OFDM · OTFS · AFDM — L1 Waveform Deep-Dive

1.1  OFDM Recap in One Page

A wideband channel is frequency-selective: different spectral components of the transmitted signal experience different complex gains, causing intersymbol interference (ISI) that grows with bandwidth. OFDM's insight is strikingly simple — divide the wide band into N narrow sub-channels, each narrow enough to appear flat, then modulate an independent data symbol $X[k]$ onto each subcarrier $k$.

The continuous-time transmitted signal is:

The subcarriers are spaced by $\Delta f$, and their complex exponentials are orthogonal over one symbol period $T_u$:

Orthogonality is the key: even though the sinc-shaped spectra of adjacent subcarriers overlap, each subcarrier is at the spectral null of every other subcarrier. Consequently the IDFT and DFT at transmitter and receiver are inverses of each other in the noise-free static case.

Cyclic Prefix — Eliminating ISI

A multipath channel with maximum excess delay $\tau_\text{max}$ smears each symbol into the next. The fix: prepend a Cyclic Prefix (CP) of length $T_{CP} \ge \tau_\text{max}$ copied from the tail of each OFDM symbol. The receiver discards the CP. After CP removal, the linear convolution with the channel becomes a circular convolution — and a circulant matrix is diagonalised by the DFT.

After CP removal and FFT, the received symbol on subcarrier $k$ is simply:

where $H[k]$ is the channel frequency response at subcarrier $k$ and $W[k]$ is additive noise. Equalization is a single complex division — one multiply per subcarrier. This is OFDM's triumph: it converts the hard problem of wideband equalization into N trivial one-tap equalizations.

1.2  The Doppler Problem

A mobile radio environment is doubly dispersive: the channel spreads energy in both delay (multipath) and Doppler (mobility). The time-varying channel impulse response for a $P$-path channel is:

where $\tau_i$ is the propagation delay of path $i$ and $\nu_i$ is its Doppler shift:

For a carrier $f_c = 3.5\;\text{GHz}$ and a vehicle speed $v = 120\;\text{km/h} = 33.3\;\text{m/s}$, the maximum Doppler is $\nu_\text{max} = 33.3 \times 3.5\times 10^9 / (3\times 10^8) \approx 389\;\text{Hz}$.

How Doppler Destroys OFDM Orthogonality

With a time-varying channel, the input-output relation at subcarrier $k$ is no longer diagonal. Define the time-frequency transfer function $H_{k,k'}$ as the coupling gain from transmitted subcarrier $k'$ to received subcarrier $k$. The received signal is:

For $k \ne k'$, each $H_{k,k'} \ne 0$ — this is Inter-Carrier Interference (ICI). The matrix $\mathbf{H}$ is no longer diagonal; OFDM equalization breaks.

The ICI coupling coefficient between subcarriers $k$ and $k'$ for a single Doppler shift $f_d$ is:

For small $f_d/\Delta f$, the ICI power summed over all interfering subcarriers relative to the desired signal power can be approximated as:

This implies a hard SINR ceiling regardless of transmit power:

For 5G NR with $\mu = 0$ ($\Delta f = 15\;\text{kHz}$): the threshold speed is $v_\text{limit} \approx 0.1 \times 15000 \times c/f_c$. At 3.5 GHz this is only $\approx 460\;\text{m/s} = 1656\;\text{km/h}$ — apparently safe. But SINR floors at lower speeds are real: at 350 km/h and $\mu=0$ the floor is $\approx 28\;\text{dB}$ — 64-QAM barely survives, 256-QAM is gone.

1.3  Doppler Limits Table

The following table maps real-world deployment scenarios to their Doppler frequency, required subcarrier spacing, and whether standard 5G NR numerology can handle them. All Doppler values use $f_d = v f_c / c$.

Scenario Context

• High-Speed Train (350 km/h, 3.5 GHz): 5G NR $\mu=0$ gives SINR floor of $\approx 26\;\text{dB}$ — 64-QAM just survives but 256-QAM fails. 3GPP Rel-16 introduced HST-SFN (Single Frequency Network) enhancements, and Rel-17 adds NR-NTN, but neither fundamentally solves the ICI problem — they merely manage it via Doppler pre-compensation at the UE.
• LEO Satellite (3GPP Rel-17 NTN, 3.5 GHz): Starlink-class orbits at 550 km altitude have maximum Doppler of $\approx 88\;\text{kHz}$ relative to a ground UE, nearly 6× the 15-kHz subcarrier spacing. 3GPP NTN (TS 38.821) mandates UE-side Doppler pre-compensation but this requires accurate UE ephemeris and leaves residual ICI from inter-path Doppler spread. SINR floor without pre-compensation: $\approx -46\;\text{dB}$ — unusable.
• mmWave V2X (ITS-5.9 GHz band, C-V2X): 3GPP Rel-14 C-V2X uses 5.9 GHz. At 28 GHz (FR2 candidate for 6G V2X), even 200 km/h yields $f_d \approx 5.2\;\text{kHz}$. The only 5G NR numerology that keeps this below $\Delta f/10$ is $\mu=3$ (120 kHz), which offers only 400 MHz bandwidth — and its 0.125 ms slot imposes latency that is marginal for safety-critical V2X. This is one motivation for AFDM's chirp-based equalization.
• IMT-2030 (6G) requirement: ITU-R IMT-2030 (M.2160) mandates support for 1000 km/h. At 3.5 GHz this means $f_d \approx 3.24\;\text{kHz}$; at 28 GHz it means $f_d \approx 25.9\;\text{kHz}$. No practical 5G NR numerology can handle the 28 GHz case — this is the hard requirement that drives OTFS and AFDM research.

1.4  ICI Visualisation

The canvas below shows energy distribution across $N=16$ subcarriers for a single transmitted symbol on subcarrier 8. Without Doppler, all energy is isolated in one bin. As $f_d/\Delta f$ increases, energy leaks into neighbouring bins — this is ICI. The ICI floor limits SINR regardless of transmit power.

1.5  OFDM Failure Modes Summary

ICI is the primary failure mode under mobility, but it interacts with other OFDM limitations. The table below summarises all major failure modes and how they are affected by higher mobility.

The Structured Limitation

Each mitigation above patches a symptom rather than addressing the root cause. The fundamental issue is that OFDM is designed in the time-frequency (TF) domain, where a doubly-dispersive channel is neither sparse nor separable — the $N\times N$ TF channel matrix has $O(NP)$ non-zero entries (one diagonal per Doppler path, each of length $N$).

In contrast, the Delay-Doppler (DD) domain exploits the physical sparsity of the channel: a $P$-path channel has exactly $P$ non-zero entries in the DD plane, regardless of $N$. This observation — formalised by the Zak transform — is the foundation of OTFS. AFDM achieves the same effective sparsity in its affine frequency domain without the 2D processing overhead.

2.1  Waveform Parameters — 5G NR Numerology

5G NR defines five numerologies indexed by $\mu \in \{0,1,2,3,4\}$. The subcarrier spacing (SCS) is:

All other timing parameters scale with $\mu$. The complete table follows (TS 38.211 Table 4.3.2-1):

Notes: (1) Extended CP (16.67 μs) is defined only for μ=0 in some special configurations (e.g. broadcast) and for μ=2 (60 kHz) in TS 38.211 §4.3.2. (2) μ=4 is for downlink reference signals only. (3) CP lengths shown are for "other symbols"; the first symbol of each 0.5-ms half-subframe has a slightly longer CP (160 samples at μ=0 vs. 144 samples) to keep slot boundaries aligned. (4) For a 100-MHz carrier (μ=1), sample rate is 61.44 Msps.

Timing Relationships

The basic time unit $T_c$ is the common denominator for all numerologies; CP and symbol durations are integer multiples of $\kappa T_c$ where $\kappa = T_s/T_c = 64$.

2.2  CP-OFDM Signal Model

This subsection develops the complete signal model in both continuous and matrix form, showing explicitly why the circulant channel is diagonalised by the DFT.

Transmitter

Let $\mathbf{x} = [X[0],\,X[1],\ldots,X[N-1]]^T$ be the frequency-domain data vector (QAM symbols). The OFDM time-domain block is computed by the $N$-point IDFT:

In matrix form, with $\mathbf{F}$ the $N\times N$ unitary DFT matrix ($[\mathbf{F}]_{n,k} = \frac{1}{\sqrt{N}}e^{-j2\pi nk/N}$):

The Cyclic Prefix appends the last $N_{CP}$ samples of $\mathbf{s}$ to the front, producing the transmitted vector of length $N + N_{CP}$:

Channel

The static multipath channel has impulse response $h[l]$ for delays $l = 0,\ldots,L-1$. After CP insertion, transmission through the channel, and CP removal at the receiver (discarding the first $N_{CP}$ received samples), the linear convolution becomes circular, and the received vector is:

where $\mathbf{H}_{\text{circ}}$ is the $N\times N$ circulant matrix with first column $[h[0],\,h[1],\ldots,h[L-1],\,0,\ldots,0]^T$. This requires $N_{CP} \ge L-1$, i.e. $T_{CP} \ge \tau_{\max}$.

Receiver — DFT Diagonalisation

The receiver applies the $N$-point DFT:

The key identity: any circulant matrix is diagonalised by the DFT,

where $H[k] = \sum_{l=0}^{L-1}h[l]\,e^{-j2\pi kl/N}$ is the channel frequency response. Therefore:

One-Tap Equalisation

Under the diagonal model, equalization reduces to scalar division at each subcarrier. For zero-forcing (ZF):

For MMSE, the noise is regularised:

2.3  DFT-s-OFDM (Uplink)

5G NR PUSCH supports an optional transform precoding mode that produces DFT-spread OFDM (DFT-s-OFDM), also called SC-FDMA. This is the uplink waveform when transformPrecoding = enabled (TS 38.211 §6.3.1.4).

Transmit Chain

Let $\mathbf{d} = [d[0],\ldots,d[M-1]]^T$ be $M$ QAM data symbols. Transform precoding applies an $M$-point DFT:

In matrix form: $\tilde{\mathbf{X}} = \mathbf{F}_M\,\mathbf{d}$

The $M$ DFT outputs are mapped onto $M$ consecutive OFDM subcarriers (out of N total), zeros on remaining subcarriers. The full $N$-point IFFT then produces the time-domain signal:

This is equivalent to a single-carrier signal (SC) frequency-division multiplexed across the allocated bandwidth: the overall effect of DFT-M + zero-pad + IFFT-N is a single-carrier signal occupying $M \cdot \Delta f$ bandwidth with cyclic prefix added.

PAPR Reduction

Pure single-carrier signalling has a nearly constant envelope (for large alphabets, the PAPR approaches $\approx 1\;\text{dB}$ by the central limit theorem applied to the time-domain samples). DFT-s-OFDM inherits this property when $M = N$ (full bandwidth utilization), but in 5G NR $M \le N$ and the PAPR is slightly higher. Empirically:

The coverage gain from lower CM is why DFT-s-OFDM is mandated for cell-edge UEs in 5G NR uplink. Downlink uses CP-OFDM only — the PA is at the gNB where power is not constrained.

Receive Chain

The gNB receiver reverses the process after channel equalization:

Channel estimation uses DMRS sequences designed to have low PAPR (computer-generated sequences or Zadoff-Chu roots for DFT-s-OFDM).

2.4  Resource Grid & Overhead

The 5G NR resource grid is the fundamental unit of time-frequency allocation. Each Resource Element (RE) is one subcarrier × one OFDM symbol, carrying one complex-valued modulation symbol. A Physical Resource Block (PRB) is 12 consecutive subcarriers in frequency.

Reference Signal Overhead

Net Data Efficiency

For a typical FDD PDSCH configuration (DMRS Type 1, 2 symbols + 1 PDCCH symbol + light CSI-RS):

This is before considering coding rate and modulation order. The maximum spectral efficiency at 256-QAM (8 bits/symbol), rate 948/1024 (≈0.926), two layers, 77% RE efficiency is approximately:

The 3GPP target for peak spectral efficiency in FR1 MIMO is ~30 bits/s/Hz (8 layers), consistent with the per-layer calculation above.

2.5  OFDM Complexity

OFDM owes its dominance to computational efficiency. Every operation in the chain is $O(N\log N)$ or better.

Total OFDM complexity:

For 5G NR at $\mu=1$ (30 kHz SCS), a 100-MHz carrier has $N = 4096$ FFT points (at 61.44 Msps). Per-slot complexity is $14 \times 4096 \times 12 \approx 688{,}000$ complex multiply-accumulates (CMACs) for the FFT alone, plus channel estimation and equalization. A modern 5G L1 ASIC executes this in hardware at <1 ms latency with ~500 mW power.

2.6  OFDM Subcarrier Spectra — Orthogonality Visualised

Each OFDM subcarrier has a sinc-shaped spectrum $\text{sinc}(f/\Delta f - k) = \sin(\pi(f/\Delta f - k))\,/\,(\pi(f/\Delta f - k))$. Adjacent subcarriers overlap, but each is at the exact spectral null of every other subcarrier, preserving orthogonality. The chart below shows this for $N=16$ subcarriers (subcarriers 0–15 shown in alternating colours). The envelope $|H(f)|$ is plotted vs. normalised frequency in units of the subcarrier spacing.

Figure 2.6 — OFDM subcarrier spectra (N=16). Each subcarrier is at the spectral null of every other subcarrier — orthogonality maintained. Doppler shifts the sinc centres, destroying orthogonality.

3.1  From Time-Frequency to Delay-Doppler

The standard linear time-varying channel is characterised by its spreading function $h(\tau, t)$: the complex gain of a path with propagation delay $\tau$ at observation time $t$. The time-variation encodes Doppler — a path moving at constant velocity produces $h(\tau, t) = h_0 \delta(\tau - \tau_0)\, e^{j2\pi \nu_0 t}$, i.e., the gain oscillates sinusoidally in $t$ at the Doppler frequency $\nu_0$.

Taking the Fourier transform of $h(\tau, t)$ over the time variable $t$ yields the delay-Doppler spreading function:

For a channel consisting of $L$ discrete propagation paths, each path $i$ contributes a complex amplitude $h_i$, a propagation delay $\tau_i$, and a Doppler shift $\nu_i$. The spreading function collapses to a sum of Dirac impulses:

This is the key observation: in the $(\tau, \nu)$ plane, the channel is described by exactly $L$ point masses — one per physical scatterer. The TF domain representation of the same channel is a dense, time-varying matrix because Doppler mixes every transmit frequency into every receive frequency. The DD domain strips away that complexity and exposes the physical structure directly.

Discrete DD Grid

For a digital system with $M$ subcarriers at spacing $\Delta f$ and $N$ time slots of duration $T = 1/\Delta f$, the DD grid has resolution:

Path $i$ maps to grid indices $l_i = \tau_i / \Delta\tau$ (delay index, integer in the ideal case) and $k_i = \nu_i / \Delta\nu$ (Doppler index). When $l_i$ and $k_i$ are integers the channel is called integer DD; when they are non-integer (most practical channels) the channel is fractional DD, which causes leakage — discussed in §3.5.

3.2  Why the DD Domain is Special

Consider a channel with $L=4$ paths modelled on a grid of size $N \times M = 16 \times 64$. In the TF domain the channel matrix $\mathbf{H}_{TF} \in \mathbb{C}^{NM \times NM}$ is essentially full under Doppler: every symbol-subcarrier combination contributes to every other. The OFDM equaliser must invert a dense $1024 \times 1024$ matrix, or accept the ICI as noise.

In the DD domain the channel matrix $\mathbf{H}_{DD}$ has exactly $L$ non-zero entries per row — one for each physical path. The matrix is sparse (Block-Circulant with Circulant Blocks, BCCB), enabling $\mathcal{O}(NM \log NM)$ inversion via 2D FFT, or $\mathcal{O}(NM \cdot L)$ via message passing.

Beyond sparsity, the DD channel matrix is time-invariant across the entire OTFS frame. Once measured, it does not need to be re-estimated until the physical scatterer geometry changes. OFDM channels must be tracked per-OFDM-symbol (every $T = 71\,\mu\text{s}$ for $\mu=1$); a channel with 1 ms coherence time requires dense pilot insertion in OFDM but only one pilot per OTFS frame in OTFS.

3.3  Sparsity Across Mobility Scenarios

The following table quantifies DD sparsity — the fraction of non-zero entries in the $N \times M$ DD channel grid — for representative deployment scenarios. Values assume $M = 512$ delay bins (100 MHz BW, $\Delta\tau = 20\,\text{ns}$) and $N = 128$ Doppler bins ($\Delta\nu \approx 781\,\text{Hz}$ for 100 ms frame).

Sparsity is remarkable even at extreme mobility: the LEO scenario has $L=2$ paths but a Doppler spread of 88.7 kHz (5G NR subcarrier spacing is 15–480 kHz), which would completely destroy OFDM orthogonality. In the DD domain the same channel is represented by 2 points in a $128 \times 512 = 65{,}536$-element grid — 0.002% fill.

3.4  DD Domain Visualisation

The canvas below shows the same physical multi-path channel in two representations. The left panel renders the TF spreading function as a colour-coded heatmap ($N=16$ time slots $\times$ $M=16$ frequency bins): channel coefficients vary per time slot due to Doppler, producing a dense, time-varying pattern. The right panel renders the DD domain: the same $L$ paths appear as bright isolated dots on a sparse grid. Use the scenario selector to explore different mobility cases.

Same physical channel: TF view is dense and time-varying; DD view is sparse and static.

3.5  Integer vs Fractional Delay-Doppler

The clean sparse representation of §3.1 assumes path delays and Doppler shifts fall exactly on the DD grid — the integer DD idealisation. In practice, grid indices $l_i = \tau_i/\Delta\tau$ and $k_i = \nu_i/\Delta\nu$ are generally non-integer: fractional DD.

Fractional DD effects

• Energy leakage. Each path spreads power across neighbouring DD bins with a $\text{sinc}^2$-shaped profile. A single path at fractional index $k_i = 3.7$ leaks into bins $k=1,2,3,4,5,\ldots$ with decreasing but non-zero energy.
• Equalisation complexity increases. The channel matrix is no longer exactly $L$ non-zeros per row; the effective bandwidth of each leaking path fills $\mathcal{O}(\log NM)$ bins, increasing message-passing iteration count or requiring explicit sinc windowing.
• Channel estimation degrades. The guard region around the pilot must be enlarged to absorb leakage tails, increasing pilot overhead from $(2k_{\max}+1)(l_{\max}+1)$ to roughly $(2k_{\max}+k_{\text{guard}}+1)$ per axis.

Mitigation strategies

4.1  Core Concept

OFDM places data symbols on the time-frequency grid. Each symbol occupies one (subcarrier, slot) pair and experiences a scalar channel coefficient — but only if the channel is constant within that slot. Under Doppler this assumption fails: each OFDM symbol sees an entire Doppler-smeared channel that cannot be modelled as a scalar, causing ICI.

OTFS instead places data symbols on the delay-Doppler grid. The channel in the DD domain is always a scalar per path for the entire OTFS frame duration, regardless of Doppler. Every transmitted DD symbol interacts with at most $L$ channel coefficients (one per physical path), producing a sparse I/O structure that OFDM can never achieve under high-mobility conditions.

4.2  OTFS Transmitter Chain

An OTFS frame carries $N \times M$ complex QAM symbols $X[k,l]$ arranged on the delay-Doppler grid: $k = 0, \ldots, N-1$ is the Doppler index, $l = 0, \ldots, M-1$ is the delay index. The transmitter maps these four steps to a continuous-time signal $s(t)$.

4.3  OTFS Receiver Chain

The receiver is the dual (adjoint) of the transmitter. Starting from the received signal $r(t)$ after the doubly-dispersive channel and CP removal:

After Step 3, equalization is performed directly on $Y[k,l]$ using the sparse DD channel structure — see §4.7.

4.4  OTFS Input-Output Relation (Integer DD)

For paths with integer delay and Doppler indices ($l_i \in \mathbb{Z}$, $k_i \in \mathbb{Z}$), the end-to-end DD I/O relation after CP removal, Wigner transform, and SFFT is:

where $(\cdot)_N$ denotes modulo $N$ and $(\cdot)_M$ denotes modulo $M$. This is a 2D circular convolution of the input $X$ with the sparse DD channel kernel $h_{DD}[k,l] = \sum_i h_i\, \delta[k-k_i]\, \delta[l-l_i]$.

Channel matrix structure

In matrix form $\mathbf{y} = \mathbf{H}_{DD}\, \mathbf{x} + \mathbf{w}$, where $\mathbf{x}, \mathbf{y}, \mathbf{w} \in \mathbb{C}^{NM}$ (lexicographic vectorisation of the $N \times M$ DD grid). The matrix $\mathbf{H}_{DD}$ has the Block-Circulant with Circulant Blocks (BCCB) structure:

where $\mathbf{\Pi}$ is the $N \times N$ cyclic shift (permutation) matrix and $\mathbf{\Delta}$ is the $M \times M$ cyclic shift. Key consequence: $\mathbf{H}_{DD}$ has exactly $L$ non-zero entries per row.

4.5  Diversity Analysis

Diversity order determines how steeply the bit error rate (BER) curve falls with increasing SNR. A system with diversity order $d$ has BER $\propto \text{SNR}^{-d}$.

OFDM under Doppler

Each OFDM subcarrier occupies a bandwidth $\Delta f \ll$ coherence bandwidth — it sees a single complex channel coefficient drawn from the small-scale fading distribution. Under Rayleigh fading, the BER for QPSK is:

where $\bar{\gamma} = E_b/N_0$. Under Doppler, ICI imposes a SINR floor so the BER curve flattens: $P_e \to \text{floor} > 0$ regardless of SNR.

OTFS diversity order

Each OTFS DD symbol $X[k,l]$ accumulates received energy from all $L$ paths through the circular convolution (Eq. 4.4). This is equivalent to transmitting on $L$ independent diversity branches. For Rayleigh fading paths the coding-theoretic argument (Raviteja et al. 2018) shows:

where $c$ is a constant depending on constellation and coding. Diversity order $d = L$.

4.6  OTFS Channel Estimation

OTFS channel estimation exploits the DD-domain sparsity: place a pilot impulse at a known grid position $(k_p, l_p)$ surrounded by guard zeros. After the channel, the pilot response reveals each path directly.

Pilot placement and guard region

The pilot $X_p[k_p, l_p] = \sqrt{E_p}$; all neighbouring positions within $|k - k_p| \le k_{\max}$, $|l - l_p| \le l_{\max}$ are set to zero (guard region) to prevent pilot-data interference. Data symbols fill the remaining DD grid positions.

Least-Squares channel estimate

By the I/O relation (§4.4), the received pilot region contains exactly the $L$ path responses shifted by $(k_i, l_i)$ from the pilot position. A simple peak search over the guard region yields $\{(k_i, l_i, \hat{h}_i)\}_{i=1}^L$ directly.

Overhead comparison

Guard region size: $(2k_{\max}+1)(l_{\max}+1)$ DD bins out of $NM$ total.

For pedestrian/urban scenarios, OTFS pilot overhead is lower than OFDM DMRS because the DD channel is sparse and time-invariant (no need to re-estimate per slot). For HST and LEO, overhead is comparable or worse than OFDM — but OFDM at those mobilities cannot estimate the channel accurately regardless of pilot density; OTFS at 26% overhead still produces a usable estimate.

4.7  Equalisation Methods

Given the received DD observations $Y[k,l]$ and estimated channel $\hat{\mathbf{H}}_{DD}$ (BCCB), several equalisation strategies are available with different complexity-performance trade-offs.

1-tap Zero-Forcing (ZF)

Valid only for $L=1$ or under the (incorrect) assumption that off-diagonal channel terms are negligible. For $L > 1$ and integer DD, 1-tap ZF produces BER floors equal to OFDM — no diversity gain is realised.

MMSE (linear)

Because $\mathbf{H}_{DD}$ is BCCB, the matrix $\mathbf{H}_{DD}^H \mathbf{H}_{DD}$ is also BCCB and can be inverted via 2D FFT in $\mathcal{O}(NM \log NM)$. This is the optimal linear estimator and achieves full $L$-th order diversity.

Message Passing (MP)

Factor-graph belief propagation exploiting the BCCB sparsity. Each DD bin is a variable node connected to $L$ factor nodes (one per path). Per-iteration complexity: $\mathcal{O}(NML)$. Typically 5–15 iterations suffice to reach near-MMSE performance. MP handles fractional Doppler naturally if the factor graph incorporates the sinc leakage coefficients, at the cost of a denser factor graph.

Krylov / Conjugate Gradient (CG)

For large $NM$, iterative Krylov methods solve $(\mathbf{H}_{DD}^H \mathbf{H}_{DD} + \sigma_w^2 \mathbf{I})\hat{\mathbf{x}} = \mathbf{H}_{DD}^H \mathbf{y}$ in $\mathcal{O}(L)$ iterations (since the matrix has $L$ distinct eigenvalue clusters), each iteration costing $\mathcal{O}(NML)$ for sparse matrix-vector multiply. Total: $\mathcal{O}(NML^2)$.

4.8  Practical Considerations

4.9  BER Performance — Doubly-Dispersive Channel (QPSK)

The chart below plots uncoded QPSK BER versus $E_b/N_0$ for five cases: AWGN reference, OFDM without Doppler, OFDM under Doppler ($f_d/\Delta f = 0.15$), OTFS with $L=1$ path (no diversity gain), and OTFS with $L=4$ paths (full 4th-order diversity). The $L=4$ OTFS curve demonstrates the dramatic SNR advantage at practical operating points.

Reading the chart. At SNR = 20 dB: OFDM without Doppler and AWGN reference achieve BER $\approx 5 \times 10^{-4}$. OFDM under Doppler ($f_d/\Delta f = 0.15$) hits the ICI floor at BER $\approx 0.02$ — uncorrectable regardless of further SNR increase. OTFS $L=1$ mirrors the AWGN curve (no diversity advantage over OFDM in a single-path channel). OTFS $L=4$ achieves BER $\approx 10^{-8}$ — five orders of magnitude below OFDM, demonstrating 4th-order diversity with its steep $\text{SNR}^{-4}$ slope.

4.10  Section Summary

§5 introduces AFDM — the chirp-based waveform that achieves the same $L$-th order diversity with 1D signal processing, single-symbol latency, and provably fractional-Doppler-immune channel structure via the DAFT parameter $c_1$.

5.1 Motivation — The OTFS Complexity Gap

OTFS achieves full diversity order $L$ (where $L$ = number of resolvable delay-Doppler paths) by mapping symbols onto a 2D delay-Doppler grid. The cost is 2D MMSE / message-passing (MP) equalization — $O(NM \cdot L \cdot \text{iter})$ per frame, roughly 1600× the complexity of OFDM's single-tap equalizer at $N = 512$, $L = 5$, 5 iterations.

The design goal of AFDM is precise: achieve the same diversity order $L$ with 1D signal processing and $O(N)$ equalization — like OFDM, but without inter-carrier interference (ICI) under Doppler.

5.2 The DAFT — Discrete Affine Fourier Transform

The DAFT (Bemani, Ksairi, Kountouris; IEEE TWC 2023, DOI: 10.1109/TWC.2023.3235526) is a length-$N$ unitary transform parameterized by two real chirp rates $c_1, c_2 \in \mathbb{R}$.

DAFT (Analysis / RX)

IDAFT (Synthesis / TX)

The parameter $c_1$ controls the pre-chirp (input-side quadratic phase); $c_2$ controls the post-chirp (output-side quadratic phase).

Matrix Factorization

The $N\times N$ DAFT matrix $\boldsymbol{\Psi}$ factors as three elementary operations:

where $\mathbf{F}$ is the standard unitary DFT matrix and the diagonal chirp matrices are

$\mathbf{D}_{c_2}$ is defined analogously on the output index $k$. Unitarity follows immediately: $\boldsymbol{\Psi}\boldsymbol{\Psi}^\dagger = \mathbf{I}$.

Special Cases of $c_1$

5.3 CPAT — Chirp-Periodic Anti-aliasing Tail

Standard CP creates a periodic extension of the time-domain block, so that the linear channel convolution appears circular — which diagonalizes the channel matrix in the DFT domain. AFDM requires a generalized prefix that enforces chirp-periodicity instead.

Chirp-Periodicity Condition

The CPAT (also called CPP — Chirp-Periodic Prefix — in some papers) ensures:

After convolution with the channel, the received signal satisfies the same chirp-periodic boundary condition, making the effective channel matrix in the DAFT domain exactly diagonal for integer delay-Doppler paths.

CPAT Construction

The CPAT of length $L_{\mathrm{CPAT}} \ge \tau_{\max}/T_s$ samples is constructed by copying the last $L_{\mathrm{CPAT}}$ samples of $s[n]$ and applying the chirp phase correction for each position $m = 1,\ldots,L_{\mathrm{CPAT}}$:

When $c_1 = 0$, the correction factor is 1 for all $m$, and CPAT reduces exactly to the standard CP copy. No new guard interval overhead type is introduced.

5.4 AFDM TX/RX Chain

Transmitter Steps

1. Input: $x[0],\ldots,x[N-1]$ — DAFT-domain QAM symbols (after channel coding, rate matching).
2. Post-chirp: $\tilde{X}_{pc}[k] = x[k]\cdot e^{+j2\pi c_2 k^2}$  [$O(N)$ LUT multiply]
3. IFFT: $x_i[n] = \mathrm{IFFT}\{\tilde{X}_{pc}\}[n]$  [$O(N\log N)$ — same engine as OFDM]
4. Pre-chirp: $s[n] = x_i[n]\cdot e^{+j2\pi c_1 n^2}$  [$O(N)$ LUT multiply]
5. CPAT insertion: prepend $L_{\mathrm{CPAT}}$ chirp-corrected tail samples forming $\tilde{s}[-L_{\mathrm{CPAT}}],\ldots,s[N-1]$.
6. D/A conversion $\to$ RF front-end.

Receiver Steps

1. RF $\to$ A/D $\to$ CPAT removal → $r[0],\ldots,r[N-1]$.
2. Pre-chirp removal: $r_{pc}[n] = r[n]\cdot e^{-j2\pi c_1 n^2}$  [$O(N)$]
3. FFT: $\tilde{Y}_{fft}[k] = \mathrm{FFT}\{r_{pc}\}[k]$  [$O(N\log N)$]
4. Post-chirp removal: $y[k] = \tilde{Y}_{fft}[k]\cdot e^{-j2\pi c_2 k^2}$  [$O(N)$]
5. Single-tap equalization: $\hat{x}[m] = y[m]/\hat{H}_{\mathrm{eff}}[m]$  [$O(N)$]
6. Decision $\to$ channel decoder.

5.5 AFDM I/O Relation

For an $L$-path doubly-dispersive channel with integer delay index $\ell_i$ (samples) and Doppler index $k_i$ (bins), and complex path gains $h_i$, the DAFT-domain received symbol at output index $m$ is:

where the per-path phase $\phi_i(m)$ is known from the channel estimate:

and $(m-\ell_i)_N$ denotes reduction modulo $N$.

Why Single-Tap Equalization Works

$e^{j\phi_i(m)}$ depends on the known indices $(\ell_i, k_i)$ and on $m$ only — not on any other output index. After phase compensation across all $L$ paths, each output $y[m]$ is a weighted sum of a single symbol $x[(m-\ell_i)_N]$:

5.6 Full Diversity Theorem

Theorem (Bemani, Ksairi, Kountouris; IEEE TWC 2023). AFDM achieves full diversity order $L$ over any $L$-path doubly-dispersive channel (integer delay-Doppler indices) if and only if the $N\times L$ matrix $\mathbf{V}$ with columns

(where $\Delta_i = 2\ell_i$) has full column rank for all distinct path pairs $(i \ne j)$.

Sufficient Condition on $c_1$

Full rank fails only when two paths produce identical phase progressions: $c_1\Delta_i + k_i/N = c_1\Delta_j + k_j/N$ for some $i \ne j$. This defines the finite forbidden set:

$|\mathcal{F}_{\mathrm{bad}}| \le \binom{L}{2}N$ — a finite set of measure zero in $\mathbb{R}$. Any $c_1 \notin \mathcal{F}_{\mathrm{bad}}$ achieves full diversity.

5.7 Channel Matrix Sparsity: OFDM vs OTFS vs AFDM

The $N\times N$ effective channel matrix $\mathbf{H}$ for $L=3$ paths, $N=16$ symbols, visualised in each waveform's modulation domain. Dark cells = non-zero entries.

$N=16$, $L=3$ paths. Left: OFDM (banded ICI). Centre: OTFS (block-sparse DD). Right: AFDM (exactly diagonal).

5.8 Complexity Comparison

5.9 AFDM PAPR

The pre-chirp multiply $e^{j2\pi c_1 n^2}$ is a deterministic unit-magnitude phase rotation applied element-wise to the IFFT output. Since $|e^{j2\pi c_1 n^2}|=1$, the envelope $|s[n]|$ is identical to the OFDM output $|x_i[n]|$. Therefore the PAPR distribution is identical to OFDM.

The PAPR CCDF for QPSK, $N$ subcarriers:

At $N=512$: $P(\mathrm{PAPR}>10\,\mathrm{dB})\approx 1\%$ — the same threshold for both OFDM and AFDM.

5.10 Backward Compatibility with OFDM

Setting $c_1 = c_2 = 0$ in the DAFT factorization:

AFDM with $c_1 = 0$ is exactly OFDM: same bit sequence, same FFT/IFFT, same CP (CPAT with $c_1=0$ reduces to standard copy), same frame structure, same 1-tap equalizer. $c_1$ is a runtime-configurable parameter — no new air interface is needed.

Firmware-Upgrade Implementation

Upgrading an OFDM baseband to AFDM requires only three software changes:

1. Pre-chirp LUT (TX, post-IFFT): $N$-entry complex table $\{e^{+j2\pi c_1 n^2}\}_{n=0}^{N-1}$ applied element-wise to IFFT output.
2. Pre-chirp LUT (RX, pre-FFT): Conjugate table $\{e^{-j2\pi c_1 n^2}\}$ applied to time-domain RX buffer.
3. CPAT block: Replace CP copy-and-prepend with copy-phase-correct-and-prepend. Same memory footprint; marginal compute overhead.

arXiv:2605.23062 (IEEE CTW 2026) demonstrates this as a firmware patch with $\sim$50–200 lines of code change, no hardware modifications, no new air interface.

5.11 AFDM for ISAC

Integrated Sensing and Communications (ISAC) is a primary IMT-2030 use case. AFDM's chirp subcarriers provide natural radar sensing capabilities with no additional waveform overhead.

Radar Ambiguity Function

The ambiguity function $\chi(\tau,\nu)$ describes a waveform's resolution in delay (range) and Doppler (velocity). For OFDM, $\chi$ has excellent delay resolution but poor Doppler sidelobes. AFDM's chirp subcarriers produce a richer $\chi(\tau,\nu)$ with more uniform delay-Doppler coverage (arXiv:2510.11216, Rou & Abreu, Oct 2025).

• Range estimation: delay shifts $\ell_i$ appear as phase offsets directly readable in the DAFT output — no separate matched filter.
• Doppler estimation: Doppler indices $k_i$ appear as linear phase progressions across AFDM symbols — standard 1D DFT across the symbol axis suffices.
• Unified TX/RX: the same waveform serves both data and sensing; no separate radar pilot bursts or mode switching.

AFDM vs OFDM for ISAC

5.12 BER Performance — AFDM vs OFDM vs OTFS

Theoretical BER vs $E_b/N_0$ for QPSK, $L=4$ path doubly-dispersive channel. Five curves: AWGN bound, OFDM with ICI floor, OTFS diversity-4, AFDM diversity-4 (same as OTFS), and AFDM with $c_1=0$ (degenerates to OFDM).

QPSK, $L=4$ path channel. OTFS and AFDM (correct $c_1$) overlay on the same diversity-4 slope. AFDM with $c_1=0$ degenerates to the OFDM ICI-floor curve — confirming that $c_1$ is the essential degree of freedom. Curves are theoretical bounds; practical BER includes pilot/guard overhead effects.

§5 Key Takeaways

• DAFT = chirp-precoded DFT. $\boldsymbol{\Psi}=\mathbf{D}_{c_2}\mathbf{F}\mathbf{D}_{c_1}$ — FFT core unchanged; only 4$\times O(N)$ LUT multiplies added. Transform complexity: $O(N\log N)$ — identical to OFDM.
• CPAT enforces chirp-periodicity. The chirp-periodic prefix is the AFDM analogue of CP; it makes the effective DAFT-domain channel matrix exactly diagonal for integer DD paths. Length constraint is identical to CP: $\ge \tau_{\max}/T_s$.
• Full diversity — provably, for almost all $c_1$. Diversity order $L$ is guaranteed for any $c_1 \notin \mathcal{F}_{\mathrm{bad}}$ (finite, measure-zero set). Standard choice $c_1 = 1/(2N)$ is sufficient for typical $L \le 10$.
• Single-tap equalization preserved. $\hat{x}[m] = y[m]/H_{\mathrm{eff}}[m]$. No 2D grid, no MP iterations, no additional latency. Same computational cost as OFDM equalization.
• Exact OFDM backward compatibility. $c_1=0$ $\Rightarrow$ AFDM $\equiv$ OFDM exactly. Upgrade is a firmware patch ($\sim$50–200 LoC); confirmed by arXiv:2605.23062 (IEEE CTW 2026).
• PAPR identical to OFDM. Deterministic unit-magnitude phase rotation preserves the envelope distribution. All OFDM PAPR reduction techniques apply unchanged.
• Natural ISAC waveform. Chirp subcarriers are discrete LFM radar waveforms. Range-Doppler estimation is unified with data demodulation — no separate radar mode, no pilot overhead penalty.
• Waveform taxonomy sweet spot. AFDM sits between OFDM (diversity-1, $O(N\log N)$) and OTFS (diversity-$L$, $O(NM\log NM + NML)$): diversity-$L$ at $O(N\log N)$ complexity — the defining characteristic.

6.1 Master Comparison Table

Full-spectrum property comparison across the three waveform families. BER slope refers to the diversity order achievable in an $L$-path doubly-dispersive channel.

6.2 BER Comparison — QPSK, Doubly-Dispersive Channel, $L = 4$ Paths

Theoretical BER vs. $E_b/N_0$ (dB) for QPSK modulation over an $L = 4$-path doubly-dispersive channel. The OTFS / AFDM curves use the high-SNR approximation for full diversity-$L$ reception: $$P_e \approx \frac{(2L-1)!}{L!\cdot(L-1)!} \cdot \left(\frac{1}{\mathrm{SNR_{lin}}}\right)^L \cdot \frac{1}{4}$$ At $L = 4$ this reduces to approximately $(5/\mathrm{SNR_{lin}})^4 / 24$. The OFDM high-Doppler trace shows the irreducible ICI error floor at $P_e \approx 0.02$.

6.3 Computational Load — $N = 512$, $M = 14$, $L = 4$

Complex multiply-accumulate (CMAC) operation counts for the three key processing stages. Modulation and demodulation costs use exact $\mathcal{O}(N \log_2 N)$ values; equalization uses $N$ for OFDM/AFDM and $NML \times 10$ iterations for OTFS Message Passing. The log scale shows the order-of-magnitude gap between OFDM/AFDM and full OTFS processing.

OFDM: $N\log_2 N = 512 \times 9 = 4608$ CMACs/stage. OTFS: $NM\log_2(NM) = 7168 \times 9 = 64512$ CMACs/stage (mod/demod); $NML \times 10 = 512 \times 14 \times 4 \times 10 = 286720$ CMACs (MP equalization shown capped for scale). AFDM: same as OFDM for mod/demod and equalization.

6.4 Waveform Selection Decision Guide

Practical flowchart for engineers choosing a waveform given deployment constraints. Start at the top and follow the path matching your scenario.

6.5 PAPR CCDF — $N = 128$ subcarriers, QPSK

Complementary CDF of Peak-to-Average Power Ratio: $\Pr(\mathrm{PAPR} > \gamma)$. For CP-OFDM with $N$ independent subcarriers, the exact CCDF is $$\Pr(\mathrm{PAPR} > \gamma) = 1 - \left(1 - e^{-\gamma_\text{lin}}\right)^N, \quad \gamma_\text{lin} = 10^{\gamma/10}.$$ DFT-s-OFDM (used in 5G UL) reduces PAPR by approximately 3 dB relative to CP-OFDM. AFDM with rectangular pulses exhibits the same CCDF as CP-OFDM (chirp subcarriers have identical envelope statistics for large $N$).

7.1 OFDM in 5G NR — The Incumbent

CP-OFDM is defined in 3GPP TS 38.211 for all NR downlink and uplink channels across all five numerologies $\mu = 0 \ldots 4$ (subcarrier spacings $15, 30, 60, 120, 240$ kHz). It has been unchanged since Rel-15 (2018) through Rel-19 (2025).

DFT-s-OFDM (transform precoding) is mandatory for PUSCH uplink in coverage-limited scenarios. It reduces PAPR by roughly 3 dB (to ~6–7 dB for $\pi/2$-BPSK), improving link budget at cell edge. Rel-20 extends DFT-s-OFDM to support uplink MIMO — previously it was restricted to single-stream transmission.

Non-Terrestrial Networks (NTN) — Rel-17, TS 38.821

3GPP's strategy for handling the extreme Doppler shifts of LEO satellites ($f_d \approx 88$ kHz at 3.5 GHz, orbital velocity ~7.5 km/s) was not to change the waveform but to add Doppler pre-compensation at the UE:

• UE estimates satellite Doppler shift from ephemeris data / GNSS position.
• UE applies a frequency offset equal and opposite to the estimated Doppler before transmission.
• Residual Doppler after pre-compensation: typically $\ll 1\%$ of $\Delta f$.
• Wider SCS ($\mu = 1$: 30 kHz, or $\mu = 2$: 60 kHz) absorbs remaining ICI.
• Timing advance (TA) pre-compensation added for delay management.

Rel-18 (FR2-NTN, CR 0129) and Rel-19 (NTN Phase 3) refined TA pre-compensation and extended NTN to mmWave bands. In all cases, the waveform remained CP-OFDM.

7.2 OTFS in 3GPP — Studied, Not Adopted

Contribution History

Cohere Technologies (the commercial OTFS spinout from Hadani et al., WCNC 2017) began submitting OTFS contributions to 3GPP RAN1 starting in 2018. A sustained campaign through the Rel-17 NTN study item (2020–2022) argued that OTFS should replace CP-OFDM for high-mobility scenarios.

At the 3GPP 6G Workshop (Incheon, March 10–11, 2025, document 6GWS-250244), Cohere explicitly objected to language describing 6G as "based on 5G NR baseline" because it implied retaining the OFDM waveform. They stated: "multiple companies indicated that at least for some use cases a new waveform is required." Despite this, the workshop produced no consensus and the chairs' summary made no prioritisation of new waveforms.

Why OTFS Was Not Adopted

7.3 AFDM — 6G Candidate

Origin and Key Paper

AFDM was proposed by Bemani, Atzeni, Leinonen, Clemente, Debbah, and Tirkkonen in "Affine Frequency Division Multiplexing for Next Generation Mobile Communication" (IEEE Transactions on Wireless Communications, 2023 — cited as IEEE TCOM vol. 71 in pre-standardisation literature). The EURECOM research group (de Abreu, Rou et al.) has since driven the academic campaign with eight major papers in 2025–2026.

As of June 2026, AFDM has no formal 3GPP work item. The closest signal of standardisation intent is a researcher co-authoring the "Towards Standardizing AFDM" paper (IEEE Communications Standards Magazine, June 2026) and the Bemani et al. (IEEE CTW 2026) demonstration that AFDM can be implemented as a firmware patch on existing OFDM FPGA hardware.

Why AFDM Is the Leading Non-OFDM 6G Candidate

The two chirp parameters $c_1, c_2$ (with $c_2 = -c_1$ for anti-aliasing) are the design freedom. Setting $c_1 = 0$ recovers the standard IDFT: AFDM subsumes OFDM as a special case. This backward compatibility means:

• Existing 5G spectrum allocations, numerologies, and scheduling frameworks are reusable.
• OFDM receivers decode AFDM signals with $c_1 = 0$ without modification.
• Phased rollout: AFDM UEs coexist with OFDM UEs on the same carrier.

6G Advantages Summary

7.4 IMT-2030 / 6G Standards Timeline

7.5 6G Use Cases — Doppler Level and Waveform Viability

Maximum Doppler shift: $f_d = v \cdot f_c / c$. At 3.5 GHz: 1 m/s $\approx 11.7$ Hz. At 28 GHz: 1 m/s $\approx 93.3$ Hz. 5G NR subcarrier spacing $\Delta f = 15$ kHz ($\mu=0$).

The IMT-2030 framework (ITU-R M.2160) mandates a mobility KPI of 500–1000 km/h (covering HST and some aerial scenarios) and includes ISAC as a first-class usage scenario. Meeting both requirements with CP-OFDM alone — without Doppler pre-compensation — is increasingly challenged.

A.1 Mathematical Symbols

A.2 Transform Acronyms

A.3 Waveform and System Acronyms