ESSAY · SIGNALS & COMMUNICATION · ~14 MIN · 7 LIVE FIGURES

AM Modulation

Modulation puts a low-frequency message onto a high-frequency wave. A microphone's signal swings a few thousand times a second; a radio wave that can cross a city swings a few million.

This article covers the simplest method: amplitude modulation, where the message sets the height of the carrier. You only need to know what a sine wave is. Every figure below is live — drag it, break it, watch what happens.

A MESSAGE, RIDING

HASSAN KESHTKAR·TEHRAN ·2026 ·← back to hkk.fyi

01 Why sound cannot be sent as sound

Start with what we actually have. A microphone turns sound into a voltage. Speech lives between about 300 Hz and 3.4 kHz; music reaches up to about 15 kHz. That voltage is the message, and a signal that sits at its natural frequencies like this is called a baseband signal.

The obvious plan is to connect that voltage straight to an antenna and let it radiate. The plan fails twice, for two completely different reasons.

Problem one: the antenna would not fit anywhere

An antenna only radiates well when its length is a decent fraction of the wavelength of the signal it is fed. The usual target is a quarter of a wavelength. And wavelength comes straight from frequency:

wavelength λ = cf c ≈ 3 × 10⁸ m/s — the speed of light

Low frequency means long wavelength. A 3 kHz voice tone has a wavelength of 100 kilometres, so a quarter-wave antenna for it is 25 kilometres tall. That is not an engineering challenge, it is a joke. Move the same signal up to 1 MHz and the antenna becomes 75 m — a normal radio mast. Move it to 2.4 GHz and it is 3 cm — a track on a circuit board.

FIG 1HOW TALL IS THE ANTENNA?

FREQUENCY3 kHz

The scale is logarithmic, so each mark is ten times the one before it. Drag the slider from the voice tone at the far left up to Wi-Fi at the far right and watch the required antenna shrink from a mountain range to a paperclip. Nothing about the physics changed — only the frequency. This single plot is the first half of the reason modulation exists.

Can I just build a smaller antenna and accept the loss?

You can, and phones do it every day, but it is not free. An antenna much shorter than a quarter wavelength has a very small radiation resistance — the part of it that turns power into radio waves. The rest of the power turns into heat in the copper and in the matching network.

A short antenna also becomes very sensitive to frequency: it works over a narrow band and needs a careful matching circuit to work at all. You can shrink an antenna by a factor of ten with clever design. You cannot shrink it by a factor of ten thousand, which is what sending audio directly would ask of you.

Problem two: everybody would be shouting in the same room

Suppose the antenna problem vanished. There is still a second wall. Every microphone in the world produces the same range of frequencies — roughly 20 Hz to 20 kHz. If two stations both transmit their audio directly, the two signals arrive on top of each other and add up.

A filter can only separate signals that are in different places in frequency. Here they are in the same place, so there is nothing to filter on. One transmitter per city, for everyone, forever.

Both problems have the same answer. Move the message up to a high frequency, and give every transmitter a different one. That is what modulation is for.

02 The carrier, which carries nothing

The high-frequency wave we move the message onto is called the carrier. It is a plain sine wave:

the carrier c(t) = A_c · cos(2π f_c t)

On its own it contains no information at all. Its height never changes, its frequency never changes, its timing never changes. If you look at it on a spectrum analyser you see a single spike and nothing else. Listening to an unmodulated carrier tells you exactly one thing: the transmitter is switched on.

That is the point. A sine wave has exactly three properties you can change, and a receiver can detect a change in any of them:

Change this	You get	What the receiver watches
Amplitude	AM	how tall the wave is
Frequency	FM	how fast it wiggles
Phase	PM	whether it arrives early or late

Modulation is the act of making one of those properties follow the message. Demodulation is the act of reading it back at the far end. This article follows the first row.

A carrier carries no information. That is precisely why we can write information onto it.

03 What amplitude modulation does

Call the message x(t), and scale it so it never leaves the range −1 to +1. Amplitude modulation replaces the constant height A_c with a height that moves:

standard AM s(t) = A_c [ 1 + m · x(t) ] · cos(2π f_c t) m is the modulation index — how deeply the message pushes the height

Read it in two parts. The right-hand factor is the carrier, untouched: it still swings at f_c, and it always will. The left-hand bracket is the new amplitude, and it goes up and down with the message.

Draw a smooth line through the peaks of the result and you get the envelope. The envelope is A_c[1 + m·x(t)] — the message itself, scaled and lifted up so that it never goes negative. The message is not inside the wiggles. It is in the shape they trace.

FIG 2THE CARRIER TAKES THE SHAPE OF THE MESSAGE

MESSAGE FREQUENCY1 kHz

CARRIER FREQUENCY40 kHz

draw the envelope

Three views of the same instant. Top is the message on its own. Middle is the bare carrier — always the same height. Bottom is the transmitted signal, and the dashed line through its peaks is the top of the message, redrawn. Raise the carrier frequency and the wiggles get denser while the shape stays exactly where it was: the envelope does not care how fast the carrier runs. Switch the message to two tones or pulse and the envelope copies that too.

A note on the drawings. In a real broadcast the carrier is thousands of times faster than the message — an AM station at 1 MHz carrying a 1 kHz tone runs a ratio of 1000 : 1. Drawn honestly, that is a solid block of ink. Every figure here uses a ratio of a few tens instead, which keeps the shape visible — and keeps the detector's ripple, in the one figure you can listen to, at a frequency a speaker can actually reproduce. The mathematics is identical.

Three of the figures further down make sound — the modulation index, the detector, and the tuning dial. Nothing plays until you ask for it, and only one plays at a time.

04 The modulation index, and how to break it

The index m sets how deep the swing is. With a single tone the envelope runs between A_c(1 − m) at its lowest and A_c(1 + m) at its highest, so you can measure m off a screen without knowing anything else about the transmitter:

modulation index, measured m = V_max − V_min V_max + V_min

Three cases matter, and only three:

m < 1 — the envelope stays above zero. Everything works. m = 0.5 means the height swings between half and one-and-a-half times the carrier.
m = 1 — full modulation. The envelope just touches zero at its lowest point. This is as loud as standard AM is allowed to go, and it is where the message gets the largest share of the power.
m > 1 — over-modulation. The bracket [1 + m·x(t)] goes negative. The envelope cannot follow it there, so it folds back up and stops being a copy of the message. The receiver hears harsh distortion, and the transmitted signal spreads far outside its channel and interferes with the neighbours.

FIG 3TURN IT UP UNTIL IT BREAKS

MODULATION INDEX m0.60

what the receiver would play

The upper plot is the transmitted signal with V_max and V_min marked; the lower plot is what a simple receiver recovers by following the envelope. Below m = 1 the two shapes match. Push past m = 1 and watch the lower plot grow sharp notches that were never in the message — that is over-modulation, and it is the most common way an AM transmitter is set up wrong. Note also what happens at m = 0: a perfect, useless carrier. Press Listen and drag the slider: the lower plot is played as sound, so m = 0 is silence, the middle is a clean tone, and everything past m = 1 turns to buzz.

Why does a negative envelope come back as a phase flip?

When [1 + m·x(t)] goes negative, the wave is still there — it is the carrier multiplied by a negative number, which is the same as the carrier turned upside down, or shifted in time by half a cycle. Mathematically nothing is lost: a receiver that also knows the carrier's phase can undo the sign and rebuild the message perfectly.

But an envelope detector does not know the phase. It only measures how tall the wave is, and height has no sign. So it reports |1 + m·x(t)| and the negative part is folded back up. Over-modulation is not a problem with the signal. It is a problem with the cheap receiver — which is the receiver everybody uses.

05 The same signal, seen in frequency

So far everything has been a picture of voltage against time. The other view — voltage against frequency — is where AM shows what it really did. Take the single-tone case, x(t) = cos(2π f_m t), and multiply the brackets out:

standard AM, term by term s(t) = A_c cos(2πf_ct) + mA_c2 cos[2π(f_c − f_m)t] + mA_c2 cos[2π(f_c + f_m)t] using cos A · cos B = ½ cos(A − B) + ½ cos(A + B)

Three pure tones, and not one of them is at f_m. The message frequency itself has disappeared from the transmitted signal. What is left is the carrier, plus one copy of the message sitting just below it and one sitting just above it. Those two copies are the sidebands: the lower sideband (LSB) and the upper sideband (USB).

A real message is not one tone but a whole band, from near zero up to some highest frequency B. Each tone in it produces its own pair, so the result is two mirrored copies of the message spectrum, one on each side of the carrier. The signal now occupies

transmission bandwidth BW = 2B twice the highest frequency in the message

This is why an AM broadcast channel is 10 kHz wide while the audio inside it stops at about 5 kHz. You pay for the second copy whether you want it or not.

FIG 4THE MESSAGE MOVES UP, AND DOUBLES

MESSAGE FREQUENCY3 kHz

MODULATION INDEX m0.70

Below is the message where it starts, near zero. Above is the transmitted signal, sitting around a 1 MHz carrier. Drag the message frequency and the two sidebands walk away from the carrier together, always the same distance out on each side, and the bandwidth readout follows at exactly twice. Drag m down to zero and the sidebands vanish while the carrier stands there unchanged — all of the information is in the sidebands, and none of it is in the carrier. Switch to speech band to see what a real message looks like: two mirrored triangles.

Modulation does not change the message. It changes where in the spectrum the message lives.

Why is the lower sideband mirrored?

The lower sideband sits at f_c − f_m. A higher message tone gives a lower frequency there, so as the message spectrum climbs, the lower sideband walks downward. It is the same information read backwards along the frequency axis.

Both sidebands carry the complete message. That is exactly the redundancy that single-sideband transmission (SSB) throws away: filter one of them off, and you send the same information in half the bandwidth — at the cost of a much more demanding receiver.

06 Where the power goes

Now for the price. With a single-tone message, the average power of the AM signal splits into a carrier term and two sideband terms:

total transmitted power P_t = P_c ( 1 + m²2 ) each sideband carries P_cm²/4 · the carrier carries P_c no matter what

The carrier term does not depend on m at all. It is there when the station is playing music and it is there when nobody is speaking. And we established one section ago that the carrier holds none of the information. So the useful fraction is

efficiency η = m²2 + m² at best, with m = 1: η = 1/3

Even at full modulation, a standard AM transmitter puts two thirds of its power into a wave that says nothing. A station running a 50 kW carrier radiates all 50 kW of it whether anyone is speaking or not, and adds at most 25 kW of sidebands on top. In practice speech and music average well below m = 1, so the real figure is worse — often only a few per cent.

FIG 5TWO THIRDS OF IT, AT BEST, IS WASTED

MODULATION INDEX m1.00

CARRIER POWER50 kW

The bar is the total transmitted power, split into the part that carries the message and the part that does not. The curve underneath is efficiency against m, with your setting marked. Push m all the way to 1 and the curve tops out at 33.3 % — that is not a limitation of the equipment, it is the arithmetic of standard AM. Everything to the left of that point is worse.

Then why send the carrier at all?

Because the carrier is what makes the envelope a faithful copy of the message. Remove it and you get DSB-SC — double sideband, suppressed carrier — which spends 100 % of its power on information, and whose envelope no longer looks like the message at all. Recovering it needs a receiver that regenerates a carrier locked to the transmitter's phase.

That is the trade at the heart of AM: the transmitter wastes power so that every receiver can be trivial. One expensive station, millions of cheap radios. For broadcasting, that is the right way round.

07 Getting the message back

Here is the reward for all that wasted power. The message is sitting in the envelope, and measuring an envelope takes three components: a diode, a resistor and a capacitor.

The diode passes only the positive half of the incoming wave. Each positive peak charges the capacitor up to that peak's height. Between peaks, the capacitor leaks slowly through the resistor. The voltage on the capacitor therefore hops up to each new peak and sags gently in between — which is a copy of the envelope, with some ripple on it.

Everything depends on the time constant τ = RC, and it is squeezed from both sides:

the envelope detector condition 1f_c ≪ RC ≪ 1f_m slow enough to ignore the carrier, fast enough to follow the message

If RC is too small, the capacitor discharges between every carrier peak and the output is a rough, rippled version of the wave. If RC is too large, the capacitor cannot fall as fast as the envelope does, and the output cuts the corner in a straight line — a failure with the excellent name diagonal clipping. Between the two, it works.

FIG 6THREE PARTS, AND ONE NUMBER TO GET RIGHT

TIME CONSTANT RC25 µs

then drag RC while it plays

compare with the original message

The faint wave is what arrives at the antenna; the solid line is the voltage on the capacitor, simulated peak by peak. Drag RC to the far left for ripple — the detector is tracking the carrier instead of the envelope. Drag it to the far right for diagonal clipping — straight sloping cuts where the output could not fall fast enough. The window in between is wide, which is exactly why this circuit has never needed to be clever. Listen plays the same detector with everything scaled into hearing range, keeping RC × f_c and RC × f_m exactly as they are on screen — so the ripple you can see becomes a ripple you can hear. Drag RC while it plays: the fast end buzzes, because the output is chasing the carrier; the middle is a clean tone, and the loudest of the three; the slow end goes quiet and harsh.

Notice what is not in that circuit: no power supply, no oscillator, nothing that has to know the transmitter's phase. A diode and two passive parts pull speech out of the air. That is the whole return on the two thirds of the power we threw away in section 6.

08 Many stations, one sky

Back to the second problem from section 1. Every station is given its own carrier frequency, so every station's pair of sidebands sits in its own slot along the frequency axis. The slots are stacked side by side and nobody overlaps. This arrangement has a name: frequency-division multiplexing.

Your antenna picks up all of them at once and adds them together, which sounds like the same mess as before — except now they are in different places in frequency, so a filter can separate them. Tune a bandpass filter to one slot, throw away everything else, and hand what survives to the envelope detector. That is what the tuning knob on a radio does.

The slot has to be at least 2B wide, because that is what one station takes. If stations are packed closer than that, their sidebands overlap and you hear both at once — adjacent-channel interference. This is why AM broadcast channels are spaced 10 kHz apart in the Americas (9 kHz in most of the rest of the world) and the audio is filtered to about 5 kHz before it ever reaches the modulator.

FIG 7TURN THE DIAL

TUNED TO995 kHz

FILTER WIDTH10 kHz

then turn the dial while it plays

Four stations share the band; each one is a carrier spike with its own two sidebands. The shaded window is your receiver's filter, and the plot underneath is the audio that comes out of the detector. Land on a station and the message is clean. Stop between two and you get almost nothing. Now widen the filter past 20 kHz and drive it up the band: the window starts swallowing two stations at once and the recovered audio becomes both of them, mixed — the interference that channel spacing exists to prevent. Watch the hiss while you do it. Cut to one channel the filter is quiet, and everything past that costs you: the receiver's own noise arrives from every frequency at once, so it grows with the width of the window, and past 2B the window stops being empty — it starts collecting the neighbours' skirts and every distant station too weak to resolve, which is mush rather than a signal. The readout counts the two together in decibels against a filter cut to one channel. That is the quieter half of why a receiver's filter is no wider than it has to be: the station is buried in hiss long before a whole neighbour arrives. Each station sends a different note with a different waveform, so press Listen and turn the dial: on a station with the filter at 10 kHz it is clean, and it gets steadily grubbier as you open the window.

09 What to remember

Six sentences, and they are the whole article:

A message at audio frequencies cannot be radiated: the antenna would be kilometres long, and every station would occupy the same band.
Modulation solves both by moving the message up to a high carrier frequency, chosen differently for each transmitter.
In AM, the message controls the height of the carrier, and the message shape appears as the envelope: s(t) = A_c[1 + m·x(t)]cos(2πf_ct).
The index m must stay at or below 1. Above it, the envelope folds and a simple receiver hears distortion.
In frequency the result is a carrier and two sidebands, so AM occupies twice the message bandwidth, and at most one third of the power does useful work.
The payment for that waste is a receiver made of three passive components.

Once those are solid, the rest of analogue modulation follows quickly. Suppress the carrier and you have DSB-SC, which is efficient but needs a synchronous receiver. Remove one sideband as well and you have SSB, which halves the bandwidth. Leave the amplitude alone and move the frequency instead and you have FM, which is far more resistant to noise — because noise adds itself to amplitude, and FM does not listen to amplitude.

All of them are the same sentence in different clothing: take a steady wave, and let the message change one thing about it.

More of this sort of thing

If you want another piece of RF explained from zero with figures you can break, the Smith chart essay next door does the same job for impedance, reflection and matching — and the chart itself runs live in the Lab.

Read the Smith chart essay Back to the Lab

Sources & further reading

S. Haykin and M. Moher, Communication Systems, 5th ed., Wiley — chapter 3, amplitude modulation and its variants.
B. P. Lathi and Z. Ding, Modern Digital and Analog Communication Systems, 4th ed., Oxford — chapter 4, for the envelope-detector conditions.
L. W. Couch, Digital and Analog Communication Systems, 8th ed., Pearson — power and efficiency of standard AM.
C. A. Balanis, Antenna Theory: Analysis and Design, 4th ed., Wiley — chapter 2, on why an electrically small antenna radiates so poorly.