Modulation, part 1: putting a signal on a carrier

A voice covers roughly 100 Hz to 3 kHz. To send it by radio, you can’t simply connect a microphone to an aerial, for two reasons:

• Aerials have to be big compared with a wavelength. A radio wave at 3 kHz is 100 km long, and a useful aerial is about a quarter of that. At 100 MHz the wavelength is 3 m, and a 75 cm rod will do.
• Everyone would be on top of each other. If every station sent its sound as it is, they would all occupy the same 0–3 kHz, and no receiver could separate them.

So a transmitter moves the sound up to a carrier frequency of its own, and the receiver moves it back down. Each station gets its own slice of the spectrum, and the aerials can be a sensible size. Moving a signal to a carrier and back is modulation and demodulation. A device that does both is a modem (modulator–demodulator).

This page starts with the simplest way to move a sound up, and finds out what goes wrong. Each fix leads to the next idea: AM, then I/Q signals, then the Hilbert transform and single sideband, then FM, and finally a receiver that handles all of them the way a modern software radio does. Part 2 moves on to sending bits.

The page leans on two earlier ones:

• The Fourier transform: spectra, and sinusoids as spinning arrows.
• Designing FIR filters: low-pass filters and windows.

Everything runs at 44.1 kHz, with carriers between 4 and 17 kHz. A real radio’s carriers are a thousand times higher, but here every signal, even one sitting on its carrier, can be listened to.

The message is the sound to be sent. Here it’s either a synthetic voice, a steady 1 kHz tone, or a note (220 Hz and its harmonics), each limited to 3 kHz. The carrier is a plain cosine at a much higher frequency, $f_c$.

The simplest way to move the message up is to multiply the two, sample by sample:

Why that moves the spectrum is one line of trigonometry. Take a single frequency in the message, $\cos(2\pi f t)$:

Every frequency $f$ in the message turns into two, at $f_c - f$ and $f_c + f$, each at half the amplitude. The whole message reappears twice around the carrier: once the right way up, above it (the upper sideband), and once mirrored, below it (the lower sideband). The carrier itself isn’t there at all. This is called double sideband, suppressed carrier, or DSB.

To bring the message back down, the receiver does the same thing again: it multiplies by the carrier. Using the same trigonometry, each sideband splits into a copy at 0 Hz and a copy around $2f_c$. A low-pass filter keeps the copy at 0 Hz and removes the other one:

mixed = 2 * received * cos(2π · fc · t + phase_error)
message = lowpass(mixed, 3500)   # keep 0–3.5 kHz

This is synchronous detection: the receiver makes its own copy of the carrier, in step with the transmitter’s. But what if it isn’t quite in step? The receiver’s oscillator is a separate piece of hardware, miles away. Even if its frequency is perfect, its phase (where it is in its cycle at a given moment) is whatever it happens to be. Working through the trigonometry with a phase error $\varphi$ gives:

So the receiver has to know the carrier’s phase exactly, and from the received signal alone, which doesn’t even contain the carrier. That can be done (Part 2 shows how), but for decades it was too hard for a cheap radio. The next two sections show two different ways round it.

The oldest fix is to send the carrier as well, and plenty of it. Amplitude modulation (AM) adds 1 to the message before multiplying:

$k$ is the modulation depth. As long as $k$ isn’t too big, $1 + k\,m(t)$ never goes negative, so it’s the outline of the signal, its envelope. Now the receiver doesn’t need its own carrier at all. It just has to trace the outline, and that’s a job for two components:

• a diode, which lets current through in one direction only, keeping the top half of every cycle
• a capacitor, which smooths away the carrier’s fast wiggles and keeps the average of those halves.

The average of the top half of a cosine is $1/\pi$ of its peak, so what comes out is the outline divided by $\pi$. Take off the steady part, and what’s left is the message. This is the envelope detector.

There’s one more part, and it comes first. An aerial picks up every station at once, and a diode fed all of them would trace the outline of the whole jumble. So before the detector there’s a band-pass filter that lets through only the station you want: its carrier and the two sidebands around it. In a simple radio, this is a tuned circuit, a coil and a capacitor that resonate at one frequency, like the two-pole resonator on the IIR design page. Turning the tuning knob changes the capacitor, and so the frequency it picks out. Aerial, tuned circuit, diode, earphone: that’s the whole of a “crystal set” radio, which needs no batteries at all, because the station’s own signal powers the earphone.

The figure below starts with one station on its own, where a tuned circuit would have nothing to remove. Then you can add a second station, and a tuned circuit to separate them.

One tuned circuit can’t be both flat across one station and steep at its edges. That’s one reason radio designers moved to the superheterodyne, in section 8.

The diode in this figure is simulated at 8 × 44.1 kHz. Chopping off half of each cycle makes harmonics far above the carrier. At 44.1 kHz, those would fold back into the audio band, as on the sampling page. A real diode has no samples, so it has no such problem.

AM is simple, robust and cheap to receive, which is why it was broadcasting’s first standard and is still in use. But it’s wasteful in two ways:

• Power: most of the transmitter’s power goes into the carrier, which carries no information at all.
• Bandwidth: it sends the message twice, once in each sideband, so it uses twice the bandwidth the message needs.

Sections 4 to 6 fix both, by finding a way to send just one sideband with no carrier.

Look again at the receiver in section 2. At a phase error of 0°, multiplying by the cosine got the whole message. At 90°, it got nothing, and the multiplier was then a sine rather than a cosine. So a receiver that multiplies by both, a cosine and a sine, can’t lose the message: whatever the phase error, it lands somewhere between the two.

The two outputs have names:

• I (“in phase”): the result of multiplying by the cosine.
• Q (“quadrature”): the result of multiplying by the sine. Two things a quarter of a turn (90°) apart are said to be “in quadrature”.

With a phase error $\varphi$, I gets $m(t)\cos\varphi$ and Q gets $-m(t)\sin\varphi$. The message is split between the two, but nothing is lost: $I^2 + Q^2 = m(t)^2$ at every phase. (The minus sign is a convention: receivers usually multiply by minus the sine, so that the arrow below turns anticlockwise for frequencies above the carrier, as on the Fourier page.)

The best way to see I and Q is as a single arrow, a complex number $I + jQ$. As on the Fourier page, a complex number is an arrow on a plane, with I across and Q up. A phase error just turns the arrow, and a frequency error makes it spin. Neither changes its length.

There’s a bonus: a transmitter can send two messages, one on the cosine and one on the sine. That can seem impossible. What arrives at the aerial is a single real wave, one number at each moment, so how can it hold two messages?

Zoom in to a few cycles of the carrier. Over that short time the messages hardly change, so what arrives is just a sinusoid at the carrier frequency. A sinusoid at a known frequency has two things you can set independently:

• its size: how high its peaks are
• its timing: where its peaks fall, earlier or later than a reference. That’s its phase.

So one wave has two knobs. “How much cosine plus how much sine” is just another way to describe the same two knobs. Adding a cosine and a sine of the same frequency doesn’t make a more complicated wave. It makes another plain sinusoid at that frequency, shifted in time:

Here $\theta$ is the angle of the arrow $(A, B)$. Turning up B doesn’t add a second wave: it slides the one wave later, and makes it bigger.

The receiver gets each knob back with the test from the Fourier page: multiply, then average. A cosine times a cosine averages to ½, but a sine times a cosine averages to 0. So multiplying by the cosine measures A and ignores B entirely, and multiplying by the sine measures B and ignores A. It’s the same orthogonality that stops one frequency in a DFT from seeing another.

You’ve seen a real signal turn into complex numbers before: in the DFT. Feed it a purely real signal and every bin comes out complex. The real part is “how much cosine” and the imaginary part is “how much sine”: size and timing again. An I/Q receiver is one DFT bin, parked at the carrier frequency, worked out again for every sample. $I + jQ$ is that bin’s value. The signal at the aerial was never complex. The complex number is just a tidy way to write down two measurements of one real wave.

The bandwidth adds up, too. A message on the cosine alone makes two sidebands that are mirror images of each other, so half the band carries nothing new. A second message on the sine makes the two sidebands different, and uses the half that was being wasted. Two messages, each 3 kHz wide, share a band 6 kHz wide.

Sending one message on each knob is quadrature amplitude modulation (QAM). Part 2 uses it to send bits.

That last point matters. A real signal’s spectrum always mirrors itself, so its negative frequencies carry nothing new. A complex signal, an arrow spinning one way or the other, can tell $+f$ from $-f$. That’s what lets a single sideband be picked out, which is the next step.

To transmit one sideband only, the transmitter needs a way to shift its message by 90°. Not just one frequency, but every frequency in it at once, without changing any of their levels.

The filter that does this is the Hilbert transform. Any filter that leaves every level alone and changes only phases is an all-pass filter. The Hilbert transform is the ideal one: exactly 90° at every frequency, so a cosine goes in and a sine comes out.

Combined with the message itself, it makes a complex signal with no negative frequencies, the analytic signal: $m(t) + j\,\mathcal{H}\{m(t)\}$. Its length is the message’s envelope: the outline traced by the AM receiver in section 3, but for any signal. The finding-signals page used it to draw smooth matched-filter outputs.

The ideal Hilbert transform’s impulse response is $h[k] = \dfrac{2}{\pi k}$ for odd $k$, and 0 for even $k$. It goes on forever in both directions, so it’s impossible to build exactly. There are two ways to approximate it:

• An FIR filter: cut the impulse response off at some length, and taper it with a window. This is the same trade-off as for the FIR low-pass on the FIR design page. Because the taps are antisymmetric, the phase difference is exactly 90°. The errors are all in the gain, which droops near 0 Hz and near the top.
• Two IIR all-pass chains: each is a series of first-order all-pass sections, as on the IIR design page, whose phase falls from 0° to −180° around a corner frequency. Neither chain is anywhere near 90° on its own. But with corners chosen carefully, staggered between the two chains, their difference stays close to 90° over a wide band. Both are all-pass, so the gain is exactly right, and the errors are all in the phase. This is how analogue radios do it, with resistors and capacitors. On this page, the corners were found by a numerical search that minimises the worst phase error between 100 Hz and 10 kHz.

For sending one sideband, both errors have the same effect: the unwanted sideband isn’t fully cancelled. A gain ratio $g$ and a phase error $e$ leave $\left|1 - g e^{je}\right| / \left|1 + g e^{je}\right|$ of it.

The IIR chains have a catch: they change the phase of every frequency, by a different amount, on both paths. The 90° difference is right, but neither path is the original signal. For a voice that hardly matters, because the ear barely notices phase. But the waveform comes out a different shape.

With a 90° shifter, one sideband can be cancelled. Send the message on the cosine and its Hilbert transform on the sine:

For each frequency in the message, the two products give the same lower sideband with opposite signs, so it cancels, and the same upper sideband twice, so it adds up. Flip the sign to keep the lower sideband instead. This is the phasing method, patented by Ralph Hartley in 1928. The other method is to make DSB, then remove one sideband with a very sharp band-pass filter. That’s the filter method, which needs a crystal filter in a real radio.

Single sideband (SSB) fixes both of AM’s problems. It uses half the bandwidth, and all of its power carries the message. Most long-distance voice radio uses it: amateur radio, aircraft over the oceans, and ships.

Speech for SSB is usually cut to 300 Hz – 3 kHz. The phase shifter then doesn’t have to work down to the lowest frequencies, and the ear still hears the voice’s pitch from its harmonics: the missing fundamental.

The receiver is the synchronous detector from section 2, called a product detector here. Its errors now behave quite differently:

• A phase error doesn’t matter at all. It shifts every frequency’s phase by the same amount, the level doesn’t change, and the voice sounds the same.
• A frequency error doesn’t make the sound fade. It moves every frequency by the same number of hertz.

That cartoon duck is a frequency shifter: everything moved by the same number of hertz. It’s a very different thing from pitch shifting, which multiplies every frequency by the same ratio, so harmonics stay harmonic. Synthesisers have both. The frequency shifter, also called a Bode shifter after the engineer Harald Bode, is the phasing method above, used as an audio effect.

Every scheme so far puts the message in the carrier’s size. So anything that changes the size of the received signal changes the sound: noise added on top, a signal that fades as a car moves, a crackle from a lightning strike.

Frequency modulation (FM) puts the message in the carrier’s frequency instead, and keeps its size constant:

$D$ is the peak deviation: how far the frequency swings when the message is at its peak. The receiver mixes down to I and Q as in section 4, and measures how fast the arrow turns from one sample to the next. That’s the frequency. The arrow’s length doesn’t come into it, so changes in level don’t matter.

FM’s bandwidth isn’t simply twice the message’s, as for AM. A rule of thumb named after John Carson, Carson’s rule, says about 98% of the power falls within $2(D + f_{\text{top}})$, where $f_{\text{top}}$ is the message’s highest frequency. The receiver’s filter has to be a bit wider than that, because cutting off the last 2% distorts the sound. This one uses 1.5 times Carson’s width.

The extra bandwidth is the price of FM’s big advantage. Noise moves the arrow’s tip a little in a random direction. If the arrow sweeps through a wide range of angles for the message, as it does with a big deviation, that small random wobble matters less.

Broadcast FM radio uses a deviation of 75 kHz for 15 kHz of audio, so Carson’s width is 180 kHz: twelve times the bandwidth of the audio, spent on a clean sound. Edwin Armstrong showed what wide deviation buys, in 1936.

FM has one more trick. If two FM stations share a frequency, the receiver locks onto the stronger one and almost completely ignores the weaker. The arrow mostly points where the stronger signal says, and the weaker one only nudges it. This is the capture effect, and it’s why aircraft radio (which uses AM) can carry two pilots talking at once, while an FM radio between two stations flips from one to the other.

Every receiver on this page started in the same way: multiply by a cosine and a sine at the station’s frequency, and low-pass the results. That gives I and Q, the arrow. Only the last step differs:

• AM: the arrow’s length, $\sqrt{I^2 + Q^2}$. No diode needed, and no phase to get right.
• SSB: a filter that keeps only positive (or negative) frequencies, then I.
• FM: how fast the arrow turns.

The low-pass filter here does the tuned circuit’s job from section 3. Only the station you’re tuned to lands near 0 Hz, so the low-pass keeps it and removes the rest, with edges much sharper than a coil and a capacitor could manage. Analogue radios do the same thing with the superheterodyne design, which Edwin Armstrong invented in 1918. They mix every station down to one fixed “intermediate” frequency. There, a filter that never has to be retuned, so it can be made both flat and steep-sided, picks out the one you want.

A software-defined radio (SDR) takes the idea all the way. Its hardware does one thing: it mixes a slice of the radio spectrum down to I and Q, and digitises them. Everything else is software, and the same samples can be heard as AM, SSB or FM, or decoded as digital data.

• Multiplying by a carrier moves a message up in frequency, as two mirror-image sidebands. Multiplying again brings it back, but only if the receiver’s carrier has the right phase: at 90° off, the output is silent.
• AM sends the carrier as well, so a receiver can just trace the envelope. It’s simple and robust, but most of its power is in the carrier, and it uses twice the bandwidth it needs.
• I/Q: a receiver that mixes with a cosine and a sine keeps everything, whatever its phase. The pair is one complex number, an arrow, which can tell positive frequencies from negative.
• The Hilbert transform shifts every frequency by 90°. As an FIR filter, its errors are in the gain. As two IIR all-pass chains, they are in the phase, and only a few sections are needed.
• Single sideband uses the Hilbert transform to cancel one sideband: half the bandwidth, and no wasted power. A tuning error moves every frequency by the same amount, which is not the same as a pitch shift.
• FM puts the message in the frequency, so changes in level don’t matter. It spends bandwidth (Carson’s rule) to buy a cleaner output, until the noise gets past a threshold. The stronger of two stations wins.
• A software-defined radio mixes to I and Q once, then demodulates any of these in software.

• J. R. Carson, “Notes on the theory of modulation”, Proceedings of the IRE 10 (1922): the first analysis of FM’s bandwidth.
• R. V. L. Hartley, “Modulation system”, US patent 1,666,206 (1928): single sideband by the phasing method.
• E. H. Armstrong, “A method of reducing disturbances in radio signaling by a system of frequency modulation”, Proceedings of the IRE 24 (1936).
• D. K. Weaver, “A third method of generation and detection of single-sideband signals”, Proceedings of the IRE 44 (1956): a way to do SSB with no wideband 90° shifter.
• R. G. Lyons, Understanding Digital Signal Processing, 3rd edition, Prentice Hall (2010): quadrature signals and Hilbert transformers, very clearly explained.
• M. Rice, Digital Communications: A Discrete-Time Approach, Pearson (2009).