Stand at the edge of your Wi-Fi's range and watch what happens. The video does not slowly go to pieces, it first drops in resolution, then stutters, then stops. Your phone's signal bars fall and the "5G" label flips to "4G". Behind them is a continuous negotiation: the radio measures how noisy the channel is right now and picks the fastest way of sending bits that will still arrive intact.
There is a limit to that negotiation, and it is not a limit of technology. In 1948 Claude Shannon proved that every noisy channel has a maximum rate, set by just two things: how wide it is and how loud the signal is compared with the noise. Below that rate, errors can be made as rare as you like. Above it, no cleverness helps. This subject is the story of getting close to that line.
Who takes it: electrical and computer engineers, usually in year 4 (sometimes year 3 as an elective).
What it's for: designing the layer that turns bits into waves and back, across a channel that adds noise, echoes and other people's signals.
Comes after: Signals & Systems (34/37) for spectra and filters, and Probability & Statistics (34/05) for noise. Electromagnetics & Waves (34/38) supplies the antenna and the wave.
Leads to: nothing in this list requires it. It is where signals, noise and waves meet, and it is the physical layer under Computer Networks (34/27), which tells the story of the layers above. The radios in Embedded Systems (34/41) use it.
Honest line: you won't be able to design a modem after this. You will see why more bits per second costs more power, why noise turns "wrong" into a probability, what a parity bit really does, and what 4G, 5G and Wi-Fi do under the hood.
A voice is made of frequencies from a few hundred hertz to a few kilohertz. You could, in principle, send it as is: a voltage on a wire. To send it through the air you meet two problems. An antenna must be a decent fraction of a wavelength long to radiate (a quarter of the wavelength of a 3 kHz tone is 25 km), and everyone sending at the same low frequencies would all be on top of each other. The fix is the oldest trick in radio: shift the message up. Generate a steady high-frequency wave, the carrier, and modulate it, which means changing one of its properties in step with the message.
Worked example: broadcast FM swings its carrier ± 75 kHz for 15 kHz audio, so the bandwidth is about 2 × (75 + 15) = 180 kHz, compared with 2 × 5 = 10 kHz for an AM station. That is FM's trade: bandwidth spent to buy noise immunity, a small version of what Shannon proved in general. In the Americas AM stations are spaced 10 kHz apart (9 kHz elsewhere), and FM stations in the US 200 kHz apart.
Drawn with 20 carrier cycles per message cycle so you can see the shape; a real carrier has thousands. For AM the strength slider sets the depth (0 to 1.5); for FM it sets the index β (0 to 8).
To send bits instead of a voice, you modulate the same properties with a short list of allowed values. Switching the carrier on and off is ASK; flipping its phase by 180° is BPSK; using four phases gives QPSK, two bits per symbol. Putting in both phase and size gives QAM: 16-QAM carries 4 bits per symbol, 64-QAM 6, 256-QAM 8. A convenient picture is the constellation: each allowed symbol is a point on a plane, whose horizontal coordinate is the in-phase part of the carrier and the vertical coordinate the quarter-cycle-shifted part. The transmitter sends a point. The receiver sees that point plus noise, and decides which allowed point is nearest.
The noise is the thermal jiggle of electrons in every receiver, spread evenly over all frequencies and bell-curve shaped (Gaussian). The noise power in a channel of bandwidth B is kTB: at room temperature this is about −174 dBm per hertz, so a 20 MHz Wi-Fi channel has a floor of about −101 dBm before anything else is added. The chance of a wrong decision is a tail probability from Probability & Statistics (34/05): the noise must push the point more than half the gap to the neighbour.
That fall is why digital wins. Below a threshold of signal to noise almost every symbol is wrong; a few dB above it, almost none. And because the receiver snaps to the nearest allowed point, noise does not accumulate along a chain of relays the way it does in an analogue copy of a copy. The price is spacing: more points in the same power shrink the gaps. For the same chance of error, each four-fold increase in points (2 more bits per symbol) needs about 6 dB more signal: going from QPSK to 16-QAM takes 7.0 dB, 16 to 64 takes 6.2 dB, 64 to 256 takes 6.1 dB.
Rings: the ideal points (average power 1). Dots: 600 received symbols, blue if decoded correctly, red if the nearest ideal point is the wrong one. Theory: the standard error formulas, checked against a long simulation. SNR here means energy per symbol over noise density.
Before 1948 engineers believed that noise forced a trade-off, and that zero errors needed a rate of zero. Shannon's paper "A Mathematical Theory of Communication" (Bell System Technical Journal, July and October 1948) said that this was wrong. For any channel there is a number, the capacity, such that below it you can make the error probability as small as you want (with clever enough coding), and above it you cannot. For the standard channel with Gaussian noise the number has a famous form, sometimes called the Shannon–Hartley theorem (after Ralph Hartley's 1928 work on counting signal levels):
Read it as two dials. Bandwidth multiplies the rate: double B, double C. Signal-to-noise sits inside a logarithm: at high S/N, doubling the signal power adds only one bit per second per hertz, about 3 dB. So more bandwidth is a straight-line gain and more power a very slow one, which is why every wireless generation hunts for new spectrum. Worked numbers: a telephone line (3.1 kHz, 35 dB) is limited to about 36 kbit/s, and that is where dial-up modems topped out. A 20 MHz Wi-Fi channel at 25 dB allows 166 Mbit/s; a 100 MHz 5G carrier at 20 dB, 666 Mbit/s (per stream; extra antennas multiply it).
Shannon proved the limit exists but not how to reach it. It took until the 1990s to get close. Plain QAM with no error correction sits some 8 dB above the limit if you demand one symbol error in 100,000, and 6 dB at one in a thousand; the orange squares in the chart show it. Coding (the next section) closes most of that gap. There is also a lower bound that no amount of bandwidth removes: the energy per bit must exceed about −1.59 dB of the noise density (ln 2 as a ratio), or nothing can be sent reliably.
Blue curve: log₂(1 + S/N), the capacity per hertz. Orange squares, left to right: BPSK (1 bit), QPSK (2), 16-QAM (4), 64-QAM (6), 256-QAM (8), each placed at the SNR where theory gives 1 symbol error per 1,000, without any error-correcting code.
At the edge of a channel a few bits come out wrong, and a flipped bit in a file is a corrupt file. The fix is to add redundancy cleverly. The crudest scheme is repetition: send every bit three times and vote. It works, but it triples the cost. The elegant idea is to add a few check bits, each of which is a parity (even or odd count of ones) of a chosen subset of the data bits. Richard Hamming, frustrated at Bell Labs in the late 1940s by a relay computer that stopped whenever it hit one bad bit, published a code in 1950 that does this for four data bits.
The Hamming (7,4) code adds three check bits so that the three parities, read as a binary number (the syndrome), spell out the position of a single flipped bit: zero means all is well, five means the fifth bit is wrong. The receiver flips it back. Two errors defeat it. That is the trade-off of all codes: more redundancy corrects more errors, at the cost of rate.
Later codes work on this idea at scale. Reed–Solomon codes (1960) correct bursts of errors and sit in CDs (together with interleaving, which can repair a scratch of a few millimetres) and in QR codes. Turbo codes (Berrou, Glavieux and Thitimajshima, 1993) came within about 0.7 dB of Shannon's limit. LDPC codes, which Robert Gallager had invented in the early 1960s and which were forgotten for three decades, did the same, and 5G uses them for user data and polar codes (Erdal Arıkan, 2009) for control signalling.
1. Pick four data bits (tap to flip). 2. Tap any received bit to add noise. Watch the checks point at the damage.
Check bits p1, p2 and p4 sit at positions 1, 2 and 4; each covers the positions whose binary number contains that bit. The table uses exact binomial probabilities for independent flips.
Many users must share the same air, and there are only a few ways to keep their signals apart. Each one slices a different dimension.
OFDM is a way of sending; OFDMA is a way of sharing: different users get different groups of subcarriers at the same moment, assigned by the base station every millisecond or so.
Each bit is spread over 8 chips with a Walsh code (A uses +−+−+−+−, B uses ++−−++−−), and the two are simply added. Each receiver correlates with its own code. Real systems also need synchronisation and power control, because codes only cancel perfectly when they are aligned.
Now every part has a place. A modern radio does four things, in a loop: it measures the signal-to-noise ratio of the channel; it picks the densest modulation and the lightest code that will still give an acceptable error rate (this is adaptive modulation and coding, and the reason your link speed changes as you walk away); it spreads users over time and frequency; and it adds an error-correcting code with retransmissions as the last resort. The generations differ mainly in how wide a channel they use, how dense a constellation they dare, which codes they choose, and how finely they share.
| System | Sharing | Densest modulation | Coding | Channel width |
|---|---|---|---|---|
| GSM (2G) | FDMA + TDMA | GMSK (low rate) | convolutional | 200 kHz |
| 3G (UMTS) | CDMA | QPSK, later 16- and 64-QAM | convolutional, turbo | 5 MHz |
| 4G LTE | OFDMA down, SC-FDMA up | to 256-QAM in later releases | turbo | 1.4 to 20 MHz |
| 5G NR | OFDMA, flexible spacing | 256-QAM | LDPC (data), polar (control) | up to 100 MHz (below 7 GHz), 400 MHz (mmWave) |
| Wi-Fi 6 | OFDMA, listen-before-talk | 1024-QAM | LDPC | 20 to 160 MHz |
| Wi-Fi 7 | OFDMA | 4096-QAM | LDPC | up to 320 MHz (6 GHz) |
Each step adds bandwidth (the straight-line gain), adds modulation density (a slow logarithmic gain, available only close to the tower or router, where S/N is high), and swaps in a code nearer to Shannon's limit. 5G adds two things outside the table: very high bands (24 to 71 GHz) where there is plenty of spectrum, and arrays of many small antennas that steer a narrow beam to each user, which raises S/N exactly where it is needed. Those millimetre waves die quickly in walls and rain, which is why they come with small cells and short range: the wave rules in Electromagnetics & Waves (34/38) set the design.
A radio link moves a message by modulating a carrier, and digital modulation puts symbols on a constellation that noise smears. Errors are tail probabilities, and denser constellations need more signal. Shannon's 1948 limit C = B·log₂(1 + S/N) says how much a channel can carry, and error-correcting codes (Hamming to LDPC) are how real links approach it. 4G, 5G and Wi-Fi are the same recipe in different proportions: wider channels, denser modulation, better codes, finer sharing.
Try this: in your phone's Wi-Fi or mobile settings, look at the signal strength and the link speed while walking away from the router. Watch the speed drop in steps. Each step is the radio dropping to a sparser constellation as the SNR falls.
Sources. Modulation, error formulas and OFDM are standard textbook material (Haykin; Proakis; Goldsmith). C. E. Shannon, "A Mathematical Theory of Communication", Bell System Technical Journal 27 (July and October 1948); Hartley, BSTJ 7 (1928) 535–563; Nyquist, Trans. AIEE 47 (1928) 617–644 (see Wikipedia, "Shannon–Hartley theorem"). Armstrong's five patents of 26 December 1933 (Wikipedia, "Edwin Howard Armstrong"). Hamming (7,4): R. W. Hamming, 1950 (Wikipedia, "Hamming code"). Reed and Solomon, 1960, CD and QR use (Wikipedia, "Reed–Solomon error correction"). Turbo codes: Berrou, Glavieux and Thitimajshima, 1993, within about 0.7 dB of the limit at a bit error rate of 10⁻⁵ for rate 1/2 (later confirmed by Divsalar and Pollara, JPL, 1995). LDPC from Gallager (early 1960s) and polar codes (Arıkan, 2009); 5G uses LDPC for data and polar for control (3GPP TS 38.212). 5G NR bands: FR1 410 to 7125 MHz, FR2 24.25 to 71 GHz (3GPP; Wikipedia, "5G NR frequency bands"); carriers up to 100 and 400 MHz. LTE: OFDMA downlink and SC-FDMA uplink, 1.4 to 20 MHz channels (Wikipedia, "LTE"); 15 kHz spacing; 256-QAM in later releases. Wi-Fi 6 (802.11ax): OFDMA, 1024-QAM; Wi-Fi 7 (802.11be): 4096-QAM, 320 MHz in 6 GHz. AM and FM band plans: ITU, Wikipedia, "Medium wave" and "FM broadcast band". Thermal noise floor −174 dBm/Hz at 290 K (kT). Computed in node: all interactive numbers, the 7.0, 6.2 and 6.1 dB steps, the 6 and 8 dB gaps, and the Hamming and repetition error rates; the simulation matched the exact formulas on 200,000 symbols.