Free-space optical (FSO) links are sensitive to channel fading caused by atmospheric turbulence, varying weather conditions, and changes in the distance between the transmitter and receiver. To mitigate FSO fading, this paper applies linear and quadratic prediction to estimate fading channel conditions and dynamically select the appropriate low-density parity check (LDPC) code rate. This adaptivity achieves reliable communication while efficiently utilizing the available channel mutual information. Protograph-based Raptor-like (PBRL) LDPC codes supporting a wide range of rates are designed, facilitating convenient rate switching. When channel state information (CSI) is known without delay, dynamically selecting LDPC code rate appropriately maximizes throughput. This work explores how such prediction behaves as the feedback delay is increased from no delay to a delay of 4 ms for a channel with a coherence time of 10 ms.

Recently, neural networks have improved MinSum message-passing decoders for low-density parity-check (LDPC) codes by multiplying or adding weights to the messages, where the weights are determined by a neural network. The neural network complexity to determine distinct weights for each edge is high, often limiting the application to relatively short LDPC codes. Furthermore, storing separate weights for every edge and every iteration can be a burden for hardware implementations. To reduce neural network complexity and storage requirements, this paper proposes a family of weight-sharing schemes that use the same weight for edges that have the same check node degree and/or variable node degree. Our simulation results show that node-degree-based weight-sharing can deliver the same performance requiring distinct weights for each node. This paper also combines these degree-specific neural weights with a reconstruction-computation-quantization (RCQ) decoder to produce a weighted RCQ (W-RCQ) decoder. The W-RCQ decoder with node-degree-based weight sharing has a reduced hardware requirement compared with the original RCQ decoder. As an additional contribution, this paper identifies and resolves a gradient explosion issue that can arise when training neural LDPC decoders.

The Consultative Committee for Space Data Systems (CCSDS) 141.11-O-1 Line Product Code (LPC) provides a rare opportunity to compare maximum-likelihood decoding and message passing. The LPC considered in this paper is intended to serve as the inner code in conjunction with a (255,239) Reed Solomon (RS) code whose symbols are bytes of data. This paper represents the 141.11-O-1 LPC as a bipartite graph and uses that graph to formulate both maximum likelihood (ML) and message passing algorithms. ML decoding must, of course, have the best frame error rate (FER) performance. However, a fixed point implementation of a Neural-Normalized MinSum (N-NMS) message passing decoder closely approaches ML performance with a significantly lower complexity.

Non-uniform message quantization techniques such as reconstruction-computation-quantization (RCQ) improve error-correction performance and decrease hardware complexity of low-density parity-check (LDPC) decoders that use a flooding schedule. Layered MinSum RCQ (L-msRCQ) enables message quantization to be utilized for layered decoders and irregular LDPC codes. We investigate field-programmable gate array (FPGA) implementations of L-msRCQ decoders. Three design methods for message quantization are presented, which we name the Lookup, Broadcast, and Dribble methods. The decoding performance and hardware complexity of these schemes are compared to a layered offset MinSum (OMS) decoder. Simulation results on a (16384, 8192) protograph-based raptor-like (PBRL) LDPC code show that a 4-bit L-msRCQ decoder using the Broadcast method can achieve a 0.03 dB improvement in error-correction performance while using 12% fewer registers than the OMS decoder. A Broadcast-based 3-bit L-msRCQ decoder uses 15% fewer lookup tables, 18% fewer registers, and 13% fewer routed nets than the OMS decoder, but results in a 0.09 dB loss in performance.

This paper uses convolutional codes (CCs) with distance-spectrum optimal (DSO) cyclic redundancy checks (CRCs) and the serial list Viterbi algorithm (S-LVA) to approach the random coding union (RCU) bound with low decoding complexity at the target FER. We show, for example, that a 64-state zero-terminated CC with a DSO CRC can achieve performance within 0.5 dB of the RCU bound for information blocklength k=64 at FER of 10-3. We also show that a tail-biting CC with a DSO CRC can achieve even better performance, within 0.05 dB of the RCU bound at FER of 10-4for a 256-state CC with k=64. This paper provides analysis of decoding complexity, which for S-LVA depends on the expected list size. We show that if the target FER is low enough, the expected list size approaches one so that the average complexity of S-LVA approaches that of standard soft Viterbi on the CC, i.e., with no list decoding. We also provide DSO CRCs for CCs with k=64 and rates of 1/2, 1/3, 1/6 and 1/12 for the 5G new radio control channel and compare their performance with polar codes.

This paper provides a reduced-complexity approach to maximum likelihood (ML) decoding of cyclic codes. A cyclic code with generator polynomial gcyclic(x) may be considered a terminated convolutional code with a nominal rate of 1. The trellis termination redundancy lowers the rate from 1 to the actual rate of the cyclic code. The proposed decoder represents gcyclic(x) as the product of two polynomials, a convolutional code (CC) polynomial gcc(x) and a cyclic redundancy check (CRC) polynomial gcrc(x), i.e., gcyclic(x) = gcc(x)gcrc(x). This representation facilitates serial list Viterbi algorithm (S-LVA) decoding. Viterbi decoding is performed on the natural trellis for gcc(x), and gcrc(x) is used as a CRC to determine when the S-LVA should conclude. At typical target frame error rates, the expected list size of S-LVA is small, and the average decoding complexity is dominated by the trellis complexity of gcc(x) rather than gcyclic(x). Some high-rate binary Bose-Chaudhuri- Hocquenghem (BCH) examples show that the proposed use of S-LVA via factorization significantly lowers complexity as compared to using the minimum-complexity trellis representation of gcyclic(x) for soft ML decoding.

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a family of easily encodable rate-compatible low-density parity-check (LDPC) codes. PBRL codes have an excellent performance across all design rates. Quasi-cyclic (QC) PBRL code families permit high-speed decoder implementations. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the PBRL family at various design rates. This paper introduces a design method to obtain better QC PBRL code families at short block-lengths (of a few hundred bits) for low frame error rate (FER) requirements. We first select a protomatrix for the highest design rate. To add a new row to lower the rate, we keep all the previously obtained rows of the PBRL protomatrix fixed and select the new row that maximizes an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protomatrix. The new QC PBRL code families outperform the original PBRL codes at short block-lengths by providing a significantly better low-FER performance. The standard approach to computing the aforementioned upper bounds requires complexity that grows exponentially with the size of the protomatrix. However, we show that the structure of the PBRL protomatrix lets us obtain the upper bounds with complexity that grows only linearly with the size of the PBRL protomatrix. Using the complexity reduction results, we also establish an equivalence between the exhaustive search to design a new row for the PBRL protomatrix according to the new design method and an integer linear program.

A rate-compatible (RC) code first transmits a set of symbols corresponding to the highest rate. These symbols form the highest-rate code (HRC). If requested by the receiver, the transmitter subsequently sends symbols that lower the rate of the code. Additional symbols are sent until the decoder decodes to a codeword or all symbols of the RC code are exhausted. Consider linear, RC low-density parity-check (LDPC) codes constructed using extending variable nodes of degree 1. That is, every symbol of incremental redundancy (IR) is a linear combination only of symbols of the HRC. We study the convergence of such codes under iterative decoding. We show that the convergence criterion considered after each iteration need only check whether the HRC variable nodes have converged to a codeword. Specifically, there is no need to consider whether the parity checks that generate the IR symbols are satisfied. We substantiate these claims with simulation results of protograph-based raptor-like LDPC (PBRL) codes, which are a family of protograph RC-LDPC codes with the extending structure under consideration. Furthermore, we demonstrate using examples that this extending structure for protograph RC codes is not very far from the optimal extension for a protograph RC code by providing examples of iterative decoding thresholds for PBRL protographs and protographs extended using the optimal degrees for incremental variable nodes.

Protograph-based Raptor-like low-density parity-check codes (PBRL codes) are a recently proposed family of easily encodable and decodable rate-compatible LDPC (RC-LDPC) codes. These codes have an excellent iterative decoding threshold and performance across all design rates. PBRL codes designed thus far, for both long and short block-lengths, have been based on optimizing the iterative decoding threshold of the protograph of the RC code family at various design rates. In this work, we propose a design method to obtain better quasi-cyclic (QC) RC-LDPC codes with PBRL structure for short block-lengths (of a few hundred bits). We achieve this by maximizing an upper bound on the minimum distance of any QC-LDPC code that can be obtained from the protograph of a PBRL ensemble. The obtained codes outperform the original PBRL codes at short block-lengths by significantly improving the error floor behavior at all design rates. Furthermore, we identify a reduction in complexity of the design procedure, facilitated by the general structure of a PBRL ensemble.

This paper uses a case study of a tail-biting convolutional code (with successful decoding indicated by the reliability output Viterbi algorithm) to present an information density approach for analyzing and optimizing the throughput of systems using incremental redundancy controlled by feedback. Polyan-skiy's normal approximation combined with a linear model for the information gap of a rate-compatible code family provides a simple and accurate characterization of the behavior of feedback systems employing practical codes, such as convolutional or low-density parity-check codes. Especially for short message lengths on the order of k <; 50 message bits, the newly proposed model is more accurate than Vakilinia's model in which the rate of first successful decoding has a Gaussian probability density function.

Recent studies on noisy decoding for LDPC codes rely on the assumption that the noise in each component is independent and perpetual. This paper examines a noisy decoding model that generalizes this approach: the noise is due to multi-state channels, where the channel states are governed by queue-like processes. This model is inspired by errors in decoders that are due to the high levels of radiation. This is an important problem, as modern non-volatile memories (NVMs) must perform well in high-radiation environments if they are to be used for deep space applications. High levels of radiation have a significant impact on floating gate-based NVMs, such as flash, and therefore, require well-tuned, powerful error-correcting codes for reliable data storage along with the decoders capable of handling radiation-induced noisy components. We introduce a noisy LDPC decoding model subsuming certain previously studied models. This model is better suited to represent transient errors-in both variable nodes and check nodes-and allows for a more refined analysis compared with older, coarser models. We perform a density evolution-like theoretical evaluation, applicable to both regular and irregular codes, optimize the voting threshold for a Gallager B/E-decoder, and analyze the resulting evaluation. We also examine the finite block length case.

This paper presents a general approach for optimizing the number of symbols in increments (packets of incremental redundancy) in a feedback communication system with a limited number of increments. This approach is based on a tight normal approximation on the rate for successful decoding. Applying this approach to a variety of feedback systems using nonbinary (NB) low-density parity-check (LDPC) codes shows that greater than 90% of capacity can be achieved with average blocklengths fewer than 500 transmitted bits. One result is that the performance with ten increments closely approaches the performance with an infinite number of increments. The paper focuses on binary-input additive-white Gaussian noise (BI-AWGN) channels but also demonstrates that the normal approximation works well on examples of fading channels as well as high-SNR AWGN channels that require larger QAM constellations. This paper explores both variable-length feedback codes with termination (VLFT) and the more practical variable length feedback (VLF) codes without termination that require no assumption of noiseless transmitter confirmation. For VLF, we consider both a two-phase scheme and CRC-based scheme.

We consider the problem of constructing (3,L) quasi-cyclic low-density parity-check (LDPC) codes from complete protographs. A complete protograph is a small bipartite graph with two disjoint vertex sets such that every vertex in the variable-node set is connected to every vertex in the check-node set by a unique edge. This paper analyzes the required lifting factor for achieving girths of six or eight in the resulting quasi-cyclic codes with constraints on lifting. The required lifting factors provide lower bounds on the block-length of such codes.

Research works exploring coding for Flash memories typically seek to correct errors taking place during normal device operation. In this paper, we study the design of codes that protect Flash devices dealing with the unusual class of errors caused by exposure to large radiation dosages. Significant radiation exposure can take place, for example, when Flash is used as on-board memory in satellites and space probes. We introduce an error model that captures the effects of radiation exposure. Such errors are asymmetric, with the additional feature that the degree (and direction) of asymmetry depends on the stored sequence. We develop an appropriate distance and an upper bound on the sizes of codes which correct such errors. We introduce and analyze several simple code constructions.

This paper uses an extension of Reciprocal Channel Approximation (RCA) to accurately and efficiently predict the frame error rate (FER) performance of polar codes by analyzing the probability density function (p.d.f) of log likelihood ratios (LLR) associated with information bits. A feedback scheme uses the RCA to predict the p.d.f of LLRs in conjunction with a repetition coding system to decrease the blocklength required for a target FER by a factor of 16. Using a rate-0.5 128-bit polar code as the initially transmitted code, the FER of the system with feedback is obtained by theoretical analysis and verified by simulation. Including the additional incremental transmissions the average blocklength for the system with feedback is 137.55 bits and the rate is 0.4653. Without feedback, a polar code with blocklength 2048 is required to achieve a comparable FER at a comparable rate. Intuitively, feedback allows the polar code to use fewer frozen bits in the initial transmission and then uses repetition codes to provide the needed reliability to resolve unreliable unfrozen bits identified by feedback.

This paper proposes protograph-based Raptor-like (PBRL) codes as a class of rate-compatible low-density parity-check codes for binary-input AWGN channels. As with the Raptor codes, exclusive-OR operations on precoded bits produce additional parity bits providing extensive rate compatibility. Unlike Raptor codes, each additional parity bit in the protograph is explicitly designed to optimize the density evolution threshold. During the lifting process, approximate cycle extrinsic message degree (ACE) and circulant progressive edge growth (CPEG) constraints are used to avoid undesirable graphical structures. Some density-evolution performance is sacrificed to obtain lower error floors, particularly at short block-lengths. Simulation results are shown for information block sizes of k = 1032 and 16 384. For a target frame error rate of 10-5, at each rate, the k = 1032 and 16 384 code families perform within 1 dB and 0.4 dB of both the Gallager bound and the normal approximation, respectively. The 16 384 code family outperforms the best known standardized code family, namely, the AR4JA codes. The PBRL codes also outperform DVB-S2 codes that have the advantages of longer blocklengths and outer BCH codes. Performance is similar to RC code families designed by Nguyen et al. that do not constrain codes to have the PBRL structure and involve simulation in the optimization process at each rate.

This paper considers finite-blocklength achievability for rate-compatible codes. For a fixed number of messages, random coding analysis determines a sequence of achievable error probabilities for a sequence of blocklengths. However, traditional random coding achievability draws each code independently so that it does not show that a family of rate-compatible codes achieves that same sequence of error probabilities. Using random code extension, this paper shows achievable error probabilities for rate-compatible channel codes with finite blocklengths. This paper also shows that for a class of input-invariant channels, the rate-compatible constraint does not affect the achievability bounds on error rates when a threshold decoder is used.

Following Poulliat et al.'s design of (2, dc) nonbinary LDPC (NB-LDPC) codes, this paper designs high-rate irregular NB-LDPC codes by addressing the problem of minimum symbol distance. The design procedure first identifies all stopping sets up to weight five in an LDPC code and enumerates them via a message passing algorithm. For each identified stopping set, careful labeling forces its corresponding parity-check sub-matrix to be full rank, thereby preventing the stopping set from being a sub-code and ensuring a minimum distance of at least six symbols. Simulation results for codes designed through this procedure show a significant improvement in the error-floor region over randomized labeling.

One advantage of feedback in a point-to-point memoryless channel is the reduction of the average blocklength required to approach capacity. This paper presents a communication system with feedback that uses carefully designed non-binary LDPC (NB-LDPC) codes and incremental transmissions to achieve 92–94% of the idealized throughput of rate-compatible sphere-packing with maximum-likelihood decoding (RCSP-ML) for average blocklengths of 150–450 bits. The system uses active feedback by carefully selecting each bit of additional incremental information to improve the reliability of the least reliable variable node. The system uses post processing in the decoder to further improve performance. The average blocklengths of 150–450 bits are small enough that feedback provides a throughput advantage but also large enough that overhead that might be associated with transmitter confirmation is more easily tolerated.

One advantage of incremental transmissions with feedback in point-to-point memoryless channels is a reduction in average blocklength required to approach capacity. This paper optimizes the size of each incremental transmission for non-binary (NB) LDPC codes to maximize throughput in VLFT and two-phase VLF settings. The optimization problem uses an approximation based on the inverse-Gaussian p.d.f. of the blocklength required for successful decoding. By using the optimized incremental transmission lengths (with an average blocklength of less than 500 bits), NB-LDPC codes for VLFT setting limited to 5 transmissions achieve a throughput greater than 96% of that obtained by an unlimited-transmission VLFT scheme with the same average blocklength. With a similar average blocklength, a two-phase VLF system limited to five transmissions (with optimized lengths) using the binary image of NB-LDPC codes achieves greater than 90% of the capacity of binary-input AWGN channel with SNR=2 dB. Two-phase VLF does not match the throughput of VLFT, but it is more practical than VLFT because it does not assume noiseless transmitter confirmation.

This paper provides a comprehensive analysis of nonbinary low-density parity check (LDPC) codes built out of protographs. We consider both random and constrained edge-weight labeling, and refer to the former as the unconstrained nonbinary protograph-based LDPC codes (U-NBPB codes) and to the latter as the constrained nonbinary protograph-based LDPC codes (C-NBPB codes). Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codewords, trapping sets, stopping sets, and pseudocodewords are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. An EXIT chart tool for computing the iterative decoding thresholds of protograph-based LDPC codes is presented, followed by several examples of finite-length U-NBPB and C-NBPB codes with high performance. Throughout this paper, we provide accompanying examples, which demonstrate the advantage of nonbinary protograph-based LDPC codes over their binary counterparts and over random constructions. The results presented in this paper advance the analytical toolbox of nonbinary graph-based codes.

Closed form upper bounds to error probabilities of nonbinary coded systems with two and higher dimensional constellation over the additive white Gaussian noise channel are derived. Computation of these bounds do not require integrations nor parameter optimizations. These bounds are the extension of bounds derived by Divsalar in 1999 for binary codes with binary modulations. The proposed bounds require only the knowledge of the average pairwise Euclidean distance enumerator of the code words when certain symmetry conditions do not hold. The bounds are tight for large block lengths. We also briefly discuss how to extend the pairwise Euclidean distance enumerators to ensembles of nonbinary protograph codes using the notion of frequency weight enumerators.

A new coding scheme for image transmission over noisy channel is proposed. Similar to standard image compression, the scheme includes a linear transform followed by successive refinement scalar quantization. Unlike conventional schemes, though, in the proposed system the quantized transform coefficients are linearly mapped into channel symbols using systematic linear encoders. This fixed-to-fixed length "linear index coding" approach avoids the use of an explicit entropy coding stage (e.g., arithmetic or Huffman coding), which is typically fragile to channel post-decoding residual errors. We use linear codes over GF(4), which are particularly suited for this application, since they are matched to the dead-zone quantizer symbol alphabet and to the QPSK modulation used on the deep-space communication channel. We optimize the proposed system where the linear codes are systematic Raptor codes over GF(4). The rateless property of Raptor encoders allows to implement a "continuum" of coding rates, in order to accurately match the channel coding rate to the transmission channel capacity and to the quantized source entropy rate for each transform subband and refinement level. Comparisons are provided with respect to the concatenation of state-of-the-art image coding and channel coding schemes used by Jet Propulsion Laboratories (JPL) for the Mars Exploration Rover (MER) Mission.

Despite encouraging advances in the design of relay codes, several important challenges remain. Many of the existing LDPC relay codes are tightly optimized for fixed channel conditions and not easily adapted without extensive re-optimization of the code. Some have high encoding complexity and some need long block lengths to approach capacity. This paper presents a high-performance protograph-based LDPC coding scheme for the half-duplex relay channel that addresses simultaneously several important issues: structured coding that permits easy design, low encoding complexity, embedded structure for convenient adaptation to various channel conditions, and performance close to capacity with a reasonable block length. The application of the coding structure to multi-relay networks is demonstrated. Finally, a simple new methodology for evaluating the end-to-end error performance of relay coding systems is developed and used to highlight the performance of the proposed codes.

This paper presents a new construction of punctured-node protograph-based Raptor-like (PN-PBRL) codes that is suitable for long-blocklength applications. As with the Raptor codes, additional parity bits can be easily produced by exclusive-OR operations on the precoded bits, providing extensive rate compatibility. The new construction provides low iterative decoding thresholds that are within 0.45 dB of the capacity for all code rates studied, and the construction is suitable for long blocklengths. Comparing at the same information block size of k = 16368 bits, the PN-PBRL codes are as good as the best known AR4JA codes in the waterfall region. The PN-PBRL codes also perform comparably to DVB-S2 LDPC codes even though the DVBi-S2 codes have longer blocklength and outer BCH codes.

This paper addresses the design of a protograph-based LDPC code which can approach the independent and uniformly distributed (i.u.d.) capacity of partial response channels. We propose a method to calculate the iterative decoding threshold of a joint graph between a protograph and the state structure of a partial response channel using the extrinsic information transfer (EXIT) chart. We then describe a simple method to search for a protograph code whose threshold is close to the i.u.d. capacity limit. This new class of codes is needed because experiments show a protograph that is capacity approaching in the AWGN channel may not perform well in partial response channels. In particular, protographs with punctured nodes are often used to produce good AWGN codes, but they perform poorly with the BCJR equalizer. Numerical results support our analysis.

This paper introduces a novel class of non-binary graph-based codes, built upon graph covers of non-binary protograph LDPC codes. This new ensemble is more restrictive than previously considered constructions, but in turn is much simpler to design and implement. The simplifications in design allow for the enumeration of codeword weight, trapping set size, stopping set size, and pseudo codeword weights, both as exact quantities in the finite-length regime and as corresponding growth rates in the asymptotic setting. The presented construction and the accompanying enumerations enhance the analytical toolbox of non-binary LDPC codes.

Coherent states achieve the Holevo capacity of a pure-loss channel when paired with an optimal measurement, but a physical realization of this measurement is unknown, and likely to be of high complexity. In this paper, we focus on the photon-counting measurement and study the photon and dimensional efficiencies attainable with modulations over classical- and nonclassical-state alphabets. We first review state-of-the-art coherent on-off-keying (OOK) and pulse-position modulation (PPM) with a photon-counting measurement, illustrating its asymptotic inefficiency relative to the Holevo limit. Then we analyze two architectures that improve upon the dimensional versus photon efficiency tradeoff achievable with conventional OOK or PPM. We show that at high photon efficiency these architectures achieve an efficiency tradeoff that differs from the best possible tradeoff by only a constant factor. The first architecture is a coherent-state transmitter that relies on feedback from the receiver to control the transmitted energy. The second architecture uses a single-photon number-state source.

This paper presents two complementary constructions of finite-length non-binary protograph-based codes with the focus on the short block-length regime. The first class is based on the existing approaches of applying the copy-and-permute operations to the constituent protograph with unweighted edges, followed by assigning non-binary scales to the edges of the derived graph. The second class is novel and is based on the so-called graph cover of a non-binary protograph: the original protograph has fixed edge scalings and copy-and-permute operations are applied to the edge-weighted protograph. The second class is arguably more restrictive, but in turn it offers simpler design and implementation. We provide design and construction of these non-binary codes for short block-lengths. Performance, cycle distribution and the minimum distance of the binary image of selected codes over AWGN is provided for information block-lengths as low as 64 bits.

Recent results by Chen et al. and Polyanskiy et al. explore using feedback to approach capacity with short blocklengths. This paper explores Chernoff bounding techniques to extend the rate-compatible sphere-packing (RCSP) analysis proposed by Chen et al. to scenarios involving numerous retransmissions and different step sizes in each incremental retransmission. Williamson et al. employ exact RCSP computations for up to six transmissions. However, exact RCSP computation with more than six retransmissions becomes unwieldy because of joint error probabilities involving numerous chi-squared distributions. This paper explores Chernoff approaches for upper and lower bounds on the error probability to provide support for computations involving more than six transmissions. We present two versions of upper and lower bounds on the error probability for the two-transmission case. One of the versions is extended to the general case of m transmissions where m ≥ 1. Computing the bounds for general m requires minimization of exponential functions with the auxiliary parameters. The numerical results, however, show that weakening the bounds by considering marginal probabilities and the case of two transmissions is already tight. These bounds also provide good estimates of the expected throughput and expected latency, which are useful for optimization purposes.

This paper presents a simple yet effective method for designing nested families of LDPC codes. Rate compatible codes are essential for many communication applications, e.g. hybrid automatic repeat request (HARQ) systems, and their design is nontrivial due to the difficulty of simultaneously guaranteeing the quality of several related codes. Puncturing can be used to generate rate-compatible LDPC codes, but it produces a gap to capacity that, in practice, often significantly exceeds the gap of the mother code. We propose an alternative method based on successively extending a high-rate protograph. The resulting codes not only inherit the advantages of protograph codes, namely low encoding complexity and efficient decoding algorithms, but also cover a wide range of rates and have very good performance with iterative decoding thresholds that are within 0.2 dB of their capacity limits.

This paper presents a new class of rate-compatible LDPC codes, protograph-based Raptor-like (PBRL) codes. The proposed PBRL codes are jointly decodable with an iterative belief propagation decoder. As with Raptor codes, additional parity bits can be easily produced by exclusive-or operations on the precoded bits, providing extensive rate compatibility. This paper provides a design procedure that optimizes this class of rate- compatible LDPC codes. The new PBRL codes outperform 3GPP rate-compatible turbo codes with the same short blocklength at high SNR and show no sign of an error floor at the FER region of 10-7.

Protograph-based bit-interleaved coded modulation (BICM) provides an elegant way of designing coded modulation over Rayleigh faded channels, however, to date the available designs have been limited to specific modulations and the corresponding decoding thresholds have not been known for Rayleigh faded channels. In this work, we present a simple method for designing protograph-based BICM that is general and applies to any modulation, and furthermore we calculate the iterative decoding thresholds of the protograph codes while mapped to higher order modulations. This general coding framework can support not only multiple rates but also adaptive modulation. We report that certain families of protograph codes achieve a threshold within a gap of approximately 0.2 - 0.4 dB of BICM capacity limit across a wide range of rates and modulations.

This paper considers the ensemble enumerators of protograph-based nonbinary (PB NB) LDPC codes. Equipped with combinatorial definitions extended to the nonbinary domain, ensemble enumerators of codeword weight, trapping set size and stopping set size are calculated. The exact enumerators are presented in the finite-length regime, and the corresponding growth rates are calculated in the asymptotic regime. Our results can provide useful analytical tools for a range of communication and storage applications employing nonbinary LDPC codes.

This paper presents a method for evaluating pseudocodeword weight enumerators of nonbinary LDPC codes built out of protographs. The ensemble enumerators are evaluated for both the finite-length and infinite-length regimes. Results of this type can be particularly useful for designing structured non-binary LDPC codes with good properties under message passing or linear programming decoding.

Protograph-based LDPC and generalized LDPC (G-LDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. The design of such “protograph-based codes” relies on what is effectively a computer-based search. As such, following Gallager, it is prudent to restrict the search to a “good ensemble,” for example, an ensemble whose minimum distance grows linearly with codeword length. A good ensemble can also mean one with good stopping set, trapping set, or pseudocodeword properties. In this paper, ensemble codeword weight enumerators for finite-length LDPC and G-LDPC codes based on protographs were derived, and then the asymptotic case was considered. The asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length. Then, the codeword weight enumerator technique is adapted to yield ensemble stopping set, trapping set, and pseudocodeword enumerators for protograph LDPC and G-LDPC codes. In this case, the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size, trapping set size, and pseudoweight grows linearly with codeword length. Trapping set enumerators for G-LDPC code ensembles represent a more complex problem which we do not consider here.

Protograph-based generalized LDPC (GLDPC) codes have the advantages of a simple design procedure and highly structured encoders and decoders. Recently, a technique for computing ensemble weight enumerators for GLDPC codes has been published. In the current paper, we investigate the existence of typical minimum distance for protograph-based GLDPC codes. That is, we first upper bound the ensemble weight enumerators for finite-length GLDPC codes based on protographs, and then we consider the sum of weight enumerators. The results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. We provide conditions on the connections of degree-2 variable nodes to constraint nodes (short block codes) to have typical minimum distance. These conditions are related to the minimum distances of the constraint nodes.

This paper presents a high-performing LDPC code for the relay channel that addresses simultaneously two important issues: a code structure that allows low encoding complexity, and a flexible rate-compatible code that allows matching to various channel conditions. Most of the previous high-performance LDPC codes for the relay channel are tightly optimized for a given channel quality and are not easily adapted, without extensive re-optimization, for various channel conditions. This paper presents a code for the relay channel that combines structured design and easy encoding with rate compatibility to allow adaptation to the three links involved in the relay channel, and furthermore offers very good performance. The proposed code is constructed by synthesizing a bilayer structure with a protograph. In addition to the contribution to relay encoding, we also produce an improved family of protograph codes for the point-to-point AWGN channel whose high-rate members enjoy thresholds that are within 0.07 dB of capacity.

This paper presents and compares two iterative coded modulation techniques for deep-space optical communications using pulse-position modulation (PPM). The first code, denoted by SCPPM, consists of the serial concatenation of an outer convolutional code, an interleaver, a bit accumulator, and PPM. The second code, denoted by LDPC-PPM, consists of the serial concatenation of an LDPC code and PPM. We employ Extrinsic Information Transfer (EXIT) charts for their analysis and design. Under conditions typical of a communications link from Mars to Earth, SCPPM is 1 dB away from capacity, while LDPC-PPM is 1.4 dB away from capacity, at a Bit Error Rate (BER) of approximately 10-5. However, LDPC-PPM lends itself naturally to low latency parallel processing in contrast to SCPPM.

Recently, pseudocodewords of Tanner graphs of LDPC codes have been used to explain the behavior of iterative decoders for these codes. In this paper, finite-length pseudocodeword weight enumerators for protograph-based generalized-LDPC code ensembles are obtained. Then asymptotic results are derived from the finite-length results by letting the block length go to infinity. The asymptotic results allow us to determine whether or not the typical minimum pseudocodeword weight grows linearly with codeword length. We give examples with Hamming component codes.

This paper discusses construction of protograph-based low-density parity-check (LDPC) codes. Emphasis is placed on protograph ensembles whose typical minimum distance grows linearly with block size. Asymptotic performance analysis for both weight enumeration and iterative decoding threshold determination is provided and applied to a series of code constructions. Construction techniques that yield both low thresholds and linear minimum distance growth are introduced by way of example throughout. The paper also examines implementation strategies for high throughput decoding derived from first principles of belief propagation on bipartite graphs.

We develop a remarkably tight upper bound on the performance of a parameterized family of bounded angle maximum-likelihood (BA-ML) incomplete decoders. The new bound for this class of incomplete decoders is calculated from the codepsilas weight enumerator, and is an extension of Poltyrev-type bounds developed for complete ML decoders. This bound can also be applied to bound the average performance of random code ensembles in terms of an ensemble average weight enumerator. We also formulate conditions defining a parameterized family of optimal incomplete decoders, defined to minimize both the total codeword error probability and the undetected error probability for any fixed capability of the decoder to detect errors. We illustrate the gap between optimal and BA-ML incomplete decoding via simulation of a small code.

Protograph-based LDPC codes have the advantages of a simple design (or search) procedure and highly structured encoders and decoders. These advantages have also been exploited in the design of protograph-based generalized LDPC (G-LDPC) codes. Recently, a technique for computing ensemble weight enumerators and stopping set enumerators for protograph-based LDPC codes has been published. In the current paper, we extend those results to protograph-based G-LDPC codes. That is, we first derive ensemble weight and stopping set enumerators for finite-length G-LDPC codes based on protographs, and then we consider the asymptotic case. In the weight enumerator case, the asymptotic results allow us to determine whether or not the typical minimum distance in the ensemble grows linearly with codeword length. In the stopping set enumerator case, the asymptotic results allows us to determine whether or not the typical smallest stopping set size grows linearly with codeword length.

The development of error-correcting codes has been closely coupled with deep-space exploration since the early days of both. Since the discovery of turbo codes in 1993, the research community has invested a great deal of work on modern iteratively decoded codes, and naturally NASA's Jet Propulsion Laboratory (JPL) has been very much involved. This paper describes the research, design, implementation, and standardization work that has taken place at JPL for both turbo and low-density parity-check (LDPC) codes. Turbo code development proceeded from theoretical analyses of polynomial selection, weight distributions imposed by interleaver designs, decoder error floors, and iterative decoding thresholds. A family of turbo codes was standardized and implemented and is currently in use by several spacecraft. JPL's LDPC codes are built from protographs and circulants, selected by analyses of decoding thresholds and methods to avoid loops in the code graph. LDPC encoders and decoders have been implemented in hardware for planned spacecraft, and standardization is under way.

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

In this paper we present protograph codes with degree-3 nodes and one high degree node. The iterative decoding threshold for proposed rate 1/2 codes are lower, by about 0.2 dB, than the best known irregular LDPC codes with degree at least 3. The main motivation is to construct rate-compatible protograph-based LDPC codes for fixed code block length (n) that simultaneously achieve low iterative decoding thresholds and guarantee minimum distance that is linearly increasing with n. We start with a rate 1/2 protograph LDPC code with degree-3 nodes and one high degree node. Higher rate codes are obtained by connecting check nodes with degree-2 non-transmitted nodes. This is equivalent to constraint combining in the protograph. The condition where all constraints are combined corresponds to the highest rate code. Through examples we show that iterative decoding thresholds as low as 0.544 dB can be achieved for small protographs with node degrees at least three. FPGA simulation results show that the proposed family of codes perform as predicted.

Recently, LDPC codes with projected graph, or protograph structures have been proposed. In this paper, finite length ensemble weight enumerators for LDPC codes with protograph structures are obtained. Asymptotic results are derived as the block size goes to infinity. In particular, we are interested in obtaining ensemble average weight enumerators for protograph LDPC codes which have typical minimum distance that grows linearly with block size. As with irregular ensembles, linear minimum distance property is sensitive to the proportion of degree-2 variable nodes. In this paper, the derived results on ensemble weight enumerators show that linear minimum distance condition on degree distribution of unstructured irregular LDPC codes is a sufficient but not a necessary condition for protograph LDPC codes

A construction method for protograph-based LDPC codes that simultaneously achieve low iterative decoding threshold and linear minimum distance is proposed. We start with a high-rate protograph LDPC code with variable node degrees of at least 3. Lower rate codes are obtained by splitting check nodes and connecting them by degree-2 nodes. This guarantees the linear minimum distance property for the lower-rate codes. Excluding checks connected to degree-1 nodes, we show that the number of degree-2 nodes should be at most one less than the number of checks for the protograph LDPC code to have linear minimum distance. Iterative decoding thresholds are obtained by using the reciprocal channel approximation. Thresholds are lowered by using either precoding or at least one very high-degree node in the base protograph. A family of high- to low-rate codes with minimum distance linearly increasing in block size and with capacity-approaching performance thresholds is presented. FPGA simulation results for a few example codes show that the proposed codes perform as predicted

In this letter, we bring to the reader's attention some interesting properties of certain line codes that appear to not have been mentioned in the literature, that, in some cases, can be used to enhance their performance when used for digital communication with particular application to radio frequency identification. In addition, we correct a previously reported result pertaining to the performance of such codes.

We propose several LDPC code constructions that simultaneously achieve good threshold and error floor performance. Minimum distance is shown to grow linearly with block size (similar to regular codes of variable degree at least 3) by considering ensemble average weight enumerators. Our constructions are based on projected graph, or protograph, structures that support high-speed decoder implementations. As with irregular ensembles, our constructions are sensitive to the proportion of degree-2 variable nodes. A code with too few such nodes tends to have an iterative decoding threshold that is far from the capacity threshold. A code with too many such nodes tends to not exhibit a minimum distance that grows linearly in block length. In this paper we also show that preceding can be used to lower the threshold of regular LDPC codes. The decoding thresholds of the proposed codes, which have linearly increasing minimum distance in block size, outperform that of regular LDPC codes. Furthermore, a family of low to high rate codes, with thresholds that adhere closely to their respective channel capacity thresholds, is presented. Simulation results for a few example codes show that the proposed codes have low error floors as well as good threshold SNR performance.

Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, in this paper we propose a construction method for LDPC codes using simple loop-free encoding modules. Such codes can be viewed as serial/parallel concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. Examples are accumulate-repeat-accumulate (ARA) codes, accumulate-repeat-accumulate-accumulate (ARAA) codes and accumulate-repeat-check-accumulate codes, and other variations. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved with low maximum variable node degrees, as the block size goes to infinity. The decoding threshold in many examples outperforms that of the best known unstructured irregular LDPC codes constrained to have the same maximum node degree. Furthermore, by puncturing the accumulator modules, any desired higher rate codes can be obtained with thresholds that stay close to their respective channel capacity thresholds uniformly.

This paper provides a construction method for low-rate low density parity check codes. Inspired by recently proposed accumulate-repeat-accumulate (ARA) codes, and hybrid concatenated codes, in this paper we extend the construction to low rates. Such codes can be viewed as hybrid concatenations of simple modules such as accumulators, repetition codes, differentiators, and punctured single parity check codes. These codes constitute a subclass of LDPC codes with very fast encoder structure. They also have a projected graph or protograph representation that allows for high-speed decoder implementation. Based on density evolution, we show through some examples that low iterative decoding thresholds close to the channel capacity limits can be achieved, as the block size goes to infinity. Iterative decoding simulation results for short blocks are provided for a few examples that show near-capacity performance and very low error floor

It is shown that a non-square (NS) 2/sup 2n+1/-ary quadrature amplitude modulation (QAM) can be decomposed into a single parity-check (SPC) block encoder and a memoryless modulator in such a way that the inherent block encoder has a recursive nature. When concatenated with a forward-error-correcting (FEC) code, iterative demodulation, demapping, and decoding of the FEC code and the inherent SPC code of NS-2/sup 2n+1/-QAM is then possible. Simulation results show that coded NS-8QAM performs nearly 2 dB better than standard 8QAM and star-8QAM, and nearly 1 dB better than 8-ary phase-shift keying when the FEC code is a rate-1/2, 16-state convolutional code, while coded NS-32QAM performs 0.75 dB better than standard 32QAM.

We propose an innovative channel coding scheme called "accumulate repeat accumulate codes" (ARA). This class of codes can be viewed as serial turbo-like codes, or as a subclass of low density parity check (LDPC) codes, thus belief propagation can be used for iterative decoding of ARA codes on a graph. The structure of encoder for this class can be viewed as precoded repeat accumulate (RA) code or as precoded irregular repeat accumulate (IRA) code, where simply an accumulator is chosen as the precoder. Thus ARA codes have simple, and very fast encoder structure when they representing LDPC codes. Based on density evolution for LDPC codes through some examples for ARA codes, we show that for maximum variable node degree 5 a minimum bit SNR as low as 0.08 dB from channel capacity for rate 1/2 can be achieved as the block size goes to infinity. Thus based on fixed low maximum variable node degree, its threshold outperforms not only the RA and IRA codes but also the best known unstructured irregular LDPC codes with the same maximum node degree. Furthermore, by puncturing the accumulators, any desired high rate codes close to code rate 1 can be obtained with thresholds that stay close to the channel capacity thresholds uniformly. Iterative decoding simulation results are provided. The ARA codes also have projected graph or protograph representation, that allows for high speed decoder implementation.

Repeat-accumulate (RA) codes are the simplest turbo-like codes that achieve good performance. However, they cannot compete with turbo codes or low-density parity-check codes (LDPC) as far as performance is concerned. The accumulate-repeat-accumulate (ARA) codes, as a subclass of LDPC codes, are obtained by adding a pre-coder in front of the RA codes with puncturing, where an accumulator is chosen as the precoder. These codes not only are very simple, but also achieve excellent performance with iterative decoding. In this paper, the performance of these codes with maximum likelihood (ML) decoding are analyzed and compared to the random codes by very tight bounds. The weight distributions of some simple ARA codes are obtained, and through existing tightest bounds we have shown that the ML SNR thresholds of the ARA codes approach very closely to the performance of random codes. We have shown that the use of a precoder improves the SNR threshold but the interleaving gain remains unchanged with respect to the RA codes with puncturing.

An innovative channel coding scheme called "accumulate repeat accumulate codes" (ARA) is proposed. ARA codes can be viewed as a subclass of low density parity check (LDPC) codes with fast encoder, and they have a projected graph or protograph representation. Using density evolution on their associated protographs, we find examples of rate 1/2 ARA codes with maximum variable node degree 5 for which a minimum bit SNR as low as 0.08 dB from channel capacity can be achieved as the block size goes to infinity. A family of high rate ARA codes with thresholds that stay uniformly close to their respective channel capacity thresholds are constructed. The ensemble weight distribution and ML threshold for rate 1/2 ARA codes were computed. For ARA with repeat 4, the ML threshold approaches within 0.005 dB of the ML threshold of random codes based on the existing tightest closed form bound.

A family of upper bounds to error probabilities of coded systems was recently proposed by D. Divsalar (see IEEE Communication Theory Workshop, 1999; JPL TMO Prog. Rep. 42-139, 1999). These bounds are valid for transmission over the additive white Gaussian noise channel, and require only the knowledge of the weight spectrum of the code words. After illustrating these bounds, we extend them to fading channels. Contrary to the union bound, our bounds maintain their effectiveness below the signal-to-noise ratio (SNR) at which the cutoff rate of the channel equals the rate of the code. Some applications are shown. First, we derive upper bounds to the minimum SNR necessary to achieve zero error probability as the code block length increases to infinity. Next, we use our bounds to predict the performance of turbo codes and low-density parity-check codes.

We derive tight exponential upper bounds on the decoding error probability of block codes which are operating over fully interleaved Rician fading channels, coherently detected and maximum-likelihood decoded. It is assumed that the fading samples are statistically independent and that perfect estimates of these samples are provided to the decoder. These upper bounds on the bit and block error probabilities are based on certain variations of the Gallager bounds. These bounds do not require integration in their final version and they are reasonably tight in a certain portion of the rate region exceeding the cutoff rate of the channel. By inserting interconnections between these bounds, we show that they are generalized versions of some reported bounds for the binary-input additive white Gaussian noise channel.

The well-known uniform error property for signal constellations and codes is extended to encompass information bits. We introduce a class of binary labelings for signal constellations, called bit geometrically uniform (BGU) labelings, for which the uniform bit error property holds, i.e., the bit error probability does not depend on the transmitted signal. Strong connections between the symmetries of constellations and binary Hamming spaces are involved. For block-coded modulation (BCM) and trellis-coded modulation (TCM) Euclidean-space codes, BGU encoders are introduced and studied. The properties of BGU encoders prove quite useful for the analysis and design of codes aimed at minimizing the bit, rather than symbol, error probability. Applications to the analysis and the design of serially concatenated trellis codes are presented, together with a case study which realizes a spectral efficiency of 2 b/s/Hz.

Based on a representation of FQPSK as a trellis-coded modulation, this paper investigates the potential improvement in power efficiency obtained from the application of simple (small number of states) outer codes to form a concatenated coding arrangement with iterative decoding. Several possible configurations for the concatenation are suggested and specific numerical results are presented for one of these in order to demonstrate the large coding gains that are achievable even when using a reduced complexity FQPSK receiver. The end result of these investigations is a system which has application in scenarios requiring a high degree of both power and bandwidth efficiency.

Based on a representation of FQPSK as a trellis-coded modulation, this paper investigates the potential improvement in power efficiency obtained from the application of simple (small number of states) outer codes to form a concatenated coding arrangement with iterative decoding. Several possible configurations for the concatenation are suggested and specific numerical results are presented for one of these in order to demonstrate the large coding gains that are achievable even when using a reduced complexity FQPSK receiver. The end result of these investigations is a system which has application in scenarios requiring a high degree of both power and bandwidth efficiency.

We track the density of extrinsic information in iterative turbo decoders by actual density evolution, and also approximate it by symmetric Gaussian density functions. The approximate model is verified by experimental measurements. We view the evolution of these density functions through an iterative decoder as a nonlinear dynamical system with feedback. Iterative decoding of turbo codes and of serially concatenated codes is analyzed by examining whether a signal-to-noise ratio (SNR) for the extrinsic information keeps growing with iterations. We define a "noise figure" for the iterative decoder, such that the turbo decoder will converge to the correct codeword if the noise figure is bounded by a number below zero dB. By decomposing the code's noise figure into individual curves of output SNR versus input SNR corresponding to the individual constituent codes, we gain many new insights into the performance of the iterative decoder for different constituents. Many mysteries of turbo codes are explained based on this analysis. For example, we show why certain codes converge better with iterative decoding than more powerful codes which are only suitable for maximum likelihood decoding. The roles of systematic bits and of recursive convolutional codes as constituents of turbo codes are crystallized. The analysis is generalized to serial concatenations of mixtures of complementary outer and inner constituent codes. Design examples are given to optimize mixture codes to achieve low iterative decoding thresholds on the signal-to-noise ratio of the channel.

A family of upper bounds to error probabilities of coded systems on the additive white Gaussian noise channel was recently proposed by Divsalar (see 1999 IEEE Communication Theory Workshop, Aptos, CA., 1999). Their calculation depends only on the weight spectrum of the code words. We first elaborate upon these bounds to show how they can be further tightened by using numerical integration instead of a Chernoff bound, and by reducing the number of code words to be included in the bound. Next, we extend them to fading channels.

We develop new, low complexity turbo codes suitable for bandwidth and power limited systems, for very low bit and word error rate requirements. Motivated by the structure of previously discovered low complexity codes such as repeat-accumulate (RA) codes with low density parity check matrix, we extend the structure to high-level modulation such as 8PSK, and 16QAM. The structure consists of a simple 4-state convolutional or short block code as an outer code, and a rate-1, 2 or 4-state inner code. Two design criteria are proposed: the maximum likelihood design criterion, for short to moderate block sizes, and an iterative decoding design criterion for very long block sizes.

A double serially concatenated code with two interleavers consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, a middle encoder, another interleaver permuting the middle codeword bits, and an inner encoder whose input words are the permuted middle codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers, where h>3. We obtain upper bounds to the average maximum likelihood bit-error probability of double serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer, middle, and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a low-complexity iterative decoding algorithm. Comparisons with parallel concatenated convolutional codes, known as "turbo codes", and with the proposed serially concatenated convolutional codes are also presented, showing that in some cases, the new schemes offer better performance.

This paper introduces an improved nonlinear parallel interference cancellation scheme for code-division multiple access (CDMA) that significantly reduces the degrading effect on the desired user of interference from the other users that share the channel. The implementation complexity of the scheme is linear in the number of users and operates on the fact that parallel processing simultaneously removes from each user a part of the interference produced by the remaining users accessing the channel the amount being proportional to their reliability. The parallel processing can be done in multiple stages. The proposed scheme uses tentative decision devices at the multiple stages to produce the most reliably estimated received data for generation and cancellation of user interference. Simulation results are given for a multitude of different situations, in particular, those cases for which the analysis is too complex.

Using an alternate form of the Gaussian probability integral discovered a number of years ago, it is shown that the solution to a number of previously considered communication problems can be simplified and, in some cases, made more accurate (i.e. exact rather than bounded). These problems include the evaluation of: (1) the bit-error probability of uncoded phase shift keying (PSK) with Costas loop tracking; (2) word-error probability of antipodal modulation in the presence of fading; (3) bit-error probability of coded M-ary PSK (MPSK) over the memoryless fading channel with given channel-state information; (4) conditional symbol-error probability of MPSK in the presence of carrier synchronization error; and (5) the average error probability for the binary additive white Gaussian noise (AWGN) intersymbol interference channel. Also obtained is a generalization of this new alternate form to the case of a two-dimensional Gaussian probability integral with arbitrary correlation which can be used to evaluate the symbol-error probability of MPSK with I-Q unbalance.

A serially concatenated code with interleaver consists of the cascade of an outer encoder, an interleaver permuting the outer codewords bits, and an inner encoder whose input words are the permuted outer codewords. The construction can be generalized to h cascaded encoders separated by h-1 interleavers. We obtain upper bounds to the average maximum-likelihood bit error probability of serially concatenated block and convolutional coding schemes. Then, we derive design guidelines for the outer and inner encoders that maximize the interleaver gain and the asymptotic slope of the error probability curves. Finally, we propose a new, low-complexity iterative decoding algorithm. Throughout the paper, extensive comparisons with parallel concatenated convolutional codes known as "turbo codes" are performed, showing that the new scheme can offer superior performance.

A serially concatenated code with interleaver consists of the cascade of an outer encoder, an interleaver permuting the outer codeword bits, and an inner encoder whose input words are the permuted outer codewords. In this paper we derive design guidelines for the outer and inner codes that maximize the interleaver gain and the asymptotic slope of the error probability curves.

This paper describes a technique related to the design of a trellis encoder, combined with the full response M-ary continuous phase frequency shift keying (CPFSK) with modulation index 1/M. A new representation of CPFSK waveforms in N signaling intervals, is proposed as a function of an (N+1)-D vector. We also decompose the generation of the proposed CPFSK waveform into two stages, an N-consecutive continuous phase encoder (NCPE) and a memoryless modulator (MM). This decomposition makes it possible to design binary convolutional encoders with various code rates, cascaded to the NCPE. Specific optimal outer convolutional encoders of two and three-consecutive full response four-ary CPFSK with modulation index 1/4 are designed following Ungerboeck's (1982) set partitioning approach. These codes achieve asymptotic coding gains up to 4.77 dB for the two consecutive case with code rate 3/4, and asymptotic coding gains up to 5.45 dB for the three-consecutive case with code rate 5/6.

This paper derives metrics for maximum-likelihood differential detection of uncoded and trellis coded MPSK and QAM transmitted over Rayleigh and Rician fading channels. Receiver structures based on these metrics are proposed and their error probability performance analyzed and/or simulated. The results represent a generalization of the notion of multiple symbol differential detection, previously introduced by the authors for MPSK over an AWGN, to the fading channel and other modulations. For the coded cases, ideal interleaving/deinterleaving is assumed and furthermore the presence or absence of channel state information. An interesting side result is that for a constant envelope modulation transmitted over a fading channel with unknown but rapidly-varying phase error (the other extreme to the slowly-varying phase error case normally assumed for differential detection), under certain practical assumptions, it is shown that the optimum receiver is of the limiter-discriminator type.>

The authors show that by using the known (or estimated) value of carrier tracking loop SNR in the decision metric, it is possible to improve the error probability performance of a partially coherent MPSK system relative to that corresponding to the commonly used ideal coherent decision rule. Using a maximum-likelihood approach, an optimum decision metric is derived and shown to take the form of a weighted sum of the ideal coherent decision metric (i.e., correlation) and the noncoherent decision metric which is optimum for differential detection of MPSK. The performance of a receiver based on this optimum decision rule is derived and shown to provide continued improvement with increasing length of observation interval (data symbol sequence length). Unfortunately increasing the observation length does not eliminate the error floor associated with the finite loop SNR. Nevertheless, in the limit of infinite observation length, the average error probability performance approaches the algebraic sum of the error floor and the performance of ideal coherent detection, i.e., at any error probability above the error floor, there is no degradation due to the partial coherence. While the above is strictly speaking only true in the limit of infinite observation length, it is shown that one can virtually achieve this limiting behavior with practical size observation lengths. Furthermore, the performance is quite insensitive to mismatch between the estimate of loop SNR (e.g., obtained from measurement) fed to the decision metric relative and its true value.>

The authors examine maximum-likelihood block detection of uncoded full response continuous phase modulation (CPM) over an additive white Gaussian noise (AWGN) channel. Both the maximum-likelihood metrics and the bit error probability performances of the associated detection algorithms are considered. The special and popular case of minimum-shift-keying (MSK) corresponding to h=0.5 and constant amplitude frequency pulse is treated separately. The many new receiver structures that result from this investigation can be compared to the traditional ones that have been used in the past both from the standpoint of simplicity of implementation and optimality of performance.>

A variety of schemes for performing differential detection in environments characterized by frequency offset are discussed. All of the schemes involve encoding the input phase information as a second-order difference and performing an analogous second-order differential detection at the receiver. Because of the back-to-back differential detection operations at the receiver, The performance of most of the schemes is considerably degraded relative to that of first-order differential detection schemes. However, the latter are quite sensitive to frequency offset and, in many instances, cannot be used at all. It is demonstrated that via a simple enhancement of using a 2 T s (instead of T s) delay in the second stage of the encoder and first stage of the decoder, the performance degradation can be significantly reduced. This result is significant in view of the fact that it comes without any penalty in implementation complexity.>

Several modulation and coding advances supported by NASA are summarized. To support long-constraint-length convolutional code, a VLSI maximum-likelihood decoder, utilizing parallel processing techniques, which is being developed to decode convolutional codes of constraint length 15 and a code rate as low as 1/6 is discussed. A VLSI high-speed 8-b Reed-Solomon decoder which is being developed for advanced tracking and data relay satellite (ATDRS) applications is discussed. A 300-Mb/s modem with continuous phase modulation (CPM) and codings which is being developed for ATDRS is discussed. Trellis-coded modulation (TCM) techniques are discussed for satellite-based mobile communication applications.>

A differential detection technique for MPSK (multiple-phase shift keying), which uses a multiple-symbol observation interval, is presented, and its performance is analyzed and simulated. The technique makes use of maximum-likelihood sequence estimation of the transmitted phases rather than symbol-by-symbol detection as in the conventional differential detection. Thus, the performance of this multiple-symbol detection scheme fills the gap between conventional (two-symbol observation) differentially coherent detection of MPSK and ideal coherent of MPSK with differential encoding. The amount of improvement gained over conventional differential detection depends on the number of phases M and the number of additional symbol intervals added to the observation. What is particularly interesting is that substantial performance improvement can be obtained for only one or two additional symbol intervals of observation. The analysis and simulation results presented are for uncoded MPSK.>

The use of interleaving/deinterleaving in trellis-coded modulation systems to reduce the SNR loss due to imperfect carrier demodulation references is demonstrated. Both the discrete carrier (phase-locked loop) and the suppressed carrier (Costas loop) cases are considered, and the differences between the two are clearly demonstrated by numerical results. The special case of convolutional codes is also treated and illustrated with an example of practical interest.>

The idea of using a multiple (more than two) symbol observation interval to improve error probability performance is applied to differential detection of trellis-coded multiple phase-shift keying (MPSK) over an additive white Gaussian noise (AWGN) channels. An equivalent Euclidean distance measure per trellis branch is determined for this detection scheme. This is used to define an augmented (larger multiplicity) trellis code whose distance measure is the conventional squared Euclidean distance typical of conventional trellis-coded modulation on the AWGN. Such an augmented multiple trellis code is a convenient mathematical tool for simplifying the analysis. Results are obtained by a combination of analysis and computer simulation. It is shown that only a slight increase (e.g. one symbol) in the length of the observation interval will provide a significant improvement in bit error probability performance.>

An open-loop technique is presented for estimating and correcting Doppler frequency shift in an M-ary differential phase-shift-keyed (MDPSK) receiver. The novelty of the scheme is based on the observation that whereas the change in phase of the received signal over a full symbol contains the sum of the data (phase) and the Doppler-induced phase shift, the same change in phase over half a symbol (within a given symbol interval) contains only the Doppler-induced phase shift. Thus, by proper processing, the latter can be estimated and removed from the former. Analytical and simulation results are given for the variance of the above estimator, and the error probability performance of the MDPSK receiver is evaluated in the presence of the Doppler correction. Next, the practical considerations associated with the application of this technique on bandlimited Nyquist channels are discussed and incorporated into the final design. It is shown that the receiver can, in the absence of timing jitter, be designed to allow combined Doppler correction and data detection with no penalty due to intersymbol interference (ISI). The effects of ISI due to timing jitter are assessed by computer simulation.>

The authors demonstrate a trellis coded modulation technique referred to as multiple trellis coded modulation (MTCM) wherein more than one channel symbol per trellis branch is transmitted. They have found simple two-state trellis codes for symmetric MPSK multiple phase-shift keying and AM modulations that can achieve 3-dB gain over uncoded modulation at very high signal-to-noise ratios without bandwidth expansion or reduction in information bit rate. The gain of these codes with respect to previously reported two-state trellis codes is between 1 and 2 dB at very high signal-to-noise ratios, depending on the number of bits per Hertz transmitted. These gains are achieved for those of the equivalent conventional trellis codes with the same number of states in the trellis diagram. The authors note that additional computations per branch are needed for the multiple trellis coding scheme. The concept can be extended to a higher number of states and other types of modulations.>

It has been well established that the appropriate criterion for optimum trellis-coded modulation design on the additive white Gaussian noise channel is maximization of the free Euclidean distance. It is shown that when the trellis-coded modulation is used on a Rician fading channel with interleaving/deinterleaving, the design of the code of optimum performance is guided by other factors, in particular, the length of the shortest error-event path, and the product of branch distances (possibly normalized by the Euclidean distance of the path) along the path. Although maximum free distance (d/sub free/) is still an important consideration, it plays a less significant role the more severe the fading is on the channel. These considerations lead to the definition of a new distance measure of optimization of trellis codes transmitted over Rician fading channels. If no interleaving/deinterleaving is used, then once again the design of the trellis code is guided by maximizing d/sub free/.>

A previous work on criteria for designing trellis-coded MPSK modulation to achieve minimum error probability performance on the Rician fading channel (see ibid., vol.36, no.9, p.1004-1012, Sep. 1988) is extended. It is demonstrated that allowing for multiple symbols per trellis branch, i.e., multiple trellis-coded modulation (MTCM), provides an additional degree of freedom for designing a code to meet the optimization on the fading channel. Diversities larger than those achievable with conventional trellis codes having the same number of trellis states are now attainable, it is under these conditions that MTCM achieves its full potential.>

The combination of trellis coding and MPSK signaling with the addition of asymmetry to the signal set is discussed with regard to its suitabllity as a modulation/coding scheme for the fading mobile satellite channel. For MPSK, introducing nonuniformity (asymmetry) into the spacing between signal points in the constellation buys a further improvement in performance over that achievable with trellis coded symmetric MPSK, all this without increasing the average or peak power, or changing the bandwidth constraints imposed on the system. Whereas previous contributions have considered the performance of trellis coded modulation transmitted over an additive white Gaussian noise (AWGN) channel, the emphasis in this paper is on the performance of trellis coded MPSK in the fading environment. The results will be obtained by using a combination of analysis and simulation. It will be assumed that the effect of the fading on the phase of the received signal is fully compensated for either by tracking it with some form of phase-locked loop or with pilot tone calibration techniques. Thus, our results will only reflect the degradation due to the effect of the fading on the amplitude of the received signal. Also, we shall consider only the case where interleaving/deinterleaving is employed to further combat the fading. This allows for considerable simplification of the analysis and is of great practical interest. Finally, the impact of the availability of channel state information on average bit error probability performance is assessed.

Traditionally symmetric signal constellations, i.e., those with uniformly spaced signal points, have been used for both uncoded and coded systems. Although symmetric signal constellations are optimum with no coding, the same is not necessarily true for coded systems. This paper shows that by designing the signal constellations to be asymmetric, one can, in many instances, obtain a performance gain over the traditional symmetric constellations combined With trellis coding. In particular, we consider the joint design ofn/(n + 1)trellis codes and asymmetric2^{n+1}-point signal constellations, which has no bandwidth expansion relative to an uncoded 2n-point symmetric signal set. The asymptotic performance gains due to coding and asymmetry are evaluated in terms of the minimum free Euclidean distance dfreeof the trellis. A comparison of the maximum value of this performance measure to the minimum distance dmin, of the uncoded system is an indication of the maxiamm reduction in requiredE_{b}/N_{0}that can be achieved for arbitrarily small system bit error rates. Bit error probability analysis is carried out for general cases. A few examples are given to show the performance gain due to the asymmetry of the signal set. It is to be emphasized that the introduction of asymmetry into the signal set does not affect the bandwidth or power requirements of the system; hence, the abovementioned improvements in performance come at little or no cost. Asymmetric signal sets in coded systems first appear in the work of Divsalar and Yuen [1], [2]. Here we expand upon these results by considering various types of asymmetric signal sets combined with the optimum (in the sense of maximum dfree) trellis code having 2, 4, 8, and 16 states. The numerical results obtained will clearly demonstrate the tradeoff between the additional savings in requiredE_{b}/N_{0}and the additional complexity (more trellis states) needed to achieve it.

Consideration is presently being given to an optical-RF relay deep space communication link that transmits optical PPM data from spacecraft to orbital relay, then retransmits the data via microwave to ground. It is generally advantageous to use Reed-Solomon (RS) encoding over the PPM optical link for improved error correction. In this paper several demodulating schemes are considered for generating the RS symbols at the relay photodetector output, and the effect of each on overall RS decoding performance is computed. Both Poisson and Gaussian optical noise models are used in the evaluation. The effect on performance of bit errors in the relay downlink is also examined.

This paper examines by analytical methods the power spectral densities of digital modulations (in particular, staggered and unstaggered quadrature modulations) passed through band-limited nonlinear channels. Previously observed (by computer simulation or hardware measurement) behavior of such spectra with regard to the suppression or restoration of its sidelobes after passing through the nonlinearity is verified analytically. Several examples corresponding to specific quadrature modulations and filter-nonlinearity combinations are presented as illustrations of the general results.

This paper considers a system that couples several phaselocked loops (PLL's) to improve carrier tracking performance. The system coherently combines the received carrier signals at geographically separated ground antennas to increase the total effective aperture. It automatically aligns the received carrier's phases to enhance received carrier signal-to-noise ratio. The tracking performance of this system is being assessed in terms of rms phase jitter.

The power spectral density of the output symbol sequence of a convolutional encoder is computed for two different input symbol stream source models, namely, an NRZ signaling format and a first-order Markov source. In the former, the two signaling states of the binary waveform are not necessarily assumed to occur with equal probability. The effects of alternate symbol inversion on this spectrum are also considered. The mathematical results are illustrated with many examples corresponding to optimal performance codes. It is demonstrated that only for the case of a purely random input source (e.g., NRZ data with equiprobable symbols) and a particular class of codes is the output spectrum identical to the input spectrum except for a frequency scaling (expansion) by the reciprocal of the code rate. In all other cases, the output spectrum is sufficiently changed relative to the input spectrum that the commonly quoted statement "a convolutional encoder produces a bandwidth expansion by a factor equal to the reciprocal of the code rate" must be exercised with care.