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.

In this paper, we propose a novel waveform index modulation orthogonal frequency-division multiplexing (WIM-OFDM) scheme to increase spectral efficiency for multicarrier systems. More specifically, the proposed WIM-OFDM scheme conveys not only the classic constellation symbols but also extra index bits by changing the subcarrier filtering shape for each symbol. As a key point, the optimization of the subcarrier filter shapes is formulated and a preliminary subcarrier filter pair is given to verify the performance of the proposed scheme. Our simulation results demonstrate that the proposed WIM-OFDM scheme exhibits superior performance in both peak-to-average power ratio and bit error ratio compared to conventional OFDM-IM and its dual-mode counterparts without increasing the out-of-band emission.

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.

In the increasingly intricate and dynamic environments, it is difficult to achieve high-precision multi-target sensing with limited bandwidth resources in integrated sensing and communication. In response to the difficulty of low-power targets detection in millimeter-wave orthogonal frequency division mul-tiplexing system, a ResN et based estimation method is proposed in this paper, which has ability to realize high-precision multi-target range and velocity estimation with limited bandwidth. We first preprocess the received signal to obtain detailed channel in delay-Doppler domain. Then we utilize ResNet who has deep network and off-grid estimation capability to achieve high-precision estimation of a single high-power target. Third, an iterative structure is used to counteract the influence of this target on other targets, thereby achieving high-precision estimation of low-power targets without the number of targets. The simulation results depict that the proposed method leads lower estimation error compared to other state-of-the-art deep learning based methods and multiple signal classification algorithm in multi-target scenario. Moreover, the proposed method boasts lower computational complexity when compared to multiple signal classification algorithm.

Faster-than-Nyquist signaling (FTNS) has already been shown to increase the communication capacity on certain channels such as additive white Gaussian noise and block flat multiple-input multiple-output (MIMO) Rayleigh fading channels. The following issues, however, remain unresolved: 1) whether FTNS enables a capacity increase in generalized MIMO Rayleigh fading channels that are selective in time, frequency, and space; and 2) how channel selectivities affect the capacity and if present, the FTN capacity gain. To address the issues, this paper firstly investigates the ergodic capacity of MIMO-FTN transmission over triply-selective fading channels. We derive a low-complexity approximate capacity formula and also show how it degenerates in other channel models, such as doubly-selective single-input single-output fading channels, which can be considered as the special cases of triply-selective fading channels. The capacity evaluation results obtained under different channel conditions show that: 1) MIMO-FTN outperforms MIMO-Nyquist in terms of capacity; 2) the FTN gains are nearly consistent, while the FTN gains obtained in the frequency-selective fading channels are slightly higher than those obtained in the flat fading channels.