The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Distance-Regular Graphs

In this letter, we address the problem of decoding nonbinary low-density parity-check (LDPC) codes over finite fields GF(q), with reasonable complexity and good performance. In the first part of the letter, we recall the original belief propagation (BP) decoding algorithm and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding. In the second part of the letter, we introduce a simplified decoder which is inspired by the min-sum decoder for binary LDPC codes. We called this decoder extended min-sum (EMS). We show that it is possible to greatly reduce the computational complexity of the check-node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance is quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very-high-order fields show that nonbinary LDPC codes are promising for applications which require low frame-error rates for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes

Decoding Algorithms for Nonbinary LDPC Codes Over GF $(q)$

国際会議開催報告:2013 IEEE International Symposium on Information Theory

We introduce a wide class of low-density parity-check (LDPC) codes, large enough to include LDPC codes over finite fields, rings, or groups, as well as some nonlinear codes. A belief-propagation decoding procedure with the same complexity as for the decoding of LDPC codes over finite fields is also presented. Moreover, an encoding procedure is developed

FFT-Based BP Decoding of General LDPC Codes Over Abelian Groups

In this paper, a new, efficient class of blind equalization algorithms is proposed for use in high-order, two-dimensional digital communication systems. We have called this family: the Constant Norm Algorithms (CNA). This family is derived in the context of Bussgang techniques. Therefore, the resulting algorithms are very simple. We show that some well-known blind algorithms such as "Sato's algorithm" or the Constant Modulus Algorithm (CMA) are particular cases in our CNA family. In addition, from this class, a new cost function, named Constant sQuare Algorithm (CQA), is derived, which is well designed for QAM. It results in a lower algorithm noise without increasing the complexity. Another advantage of this class lies in the possibility of creating new norms by combining several existing norms in order to benefit from the advantages of each original norm. For example, we present the norm resulting from the combination of the two algorithms, CMA and CQA. Moreover, we highlight that, with regard to the excess mean-square error performance, there is an optimal norm for each constellation, i.e., each modulation, in order to equalize it blindly

New Algorithms for Blind Equalization: The Constant Norm Algorithm Family

An embodiment of invention relates to a method for the transmission of a signal formed by vectors, each vector comprising N source symbols to be transmitted, using M transmission antennas, wherein M is greater than or equal to 2. The method comprises the following steps: linearly precoding the signal using a matrix product of a source matrix formed by vectors that are organized in successive lines by a linear precoding matrix, delivering a precoded matrix; and successively transmitting precoded vectors corresponding to columns of said precoded matrix, the M symbols of each precoded vector being distributed to the M antennas.

Method for the multi-antenna transmission of a linearly-precoded signal, corresponding devices, signal and reception method

Models for noncoherent error control in random linear network coding (RLNC) and store and forward (SAF) have been recently proposed. In this paper, we model different types of random network communications as the transmission of flats of matroids. This novel framework encompasses RLNC and SAF and allows us to introduce a novel protocol, referred to as random affine network coding (RANC), based on affine combinations of packets. Although the models previously proposed for RLNC and SAF only consider error control, using our framework, we first evaluate and compare the performance of different network protocols in the error-free case. We define and determine the rate, average delay, and throughput of such protocols, and we also investigate the possibilities of partial decoding before the entire message is received. We thus show that RANC outperforms RLNC in terms of data rate and throughput thanks to a more efficient encoding of messages into packets. Second, we model the possible alterations of a message by the network as an operator channel, which generalizes the channels proposed for RLNC and SAF. Error control is thus reduced to a coding-theoretic problem on flats of a matroid, where two distinct metrics can be used for error correction. We study the maximum cardinality of codes on flats in general, and codes for error correction in RANC in particular. We finally design a class of nearly optimal codes for RANC based on rank metric codes for which we propose a low-complexity decoding algorithm. The gain of RANC over RLNC is thus preserved with no additional cost in terms of complexity.

A Matroid Framework for Noncoherent Random Network Communications

Impulsive noise features in many modern communication systems—ranging from wireless to molecular—and is often modeled by the $\alpha $ -stable distribution. At present, the capacity of $\alpha $ -stable noise channels is not well understood, with the exception of Cauchy noise ( $\alpha = 1$ ) with a logarithmic constraint and Gaussian noise ( $\alpha = 2$ ) with a power constraint. In this paper, we consider additive symmetric $\alpha $ -stable noise channels with $\alpha \in (1,2]$ . We derive bounds for the capacity with an absolute moment constraint. We then compare our bounds with a numerical approximation via the Blahut–Arimoto algorithm, which provides insight into the effect of noise parameters on the bounds. In particular, we find that our lower bound is in good agreement with the numerical approximation for $\alpha $ near 2.

Alban Goupil

Papers

FFT-Based BP Decoding of General LDPC Codes Over Abelian Groups

New Algorithms for Blind Equalization: The Constant Norm Algorithm Family

Method for the multi-antenna transmission of a linearly-precoded signal, corresponding devices, signal and reception method

A Matroid Framework for Noncoherent Random Network Communications

Capacity Bounds for Additive Symmetric $\alpha $ -Stable Noise Channels