Partially systematic rate-compatible punctured SCCCs

In this letter, we propose and compare some new rate-compatible serially concatenated convolutional code (SCCC) families. To obtain them, inner coded bits only are punctured. However, and this is the novelty proposed in the letter, the puncturing is not limited to inner parity bits, but extended also to inner systematic bits, thus obtaining high rate SCCCs (i.e., beyond the outer code rate). The two main applications of this technique are its use in hybrid Automatic-Repeat-reQuest/Forward-Error Correction (ARQ/FEC) schemes and to achieve unequal error protection (UEP) of an information sequence.     

Design of punctured serially concatenated convolutional codes

An effective algorithm for the design of punctured serially concatenated convolutional codes (SCCCs) is proposed. The algorithm is based on the density evolution technique, and its main goal is to design a code matching the outer and the inner encoder in order to reduce the bit error rate (BER) in the SNR operating region ranging from the waterfall to the error floor of the designed SCCCs. The concepts are illustrated for some specific SCCC schemes. Finally, simulation results and comparisons with other approaches proposed in the literature confirm the effectiveness of the proposed algorithm.     

Design of rate-compatible structured LDPC codes for hybrid ARQ applications

In this paper, families of rate-compatible protograph-based LDPC codes that are suitable for incrementalredundancy hybrid ARQ applications are constructed. A systematic technique to construct low-rate base codes from a higher rate code is presented. The base codes are designed to be robust against erasures while having a good performance on error channels. A progressive node puncturing algorithm is devised to construct a family of higher rate codes from the base code. The performance of this puncturing algorithm is compared to other puncturing schemes. Using the techniques in this paper, one can construct a rate-compatible family of codes with rates ranging from 0.1 to 0.9 that are within 1 dB from the channel capacity and have good error floors.     

A cascaded coding scheme for error control and its performanceanalysis

A coding scheme for error control in data communication systems is investigated. The scheme is obtained by cascading two error-correcting codes, called the inner and outer codes. Its error performance is analyzed for a binary symmetric channel with a bit-error rate ϵ<1/2. It is shown that, if the inner and outer codes are chosen properly, high reliability can be attained even for a high-channel bit-error rate. Specific examples with inner codes ranging from high rates to low rates and Reed-Solomon codes as outer codes are considered, and their error probabilities evaluated. They all provide high reliability even for high bit-error rates, say 10^-1-10 ^-2. Several example schemes are being considered for such satellite and spacecraft downlink error control     

Performance analysis and design criteria for finite-alphabet source-channel codes

Efficient compression of finite-alphabet sources requires variable-length codes (VLCs). However, in the presence of noisy channels, error propagation in the decoding of VLCs severely degrades performance. To address this problem, redundant entropy codes and iterative source-channel decoding have been suggested, but to date, neither performance bounds nor design criteria for the composite system have been available. We calculate performance bounds for the source-channel system by generalizing techniques originally developed for serial concatenated convolutional codes. Using this analysis, we demonstrate the role of a recursive structure for the inner code and the distance properties of the outer code. We use density evolution to study the convergence of our decoders. Finally, we pose the question: Under a fixed rate and complexity constraint, when should we use source-channel decoding (as opposed to separable decoding)? We offer answers in several specific cases. For our analysis and design rules, we use union bounds that are technically valid only above the cutoff rate, but interestingly, the codes designed with union-bound criteria perform well even in low signal-to-noise ratio regions, as shown by our simulations as well as previous works on concatenated codes.     

Adaptive Transmission with Variable-Rate Turbo Bit-Interleaved Coded Modulation

We study an adaptive transmission scheme based on variable-rate turbo bit-interleaved coded modulation (VR- Turbo-BICM). The proposed coding scheme employs punctured turbo codes. A continuously varying transmission rate can be obtained by changing the code rate through both puncturing of the coded bits and adapting of the modulation constellation size. The main results are elaborated in two parts. First, we derive a closed-form expression for a set of achievable rate bounds (called rate thresholds) for VR-Turbo-BICM by employing recent results on the parallel channel performance of turbo code ensembles and the BICM parallel channel analysis model. The derived rate threshold is expressed as a fraction of the capacity of BICM with Gray mapping, where this fraction is a turbo code weight spectrum parameter. Simulation results illustrate that introduced rate thresholds predict well the rate versus SNR performance of VR-Turbo-BICM for a wide range of codeword error probabilities and codeword lengths. Next, based on a simplified rate threshold, we derive a power, puncturing rate, and modulation constellation size assignment policy for a slow fading channel.     

The serial concatenation of rate-1 codes through uniform random interleavers

Until the analysis of repeat accumulate codes by Divsalar et al. (1998), few people would have guessed that simple rate-1 codes could play a crucial role in the construction of "good" binary codes. We construct "good" binary linear block codes at any rate r<1 by serially concatenating an arbitrary outer code of rate r with a large number of rate-1 inner codes through uniform random interleavers. We derive the average output weight enumerator (WE) for this ensemble in the limit as the number of inner codes goes to infinity. Using a probabilistic upper bound on the minimum distance, we prove that long codes from this ensemble will achieve the Gilbert-Varshamov (1952) bound with high probability. Numerical evaluation of the minimum distance shows that the asymptotic bound can be achieved with a small number of inner codes. In essence, this construction produces codes with good distance properties which are also compatible with iterative "turbo" style decoding. For selected codes, we also present bounds on the probability of maximum-likelihood decoding (MLD) error and simulation results for the probability of iterative decoding error.     

Accumulate-Repeat-Accumulate Codes

In this paper, we propose an innovative channel coding scheme called accumulate-repeat-accumulate (ARA) codes. This class of codes can be viewed as serial turbo-like codes or as a subclass of low-density parity check (LDPC) codes, and they have a projected graph or protograph representation; this allows for high-speed iterative decoding implementation using belief propagation. An ARA code can be viewed as precoded repeat accumulate (RA) code with puncturing or as precoded irregular repeat accumulate (IRA) code, where simply an accumulator is chosen as the precoder. The amount of performance improvement due to the precoder will be called precoding gain. Using density evolution on their associated protographs, we find some rate-1/2 ARA codes, with a maximum variable node degree of 5 for which a minimum bit SNR as low as 0.08 dB from channel capacity threshold is achieved as the block size goes to infinity. Such a low threshold cannot be achieved by RA, IRA, or unstructured irregular LDPC codes with the same constraint on the maximum variable node degree. Furthermore, by puncturing the inner accumulator, we can construct families of higher rate ARA codes with thresholds that stay close to their respective channel capacity thresholds uniformly. Iterative decoding simulation results are provided and compared with turbo codes. In addition to iterative decoding analysis, we analyzed the performance of ARA codes with maximum-likelihood (ML) decoding. By obtaining the weight distribution of these codes and through existing tightest bounds we have shown that the ML SNR threshold of ARA codes also approaches very closely to that of random codes. These codes have better interleaving gain than turbo codes     

Coding for Parallel Channels: Gallager Bounds and Applications to Turbo-Like Codes

The transmission of coded communication systems is widely modeled to take place over a set of parallel channels. This model is used for transmission over block-fading channels, rate-compatible puncturing of turbo-like codes, multicarrier signaling, multilevel coding, etc. New upper bounds on the maximum-likelihood (ML) decoding error probability are derived in the parallel-channel setting. We focus on the generalization of the Gallager-type bounds and discuss the connections between some versions of these bounds. The tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes, repeat-accumulate (RA) codes, and some of their recent variations (e.g., punctured accumulate-repeat-accumulate codes). The bounds on the decoding error probability of an ML decoder are compared to computer simulations of iterative decoding. The new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels. This improvement is exemplified for relatively short block lengths, and it is pronounced when the block length is increased. In the asymptotic case, where we let the block length tend to infinity, inner bounds on the attainable channel regions of modern coding techniques under ML decoding are obtained, based solely on the asymptotic growth rates of the average distance spectra of these code ensembles.     

Performance of Concatenated Dual-k Codes on a Rayleigh Fading Channel with a Bandwidth Constraint

The error rate performance, obtained on a Rayleigh fading channel, is determined and illustrated graphically for concatenated codes in which the outer code is a dual-kconvolutional code and the inner code is either a Hadamard block code or a block orthogonal code. Comparison of the performance of these two types of concatenated codes is made on the basis of the same bandwidth utilization. It is also illustrated that the concatenated dual-kcode yields a significant improvement in performance relative to the performance achieved with the inner code alone.     

Low complexity concatenated coding schemes for digital satellite communications

A study of reduced complexity concatenated coding schemes, for commercial digital satellite systems with low-cost earth terminals, is reported. The study explored trade-offs between coding gain, overall rate and decoder complexity, and compared concatenated schemes with single codes. It concentrated on short block and constraint length inner codes, with soft decision decoding, concatenated with a range of Reed-Solomon outer codes. The dimension of the inner code was matched to the outer code symbol size, and appropriate interleaving between the inner and outer codes was used. Very useful coding gains were achieved with relatively high-rate, low-complexity schemes. For example, concatenating the soft decision decoded (9,8) single parity check inner code with the CCSDS recommended standard Reed-Solomon outer code gives a coding gain of 4.8dB at a bit error probability of 10^?5, with an overall rate of 0-78.     

A comparison of known codes, random codes, and the best codes

This paper calculates new bounds on the size of the performance gap between random codes and the best possible codes. The first result shows that, for large block sizes, the ratio of the error probability of a random code to the sphere-packing lower bound on the error probability of every code on the binary symmetric channel (BSC) is small for a wide range of useful crossover probabilities. Thus even far from capacity, random codes have nearly the same error performance as the best possible long codes. The paper also demonstrates that a small reduction k-k˜ in the number of information bits conveyed by a codeword will make the error performance of an (n,k˜) random code better than the sphere-packing lower bound for an (n,k) code as long as the channel crossover probability is somewhat greater than a critical probability. For example, the sphere-packing lower bound for a long (n,k), rate 1/2, code will exceed the error probability of an (n,k˜) random code if k-k˜>10 and the crossover probability is between 0.035 and 0.11=H^-1(1/2). Analogous results are presented for the binary erasure channel (BEC) and the additive white Gaussian noise (AWGN) channel. The paper also presents substantial numerical evaluation of the performance of random codes and existing standard lower bounds for the BEC, BSC, and the AWGN channel. These last results provide a useful standard against which to measure many popular codes including turbo codes, e.g., there exist turbo codes that perform within 0.6 dB of the bounds over a wide range of block lengths     

Design of Rate-Compatible Irregular Repeat Accumulate Codes

We consider the design of efficient rate-compatible (RC) irregular repeat accumulate (IRA) codes over a wide code rate range. The goal is to provide a family of RC codes to achieve high throughput in hybrid automatic repeat request (ARQ) scheme for high-speed data packet wireless systems. As a subclass of low-density parity-check codes, IRA codes have an extremely simple encoder and a low-complexity decoder while providing capacity approaching performance. We focus on a hybrid design method which employs both puncturing and extending. We propose a simple puncturing method based on minimizing the maximal recoverable step of the punctured nodes. We also propose a new extending scheme for IRA codes by introducing the degree-1 parity bits for the lower rate codes and obtaining the optimal proportions of extended nodes through density evolution analysis. The throughput performance of the designed RC-IRA codes in hybrid ARQ is evaluated for both AWGN and block fading channels. Simulation results demonstrate that our designed RC codes offer good error correction performance over a wide rate range and provide high throughput, especially in the high and low signal-to-noise ratio regions.     

基于分段凿孔的极化码级联方案

极化码拥有出色的纠错性能,但编码方式决定了其码长不够灵活,需要通过凿孔构造码长可变的极化码。该文引入矩阵极化率来衡量凿孔对极化码性能的影响,选择矩阵极化率最大的码字作为最佳凿孔模式。对极化码的码字进行分段,有效减小了最佳凿孔模式的搜索运算量。由于各分段的第1个码字都会被凿除,且串行抵消译码过程中主要发生1位错,因此在各段段首级联奇偶校验码作为译码提前终止标志,检测前段码字的译码错误并进行重新译码。对所提方法在串行抵消译码下的性能进行仿真分析,结果表明,相比传统凿孔方法,所提方法在10–3误码率时能获得约0.7 dB的编码增益,有效提升了凿孔极化码的译码性能。     

On the performance of hybrid FEC/ARQ systems using rate compatiblepunctured turbo (RCPT) codes

This paper introduces a hybrid forward-error correction/automatic repeat-request (ARQ) system that employs rate compatible punctured turbo (RCPT) codes to achieve enhanced throughput performance over a nonstationary Gaussian channel. The proposed RCPT-ARQ system combines the performance of turbo codes with the frugal use of incremental redundancy inherent in the rate compatible punctured convolutional codes of Hagenauer (1988). Moreover, this paper introduces the notion of puncturing the systematic code symbols of a turbo code to maximize throughput at signal-to-noise ratios (SNRs) of interest. The resulting system provides both an efficient family of achievable code rates at middle to high SNR and powerful low-rate error correction capability at low SNR     

Convolutional double accumulate codes (or double turbo DPSK)

We consider the serial concatenation of a convolutional code with two differential encoders (or accumulate codes). Simulations of such a concatenated scheme using a rate 1/2 outer convolutional code (sometimes punctured to increase the overall rate) show that, at high rates, the bit error rate (BER) and frame error rate (FER) performance of these codes is significantly better than a concatenation with only one accumulate code and a parallel concatenated convolutional code (PCCC) of similar complexity. An intuitive explanation for the performance of these codes is also given     

Improved space-time codes using serial concatenation

In this letter, a new family of space-time codes is proposed. These codes employ a serially concatenated coding scheme with a standard space-time code as the outer code and a very simple rate-1 recursive code as the inner code. Adding this simple rate-1 recursive inner code does not decrease the bit rate and introduces only negligible complexity increase to the transmitter when compared to cases with standard space-time codes. An interleaver is embedded between the inner coder and the outer coder and the size of this interleaver determines the performance gain. We also provide a relatively low complexity iterative decoding procedure. For applications which can tolerate delay, significant gain can be achieved with the proposed approach     

BER performance of turbo-coded PPM CDMA systems on optical fiber

We obtain upper bounds on the bit error rate (BER) for turbo-coded optical code-division multiple-access (CDMA) systems using pulse position modulation (PPM). We use transfer function bounding techniques to obtain these bounds, so our results correspond to the average bound over all interleavers of a given length. We consider parallel concatenated coding (PCC) schemes that use recursive convolutional codes as constituent codes. We consider systems using an avalanche photodiode (APD), and treat APD noise, thermal noise, and multi-user interference using a Gaussian approximation. We compare the performance of turbo-coded systems with that of BCH-coded systems with soft-decision decoding, and that of concatenated coding systems with outer Reed-Solomon (RS) code and inner convolutional code. We show that turbo-coded systems have better performance than BCH-coded systems. We also show that concatenated systems have better performance than turbo-coded systems when the block length is small and the received laser power is somewhat large     

The design of efficiently-encodable rate-compatible LDPC codes - [transactions papers]

We present a new class of irregular low-density parity-check (LDPC) codes for moderate block lengths (up to a few thousand bits) that are well-suited for rate-compatible puncturing. The proposed codes show good performance under puncturing over a wide range of rates and are suitable for usage in incremental redundancy hybrid-automatic repeat request (ARQ) systems. In addition, these codes are linear-time encodable with simple shift-register circuits. For a block length of 1200 bits the codes outperform optimized irregular LDPC codes and extended irregular repeat-accumulate (eIRA) codes for all puncturing rates 0.6~0.9 (base code performance is almost the same) and are particularly good at high puncturing rates where good puncturing performance has been previously difficult to achieve.     

General random coding bounds: AWGN channels to MIMO fading channels

Random coding bounds are obtained for multiple-input multiple-output (MIMO) fading channels. To derive the result in a compact and easy-to-evaluate form, a series of combinatorial codeword enumeration problems are solved for input-constrained MIMO fading channels. The bounds obtained in this paper are shown useful as performance prediction measures for MIMO systems which employ turbo-like block codes as the outer code to derive the space-time inner code. The error exponents for MIMO channels are also derived from the bounds, and then compared with the classical Gallager error exponents as well as the channel capacities. The random coding bounds associated with the maximum likelihood receiver exhibit good match with the extensive system simulation results obtained with a turbo-iterative receiver.