Learning Joint Detection, Equalization and Decoding for Short-Packet CommunicationsSebastian Dörner, Jannis Clausius, Sebastian Cammerer, Stephan ten Brink, published in journal IEEE Transactions on Communications, vol 71, no 2, pp 837-850, 2023 [doi]
Abstract–We propose and practically demonstrate a joint detection and decoding scheme for short-packet wireless communications in scenarios that require to first detect the presence of a message before actually decoding it. For this, we extend the recently proposed serial Turbo-autoencoder neural network (NN) architecture and train it to find short messages that can be, all “at once”, detected, synchronized, equalized and decoded when sent over an unsynchronized channel with memory. The conceptional advantage of the proposed system stems from a holistic message structure with superimposed pilots for joint detection and decoding without the need of relying on a dedicated preamble. This results not only in a higher spectral efficiency, but also translates into the possibility of shorter messages compared to using a dedicated preamble. We compare the detection error rate (DER), bit error rate (BER) and block error rate (BLER) performance of the proposed system with a hand-crafted state-of-the-art conventional baseline and our simulations show a significant advantage of the proposed autoencoder-based system over the conventional baseline in every scenario up to messages conveying k =96 information bits. Finally, we practically evaluate and confirm the improved performance of the proposed system over-the-air (OTA) using a software-defined radio (SDR)-based measurement testbed.
Deep Reinforcement Learning for mmWave Initial Beam AlignmentDaniel Tandler, Sebastian Dörner, Marc Gauger, Stephan ten Brink, published in proceedings of WSA & SCC 2023; 26th International ITG Workshop on Smart Antennas and 13th Conference on Systems, Communications, and Coding, pp 1-6, 2023 [doi]
Abstract–We investigate the applicability of deep reinforcement learning algorithms to the adaptive initial access beam alignment problem for mmWave communications using the state-of-the-art proximal policy optimization algorithm as an example. In comparison to recent unsupervised learning based approaches developed to tackle this problem, deep reinforcement learning has the potential to address a new and wider range of applications, since, in principle, no (differentiable) model of the channel and/or the whole system is required for training, and only agent-environment interactions are necessary to learn an algorithm (be it online or using a recorded dataset). We show that, although the chosen off-the-shelf deep reinforcement learning agent fails to perform well when trained on realistic problem sizes, introducing action space shaping in the form of beamforming modules vastly improves the performance, without sacrificing much generalizability. Using this add-on, the agent is able to deliver competitive performance to various state-of-the-art methods on simulated environments, even under realistic problem sizes. This demonstrates that through well-directed modification, deep reinforcement learning may have a chance to compete with other approaches in this area, opening up many straightforward extensions to other/similar scenarios.
Adaptive Neural Network-based OFDM ReceiversMoritz Benedikt Fischer, Sebastian Dörner, Sebastian Cammerer, Takayuki Shimizu, Hongsheng Lu, Stephan ten Brink, published in proceedings of 2022 IEEE 23rd International Workshop on Signal Processing Advances in Wireless Communication (SPAWC), 2022 [doi]
Abstract–We propose and examine the idea of continuously adapting state-of-the-art neural network (NN)-based orthogonal frequency division multiplex (OFDM) receivers to current channel conditions. This online adaptation via retraining is mainly motivated by two reasons: First, receiver design typically focuses on the universal optimal performance for a wide range of possible channel realizations. However, in actual applications and within short time intervals, only a subset of these channel parameters is likely to occur, as macro parameters, e.g., the maximum channel delay, can assumed to be static. Second, in-the-field alterations like temporal interferences or other conditions out of the originally intended specifications can occur on a practical (real-world) transmission. While conventional (filter-based) systems would require reconfiguration or additional signal processing to cope with these unforeseen conditions, NN-based receivers can learn to mitigate previously unseen effects even after their deployment. For this, we showcase on-the-fly adaption to current channel conditions and temporal alterations solely based on recovered labels from an outer forward error correction (FEC) code without any additional piloting overhead. To underline the flexibility of the proposed adaptive training, we showcase substantial gains for scenarios with static channel macro parameters, for out-of-specification usage and for interference compensation.
Adaptive NN-based OFDM Receivers: Computational Complexity vs. Achievable PerformanceMoritz Benedikt Fischer, Sebastian Dörner, Felix Krieg, Sebastian Cammerer, Stephan ten Brink, published in proceedings of 2022 56th Asilomar Conference on Signals, Systems, and Computers, pp 194-199, 2022 [doi]
Abstract–We revisit the design and retraining capabilities of neural network (NN)-based orthogonal frequency division multiplex (OFDM) receivers that combine channel estimation, equalization and soft-demapping for time-varying and frequencyselective wireless channels. Attracted by the inherent advantages of small NNs in terms of computational complexity during inference and (re-)training, we first analyze the performance of different neural receiver architectures, including versions with reduced complexity. We observe, that such neural receivers with reduced complexity show an expected but graceful degradation in terms of their achievable bit error rate (BER) performance. Further, we focus on the adaptive retraining of site-specific NN-based receivers and show that performance losses due to a reduced number of parameters can be compensated by a continuous retraining of the receiver on-the-fly and that generalization can be achieved through adaptivity. Finally, we propose an improved retraining process via data augmentation and demonstrate a better performance and faster adaptation to current channel conditions.
Introducing γ-lifting for Learning Nonlinear Pulse Shaping in Coherent Optical CommunicationTim Uhlemann, Alexander Span, Sebastian Dörner, Stephan ten Brink, published in proceedings of Photonic Networks; 23th ITG-Symposium, pp 1-8, 2022 [doi]
Abstract–Pulse shaping for coherent optical fiber communication has been an active area of research for the past decade. Most of the early schemes are based on classic Nyquist pulse shaping that was originally intended for linear channels. The best known classic scheme, the split digital back propagation, uses joint predistortion and post-equalization and hence, a nonlinear transmitter; it, however, suffers from spectral broadening on the fiber due to the Kerr-effect. With the advent of deep learning in communications, it has been realized that an “autoencoder” can learn to communicate efficiently over the optical fiber channel, jointly optimizing geometric constellations and pulse shaping – while also taking into account linear and nonlinear impairments such as chromatic dispersion and Kerr-nonlinearity. E.g., shows how an autoencoder can learn to mitigate spectral broadening due to the Kerr-effect using a trainable linear transmitter. In this paper, we extend this linear architectural template to a scalable nonlinear pulse shaping consisting of a convolutional neural network at both transmitter and receiver. By introducing a novel gamma-lifting training procedure tailored to the nonlinear optical fiber channel, we achieve stable autoencoder convergence to pulse shapes reaching information rates outperforming the classic split digital back propagation reference at high input powers.
FPGA-Based Trainable Autoencoder for Communication SystemsJonas Ney, Sebastian Dörner, Matthias Herrmann, Mohammad Hassani Sadi, Jannis Clausius, Stephan ten Brink, Norbert Wehn, published in proceedings of Proceedings of the 2022 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays, pp 154, 2022 [doi]
Abstract–In communication systems, autoencoder refers to a system that replaces parts of the traditional transmitter and receiver of the baseband processing chain with artificial neural networks (ANNs). This allows to jointly train the system for an underlying channel model by reconstructing the input symbols at the output. Since the actual behavior of a real communication channel cannot be perfectly reproduced by an abstract model, it is necessary for the autoencoder to adapt to the changing conditions at runtime. Thus, online fine-tuning, in the form of ANN-retraining is of great importance. A platform able to satisfy the low-latency and low-power requirements of embedded communication systems are Field-programmable gate arrays (FPGAs). In this paper, we present an online-trainable low-power FPGA architecture for the receiver of an autoencoder-based communication chain. The architecture is embedded into an exploration framework that automatically determines the optimal degree of parallelism to minimize latency or power consumption. Our solutions achieve 2000×higher throughput than a high-performance GPU, draw 5×less power than an embedded CPU and are 5800×more energy efficient compared to an embedded GPU, for a batch size of one. To the best of our knowledge, this is the first FPGA-based autoencoder implementation for communication systems.
Deep Learning for Uplink CSI-based Downlink Precoding in FDD massive MIMO Evaluated on Indoor MeasurementsFlorian Euchner, Niklas Süppel, Marc Gauger, Sebastian Dörner, Stephan ten Brink, published in proceedings of 2022 30th European Signal Processing Conference (EUSIPCO), pp 653-657, 2022 [doi]
Abstract–When operating massive multiple-input multiple-output (MIMO) systems with uplink (UL) and downlink (DL) channels at different frequencies (frequency division duplex (FDD) operation), acquisition of channel state information (CSI) for downlink precoding is a major challenge. Since, barring transceiver impairments, both UL and DL CSI are determined by the physical environment surrounding transmitter and receiver, it stands to reason that, for a static environment, a mapping from UL CSI to DL CSI may exist. First, we propose to use various neural network (NN)-based approaches that learn this mapping and provide baselines using classical signal processing. Second, we introduce a scheme to evaluate the performance and quality of generalization of all approaches, distinguishing between known and previously unseen physical locations. Third, we evaluate all approaches on a real-world indoor dataset collected with a 32-antenna channel sounder.
Improving Triplet-Based Channel Charting on Distributed Massive MIMO MeasurementsFlorian Euchner, Phillip Stephan, Marc Gauger, Sebastian Dörner, Stephan ten Brink, published in proceedings of 2022 IEEE 23rd International Workshop on Signal Processing Advances in Wireless Communication (SPAWC), 2022 [doi]
Abstract–The objective of channel charting is to learn a virtual map of the radio environment from high-dimensional channel state information (CSI) that is acquired by a multi-antenna wireless system. Since, in static environments, CSI is a function of the transmitter location, a mapping from CSI to channel chart coordinates can be learned in a self-supervised manner using dimensionality reduction techniques. The state-of-the-art triplet-based approach is evaluated on multiple datasets measured by a distributed massive multiple-input multiple-output (MIMO) channel sounder, with both co-located and distributed antenna setups. The importance of suitable triplet selection is investigated by comparing results to channel charts learned from a genie-aided triplet generator and learned from triplets on simulated trajectories through measured data. Finally, the transferability of learned forward charting functions to similar, but different radio environments is explored.
On End-to-End Learning of Joint Detection and Decoding for Short-Packet CommunicationsJannis Clausius, Sebastian Dörner, Sebastian Cammerer, Stephan Ten Brink, published in proceedings of 2022 IEEE Globecom Workshops (GC Wkshps), pp 377-382, 2022 [doi]
Abstract–We propose a joint detection and decoding scheme for short-packet wireless communications in scenarios that require to first detect the presence of a message before actually decoding it. For this, we extend the recently proposed serial Turbo-autoencoder neural network (NN) architecture and train it to find short messages that can be, all “at once detected, synchronized, equalized and decoded when sent over an unsynchronized channel with memory. The conceptional advantage of the proposed system stems from a holistic message structure with superimposed pilots for joint detection and decoding without the need of relying on a dedicated preamble. This results not just in a higher spectral efficiency, but also translates into shorter messages and potentially lower latency when compared to the usage of a dedicated preamble. We compare the detection error rate (DER), bit error rate (BER) and block error rate (BLER) performance of the proposed system with a hand-crafted state-of-the-art conventional baseline. Our performance simulations show a significant advantage of the proposed autoencoder-based system over the conventional baseline in every scenario up to messages conveying k=96 information bits.
Bit-wise Autoencoder for Multiple Antenna SystemsSebastian Dörner, Sarah Rottacker, Marc Gauger, Stephan ten Brink, published in proceedings of 2021 17th International Symposium on Wireless Communication Systems (ISWCS), pp 1-5, 2021 [doi]
Abstract–We propose an end-to-end learned bit-wise autoen-coder neural network (NN) for open loop multiple-input multiple-output (MIMO) antenna based communication systems. The optimized transmit vector constellations are learned based on the number of transmit and receive antennas, the number of bits conveyed per channel use and the signal-to-noise ratio. By training through an i.i.d. complex Gaussian (i.e., Rayleigh ergodic) matrix channel, the system implicitly picks up constellation shaping gains “along the way”. We evaluate and analyze these gains in comparison to the single-input single-output (SISO) results shown in [1] for different symmetric and asymmetric MIMO configurations. We first show that the NN-based receiver is able to compete with the a posteriori probability (APP) receiver performance up to certain configuration settings. Then we perform an end-to-end optimization of the transmit vector constellations using the conventional APP receiver as the decoder part. Thereby, we also investigate an edge case where the number of bits per vector symbol is not a multiple of the number of transmit antennas. Finally, we examine the performance of the system if a non-optimal zero forcing (ZF) receiver is used for inference, while the vector constellation was optimized for the APP receiver during training. We show that constellations optimized for the APP receiver are not necessarily optimal for such sub-optimal receivers that operate with reduced complexity. To fix this, we propose a detector-aware training scheme to learn constellations that have been optimized for a specific sub-optimal receiver and, thus, achieve higher performance in inference.
Wiener Filter versus Recurrent Neural Network-based 2D-Channel Estimation for V2X CommunicationsMoritz Benedikt Fischer, Sebastian Dörner, Sebastian Cammerer, Takayuki Shimizu, Bin Cheng, Hongsheng Lu, Stephan ten Brink, published in proceedings of 2021 IEEE Intelligent Vehicles Symposium (IV), pp 458-465, 2021 [doi]
Abstract–We compare the potential of neural network (NN)-based channel estimation with classical linear minimum mean square error (LMMSE)-based estimators, also known as Wiener filtering. For this, we propose a low-complexity recurrent neural network (RNN)-based estimator that allows channel equalization of a sequence of channel observations based on independent time- and frequency-domain long short-term memory (LSTM) cells. Motivated by Vehicle-to-Everything (V2X) applications, we simulate time- and frequency-selective channels with orthogonal frequency division multiplex (OFDM) and extend our channel models in such a way that a continuous degradation from line-of-sight (LoS) to non-line-of-sight (NLoS) conditions can be emulated. It turns out that the NN-based system cannot just compete with the LMMSE equalizer, but it also can be trained w.r.t. resilience against system parameter mismatch. We thereby showcase the conceptual simplicity of such a data-driven system design, as this not only enables more robustness against, e.g., signal-to-noise-ratio (SNR) or Doppler spread estimation mismatches, but also allows to use the same equalizer over a wider range of input parameters without the need of re-building (or re-estimating) the filter coefficients. Particular attention has been paid to ensure compatibility with the existing IEEE 802.11p piloting scheme for V2X communications. Finally, feeding the payload data symbols as additional equalizer input unleashes further performance gains. We show significant gains over the conventional LMMSE equalization for highly dynamic channel conditions if such a data-augmented equalization scheme is used.
A Distributed Massive MIMO Channel Sounder for "Big CSI Data"-driven Machine LearningFlorian Euchner, Marc Gauger, Sebastian Dörner, Stephan ten Brink, published in proceedings of 25th International ITG Workshop on Smart Antennas (WSA 2021), 2021 [doi]
Abstract–A distributed massive MIMO channel sounder for acquiring large CSI datasets, dubbed DICHASUS, is presented. The measured data has potential applications in the study of different machine learning algorithms for user localization, JCAS, channel charting, enabling massive MIMO in FDD operation, and many others. The proposed channel sounder architecture is distinct from similar previous designs in that each individual single-antenna receiver is completely autonomous, enabling arbitrary grouping into spatially distributed antenna deployments, and offering virtually unlimited scalability in the number of antennas. Optionally, extracted channel coefficient vectors can be tagged with ground truth position data, obtained either through a GNSS receiver (for outdoor operation) or through various indoor positioning techniques.
Serial vs. Parallel Turbo-Autoencoders and Accelerated Training for Learned Channel CodesJannis Clausius, Sebastian Dörner, Sebastian Cammerer, Stephan ten Brink, published in proceedings of 2021 11th International Symposium on Topics in Coding (ISTC), pp 1-5, 2021 [doi]
Abstract–Attracted by its scalability towards practical code-word lengths, we revisit the idea of Turbo-autoencoders for end-to-end learning of PHY-Layer communications. For this, we study the existing concepts of Turbo-autoencoders from the literature and compare the concept with state-of-the-art classical coding schemes. We propose a new component-wise training algorithm based on the idea of Gaussian a priori distributions that reduces the overall training time by almost a magnitude. Further, we propose a new serial architecture inspired by classical serially concatenated Turbo code structures and show that a carefully optimized interface between the two component autoencoders is required. To the best of our knowledge, these serial Turbo autoencoder structures are the best known neural network based learned sequences that can be trained from scratch without any required expert knowledge in the domain of channel codes.
Extended Abstract: Deep Learning of the Physical Layer for BICM SystemsFayçal Ait Aoudia, Sebastian Cammerer, Sebastian Dörner, Maximilian Stark, Jakob Hoydis, Stephan ten Brink, published in proceedings of 2020 IEEE Workshop on Signal Processing Systems (SiPS), pp 1-1, 2020 [doi]
Abstract–We demonstrate that training of autoencoder-based communication systems on the bit-wise mutual information allows seamless integration with practical bit metric decoding receivers, as well as joint optimization of constellation shaping and labeling. Additionally, we present a fully differentiable neural iterative demapping and decoding structure which achieves significant gains on additive white Gaussian noise channels. Going one step further, we show that careful code design can lead to further performance improvements. Finally, we implement the end-to-end system on software-defined radio and train it on the actual channel.
Trainable Communication Systems: Concepts and PrototypeSebastian Cammerer, Fayçal Ait Aoudia, Sebastian Dörner, Maximilian Stark, Jakob Hoydis, Stephan ten Brink, published in journal IEEE Transactions on Communications, vol 68, no 9, pp 5489-5503, 2020 [doi]
Abstract–We consider a trainable point-to-point communication system, where both transmitter and receiver are implemented as neural networks (NNs), and demonstrate that training on the bit-wise mutual information (BMI) allows seamless integration with practical bit-metric decoding (BMD) receivers, as well as joint optimization of constellation shaping and labeling. Moreover, we present a fully differentiable neural iterative demapping and decoding (IDD) structure which achieves significant gains on additive white Gaussian noise (AWGN) channels using a standard 802.11n low-density parity-check (LDPC) code. The strength of this approach is that it can be applied to arbitrary channels without any modifications. Going one step further, we show that careful code design can lead to further performance improvements. Lastly, we show the viability of the proposed system through implementation on software-defined radios (SDRs) and training of the end-to-end system on the actual wireless channel. Experimental results reveal that the proposed method enables significant gains compared to conventional techniques.
Deep-learning Autoencoder for Coherent and Nonlinear Optical CommunicationTim Uhlemann, Sebastian Cammerer, Alexander Span, Sebastian Dörner, Stephan ten Brink, published in proceedings of Photonic Networks 21th ITG-Symposium, pp 1-8, 2020 [doi]
Abstract–Motivated by the recent success of end-to-end training of communications in the wireless domain, we strive to adapt the end-to-end-learning idea from the wireless case (i.e., linear) to coherent optical fiber links (i.e., nonlinear). Although, at first glance, it sounds like a straightforward extension, it turns out that several pitfalls exist – in terms of theory but also in terms of practical implementation. This paper analyzes an autoencoder’s potential and limitations for the optical fiber under the influence of Kerr-nonlinearity and chromatic dispersion. As there is no exact capacity limit known and, hence, no analytical perfect system solution available, we set great value to the interpretability on the learnings of the autoencoder. Therefore, we design its architecture to be as close as possible to the structure of a classic communication system, knowing that this may limit its degree of freedom and, thus, its performance. Nevertheless, we were able to achieve an unexpected high gain in terms of spectral efficiency compared to a conventional reference system.
WGAN-based Autoencoder Training Over-the-airSebastian Dörner, Marcus Henninger, Sebastian Cammerer, Stephan ten Brink, published in proceedings of 2020 IEEE 21st International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), pp 1-5, 2020 [doi]
Abstract–The practical realization of end-to-end training of communication systems is fundamentally limited by its accessibility of the channel gradient. To overcome this major burden, the idea of generative adversarial networks (GANs) that learn to mimic the actual channel behavior has been recently proposed in the literature. Contrarily to handcrafted classical channel modeling, which can never fully capture the real world, GANs promise, in principle, the ability to learn any physical impairment, enabled by the data-driven learning algorithm. In this work, we verify the concept of GAN-based autoencoder training in actual over-the-air (OTA) measurements. To improve training stability, we first extend the concept to conditional Wasserstein GANs and embed it into a state-of-the-art autoencoder-architecture, including bit-wise estimates and an outer channel code. Further, in the same framework, we compare the existing three different training approaches: model-based pre-training with receiver finetuning, reinforcement learning (RL) and GAN-based channel modeling. For this, we show advantages and limitations of GAN-based end-to-end training. In particular, for non-linear effects, it turns out that learning the whole exploration space becomes prohibitively complex. Finally, we show that the training strategy benefits from a simpler (training) data acquisition when compared to RL-based training, which requires continuous transmitter weight updates. This becomes an important practical bottleneck due to limited bandwidth and latency between transmitter and training algorithm that may even operate at physically different locations.
Deep-learning Autoencoder for Coherent and Nonlinear Optical CommunicationTim Uhlemann, Sebastian Cammerer, Alexander Span, Sebastian Doerner, Stephan ten Brink, published in proceedings of Photonic Networks; 21th ITG-Symposium, pp 1-8, 2020 [doi]
Abstract–Motivated by the recent success of end-to-end training of communications in the wireless domain, we strive to adapt the end-to-end-learning idea from the wireless case (i.e., linear) to coherent optical fiber links (i.e., nonlinear). Although, at first glance, it sounds like a straightforward extension, it turns out that several pitfalls exist – in terms of theory but also in terms of practical implementation. This paper analyzes an autoencoder’s potential and limitations for the optical fiber under the influence of Kerr-nonlinearity and chromatic dispersion. As there is no exact capacity limit known and, hence, no analytical perfect system solution available, we set great value to the interpretability on the learnings of the autoencoder. Therefore, we design its architecture to be as close as possible to the structure of a classic communication system, knowing that this may limit its degree of freedom and, thus, its performance. Nevertheless, we were able to achieve an unexpected high gain in terms of spectral efficiency compared to a conventional reference system.
Towards Practical Indoor Positioning Based on Massive MIMO SystemsMark Widmaier, Maximilian Arnold, Sebastian Dörner, Sebastian Cammerer, Stephan ten Brink, published in proceedings of 2019 IEEE 90th Vehicular Technology Conference (VTC2019-Fall), pp 1-6, 2019 [doi]
Abstract–We showcase the practicability of an indoor positioning system (IPS) solely based on neural networks (NNs) and the channel state information (CSI) of a (Massive) multiple-input multiple-output (MIMO) communication system, i.e., only build on the basis of data that is already existent in today's systems. As such our IPS system promises both, a good accuracy without the need of any additional protocol/signaling overhead for the user localization task. In particular, we propose a tailored NN structure with an additional phase branch as feature extractor and (compared to previous results) a significantly reduced amount of trainable parameters, leading to a minimization of the amount of required training data. We provide actual measurements for indoor scenarios with up to 64 antennas covering a large area of 80m 2 . In the second part, several robustness investigations for real-measurements are conducted, i.e., once trained, we analyze the recall accuracy over a period of several days. Further, we analyze the impact of pedestrians walking in-between the measurements and show that finetuning and pre-training of the NN helps to mitigate effects of hardware drifts and alterations in the propagation environment over time. This reduces the amount of required training samples at equal precision and, thereby, decreases the effort of the costly training data acquisition.
On Recurrent Neural Networks for Sequence-based Processing in CommunicationsDaniel Tandler, Sebastian Dörner, Sebastian Cammerer, Stephan ten Brink, published in proceedings of 2019 53rd Asilomar Conference on Signals, Systems, and Computers, pp 537-543, 2019 [doi]
Abstract–In this work, we analyze the capabilities and practical limitations of neural networks (NNs) for sequence-based signal processing which can be seen as an omnipresent property in almost any modern communication systems. In particular, we train multiple state-of-the-art recurrent neural network (RNN) structures to learn how to decode convolutional codes allowing a clear benchmarking with the corresponding maximum likelihood (ML) Viterbi decoder. We examine the decoding performance for various kinds of NN architectures, beginning with classical types like feedforward layers and gated recurrent unit (GRU)-layers, up to more recently introduced architectures such as temporal convolutional networks (TCNs) and differentiable neural computers (DNCs) with external memory. As a key limitation, it turns out that the training complexity increases exponentially with the length of the encoding memory ν and, thus, practically limits the achievable bit error rate (BER) performance. To overcome this limitation, we introduce a new training-method by gradually increasing the number of ones within the training sequences, i.e., we constrain the amount of possible training sequences in the beginning until first convergence. By consecutively adding more and more possible sequences to the training set, we finally achieve training success in cases that did not converge before via naive training. Further, we show that our network can learn to jointly detect and decode a quadrature phase shift keying (QPSK) modulated code with sub-optimal (anti-Gray) labeling in one-shot at a performance that would require iterations between demapper and decoder in classic detection schemes.
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