Chulong Liang

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.

Obtaining extra coding gain for short codes by block Markov superposition transmission

In this paper, we present a new approach, called block Markov superposition transmission (BMST), to construct from short codes a class of convolutional codes with large constraint length. The BMST is very similar to superposition block Markov encoding (SBME), which has been widely used to prove multiuser coding theorems. We also present an iterative sliding-window decoding algorithm for the proposed transmission scheme. The extra coding gain obtained by BMST can be bounded in terms of the Markov order and with the help of the input-output weight enumerating function (IOWEF) of the BMST system, which can be computed from that of the short code by performing a trellis-based algorithm. Numerical results verify our analysis and show that an extra coding gain of 6.4 dB at bit-error rate (BER) 10-5 can be obtained by BMST of the [7, 4] Hamming code.

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 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.

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.

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=2p 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(NlogN). 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 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 10-5 over uncorrelated Rayleigh fading channels.

Unequal error protection by partial superposition transmission using low-density parity-check codes

In this study, the authors consider designing low-density parity-check (LDPC) coded modulation systems to achieve unequal error protection (UEP). They propose a new UEP approach by partial superposition transmission (PST) called UEP-by-PST. In the UEP-by-PST system, the information sequence is distinguished as two parts, the more important data (MID) and the less important data (LID), both of which are coded with LDPC codes. The codeword that corresponds to the MID is superimposed on the codeword that corresponds to the LID. The system performance can be analysed by using discretised density evolution. Also proposed in this study is a criterion from a practical point of view to compare the efficiencies of different UEP approaches. Numerical results show that, over both additive white Gaussian noise channels and uncorrelated Rayleigh fading channels, (i) UEP-by-PST provides higher coding gain for the MID compared with the traditional equal error protection approach, but with negligible performance loss for the LID; and (ii) UEP-by-PST is more efficient with the proposed practical criterion than the UEP approach in the digital video broadcasting system.

Construction of Binary LDPC Convolutional Codes Based on Finite Fields

Using a finite field approach, a novel algebraic construction of low-density parity-check (LDPC) convolutional codes with fast encoding property is proposed. According to the matrices of quasi-cyclic (QC) codes constructed based on the multiplicative groups of finite fields and the algebraic property that a binary circulant matrix is isomorphic to a finite ring, we first generate a polynomial-form parity-check matrix of an LDPC convolutional code under a given rate over a given finite field. Then some related modifications are made upon the original polynomial-form matrix to obtain the new one with fast encoding property. Simulation results show that the proposed LDPC convolutional codes have good performance with the iterative belief propagation decoding algorithm.

An Information-Spectrum Approach to the Capacity Region of the Interference Channel

In this paper, a general formula for the capacity region of a general interference channel with two pairs of users is derived, which reveals that the capacity region is the union of a family of rectangles. In the region, each rectangle is determined by a pair of spectral inf-mutual information rates. The presented formula provides us with useful insights into the interference channels in spite of the difficulty of computing it. Specially, when the inputs are discrete, ergodic Markov processes and the channel is stationary memoryless, the formula can be evaluated by the BCJR (Bahl-Cocke-Jelinek-Raviv) algorithm. Also the formula suggests that considering the structure of the interference processes contributes to obtaining tighter inner bounds than the simplest one (obtained by treating the interference as noise). This is verified numerically by calculating the mutual information rates for Gaussian interference channels with embedded convolutional codes. Moreover, we present a coding scheme to approach the theoretical achievable rate pairs. Numerical results show that the decoding gains can be achieved by considering the structure of the interference.