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).

Cyclostationary analysis has several applications in communications, e.g., spectral sensing, signal parameter estimation, and modulation classification. Most of them consider the additive white Gaussian noise (AWGN) channel model, although wireless communication systems may also be subject to non-Gaussian interference and impulsive noise. In this context, the communication channel can be better modeled by heavy-tailed distributions, such as the non-Gaussian alpha-stable one. Some applications of the cyclostationary approach based on the spatial sign cyclic correlation function (SSCCF), fractional lower-order cyclic autocorrelation function (FLOCAF), and cyclic correntropy function (CCF) demonstrate that these are promising solutions for the analysis of signals in the presence of impulsive non-Gaussian noise. However, the investigation of functions above applied to digital modulation recognition in impulsive environments, and the comparison among them are topics that did not adequately explore yet. This work demonstrates that SSCCF is a particular case of the FLOCAF. Besides, a detailed analysis of the use of the FLOCAF and CCF is presented to obtain cyclostationary descriptors for the recognition of digital modulations BPSK, QPSK, 8-QAM, 16-QAM, and 32-QAM. Automatic modulation classification (AMC) architectures, based on the functions mentioned above, are also proposed. Besides, another contribution showed is that both the FLOCAF and CCF allow the symbol rate parameter estimation. The performances of AMC architectures were evaluated in the scenario with modulated signals contaminated with additive non-Gaussian alpha-stable noise. The results demonstrate that both architectures can classify signals in different contamination scenarios. However, the architecture based on the CCF is more efficient than the FLOCAF-based one.

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Automatic Modulation Classification Architectures Based on Cyclostationary Features in Impulsive Environments

Long-range (LoRa) technology is most widely used for enabling low-power wide area networks (WANs) on unlicensed frequency bands. Despite its modest data rates, it provides extensive coverage for low-power devices, making it an ideal communication system for many internet of things (IoT) applications. In general, LoRa is considered as the physical layer, whereas LoRaWAN is the medium access control (MAC) layer of the LoRa stack that adopts a star topology to enable communication between multiple end devices (EDs) and the network gateway. The chirp spread spectrum modulation deals with LoRa signal interference and ensures long-range communication. At the same time, the adaptive data rate mechanism allows EDs to dynamically alter some LoRa features, such as the spreading factor (SF), code rate, and carrier frequency to address the time variance of communication conditions in dense networks. Despite the high LoRa connectivity demand, LoRa signals interference and concurrent transmission collisions are major limitations. Therefore, to enhance LoRaWAN capacity, the LoRa Alliance released many LoRaWAN versions, and the research community has provided numerous solutions to develop scalable LoRaWAN technology. Hence, we thoroughly examine LoRaWAN scalability challenges and state-of-the-art solutions in both the physical and MAC layers. These solutions primarily rely on SF, logical, and frequency channel assignment, whereas others propose new network topologies or implement signal processing schemes to cancel the interference and allow LoRaWAN to connect more EDs efficiently. A summary of the existing solutions in the literature is provided at the end of the paper, describing the advantages and disadvantages of each solution and suggesting possible enhancements as future research directions.

A Survey on Scalable LoRaWAN for Massive IoT: Recent Advances, Potentials, and Challenges

The number of connected devices is growing and in the near future it is expected to become extremely large in cities. As a consequence, using simulators to study and prepare a project of installing new networks before their real deployment is of great importance. They can help to predict some important information like signal overload or the feasibility of the deployment in terms of location, interferences, communication and cost. In this paper we present a new architecture for the platform CupCarbon, developed within the research project PERSEPTEUR. The main objective of this platform is to design and simulate Wireless Sensor Networks dedicated to Smart-city and IoT applications. It allows to validate distributed algorithms in a 2D/3D environment, taking account of the city buildings in which to deploy the network, the mobiles, and using accurate models of radio propagation and interferences in that environment.

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CupCarbon: A new platform for the design, simulation and 2D/3D visualization of radio propagation and interferences in IoT networks

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.

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

Internet of Things (IoT) technology has become ubiquitous in a multitude of applications and its use is growing. However, the expansion of IoT faces a major difficulty: scalability, that is, very dense deployment of communicating devices is currently limited. In long-range networks, such as LoRa, the downlink is critical because it limits the number of acknowledgements that can be sent, and consequently reliability. It also limits the possibility to update the devices, which could be critical when they are deployed for decades. To overcome those problems, we propose a solution, inspired by Non Orthogonal Multiple Access (NOMA) techniques, to increase by at least one order of magnitude the number of devices that can be addressed. While the approach differentiates the devices by the power allocated to them, it differs from the vast majority of previous works on power domain NOMA because it does not require interference cancellation. Instead, it benefits from the spectrum spreading of the modulation scheme (chirp spread spectrum), where, at the end of the decoding phase, the information carried by a symbol is found in the position of a peak in the Fourier domain. In the vast majority of cases, the information from different users results in different peak positions, not creating any interference. In that sense, we get closer to avoidance schemes such as time or frequency hopping, but without using a code. In this article, we propose a new solution for NOMA in the power domain that does not suffer from the limitations induced by interference cancellation residues. The proposed system, including preamble detection and channel estimation, is presented and evaluated by simulations. We demonstrate that our proposed scheme increases the number of devices by one order of magnitude compared to the current system which allows addressing only one user at a time and maintains full compatibility with the LoRa physical layer standard.

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Laurent Clavier

Papers

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

Multiuser Detection for Downlink Communication in LoRa- Like Networks