Communication Systems: How Much Can a Noisy Channel Carry?

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.

A message rides on a carrier

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.

• AM (amplitude modulation) changes the size of the carrier. The message shows up as the wave's outline, the envelope. It is simple to receive (a diode and a capacitor will do), but any noise that changes the size of the wave is heard. AM takes a bandwidth of twice the highest message frequency.
• FM (frequency modulation) changes how fast the carrier oscillates and leaves its size constant. Noise mostly changes size, so FM shrugs it off, at the price of more bandwidth: Carson's rule gives about 2 × (swing + highest message frequency). Edwin Armstrong's wide-band FM was granted five US patents on 26 December 1933.

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.

Digital modulation: constellations with noise

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.

Bandwidth, noise and Shannon's limit

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.

Bit errors and error-correcting codes

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.

Sharing the air: FDMA, TDMA, CDMA and OFDM

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.

• FDMA (frequency): each user gets its own slice of frequency. Radio and TV stations; each GSM phone carrier is a 200 kHz slice.
• TDMA (time): users take turns on the same frequency, in short slots. GSM puts eight users on each 200 kHz carrier.
• CDMA (code): everyone transmits at once on the same frequency, each multiplying its bits by its own pattern of +1 and −1 "chips". A receiver multiplies by the same pattern, and the others, whose patterns are orthogonal to it, average away. 3G phones and GPS satellites (all on one frequency) work this way.
• OFDM (many slow carriers): split the channel into hundreds or thousands of narrow subcarriers, each carrying a slow stream, spaced so their spectra don't interfere. It is computed with an FFT (Signals & Systems, 34/37). Slow symbols mean echoes, whose delays Electromagnetics & Waves (34/38) explains, cause little harm, particularly when a short copy of each symbol's end is added as a guard (the cyclic prefix). In LTE the subcarrier spacing is 15 kHz, so a symbol lasts 66.7 µs and the normal guard is 4.7 µs.

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.

What 4G, 5G and Wi-Fi actually do

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.

What a graduate just sees

• A radio is a message shifted up to a carrier; the bandwidth you need is set by the message and the modulation, not by the carrier.
• Noise is not an obstacle but a probability: error rates fall off a cliff as SNR rises, so there is always a threshold.
• Every extra bit per symbol costs about 3 dB, each four-fold increase in points about 6 dB.
• Capacity is B·log₂(1 + S/N): bandwidth is worth more than power, and neither can be bought past the limit.
• Redundancy, arranged cleverly, can repair errors without asking for a resend; a check-bit pattern can name the guilty bit.
• Sharing is slicing: by frequency, time, code or many slow subcarriers.

Where it shows up next

• Computer Networks (34/27): the physical layer is this subject; the layers above treat the link as a pipe with some loss and delay, and that "loss" is the leftover error after coding.
• Embedded Systems (34/41): every IoT device chooses a radio and a duty cycle; battery life is a trade between rate and range, which is Shannon's dials.
• Computer Security (34/33): jamming, eavesdropping and spoofing a radio link are attacks on this layer.
• Machine Learning (34/32): entropy, cross-entropy and compression come from the same 1948 paper.
• The real world: phones, satellites, GPS, cable and fibre, QR codes, hard drives and SSDs.

If you want to go deeper

• 📘 Textbook: Simon Haykin and Michael Moher, Communication Systems (Wiley): the long-standing standard, with analogue modulation as well as digital.
• 📘 Textbook: John G. Proakis and Masoud Salehi, Fundamentals of Communication Systems (Pearson), and Andrea Goldsmith, Wireless Communications (Cambridge University Press) for the wireless channel, OFDM and capacity.
• 📘 Gentler: Jimmy Soni and Rob Goodman, A Mind at Play (2017), a biography of Shannon; James Gleick, The Information (2011).
• ▶️ Gentler: Khan Academy, "Journey into information theory" (free).
• ▶️ Free course: MIT OpenCourseWare 6.02 Introduction to EECS II: Digital Communication Systems (Fall 2012).
• 🌐 Easier first reads in this library: Computer Networks (34/27), The Internet (29/04) and The Fourier Transform (12/12).