Block Markov Superposition Research Articles

Partially Block Markov Superposition Transmission of a Gaussian Source With Nested Lattice Codes

This paper studies the transmission of Gaussian sources through additive white Gaussian noise channels in bandwidth expansion regime, i.e., the channel bandwidth is greater than the source bandwidth. To mitigate the error propagation phenomenon of conventional digital transmission schemes, we propose in this paper a new capacity-approaching joint source channel coding (JSCC) scheme based on partially block Markov superposition transmission (BMST) of nested lattice codes. In the proposed scheme, first, the Gaussian source sequence is discretized by a lattice-based quantizer, resulting in a sequence of lattice points. Second, these lattice points are encoded by a short systematic group code. Third, the coded sequence is partitioned into blocks of equal length and then transmitted in the BMST manner. The main characteristics of the proposed JSCC scheme include: 1) entropy coding is not used explicitly and 2) only parity-check sequence is superimposed, hence, termed partially BMST. This is different from the original BMST. To show the superior performance of the proposed scheme, we present extensive simulation results which show that the proposed scheme performs within 1 dB of the Shannon limits. Hence, the proposed scheme provides an attractive candidate for transmission of Gaussian sources.

A Low-Complexity Delay-Tunable Coding Scheme for Visible Light Communication Systems

Visible light communication (VLC) systems are expected to support a variety of applications, such as common-information broadcasting, real-time multimedia streaming, and large file downloading. Typically, these applications have different delay requirements. Hence, it is desirable to design high-performance coding scheme capable of supporting a wide range of delays but with acceptable hardware complexity. To achieve this, we propose a delay-tunable coding scheme for VLC systems based on block Markov superposition transmission of short non-binary low-density parity-check (NBLDPC) codes. The proposed coding scheme includes the following advantages: 1) it is easily configurable to fulfill different delay requirements while keeping the code rate constant; 2) it requires essentially the same hardware modules to implement the encoder/decoder as the involved short NBLDPC code; and 3) it can have a larger coding gain if a longer delay is tolerated. Numerical results are presented to show the expected tradeoff between delay and energy in a VLC system.

Multiple‐rate codes from block Markov superposition transmission of first‐order Reed–Muller and extended Hamming codes

Block Markov superposition transmission (BMST) of first-order Reed–Muller and extended Hamming (RMEH) codes is investigated. Multiple code rates can be easily realised by adjusting the mixture ratio of Reed–Muller and extended Hamming codes. Compared with the original BMST of repetition and single-parity-check (RSPC) codes, the proposed BMST-RMEH codes can perform close to Shannon limits over a wide range of code rates with about only half the encoding memories of BMST-RSPC codes. As a result of the reduction of encoding memories, the decoding complexity and delay of BMST-RMEH codes are approximately half (or even lower than) those of BMST-RSPC codes.

Block Markov Superposition Transmission of RUN Codes

In this paper, we propose a simple procedure to construct (decodable) good codes with any given alphabet (of moderate size) for any given (rational) code rate to achieve any given target error performance (of interest) over additive white Gaussian noise channels. We start with constructing codes over groups for any given code rates. This can be done in an extremely simple way if we ignore the error performance requirement for the time being. Actually, this can be satisfied by repetition (R) codes and uncoded (UN) transmission along with time-sharing technique. The resulting codes are simply referred to as RUN codes for convenience. The encoding/decoding algorithms for RUN codes are almost trivial. In addition, the performance can be easily analyzed. It is not difficult to imagine that the RUN code usually performs far away from the corresponding Shannon limit. Fortunately, the performance can be improved as required by spatially coupling the RUN codes via block Markov superposition transmission (BMST), resulting in the BMST-RUN codes. Simulation results show that the BMST-RUN codes perform well (within around 1 dB away from Shannon limits) for a wide range of code rates and outperform the BMST with bit-interleaved coded modulation scheme.

Performance Analysis of Block Markov Superposition Transmission of Short Codes

In this paper, we consider the asymptotic and finite-length performance of block Markov superposition transmission (BMST) of short codes, which can be viewed as a new class of spatially coupled (SC) codes where the generator matrices of short codes (referred to as basic codes ) are coupled. A modified extrinsic information transfer (EXIT) chart analysis that takes into account the relation between mutual information (MI) and bit-error-rate (BER) is presented to study the convergence behavior of BMST codes. Using the modified EXIT chart analysis, we investigate the impact of various parameters on BMST code performance, thereby providing theoretical guidance for designing and implementing practical BMST codes suitable for window decoding. Then, we present a performance comparison of BMST codes and SC low-density parity-check (SC-LDPC) codes on the basis of equal decoding latency. Also presented is a comparison of computational complexity. Simulation results show that, under the equal decoding latency constraint, BMST codes using the repetition code as the basic code can outperform both (3,6)-regular SC-LDPC codes and (4,8)-regular SC-LDPC codes in the waterfall region but have a higher computational complexity.

Two-Layer Coded Spatial Modulation With Block Markov Superposition Transmission

This paper is concerned with the spatial modulation (SM), a multiple-input multiple-output (MIMO) transmission technique, that maps information bits not only into the conventional two-dimensional signal points but also into the indices of active transmit antennas. We present a two-layer coded SM scheme, in which the spatial bits carried by the antenna indices and the signal bits carried by the conventional signals are protected separately by two error correction codes. For the ease of decoding process, the code rates are allocated according to the chain rule of the mutual information. We choose block Markov superposition transmission (BMST) codes for each layer, since they are easily designed for any given code rate with a predictable performance lower bound. An iterative sliding-window decoding algorithm is also presented by exchanging messages iteratively between the two BMST decoders and the soft-in soft-out (SISO) demapper of the SM. To reduce the computational complexity, we propose to implement the SISO demapping algorithm by employing only partial soft inputs. Numerical results show that the BMST-SM system performs well over uncorrelated Rayleigh fading channels.

Small-Macro Cell Cooperation for HetNet Uplink Transmission: Spectral Efficiency and Reliability Analyses

We investigate the impact of small-macro cell cooperation (SMC) in improving the spectral efficiency and reliability of uplink transmission in a heterogeneous network. We consider a network of two user equipments (UEs), a macro-cell base station (BS) and a small-cell BS. Joint SMC involves macro-to-small quantized feedback and decode-forward relaying from small to macro cell. This cooperation utilizes full-duplex transmission and intra-network spectrum sharing. We first propose a transmission scheme based on superposition block Markov encoding at each UE, coherent decode-forward relaying and sliding window decoding at the small-cell BS, and quantize-forward relaying and backward decoding at the macro-cell BS. Second, we derive the optimal macro-cell quantization to maximize the whole spectral efficiency. Third, for a certain non-fading scenario, we prove that the proposed scheme asymptotically achieves the capacity (maximum spectral efficiency) by reaching the cut-set bound as macro-cell power approaches infinity. Fourth, we formulate the outage probability over block fading channels, considering the outage events at the small and macro-cell BSs and the channel variations over different blocks. Last, we generalize the proposed scheme to an $N\; (>2)$ -UE case. As macro-cell power increases, the results show that the proposed scheme achieves a full diversity order of two and outperforms all existing non-SMC schemes. These strong results suggest the utility of the proposed scheme for potential deployment in 5G cellular networks.

Cooperative Compute-and-Forward

We propose a class of signaling schemes that leverage transmitter cooperation in wireless networks employing compute-and-forward or physical-layer network coding. We devise a lattice-coding approach to superposition block Markov encoding from which we construct a cooperative lattice coding strategy. Transmitters broadcast lattice codewords, decode each other’s messages, and then cooperatively transmit resolution information which aids relays in decoding finite-field linear combinations of the incoming messages. We show that cooperation provides a substantial improvement in achievable computation rate and outage probability over noncooperative strategies. Using this strategy, we derive a new achievability scheme for the multiway relay channel, the rates of which are near capacity in many regimes and enjoy a diversity advantage over noncooperation.

A New Class of Multiple-Rate Codes Based on Block Markov Superposition Transmission

The Hadamard transform (HT) over the binary field provides a natural way to implement multiple-rate codes (referred to as HT-coset codes), where the code length $N=2^{p}$ is fixed but the code dimension $K$ can be varied from 1 to $N-1$ by adjusting the set of frozen bits. The HT-coset codes, including Reed–Muller (RM) codes and polar codes as typical examples, can share the same fundamental encoder/decoder architecture with the implementation complexity of order $O(N\log N)$ . However, to guarantee that all codes with designated rates perform well, HT-coset coding usually requires a sufficiently large code length, which, in turn, causes difficulties in the determination of which bits are better for being frozen. In this paper, we propose to transmit short HT-coset codes in the so-called block Markov superposition transmission (BMST) manner. At the transmitter, signals are spatially coupled via superposition, resulting in long codes. At the receiver, these coupled signals are recovered by a sliding-window iterative soft successive cancellation decoding algorithm. Most importantly, the performance around or below the bit-error-rate (BER) of $10^{-5}$ can be predicted by a simple genie-aided lower bound. Both these bounds and simulation results show that the BMST of short HT-coset codes performs well (within one dB away from the corresponding Shannon limits) in a wide range of code rates.

Block Markov Superposition Transmission: Construction of Big Convolutional Codes From Short Codes

A construction of big convolutional codes from short codes called block Markov superposition transmission (BMST) is proposed. The BMST is very similar to superposition block Markov encoding (SBME), which has been widely used to prove multiuser coding theorems. The BMST codes can also be viewed as a class of spatially coupled codes, where the generator matrices of the involved short codes (referred to as basic codes) are coupled. The encoding process of BMST can be as fast as that of the basic code, while the decoding process can be implemented as an iterative sliding-window decoding algorithm with a tunable delay. More importantly, the performance of BMST can be simply lower bounded in terms of the transmission memory given that the performance of the short code is available. Numerical results show that: 1) the lower bounds can be matched with a moderate decoding delay in the low bit-error-rate (BER) region, implying that the iterative sliding-window decoding algorithm is near optimal; 2) BMST with repetition codes and single parity-check codes can approach the Shannon limit within 0.5 dB at the BER of $10^{-5}$ for a wide range of code rates; and 3) BMST can also be applied to nonlinear codes.

Impact of interleavers on performance of block Markov superposition transmission

The impact of different kinds of interleavers on the performance of block Markov superposition transmission (BMST) systems is investigated. Simulation results show that the performance of a BMST system is insensitive to the types of interleavers, provided that multiple interleavers (if required) differ from each other. As a byproduct, a new interleaver called the binary linear interleaver is proposed, which can be easily implemented in hardware.

Low-Density Lattice Codes for Full-Duplex Relay Channels

We propose a class of practical efficient lattice codes for real-valued full-duplex one- and two-way relay channels. First, we investigate the problem from a theoretical perspective, proposing lattice-coding instantiations of superposition block Markov encoding. Our encoding/decoding strategies recover the well-known decode-and-foward rates for the one-way relay channel and a previously-proven rate region for the two-way relay channel. Then, we construct practical, low-complexity implementations of these schemes using low-density lattice codes. Simulations show that our schemes achieve performance as close as 2.5 dB away from theoretical limits. Finally, we show that, due to features inherent to full-duplex relaying and practical codes, the gap to theoretical limits depends on the channel gains and transmit power of the relay relative to the source(s). We characterize this gap analytically, providing insight into the design of practical full-duplex relay systems.

Block Markov Superposition Transmission of Short Codes for Indoor Visible Light Communications

This letter presents a coded modulation scheme for indoor visible light communication (VLC), which is based on the recently proposed coding scheme called block Markov superposition transmission (BMST). As the bandwidth of a single light-emitting-diode (LED) light is limited, we use parallel LED wicks to transmit simultaneously multiple coded bit-streams, resembling a multiple access system. At the transmitter, a single stream of data is first encoded by a properly designed BMST encoder and then partitioned equally into multiple streams, each of which, in the case of no precoding, is used to drive an LED wick in an on-off keying (OOK) manner. With a tailored power-allocation scheme, the noise-free received signals can form an equispaced pulse amplitude modulation (PAM) constellation. Also proposed is the use of precoding to achieve Gray mapping, which plays an important role in reducing complexity without degrading performance since no iterative processing is required between the detection and the decoding. Simulation results show that the proposed scheme achieves a significant coding gain.

Block Markov Superposition Transmission of Repetition and Single-Parity-Check Codes

This letter presents a method to construct a family of codes with rate K/N(K = 1,2,...,N - 1) for any given integer N > 1. This family of codes, referred to as RSPC codes for short, consist of the repetition (R) code, the single-parity-check (SPC) code and the codes obtained by time-sharing between the R code and the SPC code. The RSPC codes have extremely simple encoding and soft-input soft-output (SISO) decoding algorithms, hence can be integrated into the recently proposed block Markov superposition transmission (BMST) system. The BMST introduces memory across short codes by spatially coupling the generator matrices of the short codes. A distinguished feature of the BMST system is the simple relation between the asymptotic coding gain and the spatially coupling depth (encoding memory). Furthermore, the BMST-RSPC codes can share a pair of easily reconfigurable encoder and decoder. Simulation results show that the BMST-RSPC codes perform well (within one dB away from the corresponding Shannon limits) in a wide range of code rates.

Block Markov superposition transmission of convolutional codes with minimum shift keying signalling

In this study, the authors’ present a scheme, denoted as BMST-MSK, which combines the block Markov superposition transmission (BMST) with the minimum shift keying (MSK) signalling. The BMST-MSK can be implemented in two forms – the BMST with recursive MSK (BMST-RMSK) and the BMST with non-recursive MSK (BMST-NRMSK). The BMST-MSK admits a sliding-window decoding/demodulation algorithm, where two schedules with or without iterative processing between the BMST and MSK (referred to as outer iteration) are discussed. To analyse the asymptotic performance of BMST-MSK, the authors’ first assume a genie-aided decoder and then derive the union bound for the equivalent genie-aided system. Numerical results show that the performances of the BMST-MSK match well with the derived lower bounds in the low error rate regions. From simulations, the authors’ found that the outer iterations can provide performance improvement for the BMST-RMSK, but not for the BMST-NRMSK. Taking a (2,1,2) convolutional code with input length of 10 000 bits as the basic code, the BMST-NRMSK achieves a bit-error-rate of 10−5 at E b/N 0 = 0.45 dB over additive white Gaussian noise channels, which is away from the Shannon limit about 0.25 dB.

On the Reliability Function of the Discrete Memoryless Relay Channel

Bounds on the reliability function for the discrete memoryless relay channel are derived using the method of types. Two achievable error exponents are derived based on partial decode-forward and compress-forward, which are well-known superposition block-Markov coding schemes. The derivations require combinations of the techniques involved in the proofs of Csiszár-Körner-Marton's packing lemma for the error exponent of channel coding and Marton's type covering lemma for the error exponent of source coding with a fidelity criterion. The decode-forward error exponent is evaluated on Sato's relay channel. From this example, it is noted that to obtain the fastest possible decay in the error probability for a fixed effective coding rate, one ought to optimize the number of blocks in the block-Markov coding scheme assuming the blocklength within each block is large. An upper bound on the reliability function is also derived using ideas from Haroutunian's lower bound on the error probability for point-to-point channel coding with feedback.

Spatial Coupling of Generator Matrices: A General Approach to Design Good Codes at a Target BER

For any given short code (referred to as the basic code), block Markov superposition transmission (BMST) provides a simple way to obtain predictable extra coding gain by spatially coupling the generator matrix of the basic code. This paper presents a systematic design methodology for BMST systems to approach the channel capacity at any given target bit error rate (BER) of interest. To simplify the design, we choose the basic code as the Cartesian product of a short block code. The encoding memory is then inferred from the genie-aided lower bound according to the performance gap of the short block code to the corresponding Shannon limit at the target BER. In addition to the sliding-window decoding algorithm, we propose to perform one more phase decoding to remove residual (rare) errors. A new technique that assumes a noisy genie is proposed to upper bound the performance. Under some mild assumptions, these genie-aided bounds can be used to predict the performance of the proposed two-phase decoding algorithm in the extremely low BER region. Using the Cartesian product of a repetition code as the basic code, we construct a BMST system with an encoding memory 30 whose performance at the BER of $10^{-15}$ can be predicted within 1 dB away from the Shannon limit over the binary-input additive white Gaussian noise channel.

Block Markov Superposition Transmission with Spatial Modulation

Spatial modulation (SM) is a multiple-input multiple-output (MIMO) transmission technique that maps the information into both antenna indices and signals from a conventional signal constellation. Block Markov superposition transmission (BMST) is a recently proposed coding scheme which has a good performance and a predictable performance lower bound. In this letter, we propose to combine the BMST with the SM, resulting in a system, which is acronymized as BMST-SM for convenience. Also presented is a sliding-window decoding executed iteratively between the BMST decoder and the soft-in soft-out (SISO) SM demapper. To evaluate the performance of the BMST-SM system, the mutual information is derived for the SM with multiple receive antennas under uniformly distributed input constraint. Simulation results show that the BMST-SM scheme performs around one dB away from the Shannon limit at the bit-error-rate (BER) around 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> over uncorrelated Rayleigh fading channels.

Cognitive Cooperative MAC With One Primary and Two Secondary Users: Achievable Rates and Optimal Power Control

We consider a three-user fading cognitive cooperative multiple access channel (MAC) with one primary and two secondary transmitters. We propose two encoding/decoding strategies with varying levels of cooperation, based on block Markov superposition encoding and backward decoding. The first is an overlay model, where the secondary users (SUs) aid the transmission of the primary user (PU) by causally decoding the PU message and forwarding it while also cooperating among each other. The second is an underlay model, where the SUs cooperate by decoding and forwarding each other's messages while treating the signal received from the PU as noise. In either case, the PU is guaranteed to operate at its maximum achievable single user rate. We characterize the achievable SU rate region for both models and maximize this region as a function of the transmit powers. The simulation results show that the SU rate region can be significantly enlarged, particularly using the overlay model.

Block Markov Superposition Transmission with Bit-Interleaved Coded Modulation

In this letter, we construct bit-interleaved coded modulation (BICM) using a simple basic code but transmitting multiple times in a way similar to the block Markov superposition transmission (BMST). The system admits a sliding-window decoding/demapping algorithm that can be employed to trade off the performance against the delay. Motivated but different from BMST with binary phase-shift keying (BPSK) signalling, a simple lower bound on performance is derived from an equivalent genie-aided system. Numerical results show that the performances of the BMST systems with BICM using a (2, 1, 2) convolutional basic code match well with the derived lower bounds in lower error rate regions over both additive white Gaussian noise (AWGN) channels and Rayleigh fading channels. We also analyze the effect of the encoding memory and the decoding delay on the performance as well as the decoding complexity.