Frame Structure and the Resource Grid

A phone and a base station must agree on where each bit goes. They must agree very exactly: to the microsecond in time, and to the kilohertz in frequency. This note explains the map they share. It starts from zero. You do not need to know what a subcarrier or a symbol is. By the end you will know both, and you will know every channel and signal that 5G sends. Nearly all the numbers come from the 3GPP standard TS 38.211; where a channel width or a procedure is needed, they come from TS 38.101-1 and TS 38.213, and the note says which.

In one line. Think of a spreadsheet. Each column is a small piece of time. Each row is a radio frequency. Each cell holds one piece of data. This note answers two questions: how wide is a row, and how long is a column?

How to read this note. It is built as a ladder. It gets harder on purpose, and in one direction only — nothing later ever changes anything earlier, so you can stop at the end of any part and what you have learned stays true. Every fact carries a clause number from the standard, so you can always check it. Simple words are used, but the technical terms are the real ones — you will meet them again in any 5G document.

First, the picture in plain words

This section is not from the standard. It is the background knowledge that the standard expects you to have already. If you know OFDM well, go to §2.

It climbs in the same way the note does. §1.1 to §1.3 are the minimum — three ideas, no mathematics, and enough to read everything in Parts II and III. §1.4 to §1.6 go one level deeper: how the symbol is actually built, why the guard is a copy rather than silence, and the only two things that can break the whole scheme. Take them now if you want the mechanism, or come back to them from §10, where they are used.

One wide channel, or many narrow ones?

Imagine you have 20 MHz of radio spectrum. You must send data through it. The simple idea is to use all 20 MHz as one fast channel. But there is a problem: echoes.

Your signal travels to the phone by more than one path. One copy goes straight through the air. Other copies bounce off a building, a truck, or a hill. The copies that bounce arrive late, by a few microseconds.

Now imagine your symbols are very short. A late copy of symbol 5 arrives while symbol 6 is playing. The two mix together, and the receiver cannot read either one. The faster you send, the worse this problem becomes. A receiver can repair some of the damage, but this costs a lot of processing, and it gets harder when the bandwidth is larger.

OFDM does the opposite. It does not use one fast channel. It uses many slow channels, side by side. Each narrow channel is called a subcarrier. Each subcarrier sends data slowly, so its symbols are long. An echo of a few microseconds is then only a small part of a long symbol, not a whole symbol. You do not lose speed, because you send on hundreds or thousands of subcarriers at the same time.

Why the tones do not interfere

The subcarriers sit next to each other with no space between them. This sounds like a bad idea. Surely they will mix?

They do not mix, and the reason is the spacing. The spacing is chosen so that each subcarrier fits a whole number of its own waves into one symbol. When this is true, one symbol is long enough for each pair of subcarriers to cancel out completely. The receiver can then read subcarrier 7 and see nothing at all from subcarrier 8.

Engineers call this orthogonality. It is the main idea behind OFDM.

Orthogonality has one condition. It is the most important formula in this note:

Read the formula like this: wide spacing gives short symbols. Narrow spacing gives long symbols. A spacing of 15 kHz gives a symbol of 66.7 µs. A spacing of 60 kHz gives a symbol of 16.7 µs. You cannot choose both freely. If you want wide spacing, you must accept short symbols. Physics does not allow anything else.

We call $T_u$ the useful symbol time. The next part explains why: a small extra piece of time is added to the front of every symbol.

The guard interval, in one paragraph

Long symbols make echoes less harmful. But they do not make echoes safe. So OFDM adds a safety margin on purpose. Before each symbol there is a short piece of time called the cyclic prefix. The receiver ignores whatever arrives during this time.

An echo that arrives inside this window does no damage. An echo that arrives after the window damages the symbol.

The length of this guard is one choice, and the whole note depends on it. If you make it long, you waste time in every symbol. If you make it short, the system fails wherever there are far-away reflections. §10 gives the exact length in 5G, and shows what it means in metres.

The whole design in three steps. 1. Choose a subcarrier spacing. 2. The symbol length is then fixed, because $T_u = 1/\Delta f$. 3. Add a guard that is a fixed part of the symbol.

You have now decided how far an echo may travel before it causes damage. In other words, you have decided what size of cell you are building.

How a radio actually builds a symbol

A fair question at this point: if a 5G carrier has three thousand subcarriers, does the transmitter contain three thousand oscillators?

No. It contains one Fourier transform.

This is the whole trick, and it is worth saying slowly. Write the numbers you want to send into an array — one slot per subcarrier. Run an inverse discrete Fourier transform over that array. What comes out is the waveform, sample by sample, with every tone already added together. The receiver captures the samples and runs the forward transform, and the array comes back.

So "hundreds of tones" is not hundreds of pieces of hardware. It is one operation that a chip does in microseconds, and it is the same operation whether there are 12 tones or 3300.

Look at the exponent. Subcarrier $k$ turns $k$ times faster than subcarrier 1. Over one useful symbol $T_u = 1/\Delta f$, subcarrier $k$ therefore completes exactly $k$ whole turns. That is orthogonality, written as a formula — and it is why the sum can be undone exactly.

The standard does not say "IFFT" anywhere. It writes the sum instead, in clause 5.3.1, whose title is worth reading in full: OFDM baseband signal generation for all channels except PRACH and RIM-RS. Those two get their own clauses (5.3.2 and 5.3.3) because they do not use the same transform as everything else — the random-access preamble is longer and sits on its own grid.

Where this idea comes from. Multiplexing data onto orthogonal tones was proposed by Chang (1966). Doing it with a discrete Fourier transform — which is what made it buildable — is Weinstein and Ebert (1971). Both are listed in §11.2.

Why the guard is a copy, and not silence

Go back to the guard interval. It has a name that sounds odd: the cyclic prefix. Why prefix, and why cyclic?

Because it is not silence. It is a copy of the end of the symbol, pasted onto the front.

Silence would have been the obvious choice, and it does not work. Here is the reason, in one sentence: an echo that arrives late drags a piece of the previous symbol into the window the receiver is reading, and — worse — it means each tone no longer completes a whole number of turns inside that window. The moment that is untrue, orthogonality is gone, and every tone leaks into its neighbours.

Copying the tail to the front fixes both problems at once. Whatever the delay, as long as it is shorter than the guard, the receiver's window always sees a complete cycle count of every tone. The delayed copy is not a smear any more. It is the same tone, turned in phase.

Read that equation as the punchline of the whole section. Without the cyclic prefix, the channel mixes each sample with the ones before it, and undoing that costs a great deal of processing. With it, the channel becomes one number per subcarrier. The receiver measures that number using the demodulation reference signals (§9), divides, and is done.

Where this idea comes from. The cyclic prefix is Peled and Ruiz (1980). Applying the whole scheme to a moving radio channel — which is what a phone is — is Cimini (1985). See §11.2.

The two things that break it

Orthogonality is a promise with conditions. Exactly two things break it, and every number in the rest of this note is a defence against one of them.

The two fixes pull in opposite directions. That is the tension in Table 1, and it is the reason 5G has seven numerologies instead of one.

Some numbers, so this is concrete. A phone in a train at 500 km/h, on a 3.5 GHz carrier, sees a Doppler shift of about 1.6 kHz. Against a 15 kHz spacing that is 11 per cent of the gap — far too much to ignore. Against 120 kHz it is 1.3 per cent. Nothing changed except the spacing.

Phase noise behaves the same way and gets much worse with frequency, which is why millimetre wave uses the wide numerologies even where the echoes would allow narrow ones. The standard reference for how much error an OFDM link can tolerate is Pollet, Van Bladel and Moeneclaey (1995); 5G's answer at the system level is written up in Zaidi et al. (2016), listed in §11.2.

Two acronyms you will now meet everywhere. Breakage 1 is ISI — inter-symbol interference, one symbol landing on the next. Breakage 2 is ICI — inter-carrier interference, one tone landing on the next. Almost every design choice in OFDM is a trade between these two.

Why one size cannot fit all

One radio standard must work in two very different places. It must work at 700 MHz in a country cell that is 10 km wide. It must also work at 39 GHz inside one room. These two situations need almost opposite things.

The two columns ask for opposite things. No single spacing can satisfy both. 4G chose one answer, 15 kHz, and accepted the limits.

5G does not choose. It defines a family of spacings. The large time units stay the same for the whole family, so everything stays lined up in time. Then each carrier — and even each part of a carrier — uses the member of the family that fits it best.

Remember this idea. 5G did not make the radio better by picking better numbers. It made the radio flexible, and then kept one thing fixed so that the flexible parts still fit together. You will see this pattern again in every section below.

Numerology — one setting with seven values

One whole number chooses the member of the family. The standard calls it $\mu$ (the Greek letter mu).

3GPP uses the word numerology for "which member of the family we are using". Learn this word now. The standard uses it on almost every page.

Each step up in $\mu$ doubles the spacing. Because $T_u = 1/\Delta f$, each step also halves the symbol length.

The family starts at 15 kHz. This is the 4G value, and 5G keeps it so that the two systems can work side by side. After that, you simply double: 15, 30, 60, 120, 240, 480, 960 kHz.

As a rough rule: the low numerologies are for frequencies below 6 GHz, where most phones work. The high numerologies are for millimetre wave, where cells are small and echoes are short.

A check you should do once, by hand. The standard never writes $T_u = 1/\Delta f$. Instead it writes $N_u^{\mu} = 2048\kappa\cdot 2^{-\mu}$, in units of a very small constant $T_c$ (clause 5.3.1). At first this looks like a different thing. It is not. Multiply it out and you get exactly $1/\Delta f$, for all seven values of $\mu$. If your arithmetic gives the same answer, then you have understood the time constants correctly. §10 explains where $\kappa$ and $T_c$ come from.

How an operator actually chooses $\mu$

The table above lists what is possible. It does not say what anyone picks. In practice the choice is squeezed from three sides, and the answer is usually forced.

Then the band limits what is left. TS 38.101-1 Table 5.3.5-1 lists, for every FR1 band, exactly which spacings and channel widths a phone must support — and the answer for FR1 is always some subset of 15, 30 and 60 kHz. The higher numerologies are not FR1 options at all.

The 60 kHz oddity, explained. 60 kHz is allowed in FR1 but is rarely the whole story. It is the one numerology with a choice of cyclic prefix (Table 2 — normal or extended), and the extended prefix exists precisely for the case where you want short slots for latency but still face long echoes. You pay a quarter of every symbol for it.

How many blocks fit in a real channel

Everything so far has been an abstract grid. A phone tunes to a channel: a licensed slab of spectrum with a width in megahertz. So how many resource blocks fit in one?

The answer is a lookup, not a calculation — TS 38.101-1 Table 5.3.2-1. It matters because it is the first place the neat arithmetic meets a regulator.

Three things in that table are worth stopping on.

The channel is never full. A 20 MHz channel at 15 kHz spacing carries 106 blocks, which is 19.08 MHz of subcarriers. The missing 920 kHz is the guard band, split between the two edges — 460 kHz each side, less half a subcarrier, which is the 452.5 kHz that TS 38.101-1 Table 5.3.3-1 prints. The guard band is empty on purpose: a transmitter cannot stop cleanly at a hard edge, and the regulator measures what leaks out.

5G fills more of the channel than 4G did. LTE put 100 blocks in a 20 MHz channel — 18 MHz of subcarriers, 90 per cent (TS 36.101, which is not in the archive). 5G puts 106 blocks in the same channel, 95.4 per cent. Nothing changed in the arithmetic. The usual explanation is that filtering and transmitter linearity improved enough between the two standards to allow a narrower guard band, and 3GPP spent the difference on data.

There is a ceiling, and you can see it bite. A grid may be at most 275 resource blocks wide (TS 38.213 clause 12). Multiply that out and you get the widest channel each spacing can ever reach:

So the reason the mid-band workhorse runs at 30 kHz is not really a radio argument at all. A 100 MHz channel does not fit in 275 blocks of 15 kHz. You can check this against TS 38.101-1 Table 5.3.5-1: band n78 lists 100 MHz under 30 kHz and 60 kHz, and stops at 50 MHz under 15 kHz.

Note on frequency range 2. Everything above is FR1, because TS 38.101-1 is the FR1 document and it is the one in the archive. The matching FR2 requirements are in TS 38.101-2, which is not. Where this note needs an FR2 number it says so.

Frames, subframes, slots, symbols

5G divides time four times. Each unit is made of smaller units.

The top two units never change. They are the same for every value of $\mu$. This is important: it is the reason a 15 kHz carrier and a 120 kHz carrier can sit next to each other and still agree on when a frame starts.

One table has seven rows. The other has one. This difference tells you everything about the extended cyclic prefix. It is not a second family. It is one extra option, and it exists only at $\mu = 2$. Everywhere else in 5G, a slot has fourteen symbols.

Now here is the point that confuses most people. A slot at $\mu=0$ lasts a full millisecond. A slot at $\mu=6$ lasts only 15.6 µs. Both slots contain fourteen symbols.

A slot is a count, not a length of time. It is always fourteen symbols. How long those fourteen symbols take depends on the numerology. Most mistakes about 5G timing come from reading "slot" as a fixed amount of time. It is not.

This also explains the low-latency claim you hear about 5G. A transmission can start at the beginning of a slot. At $\mu=0$ you may have to wait up to 1 ms for the next slot. At $\mu=3$ you wait 125 µs at most.

So a higher numerology gives you a faster clock. The price is a shorter guard interval. Keep this trade in mind. The rest of the note measures it.

The resource grid, the element, the block

Time is done. Frequency is the easier half, because it works the same way in every release and at every numerology — the rules below do not change with $\mu$ at all.

Why twelve? Because the scheduler must give out spectrum in pieces, and a piece of one subcarrier would be far too much to keep track of. Twelve subcarriers is the piece.

Now look carefully at what stays fixed. A block is twelve subcarriers. It is not a fixed number of hertz. So the real width of a block follows the numerology: 180 kHz at 15 kHz spacing, and 11.52 MHz at 960 kHz spacing.

This choice was made on purpose, and it is a good one. The scheduler counts blocks, and it never needs to know which numerology it is using. You write the scheduler once, and it runs anywhere in the family.

From resource element to frame, in boxes

Every unit is now on the table. §3 and §4 gave you the time side, §5 the frequency side. This section is the recap, and it is a picture rather than an argument: six boxes, and each box holds a whole number of the box below it.

Notice what this means. Almost nothing in the standard is given as a length of time on its own. Each thing is a count of smaller things. Only the two largest boxes are fixed to the clock.

Four of the six boxes are time and two are frequency, which is why this recap comes after both halves and not between them. Figure 3 puts them on one ladder, because a resource element is the place where time and frequency meet — you cannot draw the bottom box without both.

The ladder — each box, an exact count of the one below

You can read this figure in two directions.

Read it downward and you are zooming in. This is the order the standard uses: it defines the frame first, then the smaller units.

Read it upward and you are building a frame out of resource elements. This is how a scheduler sees the system. It has resource elements to give away, and the boxes are the sizes it gives them away in.

Where the family splits in two. The top two boxes — frame and subframe — are the same for every device on every carrier. The bottom four boxes all change with $2^{\mu}$.

This is the whole trick. Two radios can use different numerologies on the same carrier. They will disagree about every box smaller than a millisecond. But they still agree, sample by sample, about when each millisecond begins.

The same millisecond, at every numerology

Figure 3 is drawn for one numerology only. Now look at the box that never changes — the 1 ms subframe — divided by all seven numerologies.

There are three things to read from Figure 4.

1. The number of slots doubles. The millisecond does not. There is one slot per subframe at 15 kHz, and sixty-four at 960 kHz. This is TS 38.211 Table 4.3.2-1 (Table 6) drawn as a picture. It is also why the network can make a scheduling decision every 15.6 µs at the top of the family, but only once per millisecond at the bottom.

2. The thin symbol lines disappear around $\mu = 4$. This is not a problem with the drawing. Every subframe holds $14\cdot 2^{\mu}$ symbols: fourteen at the bottom of the family, and 896 at the top. The receiver must do several jobs once per symbol — the transform, the channel estimate, the demapping. At $\mu$, it has only $2^{-\mu}$ of the time it had at 15 kHz. This shrinking time budget is what makes the high numerologies hard. The mathematics is easy; the deadline is not.

3. The dashed line at 0.5 ms is in the same place in every row. Two symbols in each subframe get the longer prefix: $l = 0$ and $l = 7\cdot 2^{\mu}$. The symbol $7\cdot 2^{\mu}$ is always the one that starts at the half-millisecond, in every numerology. §10.4 explains why the standard spends the extra $16\kappa$ exactly there.

What the boxes are worth, counted

Figure 3 gives you the shape. Figure 4 gives you the scale. This table gives you the numbers.

Every column is a count of boxes, not a measurement. You can work out each one yourself from Equation 4 and the two fixed counts: twelve subcarriers in a block, fourteen symbols in a slot.

The last column is the main lesson of this section. Take the number of resource elements that one block holds in one millisecond. Divide by the width of that block in megahertz. Every row gives the same answer: 933.

It cannot give anything else. A block is $180\cdot 2^{\mu}$ kHz wide, and it holds $168\cdot 2^{\mu}$ elements per subframe. The $2^{\mu}$ appears on both sides, so it cancels.

A higher numerology does not give you more capacity. The boxes show you why. One step up halves every box in time and doubles it in frequency. So the area of a resource element — its time multiplied by its bandwidth — is the same everywhere in the family. It is $(2048+144)/2048 = 1.07$ for a normal-prefix symbol.

You carry the same number of complex numbers per hertz per second at 15 kHz as at 960 kHz.

What changes is how fast you can start, and how much echo you survive. In other words, latency and delay spread — never throughput. §10 measures the price you pay.

One warning: a transmission does not have to fill a box. The boxes are the units the system is built from. They are not a rule about how much a transmission must use.

With PDSCH mapping type B, the transmission can start at symbol $S \in {0,\ldots,12}$ and can last $L \in {2,\ldots,13}$ symbols, with normal cyclic prefix, as long as $S+L \le 14$ (TS 38.214 Table 5.1.2.1-1). So a transmission can start in the middle of a slot and last only two symbols.

Dahlman calls this separating the transmission length from the slot length. Early papers call it the mini-slot. Dahlman argues that it is the better way to reduce latency, because it does not add any extra cyclic-prefix cost. Raising the numerology does.

Point A and the four numberings

Blocks are the unit, so blocks need numbers. 5G numbers them from four different starting points. That sounds like too much. In fact each one has a clear job.

Everything begins with point A. Point A is one agreed frequency. It is the zero of the ruler.

Every numerology on the carrier measures from the same point A. This is what stops a 15 kHz block grid and a 60 kHz block grid from disagreeing about where they are.

Point A is a frequency. It is not the edge of anything. The network tells the phone where point A is in one of two ways: as an offset from the block the phone used to find the cell (offsetToPointA), or as an absolute channel number (absoluteFrequencyPointA) (clause 4.4.4.2).

Point A does not even have to be inside the carrier. It has one job only: to be a shared zero that belongs to no numerology.

Bandwidth parts

A carrier can be much wider than a phone needs at that moment. Listening to 100 MHz uses real battery power, even when no data is arriving.

So 5G lets the network cut out a bandwidth part. This is a run of blocks next to each other, with its own numerology. The phone then works inside that part only.

In practice this works like a power switch. Put an idle device on a narrow part. Move it to a wide part when data arrives. The carrier never changes.

This is the first thing in this note that exists only to save battery. It will not be the last. Once real radio hardware enters the story, power is most of what the design is about.

A bandwidth part is four numbers

The idea sounds vague until you see how little it takes to describe one. A bandwidth part is four settings, and nothing else (TS 38.213 clause 12):

That is the whole object. Notice what is not in it: no channel width in megahertz, no band, no frequency in hertz. A bandwidth part is described entirely in blocks counted from point A (§7) — which is exactly why point A had to exist.

The length is capped at 275 blocks (clause 12), the same ceiling as the grid itself (§3.2). So a bandwidth part can be as wide as the whole carrier, and often is.

Two documents, two jobs — this pattern repeats. TS 38.211 clause 4.4.5 defines what a bandwidth part is: a set of blocks with a numerology. TS 38.213 clause 12 says how it is operated: how many you may have, which one is active, how it changes. Definitions live in 38.211; procedures live in 38.213 for control and 38.214 for data. Knowing this saves a great deal of searching.

The first one, before anything is configured

There is a chicken-and-egg problem here. The network configures bandwidth parts by sending a message. To receive a message, the phone must already know which blocks to listen to.

5G solves it with an initial bandwidth part, which the phone works out before it has been told anything personal. If the network provides initialDownlinkBWP, that is it. If it does not, the phone derives it from the control region it used to read the broadcast in the first place: the initial downlink bandwidth part is then the run of blocks spanned by the CORESET for the Type0-PDCCH common search space, with that region's spacing and cyclic prefix (TS 38.213 clause 12).

In plain words: the first bandwidth part is "however wide the door you came in through was". Everything after that is negotiated.

How the active one changes

Only one part is active per direction at a time, so switching is a real event with real cost. There are three ways it happens (TS 38.213 clause 12):

Why this is more than a power trick. A bandwidth part carries its own numerology. So switching parts can also switch $\mu$ — the same phone, the same carrier, a different spacing and a different slot length. That is the flexibility promised in §2, and this is the mechanism that actually delivers it.

What travels on the grid

Everything that 5G sends is one of two things:

• a channel carries information that comes down from the higher layers. This is your data, or control messages about your data;
• a signal carries no such information. It exists for the radio itself: to find the cell, to measure the channel, or to track the phase.

Count them. The downlink has eight kinds of signal. The uplink has three.

This difference is not a mistake in the design. Every uplink signal costs the phone battery. And, as the next note will show, it also costs radio hardware that the phone may not even have.

One term to learn correctly before you continue: an antenna port is not an antenna.

The standard defines a port by a property of the channel, not by a piece of metal. Two symbols are on the same port when you can work out the channel of one from the channel of the other (clause 4.4.1).

The same clause also defines quasi-co-location. This is the same idea, but only for the large-scale properties of the channel: delay spread, Doppler, average gain, average delay, and spatial receive parameters.

The port numbers are not in 4.4.1. They are two separate lists, one per direction: uplink (clause 6.2) — 0 and up for PUSCH demodulation reference signals, 1000 for SRS and PUSCH, 2000 for PUCCH, 4000 for PRACH; downlink (clause 7.2) — 1000 for PDSCH, 2000 for PDCCH, 3000 for CSI-RS, 4000 for the SS/PBCH block, 5000 for positioning reference signals.

Keeping "port" and "physical antenna" separate is what makes the next note possible. SRS antenna switching is exactly the case where the two do not match.

The first twenty milliseconds — how a phone finds a cell

The table is a list. It does not show the order things happen in. And the order is the part that makes the list make sense, because a phone switched on in a strange city knows nothing: not the cell, not point A, not its own identity, not even the frame boundary.

Everything it needs is bootstrapped from one small object: the SS/PBCH block, universally called the SSB (TS 38.211 clause 7.4.3).

Here is the sequence, in plain words.

1. Find the rhythm — PSS. The phone sweeps the band looking for one of three known 127-symbol sequences. Finding one gives it symbol timing, a rough frequency correction, and the first of the two numbers that identify the cell (clause 7.4.2.2).
2. Finish the identity — SSS. A second sequence, in a known position relative to the first, supplies the other number — one of 336. The two combine into the physical cell identity, $N^{\text{cell}}_{\text{ID}} = 3 N^{(1)}_{\text{ID}} + N^{(2)}_{\text{ID}}$, and $3 \times 336 = 1008$ cells can be told apart (clause 7.4.2.1).
3. Read the notice board — PBCH. Now the phone can demodulate, so it reads the broadcast channel. This tells it, among other things, where to find the full system information — and offsetToPointA, which is where the ruler starts (§7).
4. Read the rules — PDCCH, then PDSCH. With point A known, the phone can address the grid. It reads the control channel in the initial bandwidth part, and through it the first system information block.
5. Speak for the first time — PRACH. Only now does the phone transmit. The random-access preamble is the first thing it ever sends, and it is the only channel with its own signal generation clause (TS 38.211 clause 5.3.2), because it must work before timing is agreed.

Read step 5 again. Every step before it is the phone listening. The whole design assumes the network is talking constantly and the device is silent until it has understood enough to be useful. That asymmetry is the same one visible in the table above — eight downlink signals, three uplink — and it is fundamentally about battery and hardware, not about protocol taste.

Why the uplink has a second waveform

One more thing hides inside the table, and it matters more to a hardware engineer than anything else on this page. The uplink does not always use plain OFDM.

The problem is a consequence of Equation 2. Add up several thousand tones with random phases and occasionally they all line up, producing a brief spike far above the average level. The ratio of that spike to the average has a name — the peak-to-average power ratio, or PAPR.

A base station does not care much. A phone does, and the reason is the power amplifier. An amplifier must be run backed off from its maximum, with headroom left for the spikes. Back-off is wasted battery, and it is lost range. A waveform with a lower PAPR lets the amplifier run closer to its limit, so the phone shouts further on the same current.

The standard says PAPR out loud, which is unusual. 3GPP normally specifies what to send, not why. But look at the titles of TS 38.211 clauses 5.2.2 and 5.2.3: Low-PAPR sequence generation, type 1 and type 2. And clause 5.1.1 defines a modulation, π/2-BPSK, that exists for the same reason. When a specification names a hardware quantity in a clause title, that quantity was expensive.

This is the seam this folder is built around. PAPR, back-off and amplifier headroom are where the standard stops being mathematics and starts being silicon — the subject of the fe- notes, and the reason §12 points where it does.

The map is now complete, and you can use it. Everything from here to the end of §10 is the same material at one level deeper: not what the numbers are, but where they come from. Nothing later in this note changes anything earlier in it. If you only needed the map, you have it — skip to §11 and read on when you want the derivation.

The cyclic prefix, and why it decides everything

This is the deep end of the note, and it is deliberately last. Everything before it describes the map. This section derives it: it takes the guard interval from §1.3 and shows where every constant in 5G's timing actually comes from — including the two that look most arbitrary, $T_c$ and $\kappa$.

It has more constants than the rest of the note put together. At first they look random. They are not: nearly all of them are 4G's numbers, unchanged. Once you know that, they stop being a list to memorise and become one idea with a history. So we start with the smallest unit and build up.

What you get for the effort. By §10.6 you will be able to prove, in whole numbers, that a subframe is exactly one millisecond at all seven numerologies — and by §10.7 you will be able to convert any numerology into the distance an echo may travel. That second number is the one an operator actually chooses a network by.

Everything is counted in samples

A radio does not work in seconds. It works in samples. A converter ticks at a fixed rate, and every length of time in the system is a whole number of ticks.

Think about why this matters. A symbol of 66.666… µs is not a whole number of ticks. The hardware would have to round it. Small rounding errors would then add up, and after a while two radios would no longer agree on where a frame starts.

So the standard chooses one very small tick. Then it writes every length of time as a whole number of these ticks.

Read the two constants together. $\Delta f_{\max}\cdot N_f$ is a sample rate. It is the widest subcarrier spacing multiplied by the transform size that goes with it. The answer is 1.96608 GHz.

$T_c$ is one tick of that clock, about half a nanosecond. It is small enough that every other length of time in 5G is a whole number of ticks. That is the only reason it exists.

Why $\kappa = 64$, and why it matters

The standard then defines a second constant. At first it looks unnecessary:

$T_s$ is 4G's basic time unit. It comes from a 2048-point transform at 15 kHz, which gives a sample rate of 30.72 MHz. Every LTE chip in the world runs at that rate.

So $\kappa$ has a simple meaning: it is the number of 5G ticks inside one 4G tick.

This is the key to the whole clause. $\kappa$ exists so that 5G can reuse 4G's numbers without changing them.

Here is a reading rule worth keeping: wherever you see $\kappa$ in a length, the number next to it is a 4G number.

The constants come from 4G, unchanged

With that rule, the symbol and guard lengths stop looking random:

Take them one at a time.

So 5G did not invent these numbers. It took 4G's numbers and added one rule: divide by $2^{\mu}$.

Do not take this on trust. Check it. At $\mu = 0$ there is no division, so 5G should give exactly the 4G values:

Why this matters, and it is not only history. A design that reuses 4G's timing can also reuse 4G's chips, filters, test equipment, and the experience of its engineers.

The $\kappa$ in every formula is the joint where 5G was attached to 4G. It is the reason a 15 kHz 5G carrier can sit directly next to a 4G carrier and share the same radio.

Which two symbols get the longer prefix, and why

The condition in Equation 9 is $l = 0$ or $l = 7\cdot 2^{\mu}$. Here $l$ counts the symbols through the whole subframe, not inside one slot.

Those two numbers are the first symbol of each half of the millisecond. At $\mu=0$ they are symbols 0 and 7 out of fourteen. At $\mu=1$ they are symbols 0 and 14 out of twenty-eight.

This is 4G's shape showing through again. A 4G subframe of 1 ms was two half-millisecond slots of seven symbols, and the first symbol of each slot got the longer prefix.

But why does 5G make two prefixes longer at all? Because without them the numbers do not add up. And the amount that is missing is always the same.

Every ordinary symbol costs $2048 + 144 = 2192$ 4G samples. So a subframe full of ordinary symbols costs

A millisecond is $1{,}966{,}080 T_c$. So we are short by exactly 2,048 $T_c$.

That number is $32\kappa$, which is two lots of $16\kappa$. This explains two things at once:

• why there are two longer prefixes in each subframe, and
• why the $+16\kappa$ has no $2^{-\mu}$ next to it. The missing amount is the same at every numerology, so the repair must be the same too.

Two more results follow from this, and the standard never states either of them in words.

The guard is always the same slice of the symbol

Compare the guard with the symbol: $144/2048 = 7.03\%$. The $2^{-\mu}$ is in both numbers, so it cancels. The ratio is the same at every numerology.

This is more important than it looks. Going up one numerology does not just make the guard shorter. It makes the guard shorter by the same factor as everything else.

So there is nothing free anywhere in this family. If you halve your waiting time, you also halve how much echo you can survive. Always.

Checking that a subframe really is a millisecond

§10.4 said that the two longer prefixes make the arithmetic come out exactly right. Do not believe a claim like that — test it. The test is easy. Add up every symbol and every prefix in one subframe, and see whether the answer is exactly one millisecond. Then do it for all seven numerologies.

from fractions import Fraction as F
Tc, k = F(1, 480_000 * 4096), 64                    # clause 4.1
for mu in range(7):
    Nu  = 2048 * k * F(1, 2**mu)                    # clause 5.3.1
    CPl = 144 * k * F(1, 2**mu) + 16 * k            # l = 0 or l = 7*2^mu
    CPs = 144 * k * F(1, 2**mu)
    n   = 14 * 2**mu                                # symbols in a 1 ms subframe
    assert (n * Nu + 2 * CPl + (n - 2) * CPs) * Tc == F(1, 1000)   # exact, all seven

The code uses exact fractions, not decimal numbers. So this is a proof, not a lucky rounding.

What the guard is worth in metres

This is the number that really decides which numerology a network uses.

The guard protects against an echo that arrives late, up to the length of the guard itself. Radio waves travel about 300 metres in one microsecond. So multiply the guard by 300 m/µs, and the guard becomes a distance.

That distance answers a practical question: how much further can a reflection travel than the direct path before it starts to cause damage?

Now you can see the whole design in one view:

Sources, and where to read more

Every number in this note comes from the 3GPP standard. You can check any of them yourself. This section lists what to read, in the order that will help you most.

The standard itself — read this first

The 3GPP documents are free. Search for the number, for example "TS 38.211", on the 3GPP or ETSI website, and take the Release 19 version. In this project they are already downloaded, in Courses (RAW)/5G Topics/.

They are text PDFs, so you can search them from the command line:

cd ~/Documents/md-engine/"Courses (RAW)/5G Topics"
pdftotext -layout ts_138211v190400p.pdf - | grep -n "cyclic prefix"

How to read a 3GPP document without getting lost. Do not start at page 1. Take a clause number from this note, find that clause, and read only it. Each clause is short. The hard part of the standard is not the language — it is that it never explains why. That is what the books below are for.

The papers everyone cites

The standard tells you the rules. It never tells you where they came from, and the ideas in this note are much older than 5G. Below are the papers that the literature keeps returning to — the ones whose names appear in the reference list of nearly every OFDM paper written since.

Read the honesty note at the end of this section before you cite any of them.

Four more are worth knowing by name, because they are where the field worked out what OFDM costs as well as what it gives:

Then the 5G-specific literature. These are recent enough that the authors are often the same people who wrote the specifications, which makes them unusually reliable about intent:

The books

A paper argues one point. A book explains a system. For this material you want two of them: one on 5G, and one on the wireless background that 5G assumes you already have.

For the background in §1 — OFDM, fading, echoes, why any of this is necessary — the two books cited most often in the teaching literature are:

Read one of the two, not both. They cover the same ground in different styles, and the second one adds much less than the time it costs.

Three more, for particular needs:

• van Nee and Prasad, OFDM for Wireless Multimedia Communications, Artech House, 2000 — the book-length treatment of OFDM alone, including the PAPR problem behind §9.2.
• Sesia, Toufik and Baker, LTE — The UMTS Long Term Evolution, 2nd ed., Wiley, 2011 — worth having because so much of 5G is 4G's timing kept unchanged (§10.3). To understand where 144, 160 and 2048 came from, this is the book.
• Molisch, Wireless Communications, 2nd ed., Wiley, 2011 — the reference to reach for on propagation and channel models, when the question is how long the echoes actually are.

Which source answers which question

The lists above are long. This table is the short version: pick the row that matches your question and read only that.

How to check a fact in this note

This is a habit worth building, and it takes about two minutes.

1. Find the claim, and read the clause number next to it — for example (clause 4.3.1).
2. Open TS 38.211 and go to that clause.
3. Compare. If the note and the standard disagree, the standard is right.
4. If a number in the note is calculated rather than quoted, the note shows the calculation. Redo it. §10.6 is a worked example you can run as code.

One warning about section numbers. This note numbers its own sections, and so does 3GPP. They are different. In this note, "§10.4" always means a section of this page, and a 3GPP clause is always written with its document, like "TS 38.211 clause 4.3.2". Watch for this in other people's notes too, because mixing the two is a common mistake.

Where this goes next

You have climbed the whole ladder: a cell you can address, a block of twelve subcarriers, a slot of fourteen symbols, a millisecond that never moves whatever else changes — and, from §10, the reason every one of those numbers is what it is.

But the map does not yet explain one thing. How does the network learn what the radio channel is doing at frequencies where the phone is not transmitting? It must know this, because it has to decide where and how to send the next downlink burst.

That is the job of the third uplink signal: SRS. To answer the question properly, the phone must transmit from antennas that it cannot send data on.

This is the point where the standard stops being a document about mathematics and becomes a document about hardware. That is the subject of nr-srs.md, and it is the most important note in this folder.