Nonorthogonal AFDM: Chirp-Based Modulation

• Nonorthogonal AFDM is a chirp-based multicarrier modulation scheme leveraging the discrete affine Fourier transform to provide a full delay-Doppler channel representation and achieve full diversity.
• Its transform-domain formulation maps symbols to multi-chirp waveforms using pre- and post-chirp operations with FFT/IFFT, enabling efficient hardware implementation.
• Nonorthogonal variants introduce a bandwidth-compression factor to enhance spectral efficiency while employing iterative equalization to mitigate inter-carrier interference.

Nonorthogonal affine frequency division multiplexing (AFDM) is a chirp-based multicarrier modulation framework built on the discrete affine Fourier transform (DAFT) and designed for doubly dispersive, high-mobility channels. In its foundational form, AFDM maps symbols in the DAFT domain to a time-domain multi-chirp waveform through an inverse DAFT, and, with suitable parameter selection, it provides a full delay-Doppler representation of the channel and can achieve full diversity (Bemani et al., 2021). In later usage, the label non-orthogonal AFDM also denotes spectrally efficient variants—often written NO-AFDM or nAFDM—that introduce a bandwidth-compression factor to create controllable subcarrier overlap while preserving the affine-frequency structure of the waveform (Yi et al., 13 Aug 2025).

1. Origins, scope, and relation to adjacent waveforms

AFDM was introduced as a chirp-based multicarrier transceiver scheme for high-mobility communications and as a DAFT-based alternative to conventional OFDM in doubly dispersive channels (Bemani et al., 2021). Subsequent treatments described it as a modulation technique that generalizes both the discrete Fourier transform and the discrete Fresnel transform through the DAFT, and emphasized that its two real chirp parameters can be adapted to the channel delay-Doppler profile (Tao et al., 2024). A broader standardization-oriented treatment later presented AFDM as a chirp-based multi-carrier framework that generalizes OFDM and orthogonal chirp division multiplexing by introducing two tunable chirp parameters (Luo et al., 11 Jun 2026).

Its immediate research context is the comparison with OFDM, OTFS, and OCDM. OFDM uses orthogonal constant-frequency carriers and is highly sensitive to Doppler-induced inter-carrier interference in high-mobility scenarios, while OTFS uses a two-dimensional symplectic transform and achieves full delay-Doppler diversity at the cost of higher pilot and multiplexing overhead (Wu et al., 2023). AFDM was positioned as a one-dimensional transform-domain waveform that still exposes a full delay-Doppler representation of the channel when its parameters are chosen appropriately, while preserving FFT-based implementation structure and lower pilot overhead than OTFS (Bemani et al., 2022).

The expression nonorthogonal AFDM is used in two closely related senses. One concerns the intrinsic overlap of AFDM chirp waveforms in time-frequency or ambiguity space under doubly selective propagation, which leads to a sparse but non-diagonal DAFT-domain channel representation (Zheng et al., 2024). The other refers to explicit compressed-subcarrier designs that introduce a factor $\alpha<1$ to force controllable overlap between chirp subcarriers in exchange for higher spectral efficiency (Zhang et al., 23 Jul 2025).

2. Transform-domain formulation and waveform construction

In the standard discrete formulation, AFDM places information symbols in the DAFT domain and synthesizes the time-domain block by an inverse DAFT. Using the notation common in the foundational papers, if $\mathbf{x}\in\mathbb{C}^{N}$ denotes the DAFT-domain symbol vector, then the AFDM transmit vector is

$$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$

while DAFT demodulation at the receiver is

$$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$

where $\mathbf{F}$ is the normalized $N\times N$ DFT matrix and $\mathbf{\Lambda}_{c}$ is the diagonal chirp matrix with entries $e^{-j2\pi c n^2}$ (Bemani et al., 2021). Equivalent formulations in later work use $(\lambda_1,\lambda_2)$ in place of $(c_1,c_2)$ and the same cascade of pre-chirp, FFT/IFFT, and post-chirp operations (Tao et al., 2023).

Entry-wise, the AFDM basis consists of quadratic-phase chirps. One discrete synthesis form is

$\mathbf{x}\in\mathbb{C}^{N}$0

which makes explicit that each symbol is transmitted on a chirp-weighted DFT column rather than on a pure sinusoid (Bemani et al., 2021). A continuous-time counterpart writes the AFDM signal as a superposition of chirp carriers over a symbol interval, and the continuous affine Fourier transform supplies the underlying transform pair used to interpret AFDM as an affine generalization of Fourier modulation (Tao et al., 2023).

Because the waveform is chirped, the guard interval is not an ordinary cyclic prefix in the most general case. AFDM instead uses a chirp-periodic prefix (CPP) satisfying a chirp-periodicity constraint, although particular parameter choices can make the CPP reduce to a conventional cyclic prefix (Bemani et al., 2021). This property is central to later backward-compatibility arguments with 4G/5G processing chains (Luo et al., 11 Jun 2026).

3. Nonorthogonality, channel interaction, and diversity

A defining feature of AFDM is that its chirp subcarriers are not interpreted in the same way as OFDM subcarriers. Unlike OFDM’s orthogonal sinusoids, AFDM chirps span the entire bandwidth during a symbol and overlap in both time and frequency; in the delay-Doppler representation they also overlap unless the chirp parameters are chosen so that different paths map to distinct diagonals in the effective channel matrix (Luo et al., 11 Jun 2026). In the ambiguity formulation, perfect orthogonality would require all off-diagonal delay-Doppler cross-ambiguity terms to vanish, but AFDM is generally nonorthogonal (Luo et al., 11 Jun 2026).

Over a $\mathbf{x}\in\mathbb{C}^{N}$1-path linear time-varying channel, the received DAFT-domain vector takes the form

$\mathbf{x}\in\mathbb{C}^{N}$2

with $\mathbf{x}\in\mathbb{C}^{N}$3 in the DAFT basis (Bemani et al., 2021). In the integer-Doppler case, each path contributes one nonzero per row at a DAFT-domain location

$\mathbf{x}\in\mathbb{C}^{N}$4

so the channel becomes a sum of path-specific cyclic shifts with deterministic phase factors (Bemani et al., 2021). Closely related AFDM-IM formulations describe the same mechanism through index constraints of the form $\mathbf{x}\in\mathbb{C}^{N}$5 (Tao et al., 2023).

The key design objective is to prevent different paths from overlapping in the DAFT domain. A standard parameter choice is

$\mathbf{x}\in\mathbb{C}^{N}$6

together with an underspread constraint such as

$\mathbf{x}\in\mathbb{C}^{N}$7

or, in an equivalent AFDM-IM formulation,

$\mathbf{x}\in\mathbb{C}^{N}$8

Under these conditions, distinct delay-Doppler paths do not overlap in the DAFT domain, the relevant path-columns become linearly independent, and AFDM achieves full diversity order $\mathbf{x}\in\mathbb{C}^{N}$9 (Bemani et al., 2021, Tao et al., 2023).

The literature uses both orthogonality and nonorthogonality language. Some formulations describe AFDM as modulating onto orthogonal chirps over the observation interval or under AWGN, whereas ambiguity-based and channel-coupled analyses emphasize that a time-varying channel produces a sparse but non-diagonal DAFT-domain interaction (Zheng et al., 2024). This suggests that orthogonality in AFDM is context-dependent rather than a single global property.

4. Explicitly compressed non-orthogonal AFDM variants

Recent NO-AFDM and nAFDM proposals make nonorthogonality an explicit design variable. Their central mechanism is a bandwidth-compression factor $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$0 inserted into the AFDM modulation kernel, which reduces subcarrier spacing and creates controllable chirp-subcarrier overlap (Zhang et al., 23 Jul 2025). In matrix form, the compressed modulator is written as

$$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$1

where $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$2 has entries $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$3 (Zhang et al., 23 Jul 2025). The factor $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$4 directly controls the amount of overlap, and one paper states that the spectral-efficiency gain over the orthogonal case is $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$5 (Zhang et al., 23 Jul 2025).

The price of this compression is inter-carrier interference. A closed-form characterization of the modulation correlation matrix in nAFDM yields the off-diagonal ICI magnitude

$$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$6

which makes explicit that decreasing $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$7 broadens and strengthens ICI (Yi et al., 13 Aug 2025). To mitigate this effect, one line of work introduces transmitter-side ZF or MMSE precoding and a receiver-side iterative equalization and ICI-cancellation scheme (Zhang et al., 23 Jul 2025). Another develops a soft iterative detector, including low-complexity pruning based on the dominant off-diagonal terms of the correlation matrix, and an IDFT-based signal-generation method that reuses existing IFFT modules by moving to a longer transform, zero-padding, truncation, and scaling (Yi et al., 13 Aug 2025).

Reported performance gains are setup-dependent but consistently framed as a BER-versus-spectral-efficiency trade-off. One study reports that non-orthogonal AFDM with $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$8 and $$\mathbf{s}=\mathbf{\Lambda}_{c_1}^{H}\,\mathbf{F}^{H}\,\mathbf{\Lambda}_{c_2}^{H}\,\mathbf{x},$$9 provides 11% and 17.6% spectral-efficiency gains, respectively, while remaining within 0.5–1 dB of orthogonal AFDM at $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$0 when iterative detection is used (Zhang et al., 23 Jul 2025). Another reports that, with soft iterative detection, $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$1 matches the BER of conventional AFDM under MMSE while offering approximately 21.2% higher spectral efficiency, and that retaining only $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$2 or $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$3 ICI components keeps the BER within 0.1 dB of the full-ICI case while reducing cancellation complexity by 25–37.5% (Yi et al., 13 Aug 2025).

5. Detection, channel estimation, and index-modulated extensions

AFDM admits a broad range of detectors because the effective DAFT-domain channel is sparse rather than diagonal. Exhaustive maximum-likelihood detection is optimal but scales exponentially with block size. In the AFDM-IM setting, the ML rule is

$$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$4

while a lower-complexity alternative forms an MMSE estimate

$$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$5

followed by per-group ML detection (Tao et al., 2023). In sparse AFDM systems without index modulation, message passing on the observation-variable factor graph reduces the complexity of approximate MAP detection from $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$6 to $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$7, with one iteration costing $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$8 (Wu et al., 2023).

Channel estimation has been addressed through both embedded and superimposed pilots. A 2022 AFDM treatment proposed an embedded pilot-aided scheme within the same AFDM frame together with low-complexity zero-padding-based detection (Bemani et al., 2022). A later guardless approach superimposes pilots and data in the DAFT domain, chooses pilot positions spaced by $$\mathbf{y}=\mathbf{\Lambda}_{c_2}\,\mathbf{F}\,\mathbf{\Lambda}_{c_1}\,\mathbf{r},$$9 to minimize channel-estimation error, and alternates LMMSE channel estimation with message-passing data detection; the same work states that typically 2–3 iterations suffice (Zheng et al., 2024).

Index modulation extends AFDM by encoding bits in the activation pattern of DAFT-domain positions in addition to conventional constellation symbols. In AFDM-IM, $\mathbf{F}$0 DAFT-domain positions are partitioned into $\mathbf{F}$1 groups of size $\mathbf{F}$2; within each group, $\mathbf{F}$3 positions are activated, the index bits are

$\mathbf{F}$4

and the modulation bits are

$\mathbf{F}$5

for $\mathbf{F}$6-PSK (Tao et al., 2023). Both localized and distributed grouping strategies have been studied, with distributed grouping reported to provide better coding gain under MMSE equalization (Tao et al., 2023). Under the full-diversity condition, both index and modulated bits achieve diversity order $\mathbf{F}$7; when the condition is violated, the modulated bits lose diversity more rapidly, while the index bits retain stronger diversity protection (Tao et al., 2023).

These IM ideas were further extended to multiple-antenna transmission through cyclic delay diversity. The resulting CDD-AFDM-IM-I and CDD-AFDM-IM-II schemes analyze full-diversity conditions for integer and fractional Doppler, derive asymptotically tight BER upper bounds, and introduce a double-layer message passing detector whose reported complexity is only a few percent higher than single-layer MP but tens of percent lower than MMSE while delivering 1–2 dB BER gain over MP at $\mathbf{F}$8 BER (Tao et al., 2024).

6. Implementation, interoperability, robustness, and emerging roles

From an implementation standpoint, AFDM retains strong affinity with FFT-based hardware. Its core transform can be realized as one FFT/IFFT bracketed by diagonal chirp multiplications, giving $\mathbf{F}$9 complexity in the foundational formulation (Bemani et al., 2021). The 2022 AFDM framework also proposed a weighted MRC-DFE detector with complexity $N\times N$0 per iteration after zero-padding, and an embedded pilot architecture with pilot overhead substantially lower than 2D OTFS (Bemani et al., 2022). In the compressed nAFDM literature, practical generation is again tied to existing IFFT modules through longer-IDFT realization and truncation (Yi et al., 13 Aug 2025).

Backward compatibility is a major theme of the standardization literature. A standardization-focused study states that AFDM can be incorporated into 4G/5G multi-numerology frameworks and their evolution, FMCW radar waveforms, and LoRa with limited modification (Luo et al., 11 Jun 2026). In downlink-style integration, choosing

$N\times N$1

allows the CPP to reduce to a conventional cyclic prefix, after which AFDM symbols can be inserted into the NR FFT/IFFT chain while reusing the same FFT sizes, CP lengths, and resource-block mapping (Luo et al., 11 Jun 2026). In uplink-style integration, AFDM can be realized as DFT-domain spectral shaping after DFT-spreading, using configurable per-subcarrier complex weights (Luo et al., 11 Jun 2026). The same source discusses PAPR control through conventional low-PAPR methods and chirp-parameter selection, as well as MIMO and multi-user formulations based on Kronecker-structured spatial-affine processing (Luo et al., 11 Jun 2026).

AFDM has also been studied under non-white Gaussian noise. The relevant analysis argues that performance depends on how well the demodulation matrix whitens the noise and relates this to the sparsity of the demodulation operator; AFDM is reported to outperform OTFS and OFDM in the presence of non-white noise, with gains exceeding 1 dB in most application scenarios (Savaux et al., 4 Oct 2025). The same work highlights narrowband signals and coexistence with OFDM signals as representative use cases (Savaux et al., 4 Oct 2025).

Beyond data transmission, AFDM has been proposed for integrated channel sounding and communication. Because AFDM provides a full delay-Doppler representation of the channel, it has been presented as enabling simultaneous communication, channel estimation, channel tracking, channel sounding, and scatterer sensing within a unified framework (Zhou et al., 20 Sep 2025). Potential application scenarios discussed across the recent literature include non-terrestrial networks, integrated sensing and communications, vehicle-to-everything, underwater acoustic communications, and broader space-air-ground-sea integrated networks, all of which impose severe delay-Doppler dispersion and therefore directly target AFDM’s operating regime (Luo et al., 11 Jun 2026, Zhou et al., 20 Sep 2025).