Chirp-Permuted AFDM: Enabling Robust 6G Waveforms

Updated 2 August 2025

Chirp-Permuted AFDM is a multicarrier modulation framework that extends AFDM by incorporating chirp permutation to enhance delay-Doppler diversity and security.
It employs permutation matrices within quadratic phase operations to enable a sparse channel structure and maintain low-complexity detection in varying channel conditions.
CP-AFDM supports advanced 6G applications, including integrated sensing and secure communications, by improving ambiguity function resolution and physical-layer protection.

Chirp-Permuted Affine Frequency Division Multiplexing (CP-AFDM) is a multicarrier modulation framework that extends the Affine Frequency Division Multiplexing (AFDM) paradigm by introducing an additional degree of freedom: arbitrary permutation of chirp subcarriers or chirp phase sequences. This permutation operation, applied at the waveform design level, preserves the favorable core attributes of AFDM—including full delay-Doppler diversity, robustness to doubly-dispersive channels, and spectral efficiency—while enabling advanced applications such as enhanced ambiguity function resolution, index modulation over the permutation domain, and provable physical-layer security resistant to both classical and quantum adversaries. CP-AFDM generalizes the DAFT-based AFDM framework by embedding permutation matrices or functions within the chirp-phase structure, profoundly expanding the design space for 6G waveforms targeting joint communications and sensing (“ISAC”), ultra-reliability in high-mobility environments, and secure wireless transmission (Rou et al., 28 Jul 2025).

1. Mathematical Formulation and Signal Model

CP-AFDM is fundamentally defined by the combination of DAFT operations and chirp permutations. In canonical AFDM, the transmitted baseband signal is generated via

$s = \mathbf{A}^{-1} x = \Lambda_{c_1} \mathbf{F}_N \Lambda_{c_2} x$

where $\mathbf{F}_N$ is the $N$ -point DFT (unitary), $\Lambda_{c_1}$ and $\Lambda_{c_2}$ are diagonal matrices applying quadratic phase rotations with parameters $c_1$ , $c_2$ (chirp rates), and $x \in \mathbb{C}^N$ is the data symbol vector (Bemani et al., 2021).

CP-AFDM generalizes this structure by permuting one or both of the chirp matrices, yielding the “chirp-permuted” DAFT (CP-DAFT) operator:

$\mathbf{A}_{\text{CP}} = \Lambda_{c_2,i_2} \mathbf{F}_N \Lambda_{c_1,i_1}$

with

$\Lambda_{c,i} = \mathrm{diag}\left(e^{-j 2 \pi c \cdot \pi_i(n)^2}\right)$

where $\pi_i(\cdot)$ denotes a permutation over $[0, N-1]$ , controlling the ordering of the chirp phases. In practical systems, one-sided CP-AFDM (permuting only $\Lambda_{c_2}$ ) achieves nearly all the theoretical benefits with minimal implementation overhead (Rou et al., 28 Jul 2025).

The underlying modulation kernel becomes

$\kappa_n(m) = \frac{1}{\sqrt{N}} \exp\left\{ -j2\pi \left[ c_1 \cdot \pi_{i_1}(n)^2 + c_2 \cdot \pi_{i_2}(m)^2 + nm/N \right] \right\}$

illustrating how the chirp permutation controls the phase structure of each time-frequency symbol (Rou et al., 28 Jul 2025, Rou et al., 5 Feb 2025).

2. Robustness and Delay-Doppler Diversity

CP-AFDM preserves the essential delay-Doppler separability and full diversity properties of AFDM. The choice of chirp rates $(c_1,c_2)$ and permutation $\pi$ allows control over the effective channel sparsity in the DAFT domain. The system maintains a sparse channel matrix structure, where each unique propagation path appears non-overlapping in the DAFT domain, thus ensuring that the diversity order equals the number of resolvable multipath components, provided $(c_1,c_2,\pi)$ are selected so that DAFT-domain “paths” do not collide (Bemani et al., 2021, Bemani et al., 2022).

Unlike alternatives (e.g., OTFS or OCDM), CP-AFDM’s transform remains strictly one-dimensional, enabling lower-complexity detection and smaller pilot/overhead requirements while also supporting high time-frequency agility (Bemani et al., 2022). The permutation further acts as a decorrelation mechanism in the Doppler domain, effectively mitigating inter-carrier interference and enhancing diversity under time-varying channel conditions (Yin et al., 7 Feb 2025).

3. Ambiguity Function, Sensing, and ISAC Enhancements

A central metric for waveform suitability in ISAC is the ambiguity function (AF), which characterizes joint delay and Doppler resolution. CP-AFDM, by introducing a chirp permutation, achieves a modified AF whose Doppler cut exhibits a narrower mainlobe and significantly improved peak-to-sidelobe ratio (PSLR) relative to conventional AFDM and OFDM (Bedeer, 3 Apr 2025, Yin et al., 11 Jul 2025, Rou et al., 28 Jul 2025).

The theoretical AF of CP-AFDM for a data-carrying QAM symbol vector $x$ is:

$A(\tau, \nu) = \frac{1}{N} \sum_{m,m'} x_m x^*_{m'} \text{sinc}(2c_1N\tau + \nu + (m-m')) e^{j\Psi_{m,m'}(\tau,\nu)}$

With optimal sensing-oriented chirp rate $c_1$ and appropriate permutation, CP-AFDM suppresses delay/range sidelobes, enhancing both delay and Doppler estimation precision (Bedeer, 3 Apr 2025). In integrated monostatic/bistatic sensing, permutation-enabled guard-pilot isolation allows for interference-free CAF/AAF regions, which is particularly significant under high-mobility or multiuser ISAC scenarios (Yin et al., 11 Jul 2025, Zhu et al., 2023).

4. Index Modulation, Information Embedding, and Decoder Architectures

CP-AFDM supports “index modulation” over the permutation domain (CPIM), where extra information bits are mapped into the selection of chirp permutation indices ( $\pi$ ), in addition to being embedded in conventional constellation symbols (Rou et al., 3 May 2024, Rou et al., 28 Jul 2025). This expands the information rate per channel use by $\log_2K$ bits if $K$ permutations are used.

Receiver architectures for CPIM can exploit codebook-based maximum likelihood (ML) or reduced-complexity MMSE-ML detectors. The joint likelihood metric is:

$(\hat{x}, \hat{k}) = \arg\min_{x, k} \| r - H \mathbf{A}_k^{-1} x \|^2$

with $H$ the channel and $A_k$ the chirp-permuted DAFT for codeword $k$ . To manage the factorial codebook size, quantum-inspired search algorithms (e.g., Grover Adaptive Search) have been proposed to accelerate both codebook optimization and ML detection (Rou et al., 3 May 2024).

Additional schemes, such as pre-chirp index modulation (PIM), multiple-mode AFDM-IM, and dynamic constellation selection, further augment CP-AFDM’s spectral and energy efficiency while maintaining subcarrier orthogonality (Liu et al., 23 Feb 2024, Liu et al., 17 Jul 2025, Tao et al., 2023).

5. Physical-Layer Security and Quantum Resilience

A prominent feature of CP-AFDM is its capacity for physical-layer security by treating the permutation index as a cryptographic key. An eavesdropper unaware of the transmitter’s permutation faces a mathematically scrambled channel, resulting in undecodable received data even with full channel state knowledge (Rou et al., 5 Feb 2025, Rou et al., 28 Jul 2025). The attack surface comprises all $N!$ permutation candidates, yielding a random-guess breach probability of $1/N!$, which is negligible for moderate $N$ .

Classical brute-force search to recover the permutation exhibits $\mathcal{O}(N!)$ complexity; quantum adversaries leveraging Grover search are limited to $\mathcal{O}(\sqrt{N!})$ queries—still infeasible for practical system sizes ( $N \gtrsim 16$ ) (Rou et al., 5 Feb 2025). This architecture establishes CP-AFDM as a quantum-resilient secure waveform, meeting 6G security-critical ISAC requirements without extra signaling or power (Rou et al., 28 Jul 2025, Rou et al., 29 Jul 2025).

6. Complexity, Implementation, and Performance Trade-Offs

CP-AFDM maintains the low computational complexity and hardware compatibility of AFDM, requiring only the addition of pre/post-processing permutation steps in the modulator/demodulator—specifically, permutation of chirp phase sequences before chirp-phase multiplication and FFT/IFFT operations (Rou et al., 28 Jul 2025). The main computational overhead lies in index modulation scenarios with large codebooks for CPIM, which are alleviated via structured codebooks and quantum-accelerated optimization (Rou et al., 3 May 2024).

Simulation studies confirm that CP-AFDM achieves:

Identical sparse DAFT-domain channel structure and full diversity as AFDM (provided permutation indices and chirp rates avoid overlap),
No penalty in peak-to-average power ratio (PAPR) compared to AFDM or OFDM,
Superior PSLR and ISLR in the ambiguity function for Doppler/delay resolution,
Comparable or better bit error rate (BER) in both high-mobility and wideband settings, with gains pronounced when index modulation over permutation or pre-chirp assignment is used (Rou et al., 28 Jul 2025, Liu et al., 23 Feb 2024, Zhu et al., 2023, Bedeer, 3 Apr 2025).

A plausible implication is that CP-AFDM’s practical implementation will be straightforward in existing digital hardware platforms built for OFDM/AFDM, owing to its unitary transform-based structure and the backward-compatibility endowed by the DAFT.

7. Role in 6G and Future Directions

CP-AFDM is positioned as a next-generation candidate waveform for 6G standardization, excelling in emerging scenarios demanding flexible, robust, and secure joint communications and sensing. Its capacity to flexibly adapt the permutation space, chirp rates, and codebook structure enables scenario-tailored optimization for:

V2X and UAV communications (ultra-reliability under rapid delay-Doppler variability) (Yin et al., 7 Feb 2025),
ISAC waveforms with high ambiguity function resolution (Bedeer, 3 Apr 2025, Yin et al., 11 Jul 2025),
Physical-layer security against quantum and classical attacks (Rou et al., 5 Feb 2025, Rou et al., 28 Jul 2025),
Dynamic resource allocation, MIMO, multi-user, full-duplex, or integrated sensing operation (Yin et al., 2022, Zhang et al., 4 Jun 2024).

Deployment challenges include standardizing signaling and negotiation of chirp parameters and permutation indices, efficiently managing codebook complexity for index modulation, and ensuring robust receiver synchronization. Research is ongoing into adaptive permutation and chirp schedule optimization as well as application-specific CP-AFDM configurations for ISAC and quantum-resilient physical layers (Rou et al., 29 Jul 2025, Rou et al., 28 Jul 2025).

In summary, Chirp-Permuted AFDM imports a combinatorial permutation space into the DAFT-based chirp multicarrier design, delivering enhanced delay-Doppler separability, adaptability for ISAC and security applications, and new opportunities for index-modulated and physically secure waveform design in 6G and beyond (Rou et al., 28 Jul 2025, Rou et al., 5 Feb 2025, Bedeer, 3 Apr 2025, Rou et al., 29 Jul 2025).