A Tensor-Train Framework for Bayesian Inference in High-Dimensional Systems: Applications to MIMO Detection and Channel Decoding

Bayesian inference in high-dimensional discrete-input additive noise models is a fundamental challenge in communication systems, as the support of the required joint a posteriori probability (APP) mass function grows exponentially with the number of unknown variables. In this work, we propose a tensor-train (TT) framework for tractable, near-optimal Bayesian inference in discrete-input additive noise models. The central insight is that the joint log-APP mass function admits an exact low-rank representation in the TT format, enabling compact storage and efficient computations. To recover symbol-wise APP marginals, we develop a practical inference procedure that approximates the exponential of the log-posterior using a TT-cross algorithm initialized with a truncated Taylor-series. To demonstrate the generality of the approach, we derive explicit low-rank TT constructions for two canonical communication problems: the linear observation model under additive white Gaussian noise (AWGN), applied to multiple-input multiple-output (MIMO) detection, and soft-decision decoding of binary linear block error correcting codes over the binary-input AWGN channel. Numerical results show near-optimal error-rate performance across a wide range of signal-to-noise ratios while requiring only modest TT ranks. These results highlight the potential of tensor-network methods for efficient Bayesian inference in communication systems.

Towards a Unified Coding Scheme for 6G

The growing demand for higher data rates necessitates continuous innovations in wireless communication systems, particularly with the emergence of 6G. Channel coding plays a crucial role in this evolution. In 5G systems, rate-adaptive raptor-like quasi-cyclic irregular low-density parity-check codes are used for the data link, while polar codes with successive cancellation list decoding handle short messages on the synchronization channel. However, to meet the stringent requirements of future 6G systems, a versatile and unified coding scheme should be developed - one that offers competitive error-correcting performance alongside low complexity encoding and decoding schemes that enable energy-efficient hardware implementations. This white paper outlines the vision for such a unified coding scheme. We explore various 6G communication scenarios that pose new challenges to channel coding and provide a first analysis of potential solutions.

Breaking the CP Limit: Robust Long-Range OFDM Sensing via Interference Cleaning

In orthogonal frequency-division multiplexing-based radar and integrated sensing and communication systems, the sensing range is traditionally limited by the round-trip time corresponding to the cyclic prefix duration. Targets whose echoes arrive after this duration induce intersymbol interference (ISI) and associated intercarrier interference (ICI), which significantly degrade detection performance, elevate the interference-noise floor in the radar image, and reduce the useful signal power due to window mismatch. Existing methods face a trade-off between recovering useful signal and suppressing interference, particularly in multi-target scenarios. This paper proposes two frameworks to resolve this dilemma, offering a flexible trade-off between computational cost and target detection performance. First, a signal model is derived, demonstrating that ISI and ICI-oriented interference often dominates thermal noise in high-dynamic-range scenarios. To combat the ISI and ICI-based interference-noise floor increase, joint-interference cancellation with coherent compensation is proposed. This approach is an efficient evolution of the successive-interference cancellation algorithm, utilizing high-precision chirp Z-transform estimation and frequency-domain coherent compensation to recover weak distant targets. For scenarios requiring maximum precision, the full reconstruction-based sliding window scheme is presented, which shifts the receive window to capture optimal signal energy while performing full-signal reconstruction for all detected targets. Numerical results show that both methods outperform state-of-the-art benchmarks.

Precoding Design for Multi-User MIMO Joint Communications and Sensing

We investigate precoding for multi-user (MU) multiple-input multiple-output (MIMO) joint communications and sensing (JCAS) systems, taking into account the potential interference between sensing and communication channels. We derive indicators for the sensing and communication performance, i.e., the detection probability and the communication signal-to-interference-and-noise ratio (SINR) for general input signals. Our results show that the use of the communication signal for sensing can prevent a loss in communication performance if channel interference occurs, while the kurtosis of the transmit alphabet of the communication signal limits the sensing performance. We present simulation results of example setups.

Neural Network-Based Single-Carrier Joint Communication and Sensing: Loss Design, Constellation Shaping and Precoding

We investigate the impact of higher-order modulation formats on the sensing performance of single-carrier joint communication and sensing (JCAS) systems. Several separate components such as a beamformer, a modulator, a target detector, an angle of arrival (AoA) estimator and a communication demapper are implemented as trainable neural networks (NNs). We compare geometrically shaped modulation formats to a classical quadrature amplitude modulation (QAM) scheme. We assess the influence of multi-snapshot sensing and varying signal-to-noise ratio (SNR) on the overall performance of the autoencoder-based system. To improve the training behavior of the system, we decouple the loss functions from the respective SNR values and the number of sensing snapshots, using upper bounds of the sensing and communication performance, namely the Cram\'er-Rao bound for AoA estimation and the mutual information for communication. The NN-based sensing outperforms classical algorithms, such as a Neyman-Pearson based power detector for object detection and ESPRIT for AoA estimation for both the trained constellations and QAM at low SNRs. We show that the gap in sensing performance between classical and shaped modulation formats can be significantly reduced through multi-snapshot sensing. Lastly, we demonstrate system extension to multi-user multiple-input multiple-output to address the improvement of spatial efficiency when servicing multiple user equipments. Our contribution emphasizes the importance of estimation bounds for training neural networks, especially when the trained solutions are deployed in varying SNR conditions.

Novel Phase-Noise-Tolerant Variational-Autoencoder-Based Equalization Suitable for Space-Division-Multiplexed Transmission

We demonstrate the effectiveness of a novel phase-noise-tolerant, variational-autoencoder-based equalization scheme for space-division-multiplexed (SDM) transmission in an experiment over 150km of randomly-coupled multi-core fibers.

Interference Mitigation for OFDM-based Integrated Sensing and Communications with Arbitrary Modulation Formats

Integrated sensing and communication will be a key feature of future mobile networks, enabling highly efficient systems and numerous new applications by leveraging communication signals for sensing. In this paper, we analyze the impact of arbitrary modulation alphabets on the sensing performance of communication-centric OFDM systems as expected in the next-generation 6G networks. We evaluate existing interference mitigation techniques, such as coherent successive target cancellation, and propose an enhanced version of this algorithm. A systematic performance evaluation in multi-target scenarios, including the effects of scattering, demonstrates that our proposed interference mitigation methods achieve performance comparable to sensing-optimal constant modulus signals while utilizing higher order constellations for more efficient communications.

Constellation Shaping for OFDM-ISAC Systems: From Theoretical Bounds to Practical Implementation

Integrated sensing and communications (ISAC) promises new use cases for mobile communication systems by reusing the communication signal for radar-like sensing. However, sensing and communications (S&C) impose conflicting requirements on the modulation format, resulting in a tradeoff between their corresponding performance. This paper investigates constellation shaping as a means to simultaneously improve S&C performance in orthogonal frequency division multiplexing (OFDM)-based ISAC systems. We begin by deriving how the transmit symbols affect detection performance and derive theoretical lower and upper bounds on the maximum achievable information rate under a given sensing constraint. Using an autoencoder-based optimization, we investigate geometric, probabilistic, and joint constellation shaping, where joint shaping combines both approaches, employing both optimal maximum a-posteriori decoding and practical bit-metric decoding. Our results show that constellation shaping enables a flexible trade-off between S&C, can approach the derived upper bound, and significantly outperforms conventional modulation formats. Motivated by its practical implementation feasibility, we review probabilistic amplitude shaping (PAS) and propose a generalization tailored to ISAC. For this generalization, we propose a low-complexity log-likelihood ratio computation with negligible rate loss. We demonstrate that combining conventional and generalized PAS enables a flexible and low-complexity tradeoff between S&C, closely approaching the performance of joint constellation shaping.

Uncertainty Propagation in the Fast Fourier Transform

We address the problem of uncertainty propagation in the discrete Fourier transform by modeling the fast Fourier transform as a factor graph. Building on this representation, we propose an efficient framework for approximate Bayesian inference using belief propagation (BP) and expectation propagation, extending its applicability beyond Gaussian assumptions. By leveraging an appropriate BP message representation and a suitable schedule, our method achieves stable convergence with accurate mean and variance estimates. Numerical experiments in representative scenarios from communications demonstrate the practical potential of the proposed framework for uncertainty-aware inference in probabilistic systems operating across both time and frequency domain.

On the Sensing Performance of FMCW-based Integrated Sensing and Communications with Arbitrary Constellations

Integrated sensing and communications (ISAC) is expected to play a major role in numerous future applications, e.g., smart cities. Leveraging native radar signals like the frequency modulated continuous wave (FMCW) waveform additionally for data transmission offers a highly efficient use of valuable physical radio frequency (RF) resources allocated for automotive radar applications. In this paper, we propose the adoption of higher-order modulation formats for data modulation onto an FMCW waveform and provide a comprehensive overview of the entire signal processing chain. We evaluate the impact of each component on the overall sensing performance. While alignment algorithms are essential for removing the information signal at the sensing receiver, they also introduce significant dispersion to the received signal. We analyze this effect in detail. Notably, we demonstrate that the impact of non-constant amplitude modulation on sensing performance is statistically negligible when the complete signal processing chain is considered. This finding highlights the potential for achieving high data rates in FMCW-ISAC systems without compromising the sensing capabilities.