Subcarrier — One narrow tone of the OFDM grid

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The grid and its units

This page is the explanation. Subcarrier, live is the working half — pick any channel bandwidth and subcarrier spacing TS 38.101-1 defines, and a table, a chart marker and two figures follow, each number naming the clause, table, choice or formula it came from. It is computed by 5G Simulation/ while the page loads, so it cannot drift from what is written here.

Where it sits

Sits atLevel 0 of the hierarchy · The grid and its units · explained
Learn firstNothing — this is a place to start.
UnlocksOFDM · Resource element · Numerology
Primary clauseTS 38.211 §4.4.4
Used innr-frame-structure 35 · nr-pdsch 10 · index 1 · nr-srs 1
Scanned from the notes at page load and joined with terms.json; nothing on this card is typed by hand.

This is level 0 of the hierarchy — nothing comes before it, and nothing can. Every other concept in these notes is built on top of this one: a resource element is one subcarrier for one symbol; a resource block is twelve of them; a bandwidth part is a range of blocks; the transport block size is a count of elements times a code rate. All of it is arithmetic on this object, which is why the page is longer than a definition needs to be.

So this page has no prerequisites — but it does borrow. Explaining a tone properly means naming things that come later: the cyclic prefix to say why the channel goes diagonal, point A and the resource block to say how a tone is addressed, the numerology to say how wide one is. Every one of them is a link, and none of them is something you are expected to know first — they sit above this page in the hierarchy, not below it. If a borrowed idea gets in the way, follow the link, read the definition, and come back.

What it is

A subcarrier is one narrow sinusoid — one tone — and 5G transmits thousands of them side by side. Its width is set by the numerology: 15, 30, 60, 120, 240, 480 or 960 kHz (TS 38.211 Table 4.2-1). It is the finest division of frequency the standard has.

One convention, before any number appears: a frequency quoted for a subcarrier is its centre. That is how the standard defines things — point A is "the centre of subcarrier 0 of common resource block 0" (TS 38.211 clause 4.4.4.3) — and it is what every figure and table here follows.

Every exception carries the word "edge". A subcarrier's own edge sits 15 kHz from its centre, and the channel's two edges are a different kind of boundary altogether, 845 and 875 kHz beyond the outermost subcarriers. If a number is not labelled an edge, it is a centre.

Watch the verbs, because they can disagree with the number. Starts, begins and ends are edge words, so a label reading "CRB 12 starts, 3 304.32 MHz" contradicts itself — that frequency is where the block's lowest subcarrier is centred, and the figures now say so.

The scale is the surprising part. A 100 MHz carrier is not one signal. It is 3 276 separate tones, each 30 kHz wide, each carrying its own complex number every symbol, all transmitted simultaneously and all separated again at the receiver.

One tone, in frequency — 30 kHz every frequency here is a subcarrier centre The same tone, across time the 100 MHz channel 875 kHz guard 845 kHz guard guards not to scale the same span, drawn again at the same scale CRB 284 ends 3 402.57 MHz CRB 12 starts 3 304.32 MHz middle of the grid CRB 100 m = −576, below the middle point A — CRB 0 — 3 300.00 MHz offsetToCarrier 12 blocks × 12 = 144 0 1 200 1 1 201 2 1 202 3 1 203 4 1 204 5 1 205 6 1 206 7 1 207 8 1 208 9 1 209 10 1 210 11 1 211 CRB 100 12 subcarriers index k each cell becomes a row 1 200 1 201 1 202 1 203 1 204 1 205 1 206 1 207 1 208 1 209 1 210 1 211 0 1 2 3 4 5 6 7 8 9 10 11 12 13 time — 14 OFDM symbols, one slot, 0.5 ms at 30 kHz one resource element — the pair (k, l) frequency what that row actually is k = 1 206 from point A  ·  m = (1 206 − 144) − 3 276/2 = −576 x(t) = cos(2π m Δf t) — 576 turns per symbol, drawn with 3 so the shape shows
Figure 1: One subcarrier at four magnifications, with frequency running up the page in every column so the four can be read against each other. Far left is the 100 MHz channel, framed by a black rule at each of its two edges: its blue middle is the occupied part and the two red caps are the guard bands, drawn far wider than the 0.845 per cent they really are. Note that the channel is taller than the carrier beside it, and is meant to be - a guard band is exactly the amount by which the channel overhangs the carrier at each end, so only the blue middles line up. The two dashed lines from it say what the relation is - that blue middle is exactly the carrier bar beside it, same span, same scale. Down the carrier's left side runs one ruler stack, every measurement drawn the same way - extension lines out to a common column, the dimension between them, the label beside it: purple for m, measured from the middle of the grid, and red for offsetToCarrier, measured from point A and labelled with its own arithmetic - 12 resource blocks of twelve subcarriers each, so 144. Two stretches are drawn wider than they are: the guard caps, and the gap down to point A, which at the bar's own scale would be ten pixels. One block of it, CRB 100, is magnified into twelve subcarriers, and each of those twelve becomes a row of the resource grid on the right - the twelve dashed connectors are that mapping. Underneath is the tone one of those rows carries. The next figure magnifies the two ends of the leftmost column.
the blue part is the carrier of the previous figure, on its side 3 266 more subcarriers 97.98 MHz, not drawn channel edge 3 303.460 MHz channel edge 3 403.460 MHz 100 MHz channel at 30 kHz 845 kHz 875 kHz 15 kHz edge centre first subcarrier — k = 144, CRB 12 edge 3 304.305  ·  centre 3 304.32 MHz 15 kHz edge centre last subcarrier — k = 3 419, CRB 284 centre 3 402.57  ·  edge 3 402.585 MHz point A sits 3.460 MHz below this channel edge, off the bar to the left — 3.460 + 0.845 guard + 0.015 half a subcarrier = 4.32 MHz, the offsetToCarrier of the block-grid figure below To scale: the subcarriers, one cell = 30 kHz.   Compressed, and marked with a zigzag: the two guard bands and the middle of the band. The guard is measured to the subcarrier's edge, not its centre — that 15 kHz at each end is why 3 276 subcarriers occupy 98.28 MHz while their centres span 98.25.
Figure 2: The leftmost column of the previous figure, laid on its side and magnified at both ends. Same channel, same numbers, drawn this way because a guard band is 0.845 per cent of a channel and needs the room. Read it left to right: channel edge, 845 kHz of guard, the subcarriers, 875 kHz of guard, the other channel edge. Only the subcarriers are to scale, one cell being 30 kHz, and the three compressed stretches carry a zigzag to say so. The detail the magnification exists for is at each end: the guard is measured to the outermost subcarrier's edge, and its centre is another 15 kHz inside that - which is why 3 276 subcarriers occupy 98.28 MHz while their centres span only 98.25. Point A is not on this bar: it is 3.460 MHz below the lower channel edge, and adding that to the guard and the half subcarrier - 3.460 + 0.845 + 0.015 - gives the 4.32 MHz offsetToCarrier drawn in the next figure. The k indices are counted from point A, not from the channel edge, which is why the first subcarrier of this carrier is k = 144 and not k = 0.

The numbers along the bars are addresses, and they are all counted from that red mark. Point A is the zero of the block grid: the centre of subcarrier 0 of common resource block 0 sits exactly on it (TS 38.211 clause 4.4.4.3). Blocks then count upwards in frequency, from zero — and the twelve cells of the magnified stack are numbered 0 to 11 inside their own block, with 0 at the bottom because that is the lowest of the twelve. Nothing in NR is numbered from one.

Point A is a frequency, and it is usually not the edge of the carrier. This carrier begins 12 blocks above it — that gap is offsetToCarrier, from TS 38.331. Twelve blocks of twelve subcarriers is 144 subcarriers, which is where that number in the figure comes from and the only place it comes from: it is $12 \times 12$, and the 12 is a number this example chose. So the carrier's 273 blocks are CRB 12 to CRB 284 rather than 0 to 272 — which is the whole of the figure below: the grid is the ruler, and the carrier is a window onto part of it. The labels at the two ends of the carrier bar mark exactly where it starts and stops: CRB 12's lowest subcarrier is centred on 3 304.32 MHz at the bottom and CRB 284's highest on 3 402.57 MHz at the top, if point A is at 3 300 MHz — and point A itself sits below the bar, off the carrier entirely. The amber strip in the figure is the guard band — the channel's lower edge cuts through CRB 9, and everything from there up to CRB 12 is guard plus a half subcarrier, 860 kHz that no block owns. The block counts and channel widths are TS 38.101-1 Table 5.3.2-1, and the guard band has a section of its own — including why that strip is not the same thing as the offset.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 279 280 281 282 283 284 CRB 16 … 278 The common resource block grid at 30 kHz — numbered from point A, upwards in frequency, for ever every block on this line has a number; only the blue ones belong to this carrier point A 3 300.000 MHz CRB 0 begins here CRB 12 starts 3 304.32 MHz CRB 284 ends 3 402.57 MHz offsetToCarrier = 12 blocks × 12 = 144 subcarriers = 4.32 MHz this carrier — 273 blocks, CRB 12 to CRB 284 CRB 0 to CRB 11 are not missing and not spare: they are numbered, they are simply below where this carrier begins. Point A is the origin of the numbering, not an edge of anything — here it sits 4.32 MHz below the carrier, off it entirely. Every frequency shown is a subcarrier centre. the channel edge — 3 303.460 MHz the shaded strip above it is the guard band — 845 kHz, then half a subcarrier, then CRB 12 that is 860 kHz, or 2.389 blocks: a guard is never a whole number of them
Figure 3: The same carrier, seen as a window onto the block grid. The grid itself starts at point A and is numbered from zero upwards in frequency, and it does not stop where a carrier does - every block on the line has a number. What this carrier owns is the blue stretch, CRB 12 to CRB 284. CRB 0 to CRB 11 are drawn hollow because they are the point: they exist, they are numbered, and they are simply below where this carrier begins. The red dimension across them is offsetToCarrier - 12 blocks, which is 12 times twelve subcarriers, 144 of them, 4.32 MHz. The amber strip is the guard band: the channel's lower edge falls 61 per cent of the way through CRB 9, and from there to CRB 12 is 860 kHz - 845 kHz of guard, then the half subcarrier. It is 2.389 blocks wide, which is why a guard is quoted in kHz and an offset in blocks. The middle of the carrier is cut out; the blocks on either side of the break are drawn at the same width.

Those two frequencies are subcarrier centres, which is why they span 98.25 MHz and not 98.28. The first and last tones are 3 275 spacings apart, so centre to centre is $3\,275 \times 30$ kHz $= 98.25$ MHz; the occupied bandwidth quoted at the top of the figure, $3\,276 \times 30$ kHz $= 98.28$ MHz, counts the half-subcarrier that hangs off each end.

Which raises the obvious question: it is a 100 MHz carrier, so why does the top come out at 3 402.57 rather than 3 400? Three things stack up, and none of them is an approximation.

Point A is not the bottom of the channel. It is the zero of the block grid and nothing else. The example puts it at a round 3 300 MHz, and the carrier then starts 12 blocks — 4.32 MHz — above it. Adding 100 MHz to point A was never going to land on a channel edge.

A 100 MHz channel does not hold 100 MHz of subcarriers. It holds 273 blocks, which is 98.28 MHz. The missing 1.72 MHz is guard band, and it is not spare: it is what keeps the emissions inside the channel.

And the guard is not split evenly — 845 kHz at the bottom, 875 kHz at the top. Both numbers, where they come from and why they differ by exactly one subcarrier are the section after this one.

FrequencyWidth
Channel lower edge3 303.460 MHz
guard band845 kHzthe minimum of Table 5.3.3-1
Lowest subcarrier — edge, then centre3 304.305 → 3 304.32 MHz$k = 144$, CRB 12
3 276 subcarriers98.28 MHz273 blocks of twelve
Highest subcarrier — centre, then edge3 402.57 → 3 402.585 MHz$k = 3\,419$, CRB 284
guard band875 kHzone subcarrier wider than the other side
Channel upper edge3 403.460 MHz100.000 MHz in total
Table 1: Where the 100 MHz goes, for the carrier in the opening figure of this page. Read the two ends: the channel edges are 100.000 MHz apart exactly, and the subcarriers occupy 98.28 MHz of that. The two guard bands differ by one subcarrier because the grid holds one more tone below its middle than above it. The 845 kHz is the minimum of TS 38.101-1 Table 5.3.3-1 for a 100 MHz channel at 30 kHz spacing; point A at 3 300 MHz is the example's choice.

Each cell of the magnified stack carries two numbers, and they are two different questions. The left figure is the subcarrier's index inside its block, 0 to 11 — that is what a reference-signal pattern means when it says "every fourth subcarrier". The number beside it is its $k$ from point A, and those are twelve consecutive integers, 1 200 to 1 211, because 100 whole blocks of twelve come first: $100 \times 12 = 1\,200$. The one picked out, cell 6, is therefore $k = 1\,206$, at $3\,300 + 1\,206 \times 0.03 = 3\,336.18$ MHz.

And the wave in the box under the grid is drawn, not measured. A subcarrier is a sinusoid, and over one useful symbol it is

$$ x(t) = \cos\big(2\pi m \Delta f\, t\big), \qquad 0 \le t < T_u, \tag{1} $$
Equation 1: One subcarrier, written as the real sinusoid it is. Everything about it except its frequency lives in the amplitude and the phase; m is what makes it this subcarrier and not its neighbour, and because m is a whole number the tone closes exactly at the end of the symbol.

so it completes exactly $m$ cycles in that window — $\Delta f = 1/T_u$ is what makes $m$ come out whole, and it is the only reason the tones can be separated at all. $m$ is not $k$, and the figure marks both rulers so the difference is visible: $k$ is measured from point A at the far left, $m$ from the middle of the grid. For this tone $m$ is 576, and the purple arrow in the figure is that distance. The picture draws three because 576 cycles in a box that size is a solid smear — three is a property of the drawing, not of the subcarrier. A later section gives the standard's own form of that equation, works the 576 out step by step, and then says what $m$ is — an address and a frequency are not the same kind of number, and that is the whole of it.

And this is why the right-hand panel looks the way it does: a subcarrier is a frequency, not a piece of time. It exists for as long as the transmission does, so on the grid it is a row — it runs the whole width of the picture because there is nothing to make it stop. What is finite is the intersection of a subcarrier with one symbol: a single cell, which is what the dark square in the picture is, and it has a name and an address.

Three names, and they are routinely muddled.

Subcarrier — a tone. A row of the grid. It has a width in hertz and no duration.

Resource element — one subcarrier during one symbol. A cell. It carries exactly one complex number, which after modulation is one QAM point.

Resource block — twelve consecutive subcarriers. Twelve, always, at every numerology (TS 38.211 clause 4.4.4.1), which is why a block's width in hertz changes with the numerology while its width in subcarriers never does.

The guard band

A 100 MHz channel does not hold 100 MHz of subcarriers. It holds 98.28 MHz of them, and the 1.72 MHz left over is split between the two ends of the channel as guard band — spectrum that is inside the licence, inside the channel, and deliberately empty.

The answer, for the carrier this page has been working with — 100 MHz at 30 kHz spacing:

Below the subcarriers: 845 kHz. This is the tabulated value, TS 38.101-1 Table 5.3.3-1.

Above them: 875 kHz. Exactly one subcarrier more, and not by accident.

Together: 1.72 MHz, which is 1.72 % of the channel — so 98.28 % of the licensed width is actually carrying subcarriers. That fraction is as good as NR gets; at narrower channels and wider spacings it falls to 79 %, and the table below has every value.

Neither number is ever signalled. A device is told the channel bandwidth and how many resource blocks the carrier has; the guard is what is left when you subtract one from the other. It has no information element, no name in RRC, and nothing is ever transmitted in it.

0 -10 -20 -30 -40 dB relative to the in-band level the emission mask, outside the channel −24 dBm in 30 kHz — 12 dB under the in-band level when the device transmits its full 23 dBm channel edge 3 303.46 MHz channel edge 3 403.46 MHz first subcarrier its edge, not its centre last subcarrier its edge, not its centre 845 kHz 875 kHz 98.28 MHz 3 276 subcarriers not drawn −6 dB −27 dB the spectrum of the whole band −6 dB where the subcarriers stop, still only −27 dB a guard band later The two guard bands of a 100 MHz channel at 30 kHz, and the spectrum that has to come down inside them both panels are at the same scale — only the middle of the band is cut out The curve is computed, not sketched: it is the spectrum of 3 276 tones each cut off after one useful symbol, as a power density. 845 kHz below the band, 875 kHz above it — one subcarrier apart, because an even grid cannot be centred on the channel. The guard is the room the transmission is given to fall from full power to the mask. It is not spare spectrum and it carries nothing. Vertical scale is dB relative to the in-band spectral density; horizontal scale is linear and identical in both panels.
Figure 4: Both ends of the channel, at the same scale, with the spectrum that has to fit down them. The middle of the band is cut out - a guard band is under one per cent of a 100 MHz channel and cannot share a linear axis with it, so the two ends are drawn at guard scale and the zigzags say what was removed. The blue curve is computed rather than sketched: it is the power spectrum of 3 276 tones each cut off after one useful symbol, drawn as a density in dB relative to the in-band level. Read what it does at the two vertical rules. At the blue dashed line, where the subcarriers stop, the signal is only 6 dB down - a spectrum does not end where its subcarriers end. A whole guard band later, at the black rule that is the channel edge, it is 27 dB down. That fall is the guard band's entire job, and the red dashed line is what it has to fall to: the emission mask of TS 38.101-1 clause 6.5.2.2, which applies from the channel edge outwards and sits about 12 dB under the in-band level for a device transmitting its full 23 dBm. The two amber strips are the guards themselves, 845 kHz below and 875 kHz above.

Why a channel has one at all

Because a spectrum does not stop where its subcarriers stop. A channel is a licence to occupy a stretch of frequency and to stay out of everybody else's, and those are two different obligations. TS 38.101-1 enforces the second one with three requirements, and all three are measured from the channel edge, not from the last subcarrier:

Occupied bandwidth (clause 6.5.1) — 99 % of the transmitted power must lie inside the channel.

Spectrum emission mask (clause 6.5.2.2) — from the channel edge outwards, the power in any 30 kHz must not exceed −24 dBm over the first megahertz, for any channel of 50 MHz or more. Further out the mask steps down again, and past that the spurious-emission limits of clause 6.5.3 take over.

Adjacent channel leakage ratio (clause 6.5.2.4) — the power landing in the neighbouring channel must be at least 30 dB below the power in this one, for an ordinary power class 3 device (Table 6.5.2.4.1-2).

The problem those three create is visible in the figure above. Each subcarrier is a tone cut off after one useful symbol, so each one's spectrum is a sinc, and a sinc has tails. Add 3 276 of them and the tails add too — the sum falls only about 10 dB per decade of frequency, which is extraordinarily slow. At the edge of the occupied band the signal is 6 dB down; a full 845 kHz further out it is still only 27 dB down. There is nowhere for the emissions to have gone, and the guard band is the frequency in which they are given the chance to go there.

Figure 5: The same spectrum on a logarithmic frequency axis, so the whole guard band and the first few kilohertz of it can be seen at once. The horizontal axis is distance beyond the last subcarrier's edge; the vertical axis is dB under the in-band spectral density. The red line is the emission mask, which only applies once you are outside the channel edge - it is drawn all the way across so the crossing can be read. Two points are marked. The left one is where an ideal waveform, with no filter and no amplifier, already meets the mask: about 21 kHz, less than one subcarrier. The right one is where the channel edge actually is: 845 kHz, forty times further out, and by then the waveform is 16 dB under the limit. That gap is the whole answer to why the guard is as wide as it is - it is not sized for the waveform.

And here is the part worth taking away: the waveform is not what sets the number. Read the two marked points. An ideal signal — perfect tones, no filter, no amplifier, transmitting the full 23 dBm of a power class 3 device (TS 38.101-1 Table 6.2.1-1) — is already under the emission mask 21 kHz out from its last subcarrier, less than one subcarrier's width. The standard gives it 845 kHz. That is forty times more room than the mathematics needs, and the same story repeats on the other requirement: the ideal waveform's adjacent-channel leakage works out at 42 dB, against a requirement of 30.

So what is the other 824 kHz buying? Not the waveform — the transmitter.

A transmit filter needs somewhere to roll off. It has to be flat across the occupied band, or it distorts the constellation and the EVM requirement fails from the inside; and it has to be deep by the channel edge, or the mask fails from the outside. The guard band is that filter's transition band, and a transition band is the one thing a filter cannot have for free — narrow it and you pay in order, in group delay, in cost, and in insertion loss on a path the power amplifier has already paid for.

A power amplifier puts back what the filter took out. Run a signal with a high peak-to-average ratio through a non-linear device and the third and fifth-order products regrow the spectrum right next to the band — which is exactly what the 30 dB ACLR requirement is written to bound. Regrowth happens after the filter, so no amount of digital shaping removes it; only backoff and linearisation do, and both cost power.

And everything drifts. The mask has to be met over temperature, over process spread, over the whole band, at every power level, for every allocation — including a single resource block parked at the very edge of the carrier, which is the worst case and the one that decides the number.

One more thing the guard is doing, in the other direction: it protects this receiver from the neighbour. The device's own front-end filter has a transition band too, and the guard is the room in which an adjacent operator's signal is attenuated before it reaches the mixer. A guard band is not one transmitter's politeness; it is the gap two radios share.

The number, for every channel width

The tabulated guard is not derived from the emission requirements — it is what is left after the block count. TS 38.101-1 Table 5.3.2-1 fixes how many resource blocks fit in each channel bandwidth at each spacing; that is where the engineering judgement lives, and it was settled in 3GPP by asking what a real transmitter could hold to. The guard band is then arithmetic:

$$ \text{GB}_{\text{channel}} \;=\; \frac{\text{BW}_{\text{channel}} - N_{\text{RB}} \cdot \text{SCS} \cdot 12}{2} \;-\; \frac{\text{SCS}}{2} \tag{2} $$
Equation 2: The minimum guard band, TS 38.101-1 clause 5.3.3, printed there as the NOTE under Table 5.3.3-1. Take the channel, subtract the subcarriers, halve what is left, and then take off half a subcarrier - that last term is the asymmetry, and it is the reason the tabulated number is the smaller of the two guards rather than both of them.

That formula reproduces all 41 entries of Table 5.3.3-1 exactly, across the three FR1 spacings and every channel bandwidth from 3 to 100 MHz — checked entry by entry, not spot-checked, by figures/guard_band.py --check. So there is nothing to memorise: given the block count, both guards follow.

Widget not found: sim_show
ChannelBlocksOccupiedGuard belowGuard aboveUtilisationGuard, in subcarriers
Table 2: Every channel bandwidth NR defines at 30 kHz spacing, with both guard bands. <b>Every row is computed while this page loads</b>, by the same code the simulator is tested against — the block count is TS 38.101-1 Table 5.3.2-1 and everything to its right follows from the clause 5.3.3 formula, so this table cannot drift from the arithmetic. The lower guard is the number Table 5.3.3-1 publishes and the upper guard is that plus one subcarrier. Read the last two columns against each other: the share of the channel that carries subcarriers climbs steadily with channel width, while the guard itself stays within a factor of two of a megahertz - a wide channel is not given a proportionally wider guard, it simply amortises a roughly fixed one. The last column is the same guard counted in subcarriers, which is the number an implementer cares about, and it never leaves the range 17 to 36. The highlighted row is the carrier this page has been working with.
Figure 6: The minimum guard band at the lower edge, for every channel bandwidth and every FR1 subcarrier spacing, straight out of TS 38.101-1 Table 5.3.3-1. The upper guard is each point plus one subcarrier, so the three curves would simply shift by 15, 30 and 60 kHz. Two things are worth reading off it. The curves are stacked by spacing, so a wider subcarrier costs more guard at the same channel width - at 20 MHz the guard is 452.5 kHz on a 15 kHz numerology and 1 330 kHz on a 60 kHz one, three times as much spectrum given up for the same licence. And none of the three is a smooth line: the guard jumps up and down as the channel widens, because the block count has to be a whole number and the guard is only the remainder. A 60 MHz channel at 30 kHz gets a smaller guard than a 50 MHz one. The red ring marks this page's own carrier, 100 MHz at 30 kHz. On the companion page Subcarrier, live the same chart carries the same ring and it follows the picker there.

The same numbers, for any channel you like. The table above fixes 30 kHz and the chart plots the lower guard alone. Every combination TS 38.101-1 defines — with both guards, the utilisation, the block width and two figures that redraw as you pick — is on the companion page, Subcarrier, live.

Four things that table and that curve say, none of which is obvious from a single number.

It is a minimum, not a size. Clause 5.3.3 requires only that the blocks a network configures leave at least this much. Configure fewer blocks than Table 5.3.2-1 allows and the guard simply gets wider — which is what happens on a carrier deployed next to something sensitive.

A wider subcarrier costs more guard. At 20 MHz the lower guard is 452.5 kHz at 15 kHz spacing and 1 330 kHz at 60 kHz — three times the spectrum surrendered for the same licence. Both effects push the same way: one resource block is 12 subcarriers wide however wide a subcarrier is, so the rounding is coarser, and a wider tone has proportionally wider sinc tails to attenuate.

A wider channel is more efficient. 79.2 % of a 5 MHz channel carries subcarriers; 98.28 % of a 100 MHz one does. The guard is roughly a fixed cost in megahertz, so the wider the channel the less it matters — which is one of the quieter arguments for the large channel bandwidths NR introduced.

And the curve is not monotonic. A 60 MHz channel at 30 kHz spacing gets an 825 kHz guard while a 50 MHz one gets 1 045 kHz. Nothing is wrong: the block count is an integer, the guard is the remainder, and a remainder has no reason to be smooth.

There is a second kind of guard band in the same clause, and it is not this one. For shared spectrum, TS 38.101-1 Table 5.3.3-2 defines intra-cell guard bands — gaps left inside the carrier to separate it into independently usable RB sets, written as 50-6-50-6-49-6-50-6-50 for a 100 MHz channel at 30 kHz. Those are measured in resource blocks, they sit between blocks the device can use, and they exist for listen-before-talk, not for emissions. Everything on this page is the ordinary guard at the two channel edges.

Why the two guards differ

Because an even number of subcarriers has no middle one. The grid holds 3 276 of them, and the reference the waveform counts from — $m = 0$, the term the standard writes as $N^{\text{size},\mu}_{\text{grid}} N^{\text{RB}}_{\text{sc}}/2$ — sits at index 1 638. With indices running 0 to 3 275 that is not the middle of the set: the middle is 1 637.5. The reference is half a subcarrier above it.

m = -3 m = -2 m = -1 m = 0 m = +1 m = +2 the reference the waveform counts from — m = 0 the standard puts it at index N/2, here the 4th of 6 the true middle of the six half a cell 3 cells and a half below it 2 cells and a half above it An even number of cells has no middle cell, so the reference cannot sit in the middle of them. Put the reference on the channel’s centre and the whole grid hangs half a cell low — which is the entire reason the two guard bands differ. With 3 276 subcarriers instead of six: 1 638 below the reference, 1 637 above, and the guards come out 845 and 875 kHz.
Figure 7: The same situation with six subcarriers instead of 3 276, so the cells can be counted. The reference the waveform uses is at index N/2 - the fourth of the six - while the true middle of the six is the boundary half a cell to its left. Three and a half cells therefore lie below the reference and only two and a half above it. Place that reference at the centre of the channel, as the carrier does, and the occupied band hangs half a subcarrier low: the guard beneath it is one subcarrier tighter than the guard above.

So the band is not centred in the channel — it hangs 15 kHz low, and the two guards inherit the difference.

Below the referenceAbove it
Subcarriers1 6381 637
To the edge of the band$(1638 + \tfrac12) \times 30$ kHz = 49 155 kHz$(1637 + \tfrac12) \times 30$ kHz = 49 125 kHz
Half the channel50 000 kHz50 000 kHz
Guard band845 kHz875 kHz
Table 3: Both guard bands, derived rather than looked up. The distance from the reference to each edge of the occupied band is the tone count plus the half-subcarrier that hangs off the outermost tone; subtract that from the channel's half-width and the two guards fall out. They differ by exactly one subcarrier, they sum to the 1.72 MHz the channel has left over, and the smaller of them is the value tabulated in TS 38.101-1 Table 5.3.3-1.

And that is what the odd-looking $-\text{SCS}/2$ in the standard's formula is doing. TS 38.101-1 clause 5.3.3 defines the minimum guard band as $(\text{BW} - N_{\text{RB}} \cdot \text{SCS} \cdot 12)/2 - \text{SCS}/2$: split the leftover 1.72 MHz evenly — 860 kHz a side — and then take half a subcarrier off, because the band is not centred. 860 − 15 = 845 kHz, the tighter side, which is the number the table publishes. The other side is the rest: 1 720 − 845 = 875 kHz.

Which of these numbers are the standard's, and which are this example's? Worth separating, because they are easy to mistake for each other.

From the specification, and not negotiable: 273 blocks in a 100 MHz channel at 30 kHz spacing (TS 38.101-1 Table 5.3.2-1); the 845 kHz minimum guard band (Table 5.3.3-1); twelve subcarriers to a block (TS 38.211 clause 4.4.4.1); and everything computed from those — 3 276 subcarriers, 98.28 MHz, the 30 kHz between edge and centre. These are the most-quoted numbers in NR, and the 100 MHz / 30 kHz / 273 combination is the one nearly every FR1 example in the literature uses.

Chosen for this example, and arbitrary: point A at 3 300 MHz, offsetToCarrier of 12 blocks = 144 subcarriers, the block CRB 100, and cell 6 within it.

Why 144, then? For no deeper reason than that it is small, round in blocks, and keeps the arithmetic readable — the point of the figure is that the offset exists, not what it equals. offsetToCarrier is a deployment parameter: TS 38.331 allows anything from 0 up to 2 199 blocks, and what it actually is on a given carrier falls out of where the operator put point A relative to the SS/PBCH block a device finds first. The one thing 144 is not is a constant to remember. CRB 100 and cell 6 were picked for the same reason and one extra: they keep every number in the worked chain distinct, so no two of them can be confused for each other.

So the round number can be point A or the channel edges, but not both. This example rounds point A, because the figure is about counting subcarriers from it. Round the channel instead — insist its edges run 3 300.000 to 3 400.000 MHz, which are channel edges and not the point A those digits look like — and the same arithmetic drops point A on 3 296.540 MHz, which is perfectly legal (it lands on the 5 kHz raster) and thoroughly unmemorable. Real deployments look like the second: the channel is on the raster and point A falls wherever the block grid demands.

offsetToCarrier is not the guard band

They are two different measurements between two different pairs of points, and the only reason they look related is that both live at the bottom of the band. There are four levels down there, not two, and every one of them is a different frequency.

point A 3 300.000 MHz  ·  the zero of the block grid the channel edge 3 303.460 MHz  ·  where the 100 MHz begins first subcarrier, its edge 3 304.305 MHz first subcarrier, its centre 3 304.320 MHz  ·  k = 144 · CRB 12 starts 3 460 kHz — nobody chose this: it is 4 320 − 845 − 15 845 kHz — the guard band 15 kHz — half a subcarrier offsetToCarrier 4 320 kHz = 144 subcarriers = 12 resource blocks Two different measurements, between two different pairs of points — 30 kHz the guard band starts at the channel edge  ·  offsetToCarrier starts at point A Two numbers are chosen — point A and offsetToCarrier. Two are fixed by the standard — the guard band and half a subcarrier. The channel edge is then forced: 3 460 = 4 320 − 845 − 15 kHz spacing not to scale, the numbers are on the arrows
Figure 8: The four levels at the bottom of the band, and the three gaps between them. Read the two coloured spans against each other: the amber one is the guard band, measured from the channel edge to the first subcarrier, and it is an RF requirement - it exists so the transmission does not spill outside the channel. The red one is offsetToCarrier, measured from point A to the first subcarrier, and it is a numbering parameter - it exists so both ends agree which resource block is which. They share only their upper end. The spacing is not to scale; 3 460 kHz, 845 kHz and 15 kHz cannot share a linear axis and stay readable, so the numbers are on the arrows.

The guard band — 845 kHz — runs from the channel edge to the first subcarrier's edge. It is an RF requirement, from TS 38.101-1: without it the transmission would spill past the channel it was licensed for. Its size falls out of the channel width and the block count, and nothing in the protocol ever refers to it by name.

offsetToCarrier — 144 subcarriers — runs from point A to the first subcarrier's centre. It is a numbering parameter, from TS 38.331: it tells the device how far up the common block grid this carrier starts, so that both ends agree that the lowest usable block is CRB 12 and not CRB 0. It is signalled per numerology, in resource blocks, and it has no RF meaning at all.

They meet at one end and nowhere else. Both finish at the first usable subcarrier — one at its edge, one at its centre, 15 kHz apart. They start at completely different places.

Which also answers where the odd 3 460 kHz comes from. Nobody chose it. Of the four quantities stacked up at the bottom of the band, two are free and two are not:

Chosen, and arbitrary. Point A — put at 3 300.000 MHz here because a round number makes the subcarrier arithmetic legible. And offsetToCarrier — 12 blocks, small enough to draw.

Fixed by the standard, whatever you choose. The guard band, 845 kHz, from TS 38.101-1 Table 5.3.3-1 for a 100 MHz channel at 30 kHz spacing. And half a subcarrier, 15 kHz, from the geometry of a subcarrier itself.

So the channel edge is forced. Point A plus offsetToCarrier fixes the first subcarrier's centre at 3 304.320 MHz; back off the half-subcarrier to reach its edge, then the guard band to reach the channel edge, and you land on 3 303.460 MHz — 3 460 kHz above point A, because $4\,320 - 845 - 15 = 3\,460$. It is a remainder, not an input.

Change the one free number and the gap moves with it. Make offsetToCarrier 13 blocks instead of 12 and it becomes 3 820 kHz; make it 20 and it becomes 6 340. Move point A and the gap does not change at all — the whole picture just slides up or down the spectrum together, which is the sense in which point A is only a label.

The example is legal, not merely convenient. With these numbers the carrier's centre lands on 3 353.460 MHz, an exact multiple of both 15 and 30 kHz, so it sits on the channel raster band n78 uses — a real network could deploy exactly this.

And the three pieces still have to add up:

$$ \underbrace{4\,320}_{\text{offsetToCarrier}} = \underbrace{3\,460}_{\text{point A to the channel edge}} + \underbrace{845}_{\text{guard band}} + \underbrace{15}_{\text{half a subcarrier}} \ \text{kHz} \tag{3} $$
Equation 3: How the three gaps at the bottom of the band add up to offsetToCarrier. Only the middle term is the guard band; the first is wherever the example put point A, and the last is the half-subcarrier between an edge and a centre. Change point A and the first term absorbs it - the other two are fixed.

Point A is not even required to be below the channel. It is a reference frequency, free to sit below the carrier, inside it, or above it; the standard only requires that both ends agree where it is. This example puts it 3.46 MHz below the channel because that makes point A a round number and the subcarrier arithmetic legible — nothing more.

How one is addressed

A tone is no use unless both ends can name it. Three numbers do that, and mixing them up is the commonest confusion in the whole frame structure.

$k$ — the subcarrier number. Counted from point A, upwards in frequency, from zero. On a 100 MHz carrier at 30 kHz there are 3 276 of them, so $k$ runs 0 … 3 275.

$n_{\text{CRB}}$ — the common resource block number. Twelve subcarriers to a block, so the whole definition is one floor division (TS 38.211 clause 4.4.4.3):

$$ n^{\mu}_{\text{CRB}} = \left\lfloor \frac{k}{12} \right\rfloor \tag{4} $$
Equation 4: The block a subcarrier belongs to, TS 38.211 clause 4.4.4.3. Twelve subcarriers to a block, so the whole definition is one floor division - and k is counted from point A, which is why the answer is a common resource block number rather than a position in the carrier.

$l$ — the OFDM symbol number within the slot, 0 … 13. A subcarrier plus a symbol is a resource element, and the pair $(k, l)$ is its address (clause 4.4.2).

So the figure above reads as arithmetic. Block 100 is not the 100th block from the bottom edge of the channel; it is the block holding subcarriers $100 \times 12 = 1\,200$ through 1 211, and its seventh cell — index 6, because the count starts at zero — is subcarrier 1 206.

Inside its blockFrom point A, $k$Its block, $n_{\text{CRB}}$Above point ACentre frequency
01 20010036.00 MHz3 336.00 MHz
……100……
61 20610036.18 MHz3 336.18 MHz
……100……
111 21110036.33 MHz3 336.33 MHz
01 212101 — the next block begins36.36 MHz3 336.36 MHz
Table 4: The same twelve subcarriers, addressed four ways. The first column is the index printed inside each cell of the figure's magnified stack; the second is the same tone counted from point A; the third is the block it belongs to, the same for all twelve because a block is a floor division by twelve. The last two are frequency - relative to point A, which is k times 30 kHz and needs nothing else, and absolute, which needs to know where point A is. The highlighted row is the tone drawn in the box under the grid, and the absolute column assumes the example point A of 3 300 MHz.

So where is point A? Nowhere in particular — that is the point of it. It is one agreed frequency, and the network tells the device where it is in one of two ways (TS 38.211 clause 4.4.4.2): as absoluteFrequencyPointA, a channel number naming it outright, or as offsetToPointA, a distance in 15 kHz blocks down from the SS/PBCH block the device used to find the cell — because that block is the only thing it had found so far.

It is a reference, not an edge, and it is normally below the carrier. How far below is offsetToCarrier, defined in TS 38.331 as "offset in frequency domain between Point A (lowest subcarrier of common RB 0) and the lowest usable subcarrier on this carrier, in number of PRBs" — per numerology, and it may be as large as 2 199 blocks. In the figure it is 12, which is why the carrier holds CRB 12 to CRB 284.

And that is what buys the shared ruler. Every numerology on the carrier measures from the same point A, so a 15 kHz block grid and a 60 kHz block grid cannot disagree about where they are: one 60 kHz block spans exactly four 15 kHz blocks, aligned, because both counts start at the same frequency. Put the zero at the carrier edge instead and the grids would only line up by luck. See Point A.

A block number alone is ambiguous, and that is why the figure says CRB. There are four numberings (clause 4.4.4): common blocks count from point A and are the absolute ruler; physical blocks count from the start of the bandwidth part in use and are what a scheduler hands out; virtual blocks are what a grant names before interleaving; and interlaced blocks exist for shared spectrum. Converting between the first two is one addition — $n_{\text{CRB}} = n_{\text{PRB}} + N^{\text{start}}_{\text{BWP}}$ — and forgetting it is how an allocation ends up in the wrong part of the band. The four are laid out in nr-frame-structure.

The condition: a whole number of cycles

The tones sit directly against each other with no guard band between them, and they still do not interfere. That is the claim, and it has exactly one condition.

Each subcarrier must fit a whole number of its own cycles into one symbol. Subcarrier 1 does one cycle, subcarrier 2 does two, subcarrier 17 does seventeen — all in the same useful symbol time $T_u$.

Figure 9: The first three subcarriers of a 15 kHz numerology, over one useful symbol of 66.67 microseconds. Subcarrier k completes exactly k cycles in that window and comes back to where it started. Nothing about the picture is a coincidence: the symbol length was chosen to make it true. Because it is true, the integral of any two different tones multiplied together, over exactly this window, is zero - which is the whole of OFDM in one sentence.

So how many cycles does a given subcarrier do, and where does that number come from? TS 38.211 clause 5.3.1 writes a whole OFDM symbol as a sum of tones — this is the standard's own definition of the transmitted signal, with the cyclic prefix and symbol-start terms dropped for clarity:

$$ s(t) \;=\; \sum_{k} a_{k}\, e^{\,j2\pi m_k \Delta f\, t}, \qquad m_k \;=\; k + k^{\mu}_{0} - \frac{N^{\text{size},\mu}_{\text{grid}} N^{\text{RB}}_{\text{sc}}}{2} \tag{5} $$
Equation 5: A whole OFDM symbol as TS 38.211 clause 5.3.1 defines it, with the cyclic prefix and symbol-start terms dropped for clarity. Read what m_k counts: the subcarrier index measured from the middle of the grid, not from point A.

One subcarrier is one term of that sum. Its complex form and, if you prefer real signals, its real part:

$$ x_k(t) \;=\; a_k\, e^{\,j2\pi m_k \Delta f\, t} \qquad\Longrightarrow\qquad \Re\, x_k(t) \;=\; |a_k|\,\cos\big(2\pi m_k \Delta f\, t + \arg a_k\big), \qquad 0 \le t < T_u \tag{6} $$
Equation 6: One subcarrier is one term of that sum, in complex form and as the real signal it becomes. The complex number a_k that this subcarrier carries sets the amplitude and the starting phase, and never the frequency.

$m_k$ is the whole number of cycles, and the formula says what it counts: the subcarrier's distance from the middle of the grid, not from point A. With one numerology configured $k^{\mu}_0 = 0$, so $m_k$ is simply the subcarrier's index within the carrier minus half the carrier's subcarriers. Because $\Delta f = 1/T_u$, the tone completes exactly $|m_k|$ cycles in one useful symbol — and $a_k$, the one complex number this subcarrier carries, sets only its amplitude and starting phase, never its frequency.

Which answers the question the figure raises: where does the $m$ come from? Three numbers and two subtractions, and the figure marks all of them. The tone drawn there is $k = 1\,206$ from point A. The carrier's grid does not begin at point A — it begins 144 subcarriers above it — so within the grid the tone is number $1\,206 - 144 = 1\,062$. The grid holds 3 276 subcarriers, so its middle is number $3\,276/2 = 1\,638$. Subtract the second from the first:

$$ m_k = \underbrace{(1\,206 - 144)}_{\text{index within the grid}} - \underbrace{\tfrac{3\,276}{2}}_{\text{middle of the grid}} = 1\,062 - 1\,638 = -576 \tag{7} $$
Equation 7: Where m comes from, worked out for the tone in the opening figure of this page. Subtract the grid start to turn a number counted from point A into a number counted from the bottom of the carrier, then subtract half the grid to count from its middle instead. The tone lands 576 subcarriers below the middle, so it turns 576 times in every useful symbol.

It does 576 cycles per useful symbol, the minus sign meaning only that it sits below the middle — $576 \times 30$ kHz $= 17.28$ MHz below it. The picture draws three because 576 cycles in a box that size would be a solid smear. Three is not a property of the subcarrier; the shape is.

What $m$ actually is

$k$ is an address; $m$ is a frequency. That is the whole distinction, and every part of the formula follows from it.

$k$ answers which slot. It is an index into an array — the grid — running from 0 upwards, and it is what signalling uses, because a scheduler has to be able to say this block, not that one. Nothing about it says how fast anything oscillates.

$m$ answers how far from the middle, in units of $\Delta f$ — and therefore how fast. It is the tone's frequency, written as a count of subcarrier spacings, signed, with zero at the centre of the transmitted band. The transmitter builds one OFDM symbol by adding up $N$ complex sinusoids at frequencies $m \Delta f$ around that centre, and then a mixer shifts the whole block up to the carrier frequency. $m$ is a property of the waveform; $k$ is a property of the paperwork.

So the subtraction is a change of origin, nothing more. An array index starts at 0 at the bottom; a frequency measured from the middle has to be negative below it and positive above. Turning the first into the second is exactly subtract half the length.

Index within the grid$m$Offset from the middleWhat it is
0−1 638−49.14 MHzthe lowest subcarrier of the carrier
1 062−576−17.28 MHzthe tone in the opening figure
1 637−1−30 kHzone step below the middle
1 63800the middle itself — a tone that does not turn at all
1 639+1+30 kHzone step above
3 275+1 637+49.11 MHzthe highest subcarrier of the carrier
Table 5: The same 3 276 subcarriers under both rulers. The left column is the array index a receiver would store them at, the middle column is the frequency each one really carries, in units of the subcarrier spacing, and the right column is that frequency in megahertz relative to the middle of the band. Subtracting half the grid is what converts one column into the other. Note the asymmetry at the two ends: an even-length grid has one more tone below the middle than above it, because zero has to live somewhere and it lives in the lower half.

And this is why $m$, not $k$, is the number of turns. Over one useful symbol the phase of $e^{\,j2\pi m \Delta f t}$ advances by $2\pi m \Delta f T_u$, and $\Delta f T_u = 1$ by construction, so the advance is exactly $2\pi m$ — $m$ whole turns, ending where it began. Put $k$ in that expression instead and the number is meaningless: it would be counting turns against an origin, point A, that no radio has ever measured anything from.

Two things in the standard that make this harder than it is.

The letter $k$ means two different things in two clauses, and that single fact is the whole reason the worked example subtracts 144 before it does anything else. In clause 4.4.4.3 — the numbering clause — $k$ is counted from point A. In clause 5.3.1 — the waveform clause — the sum runs $k = 0$ to $N^{\text{size},\mu}_{\text{grid}} N^{\text{RB}}_{\text{sc}} - 1$, so $k$ is counted from the start of the grid. Same letter, two origins, 144 subcarriers apart on this carrier.

$k^{\mu}_0$ is zero unless the carrier runs more than one numerology. When it does, that term re-centres each numerology's grid on the same physical frequency, so a 15 kHz waveform and a 60 kHz waveform built side by side have a common middle as well as a common point A. With a single numerology configured the two halves of its definition are equal and it vanishes — which is why it can be ignored while learning, and must not be ignored when implementing.

In code, nobody subtracts anything. An $N$-point inverse FFT numbers its bins $0$ to $N-1$ and treats the upper half as the negative frequencies, so a tone with $m = -576$ is written into bin $N - 576$ and the transform does the rest. The standard's $-N/2$ and an implementation's wrap-around are the same statement about the same tone; only the bookkeeping differs.

Which is also why the middle has to be agreed, not merely known. If the two ends disagree about where the centre of the band sits, every $m$ is wrong by the same constant — and that is precisely the frequency offset $\varepsilon$ of the section on what breaks orthogonality. Being wrong by a whole subcarrier is $\varepsilon = 1$: not a degraded link, a destroyed one.

Two things that formula quietly settles.

The carrier frequency is not in it. 3 336.18 MHz never appears — the sum is a baseband signal, and the radio shifts the whole thing up to 3.3 GHz afterwards. At the antenna that tone really does turn about 111 000 times per useful symbol; the 576 is what matters, because orthogonality is a statement about the tones' spacing, not about their absolute frequency.

The middle of the grid is subcarrier $m = 0$ — a tone that does no cycles in a symbol, a constant. It is often left empty (the "DC subcarrier"), because a real receiver's own local oscillator leaks a constant into exactly that bin.

That requirement is a single equation, and it is the most consequential one in these notes.

$$ \Delta f = \frac{1}{T_u} \qquad\Longleftrightarrow\qquad \frac{1}{T_u}\int_{0}^{T_u} e^{\,j2\pi k \Delta f t}\, e^{-j2\pi l \Delta f t}\, dt = \delta_{kl} \tag{8} $$
Equation 8: The orthogonality condition of OFDM. The spacing between the tones and the length of a symbol are the same number, upside down. You cannot choose the two separately - fixing either one fixes the other, and with it the guard length, the slot length and the size of cell you can build.

Why the integral is zero. Substitute $\Delta f = 1/T_u$ and the two exponentials merge into one:

$$ \frac{1}{T_u}\int_{0}^{T_u} e^{\,j2\pi (k-l)t/T_u}\, dt = \frac{e^{\,j2\pi (k-l)} - 1}{\,j2\pi (k-l)\,} \tag{9} $$
Equation 9: The orthogonality integral with the spacing substituted in. Two exponentials become one, and the whole question turns into whether e to the j2pi times an integer is 1 - which it always is.

For any integers $k \neq l$, the difference $k-l$ is a non-zero integer, so $e^{\,j2\pi(k-l)} = 1$ and the numerator is exactly zero. For $k = l$ the integrand is 1 and the average is 1. There is no approximation anywhere in that line — the tones are not nearly separable, they are separable exactly, and only because the spacing is exactly the reciprocal of the symbol.

So the receiver's job on one subcarrier is one integral. Multiply the received symbol by $e^{-j2\pi k \Delta f t}$, average over $T_u$, and everything that was not subcarrier $k$ cancels to nothing. Doing that for all the tones at once is a discrete Fourier transform, which is why every OFDM receiver ends in an FFT and why one subcarrier is exactly one FFT bin.

How large is that transform? The standard's own time base makes the useful symbol $N_u = 2048\,\kappa\,2^{-\mu}$ in units of $T_c$ (TS 38.211 clause 5.3.1) — 2 048 samples at the reference rate of 30.72 MHz. A real 100 MHz radio at 30 kHz spacing runs a 4 096-point FFT at 122.88 MHz, of which 3 276 bins carry subcarriers and the rest are the guard band.

This is also the answer to "does the transmitter contain three thousand oscillators". It does not. It contains one inverse FFT, and the tones fall out of it — the reason OFDM became practical in the 1990s rather than the 1960s, when it was first described.

The same fact, seen in frequency

A tone that is switched on for a finite window is not a spike in the frequency domain. Cutting a sinusoid off after $T_u$ smears it into a $\operatorname{sinc}$ shape whose main lobe is $2\Delta f$ wide — twice the spacing. So the tones do overlap, heavily, and the picture looks at first like a design that cannot work.

Figure 10: Five neighbouring subcarriers, drawn as the spectra they really have. Each is a sinc, because each is a tone cut off after a finite window, and each main lobe is two spacings wide - so every subcarrier overlaps both of its neighbours and most of theirs. Look instead at the centres. At the peak of any one tone, every other tone is passing through exactly zero. That is the same condition as the whole-number-of-cycles picture above, seen from the other side of the transform: the receiver samples each tone at its own peak, where all the others contribute nothing, so the overlap costs nothing at all. This is the shape for a large number of subcarriers; for a finite transform the true curve is a Dirichlet kernel, which has the same zeros.

The two pictures are one fact. A whole number of cycles per symbol in time is the same statement as a null at every neighbour's centre in frequency, because the Fourier transform of a rectangular window of length $T_u$ has its zeros spaced exactly $1/T_u$ apart. Spacing the tones at $1/T_u$ therefore puts every neighbour on a zero, by construction.

Why this makes a receiver simple

Here is the payoff, and it is the reason every modern radio standard is built this way.

A radio channel is a sum of echoes: the signal arrives several times, at several delays, with several strengths. In the time domain that is a convolution, and undoing it means an equaliser that tracks a filter with many taps — the expensive, fragile machinery that dominated 2G and 3G receivers.

Add a cyclic prefix longer than the longest echo, and the convolution becomes circular. A circulant matrix is diagonalised by the Fourier basis — and the subcarriers are that basis. So each tone comes out of the channel multiplied by one complex number and nothing else:

$$ Y_k = H_k X_k + N_k \tag{10} $$
Equation 10: What a cyclic prefix buys, in one line: each subcarrier arrives multiplied by a single complex number and added to noise. No filter, no memory of the previous symbol - which is why equalising a 100 MHz channel is a few thousand complex divisions.

Equalising a 100 MHz channel is therefore 3 276 complex divisions, one per subcarrier, and each one is a division by a number the reference signals measured. No filter, no tracking loop, no matrix inverse.

That is what a subcarrier buys. It is not merely a way of slicing spectrum; it is the choice of a basis in which a hostile channel becomes diagonal. Every reference signal in these notes exists to estimate one of those numbers, and every CSI report is a compressed description of a few thousand of them.

How wide is one, and how many are there

Every number on this page is computed rather than transcribed, and each one carries the clause, table or formula it came from. That provenance, and proofs of the three facts that are identities rather than arithmetic, are on Subcarrier, live.

Because $\Delta f = 1/T_u$, choosing the width of a subcarrier chooses everything else. The standard offers seven widths, each twice the last, and the whole rest of the frame structure follows mechanically.

$\mu$$\Delta f$$T_u = 1/\Delta f$Cyclic prefixEcho reachBlock width1.6 kHz Doppler is…
015 kHz66.67 µs4.688 µs1 405 m180 kHz10.8 % of the spacing
130 kHz33.33 µs2.344 µs703 m360 kHz5.4 %
260 kHz16.67 µs1.172 µs351 m720 kHz2.7 %
3120 kHz8.33 µs0.586 µs176 m1.44 MHz1.35 %
4240 kHz4.17 µs0.293 µs88 m2.88 MHz0.68 %
5480 kHz2.08 µs0.146 µs44 m5.76 MHz0.34 %
6960 kHz1.04 µs0.073 µs22 m11.52 MHz0.17 %
Table 6: The seven subcarrier widths of TS 38.211 Table 4.2-1, and what each one decides. The useful symbol is 1/Δf. The cyclic prefix is the 144-sample constant of clause 5.3.1 scaled by the numerology, which comes to 7.03 per cent of the symbol at every one of them - the guard is always the same slice, never a fixed number of microseconds. The reach is that guard multiplied by the speed of light: how much further an echo may travel than the direct path before it does damage. The last column is the same Doppler shift - 1.6 kHz, a train at 500 km/h on a 3.5 GHz carrier - expressed as a fraction of each spacing. Read the last two columns against each other and the entire numerology table is explained: going down the table buys tolerance of movement and spends tolerance of distance.

Seven here, but only three channel spacings in TS 38.101-1 — and both are right. This table is TS 38.211's, which defines the physical layer without reference to any band; the channel bandwidths are TS 38.101-1's, and that document says of itself "The present specification covers FR1 operating bands." The rest are FR2's, and its channels and bands are in TS 38.101-2 — over channels from 50 MHz to 2 GHz. The two ranges are set against each other, with all 78 operating bands, on Subcarrier, live.

Line the two tables up against Table 4.2-1 and one row is left over. It is worth doing explicitly, because the leftover is the most-misunderstood entry in the numerology table:

ΔfFR1 · TS 38.101-1FR2 · TS 38.101-2what it carries
15 kHz✓—data carrier
30 kHz✓—data carrier
60 kHz✓✓data carrier — the one overlap, and the two tables disagree
120 kHz—✓data carrier
240 kHz——SS/PBCH block only
480 kHz—✓ optionaldata carrier
960 kHz—✓ optionaldata carrier
Table 7: Which of the seven numerologies a carrier may actually use, and where that is written. The middle two columns are the subcarrier-spacing rows of each range's Table 5.3.2-1; the physical layer defines all seven regardless. 60 kHz is the only spacing both ranges tabulate, and they disagree there - at 50 and 100 MHz FR1 gives 65 and 135 resource blocks where FR2 gives 66 and 132, because the two ranges settle the filter and emission trade-off differently. A carrier is in one range or the other and never both.

240 kHz is the odd one out, and it is not an omission. It is the only numerology of the seven for which neither range tabulates a UE channel bandwidth — so no data carrier can be configured with it. It exists for the synchronisation signal block: TS 38.213 clause 4.1 lists it as Case E, the 240 kHz SS/PBCH case, for carrier frequencies within FR2-1. And because an SS/PBCH block at 240 kHz sits inside a channel whose carrier runs at some other spacing, it needs a guard band of its own — TS 38.104 gives it a separate table, 5.3.3-3, apart from the carrier tables in clause 5.3.3:

Δf100 MHz200 MHz400 MHz
240 kHz3 800 kHz7 720 kHz15 560 kHz
Table 8: TS 38.104 Table 5.3.3-3 - the minimum guard band for a 240 kHz SS/PBCH block in FR2, quoted rather than derived. The clause 5.3.3 formula used everywhere else on this page does not produce these: it sizes the guard around a carrier of N_RB resource blocks, and this is a fixed 20-block block placed against the channel edge. The specification's note is explicit that the values apply only when the SS/PBCH block is adjacent to the edge of the channel it sits in.

Why this matters for reading the table above. A reader who takes Table 4.2-1 as a menu of carriers will look for a 240 kHz channel bandwidth, fail to find one in either document, and conclude the specification is incomplete. It is not: Table 4.2-1 describes the numerologies the physical layer supports, and carrying a shared channel is only one of the things a numerology can be used for. The physical layer defines what is possible; the RF specifications decide what a carrier may be.

And how many tones is a real carrier? Multiply the resource block count of TS 38.101-1 by twelve:

20 MHz at 15 kHz — 106 blocks, 1 272 subcarriers, 19.08 MHz occupied.

100 MHz at 30 kHz — 273 blocks, 3 276 subcarriers, 98.28 MHz occupied.

400 MHz at 120 kHz — 264 blocks, 3 168 subcarriers, 380.16 MHz occupied.

The count barely changes. Three carriers, twenty times apart in bandwidth, all land near three thousand tones — because the FFT size a receiver can afford is the real constraint, and going wider is done by making each tone wider rather than by adding more of them.

The same channel at 60 kHz

Every figure on this page was drawn for one carrier: 100 MHz at 30 kHz, 273 blocks, point A at 3 300 MHz. Double the spacing and not one of its numbers survives. The four figures below are the same four drawings for 100 MHz at 60 kHz — same channel width, same point A, same TS 38.101-1 tables, one row further down them.

One tone, in frequency — 60 kHz every frequency here is a subcarrier centre The same tone, across time the 100 MHz channel 1 430 kHz guard 1 370 kHz guard guards not to scale the same span, drawn again at the same scale CRB 141 ends 3 402.18 MHz CRB 7 starts 3 305.04 MHz middle of the grid CRB 51 m = −276, below the middle point A — CRB 0 — 3 300.00 MHz offsetToCarrier 7 blocks × 12 = 84 0 612 1 613 2 614 3 615 4 616 5 617 6 618 7 619 8 620 9 621 10 622 11 623 CRB 51 12 subcarriers index k each cell becomes a row 612 613 614 615 616 617 618 619 620 621 622 623 0 1 2 3 4 5 6 7 8 9 10 11 12 13 time — 14 OFDM symbols, one slot, 0.25 ms at 60 kHz one resource element — the pair (k, l) frequency what that row actually is k = 618 from point A  ·  m = (618 − 84) − 1 620/2 = −276 x(t) = cos(2π m Δf t) — 276 turns per symbol, drawn with 3 so the shape shows
Figure 11: The same drawing as the first figure on this page, for a 60 kHz carrier. Four magnifications of one tone: the channel it sits in, the carrier as a bar, one block of twelve subcarriers, and that block as a row of the resource grid. Every number has moved. The carrier is 135 blocks, not 273, so the block numbers run CRB 7 to CRB 141; the guards are 1 370 and 1 430 kHz instead of 845 and 875; a slot is 0.25 ms instead of 0.5, because a slot is always fourteen symbols and the symbols are half as long. The tone drawn is the seventh subcarrier of CRB 51, k = 618 counted from point A, which is 276 subcarriers below the middle of the grid - so it turns 276 times in a useful symbol that is itself half as long.

The block count halves and the guard band nearly doubles. Those are the two numbers a reader should take from the budget: a 60 kHz carrier fits 135 blocks in the same 100 MHz where a 30 kHz carrier fits 273, and it has to leave 1 370 kHz clear at the bottom where the 30 kHz carrier left 845. Wider tones spill further, so the channel has to keep more of itself empty — and the carrier occupies 97.2 MHz where the 30 kHz one occupied 98.28.

the blue part is the carrier of the previous figure, on its side 1 610 more subcarriers 96.6 MHz, not drawn channel edge 3 303.640 MHz channel edge 3 403.640 MHz 100 MHz channel at 60 kHz 1 370 kHz 1 430 kHz 30 kHz edge centre first subcarrier — k = 84, CRB 7 edge 3 305.010  ·  centre 3 305.04 MHz 30 kHz edge centre last subcarrier — k = 1 703, CRB 141 centre 3 402.18  ·  edge 3 402.210 MHz point A sits 3.640 MHz below this channel edge, off the bar to the left — 3.640 + 1.370 guard + 0.030 half a subcarrier = 5.04 MHz, the offsetToCarrier of the block-grid figure below To scale: the subcarriers, one cell = 60 kHz.   Compressed, and marked with a zigzag: the two guard bands and the middle of the band. The guard is measured to the subcarrier's edge, not its centre — that 30 kHz at each end is why 1 620 subcarriers occupy 97.2 MHz while their centres span 97.14.
Figure 12: Where the 100 MHz goes at 60 kHz. The channel is the same width and sits 180 kHz higher than the 30 kHz one, for the reason given below; the guards are 1 370 kHz at the bottom and 1 430 at the top, and the 1 620 subcarriers between them occupy 97.2 MHz edge to edge while their centres span 97.14. The half-subcarrier at each end is now 30 kHz rather than 15, which is the whole of the difference between those two spans.

The two channels cannot start at the same frequency, and that is not an accident. Point A is shared by every numerology of a cell, and offsetToCarrier is a whole number of blocks of that numerology — 360 kHz each at 30 kHz, 720 kHz each at 60. From point A at 3 300 MHz, the lower edge of a legal 100 MHz channel would have to sit $0.72n - 3.49$ MHz above it, and the minimum guard of TS 38.101-1 confines that to between 1.370 and 1.430 MHz — which needs $n$ between 6.75 and 6.83. There is no whole number there. So the 60 kHz channel takes $n = 7$ and starts at 3 303.640 MHz, 180 kHz above the 30 kHz one. Either the channel moves or point A does; the grid does not bend.

0 1 2 3 4 5 6 7 8 9 10 136 137 138 139 140 141 CRB 11 … 135 The common resource block grid at 60 kHz — numbered from point A, upwards in frequency, for ever every block on this line has a number; only the blue ones belong to this carrier point A 3 300.000 MHz CRB 0 begins here CRB 7 starts 3 305.04 MHz CRB 141 ends 3 402.18 MHz offsetToCarrier = 7 blocks × 12 = 84 subcarriers = 5.04 MHz this carrier — 135 blocks, CRB 7 to CRB 141 CRB 0 to CRB 6 are not missing and not spare: they are numbered, they are simply below where this carrier begins. Point A is the origin of the numbering, not an edge of anything — here it sits 5.04 MHz below the carrier, off it entirely. Every frequency shown is a subcarrier centre. the channel edge — 3 303.640 MHz the shaded strip above it is the guard band — 1 370 kHz, then half a subcarrier, then CRB 7 that is 1 400 kHz, or 1.944 blocks: a guard is never a whole number of them
Figure 13: The 60 kHz carrier as a window onto its own block grid. CRB 0 to CRB 6 exist and are numbered; the carrier owns CRB 7 to CRB 141. The offset is 7 blocks, which is 84 subcarriers and 5.04 MHz - a larger offset in megahertz than the 30 kHz carrier's 4.32, out of fewer blocks, because each block is twice as wide. The amber strip is the guard band: 1 370 kHz, then half a subcarrier, then CRB 7 - 1 400 kHz in all, which is 1.944 blocks and again not a whole number of them.
point A 3 300.000 MHz  ·  the zero of the block grid the channel edge 3 303.640 MHz  ·  where the 100 MHz begins first subcarrier, its edge 3 305.010 MHz first subcarrier, its centre 3 305.040 MHz  ·  k = 84 · CRB 7 starts 3 640 kHz — nobody chose this: it is 5 040 − 1 370 − 30 1 370 kHz — the guard band 30 kHz — half a subcarrier offsetToCarrier 5 040 kHz = 84 subcarriers = 7 resource blocks Two different measurements, between two different pairs of points — 60 kHz the guard band starts at the channel edge  ·  offsetToCarrier starts at point A Two numbers are chosen — point A and offsetToCarrier. Two are fixed by the standard — the guard band and half a subcarrier. The channel edge is then forced: 3 640 = 5 040 − 1 370 − 30 kHz spacing not to scale, the numbers are on the arrows
Figure 14: The four levels at the bottom of the 60 kHz band. Two numbers are chosen - point A and offsetToCarrier - and two are fixed by the standard - the guard band and half a subcarrier - and the channel edge is then forced: 3 640 = 5 040 minus 1 370 minus 30 kHz. It is the same sentence as the 30 kHz ladder with every number replaced, which is the point of drawing it twice.
$\mu = 1$ · 30 kHz$\mu = 2$ · 60 kHz
Blocks in 100 MHz (Table 5.3.2-1)273135
Subcarriers3 2761 620
One block360 kHz720 kHz
Occupied, edge to edge98.28 MHz97.2 MHz
Guard, lower · upper (Table 5.3.3-1)845 · 875 kHz1 370 · 1 430 kHz
offsetToCarrier from point A12 blocks = 4.32 MHz7 blocks = 5.04 MHz
Channel lower edge3 303.460 MHz3 303.640 MHz
First · last subcarrier centre3 304.32 · 3 402.57 MHz3 305.04 · 3 402.18 MHz
Useful symbol $T_u$33.33 µs16.67 µs
Slot, fourteen symbols0.5 ms0.25 ms
1 622 Hz Doppler is…5.4 % of the spacing2.7 %
Table 9: The same 100 MHz channel under both numerologies, every row computed rather than quoted. Read the last two rows together: the 60 kHz carrier gives up 1.08 MHz of spectrum and half its tones, and buys a symbol that is half as long and a Doppler tolerance that is twice as good. Nothing here is a preference - each number follows from the spacing.

What the pair of drawings is for. Nothing in the second set was retyped: the figures come from the same five scripts as the first set, given --scs 60, and every number in them is derived from the numerology, the block count and the two chosen values. That is the only way to be sure the two sets agree with each other — and it is why a reader can trust that the differences between them are real differences, not drafting.

What knocks a tone off its slot

Orthogonality is a promise with conditions, and there are exactly two ways to break it. Every number in the table above is a defence against one of them.

Breakage one — the frequency is wrong. If the receiver's idea of where the tones are differs from the transmitter's by $\varepsilon$ spacings, every tone lands off its neighbours' nulls and each one leaks into all the others. The leaked power has a closed form: the wanted tone keeps $\operatorname{sinc}^2(\varepsilon)$ of its energy, and since the sinc-squared family sums to one at every offset, everything else — $1 - \operatorname{sinc}^2(\varepsilon)$ — becomes interference on the other tones.

Figure 15: Signal-to-interference ratio against carrier frequency offset, measured as a fraction of the subcarrier spacing. The curve is 10 log of sinc squared over one minus sinc squared, exact for a large number of subcarriers. It falls fast: one per cent of the spacing already caps the link at about 35 dB, ten per cent at just under 15 dB, and at ten per cent the link is capped near 16QAM no matter how good the signal-to-noise ratio is, because this interference does not fall when the transmitter shouts louder. The five marked points are rows of the table below - the same physical situations at different subcarrier spacings, moving up and down the same curve.

The offset comes from three places: the local oscillator is not exactly on frequency, the oscillator is noisy — worse at higher carriers — and the device is moving, which shifts every tone by $f_d = v f_c / c$.

MovementCarrier$f_d$15 kHz30 kHz120 kHz
3 km/h — walking3.5 GHz9.7 Hz0.06 % · 59 dB0.03 % · 65 dB0.01 % · 77 dB
120 km/h — motorway3.5 GHz389 Hz2.6 % · 26.5 dB1.3 % · 32.6 dB0.3 % · 44.6 dB
500 km/h — high-speed rail3.5 GHz1 622 Hz10.8 % · 14.1 dB5.4 % · 20.1 dB1.35 % · 32.2 dB
3 km/h — walking28 GHz78 Hz0.5 % · 40.5 dB0.3 % · 46.5 dB0.06 % · 59 dB
120 km/h — motorway28 GHz3 113 Hz20.8 % · 8.1 dB10.4 % · 14.4 dB2.6 % · 26.5 dB
Table 10: The Doppler shift of a moving device, and what it costs at three subcarrier spacings. Each cell is the offset as a fraction of the spacing, and the resulting signal-to-interference ratio. The physics is identical in every row of a pair - the same speed, the same carrier, the same shift in hertz - and only the spacing changes. Read the 120 km/h line at 28 GHz: on a 15 kHz numerology the link is drowning at 8 dB, and on 120 kHz the same journey is a 26.5 dB link. This one table is most of the argument for having seven numerologies instead of one.

Breakage two — the echo is too late. A copy of the signal arriving after the cyclic prefix has ended drags a piece of the previous symbol into the integration window, and the whole-number-of-cycles condition fails for every tone at once. The fix is a longer guard, which means a longer symbol, which means a narrower subcarrier.

The two fixes pull in opposite directions, and that is the whole design.

Movement wants wide subcarriers, so that a Doppler shift in hertz is a small fraction of the spacing.

Distance wants narrow subcarriers, so that the guard — always 7.03 % of the symbol — is long enough in microseconds to cover the echoes.

A rural cell at 700 MHz has kilometre-long echoes and slow-moving users: it takes 15 kHz and its 1 405 m of reach. A millimetre-wave cell has metre-long echoes and vicious phase noise: it takes 120 kHz and accepts 176 m. Neither is a compromise — each is the only workable answer to its own situation, which is why the standard ships seven and lets the network choose.

Why fifteen kilohertz, and why twelve of them

Neither number is derived from physics. Both are inherited, and kept on purpose.

15 kHz is LTE's spacing, and 5G starts there so that the two systems can share spectrum and sit in the same band without their grids fighting. Everything above it is $15 \times 2^{\mu}$, so the symbol boundaries of every numerology line up with each other: two 30 kHz symbols fit exactly inside one 15 kHz symbol, and the cyclic prefixes line up too. A network can therefore run two numerologies side by side in one carrier without either one's symbols straddling the other's.

Twelve subcarriers to a block is also from LTE, and twelve is a friendly number: it divides by 2, 3, 4 and 6, so a block can be split evenly in more ways than a power of two would allow, which matters for reference signal patterns that repeat every 2, 3, 4 or 6 subcarriers — the DM-RS combs, CSI-RS densities and PT-RS spacings all use that divisibility.

What is not inherited is the ladder above 120 kHz. 480 and 960 kHz arrived in Release 17 for the 52.6–71 GHz bands, where phase noise makes anything narrower unusable — the same argument as the Doppler table, driven by the oscillator instead of by the vehicle.

Read on

The full treatment is nr-frame-structure — this page defines the object and shows why it works; that note builds the whole frame on top of it.

To go deeper on…Read
Why many narrow tones beat one wide channel at allOne wide channel, or many narrow ones?
Orthogonality in plain words, with no integralWhy the tones do not interfere
How a radio actually builds a symbol, and the IFFTHow a radio builds a symbol
Why the guard is a copy of the tail rather than silenceWhy the guard is a copy
The seven numerologies, and how an operator picks oneNumerology — one setting with seven values
The grid, the element, the block, and the four numberingsThe resource grid, the element, the block
$\kappa$, $T_c$, and why every duration is counted in samplesEverything is counted in samples
The guard in metres, at every numerologyWhat the guard is worth in metres
Table 11: Where each part of this page is developed further. The right-hand column is the section of nr-frame-structure.md that owns the subject.

Sideways, to what is built directly on this: OFDM · OFDM symbol · resource element · resource block · cyclic prefix · numerology · bandwidth part · Point A · FR1 and FR2 · PAPR

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Before this concept, the hierarchy says to learn the following — the full chain, in order:

Subcarrier needs nothing first — it is one of the places to start reading.

Keep going — where this sits on the route
the route starts here
Step 1 of 118level 0 · The grid and its units
Logical channelstep 2 · level 0▶
Level 0 → 1 · this unlocks
OFDM · Resource element level 3 · Numerology level 4
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