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In late 1950s and early 1960s, Gilbert and Elliott at Bell Labs were modeling burst-noise telephone circuits with a very simple two-state channel model with memory. This simple model allowed them to evaluate channel capacity and error rate performance through bursty wireline telephone circuits. However, it took another 30 years for the so-called Gilbert-Elliott channel (GEC) and its generalized finite-state Markov channel (FSMC) to be applied in the design of second-generation (2G) wireless communication systems. Since the mid 1990s, the GEC and FSMC models have been widely used for modeling wireless flat-fading channels in a variety of applications, ranging from modeling channel error bursts to decoding at the receiver. FSMC models are versatile, and with suitable choices of model parameters, can capture the essence of time-varying fading channels. This article's goal is to provide an in-depth understanding of the principles of FSMC modeling of fading channels with its applications in wireless communication systems. Digital Object Identifier 10.1109/MSP.2008 [ Parastoo Sadeghi, Rodney A. Kennedy, Predrag B. Rapajic, and Ramtin Shams ] While the emphasis is on frequency nonselective or flat-fading channels, this understanding will be useful for future generalizations of FSMC models for frequency-selective fading channels. The target audience of this article include both theory-and practice-oriented researchers who would like to design accurate channel models for evaluating the performance of wireless communication systems in the physical or media access control layers, or those who would like to develop more efficient and reliable transceivers that take advantage of the inherent memory in fading channels. Both FSMC models and flat-fading channels will be formally introduced. However, a background in time-varying fading communication channels is beneficial.We consider the FSMC modeling of fading channels from five distinct viewpoints. First, we provide a brief history of FSMC models and the FSMC modeling of flat-fading channels. Second, we define the parameters of FSMC models and discuss how these parameters can be derived from flat-fading channel statistics. We point out the trade-offs between model accuracy and complexity. Third, we categorize applications of FSMC models for fading channels into four categories and discuss the FSMC model applicability and accuracy in each application. We pay special attention to the effect of FSMC memory order on the model accuracy. Fourth, we consider the information-theoretical aspects of FSMC models and the FSMC modeling of fading channels. Finally, we present open questions and directions for future research. For easier access to the technical contents, the reader can refer to the "List of Acronyms" and "Notational Conventions." THE HISTORY OF FSMC 1957-1968: DEVELOPMENT OF FSMC MODELSThe study of finite-state communication channels with memory dates back to the work by Shannon in 1957 [1]. In [1], Shannon proved coding theorems for finite-state channels (FSCs) with discret...

We consider wireless-powered amplify-and-forward and decode-and-forward relaying in cooperative communications, where an energy constrained relay node first harvests energy through the received radio-frequency signal from the source and then uses the harvested energy to forward the source information to the destination node. We propose time-switching based energy harvesting (EH) and information transmission (IT) protocols with two modes of EH at the relay. For continuous time EH, the EH time can be any percentage of the total transmission block time. For discrete time EH, the whole transmission block is either used for EH or IT. The proposed protocols are attractive because they do not require channel state information at the transmitter side and enable relay transmission with preset fixed transmission power. We derive analytical expressions of the achievable throughput for the proposed protocols. The derived expressions are verified by comparison with simulations and allow the system performance to be determined as a function of the system parameters. Finally, we show that the proposed protocols outperform the existing fixed time duration EH protocols in the literature, since they intelligently track the level of the harvested energy to switch between EH and IT in an online fashion, allowing efficient use of resources.Radio frequency (RF) or wireless energy harvesting has recently emerged as an attractive solution to power nodes in future wireless networks [2]- [6]. Wireless energy harvesting techniques are now evolving from theoretical concepts into practical devices for low-power electronic applications [7]. The feasibility of wireless energy harvesting for low-power cellular applications has been studied using experimental results, which have been summarised in [2]. With wireless energy harvesting, there is a choice between harvesting energy from ambient sources or by carefully designing wireless power transfer links. For instance, a power density of around 1 mW/m 2 is reported around 50 meter distance from the base station in the GSM band (935 MHz -960 MHz) [4], which means that a wireless device with a typical size of around 100 cm 2 can harvest power in the range of tens of µW. Such an amount of harvested power could be sufficient for relaying nodes in sensor networks with sporadic activities. For devices that need to support frequent communication activities, harvesting ambient RF energy is not sufficient. Instead, harvesting wireless energy from carefully designed power transfer links is needed. In addition, the energy conversion efficiency of the wireless energy harvesting plays an important role in determining the amount of energy that can be harvested. Employing different circuit design technologies, wireless energy harvesting with energy conversion efficiency in the range of 10%-80% has been reported over a wide range of frequencies, e.g., 15 MHz -2.5 GHz [2], [8]. More specifically, energy conversion efficiency of around 65% has been reported in the ISM band (900 MHz, 2.4 GHz) with 13 nm CMOS techno...

This lively and accessible book describes the theory and applications of Hilbert spaces and also presents the history of the subject to reveal the ideas behind theorems and the human struggle that led to them. The authors begin by establishing the concept of 'countably infinite', which is central to the proper understanding of separable Hilbert spaces. Fundamental ideas such as convergence, completeness and dense sets are first demonstrated through simple familiar examples and then formalised. Having addressed fundamental topics in Hilbert spaces, the authors then go on to cover the theory of bounded, compact and integral operators at an advanced but accessible level. Finally, the theory is put into action, considering signal processing on the unit sphere, as well as reproducing kernel Hilbert spaces. The text is interspersed with historical comments about central figures in the development of the theory, which helps bring the subject to life.

In this paper, we consider a decode-and-forward (DF) relaying network based on wireless energy harvesting. The energy constrained relay node first harvests energy through radio-frequency (RF) signals from the source node. Next, the relay node uses the harvested energy to forward the decoded source information to the destination node. The source node transfers energy and information to the relay node through two mechanisms, i) time switching-based relaying (TSR) and ii) power splitting-based relaying (PSR). Considering wireless energy harvesting constraint at the relay node, we derive the exact analytical expressions of the achievable throughput and ergodic capacity of a DF relaying network for both TSR and PSR schemes. Through numerical analysis, we study the throughput performance of the overall system for different system parameters, such as energy harvesting time, power splitting ratio, and signal-tonoise-ratio (SNR). In particular, the throughput performance of the PSR scheme outperforms the throughput performance of the TSR scheme for a wide range of SNRs.

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We study the dimensions or degrees of freedom of farfield multipath that is observed in a limited, source-free region of space. The multipath fields are studied as solutions to the wave equation in an infinite-dimensional vector space. We prove two universal upper bounds on the truncation error of fixed and random multipath fields. A direct consequence of the derived bounds is that both fixed and random multipath fields have an effective finite dimension. For circular and spherical spatial regions, we show that this finite dimension is proportional to the radius and area of the region, respectively. We use the Karhunen-Loeve (KL) expansion of random multipath fields to quantify the notion of multipath richness. The multipath richness is defined as the number of significant eigenvalues in the KL expansion that achieves 99% of the total multipath energy.We prove a lower bound on the largest eigenvalue. This lower bound quantifies, to some extent, the well-known reduction of multipath richness with reducing the angular spread of multipath angular power spectrum. We also provide a numerical algorithm to find multipath eigenvalues, which unlike the Fredholm equation method, does not require selecting quadrature points.