Analysis of convolutional codes on the erasure channel

We study a serially interleaved concatenated code construction, where the outer code is a standard convolutional code, and the inner code is a recursive convolutional code of rate 1. We focus on the ubiquitous inner differential encoder (used, in particular, to resolve phase ambiguities), double differential encoder (used to resolve both phase and frequency ambiguities), and another rate 1 recursive convolutional code of memory 2. We substantiate analytically the rather surprising result, that the error probabilities corresponding to a maximum-likelihood (ML) coherently detected antipodal modulation over the additive white Gaussian noise (AWGN) channel for this construction are advantageous as compared to the stand-alone outer convolutional code. This is in spite of the fact that the inner code is of rate 1. The analysis is based on the tangential sphere upper bound of an ML decoder, incorporating the ensemble weight distribution (WD) of the concatenated code, where the ensemble is generated by all random and uniform interleavers. This surprising result is attributed to the WD thinning observed for the concatenated scheme which shapes the WD of the outer convolutional code to resemble more closely the binomial distribution (typical of a fully random code of the same length and rate). This gain is maintained regardless of a rather dramatic decrease, as demonstrated here, in the minimum distance of the concatenated scheme as compared to the minimum distance of the outer stand-alone convolutional code. The advantage of the examined serially interleaved concatenated code, given in terms of bit and/or block error probability which is decoded by a practical suboptimal decoder, over the optimally decoded standard convolutional code is demonstrated by simulations, and some insights into the performance of the iterative decoding algorithm are also discussed. Though we have investigated only specific constructions of constituent inner (rate 1) and outer codes, we trust, hinging on the rational of the arguments here, that these results extend to many other constituent convolutional outer codes and rate 1 inner recursive convolutional codes.

We study the maximum a posteriori (MAP) decoding of memoryless non-uniform sources over multiple-antenna channels. Our model is general enough to include space-time coding, BLAST architectures, and single-transmit multi-receive antenna systems which employ any type of channel coding. We derive a closed-form expression for the codeword pairwise error probability (PEP) of general multi-antenna codes using moment generating function and Laplace transform arguments. We then consider space-time orthogonal block (STOB) coding and prove that, similar to the maximum likelihood (ML) decoding case, detection of symbols is decoupled in MAP decoding. We also derive the symbol PEP in closed-form for STOB codes. We apply these results in several scenarios. First, we design a binary antipodal signaling scheme which minimizes the system bit error rate (BER) under STOB coding. At a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> , this constellation has a channel signal-to-noise ratio (CSNR) gain of 4.7 dB over conventional BPSK signaling for a binary nonuniform source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> Delta <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">=</sub> P(0) = 0.9. We next design space-time linear dispersion (LD) codes which are optimized for the source distribution under the criterion of minimizing the union upper bound on the frame error rate (FER). Two codes are given here: one outperforms V-BLAST by 3.5 dB and Alamouti's code by 12.3 dB at an FER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> for a binary source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9, and the other outperforms V-BLAST by 4.2 dB at an FER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> for a uniform source. These codes also outperform the LD codes of constructed under a different criteria. Finally, the problem of bit-to-signal mapping is studied. It is shown that for a binary source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9, 64-QAM signaling, and SER = 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> , a gain of 3.7 dB can be achieved using a better-than-Gray mapping. For a system with one transmit and two receive antennas that uses trellis coding with 16-QAM signaling, a 1.8 dB gain over quasi-Gray mapping and ML decoding is observed when MAP decoding is used for binary sources with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9.

This work deals with coding systems based on sparse graph codes. The key issue we address is the relationship between iterative (in particular belief propagation) and maximum a posteriori decoding. We show that between the two there is a fundamental connection, which is reminiscent of the Maxwell construction in thermodynamics. The main objects we consider are EXIT-like functions. EXIT functions were originally introduced as handy tools for the design of iterative coding systems. It gradually became clear that EXIT functions possess several fundamental properties. Many of these properties, however, apply only to the erasure case. This motivates us to introduce GEXIT functions that coincide with EXIT functions over the erasure channel. In many aspects, GEXIT functions over general memoryless output-symmetric channels play the same role as EXIT functions do over the erasure channel. In particular, GEXIT functions are characterized by the general area theorem. As a first consequence, we demonstrate that in order for the rate of an ensemble of codes to approach the capacity under belief propagation decoding, the GEXIT functions of the component codes have to be matched perfectly. This statement was previously known as the matching condition for the erasure case. We then use these GEXIT functions to show that in the limit of large blocklengths a fundamental connection appears between belief propagation and maximum a posteriori decoding. A decoding algorithm, which we call Maxwell decoder, provides an operational interpretation of this relationship for the erasure case. Both the algorithm and the analysis of the decoder are the translation of the Maxwell construction from statistical mechanics to the context of probabilistic decoding. We take the first steps to extend this construction to general memoryless output-symmetric channels. More exactly, a general upper bound on the maximum a posteriori threshold for sparse graph codes is given. It is conjectured that the fundamental connection between belief propagation and maximum a posteriori decoding carries over to the general case.

In this paper, we study the decoding error probability of linear codes over the erasure channel under the list decoding. The notion of the $q^\ell $ -incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding. By calculating the $q^\ell $ -incorrigible set distributions, the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions. For the ensemble of all $[n,k]$ linear codes, the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined. Furthermore, the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all $[n,nR]$ linear codes.

Irregular LDPC codes can achieve better error rate performance than regular LDPC codes. However, irregular LDPC codes have higher error floors than regular LDPC codes. The Ordered Statistic Decoding (OSD) algorithm achieves approximate Maximum Likelihood (ML) decoding. ML decoding is effective to lower error floors. However, the OSD estimates satisfy the parity check equation of the LDPC code even the estimates are wrong. Hybrid decoder combining LLR-BP decoding algorithm and the OSD algorithm cannot also lower error floors, because wrong estimates also satisfy the LDPC parity check equation. We proposed the concatenated code constructed with an inner irregular LDPC code and an outer Cyclic Redundancy Check (CRC). Owing to CRC, we can detect wrong codewords from OSD estimates. Our CRC-LDPC code with hybrid decoder can lower error floors in an AWGN channel. In wireless communications, we cannot neglect the effects of the channel. The OSD algorithm needs the ordering of each bit based on the reliability. The Channel State Information (CSI) is used for deciding reliability of each bit. In this paper, we evaluate the Block Error Rate (BLER) of the CRC-LDPC code with hybrid decoder in a fast fading channel with perfect and imperfect CSIs where 'imperfect CSI' means that the distribution of channel and those statistical average of the fading amplitudes are known at the receiver. By computer simulation, we show that the CRC-LDPC code with hybrid decoder can lower error floors than the conventional LDPC code with hybrid decoder in the fast fading channel with perfect and imperfect CSIs. We also show that combining error detection with the OSD algorithm is effective not only for lowering the error floor but also for reducing computational complexity of the OSD algorithm.

We investigate the performance of fountain codes on erasure channels with message passing and maximum likelihood decoding algorithms. Fountain codes under message passing decoding are efficient but, for short data length, the decoding overhead may not be optimal. We focus on reducing overhead without losing the code's efficiency and universality, and propose a hybrid message passing and fast maximum likelihood decoding algorithm for LT codes. We also design Raptor codes with one stage pre-codes which are remarkable LDPC codes in the IEEE 802.16e standard. Simulation results show that small dimension fountain codes can save impressive decoding overhead with the maximum likelihood algorithm and the hybrid scheme.

In this paper, an integrated Orthogonal Frequency Division Multiplexing (OFDM) system is developed to improve Bit Error Rate (BER) performance of communication systems in order to meet the requirements of advanced communication systems like long term evaluation advanced (LTE –A). The proposed OFDM system consists of channel estimation using comb type pilot arrangement and convolution encoder with the best known Coding Bounds and Maximum A Posteriori (MAP) decoder. Using pilot symbols, the channel quality is estimated. The BER performance of OFDM system is improved using pilot carriers, channel estimation schemes and MAP decoder at the best known coding bounds. The system performance is analyzed with recursive systematic and non-systematic convolution codes, ML and MAP decoders. From the results, it is evident that the proposed OFDM system with channel estimation and MAP decoder at the best known coding bounds improves the system performance significantly.

The error probability of maximum-likelihood (ML) soft-decision decoded binary block codes rarely accepts exact closed forms. In addition, for long codes ML decoding becomes prohibitively complex. Nevertheless, bounds on the performance of ML decoded systems provide insight into the effect of system parameters on the overall system performance in addition to a measure of efficiency of the sub-optimum decoding methods used in practice. In the article, a comprehensive study of a number of lower and upper bounds on the error probability of ML decoding of binary codes over AWGN channel is provided. Bounds considered here are bounds based on the so-called Bonferroni-type inequalities and bounds developed primarily in the light of the geometrical structure of the underlying signal constellations. The interrelationships among the bounds are explored and current tightest bounds at different noise levels are pointed out.

We present two bounds (one lower bound and one upper bound) on the probability of a union of a finite number of events. The bounds-which are expressed in terms of only the individual event probabilities and the pairwise event probabilities-are applied to examine the symbol error (P/sub s/) and bit error (P/sub b/) probabilities of an uncoded communication system used in conjunction with M-ary PSK/QAM modulations and maximum a posteriori (MAP) decoding over AWGN channels. It is shown that both bounds provide a good, and often excellent, estimate of the error probabilities over the entire range of the signal-to-noise ratio E/sub b//N/sub 0/. The upper bound on P/sub s/ and the lower bound on P/sub b/ are particularly impressive as they agree with the simulation results even during very severe channel conditions.

In this paper, we propose a near maximum likelihood (ML) decoding technique, which reduces the computational complexity of the exact ML decoding algorithm. The computations needed for the tree search in the ML decoding is simplified by reducing the dimension of the search space prior to the tree search. In order to compensate performance loss due to the dimension reduction, a list stack algorithm (LSA) is considered, which produces a list of the top K closest points. The combination of both approaches, called reduced dimension list stack algorithm (RD-LSA), is shown to provide flexibility and offers a performance-complexity trade-off. Simulations performed for V-BLAST transmission demonstrate that significant complexity reduction can be achieved compared to the sphere decoding algorithm (SDA) while keeping the performance loss below an acceptable level.

Recently, it has been verified that the Reed-Muller (RM) code, which has been predicted for a long time in the field of code theory, achieves capacity in a binary erasure channel (BEC) with a MAP (maximum a posteriori) decoder. Through weight distribution interpretation, it was also discovered that the RM code achieves a binary memoryless symmetric (BMS) channel capacity. A successive cancellation list (SCL) technique has been studied as a practical decoder instead of a MAP decoder. In this paper, the performance improvement potential that can be improved when a generalized concatenated code (GCC) with an outer code is designed to improve the practical SCL decoder performance is analyzed. When information bits are used as frozen bits in these code-word designs, we analyze an inter-code-word distance and the decoding outage probability of the designed code. We expect that our results can be used as a conceptual base for improving communication methods in the future development of channel encoding technology.

Updating cover type maps using sequential indicator simulation

Convolutional low-density parity-check (LDPC) ensembles, introduced by Felström and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing functions of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles. We describe the fundamental mechanism that explains why “convolutional-like” or “spatially coupled” codes perform so well. In essence, the spatial coupling of individual codes increases the belief-propagation (BP) threshold of the new ensemble to its maximum possible value, namely the maximum a posteriori (MAP) threshold of the underlying ensemble. For this reason, we call this phenomenon “threshold saturation.” This gives an entirely new way of approaching capacity. One significant advantage of this construction is that one can create capacity-approaching ensembles with an error correcting radius that is increasing in the blocklength. Although we prove the “threshold saturation” only for a specific ensemble and for the binary erasure channel (BEC), empirically the phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the “dynamical” threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms and new techniques for analysis.

The design of moderate-length erasure correcting low-density parity-check (LDPC) codes over correlated erasure channels is considered. Although the asymptotic LDPC code design remains the same as for a memoryless erasure channel, robustness to the channel correlation shall be guaranteed for the finite length LDPC code. This further requirement is of great importance in several wireless communication scenarios where packet erasure correcting codes represent a simple countermeasure for correlated fade events (e.g., in mobile wireless broadcasting services) and where the channel coherence time is often comparable with the code length. In this paper, the maximum tolerable erasure burst length (MTBL) is adopted as a simple metric for measuring the code robustness to the channel correlation. Correspondingly, a further step in the code construction is suggested, consisting of improving the LDPC code MTBL. Numerical results conducted over a Gilbert erasure channel, under both iterative and maximum likelihood decoding, highlight both the importance of the MTBL improvement in the finite-length code construction and the possibility to tightly approach the performance of maximum distance separable codes.

Tail-biting codes are efficient for short data packets by eliminating the rate loss in conventional known-tail codes. The existing maximum-likelihood (ML) decoding algorithms, such as the Viterbi heuristic ML decoder (VH ML decoder) and the trap detection-based ML decoder (TD-ML decoder), have to visit each state of the tail-biting trellis at least once. The decoding efficiency of these decoding algorithms can be improved further. In this paper, we propose an ML decoder for tail-biting trellises with bounded searches (BSs). We first perform a unidirectional bounded searching algorithm to estimate the lower bound of path metric of tail-biting paths on each sub-tail-biting trellis and exclude impossible candidates of starting states. In the second phase, a bidirectional searching algorithm is applied to find the ML tail-biting (MLTB) path on the survivor sub-tail-biting trellises. The proposed algorithm exhibits lower decoding complexity floor than other existing algorithms on tail-biting trellis. Simulation results for the (24, 12, 8) Golay codes and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">$E_{b}/N_{0}=\hbox{5}\ \hbox{dB} $</tex-math></inline-formula> show that, for the proposed ML decoder, the average number of visited states per decoded bit is less than 2, whereas the average number of visited states per decoded bit is more than 12 for the VH ML decoder.