OFDM — Orthogonal frequency division multiplexing — the waveform itself

The grid and its units

Where it sits

In one line. OFDM carries a channel as thousands of narrow tones side by side. They overlap and still do not interfere, because each is spaced from its neighbours by exactly one over the symbol length. A guard made from each symbol's own tail then turns the whole radio channel into one complex number per tone, and that is the property everything else on this page is about.

What it is, and what it is for

OFDM divides a channel in frequency so that the pieces can overlap without interfering. Instead of one wide, fast stream of symbols, the band carries hundreds or thousands of narrow tones — subcarriers — side by side. Each one is modulated slowly, and each sits exactly one over the symbol length away from its neighbours.

In NR this is not one option among several. It is the waveform, downlink and uplink both (TS 38.211 clause 5.3.1). The uplink has one variant, transform precoding (clause 6.3.1.4), which this page reaches in §11.

The problem it solves is the equaliser, not the echo

The usual one-line explanation — long symbols survive echoes — is true, but it is not the argument. A single-carrier receiver survives echoes too. What it cannot do is survive them cheaply.

Take the running carrier: 100 MHz of spectrum, and a radio channel with a reflection arriving 472 ns after the direct path. That is 142 m of extra path length, an ordinary urban number.

That is the trade, stated plainly: OFDM does not beat a single carrier on theory. It beats it on arithmetic. A single-carrier system with frequency-domain equalisation gets much of the same benefit — Sari, Karam and Jeanclaude (1995) made the point early — and that is roughly what NR's uplink does when transform precoding is switched on. What OFDM adds is that the frequency view is the native one. The transmitter already works in bins, so the scheduler can hand different bins to different devices, at different powers and with different modulations, with no extra machinery. That is OFDMA, and it is the reason the resource grid exists at all.

The equation the standard actually writes

TS 38.211 clause 5.3.1 is titled OFDM baseband signal generation for all channels except PRACH and RIM-RS, and this is what it says. The signal on antenna port $p$ at subcarrier spacing configuration $\mu$, for OFDM symbol $l$ of a subframe, is zero outside that symbol's own interval. Inside it, it is:

Nothing in it is decoration. The same document defines every symbol in it, and every one of them has a value on the running carrier:

Why the tones may overlap

Orthogonality is one condition, but it turns up twice in different clothes. Once in continuous time, where it explains the spacing, and once in the discrete domain, where it explains the transform. Both are worth having: most confusion about OFDM comes from meeting only one of them.

In continuous time

Two tones $e^{j2\pi k \Delta f t}$ and $e^{j2\pi m \Delta f t}$ integrate to zero over one useful symbol whenever $\Delta f = 1/T_u$ and $k \neq m$. That is the whole spacing rule, and it is why the useful symbol length is not a free choice. Subcarrier — the condition: a whole number of cycles works the integral out in full and shows what "a whole number of cycles" means on the grid, so it is not repeated here.

What matters for this page is the consequence. The spectra of the tones overlap heavily — each is a sinc whose main lobe is two spacings wide — and they still do not interfere, because each one is sampled at its own peak, where every other tone passes through zero. Sharing the band with no guard bands between the tones is the whole economic case for the scheme, and it is Chang's (1966) idea.

In the discrete domain

Now sample the clause 5.3.1 sum. Set $k_0^{\mu} = 0$, ignore the prefix and the symbol start, and read the signal at $N$ evenly spaced instants $t = nT_u/N$. The exponent becomes $j2\pi kn/N$, and the sum is exactly an inverse discrete Fourier transform of the grid values. Orthogonality then becomes a finite sum — and an exact identity, not an approximation:

Read the "mod $N$" in that condition, because it is a warning. After sampling, tones whose indices differ by a multiple of $N$ are indistinguishable. That is aliasing, and it is why a real transmitter runs the transform larger than the occupied band — on the running carrier, 4 096 bins for 3 276 subcarriers — and filters what falls in between.

From a sum to a transform

This is the step that made OFDM affordable, and it is why the idea waited five years for the paper that made it buildable. Chang (1966) proposed orthogonal overlapping tones. Building them meant one oscillator per tone, and for thousands of tones that is not an engineering proposal. Weinstein and Ebert (1971) saw that the sum is a discrete Fourier transform, and that a transform is one block of arithmetic. The fast algorithm for it was already in print: Cooley and Tukey (1965), six years earlier.

So the standard specifies a sum, and every implementation computes a transform. The two agree exactly at the sample instants, which is all a sampled system needs. Here is the whole transmit path, with the running carrier's numbers on it.

Why 4 096 and 122.88 MHz, when the standard names neither. The transform has to be at least as large as the occupied band, 3 276 bins, and a power of two for a radix-2 algorithm. 4 096 is the first number that is both. The sample rate then follows: $4096 \times 30$ kHz $= 122.88$ MHz, which is $1/T_{\text{c}}$ divided by 16 — so both the useful part and the prefix land on whole numbers of samples, 4 096 and 288. Another implementer may choose differently and still be just as compliant. What nobody may change is the durations, because the clause counts those in $T_{\text{c}}$.

The same page at every channel width

100 MHz at 30 kHz is an example, not a special case, and it is worth being exact about what a different channel width changes and what it leaves alone.

Look at where the channel enters clause 5.3.1. $N^{\text{size},\mu}_{\text{grid},x}$ appears in exactly two places: the upper limit of the sum, and the term that centres the grid. Everything else in the clause is a function of $\mu$ alone — $N^{\mu}_{\text{u}} = 2048\kappa 2^{-\mu}$ and $N^{\mu}_{\text{CP},l} = 144\kappa 2^{-\mu}$ carry no bandwidth term at all.

The other two FR1 numerologies are the same lookup. At 15 kHz the defined widths run from 3 to 50 MHz — 15 blocks to 270 — and 60 MHz upwards is not defined at all; at 60 kHz they run from 10 to 100 MHz, 11 blocks to 135. All three rows are TS 38.101-1 Table 5.3.2-1, reproduced for four common widths in frame structure — how many blocks fit in a real channel. FR2 is a different document, TS 38.101-2, with its own widths of 50 to 400 MHz at 60 and 120 kHz; it is not in this folder's specification snapshot, so no block count for it is quoted here.

What the cyclic prefix actually buys

Everything above assumed the channel does nothing. It does. The signal arrives by several paths, so the receiver gets a convolution of the transmitted samples with the channel's impulse response. A convolution destroys the whole-number-of-turns condition, because the delayed copies do not start where the receiver's window starts.

The cyclic prefix repairs exactly that, and it repairs it by an identity, not by an approximation. Copy the last $N_{\text{CP}}$ samples of the block to its front and send them first. Then, so long as every path arrives within the prefix, the $N$ samples the receiver keeps are the circular convolution of the block with the channel instead of the linear one. The figure counts it out cell by cell.

Once the convolution is circular, the transform diagonalises it. That is the convolution theorem for the DFT, and it turns the whole radio channel into one complex multiplication per subcarrier:

This is the payoff, and it is worth seeing as a picture rather than as a claim. $H[k]$ is not constant across a wide carrier: a channel with echoes is frequency selective, and over 100 MHz it varies enormously. But however much it varies, it is still one number per subcarrier.

What it costs

The prefix carries no new information — it is a copy, transmitted and then thrown away. At $\mu = 1$ that is 4 608 time units against a useful 65 536:

And this is where the numerology comes from. A longer prefix buys reach and costs airtime, and a wider subcarrier spacing shortens the symbol and the prefix in the same proportion. NR did not pick one balance. It defined seven, $\mu = 0$ to $6$, in TS 38.211 Table 4.2-1, and numerology is the page about that choice.

The whole thing in matrices

§6 claimed that the transform diagonalises the channel. This section proves it, and the proof is short enough to do rather than cite. It is also the point where OFDM stops being a collection of tricks and becomes a single fact of linear algebra.

The receiver's whole job, in three lines. Send $\mathbf{x} = \mathbf{F}^{\mathsf H}\mathbf{X}$ with the prefix in front, throw the prefix away, and take the transform:

That middle line rests on one fact, and this is it.

Theorem. The DFT diagonalises every circulant matrix. Its eigenvectors do not depend on the matrix at all — they are the same $N$ vectors for every circulant matrix of that size — and its eigenvalues are the DFT of its first column.

Proof. Let $\mathbf{v}_k$ be the vector with entries $v_k[n] = \tfrac{1}{\sqrt N} e^{j2\pi kn/N}$, which is subcarrier $k$, sampled. A circulant matrix acts by circular convolution, so row $n$ of $\mathbf{H}_{\text{c}}\mathbf{v}_k$ is

so $\mathbf{v}_k$ is an eigenvector with eigenvalue $H[k]$, for every $k = 0,\dots,N-1$. Those $N$ vectors are exactly the columns of $\mathbf{F}^{\mathsf H}$, and they are orthonormal, so $\mathbf{H}_{\text{c}} = \mathbf{F}^{\mathsf H}\boldsymbol{\Lambda}\mathbf{F}$. $\blacksquare$

The sentence worth keeping. The subcarriers are the eigenvectors of every channel shorter than the cyclic prefix. Not approximately, and not for one particular channel, but for all of them at once — because the eigenvectors of a circulant matrix do not depend on its entries. That is why the receiver can equalise while knowing nothing about the channel except $N$ numbers.

And here is the same statement as a picture, computed rather than drawn.

The two things that break it

Orthogonality is a promise with exactly two conditions, and every parameter in the frame structure defends one of them. The two pull in opposite directions, which is why one numerology could never have been enough.

One: the echo arrives after the prefix

A path later than $N_{\text{CP}}$ drags the tail of the previous symbol into the window, and the wrap in the figure above no longer closes. Two things then go wrong at once. Inter-symbol interference: one symbol lands on the next. And inter-carrier interference, because the whole-turns condition has been broken as well. The fix is a longer prefix, which means a longer symbol, which means a narrower spacing.

Two: the frequency is wrong

If transmitter and receiver disagree about where the tones are — oscillator error, oscillator phase noise, or Doppler because the device is moving — then each tone is sampled off its own peak, and there its neighbours are no longer zero. Moose (1994) derived the inter-carrier interference that results. The compact form used across this folder gives the signal-to-interference ratio for a normalised offset $\varepsilon$, measured in subcarrier spacings, as $\text{sinc}^2(\varepsilon)/(1-\text{sinc}^2(\varepsilon))$, and subcarrier — what knocks a tone off its slot plots it. The next section derives it, because it takes four lines and it explains why the ratio is a floor rather than a penalty.

Pollet, Van Bladel and Moeneclaey (1995) turned the same effect into the number a link budget wants: the extra signal-to-noise ratio needed to hold the same error rate.

Where the interference formula comes from

The sinc expression is not a rule of thumb. It falls out of the transform in four lines, and doing it once is worth more than quoting it ten times.

Suppose the receiver's oscillator is off by $\varepsilon$ subcarrier spacings. Every sample it takes is multiplied by $e^{j2\pi\varepsilon n/N}$, so its transform gives, for subcarrier $k$:

That expression says two things. With $\varepsilon = 0$, every $I_d$ with $d \neq 0$ is exactly zero, because the numerator vanishes at every integer. That is orthogonality, and it is why the figure below is a single spike. With $\varepsilon \neq 0$, nothing is zero any more, and every other subcarrier on the grid contributes.

One identity finishes the derivation. The receiver multiplies by something of unit modulus and then applies a unitary transform, so it conserves energy — which means

so the wanted power is $|I_0|^2$, the interference is everything else, $1 - |I_0|^2$, and the ratio of the two is the expression this folder uses everywhere:

So the tension, in one line. The first breakage wants narrow spacing, the second wants wide spacing, and the delay spread and Doppler of a deployment decide which one is pressing. Frame structure §2 is that argument at length.

The cost nobody can design away

An OFDM symbol is the sum of thousands of independently modulated tones, and independent things occasionally line up. When they do, the instantaneous power is far above the average — and the power amplifier has to stay linear all the way up to that peak while delivering only the average.

The statistics are the useful part, because the worst case is meaningless. If all 3 276 tones lined up in phase, the peak would be 35.15 dB above the average — and that never happens. Under the standard complex-Gaussian approximation, the probability that a symbol's peak-to-average ratio exceeds $\gamma$ is $1-(1-e^{-\gamma})^{N}$, and the picture that formula draws is far less alarming than the worst case:

Where the distribution comes from

Two lines, and then one honest correction. An OFDM sample is a sum of $N$ independent modulation symbols, each multiplied by a unit-magnitude exponential. For $N$ in the hundreds, the central limit theorem applies to the real and imaginary parts separately, so $x[n]$ is very nearly complex Gaussian. Its magnitude is then Rayleigh and its power $|x[n]|^2$ is exponential. Treat the $N$ samples as independent, and the peak of the symbol is the maximum of $N$ exponentials:

That is the curve plotted above, and it is optimistic — measurably so. The model counts only the Nyquist-rate samples, and the continuous waveform peaks between them. Oversampling by about four is the usual way to catch that, and this page did it rather than claiming it:

So the model is worth having, and worth distrusting by half a decibel — which is what the caveat in the previous section said, now with a number attached to it.

What the grid is worth: parallel channels

The algebra of §7 has a payoff in information theory, and it is the reason OFDM is not merely convenient. One frequency-selective channel of bandwidth $B$ is a hard object: its capacity is an integral over a spectrum, and reaching it needs a transmitter that shapes the signal across the whole band. After the transform, that same channel is $N$ parallel Gaussian channels, each with its own gain $|H[k]|^2$ and its own noise — and parallel Gaussian channels are one of the few things information theory answers completely.

Drawn, it is a bathtub. Plot noise-over-gain against subcarrier index. The good subcarriers are the low ones, the fades are the peaks, and the best allocation is what happens if you pour a fixed volume of water into that shape.

The uplink's second waveform

TS 38.211 clause 6.3.1.4, Transform precoding, is one paragraph and one equation, and it changes what the waveform is. Before the values reach the subcarriers, each symbol's block of $M^{\text{PUSCH}}_{\text{sc}}$ modulation symbols goes through a DFT of its own length:

This is DFT-spread OFDM, and LTE used it for the entire uplink. NR made it optional, switched on per TS 38.214 clause 6.1.3. The reasons for not making it the default are set out in Dahlman, Parkvall and Sköld §7.1 — a book written by people who were in the room:

And one constraint that is pure implementation, written into the standard. Clause 6.3.1.4 requires the allocation size in resource blocks to satisfy $M^{\text{PUSCH}}_{\text{RB}} = 2^{\alpha_2} 3^{\alpha_3} 5^{\alpha_5}$ with non-negative integer exponents. The DFT length is therefore always 5-smooth, and a mixed-radix algorithm is always cheap. The consequence is concrete: 273 resource blocks is $3\times 7\times 13$, so a full 100 MHz allocation cannot use transform precoding at all. The largest legal allocation below it is 270 blocks, then 256, 250, 243 and 240.

So the asymmetry in NR is deliberate. PDSCH is CP-OFDM only, because a base station can afford the back-off. The uplink keeps a switch, and the network throws it when the device is at the edge of the cell and the amplifier, not the scheduler, is the binding constraint.

What it is not

Where each part of this is developed

This page defines the waveform. It is not where any of these subjects is finished.

Sources

Two kinds of source, and they answer different questions. The specification says what the rule is. It never says where the rule came from, and every idea on this page is decades older than 5G. Both kinds are below, each with its venue in full — because "the OFDM paper" is ambiguous, and the papers people mean sit in four different kinds of publication.

Read the note on what was and was not verified before citing any of them.

The specification

The papers, by venue

The books

Two are in the archive and were read for this page. The rest are pointers.

For the background this page assumes — fading, delay spread, and why any of it is necessary — here are the two books the teaching literature reaches for most often:

Three more, for particular needs:

• Cover and Thomas, Elements of Information Theory, 2nd ed., Wiley, 2006 — chapter 9 is parallel Gaussian channels and water-filling, which is §10 done in general. Tse and Viswanath cover the same ground for fading channels in their chapter 5.
• Proakis and Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008 — the textbook treatment of multicarrier modulation, and the place to go if the question is about the equaliser that §2.1 claims is expensive.
• van Nee and Prasad, OFDM for Wireless Multimedia Communications, Artech House, 2000 — the book-length treatment of OFDM alone, PAPR included.
• Sesia, Toufik and Baker, LTE — The UMTS Long Term Evolution, 2nd ed., Wiley, 2011 — because the constants 144, 160 and 2048 in clause 5.3.1 are 4G's, unchanged.

What was verified, and what was not

Read on

The subject is developed in nr-frame-structure, where OFDM stops being a waveform and becomes a grid with addresses on it — and in subcarrier, which is this page's one prerequisite, taken slowly.