Anti-Interference AFDM: A DAFT-Domain Approach

• Anti-Interference AFDM is a chirp-based multicarrier waveform derived from the DAFT that transforms delay and Doppler distortions into sparse, quasi-diagonal channel interactions.
• The design leverages chirp parameter selection, chirp-periodic prefixes, and structured detection to mitigate inter-carrier interference typical of OFDM in doubly dispersive channels.
• Extensions such as index modulation, pilot-efficient channel estimation, and integrated sensing and communication (ISAC) further enhance its performance in high-mobility and interference-prone environments.

Searching arXiv for recent AFDM papers and overviews relevant to anti-interference properties. Search query: "Affine Frequency Division Multiplexing anti interference high mobility DAFT index modulation channel estimation". Affine Frequency Division Multiplexing (AFDM) is a chirp-based multicarrier waveform built on the discrete affine Fourier transform (DAFT) and designed for doubly dispersive, high-mobility channels. In the AFDM literature, the “anti-interference” characterization refers to a family of design choices—chirp parameter selection, chirp-periodic prefixing, sparse DAFT-domain channel representation, structured detection, and, in later variants, index modulation, spread spectrum, and sensing-oriented dechirping—that convert delay- and Doppler-induced distortion into sparse or quasi-diagonal interactions rather than the dense inter-carrier interference typical of OFDM (Bemani et al., 2021, Yin et al., 7 Feb 2025). Related developments extend this viewpoint to AFDM with index modulation, pilot-efficient channel estimation, integrated sensing and communication (ISAC), and adaptive chirp control (Tao et al., 2023, Zheng et al., 2024, Bemani et al., 6 Nov 2025).

1. DAFT-based waveform structure

AFDM maps a DAFT-domain symbol vector $\mathbf{x}\in\mathbb{C}^{N\times 1}$ to the time domain through an inverse DAFT. In the notation common to the AFDM overview literature, the modulation can be written as

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$

or equivalently as a chirp–DFT–chirp cascade,

$$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$

with diagonal chirp matrices determined by $c_1$ and $c_2$ (Yin et al., 7 Feb 2025). In related AFDM formulations, the same structure appears as

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

emphasizing that AFDM is a DFT generalized by pre- and post-multiplication with quadratic-phase factors (Bemani et al., 2021).

The discrete-time kernel shows that AFDM subcarriers are chirps rather than pure tones. The $c_1 n^2$ term imposes a common digital chirp rate in time, the $c_2 m^2$ term assigns different initial phases across chirp indices, and the $\frac{mn}{N}$ term preserves the uniformly spaced initial frequencies familiar from OFDM (Yin et al., 7 Feb 2025). OFDM is recovered at $c_1=c_2=0$, while OCDM appears as the special case $$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$0 (Yin et al., 7 Feb 2025).

To preserve blockwise structure under multipath, AFDM uses a chirp-periodic prefix (CPP). When $$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$1 is an integer and $$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$2 is even, the CPP reduces to the conventional cyclic prefix, which is one reason AFDM can be implemented efficiently by placing diagonal chirp multipliers around standard FFT/IFFT modules (Yin et al., 7 Feb 2025, Bemani et al., 2021). This implementation compatibility has been treated as one of AFDM’s practical links to legacy OFDM hardware.

2. Delay–Doppler separability and full diversity

The core anti-interference claim for AFDM is that, in doubly dispersive channels, the DAFT-domain input–output relation becomes sparse and structured. For a channel modeled as

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$3

AFDM demodulation yields an effective DAFT-domain model

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$4

or, pathwise,

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$5

with each path mapped into a shifted DAFT-domain support determined by delay and Doppler (Bemani et al., 2021, Yin et al., 7 Feb 2025).

A recurring interpretation in the literature is that a unit Doppler shift corresponds to a unit DAFT-domain shift, whereas a unit delay shift corresponds to a $$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$6-step shift. This establishes a bijective relationship between delay–Doppler coordinates and DAFT-domain shifts when

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$7

or, in the notation used by the original full-diversity AFDM derivation,

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$8

with the underspread condition

$$\mathbf{s}=\mathbf{A}\mathbf{x},\qquad A_{n,m}=\frac{1}{\sqrt{N}}\exp\!\Big\{j2\pi\big(c_1 n^2+\tfrac{mn}{N}+c_2 m^2\big)\Big\},$$9

or equivalently

$$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$0

ensuring non-overlapping path locations in the DAFT domain (Bemani et al., 2021, Yin et al., 7 Feb 2025, Tao et al., 2023).

Under these conditions, AFDM achieves full diversity order $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$1, and the pairwise error probability decays approximately as $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$2 at high SNR (Bemani et al., 2021, Tao et al., 2023). In anti-interference terms, this means that deep fades on individual paths are averaged through path diversity, while Doppler-induced mixing is confined to structured shifts rather than the dense, frequency-dependent ICI matrix characteristic of OFDM in doubly dispersive channels (Yin et al., 7 Feb 2025). AFDM is therefore often described as transforming a channel that is interference-laden in the Fourier domain into one that is sparse, quasi-static, and compact in the DAFT domain (Yin et al., 7 Feb 2025).

The comparison with adjacent waveforms follows directly from this structure. OFDM loses orthogonality under Doppler and develops severe ICI; OCDM improves robustness but cannot generally guarantee full diversity; OTFS attains comparable robustness but does so through a two-dimensional delay–Doppler modulation and equalization framework, with correspondingly higher pilot and equalization overhead (Bemani et al., 2021, Yin et al., 7 Feb 2025).

3. Index modulation and spread-spectrum extensions

A major strand of anti-interference AFDM research introduces index modulation. In AFDM-IM, bits are conveyed both by conventional constellation symbols and by the activation pattern of DAFT-domain subsymbols. For each group of $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$3 DAFT-domain positions, $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$4 positions are activated, and the total bits per AFDM frame are

$$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$5

The resulting group vector contains exactly $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$6 non-zero entries and zeros elsewhere (Tao et al., 2023).

Two grouping strategies have been studied. Localized grouping uses contiguous DAFT-domain indices per group, whereas distributed grouping interleaves group positions across the DAFT domain. Distributed grouping improves coding gain and diversity because different indices within a group experience more independent fades (Tao et al., 2023). In linear time-varying channels, AFDM-IM was reported to outperform OFDM, OFDM-IM, and conventional AFDM under equalized spectral efficiency, including gains of about $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$7 dB over OFDM at BER $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$8, about $$\mathbf{s}=\mathbf{D}_1\mathbf{F}^H\mathbf{D}_2\mathbf{x},$$9 dB over OFDM-IM at BER $c_1$0, and about $c_1$1 dB over AFDM at BER $c_1$2 under the cited full-diversity ML-detection setup (Tao et al., 2023).

The anti-interference interpretation of AFDM-IM is twofold. First, sparse activation reduces effective interference in the DAFT domain, especially when path overlap or residual mismatch occurs. Second, the index bits are decided from a joint activation pattern rather than from one symbol value, which gives them stronger diversity protection. In the non-full-diversity regime $c_1$3, the modulated bits were reported to degrade significantly, whereas the BER of the index bits remained almost unchanged and retained a higher diversity slope (Tao et al., 2023). A common design implication is that reliability-sensitive control information can be mapped into the index domain in severe mobility scenarios.

A related extension, IM-AFDM-SS, combines AFDM, index modulation, and spread spectrum. In that scheme, information bits are carried by both constellation symbols and the indices of selected spreading codes, and a low-complexity maximal ratio combining detector first recovers the spreading-code indices and then demodulates the symbols. The stated motivation is to combat interference caused by doubly dispersive channels, and the numerical results reported superiority over classical AFDM spread spectrum and over existing index-modulated AFDM systems (Qian et al., 14 May 2025).

4. Pilot design, channel estimation, and structured detection

Because AFDM renders the effective channel sparse in the DAFT domain, pilot-aided channel estimation can be performed with unusually small pilot support. The earliest AFDM pilot schemes used a single pilot aided (SPA) or multiple pilots aided (MPA) arrangement with guard symbols in the DAFT domain and a threshold-plus-mapping-table estimator. The reported BER loss relative to ideal CSI was only marginal, and the SPA construction was extended to MIMO and multi-user uplink/downlink settings (Yin et al., 2022).

The main drawback of embedded-guard pilots is spectral-efficiency loss. Two subsequent directions remove or relax this cost. One is superimposed pilot design: pilots are added to data in the DAFT domain, with equally spaced pilot locations separated by $c_1$4, where

$c_1$5

The paper shows that this spacing makes the pilot-induced columns orthogonal in the MMSE sense, and it develops an iterative channel estimator and signal detector to mitigate pilot–data interference (Zheng et al., 2024). The other is GI-free pilot-aided channel estimation, which eliminates guard intervals entirely and relies on iterative joint interference cancellation, channel estimation, and signal detection; its BER was reported to approach the ideal case with perfect channel estimation (Zhou et al., 2024).

On the receiver side, AFDM detection exploits the same sparsity. AFDM-IM studies used both full ML and MMSE-plus-per-group-ML architectures (Tao et al., 2023). For conventional AFDM communications, a message passing detector was proposed on the sparse DAFT-domain factor graph; it performs joint interference cancellation and detection and was reported to outperform MMSE and MRC detectors while maintaining low complexity (Wu et al., 2023). The broader AFDM overview also surveys embedded pilot-aided estimation and EPA-DR, which reconstructs the effective channel matrix directly from pilot observations by exploiting its quasi-diagonal structure (Yin et al., 7 Feb 2025).

Taken together, these estimation and detection schemes show that AFDM’s anti-interference behavior is not solely a waveform property. It also depends on receiver architectures that preserve and exploit the DAFT-domain sparsity rather than ignoring it.

5. Sensing, self-interference cancellation, and multiradar operation

AFDM’s chirp structure has made it a candidate waveform for ISAC. An important early result is that, in AFDM-based ISAC, either the full AFDM frame or only the pilot part consisting of one DAFT-domain symbol and its guard interval can be used to identify all delay and Doppler components associated with the propagation medium, and that using one pilot achieves almost the same sensing performance as using the entire AFDM frame (Bemani et al., 2024). The same work emphasizes a distinctive AFDM property: because the sensing pilot is itself a chirp, simple self-interference cancellation can be performed by dechirping, DC blocking, and analog filtering, avoiding expensive full-duplex methods (Bemani et al., 2024).

The later ISAC overview generalizes this argument. In monostatic sensing, AFDM supports analog dechirping so that the strong self-interference term collapses to DC or near-DC and can be removed before the ADC, while useful echoes remain in narrow bands around DC. This allows reduced sampling rates and low-complexity self-interference cancellation (Bemani et al., 6 Nov 2025). In bistatic sensing, AFDM supports sub-Nyquist sampling without hardware modification while preserving delay resolution, and in dense multiradar settings its DAFT-domain resource assignment allows pilot differentiation through code or index allocation rather than through chirp-rate proliferation (Bemani et al., 6 Nov 2025).

Sensing-oriented ambiguity-function analysis reaches a parallel conclusion. For AFDM modulated by random $c_1$6-QAM symbols, the delay sidelobes can be minimized at zero Doppler by choosing the chirp rate so that, for nonzero delay bins of interest,

$c_1$7

thereby forcing the relevant sinc terms in the ambiguity function to vanish. The same study reported lower PSLR and ISLR than OFDM and found that increasing the number of chirps $c_1$8 improves both metrics, while modulation order has little effect on sidelobe behavior (Bedeer, 3 Apr 2025). A related AFDM-based ISAC study also showed that DAFT-domain sensing processing decouples delay and Doppler in the fast-time axis and maintains good sensing performance in large-Doppler scenarios (Ni et al., 2022).

In anti-interference language, AFDM’s ISAC contribution is therefore not limited to communications robustness. It also provides waveform-level mechanisms for self-interference suppression, multiradar interference management, and sidelobe control in sensing.

6. Limitations, misconceptions, and evolving directions

A common misconception is that “anti-interference AFDM” implies universal robustness. The literature does not support that interpretation. AFDM is primarily robust to channel-induced interference in doubly dispersive propagation; it is not automatically robust to every impairment or every operating regime.

One important limitation is parameter feasibility. Full diversity depends on the non-overlap condition $c_1$9; when that condition fails, overlapping taps appear and the modulated bits in AFDM-IM can lose full diversity, even though the index bits remain more robust (Tao et al., 2023). Another is fractional delay and Doppler. The AFDM overview notes that fractional components broaden the quasi-diagonal effective channel matrix and cause symbol spreading; pulse shaping in the DAFT domain, using windows such as Hamming or Dolph–Chebyshev, reduces sidelobes and restores sparsity more effectively than a rectangular window (Yin et al., 7 Feb 2025).

Hardware impairments are a sharper warning. A recent study of receiver IQ imbalance derives the resulting DAFT-domain inter-carrier interference explicitly and reports that, under identical IQ imbalance conditions, AFDM exhibits more pronounced BER degradation than OFDM (Huang et al., 15 Jun 2026). This does not contradict the anti-interference literature; it qualifies it by separating propagation robustness from hardware robustness.

Detection complexity also remains a central issue. AFDM often requires sequence-wise or sparse-graph detection rather than trivial one-tap demodulation, and near-optimal detection with fractional delay–Doppler spreads, MIMO coupling, or many users remains an open problem (Yin et al., 7 Feb 2025). Channel estimation, pulse design, and multiuser resource assignment in the DAFT domain remain active topics as well.

A further direction is chirp agility. Agile-AFDM reinterprets $c_2$0 and $c_2$1 as blockwise optimization variables and reports that they can be tuned per transmission block to minimize PAPR, suppress ICI, or reduce CRLB, yielding performance gains over both OFDM and static AFDM (Cao et al., 16 Dec 2025). This suggests a plausible evolution of anti-interference AFDM from a fixed waveform into an adaptive, context-aware signal design framework.

In that broader sense, anti-interference AFDM denotes not a single standardized format but a research program: use DAFT-domain chirp modulation to expose delay–Doppler structure, then exploit that structure through pilot placement, sparse detection, index or code-domain signaling, and, increasingly, adaptive waveform control.