, who had been manufacturing FM equipment using his patented methods, Armstrong was left near bankruptcy in 1953. All his patents and licence agreements had expired, his wife had left him, and there seemed little hope of success for his greatest invention. On January 31st, 1954, he put on his hat, scarf, and gloves, and walked out of his thirteenth floor apartment window to his death.

### How FM Works

The process of modulation is the use of a higher frequency carrier signal to transmit a lower frequency information or modulating signal. This is done primarily to allow the information signal to be transmitted further and received more easily. Amplitude Modulation (AM) works by varying the amplitude of the carrier signal in sympathy with the amplitude of the modulation signal; this is very simple to do, but is incredibly power inefficient, having a maximum power efficiency of only 17%, and very poor at tolerating noise.

Frequency Modulation is a more complicated but otherwise far superior method of modulation. Its noise performance is two to three orders of magnitude higher than that of Amplitude Modulation, as wideband FM (Armstrong's real innovation) has an inherently higher signal-to-noise ratio, while also increasing the quality of the information signal due to increased spectrum usage. This larger bandwidth, along with FM stereo transmission, gives the vastly increased sound quality in FM radio as opposed to AM radio.

As one of two very similar variations of Angle Modulation (the other being Phase Modulation), FM varies the instantaneous frequency of the carrier signal in sympathy with the amplitude of the modulation signal. Unfortunately, in order to find the instantaneous frequency of the carrier signal, we have to do some maths.

### Derivation of FM Equations

Consider the carrier signal with amplitude A_{c}, frequency f_{c}, and phase offset φ:

f_{c}(t) = A_{c} cos(2πf_{c}t + φ)

The parameter of the `cos` function is the phase angle of the signal, which is considered as a linear function of time:

θ(t) = 2πf_{c}t + φ

Then the instantaneous frequency (f_{i}) of the carrier signal is the derivative of this function (f_{i} is not necessarily equal to f_{c} because of the possible phase offset φ):

dθ(t)/dt = 2πf_{i}

Therefore, FM varies the derivative of θ(t) with the amplitude of the modulating signal. For a modulating signal f_{m}(t) and a modulation constant of k_{f}, this gives us the FM characteristic equation:

f_{o}(t) = A_{c} cos(2πf_{c}t + 2πk_{f}._{0}∫ ^{t}f_{m}(t) dt)

From this, equations for both narrowband and wideband FM can be derived. In particular, assuming that f_{m}(t) is of the form A_{m} cos(2πf_{m}t), then the FM signal is:

f_{o}(t) = A_{c} cos(2πf_{c}t + β.sin(2πf_{m}t))

(where the modulation index, β = (k_{f}.A_{m})/f_{m} = Δf/f_{m}

This modulation index is what determines whether or not the FM signal is narrowband (β << 1) or wideband. In addition, Carson's Rule approximates the FM bandwidth as:

BW = 2.f_{m}(1 + β)

which, in true engineering style, is simplified to:

BW = 2.f_{m} (*for β << 1*)

BW = 2.f_{m}.β (* forβ > 10*)

Sources:

Doomed Engineers, John Redford

Armstrong Memorial Research Foundation, http://www.armstrongfoundation.org/

Introduction to Communication Systems, Ferrel G. Stremler, Addison-Wesley

Communications Systems, David Harle, University of Strathclyde