Laurent Schmalen

Vincent Lauinger, Fred Buchali, Laurent Schmalen

We investigate the potential of adaptive blind equalizers based on variational inference for carrier recovery in optical communications. These equalizers are based on a low-complexity approximation of maximum likelihood channel estimation. We generalize the concept of variational autoencoder (VAE) equalizers to higher order modulation formats encompassing probabilistic constellation shaping (PCS), ubiquitous in optical communications, oversampling at the receiver, and dual-polarization transmission. Besides black-box equalizers based on convolutional neural networks, we propose a model-based equalizer based on a linear butterfly filter and train the filter coefficients using the variational inference paradigm. As a byproduct, the VAE also provides a reliable channel estimation. We analyze the VAE in terms of performance and flexibility over a classical additive white Gaussian noise (AWGN) channel with inter-symbol interference (ISI) and over a dispersive linear optical dual-polarization channel. We show that it can extend the application range of blind adaptive equalizers by outperforming the state-of-the-art constant-modulus algorithm (CMA) for PCS for both fixed but also time-varying channels. The evaluation is accompanied with a hyperparameter analysis.

Vincent Lauinger, Manuel Hoffmann, Jonas Ney et al.

Due to the rapid development of autonomous driving, the Internet of Things and streaming services, modern communication systems have to cope with varying channel conditions and a steadily rising number of users and devices. This, and the still rising bandwidth demands, can only be met by intelligent network automation, which requires highly flexible and blind transceiver algorithms. To tackle those challenges, we propose a novel adaptive equalization scheme, which exploits the prosperous advances in deep learning by training an equalizer with an adversarial network. The learning is only based on the statistics of the transmit signal, so it is blind regarding the actual transmit symbols and agnostic to the channel model. The proposed approach is independent of the equalizer topology and enables the application of powerful neural network based equalizers. In this work, we prove this concept in simulations of different -- both linear and nonlinear -- transmission channels and demonstrate the capability of the proposed blind learning scheme to approach the performance of non-blind equalizers. Furthermore, we provide a theoretical perspective and highlight the challenges of the approach.

Jonas Ney, Vincent Lauinger, Laurent Schmalen et al.

In recent years, communication engineers put strong emphasis on artificial neural network (ANN)-based algorithms with the aim of increasing the flexibility and autonomy of the system and its components. In this context, unsupervised training is of special interest as it enables adaptation without the overhead of transmitting pilot symbols. In this work, we present a novel ANN-based, unsupervised equalizer and its trainable field programmable gate array (FPGA) implementation. We demonstrate that our custom loss function allows the ANN to adapt for varying channel conditions, approaching the performance of a supervised baseline. Furthermore, as a first step towards a practical communication system, we design an efficient FPGA implementation of our proposed algorithm, which achieves a throughput in the order of Gbit/s, outperforming a high-performance GPU by a large margin.

Eike-Manuel Bansbach, Alexander von Bank, Laurent Schmalen

In the past years, artificial neural networks (ANNs) have become the de-facto standard to solve tasks in communications engineering that are difficult to solve with traditional methods. In parallel, the artificial intelligence community drives its research to biology-inspired, brain-like spiking neural networks (SNNs), which promise extremely energy-efficient computing. In this paper, we investigate the use of SNNs in the context of channel equalization for ultra-low complexity receivers. We propose an SNN-based equalizer with a feedback structure akin to the decision feedback equalizer (DFE). For conversion of real-world data into spike signals we introduce a novel ternary encoding and compare it with traditional log-scale encoding. We show that our approach clearly outperforms conventional linear equalizers for three different exemplary channels. We highlight that mainly the conversion of the channel output to spikes introduces a small performance penalty. The proposed SNN with a decision feedback structure enables the path to competitive energy-efficient transceivers.

Jonathan Mandelbaum, Paul Bezner, Holger Jäkel et al.

In the short block length regime, ensemble decoding schemes with their inherently parallel structure can improve error correction performance and reduce latency compared to stand-alone suboptimal decoders such as belief propagation (BP). In this work, we introduce affine subcode ensemble decoding (aSCED), which uses an ensemble of decoders operating on linear block codes and both linear and strictly affine subcodes. This generalizes the recently proposed subcode ensemble decoding (SCED), which is restricted to linear subcodes. We derive BP update rules for affine subcodes and show that aSCED simplifies ensemble design compared to SCED, multiple bases BP, and automorphism ensemble decoding. Monte-Carlo simulations of two low-density parity-check codes and two Bose-Chaudhuri-Hocquenghem (BCH) codes demonstrate improved error correction performance of aSCED over competing existing ensemble schemes. Notably, for one BCH code, when combining ensemble design with algorithms for constructing high-performance parity-check matrices, aSCED achieves near-maximum likelihood performance using only 64 BP decoding paths.

Vincent Lauinger, Fred Buchali, Laurent Schmalen

We evaluate the start-up of blind equalizers at critical working points, analyze the advantages and obstacles of commonly-used algorithms, and demonstrate how the recently-proposed variational autoencoder (VAE) based equalizers can improve bootstrapping.

Alexander von Bank, Eike-Manuel Edelmann, Laurent Schmalen

A spiking neural network (SNN) equalizer with a decision feedback structure is applied to an IM/DD link with various parameters. The SNN outperforms linear and artificial neural network (ANN) based equalizers.

Luca Schmid, Tomer Raviv, Nir Shlezinger et al.

We study iterative blind symbol detection for block-fading linear inter-symbol interference channels. Based on the factor graph framework, we design a joint channel estimation and detection scheme that combines the expectation maximization (EM) algorithm and the ubiquitous belief propagation (BP) algorithm. Interweaving the iterations of both schemes significantly reduces the EM algorithm's computational burden while retaining its excellent performance. To this end, we apply simple yet effective model-based learning methods to find a suitable parameter update schedule by introducing momentum in both the EM parameter updates as well as in the BP message passing. Numerical simulations verify that the proposed method can learn efficient schedules that generalize well and even outperform coherent BP detection in high signal-to-noise scenarios.

Luca Schmid, Laurent Schmalen

We study the application of the factor graph framework for symbol detection on linear inter-symbol interference channels. Cyclic factor graphs have the potential to yield low-complexity symbol detectors, but are suboptimal if the ubiquitous sum-product algorithm is applied. In this paper, we present and evaluate strategies to improve the performance of cyclic factor graph-based symbol detection algorithms by means of neural enhancement. In particular, we apply neural belief propagation as an effective way to counteract the effect of cycles within the factor graph. We further propose the application and optimization of a linear preprocessor of the channel output. By modifying the observation model, the preprocessing can effectively change the underlying factor graph, thereby significantly improving the detection performance as well as reducing the complexity.

Paul Bezner, Erdem Eray Cil, Jannis Clausius et al.

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.

Luca Schmid, Joshua Brenk, Laurent Schmalen

Message passing on factor graphs is a powerful framework for probabilistic inference, which finds important applications in various scientific domains. The most wide-spread message passing scheme is the sum-product algorithm (SPA) which gives exact results on trees but often fails on graphs with many small cycles. We search for an alternative message passing algorithm that works particularly well on such cyclic graphs. Therefore, we challenge the extrinsic principle of the SPA, which loses its objective on graphs with cycles. We further replace the local SPA message update rule at the factor nodes of the underlying graph with a generic mapping, which is optimized in a data-driven fashion. These modifications lead to a considerable improvement in performance while preserving the simplicity of the SPA. We evaluate our method for two classes of cyclic graphs: the 2x2 fully connected Ising grid and factor graphs for symbol detection on linear communication channels with inter-symbol interference. To enable the method for large graphs as they occur in practical applications, we develop a novel loss function that is inspired by the Bethe approximation from statistical physics and allows for training in an unsupervised fashion.

Lukas Rapp, Luca Schmid, Andrej Rode et al.

We propose a novel method to optimize the structure of factor graphs for graph-based inference. As an example inference task, we consider symbol detection on linear inter-symbol interference channels. The factor graph framework has the potential to yield low-complexity symbol detectors. However, the sum-product algorithm on cyclic factor graphs is suboptimal and its performance is highly sensitive to the underlying graph. Therefore, we optimize the structure of the underlying factor graphs in an end-to-end manner using machine learning. For that purpose, we transform the structural optimization into a clustering problem of low-degree factor nodes that incorporates the known channel model into the optimization. Furthermore, we study the combination of this approach with neural belief propagation, yielding near-maximum a posteriori symbol detection performance for specific channels.

Eike-Manuel Bansbach, Yigit Kiyak, Laurent Schmalen

For orthogonal multiple access (OMA) systems, the number of served user equipments (UEs) is limited to the number of available orthogonal resources. On the other hand, non-orthogonal multiple access (NOMA) schemes allow multiple UEs to use the same orthogonal resource. This extra degree of freedom introduces new challenges for resource allocation. Buffer state information (BSI), like the size and age of packets waiting for transmission, can be used to improve scheduling in OMA systems. In this paper, we investigate the impact of BSI on the performance of a centralized scheduler in an uplink multi-carrier NOMA scenario with UEs having various data rate and latency requirements. To handle the large combinatorial space of allocating UEs to the resources, we propose a novel scheduler based on actor-critic reinforcement learning incorporating BSI. Training and evaluation are carried out using Nokia's "wireless suite". We propose various novel techniques to both stabilize and speed up training. The proposed scheduler outperforms benchmark schedulers.

Marco Origlia, Erdem Eray Cil, Laurent Schmalen et al.

We recently introduced a reverse reconciliation scheme with soft information. In this paper, we assess its performance at ultra-low SNR, thus proving that such scheme is a versatile solution to the reverse reconciliation problem.

Leo Wursthorn, Jonathan Mandelbaum, Sisi Miao et al.

Quantum low-density parity-check codes are promising candidates for low-overhead fault-tolerant quantum computing, but degeneracy is known to impair the convergence of belief-propagation (BP) decoding of these codes. In this work, we show that appending linearly independent rows to a check matrix of a stabilizer code can reduce the search space for a valid degenerate solution. Motivated by this, we extend the recently proposed affine subcode ensemble decoding technique from the classical to the quantum setting. Moreover, we employ overcomplete matrices for each decoding path. Monte-Carlo simulations on toric and generalized bicycle codes demonstrate improved convergence and reduced logical error rate.

Luca Schmid, Tomer Raviv, Nir Shlezinger et al.

We investigate the application of the factor graph framework for blind joint channel estimation and symbol detection on time-variant linear inter-symbol interference channels. In particular, we consider the expectation maximization (EM) algorithm for maximum likelihood estimation, which typically suffers from high complexity as it requires the computation of the symbol-wise posterior distributions in every iteration. We address this issue by efficiently approximating the posteriors using the belief propagation (BP) algorithm on a suitable factor graph. By interweaving the iterations of BP and EM, the detection complexity can be further reduced to a single BP iteration per EM step. In addition, we propose a data-driven version of our algorithm that introduces momentum in the BP updates and learns a suitable EM parameter update schedule, thereby significantly improving the performance-complexity tradeoff with a few offline training samples. Our numerical experiments demonstrate the excellent performance of the proposed blind detector and show that it even outperforms coherent BP detection in high signal-to-noise scenarios.

Vincent Lauinger, Patrick Matalla, Jonas Ney et al.

We demonstrate and evaluate a fully-blind digital signal processing (DSP) chain for 100G passive optical networks (PONs), and analyze different equalizer topologies based on neural networks with low hardware complexity.

Sisi Miao, Mert Birincioglu, Laurent Schmalen

We propose an enhanced Chase-Pyndiah decoder that scales extrinsic messages based on decoder confidence of the component decoder, achieving a 0.1 dB gain over the original with negligible complexity increase.

Alexander von Bank, Eike-Manuel Edelmann, Laurent Schmalen

We propose an energy-efficient equalizer for IM/DD systems based on spiking neural networks. We optimize a neural spike encoding that boosts the equalizer's performance while decreasing energy consumption.

Laurent Schmalen, Vincent Lauinger, Jonas Ney et al.

In this paper, we highlight recent advances in the use of machine learning for implementing equalizers for optical communications. We highlight both algorithmic advances as well as implementation aspects using conventional and neuromorphic hardware.

Luca Schmid, Dominik Sulz, Shrinivas Chimmalgi et al.

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.

Jonas Ney, Christoph Füllner, Vincent Lauinger et al.

To satisfy the growing throughput demand of data-intensive applications, the performance of optical communication systems increased dramatically in recent years. With higher throughput, more advanced equalizers are crucial, to compensate for impairments caused by inter-symbol interference (ISI). The latest research shows that artificial neural network (ANN)-based equalizers are promising candidates to replace traditional algorithms for high-throughput communications. On the other hand, not only throughput but also flexibility is a main objective of beyond-5G and 6G communication systems. A platform that is able to satisfy the strict throughput and flexibility requirements of modern communication systems are field programmable gate arrays (FPGAs). Thus, in this work, we present a high-performance FPGA implementation of an ANN-based equalizer, which meets the throughput requirements of modern optical communication systems. Further, our architecture is highly flexible since it includes a variable degree of parallelism (DOP) and therefore can also be applied to low-cost or low-power applications which is demonstrated for a magnetic recording channel. The implementation is based on a cross-layer design approach featuring optimizations from the algorithm down to the hardware architecture, including a detailed quantization analysis. Moreover, we present a framework to reduce the latency of the ANN-based equalizer under given throughput constraints. As a result, the bit error ratio (BER) of our equalizer for the optical fiber channel is around four times lower than that of a conventional one, while the corresponding FPGA implementation achieves a throughput of more than 40 GBd, outperforming a high-performance graphics processing unit (GPU) by three orders of magnitude for a similar batch size.

Vincent Lauinger, Lennart Schmitz, Patrick Matalla et al.

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.

Luca Schmid, Charlotte Muth, Laurent Schmalen

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.

Rodrigo Fischer, Patrick Matalla, Sebastian Randel et al.

We investigate Kolmogorov-Arnold networks (KANs) for non-linear equalization of 112 Gb/s PAM4 passive optical networks (PONs). Using pruning and extensive hyperparameter search, we outperform linear equalizers and convolutional neural networks at low computational complexity.

Jonas Ney, Patrick Matalla, Vincent Lauinger et al.

In this work, we present a high-throughput field programmable gate array (FPGA) demonstrator of an artificial neural network (ANN)-based equalizer. The equalization is performed and illustrated in real-time for a 30 GBd, two-level pulse amplitude modulation (PAM2) optical communication system.

Luca Schmid, Laurent Schmalen

We consider the application of the factor graph framework for symbol detection on linear inter-symbol interference channels. Based on the Ungerboeck observation model, a detection algorithm with appealing complexity properties can be derived. However, since the underlying factor graph contains cycles, the sum-product algorithm (SPA) yields a suboptimal algorithm. In this paper, we develop and evaluate efficient strategies to improve the performance of the factor graph-based symbol detection by means of neural enhancement. In particular, we consider neural belief propagation and generalizations of the factor nodes as an effective way to mitigate the effect of cycles within the factor graph. By applying a generic preprocessor to the channel output, we propose a simple technique to vary the underlying factor graph in every SPA iteration. Using this dynamic factor graph transition, we intend to preserve the extrinsic nature of the SPA messages which is otherwise impaired due to cycles. Simulation results show that the proposed methods can massively improve the detection performance, even approaching the maximum a posteriori performance for various transmission scenarios, while preserving a complexity which is linear in both the block length and the channel memory.

Eike-Manuel Bansbach, Victor Eliachevitch, Laurent Schmalen

As the number of user equipments (UEs) with various data rate and latency requirements increases in wireless networks, the resource allocation problem for orthogonal frequency-division multiple access (OFDMA) becomes challenging. In particular, varying requirements lead to a non-convex optimization problem when maximizing the systems data rate while preserving fairness between UEs. In this paper, we solve the non-convex optimization problem using deep reinforcement learning (DRL). We outline, train and evaluate a DRL agent, which performs the task of media access control scheduling for a downlink OFDMA scenario. To kickstart training of our agent, we introduce mimicking learning. For improvement of scheduling performance, full buffer state information at the base station (e.g. packet age, packet size) is taken into account. Techniques like input feature compression, packet shuffling and age capping further improve the performance of the agent. We train and evaluate our agents using Nokia's wireless suite and evaluate against different benchmark agents. We show that our agents clearly outperform the benchmark agents.

Kadir Gümüs, Laurent Schmalen

Error correction plays a major role in the reconciliation of continuous variable quantum key distribution (CV-QKD) and greatly affects the performance of the system. CV-QKD requires error correction codes of extremely low rates and high reconciliation efficiencies. There are only very few code designs available in this ultra low rate regime. In this paper, we introduce a method for designing protograph-based ultra low rate LDPC codes using differential evolution. By proposing type-based protographs, a new way of representing low rate protograph-based LDPC codes, we drastically reduce the complexity of the protograph optimization, which enables us to quickly design codes over a wide range of rates. We show that the codes resulting from our optimization outperform the codes from the literature both in regards to the threshold and in finite-length performance, validated by Monte-Carlo simulations, showing gains in the regime relevant for CV-QKD.

Boris Karanov, Mathieu Chagnon, Vahid Aref et al.

We investigate methods for experimental performance enhancement of auto-encoders based on a recurrent neural network (RNN) for communication over dispersive nonlinear channels. In particular, our focus is on the recently proposed sliding window bidirectional RNN (SBRNN) optical fiber autoencoder. We show that adjusting the processing window in the sequence estimation algorithm at the receiver improves the reach of simple systems trained on a channel model and applied "as is" to the transmission link. Moreover, the collected experimental data was used to optimize the receiver neural network parameters, allowing to transmit 42 Gb/s with bit-error rate (BER) below the 6.7% hard-decision forward error correction threshold at distances up to 70km as well as 84 Gb/s at 20 km. The investigation of digital signal processing (DSP) optimized on experimental data is extended to pulse amplitude modulation with receivers performing sliding window sequence estimation using a feed-forward or a recurrent neural network as well as classical nonlinear Volterra equalization. Our results show that, for fixed algorithm memory, the DSP based on deep learning achieves an improved BER performance, allowing to increase the reach of the system.

Boris Karanov, Mathieu Chagnon, Vahid Aref et al.

We investigate end-to-end optimized optical transmission systems based on feedforward or bidirectional recurrent neural networks (BRNN) and deep learning. In particular, we report the first experimental demonstration of a BRNN auto-encoder, highlighting the performance improvement achieved with recurrent processing for communication over dispersive nonlinear channels.

Andreas Buchberger, Christian Häger, Henry D. Pfister et al.

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 Reed-Muller and short low-density parity-check codes, we achieve performance within 0.27 dB and 1.5 dB of the ML performance while reducing the complexity of the decoder.

Boris Karanov, Mathieu Chagnon, Vahid Aref et al.

We perform an experimental end-to-end transceiver optimization via deep learning using a generative adversarial network to approximate the test-bed channel. Previously, optimization was only possible through a prior assumption of an explicit simplified channel model.

Boris Karanov, Domaniç Lavery, Polina Bayvel et al.

We propose an autoencoding sequence-based transceiver for communication over dispersive channels with intensity modulation and direct detection (IM/DD), designed as a bidirectional deep recurrent neural network (BRNN). The receiver uses a sliding window technique to allow for efficient data stream estimation. We find that this sliding window BRNN (SBRNN), based on end-to-end deep learning of the communication system, achieves a significant bit-error-rate reduction at all examined distances in comparison to previous block-based autoencoders implemented as feed-forward neural networks (FFNNs), leading to an increase of the transmission distance. We also compare the end-to-end SBRNN with a state-of-the-art IM/DD solution based on two level pulse amplitude modulation with an FFNN receiver, simultaneously processing multiple received symbols and approximating nonlinear Volterra equalization. Our results show that the SBRNN outperforms such systems at both 42 and 84\,Gb/s, while training fewer parameters. Our novel SBRNN design aims at tailoring the end-to-end deep learning-based systems for communication over nonlinear channels with memory, such as the optical IM/DD fiber channel.

Boris Karanov, Mathieu Chagnon, Félix Thouin et al.

In this paper, we implement an optical fiber communication system as an end-to-end deep neural network, including the complete chain of transmitter, channel model, and receiver. This approach enables the optimization of the transceiver in a single end-to-end process. We illustrate the benefits of this method by applying it to intensity modulation/direct detection (IM/DD) systems and show that we can achieve bit error rates below the 6.7\% hard-decision forward error correction (HD-FEC) threshold. We model all componentry of the transmitter and receiver, as well as the fiber channel, and apply deep learning to find transmitter and receiver configurations minimizing the symbol error rate. We propose and verify in simulations a training method that yields robust and flexible transceivers that allow---without reconfiguration---reliable transmission over a large range of link dispersions. The results from end-to-end deep learning are successfully verified for the first time in an experiment. In particular, we achieve information rates of 42\,Gb/s below the HD-FEC threshold at distances beyond 40\,km. We find that our results outperform conventional IM/DD solutions based on 2 and 4 level pulse amplitude modulation (PAM2/PAM4) with feedforward equalization (FFE) at the receiver. Our study is the first step towards end-to-end deep learning-based optimization of optical fiber communication systems.

Mohamad Dia, Vahid Aref, Laurent Schmalen

In this work, we formulate the fixed-length distribution matching as a Bayesian inference problem. Our proposed solution is inspired from the compressed sensing paradigm and the sparse superposition (SS) codes. First, we introduce sparsity in the binary source via position modulation (PM). We then present a simple and exact matcher based on Gaussian signal quantization. At the receiver, the dematcher exploits the sparsity in the source and performs low-complexity dematching based on generalized approximate message-passing (GAMP). We show that GAMP dematcher and spatial coupling lead to asymptotically optimal performance, in the sense that the rate tends to the entropy of the target distribution with vanishing reconstruction error in a proper limit. Furthermore, we assess the performance of the dematcher on practical Hadamard-based operators. A remarkable feature of our proposed solution is the possibility to: i) perform matching at the symbol level (nonbinary); ii) perform joint channel coding and matching.