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.
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.
This paper considers the fundamental limit of random linear estimation for i.i.d. signal distributions and i.i.d. Gaussian measurement matrices. Its main contribution is a rigorous characterization ...of the asymptotic mutual information (MI) and minimum mean-square error (MMSE) in this setting. Under mild technical conditions, our results show that the limiting MI and MMSE are equal to the values predicted by the replica method from statistical physics. This resolves a well-known problem that has remained open for over a decade.
This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is ...that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.
We consider a communication system where two transmitters wish to exchange information through a central relay. The transmitter and relay nodes exchange data over synchronized, average power ...constrained additive white Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of snr. An upper bound on the capacity is 1/2 log(1 + snr) bits per transmitter per use of the multiple access phase and broadcast phase of the bidirectional relay channel. We show that, using lattice codes and lattice decoding, we can obtain a rate of 1/2 log(1/2 + snr) bits per transmitter, which is essentially optimal at high SNR. The main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs Slepian-Wolf coding. We also show that if the two transmitters use identical lattices with minimum angle decoding, we can achieve the same rate of 1/2 log(1/2 + snr). The proposed scheme can be thought of as a joint physical-layer network-layer code which outperforms other recently proposed analog network coding schemes.
We introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding. Rather than relying on the precise ...structure of the codes, our method exploits code symmetry. In particular, the technique applies to any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive. In other words, we show that symmetry alone implies near-optimal performance. An important consequence of this result is that a sequence of Reed-Muller codes with increasing block length and converging rate achieves capacity. This possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1. Moreover, these results extend naturally to all affine-invariant codes and, thus, to extended primitive narrow-sense BCH codes. This also resolves, in the affirmative, the existence question for capacity-achieving sequences of binary cyclic codes. The primary tools used in the proof are the sharp threshold property for symmetric monotone Boolean functions and the area theorem for extrinsic information transfer functions.
Low-density parity-check (LDPC) convolutional codes (or spatially coupled codes) were recently shown to approach capacity on the binary erasure channel (BEC) and binary-input memoryless symmetric ...channels. The mechanism behind this spectacular performance is now called threshold saturation via spatial coupling. This new phenomenon is characterized by the belief-propagation threshold of the spatially coupled ensemble increasing to an intrinsic noise threshold defined by the uncoupled system. In this paper, we present a simple proof of threshold saturation that applies to a wide class of coupled scalar recursions. Our approach is based on constructing potential functions for both the coupled and uncoupled recursions. Our results actually show that the fixed point of the coupled recursion is essentially determined by the minimum of the uncoupled potential function and we refer to this phenomenon as Maxwell saturation. A variety of examples are considered including the density-evolution equations for: irregular LDPC codes on the BEC, irregular low-density generator-matrix codes on the BEC, a class of generalized LDPC codes with BCH component codes, the joint iterative decoding of LDPC codes on intersymbol-interference channels with erasure noise, and the compressed sensing of random vectors with independent identically distributed components.
We propose a spatial multiplexing system using reconfigurable cavity-backed metasurface antennas. The metasurface antennas consist of a printed cavity with dynamically tunable metamaterial radiators ...patterned on one side and fed by multiple radio frequency ports on the other side (each port representing one communication node), forming a shared aperture. By individual tuning of the radiators, the antennas can generate steerable, concurrent beams that can be adapted to the properties of multiple-input-multiple-output (MIMO) channels. In this paper, we present a <inline-formula> <tex-math notation="LaTeX">2\times 2 </tex-math></inline-formula> MIMO system with simulated metasurface antennas as transmit and receive antennas operating at 5.9 GHz. We demonstrate that the flexibility in beamforming supported by the metasurface antennas can be used to achieve low spatial correlation and high SNR gain in clustered MIMO channels, leading to a significant improvement of the channel capacity. Numerical studies show 2.36-fold, 2.11-fold enhancements of capacity in MIMO channels with one and two clusters, respectively, compared with an MIMO system consisting of linear dipoles. The MIMO system based on the metasurface antennas can be low cost, low profile, and low power. The metasurface antenna thus has potential applications in small cell networks requiring high data rate under bandwidth, energy, and cost constraints.
We consider near maximum-likelihood (ML) decoding of short linear block codes. In particular, we propose a novel decoding approach based on neural belief propagation (NBP) decoding recently ...introduced by Nachmani et al. in which we allow a different parity-check matrix in each iteration of the algorithm. The key idea is to consider NBP decoding over an overcomplete parity-check matrix and use the weights of NBP as a measure of the importance of the check nodes (CNs) to decoding. The unimportant CNs are then pruned. In contrast to NBP, which performs decoding on a given fixed parity-check matrix, the proposed pruning-based neural belief propagation (PB-NBP) typically results in a different parity-check matrix in each iteration. For a given complexity in terms of CN evaluations, we show that PB-NBP yields significant performance improvements with respect to NBP. We apply the proposed decoder to the decoding of a Reed-Muller code, a short low-density parity-check (LDPC) code, and a polar code. PB-NBP outperforms NBP decoding over an overcomplete parity-check matrix by 0.27-0.31 dB while reducing the number of required CN evaluations by up to 97%. For the LDPC code, PB-NBP outperforms conventional belief propagation with the same number of CN evaluations by 0.52 dB. We further extend the pruning concept to offset min-sum decoding and introduce a pruning-based neural offset min-sum (PB-NOMS) decoder, for which we jointly optimize the offsets and the quantization of the messages and offsets. We demonstrate performance 0.5 dB from ML decoding with 5-bit quantization for the Reed-Muller code.
In this article, we propose a model-based machine-learning approach for dual-polarization systems by parameterizing the split-step Fourier method for the Manakov-PMD equation. The resulting method ...combines hardware-friendly time-domain nonlinearity mitigation via the recently proposed learned digital backpropagation (LDBP) with distributed compensation of polarization-mode dispersion (PMD). We refer to the resulting approach as LDBP-PMD. We train LDBP-PMD on multiple PMD realizations and show that it converges within 1% of its peak dB performance after 428 training iterations on average, yielding a peak effective signal-to-noise ratio of only 0.30 dB below the PMD-free case. Similar to state-of-the-art lumped PMD compensation algorithms in practical systems, our approach does not assume any knowledge about the particular PMD realization along the link, nor any knowledge about the total accumulated PMD. This is a significant improvement compared to prior work on distributed PMD compensation, where knowledge about the accumulated PMD is typically assumed. We also compare different parameterization choices in terms of performance, complexity, and convergence behavior. Lastly, we demonstrate that the learned models can be successfully retrained after an abrupt change of the PMD realization along the fiber.
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