Abstract-We study the limits of performance of Gallager codes (low-density parity-check (LDPC) codes) over binary linear intersymbol interference (ISI) channels with additive white Gaussian noise (AWGN). Using the graph representations of the channel, the code, and the sum-product message-passing detector/decoder, we prove two error concentration theorems. Our proofs expand on previous work by handling complications introduced by the channel memory. We circumvent these problems by considering not just linear Gallager codes but also their cosets and by distinguishing between different types of message flow neighborhoods depending on the actual transmitted symbols. We compute the noise tolerance threshold using a suitably developed density evolution algorithm and verify, by simulation, that the thresholds represent accurate predictions of the performance of the iterative sum-product algorithm for finite (but large) block lengths. We also demonstrate that for high rates, the thresholds are very close to the theoretical limit of performance for Gallager codes over ISI channels. If C denotes the capacity of a binary ISI channel and if C i i d denotes the maximal achievable mutual information rate when the channel inputs are independent and identically distributed (i.i.d.) binary random variables (C i i d C), we prove that the maximum information rate achievable by the sum-product decoder of a Gallager (coset) code is upper-bounded by C i i d . The last topic investigated is the performance limit of the decoder if the trellis portion of the sum-product algorithm is executed only once; this demonstrates the potential for trading off the computational requirements and the performance of the decoder.Index Terms-Bahl-Cocke-Jelinek-Raviv (BCJR)-once bound, channel capacity, density evolution, Gallager codes, independent and identically distributed (i.i.d.) capacity, intersymbol interference (ISI) channel, low-density parity-check (LDPC) codes, sumproduct algorithm, turbo equalization.

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.

Abstract-In this correspondence, we investigate in a comprehensive fashion a one-layer coding/shaping scheme resembling a perfectly cooperated multiple-access system. At the transmitter, binary data are encoded by either single-level or multilevel codes. The coded bits are first randomly interleaved and then entered into a signal mapper. At each time, the signal mapper accepts as input multiple binary digits and delivers as output an amplitude signal, where the input are first independently mapped into 2-PAM signals (possibly having different amplitudes) and then superimposed to form the output. The receiver consists of an iterative decoding/demapping algorithm with an entropy-based stopping criterion. In the special cases when all the 2-PAM signals have equal amplitudes, based on an irregular trellis, we propose an optimal soft-input-soft-output (SISO) demapping algorithm with quadratic rather than exponential complexity. In the general cases, when multilevel codes are employed, we propose power-allocation strategies to facilitate the iterative decoding/dempaping algorithm. Using the unequal power-allocations and the Gaussian-approximation-based suboptimal demapping algorithm (with linear complexity), coded modulation with high bandwidth efficiency can be implemented.Index Terms-Coded modulation, coding/shaping scheme, iterative decoding/demapping algorithm, iterative multistage decoding, multilevel coding, sigma-mapping, soft-input-soft-output (SISO) demapping algorithm.

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In status update scenarios, the freshness of information is measured in terms of age-of-information (AoI), which essentially reflects the timeliness for real-time applications to transmit status update messages to a remote controller. For some applications, computational expensive and time consuming data processing is inevitable for status information of messages to be displayed. Mobile edge servers are equipped with adequate computation resources and they are placed close to users. Thus, mobile edge computing (MEC) can be a promising technology to reduce AoI for computation-intensive messages. In this paper, we study the AoI for computation-intensive messages with MEC, and consider three computing schemes: local computing, remote computing at the MEC server, and partial computing, i.e., some part of computing tasks are performed locally, and the rest is executed at the MEC server. Zero-wait policy is adopted in all three schemes. Specifically, in local computing, a new message is generated immediately after the previous one is revealed by computing. While in remote computing and partial computing, a new message is generated once the previous one is received by the remote MEC server. With infinite queue size and exponentially distributed transmission time, closed-form average AoI for exponentially distributed computing time is derived for the three computing schemes. For deterministic computing time, the average AoI is analyzed numerically. Simulation results show that by carefully partitioning the computing tasks, the average AoI in partial computing is the smallest compared to local computing and remote computing. The results also indicate numerically the conditions on which remote computing attains smaller average AoI compared with local computing.

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