Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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

Massive multiple-input multiple-output MIMO is one of themost promising technologies for the next generation of wirelesscommunication networks because it has the potential to providegame-changing improvements in spectral efficiency SE and energyefficiency EE. This monograph summarizes many years ofresearch insights in a clear and self-contained way and providesthe reader with the necessary knowledge and mathematical toolsto carry out independent research in this area. Starting froma rigorous definition of Massive MIMO, the monograph coversthe important aspects of channel estimation, SE, EE, hardwareefficiency HE, and various practical deployment considerations.From the beginning, a very general, yet tractable, canonical systemmodel with spatial channel correlation is introduced. This modelis used to realistically assess the SE and EE, and is later extendedto also include the impact of hardware impairments. Owing tothis rigorous modeling approach, a lot of classic "wisdom" aboutMassive MIMO, based on too simplistic system models, is shownto be questionable.

https://www.nowpublishers.com/article/DownloadSummary/SIG-093

Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency

Intelligent reflecting surface (IRS) is an enabling technology to engineer the radio signal propagation in wireless networks. By smartly tuning the signal reflection via a large number of low-cost passive reflecting elements, IRS is capable of dynamically altering wireless channels to enhance the communication performance. It is thus expected that the new IRS-aided hybrid wireless network comprising both active and passive components will be highly promising to achieve a sustainable capacity growth cost-effectively in the future. Despite its great potential, IRS faces new challenges to be efficiently integrated into wireless networks, such as reflection optimization, channel estimation, and deployment from communication design perspectives. In this paper, we provide a tutorial overview of IRS-aided wireless communications to address the above issues, and elaborate its reflection and channel models, hardware architecture and practical constraints, as well as various appealing applications in wireless networks. Moreover, we highlight important directions worthy of further investigation in future work.

Intelligent Reflecting Surface-Aided Wireless Communications: A Tutorial

Reconfigurable intelligent surfaces (RISs) are an emerging transmission technology for application to wireless communications. RISs can be realized in different ways, which include (i) large arrays of inexpensive antennas that are usually spaced half of the wavelength apart; and (ii) metamaterial-based planar or conformal large surfaces whose scattering elements have sizes and inter-distances much smaller than the wavelength. Compared with other transmission technologies, e.g., phased arrays, multi-antenna transmitters, and relays, RISs require the largest number of scattering elements, but each of them needs to be backed by the fewest and least costly components. Also, no power amplifiers are usually needed. For these reasons, RISs constitute a promising software-defined architecture that can be realized at reduced cost, size, weight, and power (C-SWaP design), and are regarded as an enabling technology for realizing the emerging concept of smart radio environments (SREs). In this paper, we (i) introduce the emerging research field of RIS-empowered SREs; (ii) overview the most suitable applications of RISs in wireless networks; (iii) present an electromagnetic-based communication-theoretic framework for analyzing and optimizing metamaterial-based RISs; (iv) provide a comprehensive overview of the current state of research; and (v) discuss the most important research issues to tackle. Owing to the interdisciplinary essence of RIS-empowered SREs, finally, we put forth the need of reconciling and reuniting C. E. Shannon’s mathematical theory of communication with G. Green’s and J. C. Maxwell’s mathematical theories of electromagnetism for appropriately modeling, analyzing, optimizing, and deploying future wireless networks empowered by RISs.

/pdf/smart-radio-environments-empowered-by-reconfigurable-2uovma9m3w.pdf

Smart Radio Environments Empowered by Reconfigurable Intelligent Surfaces: How It Works, State of Research, and The Road Ahead

In this letter, a multiuser relay network with massive multiple-input multiple-output is investigated with mixed-analog-to-digital converter (ADC) at receiver. We first characterize the uplink achievable rate by deriving a tight approximation, which embraces the conventional unquantized system as a special case. Both power scaling laws at sources and relay are presented. It is validated that the performance loss due to low-precision ADCs can be compensated by increasing the number of antennas $M$ , obeying a logarithmical scaling law, rather than by increasing the transmit power at sources and/or relay. We show that the performance suffers from a loss factor interpreted as a nominal effective resolution of the entire mixed-ADC structure. Simulation results verify our observations.

Multiuser Massive MIMO Relaying With Mixed-ADC Receiver

In this paper, we investigate secure transmission in a massive multiple-input multiple-output system adopting low-resolution digital-to-analog converters (DACs). Artificial noise (AN) is deliberately transmitted simultaneously with the confidential signals to degrade the eavesdropper’s channel quality. By applying the Bussgang theorem, a DAC quantization model is developed which facilitates the analysis of the asymptotic achievable secrecy rate. Interestingly, for a fixed power allocation factor $\phi $ , low-resolution DACs typically result in a secrecy rate loss, but in certain cases, they provide superior performance, e.g., at low signal-to-noise ratio (SNR). Specifically, we derive a closed-form SNR threshold which determines whether low-resolution or high-resolution DACs are preferable for improving the secrecy rate. Furthermore, a closed-form expression for the optimal $\phi $ is derived. With AN generated in the null-space of the user channel and the optimal $\phi $ , low-resolution DACs inevitably cause secrecy rate loss. On the other hand, for random AN with the optimal $\phi $ , the secrecy rate is hardly affected by the DAC resolution because the negative impact of the quantization noise can be compensated by reducing the AN power. All the derived analytical results are verified by numerical simulations.

/pdf/secure-massive-mimo-communication-with-low-resolution-dacs-2b8wsyphj2.pdf

Secure Massive MIMO Communication With Low-Resolution DACs

Low-resolution digital-to-analog converters (DACs) and analog-to-digital converters (ADCs) are considered to reduce cost and power consumption in multiuser massive multiple-input multiple-output. Using the Bussgang theorem, we derive the asymptotic downlink achievable rate with respect to the resolutions of both DACs and ADCs, i.e., $b_{\mathrm{ DA}}$ and $b_{\mathrm{ AD}}$ , under the assumption of large antenna number ${N}$ , and fixed user load ratio ${\beta }$ . We characterize the rate loss caused by finite-bit-resolution converters and reveal that the quantization distortion is ignorable at low signal-to-noise ratio (SNR) even with low-resolution converters at both sides. While for maintaining the same rate loss at high SNR, it is discovered that one-more-bit DAC resolution is needed when more users are scheduled with $\beta $ increased by four times. More specifically for one-bit rate loss requirement, $b_{\mathrm{ DA}}$ can be set by $\lceil b_{\mathrm{ AD}}+({1}/{2})\log \beta \rceil$ given $b_{\mathrm{ AD}}$ . Similar observations on ADCs are also obtained with numerical verifications.

On Performance of Quantized Transceiver in Multiuser Massive MIMO Downlinks

Intelligent reflecting surface (IRS) has recently been identified as a prominent technology with the ability of enhancing wireless communication by dynamically manipulating the propagation environment. This letter investigates a multiple-input single-output (MISO) system deploying distributed IRSs. For practical considerations, we propose an efficient design of passive reflecting beamforming for the IRSs to exploit statistical channel state information (CSI) and analyze the achievable rate of the network taking into account the impact of CSI estimation error. The ergodic achievable rate is derived in a closed form, which provides insightful system design guidelines. Numerical results confirm the accuracy of the derived results and unveil the performance superiority of the proposed distributed IRS deployment over the conventional centralized deployment.

Distributed IRS With Statistical Passive Beamforming for MISO Communications

In this paper, we investigate a downlink channel of a large intelligent surface (LIS) communication system. The LIS is equipped with B-bit discrete phase shifts while base station (BS) exploits low-resolution digital-to-analog converters (DACs). Without the knowledge of channel state information (CSI) related to the LIS, we propose a practical phase shift design method, whose computational complexity increases by 2B independent of the number of reflecting elements N. A tight lower bound for the asymptotic rate of the user is obtained in closed form. As N increases, we observe that the asymptotic rate becomes saturated because both the received signal power and the DAC quantization noise increase. Compared to the optimal continuous phase shift design with perfect CSI, our proposed method asymptotically approaches the ideal benchmark performance for moderate to high values of B. The derived results and observations are verified by simulation results.

Jindan Xu

Papers

Multiuser Massive MIMO Relaying With Mixed-ADC Receiver

Secure Massive MIMO Communication With Low-Resolution DACs

On Performance of Quantized Transceiver in Multiuser Massive MIMO Downlinks

Distributed IRS With Statistical Passive Beamforming for MISO Communications

Discrete Phase Shift Design for Practical Large Intelligent Surface Communication