Reed-Solomon error control coding for Rayleigh fading channels withfeedback

The use of nonbinary block error control codes over Rayleigh fading channels with feedback is examined. It is assumed that the fading is slow with respect to the rate of symbol transmission. Expressions are derived for the probabilities of channel symbol error and erasure, which are in turn used to develop expressions for code symbol error and erasure. Two erasure generation mechanisms are considered, one based on the existence of channel amplitude side information, the other not. This analytical framework is used to evaluate the performance of the Reed-Solomon/hybrid-ARQ protocol (RS/HARQ) over fading channels with feedback. The RS/HARQ system uses erasure decoding in a hybrid-ARQ protocol to provide excellent reliability performance at the expense of a reduction in throughput. The RS/HARQ protocol allows for the variation of the erasure threshold and the effective diameter of the decoding operation     

Exact performance analysis of linear block codes in frequency-hopspread spectrum communication systems

We present a framework for determining the exact decoder error and failure probabilities for linear block codes in a frequency-hop communication channel with an arbitrary number of conditional symbol error and erasure probabilities. Applications are demonstrated for type-I hybrid ARQ systems by deriving equations for the packet error probability and throughput. Because these quantities are too small to be obtained by simulation, the framework provides exact results which are unobtainable by previous work     

Correction to `Packet error probabilities in frequency hoppedspread-spectrum packet radio networks-memoryless frequency hoppingpatterns considered'

The author corrects the approach followed in the above-titled paper (see ibid., vol.36, no.6, p.720-3, 1988) to compute packet error probabilities in frequency-hopped spread-spectrum packet radio networks with memoryless frequency-hopping patterns. The nature of the error is that the sequence of symbol errors is not Markov, so that a formula which is instrumental for the computation of packet error probabilities, is incorrect. He obtains exact and easy to compute expressions for the evaluation of the packet error probabilities     

Simplification of Packet-Symbol Decoding With Errors, Deletions, Misordering of Packets, and No Sequence Numbers

In this paper, a new method is described which builds on Mitzenmacher's idea of adding a different pseudorandom number to each packet to help decode packet-symbol low-density codes, with deletions, errors, and out-of-order reception, without sequence numbers. The new method has lower decoding complexity than the original method. The most basic form of the new method applies to any parity-check code structure, but is limited to a rather small number of packets in the code. Decoding success is slightly inferior to an ideal erasure channel, which would require sequence numbering and error detection in each packet. Error detection is needed only for the whole code, amounting to usually less than one bit per packet symbol. Moreover, if error detection can resolve one of a small number of alternatives, the ordered case performs almost as well as the ideal erasure channel. Ways are shown to modify the basic algorithm for use with long codes, possibly approaching the erasure channel capacity limit.      

Variable-rate coding for meteor-burst communications

The performance of variable-rate Reed-Solomon error-control coding for meteor-burst communications is considered. The code rate is allowed to vary from codeword to codeword within each packet, and the optimum number of codewords per packet and optimum rates for the codewords are determined as a function of the length of the message and the decay rate for the meteor trail. The resulting performance is compared to that obtained from, fixed-rate coding. Of central importance is the derivation of tractable expressions for the probability of correct decoding for bounded-distance decoding on a memoryless channel with a time-varying symbol error probability. A throughout measure is developed that is based on the probability distribution of the initial signal-to-noise ratio     

Errors and erasures decoding of BCH and Reed-Solomon codes for reduced M-ary orthogonal signaling

The paper presents a comparison of communication systems using different signal constellation sizes and Reed-Solomon or Bose-Chaudhuri-Hocquengem codes with different rates so that the overall required bandwidth is the same for each system. In these comparisons, the channel symbol size is smaller than the code symbol size, so that a code symbol contains parts of multiple channel symbols. Thus, the normal assumption of independent code symbols does not apply. Instead, consideration must be taken to obtain the best arrangement of channel symbols in each code symbol. Analytical expressions are developed to compare the bit error probability performance of comparable systems, based on individual codewords using errors-only decoding and errors and erasures decoding with transmission over a Rayleigh fading channel.     

Error probability of Q-ary symbol consisting of multiple channel symbols under Rayleigh fading

Closed-form expressions for symbol error probabilities are derived for the case when modulation alphabet size M is smaller than symbol size Q. We consider slow Rayleigh fading channel where fading amplitudes and phases remain constant during one Q-ary symbol duration. An exact expression is obtained for noncoherent M-ary frequency-shift keyed (MFSK) modulation and the simple and accurate approximate expressions are obtained for M-ary phase-shift keyed (MPSK) and quadrature-amplitude modulation. Simulation results showed that the derived error probability are well agreed.     

On the computation of the performance probabilities for block codeswith a bounded-distance decoding rule

When a block code is used on a discrete memoryless channel with an incomplete decoding rule that is based on a generalized distance, the probability of decoding failure, the probability of erroneous decoding, and the expected number of symbol decoding errors can be expressed in terms of the generalized weight enumerator polynomials of the code. For the symmetric erasure channel, numerically stable methods to compute these probabilities or expectations are proposed for binary codes whose distance distributions are known, and for linear maximum distance separable (MDS) codes. The method for linear MDS codes saves the computation of the weight distribution and yields upper bounds for the probability of erroneous decoding and for the symbol error rate by the cumulative binomial distribution. Numerical examples include a triple-error-correcting Bose-Chaudhuri-Hocquenghem (BCH) code of length 63 and a Reed-Solomon code of length 1023 and minimum distance 31     

Performance evaluation of HDAF relaying systems for pilot and data symbol burst transmission over quasi‐static Rayleigh fading channels

This paper analyzes averaged symbol error probabilities of burst transmission consisting of pilot and data symbols for hybrid adaptive decode‐or‐amplify‐forward (HDAF) relaying systems. Under the assumption of quasi‐static Rayleigh fading channels with independent and non‐identically distribution, we consider a channel estimation scheme based on pilot symbols and show how channel estimation error affects received signal‐to‐noise ratio (SNR) and symbol error probability (SEP). Firstly, all the possible detection error‐events are presented for all the relay nodes, and their probabilities are derived as forms related with data symbol burst transmission. For the given error event, we analyze the conditional SEP as an exact form and then, the averaged SEP (ASEP) is approximately derived as a closed‐form. The simulation results verify that our derived ASEP expression is accurate over all the regions of SNR. Utilizing the proposed expressions, we can evaluate ASEP performance of HDAF relay systems easily and fast. Copyright © 2015 John Wiley & Sons, Ltd.     

Symbol error probability for guided scrambling over a recording channel

We derive expressions for the symbol error probabilities for a recording code concatenated with a Reed-Solomon (RS) code. The recording code is structured by a guided scrambling (GS) code for the direct current (DC) suppression in conjunction with a runlength-limited (RLL) code. As for the GS codes, convolutional GS and Galois field (GF) addition GS schemes are examined. As for the RLL codes, two types of RLL codes are investigated. One is a traditional RLL code where a bit length m of an RS symbol is an integer multiple of a bit length p of an RLL source symbol. The other is a new type of high-rate RLL code where p>m. We compute the RS symbol error rates when these RLL codes are combined with the two GS schemes.     

LT码和q-LDPC码级联方案在深空通信中的应用

该文针对深空通信对长纠删码的需求,提出了LT (Luby Transform)码和q-LDPC码的级联方案。在综合考虑性能和复杂度的情况下,选取8-LDPC码和8PSK的级联作为等效的删除信道,长度选择灵活、编译码简单的LT码实现纠删功能。文中设计了两种短8-LDPC码,并对整个级联系统的纠错性能进行了仿真。仿真结果表明8-LDPC码的性能优于信源信息速率和码率相同的二进制LDPC码,级联系统在等效包删除概率不超过0.1时,系统误比特率以概率1趋于0。     

Packet combining in frequency-hop spread-spectrum communicationsystems

The performance of frequency-hop spread-spectrum (FHSS) communication systems using hybrid automatic-repeat-request (ARQ) can be improved by combining current and prior transmissions at the receiver. Two methods for combining packets in systems that employ interleaving and Reed-Solomon (RS) coding are presented and analyzed for the partial-band interference channel. These methods use majority logic combining at the codeword level to make retransmission decisions. Bounded distance errors-and-erasures decoding and erasure generation by means of Viterbi's ratio threshold test (RTT) are incorporated in the analysis. Results of the analysis show that, with comparable packet error probabilities, the packet-combining schemes provide significant gains in throughput when compared with systems that do not employ combining     

Hermitian codes for frequency-hop spread-spectrum packet radio networks

Hermitian codes are an attractive alternative to Reed-Solomon codes for use in frequency-hop spread-spectrum packet radio networks. For a given alphabet size, a Hermitian code has a much longer block length than a Reed-Solomon code. This and other considerations suggest that Hermitian codes may be superior for certain applications. Analytical results are developed for the evaluation of the packet error probability for frequency-hop transmissions using Hermitian coding. We find there are several situations for which Hermitian codes provide much lower packet error probabilities than can be obtained with Reed-Solomon codes. In general, as the code rate decreases or the symbol alphabet size increases, the relative performance of Hermitian codes improves with respect to Reed-Solomon codes. Performance evaluations are presented for an additive white Gaussian noise channel and for certain partial-band interference channels, and the packet error probability is evaluated for both errors-only and errors-and-erasures decoding.     

Performance of frame synchronization in packet transmission usingbit erasure information

The probability of successful frame sync in packet transmission using bit erasure information is compared to the same probability when erasure information is ignored. For a given bound on the false alarms, a significant improvement is found on additive white Gaussian noise (AWGN) channels if an erasure zone is added to the receiver. An improvement is also found for channels in which the presence of jamming or other interference is detected and erased by the receiver. An expression for the probability of frame synchronization in packet transmission is derived as a function of the bit error and erasure probability, the amount of overlap, and the number of bit errors/erasures which are tolerated in the frame sync word     

On the error probability of quasi‐orthogonal space–time block codes

In this paper, we derive the exact pairwise error probabilities (PEPs) of various quasi‐orthogonal space–time block codes (QO‐STBCs) using the moment generating function. By classifying the exact PEPs of QO‐STBCs into three types, we derive the closed‐form expression for each type of PEP. Based on these closed‐form expressions, we obtain the union bounds on the symbol error probability and bit error probability for QPSK modulation. Through simulation, it is shown that these union bounds are quite tight. Copyright © 2008 John Wiley & Sons, Ltd.     

Closed-form expressions for the symbol error probability of 3-D OFDM

In this letter, closed-form expressions for the symbol error probability of 3-D orthogonal frequency division multiplexing in additive white Gaussian noise channel are derived. When a 3-D 4-ary constellation is used as signal mapper, we calculate the upper and the lower bound for the error probability, and also provide its approximation. Since decision boundaries of 8- ary constellation form an extended regular hexahedron, an exact symbol error probability can be computed. It is verified that the theoretical error probabilities are very close to or almost the same as simulation results.     

Exact Performance Analysis of Dual-Branch Coherent Equal-Gain Combining in Nakagami- $m$, Rician, and Hoyt Fading

The exact symbol error rates (SERs) of several M-ary modulation schemes are obtained in a unified manner for mobile radio receivers employing dual-branch coherent equal-gain combining (EGC), where the received signals on each branch follow either similar or different fading distributions. Specifically, the cases considered are Nakagami-m, Rice, and Hoyt fading on both branches, but with not necessarily identical parameters, and mixed Nakagami-m/Rice and Nakagami-m/Hoyt fading; only the integer or half-integer values of the Nakagami-parameter are considered. While the previous exact expressions for the SERs of general M-ary modulations with L-branch EGC contained double integrals, the error probability expressions derived in this paper for the case of L=2 contain only a single integral. In addition, the derived probability distributions also allow the direct obtaining of other useful performance measures, such as the exact level crossing rates, average fade durations, and outage probabilities.     

On Broadcast Stability of Queue-Based Dynamic Network Coding Over Erasure Channels

The transmission of packets is considered from one source to multiple receivers over single-hop erasure channels. The objective is to evaluate the stability properties of different transmission schemes with and without network coding. First, the throughput limitation of retransmission schemes is discussed and the stability benefits are shown for randomly coded transmissions, which, however, need not optimize the stable throughput for finite coding field size and finite packet block size. Next, a dynamic scheme is introduced for distributing packets among virtual queues depending on the channel feedback and performing linear network coding based on the instantaneous queue contents. The difference of the maximum stable throughput from the min-cut rate is bounded as function of the order of erasure probabilities depending on the complexity allowed for network coding and queue management. This queue-based network coding scheme can asymptotically optimize the stable throughput to the max-flow min-cut bound, as the erasure probabilities go to zero. This is realized for a finite coding field size without accumulating packet blocks at the source to start network coding. The comparison of random and queue-based dynamic network coding with plain retransmissions opens up new questions regarding the tradeoffs of stable throughput, packet delay, overhead, and complexity.      

Computing the Average Symbol Error Probability of the MPSK System Having Quadrature Error

When quadrature error exists, the shape of the M‐ary phase shift keying (MPSK) signal constellation becomes skewed‐elliptic. Each MPSK symbol takes on a different symbol error probability (SEP) value. The analytical results presented thus far have been derived from studies which examined the SEP problem assuming that the SEP of each MPSK symbol is equally likely; therefore, those results should not be treated as offering a complete solution. In this letter, we present a new and more complete solution to the SEP problem of MPSK by relaxing the above assumption and finding the expressions for the average as well as individual SEP in the presence of quadrature error.     

Performance study of hybrid spread-spectrum random-accesscommunications

A numerical study of the performance of hybrid direct-sequence slow-frequency-hopped spread-spectrum random-access schemes using Reed-Solomon forward-error control coding is performed. The performance measures of interest are the maximum numbers of simultaneous transmitters that can be tolerated in the vicinity of a receiver at a desirable level of the bit (or symbol) error probability, the normalized throughput, and the packet error probability. For the case where all simultaneous signals have the same received power levels, a critical comparison of the performance of direct-sequence, frequency-hopped, and hybrid spread-spectrum systems with the same overall bandwidth expansion is conducted with respect to the above performance measures, and ranges of parameters, for which each system outperforms the others, are identified. It is also established that hybrid spread-spectrum random-access schemes have considerably higher throughput than the uncoded nonspread-spectrum ones, for the desired range of values of the packet error probability