Anti-Interference AFDM: GI-Free Multicarrier System

• Anti-Interference AFDM is a multicarrier waveform using the discrete affine Fourier transform with tuned chirp parameters to uniquely map delay-Doppler paths.
• It achieves full path diversity and suppresses inter-symbol and inter-carrier interference by operating without traditional guard intervals while iteratively canceling pilot-data interference.
• Iterative GI-free channel estimation with LMMSE detection demonstrates near-ideal BER performance and rapid convergence, significantly boosting spectral efficiency.

Anti-Interference Affine Frequency Division Multiplexing (AFDM) systems are a class of multicarrier waveforms that leverage the discrete affine Fourier transform (DAFT) to achieve robust interference rejection in highly doubly-selective wireless channels. By judicious selection of two underlying chirp parameters, AFDM systems decorrelate multi-path and Doppler-induced channel impairments, enabling full diversity order, efficient pilot-aided channel estimation, and high spectral efficiency. The "anti-interference" property refers specifically to schemes that operate without traditional guard intervals (GI), and which iteratively cancel pilot-data interference while retaining tractable receiver complexity and near-ideal error rates.

1. Principles of AFDM and Anti-Interference Mechanism

AFDM operates by mapping data and pilot symbols onto a bank of unitary, parameterized chirp functions via the inverse DAFT (IDAFT). The time-domain transmit vector $s[n]$ is constructed as

$$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$

where $x[\cdot]$ is the DAFT-domain vector (pilot at $m=0$, data at $m=1\dots N-1$), and $(c_1, c_2)$ are the chirp parameters controlling time/frequency spreading. After a chirp-periodic prefix (CPP) is appended to absorb maximum channel delay, the waveform is transmitted over a doubly-selective channel with $P$ distinct delay-Doppler paths.

At the receiver, removal of CPP and application of the forward DAFT recovers the DAFT-domain observations:

$$y[m] = \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} r[n] e^{-j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$

The effective input-output model is $y = H_{\mathrm{eff}} x + w$, where $H_{\mathrm{eff}}$ is a sparse, cyclic-shift/coupling matrix whose structure is determined by the channel's delay-Doppler support and the chosen $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$0.

The intrinsic anti-interference mechanism arises from tunable chirp parameters that map each delay-Doppler path to a unique (non-overlapping) index in the DAFT domain. This suppresses both inter-symbol interference (ISI) and inter-carrier interference (ICI) so long as $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$1 is chosen such that all shifted indices $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$2 (for path $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$3 with delay $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$4 and Doppler $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$5) are distinct (Zhou et al., 2024, Rou et al., 29 Jul 2025, Bemani et al., 2021). This ensures the DAFT-domain channel is maximally sparse (one nonzero per row per path), and the system achieves the full path diversity $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$6.

2. Interference Representation in the Absence of Guard Intervals

In conventional pilot-aided architectures, a guard interval (GI) is reserved around the pilot to prevent data-induced interference during channel estimation, sacrificing spectral efficiency. The anti-interference AFDM scheme instead eliminates the GI entirely, packing the DAFT domain as densely as possible (pilot at $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$7, data on $$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$8), and explicitly models two interference types (Zhou et al., 2024):

• Data-to-pilot interference (ID2P): Data symbols leak into the pilot's DAFT output:

$$s[n] = \frac{1}{\sqrt{N}} \sum_{m=0}^{N-1} x[m] \, e^{j2\pi(c_1 n^2 + (m n)/N + c_2 m^2)}$$9

• Pilot-to-data interference (IP2D): The pilot leaks into data symbol outputs:

$x[\cdot]$0

where $x[\cdot]$1, and $x[\cdot]$2.

Without a GI, these interference terms manifest in each output, obfuscating direct recovery of channel taps and data symbols.

3. Iterative GI-Free Interference Cancellation and Channel Estimation

To resolve mutual interference between pilot and data, GI-free AFDM introduces an iterative loop comprising four steps:

1. Interference cancellation: At iteration $x[\cdot]$3, subtract estimated ID2P or IP2D terms using the previous data and channel estimates.
2. Channel estimation: Use a threshold $x[\cdot]$4 to detect new physical paths from residual peaks, estimating $x[\cdot]$5 indices and gains $x[\cdot]$6 via:

$x[\cdot]$7

The threshold is updated per step to accommodate missed weak paths and limit false alarms.

1. Data detection: Perform linear minimum mean-square error (LMMSE) detection using current $x[\cdot]$8.
2. Channel update: Reconstruct $x[\cdot]$9 from new path estimates. Repeat until convergence or a small maximum number of rounds $m=0$0 (empirically $m=0$1 suffices; further iterations give diminishing returns).

The procedure is initiated with a coarse path search and LMMSE estimate, and refined iteratively, alternately cancelling ID2P and IP2D based on up-to-date symbol and channel beliefs (Zhou et al., 2024).

4. Complexity and Spectral Efficiency

The principal advantage of this GI-free approach is a substantial gain in spectral efficiency:

• Traditional GI-based: $m=0$2
• GI-free: $m=0$3

For practical system parameters (e.g., $m=0$4, $m=0$5 for max delay $m=0$6, max Doppler $m=0$7), this translates to $m=0$8, compared to $m=0$9—a $m=1\dots N-1$0 improvement (Zhou et al., 2024).

The computational burden is dominated by LMMSE inversion per iteration ($m=1\dots N-1$1), but the number of necessary iterations $m=1\dots N-1$2 is very small (typically 2), making the total complexity $m=1\dots N-1$3, equivalent in order to classical GI-based methods. For large block sizes and highly sparse $m=1\dots N-1$4, further acceleration via message-passing or sparse solvers is possible.

5. Performance Evaluation and Convergence Behavior

Extensive simulations in a canonical 3-path Jakes channel (e.g., $m=1\dots N-1$5, BPSK, $m=1\dots N-1$6, $m=1\dots N-1$7) demonstrate:

• Zero-GI, non-iterative (coarse) detection suffers over 10 dB loss at fixed BER compared to perfect-CSI detection.
• A single joint iteration recovers approximately 4 dB, closely approaching the perfect CSI bound for SNR $m=1\dots N-1$8 10 dB.
• A second iteration provides further minor (sub-1 dB) gain; additional iterations give negligible benefit.
• At SNR = 12 dB, BER$m=1\dots N-1$9 for GI-free AFDM vs. $(c_1, c_2)$0 for the perfect-CSI case—a gap of roughly 3 dB.

The algorithm exhibits rapid convergence: nearly all benefit is obtained after one iteration, and the residual BER curve closely tracks the ideal channel performance (Zhou et al., 2024).

6. Practical Trade-offs and Design Guidelines

Several system-level trade-offs characterize the practical deployment of GI-free, anti-interference AFDM:

• Pilot Power and PAPR: Increasing pilot power $(c_1, c_2)$1 sharpens ID2P suppression but increases peak-to-average power ratio (PAPR); simulations with $(c_1, c_2)$2 dB and $(c_1, c_2)$3 dB yield a $(c_1, c_2)$4 PAPR increase but gain $(c_1, c_2)$5 in spectral efficiency.
• Threshold and Path Redetection: Setting $(c_1, c_2)$6 controls the threshold for new path detection; higher $(c_1, c_2)$7 avoids missed weak paths but may induce more false alarms.
• Pilot Placement: Positioning the single pilot on a unique DAFT bin enables straightforward path indexing but necessitates the joint ID2P/IP2D cancellation described above.
• Channel Sparsity Exploitation: The (delay, Doppler) sparsity inherent to AFDM’s DAFT domain can be leveraged for further algorithmic simplification, particularly in large-$(c_1, c_2)$8 regimes.

7. Context and Impact within the AFDM Literature

The GI-free anti-interference AFDM architecture refines classical DAFT/AFDM waveforms by systematizing pilot-aided channel estimation, achieving near-orthogonal path separation even under severe spectral reuse (Rou et al., 29 Jul 2025, Bemani et al., 2021). This approach directly addresses the efficiency–robustness trade-off that handicaps traditional guard-interval schemes, especially in high-mobility and high-path-count environments.

It is distinct from alternative DAFT-based estimation methods (e.g., embedded-pilot (Bemani et al., 2022) or superimposed-pilot (Zheng et al., 2024) AFDM), which rely on explicit guard symbol overhead or iterative pilot separation, but share the underlying principle of exploiting DAFT-channel sparsity and the non-overlapping property enabled by $(c_1, c_2)$9 tuning.

GI-free anti-interference AFDM occupies a central role in the drive toward spectrally efficient, robust, and tractable transceiver designs for 6G and beyond, providing a template for advanced channel-coded, index-modulated, and sensing-integrated architectures (Zhou et al., 2024, Rou et al., 29 Jul 2025).