Appearance
7.2 — Analog Modulation: AM, FM and PM
A carrier is a sine wave:
c(t)=A_c\cos(\omega_ct+\phi)
It has exactly three things you can vary: amplitude, frequency and phase. Vary the amplitude with the message and you have AM. Vary the frequency and you have FM. Vary the phase and you have PM.
That is the entire menu, and every analog modulation scheme in history is one of the three or a refinement of one.
Modulation is multiplication, and Chapter 4.4's shifting property is what makes it work: multiplying a signal by a sine moves its whole spectrum up to that sine's frequency, unchanged in shape. That one line is the reason this chapter exists.
The letters this chapter uses, said out loud
f_c and \omega_c — the carrier frequency, in hertz and in radians per second. The subscript c is for carrier: the high-frequency sine that does the travelling and carries no information of its own.
f_m and \omega_m — the modulating frequency: the signal you actually want to send, usually audio, usually thousands of times lower than the carrier.
m — the modulation index. In AM it is the fraction by which the carrier's amplitude swings, from 0 (no modulation) to 1 (full modulation). Above 1 is overmodulation, and section 2 shows what it does to the signal. In FM it means something related but different, defined in section 3, and the reuse of the letter across two definitions is a genuine trap.
\Delta f — "delta f", the frequency deviation: how far an FM carrier swings away from its resting frequency at full modulation. The capital delta means "change in", as everywhere else in this book. Broadcast FM uses \Delta f = 75 kHz.
\beta — in this chapter only, the FM modulation index, \beta = \Delta f / f_m. It collides with the transistor's current gain in Chapter 2.3 and with several other uses elsewhere; the Greek alphabet is small and engineering is large.
B — bandwidth, in hertz: how much of the radio spectrum a transmission occupies. This is the quantity that costs money, because spectrum is licensed and finite, and almost every design trade in Part 7 is really a trade against it.
1. Amplitude modulation
s(t)=\left[A_c+m(t)\right]\cos(\omega_ct)
The carrier's amplitude follows the message. For a single tone message m(t)=A_m\cos(\omega_mt):
s(t)=A_c\left[1+\mu\cos(\omega_mt)\right]\cos(\omega_ct)
where
\mu=\frac{A_m}{A_c}
is the modulation index.
The spectrum
Expand using the product-to-sum identity \cos A\cos B=\tfrac12[\cos(A-B)+\cos(A+B)]:
s(t)=A_c\cos\omega_ct+\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t+\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t
Three components: the carrier, and two sidebands at f_c\pm f_m.
\boxed{B_{AM}=2f_m}
Bandwidth is twice the highest message frequency. For AM broadcast with a 5 kHz audio limit, that is 10 kHz per station, which is exactly the channel spacing used.
Power, and AM's fundamental waste
Total power for a tone-modulated AM signal:
P_{total}=P_c\left(1+\frac{\mu^2}{2}\right)
Derivation: the carrier has power A_c^2/2; each sideband has (\mu A_c/2)^2/2 = \mu^2A_c^2/8; two of them give \mu^2A_c^2/4. The ratio to the carrier is \mu^2/2.
At full modulation, \mu=1:
P_{total}=1.5P_c
So the sidebands carry 0.5P_c out of 1.5P_c — one third of the total. Two thirds of the transmitter's output is in the carrier, which contains no information whatsoever.
Efficiency:
\eta=\frac{\mu^2}{2+\mu^2}
Maximum 33% at \mu=1, and speech with its low average level typically gives \mu\approx0.3, so efficiency in practice is around 4%. A 50 kW AM transmitter puts about 2 kW into the information.
Overmodulation
If \mu>1, the term A_c[1+\mu\cos\omega_mt] goes negative, so the envelope no longer follows the message — it folds over. A simple envelope detector produces badly distorted audio, and the sharp corners generate splatter across neighbouring channels.
Broadcast transmitters therefore have hard limiters preventing \mu from exceeding about 0.95, and the "loudness" processing on AM radio is largely about maximising average modulation without ever crossing that line.
Demodulation
Envelope detection. A diode, a capacitor and a resistor. The diode passes the positive half-cycles, the capacitor holds the peak, and the resistor lets it decay slowly enough to track the audio but fast enough to follow it down.
\frac{1}{f_c}\ll RC\ll\frac{1}{f_m}
Read the condition: the time constant must be long compared with a carrier cycle and short compared with an audio cycle. Too short and you get carrier ripple; too long and the envelope cannot fall fast enough, producing diagonal clipping.
This is why AM won in 1920. The receiver is a diode and two components, and the very first ones used a crystal of galena with a wire touching it — a point-contact diode discovered by accident. A crystal radio needs no power supply at all, extracting everything it needs from the transmitted wave.
Synchronous detection multiplies by a locally generated carrier at exactly the right frequency and phase, then low-pass filters. Better performance, especially at low SNR, and essential for the suppressed-carrier variants below. It requires a phase-locked loop (Chapter 2.6) to recover the carrier phase.
The efficient variants
DSB-SC — double sideband suppressed carrier. Simply transmit m(t)\cos\omega_ct with no added carrier. 100% of the power carries information. The cost is that envelope detection no longer works, because the envelope now follows |m(t)| and loses the sign, so synchronous detection is compulsory.
SSB — single sideband. The two sidebands are mirror images and carry identical information, so transmit only one. Halves the bandwidth and needs a quarter of the power for the same range.
Generating it requires either a very sharp filter, or the phasing method: split the message into two paths 90° apart, do the same to the carrier, multiply and subtract. The 90° network across the whole audio band is the hard part.
SSB is what amateur radio and long-distance voice services use, and its characteristic sound — voices that go squeaky or growly if the receiver is tuned slightly off — is because there is no carrier to lock onto, so any frequency error shifts every audio component by the same absolute amount rather than the same ratio.
VSB — vestigial sideband. One full sideband plus a trace of the other, which makes the filter buildable while keeping most of the bandwidth saving. Analog television used it for the picture, because the video signal extends down to DC and a true SSB filter at DC is impossible.
2. Frequency modulation
s(t)=A_c\cos\left(\omega_ct+k_f\int_{-\infty}^tm(\tau)d\tau\right)
The instantaneous frequency follows the message:
f_i(t)=f_c+\frac{k_f}{2\pi}m(t)
The integral in the phase is there because frequency is the derivative of phase, so to make frequency proportional to the message, the phase must be proportional to its integral.
The amplitude never changes. That single fact is where all of FM's advantages come from.
For a tone message:
s(t)=A_c\cos\left(\omega_ct+\beta\sin\omega_mt\right), \qquad \beta=\frac{\Delta f}{f_m}
\Delta f is the frequency deviation — how far the carrier swings — and \beta is the modulation index.
The spectrum, and why it is infinite
Expanding the FM signal gives an infinite series of sidebands whose amplitudes are Bessel functions:
s(t)=A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos\left[(\omega_c+n\omega_m)t\right]
FM has infinitely many sidebands, spaced f_m apart, with amplitudes given by J_n(\beta) — the Bessel function of the first kind, order n.
In practice most of them are negligible. Carson's rule keeps those carrying 98% of the power:
\boxed{B_{FM}\approx2(\Delta f+f_m)=2f_m(\beta+1)}
Worked example — FM broadcast. \Delta f=75 kHz (the standard deviation), f_m=15 kHz (the audio limit):
B=2(75+15)=180\ \text{kHz}
Channels are spaced 200 kHz apart, which matches with a little guard band.
Compare with AM's 2f_m=30 kHz for the same audio. FM uses six times the bandwidth. Section 3 explains what it buys with it.
A curiosity from the Bessel functions worth knowing: J_0(\beta) is zero at \beta = 2.405. At that exact modulation index the carrier disappears completely, with all the power in the sidebands. This is used to calibrate deviation meters — increase the modulation until the carrier nulls, and you know \beta exactly.
Narrowband and wideband
Narrowband FM, \beta \ll 1: only the first sideband pair matters, so B\approx2f_m — the same as AM. Used for two-way radio, where spectrum matters more than fidelity.
Wideband FM, \beta \gg 1: B\approx2\Delta f, set by the deviation rather than the audio bandwidth. Used for broadcast.
3. Why FM sounds better
Noise, and the triangular spectrum
Noise adds to the signal as a small vector at a random phase. In AM, that changes the amplitude — which is the message, so the noise appears directly in the output.
In FM the amplitude carries nothing. A limiter in the receiver clips the signal to a constant amplitude, removing amplitude noise entirely. What remains is the phase perturbation the noise causes, and that is much smaller.
The key result:
\text{SNR}_{out}=3\beta^2(\beta+1)\,\text{SNR}_{in}
Worked example. Broadcast FM has \beta = 75/15 = 5:
\text{gain}=3\times25\times6=450 = 26.5\ \text{dB}
Twenty-six decibels better than AM for the same received power — a factor of 450 in noise power. That is the whole answer to "why does FM sound better", and it is bought entirely with the six times bandwidth.
The trade is quantitative and it is a good one: SNR improves as \beta^2 while bandwidth grows only as \beta. Spend bandwidth, gain quality faster than linearly. Chapter 7.5 shows this is a general principle and gives its limit.
There is a second effect. The noise power at the FM detector's output rises as the square of the frequency offset from the carrier, giving a triangular noise spectrum. So high audio frequencies suffer more noise than low ones.
The fix, and it is elegant: pre-emphasis and de-emphasis. Boost the high frequencies before transmitting and cut them by exactly the same amount in the receiver. The signal comes out flat, and the noise added in between gets only the cut, so high-frequency noise is reduced.
The time constant is 50 µs in Europe and 75 µs in North America, corresponding to a corner at 3.2 kHz and 2.1 kHz. This is the same idea as Dolby noise reduction on cassette tape, and it buys around 13 dB at the top of the audio band for nothing but two RC networks.
The threshold effect
FM's advantage vanishes suddenly below about 10 dB of input SNR. Above threshold, FM is far better than AM; below it, FM fails completely while AM degrades gracefully.
The mechanism is that a noise vector larger than the signal can swing the phase right around, producing a large click in the output. That is the crackling you hear as an FM station fades, and it is why FM coverage has a definite edge while AM fades away gradually over hundreds of kilometres.
This threshold behaviour is the analog ancestor of the digital cliff described in Chapter 7.1 — both come from a system that trades graceful degradation for better performance while it works.
4. Phase modulation
s(t)=A_c\cos\left(\omega_ct+k_pm(t)\right)
The phase follows the message directly, with no integral.
PM and FM are near-identical, and the relationship is exactly the calculus you would expect:
- FM a carrier with m(t) = PM it with \int m\,dt
- PM a carrier with m(t) = FM it with dm/dt
So a practical FM transmitter is often built as a phase modulator with an integrator in front of it, because phase modulation is easier to do with a crystal-controlled oscillator, and crystal stability is what a broadcast transmitter needs.
Where PM is used directly: almost nowhere in analog communication, and everywhere in digital, where it becomes PSK (Chapter 7.3). Its noise behaviour differs slightly from FM — the noise spectrum is flat rather than triangular, so no pre-emphasis is needed.
5. Generating and receiving
Generating FM
Direct: a voltage-controlled oscillator, usually an LC oscillator with a varactor diode (Chapter 2.2) in the tank. Simple, and its frequency stability is only as good as the LC circuit's, so it needs a phase-locked loop to hold it on frequency.
Indirect (Armstrong): phase-modulate a crystal oscillator by a small amount, then multiply the frequency repeatedly to increase the deviation. Excellent stability, and it is how Edwin Armstrong built the first practical FM system in 1933.
The superheterodyne receiver
Almost every radio receiver made since 1920 uses this architecture, and it deserves explaining because the reason for it is not obvious.
The problem: a receiver must select one station from many, and the selectivity filter must be sharp. Building a sharp filter that tunes across a band is extremely difficult, because its bandwidth changes as it tunes.
Armstrong's solution: do not tune the filter. Move the signal to the filter.
- Mix the incoming signal with a local oscillator, producing sum and difference frequencies.
- The difference is the intermediate frequency, always the same value regardless of which station you selected — 455 kHz for AM, 10.7 MHz for FM.
- A fixed, sharp filter at the IF does all the selectivity. It never has to tune, so it can be as good as you like.
- Amplify and demodulate at the IF.
Tuning is done by changing the local oscillator frequency, which is easy, rather than by changing a filter, which is hard.
The image problem. Two input frequencies produce the same IF: f_{LO}+f_{IF} and f_{LO}-f_{IF}. One is the station you want; the other is the image, and it will come through just as strongly.
Worked example. AM broadcast, wanting 1000 kHz with a 455 kHz IF. The local oscillator runs at 1455 kHz. The image is at 1455+455 = 1910 kHz. A station at 1910 kHz would be received simultaneously.
The fix is a filter before the mixer — the RF preselector — which need only be sharp enough to reject something 910 kHz away, which is easy. This is why the IF is chosen to be reasonably high: a higher IF pushes the image further away and makes the preselector's job easier.
Double conversion uses two IF stages: a high first IF for good image rejection, and a low second IF for good selectivity. You get both, and it is what every good communications receiver does.
6. Where each is used, and why
| Scheme | Where | Why that choice |
|---|---|---|
| AM | broadcast 530–1700 kHz, aviation | simplest receiver; aviation needs simultaneous transmissions to be audible |
| SSB | amateur, marine, military HF | best power and bandwidth efficiency |
| FM | broadcast 88–108 MHz, two-way radio | noise immunity; capture effect |
| PM | analog: rare | superseded by digital PSK |
| VSB | analog TV picture | video extends to DC |
Aviation deserves its note. Air traffic control uses AM at 118 to 137 MHz, and it is deliberate. If two aircraft transmit at once on AM, both are heard as a heterodyne and the controller knows something is wrong. On FM, the capture effect means the stronger signal completely suppresses the weaker one — and a controller would never know that a second aircraft had transmitted. In aviation, hearing the interference is a safety feature.
And the capture effect is why FM broadcast has such clean coverage boundaries. A receiver locks onto the strongest signal and rejects a weaker co-channel one almost entirely, so you either hear one station cleanly or the other, rather than both mixed. AM has no such property, which is why distant AM stations bleed through each other at night.
Chapter 7.3 does the same three modulations with a message that has only discrete values, and finds that the mathematics gets simpler while the performance gets dramatically better.
Every formula above, built from scratch
None of the results in this chapter are worth memorising, because each one can be rebuilt in under a minute from something simpler. What follows is that rebuilding, one result at a time, so the formula and the reason for it sit on the same page as the explanation that needed them.
Amplitude modulation
s(t)=A_c[1+\mu\cos\omega_mt]\cos\omega_ct, \qquad \mu=\frac{A_m}{A_c}
Expanding with \cos A\cos B=\tfrac12[\cos(A-B)+\cos(A+B)]:
s(t)=A_c\cos\omega_ct+\frac{\mu A_c}{2}\cos(\omega_c-\omega_m)t+\frac{\mu A_c}{2}\cos(\omega_c+\omega_m)t
B_{AM}=2f_m
P_{total}=P_c\left(1+\frac{\mu^2}{2}\right), \qquad \eta=\frac{\mu^2}{2+\mu^2}
Maximum efficiency 33% at \mu=1, and about 4% for typical speech.
From two-tone or complex modulation, the effective index is \mu_{eff}=\sqrt{\mu_1^2+\mu_2^2+\cdots}
Envelope detector time constant:
\frac{1}{f_c}\ll RC\ll\frac{1}{f_m}
Too long and you get diagonal clipping; too short and carrier ripple appears.
Frequency modulation
s(t)=A_c\cos\left(\omega_ct+k_f\int m\,dt\right), \qquad f_i=f_c+\frac{k_f}{2\pi}m(t)
\beta=\frac{\Delta f}{f_m}
Bessel expansion:
s(t)=A_c\sum_{n=-\infty}^{\infty}J_n(\beta)\cos[(\omega_c+n\omega_m)t]
Carson's rule (98% of the power):
B\approx2(\Delta f+f_m)=2f_m(\beta+1)
Noise improvement over AM:
\text{SNR}_{out}=3\beta^2(\beta+1)\text{SNR}_{in}
Read the exponents: SNR improves as \beta^2 while bandwidth grows as \beta. Spending bandwidth buys quality faster than linearly, which is the whole argument for FM.
Broadcast FM: \Delta f=75 kHz, f_m=15 kHz, \beta=5, B=180 kHz, improvement 26.5 dB.
Carrier null: J_0(\beta)=0 at \beta=2.405, used to calibrate deviation.
Pre-emphasis time constant: 50 µs (Europe), 75 µs (North America).
PM relationship: FM with m(t) = PM with \int m\,dt, and PM with m(t) = FM with dm/dt.
Superheterodyne image frequency:
f_{image}=f_{signal}+2f_{IF}
A higher IF pushes the image further away and eases the preselector.
What the next chapter fixes
Analog modulation carries a copy of the waveform, so every bit of noise the channel adds arrives as noise you can hear. Chapter 7.3 sends numbers instead, which can be restored perfectly right up to the moment they cannot be restored at all — and explains why that cliff edge is a feature rather than a flaw.