Probability of Block Error for Very Slow Rayleigh Fading in Gaussian Noise

Expressions for probability of block error,P_{f}(M,N), the probability of more thanMerrors in a block ofNdigits, are derived for binary signaling over a channel with very slow nonselective Reyleigh fading and additive Gaussian noise. The analysis is applicable to noncoherent FSK, coherent FSK, and coherent PSK signaling. An asymptotic expression for the block error probability, suitable for high signal-to-noise ratio, is also derived. Computations ofP_{f}(0,N)are presented for a broad range of error probabilities and block sizes, and the characteristics of block error probabilities for the fading and nonfading cases are compared.     

Bounds on the probability of error for convolutional codes (Corresp.)

For the single error correcting convolutional codes introduced by Wyner and Ash, it is shown that if sufficiently few errors occur in an appropriate neighborhood of a block, the probability of correctly decoding that block is independent of errors outside that neighborhood. This fact is used to derive bounds on the bit error probability and the mean time to first error.     

On weak convergence of probability measures, channel capacity andcode error probabilities

The relationship between weak convergence of channel probability measures, channel capacity, and error probability of block codes is examined for memoryless channels with general input and output alphabets. It is shown that channel capacity is a lower semi-continuous function and that every block code with maximal probability of error δ for a nominal channel for any ϵ>0 can be modified such that the modification has a probability of error less than δ+ϵ for all channels in a sufficiently small neighborhood of the nominal channel     

Bit-to-bit error dependence in slotted DS/SSMA packet systems withrandom signature sequences

A technique is developed to find an accurate approximation to the probability of data bit error and the probability of packet success in a direct-sequence spread-spectrum multiple-access (DS/SSMA) packet radio system with random signature sequences. An improved Gaussian approximation to the probability of data bit error is performed. Packet performance is analyzed by using the theory of moment spaces to gain insight into the effect of bit-to-bit error dependence caused by interfering signal relative delays and phases which are assumed constant over the duration of a desired packet. Numerical results show that if no error control exists in the desired packet or if block error control is used when multiple-access interference is high, the error dependence increases the average probability of packet success beyond that predicted by models which use independent bit errors. However, when block error control is used and the multiple-access interference is low, the bit error dependencies cause a reduction in packet error performance     

On the robustness of space-time coding techniques based on a general space-time covariance model

The robustness of space-time coding techniques for wireless channels that exhibit both temporal and spatial correlation is investigated. A general space-time covariance model is developed and employed to evaluate the exact pairwise error probability for space-time block codes. The expressions developed for the pairwise error probability are used in conjunction with the union bound to determine an upper bound for the probability of a block error. The block error probability is evaluated for several space-time codes and for wireless channels that exhibit varying degrees of spatial and temporal correlation. Numerical results are presented for a two-dimensional Gaussian scatterer model which has been shown to be consistent with recent field measurements of wireless channels. The results demonstrate that the best-case wireless channel is uncorrelated in both space and time. Correlation between transmission paths, due to insufficient spacing of the transmit antennas or scatterers located in close proximity to the mobile, can result in a significant performance degradation. The conditions that result in uncorrelated transmission paths are quantified in terms of the effective scattering radius and the spacing of the transmit and receive antennas.     

A method to analyze performance of digital connections

A comprehensive performance analysis method that models, at bit level, the error performance of individual links in an end-to-end connection is presented. The link model accounts for the burst-error behaviour of each individual link. A method to concatenate several individual links and extract a model for the end-to-end connection is given. This resulting end-to-end model can be used to calculate performance measures such as bit error rate and block error rate for any given block size. A procedure to compute the probability distribution of errors within a specific block is also developed. Finally, a method to compute the probability distribution of blocks having a certain error rate over a given period of time is presented. The utility and power of the model are illustrated with the help of an example connection     

Error probability and free distance bounds for two-user tree codes on multiple-access channels

Two-user tree codes are considered for use on an arbitrary two-user discrete memoryless multiple-access channel (MAC). A two-user tree Is employed to achieve true maximum likelihood (ML) decoding of two-user tree codes on MAC's. Each decoding error event has associated with it a configuration indicating the specific time slots in which a decoding error has occurred for the first user alone, for the second user alone, or for both users simultaneously. Even though there are many possible configurations, it is shown that there are five fundamental configuration types. An upper bound on decoding error probability, similar to Liao's result for two-user block codes, is derived for sets of error events having a particular configuration. The total ML decoding error probability is bounded using a union bound first over all configurations of a given type and then over the five configuration types. A two-user tree coding error exponent is defined and compared with the corresponding block coding result for a specific MAC. It is seen that the tree coding error exponent is larger than the block coding error exponent at all rate pairs within the two-user capacity region. Finally, a new lower bound on free distance for two-user codes is derived using the same general technique used to bound the error probability.     

The random coding bound is tight for the average code (Corresp.)

The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upper-bounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.     

On the Probability of Undetected Error for Linear Block Codes

The problem of computing the probability of undetected error is considered for linear block codes used for error detection. The recent literature is first reviewed and several results are extended. It is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual. Using the dual code formulation, the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes. Several bounds on the undetected error probability are then presented. We conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained. An efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results.     

Average digit error probability in decoding a linear block code

An upper bound on the average digit error probability of a linear block code is given which is dependent only upon the minimum distance of the code. The tightness of this bound is also demonstrated, and an example is given where knowledge of the average digit error probability is important.     

The Information-Bit Error Rate for Block Codes

The relations between the word error probability and the decoding algorithms for block codes are reviewed. A simple approximation that does not depend upon the code weight structure or the decoding details is derived for the information-bit error rate in terms of the channel-symbol error probability.     

Constructing LDPC codes by error minimization progressive edge growth

A novel approach to constructing Tanner graphs using progressive edge growth (PEG) is introduced. It yields LDPC codes providing minimized block error probability in Binary Erasure Channels (BEC). The constructed codes exhibit superior performance over codes generated by previously known algorithms, both for BEC and AWGN channels. Furthermore, an upper bound on the expected block error probability in the error floor region of the generated codes is derived. This allows analytical prediction of the codes' error floor performance. Finally, the method is generalized for generating simple in implementation LDPC codes based on lifted graphs.     

A Worst-Case Robust Combination Method of Beam-Forming and Orthogonal Space–Time Block Coding

In this paper, a new robust multi-input multi-output system is proposed in a perturbed wireless channel which is to model imperfect channel information at the source side when beam-forming and orthogonal space–time block coding is utilized. The channel perturbation is bounded by a predefined variation based on worst-case robust design. Beam-forming is used to improve the performance of the system expressed by the upper bound of pairwise error probability of symbols. In this paper firstly, the maximum value of pairwise error probability is obtained in a closed form when channel perturbation is kept below a threshold. Then the beam-forming matrix is designed to minimize the pairwise error probability subject to a predefined maximum transmitting power. This approach provides near optimal results due to using the upper bound of pairwise error probability. It shows good performance based on the symbol error rate criterion compared with other existing methods of the multiple input multiple output system.     

The probability of undetected error when a code is used for error correction and detection (Corresp.)

We study the probability of having an undetected error when a linear block code is used to correct up toterrors on a symmetric channel, and the remaining power of the code is used for error detection.     

A performance bound on random-coded MIMO systems

A closed-form upper bound on the average error probability is proposed for random block codes operating in multi-input multi-output (MIMO) systems. The bound exponentially decays to zero with increasing block length, and the obtained error exponent proves consistent with Gallager's random coding exponent and the information-theoretic channel capacity.     

非理想连续干扰消除下非正交多址接入上行传输系统性能分析

非正交多址接入(NOMA)技术允许多个发送方共用同一个资源块,接收方通过连续干扰消除(SIC)解码出不同发送方的信息。然而,目前针对NOMA系统的研究大多基于理想SIC的假设,而没有考虑非理想SIC对系统性能带来的影响。针对此问题,该文在非理想SIC的假设下,针对单小区上行NOMA系统提出一套性能分析框架。首先,采用二项式点过程(BPP)对上行NOMA系统中基站和用户设备的空间分布进行建模。基于此模型,采用基于大尺度衰落的干扰消除顺序,对干扰消除的误差情况进行分析。进一步,基于随机几何理论和次序统计理论,推导出距基站由近至远次序为k的用户设备的覆盖概率,并采用平均覆盖概率衡量整个NOMA传输系统的可靠性。理论和仿真结果分析了远近次序、基站半径和发射功率等系统参数对传输可靠性的影响,并验证了理论推导的准确性。     

Waterfall performance analysis of finite-length LDPC codes on symmetric channels

An efficient method for analyzing the performance of finite-length low-density parity-check (LDPC) codes in the waterfall region, when transmission takes place on a memoryless binary-input output-symmetric channel is proposed. This method is based on studying the variations of the channel quality around its expected value when observed during the transmission of a finite-length codeword. We model these variations with a single parameter. This parameter is then viewed as a random variable and its probability distribution function is obtained. Assuming that a decoding failure is the result of an observed channel worse than the code?s decoding threshold, the block error probability of finite-length LDPC codes under different decoding algorithms is estimated. Using an extrinsic information transfer chart analysis, the bit error probability is obtained from the block error probability. Different parameters can be used for modeling the channel variations. In this work, two of such parameters are studied. Through examples, it is shown that this method can closely predict the performance of LDPC codes of a few thousand bits or longer in the waterfall region.     

Markov characterization of digital fading mobile VHF channels

We apply techniques for the modeling of error sequences on digital communication channels to results of experiments undertaken on mobile VHF channels. The experiments were carried out using four different modulation schemes at some of the different standardized signaling rates. The modulation schemes used were: FSK @ 300 baud, DPSK @ 1200 baud, QPSK @ 1200 baud, and 8-ary PSK @ 1600 baud, and in each case, subcarrier modulation was used. The experiments were undertaken for urban as well as freeway environments. Fritchman-partitioned Markov chain models were derived throughout, and from the models, block error probability distributions were derived. These block error probability distributions or P(m,n) give the probability that a block of n bits will contain exactly m errors. We present P(⩾m,n) for 7-, 15-, 31-, 63-, 127-, and 255-b blocks, for the above-mentioned modulation schemes, in the mobile VHF environments mentioned. P(⩾m,n) denotes the probability that at least m errors will occur in a block of n bits. Furthermore, the P(⩾m,n) information presented here, should give some indication of the performance to be expected from block error-correcting schemes     

Linear block codes for error detection (Corresp.)

The probability of undetected error of linear block codes for use on a binary symmetric channel is investigated. Upper hounds are derived. Several classes of linear block codes are proved to have good error-detecting capability.     

On the error probability of coded frequency-hopped spread-spectrummultiple-access systems

A simple, exact calculation is presented of the probability distribution of the number of hits in a block of n symbols in a frequency-hopped, spread-spectrum, multiple-access communication system. While the sequence of hits is not Markovian, there is an underlying Markovian structure that allows the probability distribution of the number of hits to be calculated in a recursive fashion. Knowing the probability distribution of the number of hits makes it possible to calculate the probability of error for a system employing error correcting codes for several different types of receivers, including receivers with both errors and erasures. The numerical results show that both the approximation obtained by assuming the actual sequence of hits is Markovian and the approximation obtained by assuming the hits are independent are very good. When the number of frequency slots is not too small (less than five), calculations show that assuming the independence of hits gives an error probability accurate to within 1% of the actual error probability. Assuming the hits are Markovian gives error probabilities which are accurate to within 0.001%