Channel charting (CC) is a self-supervised positioning technique whose main limitation is that the estimated positions lie in an arbitrary coordinate system that is not aligned with true spatial coordinates. In this work, we propose a novel method to produce CC locations in true spatial coordinates with the aid of a digital twin (DT). Our main contribution is a new framework that (i) extracts large-scale channel-state information (CSI) features from estimated CSI and the DT and (ii) matches these features with a cosine-similarity loss function. The DT-aided loss function is then combined with a conventional CC loss to learn a positioning function that provides true spatial coordinates without relying on labeled data. Our results for a simulated indoor scenario demonstrate that the proposed framework reduces the relative mean distance error by 29% compared to the state of the art. We also show that the proposed approach is robust to DT modeling mismatches and a distribution shift in the testing data.

Accurate and robust localization is a critical enabler for emerging 5G and 6G applications, including autonomous driving, extended reality (XR), and smart manufacturing. While data-driven approaches have shown promise, most existing models require large amounts of labeled data and struggle to generalize across deployment scenarios and wireless configurations. To address these limitations, we propose a foundation-model-based solution tailored for wireless localization. We first analyze how different self-supervised learning (SSL) tasks acquire general-purpose and task-specific semantic features based on information bottleneck (IB) theory. Building on this foundation, we design a pretraining methodology for the proposed Large Wireless Localization Model (LWLM). Specifically, we propose an SSL framework that jointly optimizes three complementary objectives: (i) spatial-frequency masked channel modeling (SF-MCM), (ii) domain-transformation invariance (DTI), and (iii) position-invariant contrastive learning (PICL). These objectives jointly capture the underlying semantics of wireless channel from multiple perspectives. We further design lightweight decoders for key downstream tasks, including time-of-arrival (ToA) estimation, angle-of-arrival (AoA) estimation, single base station (BS) localization, and multiple BS localization. Comprehensive experimental results confirm that LWLM consistently surpasses both model-based and supervised learning baselines across all localization tasks. In particular, LWLM achieves 26.0%--87.5% improvement over transformer models without pretraining, and exhibits strong generalization under label-limited fine-tuning and unseen BS configurations, confirming its potential as a foundation model for wireless localization.

Gain-phase impairments (GPIs) affect both communication and sensing in 6G integrated sensing and communication (ISAC). We study the effect of GPIs in a single-input, multiple-output orthogonal frequency-division multiplexing ISAC system and develop a model-based unsupervised learning approach to simultaneously (i) estimate the gain-phase errors and (ii) localize sensing targets. The proposed method is based on the optimal maximum a-posteriori ratio test for a single target. Results show that the proposed approach can effectively estimate the gain-phase errors and yield similar position estimation performance as the case when the impairments are fully known.

Direction of arrival (DoA) estimation is a common sensing problem in radar, sonar, audio, and wireless communication systems. It has gained renewed importance with the advent of the integrated sensing and communication paradigm. To fully exploit the potential of such sensing systems, it is crucial to take into account potential hardware impairments that can negatively impact the obtained performance. This study introduces a joint DoA estimation and hardware impairment learning scheme following a model-based approach. Specifically, a differentiable version of the multiple signal classification (MUSIC) algorithm is derived, allowing efficient learning of the considered impairments. The proposed approach supports both supervised and unsupervised learning strategies, showcasing its practical potential. Simulation results indicate that the proposed method successfully learns significant inaccuracies in both antenna locations and complex gains. Additionally, the proposed method outperforms the classical MUSIC algorithm in the DoA estimation task.

Reconfigurable intelligent surfaces (RISs) have emerged as a transformative technology for sixth-generation (6 G) communication networks, offering the ability to dynamically shape wireless propagation environments and thus efficiently enhance received signal quality. However, practical implementation of RIS faces challenges, including potential failures of individual elements (pixels), which can degrade the performance significantly. This paper leverages autoencoders and end-to-end (E2E) learning in RIS-aided systems to jointly optimize the RIS phase profiles and receiver angle-of-departure (AoD) estimation in the presence of pixel failures. The proposed E2E approach demonstrates resilience against practical pixel errors while is shown to achieve performance close to the fundamental bounds, thereby advancing the state-of-the-art in RIS-aided systems towards the 6 G era.

We study model-based end-to-end learning in the context of integrated sensing and communication (ISAC) under hardware impairments. Hardware impairments are usually addressed by means of array calibration with a focus on communication performance. However, residual impairments may exist that affect sensing performance. This paper proposes a data-driven framework for mitigating such impairments. A monostatic orthogonal frequency-division multiplexing (OFDM) sensing and multiple-input single-output (MISO) communication scenario is considered, incorporating hardware imperfections at the ISAC transceiver antenna array. Since conventional ISAC signal processing algorithms rely on mathematical models of the wireless channel, a mismatch occurs between the assumed mathematical models and the underlying reality in the presence of hardware impairments. We first study the detrimental effects of such impairments at the transmitter and receiver side of the proposed scenario, showcasing different levels of degradation on communication and sensing performances. As the core contribution of this work, we propose a novel differentiable version of the orthogonal matching pursuit (OMP) algorithm that is suitable for multi-target sensing and allows for efficient end-to-end learning of the hardware impairments. Based on the differentiable OMP, we devise two model-based parameterization strategies of the ISAC beamformer and sensing receiver to account for hardware impairments: (i) learning a dictionary of steering vectors for different angles and (ii) learning the parameterized hardware impairments. We carry out a comprehensive performance analysis of the proposed model-based learning approaches and a strong baseline consisting of least-squares beamforming, conventional OMP, and maximum-likelihood symbol detection for communication. Results show that by parameterizing the hardware impairments, learning approaches offer gains in terms of higher detection probability, position estimation accuracy, and lower symbol error rate (SER) compared to the baseline. We demonstrate that learning the parameterized hardware impairments outperforms learning a dictionary of steering vectors, also exhibiting the lowest complexity.

In this paper, we propose a novel decoding method for Quantum Low-Density Parity-Check (QLDPC) codes based on Graph Neural Networks (GNNs). Similar to the Belief Propagation (BP)-based QLDPC decoders, the proposed GNN-based QLDPC decoder exploits the sparse graph structure of QLDPC codes and can be implemented as a message-passing decoding algorithm. We compare the proposed GNN-based decoding algorithm against selected classes of both conventional and neural–enhanced QLDPC decoding algorithms across several QLDPC code designs. The simulation results demonstrate excellent performance of GNN-based decoders along with their low complexity compared to competing methods.

Quantum low-density parity-check (QLDPC) codes have emerged as a promising technique for quantum error correction. A variety of decoders have been proposed for QLDPC codes and many utilize belief propagation (BP) decoding in some fashion. However, the use of BP decoding for degenerate QLDPC codes is known to have issues with convergence. These issues are typically attributed to short cycles in the Tanner graph and error patterns with the same syndrome due to code degeneracy. In this work, we propose a decoder for QLDPC codes based on BP guided decimation (BPGD), which has been previously studied for constraint satisfaction and lossy compression problems. This decimation process is applicable to both binary and quaternary BP and it involves sequentially freezing the value of the most reliable qubits to encourage BP convergence. We find that BPGD significantly reduces the BP failure rate due to non-convergence, achieving performance on par with BP with ordered statistics decoding and BP with stabilizer inactivation, without the need to solve systems of linear equations. To explore how and why BPGD improves performance, we discuss several interpretations of BPGD and their connection to BP syndrome decoding.

Integrated sensing and communication (ISAC) is envisioned to be one of the pillars of 6G. However, 6G is also expected to be severely affected by hardware impairments. Under such impairments, standard model-based approaches might fail if they do not capture the underlying reality. To this end, data-driven methods are an alternative to deal with cases where imperfections cannot be easily modeled. In this paper, we propose a model-driven learning architecture for joint single-target multi-input multi-output (MIMO) sensing and multi-input single-output (MISO) communication. We compare it with a standard neural network approach under complexity constraints. Results show that under hardware impairments, both learning methods yield better results than the model-based standard baseline. If complexity constraints are further introduced, model-driven learning outperforms the neural-network-based approach. Model-driven learning also shows better generalization performance for new unseen testing scenarios.

Most of today's communication systems are designed to target reliable message recovery after receiving the entire encoded message (codeword). However, in many practical scenarios, the transmission process may be interrupted before receiving the complete codeword. This paper proposes a novel rateless autoencoder (AE)-based code design suitable for decoding the transmitted message before the noisy codeword is fully received. Using particular dropout strategies applied during the training process, rateless AE codes allow to trade off between decoding delay and reliability, providing a graceful improvement of the latter with each additionally received codeword symbol. The proposed rateless AEs significantly outperform the conventional AE designs for scenarios where it is desirable to trade off reliability for lower decoding delay.

Integrated sensing and communication (ISAC) aims to unify radar and communication systems through a combination of joint hardware, joint waveforms, joint signal design, and joint signal processing. At high carrier frequencies, where ISAC is expected to play a major role, joint designs are challenging due to several hardware limitations. Model-based approaches, while powerful and flexible, are inherently limited by how well the models represent reality. Under model deficit, data-driven methods can provide robust ISAC performance. We present a novel approach for data-driven ISAC using an auto-encoder (AE) structure. The approach includes the proposal of the AE architecture, a novel ISAC loss function, and the training procedure. Numerical results demonstrate the power of the proposed AE, in particular under hardware impairments.

We propose an over-the-air digital predistortion optimization algorithm using reinforcement learning. Based on a symbol-based criterion, the algorithm minimizes the errors between downsampled messages at the receiver side. The algorithm does not require any knowledge about the underlying hardware or channel. For a generalized memory polynomial power amplifier and additive white Gaussian noise channel, we show that the proposed algorithm achieves performance improvements in terms of symbol error rate compared with an indirect learning architecture even when the latter is coupled with a full sampling rate ADC in the feedback path. Furthermore, it maintains a satisfactory adjacent channel power ratio.

End-to-end autoencoder (AE) learning has the potential of exceeding the performance of human-engineered transceivers and encoding schemes, without a priori knowledge of communication-theoretic principles. In this work, we aim to understand to what extent and for which scenarios this claim holds true when comparing with fair benchmarks. Our particular focus is on memoryless multiple-input multiple-output (MIMO) and multi-user (MU) systems. Four case studies are considered: two point-to-point (closed-loop and open-loop MIMO) and two MU scenarios (MIMO broadcast and interference channels). For the point-to-point scenarios, we explain some of the performance gains observed in prior work through the selection of improved baseline schemes that include geometric shaping as well as bit and power allocation. For the MIMO broadcast channel, we demonstrate the feasibility of a novel AE method with centralized learning and decentralized execution. Interestingly, the learned scheme performs close to nonlinear vector-perturbation precoding and significantly outperforms conventional zero-forcing. Lastly, we highlight potential pitfalls when interpreting learned communication schemes. In particular, we show that the AE for the considered interference channel learns to avoid interference, albeit in a rotated reference frame. After de-rotating the learned signal constellation of each user, the resulting scheme corresponds to conventional time sharing with geometric shaping.

We introduce a two-stage decimation process to improve the performance of neural belief propagation (NBP), recently introduced by Nachmani et al., for short low-density parity-check (LDPC) codes. In the first stage, we build a list by iterating between a conventional NBP decoder and guessing the least reliable bit. The second stage iterates between a conventional NBP decoder and learned decimation, where we use a neural network to decide the decimation value for each bit. For a (128,64) LDPC code, the proposed NBP with decimation outperforms NBP decoding by 0.75dB and performs within 1dB from maximum-likelihood decoding at a block error rate of 10−4.

We propose a new machine-learning approach for fiber-optic communication systems whose signal propagation is governed by the nonlinear Schrödinger equation (NLSE). Our main observation is that the popular split-step method (SSM) for numerically solving the NLSE has essentially the same functional form as a deep multi-layer neural network; in both cases, one alternates linear steps and pointwise nonlinearities. We exploit this connection by parameterizing the SSM and viewing the linear steps as general linear functions, similar to the weight matrices in a neural network. The resulting physics-based machine-learning model has several advantages over “black-box” function approximators. For example, it allows us to examine and interpret the learned solutions in order to understand why they perform well. As an application, low-complexity nonlinear equalization is considered, where the task is to efficiently invert the NLSE. This is commonly referred to as digital backpropagation (DBP). Rather than employing neural networks, the proposed algorithm, dubbed learned DBP (LDBP), uses the physics-based model with trainable filters in each step and its complexity is reduced by progressively pruning filter taps during gradient descent. Our main finding is that the filters can be pruned to remarkably short lengths-as few as 3 taps/step-without sacrificing performance. As a result, the complexity can be reduced by orders of magnitude in comparison to prior work. By inspecting the filter responses, an additional theoretical justification for the learned parameter configurations is provided. Our work illustrates that combining data-driven optimization with existing domain knowledge can generate new insights into old communications problems.

End-to-end data-driven machine learning (ML) of multiple-input multiple-output (MIMO) systems has been shown to have the potential of exceeding the performance of engineered MIMO transceivers, without any a priori knowledge of communication-theoretic principles. In this work, we aim to understand to what extent and for which scenarios this claim holds true when comparing with fair benchmarks. We study closed-loop MIMO, open-loop MIMO, and multi-user MIMO (MU-MIMO) and show that the gains of ML-based communication in the former two cases can be to a large extent ascribed to implicitly learned geometric shaping and bit and power allocation, not to learning new spatial encoders. For MU-MIMO, we demonstrate the feasibility of a novel method with centralized learning and decentralized executing, outperforming conventional zero-forcing. For each scenario, we provide explicit descriptions as well as open-source implementations of the selected neural-network architectures.

We consider near maximum-likelihood (ML) decoding of short linear block codes based on neural belief propagation (BP) decoding recently introduced by Nachmani et al.. While this method significantly outperforms conventional BP decoding, the underlying parity-check matrix may still limit the overall performance. In this paper, we introduce a method to tailor an overcomplete parity-check matrix to (neural) BP decoding using machine learning. We consider the weights in the Tanner graph as an indication of the importance of the connected check nodes (CNs) to decoding and use them to prune unimportant CNs. As the pruning is not tied over iterations, the final decoder uses a different parity-check matrix in each iteration. For ReedMuller and short low-density parity-check codes, we achieve performance within 0.27dB and 1.5dB of the ML performance while reducing the complexity of the decoder.

The recursive projection-aggregation (RPA) decoding algorithm for Reed-Muller (RM) codes was recently introduced by Ye and Abbe. We show that the RPA algorithm is closely related to (weighted) belief-propagation (BP) decoding by interpreting it as a message-passing algorithm on a factor graph with redundant code constraints. We use this observation to introduce a novel decoder tailored to high-rate RM codes. The new algorithm relies on puncturing rather than projections and is called recursive puncturing-aggregation (RXA). We also investigate collapsed (i.e., non-recursive) versions of RPA and RXA and show some examples where they achieve similar performance with lower decoding complexity.

Data-driven optimization of transmitters and receivers can reveal new modulation and detection schemes and enable physical-layer communication over unknown channels. Previous work has shown that practical implementations of this approach require a feedback signal from the receiver to the transmitter. In this paper, we study the impact of quantized feedback on data-driven learning of physical-layer communication. A novel quantization method is proposed, which exploits the specific properties of the feedback signal and is suitable for non-stationary signal distributions. The method is evaluated for linear and nonlinear channels. Simulation results show that feedback quantization does not appreciably affect the learning process and can lead to similar performance as compared to the case where unquantized feedback is used for training, even with 1-bit quantization. In addition, it is shown that learning is surprisingly robust to noisy feedback where random bit flips are applied to the quantization bits.

We consider the weighted belief-propagation (WBP) decoder recently proposed by Nachmani et al. where different weights are introduced for each Tanner graph edge and optimized using machine learning techniques. Our focus is on simple-scaling models that use the same weights across certain edges to reduce the storage and computational burden. The main contribution is to show that simple scaling with few parameters often achieves the same gain as the full parameterization. Moreover, several training improvements for WBP are proposed. For example, it is shown that minimizing average binary cross-entropy is suboptimal in general in terms of bit error rate (BER) and a new "soft-BER" loss is proposed which can lead to better performance. We also investigate parameter adapter networks (PANs) that learn the relation between the signal-to-noise ratio and the WBP parameters. As an example, for the (32, 16) Reed-Muller code with a highly redundant parity-check matrix, training a PAN with soft-BER loss gives near-maximum-likelihood performance assuming simple scaling with only three parameters.

An important problem in fiber-optic communications is to invert the nonlinear Schrödinger equation in real time to reverse the deterministic effects of the channel. Interestingly, the popular split-step Fourier method (SSFM) leads to a computation graph that is reminiscent of a deep neural network. This observation allows one to leverage tools from machine learning to reduce complexity. In particular, the main disadvantage of the SSFM is that its complexity using M steps is at least M times larger than a linear equalizer. This is because the linear SSFM operator is a dense matrix. In previous work, truncation methods such as frequency sampling, wavelets, or least-squares have been used to obtain “cheaper” operators that can be implemented using filters. However, a large number of filter taps are typically required to limit truncation errors. For example, Ip and Kahn showed that for a 10 Gbaud signal and 2000 km optical link, a truncated SSFM with 25 steps would require 70-tap filters in each step and 100 times more operations than linear equalization. We find that, by jointly optimizing all filters with deep learning, the complexity can be reduced significantly for similar accuracy. Using optimized 5-tap and 3-tap filters in an alternating fashion, one requires only around 2-6 times the complexity of linear equalization, depending on the implementation.

Reed-Muller (RM) codes exhibit good performance under maximum-likelihood (ML) decoding due to their highly-symmetric structure. In this paper, we explore the question of whether the code symmetry of RM codes can also be exploited to achieve near-ML performance in practice. The main idea is to apply iterative decoding to a highly-redundant parity-check (PC) matrix that contains only the minimum-weight dual codewords as rows. As examples, we consider the peeling decoder for the binary erasure channel, linear-programming and belief propagation (BP) decoding for the binary-input additive white Gaussian noise channel, and bit-flipping and BP decoding for the binary symmetric channel. For short block lengths, it is shown that near-ML performance can indeed be achieved in many cases. We also propose a method to tailor the PC matrix to the received observation by selecting only a small fraction of useful minimum-weight PCs before decoding begins. This allows one to both improve performance and significantly reduce complexity compared to using the full set of minimum-weight PCs.

Rapid improvements in machine learning over the past decade are beginning to have far-reaching effects. For communications, engineers with limited domain expertise can now use off-the-shelf learning packages to design high-performance systems based on simulations. Prior to the current revolution in machine learning, the majority of communication engineers were quite aware that system parameters (such as filter coefficients) could be learned using stochastic gradient descent. It was not at all clear, however, that more complicated parts of the system architecture could be learned as well. In this paper, we discuss the application of machine-learning techniques to two communications problems and focus on what can be learned from the resulting systems. We were pleasantly surprised that the observed gains in one example have a simple explanation that only became clear in hindsight. In essence, deep learning discovered a simple and effective strategy that had not been considered earlier.

Product codes (PCs) protect a 2-D array of bits using short component codes. Assuming transmission over the binary symmetric channel, the decoding is commonly performed by iteratively applying bounded-distance decoding to the component codes. For this coding scheme, undetected errors in the component decoding-also known as miscorrections-significantly degrade the performance. In this paper, we propose a novel iterative decoding algorithm for PCs which can detect and avoid most miscorrections. The algorithm can also be used to decode many recently proposed classes of generalized PCs, such as staircase, braided, and half-product codes. Depending on the component code parameters, our algorithm significantly outperforms the conventional iterative decoding method. As an example, for double-error-correcting Bose-Chaudhuri-Hocquenghem component codes, the net coding gain can be increased by up to 0.4 dB. Moreover, the error floor can be lowered by orders of magnitude, up to the point where the decoder performs virtually identical to a genie-aided decoder that avoids all miscorrections. We also discuss post-processing techniques that can be used to reduce the error floor even further.

Generalized product codes (GPCs) are extensions of product codes (PCs), where code symbols are protected by two component codes but not necessarily arranged in a rectangular array. We consider a deterministic construction of GPCs (as opposed to randomized code ensembles) and analyze the asymptotic performance over the binary erasure channel under iterative decoding. Our code construction encompasses several classes of GPCs previously proposed in the literature, such as irregular PCs, blockwise braided codes, and staircase codes. It is assumed that the component codes can correct a fixed number of erasures and that the length of each component code tends to infinity. We show that this setup is equivalent to studying the behavior of a peeling algorithm applied to a sparse inhomogeneous random graph. Using a convergence result for these graphs, we derive the density evolution equations that characterize the asymptotic decoding performance. As an application, we discuss the design of irregular GPCs, employing a mixture of component codes with different erasure-correcting capabilities.

Several authors have proposed spatially-coupled (or convolutional-like) variants of product codes (PCs). In this paper, we focus on a parametrized family of generalized PCs that recovers some of these codes (e.g., staircase and block-wise braided codes) as special cases and study the iterative decoding performance over the binary erasure channel. Even though our code construction is deterministic (and not based on a randomized ensemble), we show that it is still possible to rigorously derive the density evolution (DE) equations that govern the asymptotic performance. The obtained DE equations are then compared to those for a related spatially-coupled PC ensemble. In particular, we show that there exists a family of (deterministic) braided codes that follows the same DE equation as the ensemble, for any spatial length and coupling width.

We present a comprehensive study of the information rate loss of the max-log approximation for M-ary pulse-amplitude modulation (PAM) in a bit-interleaved coded modulation (BICM) system. It is widely assumed that the calculation of L-values using the max-log approximation leads to an information loss. We prove that this assumption is correct for all M-PAM constellations and labelings with the exception of a symmetric 4-PAM constellation labeled with a Gray code. We also show that for max-log L-values, the BICM generalized mutual information (GMI), which is an achievable rate for a standard BICM decoder, is too pessimistic. In particular, it is proved that the so-called harmonized GMI, which can be seen as the sum of bit-level GMIs, is achievable without any modifications to the decoder. We then study how bit-level channel symmetrization and mixing affect the MI and the GMI for max-log L-values. Our results show that these operations, which are often used when analyzing BICM systems, preserve the GMI. However, this is not necessarily the case when the MI is considered. Necessary and sufficient conditions under which these operations preserve the MI are provided.

In many practical communication systems, one binary encoder/decoder pair is used to communicate over a set of parallel channels. Examples of this setup include multi-carrier transmission, rate-compatible puncturing of turbo-like codes, and bit-interleaved coded modulation (BICM). A bit mapper is commonly employed to determine how the coded bits are allocated to the channels. In this paper, we study spatially coupled low-density parity check codes over parallel channels and optimize the bit mapper using BICM as the driving example. For simplicity, the parallel bit channels that arise in BICM are replaced by independent binary erasure channels (BECs). For two parallel BECs modeled according to a 4-PAM constellation labeled by the binary reflected Gray code, the optimization results show that the decoding threshold can be improved over a uniform random bit mapper, or, alternatively, the spatial chain length of the code can be reduced for a given gap to capacity. It is also shown that for rate-loss free, circular (tail-biting) ensembles, a decoding wave effect can be initiated using only an optimized bit mapper.

We study the design of amplitude phase-shift keying (APSK) constellations for a coherent fiber-optical communication system where nonlinear phase noise (NLPN) is the main system impairment. APSK constellations can be regarded as a union of phase-shift keying (PSK) signal sets with different amplitude levels. A practical two-stage (TS) detection scheme is analyzed, which performs close to optimal detection for high enough input power. We optimize APSK constellations with 4, 8, and 16 points in terms of symbol error probability (SEP) under TS detection for several combinations of input power and fiber length. For 16 points, performance gains of 3.2 dB can be achieved at a SEP of 10^{-2} compared to 16-QAM by choosing an optimized APSK constellation. We also demonstrate that in the presence of severe nonlinear distortions, it may become beneficial to sacrifice a constellation point or an entire constellation ring to reduce the average SEP. Finally, we discuss the problem of selecting a good binary labeling for the found constellations.

We consider the design of amplitude phase-shift keying (APSK) constellations, targeting their application to coherent fiber-optical communications. Phase compensation is used at the receiver to combat nonlinear phase noise caused by the Kerreffect. We derive the probability density function of the post-compensated observation for multilevel constellations. Optimal APSK constellations in terms of symbol error probability (SEP) are found assuming a two-stage detector. Performance gains of 3:2 dB can be achieved compared to 16-QAM at a SEP of 10-2. We optimize the number of rings, the number of points per ring, as well as the radius distribution of the constellation. For low to moderate nonlinearities, radius optimization only yields minor improvements over an equidistant spacing of rings. In the highly nonlinear regime, however, a smaller SEP can be achieved by “sacrificing” the outer ring of the constellation, in favor of achieving good SEP in the remaining rings.