Appearance
7.4 — Multiplexing, Multiple Access and OFDM
One cable, one radio band, one fibre — and thousands of users who all want it at once. Multiplexing is how they share.
There are only four ways to divide a channel: by frequency, by time, by code, or by space. Everything else is a combination.
1. Frequency division
FDM gives each user their own frequency band, all transmitting simultaneously.
This is the oldest and most obvious method. Every radio station in a city transmits at the same moment on a different frequency, and your receiver's tuned circuit (Chapter 1.6) picks out one.
Guard bands are needed between channels, because no filter is perfect and no oscillator is exactly on frequency. They are wasted spectrum, typically 5 to 10% of the total.
Where it is used: broadcast radio and television, cable television (each channel on its own 6 or 8 MHz slot), and the original analog telephone trunk system, which stacked twelve 4 kHz voice channels into a 48 kHz group, five groups into a supergroup, and so on up to thousands of calls on one coaxial cable.
WDM — wavelength division multiplexing — is FDM for optical fibre, with each channel on a different colour of light. Dense WDM packs 80 or more channels 50 GHz apart into a single fibre, each carrying 100 Gbit/s or more. That is how a single strand of glass carries tens of terabits per second (Chapter 8.5).
2. Time division
TDM gives each user the whole bandwidth, in turn, for a brief slot.
The digital telephone system is the clearest example. Each voice channel is sampled at 8 kHz and encoded to 8 bits, giving 64 kbit/s. Thirty-two such channels — thirty for voice, two for signalling and framing — are interleaved into the 2.048 Mbit/s E1 stream used across Europe, or twenty-four into the 1.544 Mbit/s T1 used in North America.
The 8 kHz sampling rate is Chapter 4.7's theorem applied to a 3.4 kHz voice band, with the remaining 600 Hz as a guard for the anti-alias filter's transition.
Framing is the essential overhead: the receiver must know where each frame starts, so a known pattern is inserted periodically and the receiver hunts for it. Losing frame alignment scrambles every channel at once, which is why telephone equipment reports "loss of frame" as a major alarm.
Statistical TDM allocates slots only to users who actually have data, rather than reserving them. This is packet switching, and it is why the internet carries far more users than a circuit-switched network of the same capacity — most users are idle most of the time. Volume I, Chapter 5.1 develops the networking consequences.
3. Code division
CDMA lets everyone transmit at the same time on the same frequency, distinguished by a code.
The mechanism. Each user's data is multiplied by a fast pseudorandom sequence — the spreading code or chip sequence. If the code runs at 128 times the data rate, the signal's bandwidth expands 128-fold and its power spectral density drops correspondingly.
At the receiver, multiplying by the same code again recovers the data, because the code multiplied by itself is 1. Multiplying by a different user's code leaves the signal spread out and looking like noise, provided the codes are orthogonal or nearly so.
Processing gain:
G_p=\frac{B_{spread}}{B_{data}}=10\log_{10}(128)=21\ \text{dB}
That gain is what lets the wanted signal be pulled out from under all the others.
The properties that follow, and each is genuinely useful:
- The signal can be below the noise floor and still be recovered. This is exactly how GPS works (Chapter 8.2) — the satellite signal arrives around 20 dB below thermal noise.
- Resistant to narrowband interference, because despreading spreads the interferer out while gathering the wanted signal together.
- Hard to detect or jam, which is why it was developed by the military. The frequency-hopping variant was patented in 1942 by the actress Hedy Lamarr and the composer George Antheil, using a mechanism based on player-piano rolls to synchronise the hops.
- Soft handover. A phone can talk to two base stations at once, since they share the frequency, and combine both signals.
The problem that dominates CDMA engineering: the near-far problem. A nearby phone's signal swamps a distant one, because the codes are not perfectly orthogonal and the processing gain is finite. Every CDMA system therefore needs fast, precise power control — 3G systems adjust each phone's transmit power 1500 times per second, so that all signals arrive at the base station at nearly equal power.
4. Space division
Reuse the same frequency in different physical places.
The cellular concept. Divide the coverage area into cells, each with its own base station, and reuse frequencies in cells far enough apart that interference is acceptable.
A frequency reuse factor of 7 means each frequency is used in one cell out of every seven, and the reuse distance is
D=R\sqrt{3N}
for cell radius R and cluster size N. With N=7, D=4.58R — the nearest cell using the same frequency is four and a half cell radii away.
The capacity consequence is the entire history of mobile telephony. Halving the cell radius quadruples the number of cells in a given area, and therefore quadruples the capacity, using no extra spectrum at all. Every generation of mobile network has got its capacity mainly from smaller cells, and the femtocell in a home is that trend taken to its conclusion.
MIMO — multiple input, multiple output — is space division inside a single link. Several antennas at each end create several independent spatial paths through the multipath environment, and the receiver separates them by solving a matrix equation.
\text{capacity} \propto \min(N_{tx},N_{rx})
A 4×4 MIMO system carries roughly four times the data of a single antenna pair in the same bandwidth. And it works because of multipath, not despite it — a perfectly clear line-of-sight channel offers no independent paths and MIMO gains nothing. That inversion is worth noting: the thing that ruined earlier systems became the resource that later ones exploit.
5. OFDM
The technique behind Wi-Fi, 4G, 5G, digital television and DSL — and the reason all of them work in environments that would have defeated earlier designs.
The problem it solves
A high-rate single-carrier signal has a short symbol period. Multipath causes echoes delayed by microseconds. If the delay spread exceeds the symbol period, each symbol smears into several following ones, and the equaliser needed to undo it becomes impossibly complex.
Worked numbers. A 20 Mbit/s single-carrier link has a symbol period of 50 ns. Typical indoor delay spread is 100 to 500 ns. The echo covers ten symbols, so the equaliser must untangle ten overlapping symbols simultaneously — thousands of taps, adapting continuously.
The idea
Split the data across many slow subcarriers instead of one fast one.
With 64 subcarriers, each carries 1/64 of the rate, so each symbol lasts 64 times longer — 3.2 µs instead of 50 ns. Now the 500 ns echo is a small fraction of one symbol rather than spanning ten, and the equalisation becomes trivial: one complex multiplication per subcarrier.
Orthogonality
The subcarriers overlap in frequency, which would normally be a disaster. They are chosen so that each one's peak falls exactly on every other one's null:
\Delta f=\frac{1}{T_{symbol}}
Then they are orthogonal — integrating one subcarrier against another over a symbol period gives exactly zero, by the same result as Chapter 4.1's Fourier orthogonality.
So no guard bands are needed at all, and the spectral efficiency is about 50% better than a scheme that separates its carriers properly.
The FFT connection
Generating dozens of carriers with dozens of oscillators would be absurd. The insight, from Weinstein and Ebert in 1971: the whole OFDM signal is an inverse DFT of the subcarrier data.
s[n]=\sum_{k=0}^{N-1}X[k]e^{j2\pi kn/N}
That is exactly the inverse DFT of Chapter 5.1. So:
- Transmitter: map bits to constellation points, one per subcarrier, then take an inverse FFT.
- Receiver: take an FFT, then decode each subcarrier's constellation point.
OFDM became practical only when the FFT became cheap, which is why it was described in the 1960s and deployed in the 1990s. It is the single most important application of the FFT in engineering.
The cyclic prefix
Even with long symbols, an echo from the previous symbol still intrudes at the start of the current one.
The fix is elegant: copy the last portion of each symbol and paste it at the front.
T_{total}=T_{useful}+T_{cp}
Two things happen. Any echo arriving within T_{cp} lands entirely inside the copied portion, which the receiver discards. And because the prefix is a copy of the end, the received signal within the useful part looks like a circular convolution with the channel — which, by Chapter 5.1's convolution theorem, becomes a simple multiplication in the frequency domain.
That is why one complex multiplication per subcarrier equalises the channel. The cyclic prefix is what converts a linear convolution into a circular one, and the circular one is the easy case.
The cost: T_{cp} is pure overhead. Wi-Fi uses 0.8 µs on a 3.2 µs symbol — 20% of the capacity thrown away to make the equalisation trivial. It is a bargain.
Wi-Fi's numbers
For 802.11a/g in a 20 MHz channel:
- 64 subcarriers, spaced 20\ \text{MHz}/64 = 312.5 kHz.
- 48 carry data, 4 are pilots for tracking phase, 12 are unused at the band edges as a guard.
- Symbol period 3.2 µs, plus 0.8 µs cyclic prefix = 4 µs.
- With 64-QAM (6 bits) and rate-3/4 coding:
48\times6\times0.75=216\ \text{bits per symbol}
\frac{216}{4\ \mu\text{s}}=54\ \text{Mbit/s}
Exactly the headline rate of 802.11g, derived from first principles.
The two real problems with OFDM
Peak-to-average power ratio. Adding many subcarriers with random phases occasionally produces a large peak — the sum of N signals can momentarily reach N times the average amplitude.
\text{PAPR}\le10\log_{10}N \ \text{dB}
For 64 subcarriers that is up to 18 dB, though the statistics make 8 to 12 dB the practical figure.
The consequence is expensive. The transmitter's amplifier must stay linear up to the peak, so it must be backed off and runs well below its maximum efficiency. A power amplifier operating 10 dB below saturation might be 15% efficient instead of 60%. In a base station that is a large electricity bill; in a handset it is battery life.
This is why 4G uplink uses SC-FDMA instead — a variant with a lower peak-to-average ratio, chosen specifically because the handset's battery matters more than the base station's.
Sensitivity to frequency offset. The orthogonality depends on the subcarriers being at exactly the right frequencies. A frequency error destroys it and every subcarrier leaks into its neighbours, which is called inter-carrier interference.
\text{tolerable offset} \approx 1\% \text{ of the subcarrier spacing}
For Wi-Fi's 312.5 kHz spacing that works out at 3 kHz when the carrier is 2.4 GHz, which is about 1.2 parts per million. The frequency has to be that accurate or the subcarriers stop being orthogonal and start interfering with each other.
Doppler shift produces exactly that much error. A vehicle moving fast enough shifts every received frequency by the amount Volume IV Chapter 2.5 computes, and at these tolerances that is enough to break the scheme. It is why high-speed trains were a genuine engineering problem for 4G, and why 5G uses wider subcarrier spacing at its higher frequencies — wider spacing means a larger error is tolerable.
6. Multiple access — who transmits when
Multiplexing is dividing a channel. Multiple access is the protocol deciding who gets which division, and it is a different problem.
FDMA, TDMA, CDMA, SDMA are the four divisions applied to users.
OFDMA — the OFDM version — assigns different subcarriers to different users, and can reassign them every millisecond. This is what makes 4G and 5G efficient: a user with a good channel and lots of data gets many subcarriers; a user with a poor channel gets few. The scheduler can even give each user the subcarriers where their particular channel happens to be strong, which is called frequency-selective scheduling and can double a cell's capacity.
Contention-based access — nobody assigns anything, and stations transmit when they judge the channel free.
ALOHA, from the University of Hawaii in 1971, is the ancestor: transmit whenever you have data, and if there is a collision, wait a random time and retry. Maximum throughput 18% — most of the channel is wasted on collisions.
Slotted ALOHA restricts transmissions to slot boundaries, halving the collision window and doubling throughput to 37%.
CSMA/CD — listen before transmitting, and abort if you detect a collision. Classic Ethernet, achieving 60 to 90% on a lightly loaded network.
CSMA/CA — listen before transmitting, and use random backoff to avoid collisions rather than detecting them. Wi-Fi, because a radio cannot listen while transmitting — its own signal is billions of times stronger than any it might hear — so collision detection is impossible and only avoidance is available.
The hidden node problem is Wi-Fi's structural weakness: two stations that can both hear the access point but not each other will transmit simultaneously and collide, and neither will know why. The RTS/CTS handshake solves it by having the access point announce that the channel is reserved, at the cost of extra overhead.
7. Duplexing — talking both ways
FDD — frequency division duplex. Uplink and downlink on different frequencies, so both can be continuous. Needs a duplexer, a pair of sharp filters, and paired spectrum allocations.
TDD — time division duplex. The same frequency, alternating in time. No duplexer needed, and the split between uplink and downlink can be adjusted to match the traffic — which matters enormously now that downloads vastly exceed uploads.
TDD's requirement is tight synchronisation between cells. If one base station transmits while a neighbour receives, on the same frequency, the neighbour is deafened. So all base stations in a TDD network must be time-aligned, usually from GPS, which is a genuine dependency worth knowing: a large fraction of the world's mobile networks stop working properly if GPS timing is lost.
5G uses TDD in its higher bands for exactly the flexibility reason, and this has made GPS timing a critical infrastructure concern.
Chapter 7.5 asks the question all of this has been circling: given a bandwidth and a noise level, what is the absolute maximum rate, and how close can anything get to it.
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.
Multiplexing
Spread spectrum processing gain:
G_p=\frac{B_{spread}}{B_{data}}=\frac{R_{chip}}{R_{bit}}
Cellular reuse:
D=R\sqrt{3N}
Halving the cell radius quadruples the capacity in a given area with no extra spectrum.
MIMO capacity:
C\approx\min(N_{tx},N_{rx})\cdot B\log_2(1+\text{SNR})
OFDM
\Delta f=\frac{1}{T_{useful}} \qquad\text{(the orthogonality condition)}
T_{symbol}=T_{useful}+T_{cp}
R_b=\frac{N_{data}\times\log_2M\times r_{code}}{T_{symbol}}
Worked, for 802.11a in 20 MHz: 48 data subcarriers × 6 bits (64-QAM) × 0.75 (code rate) ÷ 4 µs = 54 Mbit/s.
Peak-to-average power ratio:
\text{PAPR}\le10\log_{10}N \ \text{dB}
Frequency offset tolerance: about 1% of the subcarrier spacing before inter-carrier interference becomes significant.
Why the cyclic prefix works: it makes the received signal within the useful window equal to a circular convolution with the channel, and circular convolution becomes multiplication under the DFT — so one complex multiply per subcarrier equalises everything.
ALOHA throughput: 18% pure, 37% slotted.
What the next chapter fixes
Every scheme so far has been a way of using a channel well. None of them says how well a channel can be used. Chapter 7.5 gives the limit, proved by Shannon in 1948, which no modulation scheme however clever can exceed — and which tells you immediately whether an engineering problem is hard or impossible.