FM Modulation

01 What noise does to a height

Everything a receiver picks up arrives as a voltage on a wire, and the wanted signal is not the only thing arriving. Lightning hundreds of kilometres away, a car ignition in the next street, the thermal agitation of the electrons in the receiver's own first transistor — all of it adds itself to the antenna voltage. It is a sum, not a mixture: what reaches the detector is signal plus noise, one waveform, with no label on either part.

Now recall what an AM receiver does with that waveform. It measures the height. The envelope detector follows the peaks, and the peaks it follows are the peaks of the sum. A wave that should have been 1.00 volts tall arrives 1.04 volts tall, and the detector reports 1.04, because that is genuinely how tall it is. Nothing has malfunctioned. The noise simply landed in the same place the message was kept.

You can beat this with brute force. Noise power in a given bandwidth is roughly fixed, so doubling the signal-to-noise ratio means quadrupling the transmitter power, and a broadcaster who wants a quiet signal over a wide area ends up paying an electricity bill for it. That is the whole remedy AM offers. There is no clever detector that helps, because there is nothing to be clever about: the detector is reading a height, and the height is genuinely wrong.

Which points at the way out. The noise adds a voltage. If the message is not stored in the voltage, the noise has nothing to corrupt.

02 Put the message in the frequency

A sine wave has three things you can change: its height, its frequency, and its phase. AM takes the first. FM takes the second, and leaves the height strictly alone.

The quantity being controlled is the instantaneous frequency — how fast the wave is going right now. At rest it sits at the carrier frequency f_c. The message pushes it above and below:

Δf is the deviation: the furthest the frequency ever gets from the carrier, reached when the message is at full volume. It is a property of the transmitter, chosen by whoever designed it and fixed by regulation. Broadcast FM in most of the world uses Δf = 75 kHz. A hand-held two-way radio uses 2.5 kHz.

Notice what is missing from that equation: any mention of amplitude. The transmitted wave is A_c tall when the message is loud, A_c tall when it is quiet, and A_c tall in the silence between words. The envelope of an FM signal is a flat line, and it carries nothing. That is not a curiosity. It is the entire point, and section 7 collects on it.

03 The phase is the integral

There is an obvious way to write the transmitted wave from the equation above, and it is wrong. It is worth seeing why, because the correct version looks stranger than it should and the reason is easy to miss.

The tempting thing to write is A_c cos(2π f_i(t) · t). Put the instantaneous frequency in the place the carrier frequency used to be. But a cosine does not read the number in front of t and go at that speed; it reads the whole angle inside the brackets, and the speed is how fast that angle is turning. Frequency is not a term in the phase — it is the rate of change of the phase:

So if you want the frequency to follow the message, the phase has to follow the running total of the message. Integrate f_i to get φ, and the transmitted wave is:

The integral is not decoration. It changes what the transmitted wave looks like in ways you would not guess from the message alone — a message that jumps abruptly produces a phase that only ever bends, because an integral cannot jump. It also explains a fact from the next section: the deviation depends on how loud the message is, and the resulting bandwidth depends on how fast it is, and those are two different knobs.

04 Deviation, and the index β

Two numbers describe an FM signal, and confusing them is the usual source of trouble.

• The deviation Δf is how far the frequency swings, in hertz. It is set by the loudness of the message and capped by the transmitter. Shout into the microphone and Δf grows; whisper and it shrinks.
• The message frequency f_m is how often the frequency swings. Sing a higher note and f_m grows, while Δf stays exactly where the volume put it.

The ratio of the two is the modulation index:

β is where FM stops resembling AM. In AM the index m has a hard ceiling: at m = 1 the envelope touches zero, and past it the signal is broken and a simple receiver hears distortion. β has no such ceiling. β = 0.2, β = 5, β = 40 — nothing folds over, nothing clips, and the recovered message is perfect in every case. What changes is how much room the signal takes up, and that is the only thing that changes.

Two regimes get their own names, and the boundary is conventionally put at β ≈ 0.3:

• Narrowband FM (β ≪ 1). The deviation is small compared with the message frequency. The spectrum is a carrier and essentially one pair of sidebands — the same shape as AM, and the same bandwidth, 2f_m. This is what two-way radios, aircraft marker beacons and the FM sub-carriers inside other systems use, and it buys almost none of FM's noise advantage.
• Wideband FM (β ≫ 1). The deviation dominates. The spectrum spreads out over many sideband pairs and the bandwidth approaches 2Δf, independent of the message frequency. Broadcast FM lives here: Δf = 75 kHz against a 15 kHz top note is β = 5.

05 A spectrum that never ends

Take the single-tone case, x(t) = cos(2π f_m t), and put it through the integral from section 3. The phase comes out as β sin(2π f_m t), so:

In AM the equivalent step was a line of algebra and out fell three tones. Here the message sits inside a cosine, inside another cosine, and no amount of rearranging gets it out. The expansion that does work has been known since the 1820s and produces something unsettling:

One tone in, and out come infinitely many sidebands, spaced f_m apart, running away from the carrier in both directions forever. Their heights are Bessel functions of β, which is the only place in this article where a named special function is unavoidable — it is simply what the integral of a cosine-inside-a-cosine evaluates to.

Three things about that sum are worth more than the formula itself:

• The total power never changes. Σ J_n(β)² = 1 for every β, so the transmitted power is A_c²/2 whatever the message does — which it must be, since the amplitude is constant. Modulating an FM transmitter does not add power. It only moves power out of the carrier and into the sidebands.
• The carrier can disappear completely. J₀(β) is zero at β = 2.405, again at 5.520, again at 8.654. At those settings the transmitter is putting out full power and nothing at all on its own centre frequency. This is not a curiosity either — it is the classic way to calibrate a deviation meter, because you can hear or see the carrier null exactly.
• The sidebands do die out. J_n(β) stays small until n approaches β and falls away sharply past it, so although the sum is infinite, only about β + 1 pairs matter. That observation is the whole of the next section.

06 Carson's rule

An infinite spectrum is not something you can license, so the practical question is where to cut it off. The answer everyone uses dates from 1922 and is a single line:

It is a rule of thumb with a real justification: keep every sideband whose amplitude exceeds about 1 % of the unmodulated carrier, and you have kept 98 % or more of the power, and what you have kept is β + 1 pairs on each side. Both limits check out against what you already know:

• As β → 0 it gives BW → 2f_m, which is exactly AM's answer. Narrowband FM occupies the same room as AM, because it has the same shape.
• As β grows large it gives BW → 2Δf. The bandwidth stops caring about the message frequency and is set purely by how hard the transmitter swings.

Put broadcast numbers in: Δf = 75 kHz, top audio frequency 15 kHz, so BW ≈ 2(75 + 15) = 180 kHz. FM broadcast channels are spaced 200 kHz apart, which is that number plus a guard band. Compare with an AM broadcast channel at 10 kHz. One FM station occupies the room of twenty AM stations, and that is the bill for everything section 8 is about.

07 Getting the message back

An FM receiver does two things in order, and the first one is the reason the whole scheme works.

Step one: throw the amplitude away

The wanted signal has a constant envelope. Therefore any variation in the envelope is, by definition, not the message — it is noise, fading, or interference. So the receiver limits: it amplifies hard into saturation until the wave is clipped flat top and bottom, closer to a square wave than a sine. Every trace of amplitude information is destroyed, deliberately, along with the noise that was carried in it. The zero crossings survive untouched, and the zero crossings are where the message is.

An AM receiver can never do this, at any price, with any circuit. Its amplitude is its message. This single step is what FM buys with its 180 kHz, and it is why the comparison in figure 1 looks the way it does.

Step two: turn frequency into voltage

Now something has to convert "how fast is this wave going" into a voltage. The tidiest way to see it is to differentiate. If s(t) = A_c cos φ(t), then

Read the right-hand side as a wave whose height is proportional to the instantaneous frequency. The differentiator has converted FM into AM — and an AM signal is something we already know how to demodulate, with the diode, resistor and capacitor from the previous essay. Differentiate, then envelope-detect. That pair is a slope detector, and in practice the differentiator is just a tuned circuit operated off to one side of resonance, where its response happens to be a rising line.

Real receivers mostly use one of three refinements — the Foster-Seeley discriminator, the ratio detector, or, in anything built in the last forty years, a phase-locked loop, which steers an oscillator to follow the incoming frequency and reads the message straight off the steering voltage. All three do the job of the two lines above with better linearity.

08 What the bandwidth buys

FM spends twenty times the room of an AM station. Three things come back for it.

Quieting, and why it grows as β²

After the discriminator, the noise that survives is the phase noise differentiated — and differentiating multiplies each frequency component by that frequency, so the output noise density rises as f². Integrating that across the message band and comparing with the recovered signal gives a clean result for a single tone:

The improvement grows with the square of the index, while the bandwidth grows only linearly with it — so every doubling of bandwidth buys about 6 dB. At broadcast settings, β = 5 gives a factor of 37.5. Standard AM at full modulation manages a figure of merit of 1/3. The ratio of the two is about 112, which is roughly 20 dB: the same transmitter power, into the same noise, arriving a hundred times quieter.

Capture: the stronger station wins outright

Two AM stations inside one receiver's filter give you both, mixed, at their relative strengths — that was figure 7 of the AM essay. Two FM stations do not mix. The limiter and the discriminator between them track whichever signal is momentarily larger, so the stronger station takes the output completely and the weaker one disappears. A few decibels of advantage is enough. It is why driving between two transmitters on the same frequency gives you one station, then a band of noise, then the other station, instead of a long stretch of both at once.

The threshold, where it all stops working

None of this is free below a certain carrier-to-noise ratio. The advantage rests on the noise being small enough that it only nudges the phase. Once the noise vector is comparable with the signal vector it can swing the sum all the way around zero, adding or removing a whole cycle — and a discriminator reads that as an enormous momentary frequency, which you hear as a click. The clicks arrive slowly at first, then all at once.

The collapse happens near a carrier-to-noise ratio of about 10 dB, and it is steep: a decibel of extra noise below threshold costs far more than a decibel of output. So FM is better than AM above threshold and worse below it. AM degrades into something hissy but audible; FM degrades into nothing. It is the reason a distant AM station is still just about listenable while a distant FM station is silent.

Pre-emphasis, the last trick

One consequence of that f² noise slope is that the treble suffers most, even though music has least energy there. So broadcast FM tilts the deal: before transmission the high frequencies are boosted by a simple RC network, and the receiver applies the exact inverse afterwards. The music comes out flat, the noise comes out cut. The time constant is 50 µs in Europe and most of the world, 75 µs in the Americas — which is why a radio set for the wrong one sounds either dull or harsh.

09 What to remember

Six sentences, and they are the whole article:

• Noise arrives as a voltage added to the signal, so it lands on the amplitude — which is exactly where AM keeps its message.
• FM keeps the amplitude constant and puts the message in the instantaneous frequency, f_i(t) = f_c + Δf·x(t), which means the message enters the wave through an integral: s(t) = A_ccos(2πf_ct + 2πΔf∫x dτ).
• The index β = Δf/f_m has no ceiling — nothing breaks as it grows, only the bandwidth.
• One tone produces infinitely many sidebands with Bessel amplitudes J_n(β), total power constant, and a carrier that vanishes at β = 2.405.
• Carson's rule cuts the infinity down to BW ≈ 2(Δf + f_m) — 180 kHz for broadcast FM, twenty AM channels.
• The receiver limits first, destroying the amplitude and the noise in it, then differentiates to turn frequency back into amplitude — and that buys (3/2)β² in noise, plus the capture effect, down to a threshold below which it all collapses at once.

AM and FM are the same sentence with one word changed: take a steady wave, and let the message change one thing about it. AM changes the height and pays in noise. FM changes the pace and pays in bandwidth. Which is the better bargain depends entirely on which of the two you have to spare — and in 1930, when the spectrum was empty and amplifiers were noisy, everyone had bandwidth to spare.

Sources & further reading

• S. Haykin and M. Moher, Communication Systems, 5th ed., Wiley — chapter 4, angle modulation, and chapter 9 for the noise analysis and the threshold effect.
• B. P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems, 4th ed., Oxford — chapter 5, for the Bessel expansion and Carson's rule.
• J. R. Carson, “Notes on the theory of modulation”, Proc. IRE 10(1), 1922 — the original, and shorter than you would expect.
• E. H. Armstrong, “A method of reducing disturbances in radio signalling by a system of frequency modulation”, Proc. IRE 24(5), 1936 — wideband FM, and the paper that overturned Carson's own conclusion that FM could not help.
• L. W. Couch, Digital and Analog Communication Systems, 8th ed., Pearson — pre-emphasis, de-emphasis and practical discriminators.