M8-L3 ended with a promise. It measured the damage that delay spread does to a wideband signal, defined the coherence bandwidth that separates flat fading from frequency-selective fading, and then said that the answer was a technique called OFDM — and deferred it. This is that lesson. Orthogonal frequency-division multiplexing is the modulation of WiFi, LTE, 5G, digital television, digital radio and every wired DSL line, and it earned that position by turning multipath from the hazard M8-L3 described into something a receiver can undo with one complex multiplication per subcarrier.
Take the case M8-L3 worked. An urban macrocell has an RMS delay spread of about στ = 1 µs, because echoes from buildings a few hundred metres away arrive a few microseconds after the direct ray. Now send a single-carrier signal through it at a symbol rate of 5 MBd, so that by M6-L2’s definition each symbol lasts Ts = 1/5×10⁶ = 200 ns. Every symbol’s echoes are still arriving while the next several symbols are being transmitted:
An equaliser can undo it. A five-tap equaliser, adapting continuously as the channel changes, is exactly what GSM receivers do — and it is expensive, it needs training sequences, and its complexity grows sharply as the channel gets longer. Push the symbol rate to 50 MBd for a WiFi-class data rate and the same 1 µs of delay spread spans 50 symbols; a 50-tap adaptive equaliser tracked in real time is not a component you casually put in a battery-powered device. Single-carrier transmission had hit a wall, and the wall was not noise, not power and not bandwidth. It was the channel’s memory.
The OFDM idea is a change of variables, not a new physics. Instead of one stream at 5 MBd, send N parallel streams at 5/N MBd each, on N carriers stacked side by side across the same total bandwidth. Nothing about the total data rate changes — you are still delivering 5 million symbols per second — but each individual symbol now lasts N times as long. With N = 128 the per-subcarrier symbol lasts 128 × 200 ns = 25.6 µs, against a delay spread of 1 µs, so the echo occupies 1/25.6 = 3.9% of a symbol instead of five whole symbols. The ISI has not been cancelled; it has been made small enough to fence off, which is what the cyclic prefix does below.
There is a second, equivalent way to see the same win, and it is the frequency-domain view M8-L3 set up. That lesson gave the coherence bandwidth as Bc ≈ 1/(5στ), so at στ = 1 µs the channel is flat over about 1/(5 × 10−6) = 200 kHz. A 5 MHz single-carrier signal is 25 times wider than that, so it is deep in frequency-selective fading: some of its spectrum is amplified and some is nulled, and the receiver has to repair a distorted spectrum. Slice the same 5 MHz into subcarriers 15 kHz apart, and each subcarrier is 200/15 = 13 times narrower than the coherence bandwidth. Each one therefore experiences flat fading — a single complex gain, one number. The wideband problem has been decomposed into a few thousand narrowband problems, each of which is trivial.
The two views are the same statement. A channel with memory in time is a channel with structure in frequency, and the reason a long symbol sees flat fading is that a long symbol is a narrow spectrum. M6-L2’s reciprocal relationship between symbol duration and bandwidth is doing all the work here: making symbols longer to dodge time-domain ISI is identical to making them narrower to dodge frequency-selective fading. Every OFDM design decision in this lesson is one of those two sentences.
Stacking N carriers side by side is an old idea — it is just FDMA from M10-L1, applied inside one transmitter instead of across users. Done naively it is wasteful, because M10-L1’s guard bands would have to sit between every pair of subcarriers, and with a thousand subcarriers you would spend most of your spectrum on guard bands. OFDM’s contribution is that it needs no guard band at all between subcarriers. Their spectra overlap heavily, and they are still perfectly separable. The condition that makes this true is a single choice of spacing:
Read that concretely. If Tu = 66.67 µs and Δf = 15 kHz, then subcarrier 1 fits 15000 × 66.67×10−6 = 1.0 cycle in the symbol, subcarrier 2 fits 2.0 cycles, subcarrier 1200 fits 1200.0 cycles. Not 1199.7, not 1200.3 — exactly 1200. Multiply subcarrier 7 by subcarrier 12 and average over the symbol and you are averaging five complete cycles of a sinusoid, which is zero. That is the whole mechanism, and it is worth pausing on how little it demands: no filters, no guard bands, no separation in frequency. Just a period and a spacing that are reciprocals.
The frequency-domain picture is the one that makes OFDM feel like a conjuring trick. Because each subcarrier is a sinusoid truncated to a rectangular window of length Tu, its spectrum is a sinc function — a main lobe of width 2Δf and sidelobes decaying slowly on both sides. The subcarriers therefore overlap enormously: at the centre of subcarrier 5, subcarrier 4 and subcarrier 6 are both still present as spectra. But the sinc has zeros at every multiple of Δf, so at the exact centre frequency of subcarrier 5, every other subcarrier contributes exactly zero. The receiver does not filter subcarrier 5 out; it samples the spectrum at subcarrier 5’s centre, where all its neighbours happen to vanish. Overlapping and separable at once, and the interactive panel below plots exactly this.
Orthogonality is not a robust property; it is a knife edge. Shift the receiver’s frequency reference by a fraction ε of the subcarrier spacing, and it no longer samples at the nulls. Each subcarrier now picks up energy from all the others, and that leakage has a name: inter-carrier interference, ICI. It is a floor that no amount of transmit power removes, because the interference scales with the signal. Two mechanisms create the offset, and M8-L3 introduced both:
So OFDM trades one sensitivity for another. It is remarkably tolerant of multipath, which is what kills single-carrier systems, and remarkably intolerant of frequency error, which single-carrier systems barely notice. That trade was the right one to make because multipath is a property of the world and frequency error is a property of your hardware — and hardware got better.
Here is the part that turned OFDM from a curiosity into an industry. Read the orthogonality condition again as an instruction for building a transmitter and it says: generate N sinusoids at spacings of 1/Tu, multiply each by its own QAM symbol from M5-L4, and add them. That is N oscillators, N multipliers and N filters, all of which must hold their relative phases to a fraction of a degree. For N = 1200 it is not a radio, it is a building.
But the sum of N complex exponentials at integer multiples of a fundamental, weighted by N coefficients, sampled at N points, is not merely like an inverse discrete Fourier transform. It is one, written out:
The cost saving is the reason OFDM exists in products. Computing that sum directly, subcarrier by subcarrier, takes N complex multiplications per output sample and N samples, so N² complex multiplications per symbol. A radix-2 FFT takes (N/2) log₂N. For LTE’s 20 MHz configuration, N = 2048:
That last line is the history of this subject. The multicarrier idea was published by Robert Chang at Bell Labs in 1966; Weinstein and Ebert showed in 1971 that the DFT implements it; Peled and Ruiz added the cyclic prefix in 1980. Every theoretical piece of OFDM was in the literature before the first mobile phone call, and it went unused for roughly two decades — not because anyone doubted the mathematics, but because nobody could afford the arithmetic. OFDM reached consumers only when cheap digital signal processing did: European digital audio broadcasting in 1995, ADSL in the mid-1990s, IEEE 802.11a in 1999, LTE in 2009. It is the clearest example in this course of a communications technique whose adoption date was set by Moore’s law rather than by insight.
Long symbols shrink the ISI but do not remove it, and they do something else too: an echo delayed by τ destroys the integer-cycle property that orthogonality depends on, because the delayed copy is no longer aligned with the FFT window. So OFDM adds one more trick, and it is the most elegant idea in the lesson. Before transmitting a symbol, copy the last TCP seconds of it and paste that copy onto the front. The transmitted symbol is now Tu + TCP long and starts with a redundant repetition of its own tail.
Two things follow, and they are worth stating separately because they are usually blurred together. First, the prefix is a guard interval: an echo arriving up to TCP late spills only into the prefix, and the receiver simply discards the prefix before running the FFT, so no energy from the previous symbol reaches the transform. That kills the ISI. Second, and less obviously, within the retained window the delayed copy looks exactly like a cyclic shift of the symbol rather than a truncated one — because the piece that would have been missing from the front was pasted there on purpose. A cyclic shift in time is a pure phase rotation in frequency. So the FFT sees each subcarrier multiplied by one complex number:
The prefix is, however, pure overhead. It carries no new information — by construction it is a copy — so every second spent transmitting it is a second not spent sending data. That gives the two design constraints that decide every OFDM numerology ever standardised, and they pull in opposite directions:
LTE also defines an extended cyclic prefix of 16.67 µs, used in large cells and hilly terrain where echoes come from further away. Its reach is 16.67 µs × 3×10⁸ = 5001 m, five kilometres of excess path — and it costs 16.67/(66.67 + 16.67) = 16.67/83.34 = 20.0% overhead, three times the normal prefix. That is the trade priced exactly: a fifth of your capacity, in exchange for surviving a channel three and a half times longer. A network turns it on where the terrain demands it and nowhere else.
There is a second overhead in that channel, and it has nothing to do with the prefix. Of the 64 subcarriers only 52 are used: 48 carry data and 4 are pilots for estimating Hk. The remaining 12 are left empty — 11 at the band edges as a guard against adjacent channels, plus the DC bin, which is discarded because receiver DC offset lands exactly there. So 52 × 312.5 kHz = 16.25 MHz of the 20 MHz channel is actually occupied, and the fraction of subcarriers carrying payload is 48/64 = 75%. Multiply the two overheads and the honest figure is 0.75 × 0.80 = 60% of the raw channel-time-bandwidth product delivering payload. M6-L3 called 20–30% a typical system overhead; 802.11a spends 40%, and that comparison is fair criticism of a 1999 design rather than a defence of it.
The payoff, and the check that all of this is right: 48 data subcarriers, one symbol every 4.0 µs, is 48/4.0×10−6 = 12×10⁶ = 12 million data symbols per second. At 64-QAM each carries 6 bits (M5-L4), giving 72 Mbit/s raw, and with the rate-3/4 code that mode uses, 72 × 3/4 = 54 Mbit/s ✓ — the top rate on the 802.11a specification sheet, reconstructed from the numerology alone.
Both worked examples had comfortable margin, so it is worth saying plainly what happens when they do not, because the failure is worse than a gentle degradation. If an echo arrives later than TCP, two things break at once. The tail of the previous symbol reaches into the retained window, which is ISI returning exactly as the single-carrier case had it. And the delayed copy inside the window is no longer a clean cyclic shift, so the channel is no longer diagonalised by the FFT — Hk stops being a single complex number per subcarrier and the subcarriers start leaking into each other, which is ICI again. The one-multiply equaliser of the equation above is then solving the wrong problem, and no amount of transmit power helps, because both impairments scale with the signal. A WiFi link taken outdoors into a courtyard with 1 µs of delay spread does not get slower and noisier; it hits an error floor and stops. This is what the verdict readout in the panel below is telling you.
A technique this dominant attracts uncritical description, so here is the bill. Three items, and the first is serious enough to have changed a standard.
Everything so far concerned one link. M10-L1 gave three ways to share a channel among users, and each had a characteristic weakness: FDMA wastes spectrum on guard bands and gives each user a fixed slice whether they need it or not; TDMA gives each user the whole channel but only sometimes, and needs guard times; CDMA lets everyone transmit at once but pays for it with near–far power control and multiple-access interference. OFDMA — orthogonal frequency-division multiple access — is the observation that an OFDM transmitter has already built a grid that solves all three.
The N subcarriers are indexed in frequency and the symbols are indexed in time, so an OFDM signal is a two-dimensional resource grid: subcarriers by symbols. Nothing requires all the cells in that grid to belong to the same user. Hand a rectangle of it — some subcarriers, for some symbols — to each user, and you have multiple access with no guard bands (orthogonality already separates the subcarriers), no guard times (the cyclic prefix already absorbs timing error), and no power-control tightrope (users are orthogonal, not merely uncorrelated). It is FDMA and TDMA at the same time, on a grid whose separation is exact by construction, with a scheduler choosing the rectangles.
The scheduler is where the real gain is, and it is worth being precise about why OFDMA beats all three of M10-L1’s schemes rather than merely tying with them. Because each subcarrier fades independently over a bandwidth of Bc (M8-L3), at any instant a given user’s channel is good on some subcarriers and poor on others — and different users have different good subcarriers. A scheduler that knows this can give every user the part of the spectrum where that user is strong, every millisecond. FDMA cannot: its slices are fixed. TDMA cannot: it hands over the whole band or none of it. CDMA cannot: it spreads every user across everything by design. This is called frequency-selective or multi-user scheduling gain, and it converts M8-L3’s frequency-selective fading from a defect into a resource — the same inversion the closing paragraph of M9-L4 promised.
The grid needs a unit of allocation, because scheduling individual subcarriers would drown the system in signalling. LTE’s unit is the resource block: 12 consecutive subcarriers, for one 0.5 ms slot.
5G NR keeps the resource block at 12 subcarriers but makes the spacing configurable, in powers of two: Δf = 15, 30, 60, 120 or 240 kHz. A resource block is therefore 180 kHz at 15 kHz spacing and 12 × 30 = 360 kHz at 30 kHz spacing. Because the prefix scales down with Tu, the overhead does not change — at 30 kHz spacing Tu = 33.33 µs and the normal prefix is 4.69/2 = 2.34 µs, and 2.34/(33.33 + 2.34) = 6.6%, the same fraction. What does change is the pair of physical limits: wider spacing tolerates more Doppler, which is why millimetre wave uses 120 kHz, and narrower spacing tolerates more delay spread, which is why the low bands stay at 15 or 30 kHz. Scalable numerology is not a new idea in this lesson; it is the same three-way argument, made adjustable.
WiFi 6 does the same thing under a different name. 802.11ax quadrupled the symbol length: a 20 MHz channel now takes a 256-point FFT, so Δf = 20×10⁶/256 = 78.125 kHz and Tu = 1/78125 = 12.8 µs. With a 0.8 µs prefix the symbol is 13.6 µs and the overhead is 0.8/13.6 = 5.9%, down from 802.11a’s 20%. There are 234 data and 8 pilot subcarriers, so 234/256 = 91.4% of the bins carry payload against 802.11a’s 75%, and the combined useful fraction rises to 0.914 × (12.8/13.6) = 0.914 × 0.941 = 86%, against 60%. Two decades of OFDM engineering, and the mechanism of the improvement is entirely contained in the design rule above: longer symbols make the guard interval cheap. 802.11ax also added OFDMA proper, in units called resource units — the smallest is 26 tones, and 26 × 78.125 kHz = 2.03 MHz, with nine of them filling the 234 data subcarriers of a 20 MHz channel (9 × 26 = 234 ✓).
| System | Channel / FFT | Δf | Tu | TCP | CP overhead |
|---|---|---|---|---|---|
| 802.11a/g | 20 MHz / 64 | 312.5 kHz | 3.2 µs | 0.8 µs | 0.8/4.0 = 20.0% |
| 802.11n/ac, short GI | 20–160 MHz / 64–512 | 312.5 kHz | 3.2 µs | 0.4 µs | 0.4/3.6 = 11.1% |
| 802.11ax (WiFi 6) | 20 MHz / 256 | 78.125 kHz | 12.8 µs | 0.8 µs | 0.8/13.6 = 5.9% |
| LTE, normal CP | 20 MHz / 2048 | 15 kHz | 66.67 µs | 4.69 µs | 4.69/71.36 = 6.6% |
| LTE, extended CP | 20 MHz / 2048 | 15 kHz | 66.67 µs | 16.67 µs | 16.67/83.34 = 20.0% |
| 5G NR, mid-band | 100 MHz / 4096 | 30 kHz | 33.33 µs | 2.34 µs | 2.34/35.67 = 6.6% |
| 5G NR, millimetre wave | 400 MHz / 4096 | 120 kHz | 8.33 µs | 0.59 µs | 0.59/8.92 = 6.6% |
Read the last column, not the first. Six of the seven rows use a completely different subcarrier spacing and land on 6.6%, 5.9% or 20.0% — and the three systems that chose short symbols are the ones paying 11% and 20%. The spacing is chosen by the physics of the deployment: delay spread sets the floor under TCP, Doppler sets the ceiling over Tu, and the overhead is what falls out. Nobody picked 15 kHz or 78.125 kHz because it was elegant. They picked the guard interval their channel needed and then made the symbol long enough to afford it.
It is easier to list what does not use OFDM. Digital audio broadcasting, digital terrestrial television, ADSL and VDSL, powerline networking, every WiFi generation from 802.11a onward, LTE, 5G NR in both frequency ranges, and most satellite return links: all OFDM or a close relative. The exceptions are instructive rather than embarrassing — LTE’s uplink chose SC-FDMA for battery reasons, Bluetooth stayed single-carrier because its channel is short and its power budget tiny, and GPS uses spread spectrum because its problem is not multipath but hiding a signal under the noise floor (M10-L1).
Three things this lesson deliberately did not do. It treated the multipath channel as something to be neutralised, when several antennas at both ends can use the same multipath to send parallel data streams — that is MIMO, M10-L3, and it composes with OFDM rather than competing with it: real systems run MIMO per subcarrier, which is only tractable because the cyclic prefix made each subcarrier a single complex gain. It said nothing about how the two directions of a link share the medium in time or frequency, which is duplexing, M10-L4. And it named WiFi, LTE and 5G numerologies without describing the standards that contain them, which is Module 11. Every one of those three builds on the grid this lesson constructed.