Frequency Modulation

Frequency and phase modulation, the modulation index, Bessel sidebands, Carson's rule, the noise advantage, and demodulation

26 December 1933 — five patents from a basement

Static bedevilled early radio. In 1922 John Carson of AT&T had published a detailed mathematical analysis showing that frequency modulation offered no improvement over AM — correct, it later turned out, but only for what is now called narrow-band FM. Working in secret in the basement laboratory of Columbia's Philosophy Hall, Edwin Armstrong made the carrier's frequency swing far more widely than the audio frequencies it carried, and found that this "wide-band" FM rejected noise. On 26 December 1933 he was granted five US patents on the system (Edwin Howard Armstrong). In June 1936 he demonstrated it to the Federal Communications Commission, playing a jazz record first over AM, then over FM. A United Press reporter wrote that if the audience of 500 engineers had shut their eyes, "they would have believed the jazz band was in the same room. There were no extraneous sounds" (Edwin Howard Armstrong). This lesson is how it works — and why spending bandwidth buys silence.

Derivation

Frequency and phase modulation

Keep the carrier's amplitude constant and put the message into its angle:

x(t) = A\cos\big(\omega_c t + \phi(t)\big)

. Phase modulation sets

\phi = k_p m(t)

. Frequency modulation makes the instantaneous frequency (Lesson 6-4) follow the message,

f_i(t) = f_c + \frac{1}{2\pi}\frac{d\phi}{dt} = f_c + k_f\,m(t) \quad\Longrightarrow\quad \phi(t) = 2\pi k_f\int_0^t m(\tau)\,d\tau .

For a tone

m(t) = \cos\omega_m t

the frequency swings by the peak deviation

\Delta f = k_f

either side of

f_c

, and

\phi(t) = \beta\sin\omega_m t

with the modulation index

\beta = \Delta f/f_m

. The amplitude never changes: whatever the message, the transmitter runs at full power.

Derivation

Bessel sidebands

The periodic function

e^{j\beta\sin\theta}

has a Fourier series whose coefficients are Bessel functions of the first kind (the Jacobi–Anger expansion):

e^{j\beta\sin\theta} = \sum_{n=-\infty}^{\infty} J_n(\beta)\,e^{jn\theta} .

With

\theta = \omega_m t

, taking the real part of

Ae^{j\omega_c t}e^{j\beta\sin\omega_m t}

x(t) = A\sum_{n=-\infty}^{\infty} J_n(\beta)\cos\big((\omega_c + n\omega_m)t\big):

sidebands at every multiple of

f_m

on both sides of the carrier, with amplitudes

J_n(\beta)

. The carrier's own amplitude,

J_0(\beta)

, vanishes at

\beta = 2.405

— the lab measures the lines and finds them equal to

|J_n|

to three decimals.

💡Carson's rule

In principle the sidebands go on for ever; in practice

J_n(\beta)

dies off quickly once

n

exceeds

\beta + 1

. Carson's rule estimates the band that carries about 98 % of the power:

B \approx 2(\Delta f + f_m) = 2(\beta + 1)f_m .

Broadcast-style FM, with 75 kHz deviation and 15 kHz audio (

\beta = 5

), needs about 180 kHz, and the lab measures exactly 180 kHz for the band holding 98 % of the power.

💡Why wide-band FM beats noise

Noise added to an FM signal changes its amplitude — which a limiter simply removes — and nudges its phase a little. The demodulator reads the message from the rate of change of phase, which the full deviation

\Delta f

drives, so a larger deviation makes the message stand far above the small phase jitter. Above a threshold carrier-to-noise ratio, the output signal-to-noise ratio grows as

\beta^2

: doubling the deviation buys about 6 dB, paid for with roughly twice the bandwidth. Below the threshold the advantage collapses into clicks.

Demodulation. An FM receiver must turn frequency back into amplitude. Digitally, form the analytic signal with the Hilbert transform, unwrap its phase and differentiate:

f_i = \frac{1}{2\pi}\,d\phi/dt

. The lab recovers a two-tone message to within 0.0000 this way, with a peak deviation of 4965 Hz — exactly

k_f

times the message's largest value. Hardware does the same with frequency discriminators and phase-locked loops.

I thought Armstrong would invent some kind of a filter to remove static from our AM radio. I didn't think he'd start a revolution—start up a whole damn new industry to compete with RCA.

– David Sarnoff, president of RCA, quoted in Lawrence Lessig, Free Culture (2004), quoted on Wikiquote

🕰Carson was right — about narrow-band FM

Carson's 1922 review analysed only small deviations, where FM does behave much like AM; it missed the wide-deviation regime that Armstrong exploited. Carson's name survives where he was right: the bandwidth rule for FM is his, and so is single-sideband modulation (Edwin Howard Armstrong; Carson bandwidth rule).

⚠Bandwidth is the price

Broadcast FM's 180 kHz is six times the 30 kHz that double-sideband AM needs for the same 15 kHz of audio. Wide-band FM makes sense where spectrum is plentiful — the VHF band where FM broadcasting lives — and not where it is scarce.

💡Where you will use this

FM broadcasting, frequency-shift keying in data links and the chirped pulses of radar (Lesson 4-2) are all angle modulation; phase-locked loops (a feedback system, Lesson 7-4) demodulate them. The bandwidth-for-noise trade returns, in its purest form, in the spread spectrum of Lesson 10-3.

📚Further reading

Oppenheim and Willsky, Signals and Systems, section 8.7 — sinusoidal frequency modulation.
Edwin Howard Armstrong on Wikipedia — wide-band FM and its long battle for acceptance.

↓But there is a problem

FM spends bandwidth to beat noise. Push the idea to its extreme and something remarkable happens: a signal spread over a band a thousand times wider than its message can arrive far below the noise and still be found. Every GPS receiver does it for every satellite. The idea's most famous early patent was filed in the Second World War by a Hollywood film star and a composer.