There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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

Convergence of Probability Measures

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Table Of Integrals Series And Products

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

Communication at millimeter wave (mmWave) frequencies is defining a new era of wireless communication. The mmWave band offers higher bandwidth communication channels versus those presently used in commercial wireless systems. The applications of mmWave are immense: wireless local and personal area networks in the unlicensed band, 5G cellular systems, not to mention vehicular area networks, ad hoc networks, and wearables. Signal processing is critical for enabling the next generation of mmWave communication. Due to the use of large antenna arrays at the transmitter and receiver, combined with radio frequency and mixed signal power constraints, new multiple-input multiple-output (MIMO) communication signal processing techniques are needed. Because of the wide bandwidths, low complexity transceiver algorithms become important. There are opportunities to exploit techniques like compressed sensing for channel estimation and beamforming. This article provides an overview of signal processing challenges in mmWave wireless systems, with an emphasis on those faced by using MIMO communication at higher carrier frequencies.

An Overview of Signal Processing Techniques for Millimeter Wave MIMO Systems

Massive multiple-input multiple-output (MIMO) systems rely on channel state information (CSI) feedback to perform precoding and achieve performance gain in frequency division duplex networks. However, the huge number of antennas poses a challenge to the conventional CSI feedback reduction methods and leads to excessive feedback overhead. In this letter, we develop a real-time CSI feedback architecture, called CsiNet-long short-term memory (LSTM), by extending a novel deep learning (DL)-based CSI sensing and recovery network. CsiNet-LSTM considerably enhances recovery quality and improves tradeoff between compression ratio (CR) and complexity by directly learning spatial structures combined with time correlation from training samples of time-varying massive MIMO channels. Simulation results demonstrate that CsiNet-LSTM outperforms existing compressive sensing-based and DL-based methods and is remarkably robust to CR reduction.

Deep Learning-Based CSI Feedback Approach for Time-Varying Massive MIMO Channels

In this paper, we propose a model-driven deep learning network for multiple-input multiple-output (MIMO) detection. The structure of the network is specially designed by unfolding the iterative algorithm. Some trainable parameters are optimized through deep learning techniques to improve the detection performance. Since the number of trainable variables of the network is equal to that of the layers, the network can be easily trained within a very short time. Furthermore, the network can handle time-varying channel with only a single training. Numerical results show that the proposed approach can improve the performance of the iterative algorithm significantly under Rayleigh and correlated MIMO channels.

A Model-Driven Deep Learning Network for MIMO Detection

In this letter, we propose a model-driven deep learning (DL) approach that combines DL with the expert knowledge to replace the existing orthogonal frequency-division multiplexing receiver in wireless communications. Different from the data-driven fully connected deep neural network (FC-DNN) method, we adopt the block-by-block signal processing method that divides the receiver into channel estimation subnet and signal detection subnet. Each subnet is constructed by a DNN and uses the existing simple and traditional solution as initialization. The proposed model-driven DL receiver offers more accurate channel estimation comparing with the linear minimum mean-squared error method and exhibits higher data recovery accuracy comparing with the existing methods and FC-DNN. Simulation results further demonstrate the robustness of the proposed approach in terms of signal-to-noise ratio and its superiority to the FC-DNN approach in the computational complexities or the memory usage.

ComNet: Combination of Deep Learning and Expert Knowledge in OFDM Receivers

Massive multiple-input multiple-output (MIMO) systems rely on channel state information (CSI) feedback to perform precoding and achieve performance gain in frequency division duplex (FDD) networks. However, the huge number of antennas poses a challenge to conventional CSI feedback reduction methods and leads to excessive feedback overhead. In this article, we develop a real-time CSI feedback architecture, called CsiNet-long short-term memory (LSTM), by extending a novel deep learning (DL)-based CSI sensing and recovery network. CsiNet-LSTM considerably enhances recovery quality and improves trade-off between compression ratio (CR) and complexity by directly learning spatial structures combined with time correlation from training samples of time-varying massive MIMO channels. Simulation results demonstrate that CsiNet- LSTM outperforms existing compressive sensing-based and DLbased methods and is remarkably robust to CR reduction.

Deep Learning-based CSI Feedback Approach for Time-varying Massive MIMO Channels

Massive multiple-input multiple-output (MIMO) is a promising technology to increase link capacity and energy efficiency. However, these benefits are based on available channel state information (CSI) at the base station (BS). Therefore, user equipment (UE) needs to keep on feeding CSI back to the BS, thereby consuming precious bandwidth resource. Large-scale antennas at the BS for massive MIMO seriously increase this overhead. In this paper, we propose a multiple-rate compressive sensing neural network framework to compress and quantize the CSI. This framework not only improves reconstruction accuracy but also decreases storage space at the UE, thus enhancing the system feasibility. Specifically, we establish two network design principles for CSI feedback, propose a new network architecture, CsiNet+, according to these principles, and develop a novel quantization framework and training strategy. Next, we further introduce two different variable-rate approaches, namely, SM-CsiNet+ and PM-CsiNet+, which decrease the parameter number at the UE by 38.0% and 46.7%, respectively. Experimental results show that CsiNet+ outperforms the state-of-the-art network by a margin but only slightly increases the parameter number. We also investigate the compression and reconstruction mechanism behind deep learning-based CSI feedback methods via parameter visualization, which provides a guideline for subsequent research.

Shi Jin

Papers

Deep Learning-Based CSI Feedback Approach for Time-Varying Massive MIMO Channels

A Model-Driven Deep Learning Network for MIMO Detection

ComNet: Combination of Deep Learning and Expert Knowledge in OFDM Receivers

Deep Learning-based CSI Feedback Approach for Time-varying Massive MIMO Channels

Convolutional Neural Network-Based Multiple-Rate Compressive Sensing for Massive MIMO CSI Feedback: Design, Simulation, and Analysis