Affine frequency division multiplexing (AFDM) has emerged as a promising technology for high-mobility scenarios, offering reliable performance in doubly selective channels. However, the computational complexity of the maximum likelihood (ML) detection scheme renders it impractical for real-time AFDM applications. To address this, we propose a low-complexity AFDM symbol detection algorithm based on expectation propagation (EP) in this paper. The proposed EP-based detection scheme iteratively updates messages to approximate the ML result, reducing computational complexity from exponential to cubic order. By exploiting the sparse and quasi-banded structure of the channel in the discrete affine Fourier transform (DAFT) domain and employing matrix block decomposition, lower-upper factorization, and upper triangular matrix forward substitution, we further reduce the complexity of the EP algorithm to linear order. Additionally, we optimize the EP algorithm’s performance by incorporating deep learning-based moment matching, making the algorithm more adaptive with trainable parameters for both positive and negative components. Moreover, we propose a DAFT-domain iterative detection and decoding scheme, where external information from the decoder is fed back to the detector, resulting in improved system reliability. Simulation results show that the proposed scheme achieves near-ML performance while reducing complexity by dozens of orders of magnitude compared to the ML detector, striking a balance between performance enhancement and computational complexity.

In affine frequency division multiplexing (AFDM) systems, the severe peak-to-average power ratio (PAPR) signals exist in the time domain due to the coherent superposition of numerous modulated symbols. Eventually, high PAPR signals require sophisticated and expensive power amplifiers with a very large linear range. To this end, a neural network (NN) aided intelligent transceiver optimization framework is proposed for suppressing PAPR based on the spreading AFDM structure. Specifically, the transceiver jointly optimizes the constellation geometry and associated bit labeling, the precoding NN, as well as the NN based detector. Moreover, the precoding NN is learned from a precoding approach which minimizes the variance of the instantaneous power of output signals at the transmitter. The joint optimization framework aims to achieve maximum PAPR reduction under the constraints of unit energy and spectral emission mask. Besides, to mitigate the potential inter-carrier interference during the offline training, a long short term memory based detector is designed within the optimization framework. Simulation results demonstrate that the conceived NN based optimization method achieves a significant enhancement on PAPR reduction compared with conventional approaches, while slightly improving the bit error ratio performance.

Affine frequency division multiplexing (AFDM) achieves full diversity but faces multiple-access challenges due to signal dispersion. To address this issue, we propose a power domain non-orthogonal multiple access AFDM (PD-NOMA-AFDM) system, which enables parallel transmission of multi-user signals on the same resource block through power-domain multiplexing. Furthermore, we design a binary-loop maximal ratio combining-message passing (BLMM)-based successive interference cancellation (SIC) scheme. Specifically, the inner loop fully leverages the sparsity of the AFDM equivalent channel to effectively eliminate inter-symbol interference and achieve reliable initial symbol estimation; the outer loop iteratively updates extrinsic information to compensate for performance degradation caused by banded-matrix approximation. We prove the convergence of the inner loop to the MMSE fixed point and the local convergence of the outer loop. Subsequently, by combining Lipschitz continuity and perturbation theory, we demonstrate the convergence of the overall BLMM detector to a neighborhood of the exact fixed point. The pairwise error probability analysis is then used to characterize its diversity gain and performance gap to maximum likelihood (ML) detection. To further narrow this gap, a graph neural network (GNN) is incorporated into the BLMM multi-user detection framework. This approach dynamically captures the multi-user interference (MUI) characteristics through node message interactions, thereby improving the accuracy of the approximatea posterioriprobability distribution. Simulation results show that the proposed BLMM-GNN achieves near-ML performance with strong robustness.

Index modulation (IM) techniques have been widely studied over the past decade for their ability to enhance spectral and energy efficiency by exploiting additional degrees of freedom (DoF) in waveforms. In this paper, we propose a novel subcarrier filtering orthogonal frequency-division multiplexing (OFDM) scheme, named waveform index modulation (WIM), to boost the spectral efficiency (SE) of OFDM systems without compromising other performance metrics. In WIM-OFDM, information is conveyed not only by the modulated constellation symbols but also by altering the subcarrier filter shapes, thereby utilizing an additional DoF in the OFDM signaling process. An SE-enhanced version, referred to as generalized WIM-OFDM (GWIM-OFDM), is also designed to further boost the index transmission rate by maximizing the DoF for filter selection on each subcarrier. Additionally, the optimization of subcarrier filter shapes is formulated, and a special case of subcarrier filter pair can be optimized by utilizing the proposed non-convex to convex scaling method. At the receiver, a low-complexity interference cancellation algorithm is proposed to eliminate the introduced inter-carrier interference caused by the non-orthogonal subcarrier shapes. Finally, to validate the proposed scheme, closed-form expressions for the achievable rates and the upper bound on the average bit error rate are derived to prove the superiority of our WIM-OFDM and GWIM-OFDM schemes theoretically. Monte Carlo simulation results corroborate the benefits of the proposed scheme, that is, the GWIM-OFDM scheme exhibits 4.7 to 6.1 dB performance gain, considering both bit error rate and peak-to-average power ratio, compared with the traditional OFDM and other IM benchmarking schemes under the same spectrum mask at different transmission rate scenarios.

To fully obtain the time-frequency diversity gain of the affine frequency division multiplexing (AFDM) system, detection algorithms that offer high performance and low complexity are essential. The maximum likelihood (ML) algorithm can achieve theoretically optimal performance. However, the exponential complexity limits its practical application. This paper designs an AFDM signal detection algorithm based on expectation propagation (EP). The proposed EP-based scheme achieves effective AFDM signal detection by iteratively updating messages to approximate the true a posterior distribution. In addition, this paper further reduces the complexity of the proposed EP-based algorithm by utilizing the characteristics of the discrete affine Fourier transform (DAFT) domain equivalent channel. Specifically, the sparsity and quasi-banded structure of the DAFT domain channel are first utilized for block processing. Subsequently, a low complexity matrix inversion operation is realized by combining the lower-upper (LU) factorization and the upper triangular matrix forward substitution algorithm. With typical AFDM system parameters, the proposed scheme reduces the complexity by 35.6 times compared to the traditional EP algorithm, while the performance is virtually unaffected. Simulation results show that the proposed scheme has a performance gain of up to 5 dB over the conventional algorithm.

This paper proposes a novel joint transceiver optimization framework for multi-carrier (MC) waveform design. Unlike conventional orthogonal frequency division multiplexing, which employs memoryless modulation and fixed inverse discrete Fourier transform-based waveform generation, our approach utilizes neural network (NN)-based modulation with memory and NN-driven waveform generation at the transmitter. On the receiver side, a large-kernel attention-based NN replaces the traditional demodulation process, effectively mitigating large-span inter-carrier interference. This architecture provides enhanced flexibility for MC waveform optimization, allowing better adaptation to spectral emission mask constraints and maximizing the utilization of allocated spectrum resources. Additionally, it achieves significant spectral efficiency gains across diverse channel conditions, including additive white Gaussian noise (AWGN) and linear time-varying (LTV) channels with delay and Doppler spread. Numerical evaluations demonstrate significant bit error rate performance improvements, with up to 10 dB signal-to-noise ratio gain in LTV channels and approximately 6 dB gain in AWGN channels, underscoring the superiority of the proposed framework over state-of-the-art schemes.

Non-linear Multiple-Input Multiple-Output (MIMO) has attracted considerable attention because of its high power-efficiency characteristic, particularly in the fifth generation (5G) and beyond. This paper focuses on the non-linear MIMO baseband algorithms in power-efficiency networks. In previous works, Generalized Approximate Message Passing (GAMP) and importance sampling technique were used to solve the non-linear distortion in Halved Phase-Only (HPO-) MIMO system. However, its convergence rate becomes unstable, and it’s converge is not guaranteed in some cases. In this paper, to improve the efficiency of convergence rate, we propose Gaussian Mixture Model (GMM) based ExpectationMaximization (EM) signal processing algorithm in HPO MIMO system. We first transforme channel estimation and multiuser detection problems into generalized linear mixed problems under π-phase observations. Then, the GMM algorithm is used to estimate the distribution of π-phase observation. Meanwhile, the EM algorithm is used to estimate the recovered signal. Simulation results show that the proposed method achieves high convergence and has better performance than the reference GAMP algorithm.