Introduction

The previous chapter shows that the performance analysis of wireless communication systems requires the statistics of the signal-to-noise ratio (SNR) at the receiver. In the analysis of many advanced wireless communication techniques in later chapters, we will make use of some order statistical results. This chapter summarizes these results and their derivations for easy reference. Specifically, we first review the basic distribution functions of ordered random variables. After that, we derive some new order statistics results, including the joint distribution functions of partial sums of ordered random variables, for which we also present a novel analytical framework based on the moment generating function (MGF). The chapter is concluded with a discussion on the limiting distributions of extremes. Whenever appropriate, we use the exponential random variable special case as an illustrative example. Note that we focus on those order statistics results that will be employed in later chapters in the performance and complexity analysis of different wireless technologies. For a more thorough treatment of order statistics, the readers are referred to [1, 2].

Basic distribution functions

Order statistics deals with the distributions and statistical properties of the new random variables obtained after ordering the realizations of some random variables. Let γj's, j = 1, 2,…, L denote L independent and identically distributed (i.i.d.) nonnegative random variables with common PDF pγ (·) and CDF Fγ(·). Let γl:L denote the random variable corresponding to the lth largest observation of the L original random variables, such that γ1:L ≥ γ2:L ≥ … ≥ γL:L · γl:L is also called lth order statistics. The ordering process is illustrated in Fig. 3.1.

Order statistics is an important sub-discipline of statistical theory and finds applications in a vast variety of fields, with life science as the most notable example [1]. Over the years, order statistics has made an increasing number of appearances in design and analysis wireless communication systems, primarily because of the simple but effective engineering principle – ‘pick the best’. For example, the diversity combining technique is an effective solution to improve the performance of wireless communication systems operating over fading channels by generating differently faded replicas of the same information-bearing signal. Selection combining (SC) [2, 3], which selects the replica with the best quality for further processing, is an attractive practical combining scheme and has been researched extensively in the literature. The performance analysis of the SC scheme entails the distribution functions of the largest random variables among multiple ones, which is available in conventional order statistics literature.

More recently, order statistics has also found application in the analysis and design of many emerging wireless transmission and reception techniques, such as advanced diversity combining techniques, channel adaptive transmission techniques, and multiuser scheduling techniques. These techniques are becoming the essential building blocks of future wireless systems for the delivery of multimedia services with high spectrum efficiency [4]. In particular, order statistics results have allowed for the accurate quantification of the trade-off of performance versus complexity among different design options, which will greatly facilitate the applications of these technologies in future wireless systems.

Introduction

We present a brief summary of digital wireless communications in this chapter. The material will serve as a useful background for the advanced wireless technologies in later chapters. We first review the statistical fading channel models commonly used in wireless system analysis. After that, we discuss digital modulation schemes and their performance analysis over fading channels, including the well-known moment generating function (MGF)-based approach [1]. The basic concept of adaptive modulation and diversity combining will also be presented. While most of the materials of this chapter are reviews of classical results, which can be found in other wireless textbooks, this chapter provides for the first time a thorough treatment of various conventional threshold-based combining schemes, a class of combining scheme enjoying even lower complexity than selection combining (SC). The chapter concludes with a brief discussion of the transmit diversity technique. The discussion of this chapter is by no means comprehensive. The main objective is to introduce some common notation and system models for later chapters. For a thorough treatment of these subjects, the reader may refer to [1,2].

Statistical fading channel models

Wireless channels rely on the physical phenomenon of electromagnetic wave propagation, due to the pioneering discoveries of Maxwell and Hertz. Radio waves propagate through several mechanisms, including direct line of sight (LOS), reflection, diffiraction, scattering, etc. As such, there usually exist multiple propagation paths between the transmitters and the receivers, as illustrated in Fig. 2.1.

Introduction

Multiple-antenna transmission and reception (i.e. MIMO) techniques can considerably improve the performance and/or effciency of wireless communication systems. First introduced in mid 1990s [1, 2], MIMO technique has been an area of active research and has found applications in various emerging wireless systems. Most of early MIMO designs focus on point-to-point link where both transmitter and receiver have multiple antennas and demonstrate the huge potential of MIMO techniques in terms of providing array gain, spatial diversity gain, spatial multiplexing gain and interference reduction capability, among many others [3–5]. Meanwhile, most mobile receivers will possess less antennas than the base stations in the near future due to their size/cost constraints. In such scenarios, the capacity of a point-to-point link between the base station and a mobile will be limited by the number of antennas at the mobile. On the other hand, if we consider the antennas of different receivers together, a virtual MIMO system is formed with huge capacity potential [6–10]. The design and analysis of effcient transmission strategy for resulting multiuser MIMO systems is the subject of this chapter.

A rich literature on MIMO wireless communications already exists. There has already been a rich literature on MIMO wireless communications. Several books have been published on this general subject (see for example [11,12]). This chapter complements existing literature on MIMO wireless communications by focusing on the different multiuser scheduling schemes for multiuser MIMO systems. In general, the downlink transmission from the base station to mobile receivers is more challenging in the multiuser MIMO system, as mobile users are randomly distributed and cannot perform joint detection.

Recall from Chapter 3 that a stochastic equation is a differential equation for a quantity whose rate of change contains a random component. One often refers to a quantity like this as being “driven by noise”, and the technical term for it is a stochastic process. So far we have found the probability density for a stochastic process by solving the stochastic differential equation for it. There is an alternative method, where instead one derives a partial differential equation for the probability density for the stochastic process. One then solves this equation to obtain the probability density as a function of time. If the process is driven by Gaussian noise, the differential equation for the probability density is called a Fokker–Planck equation.

Describing a stochastic process by its Fokker–Planck equation does not give one direct access to as much information as the Ito stochastic differential equation, because it does not provide a practical method to obtain the sample paths of the process. However, it can be used to obtain analytic expressions for steady-state probability densities in many cases when these cannot be obtained from the stochastic differential equation. It is also useful for an alternative purpose, that of describing the evolution of many randomly diffusing particles. This is especially useful for modeling chemical reactions, in which the various reagents are simultaneously reacting and diffusing.

In this book we will study dynamical systems driven by noise. Noise is something that changes randomly with time, and quantities that do this are called stochastic processes. When a dynamical system is driven by a stochastic process, its motion too has a random component, and the variables that describe it are therefore also stochastic processes. To describe noisy systems requires combining differential equations with probability theory. We begin, therefore, by reviewing what we will need to know about probability.

Random variables and mutually exclusive events

Probability theory is used to describe a situation in which we do not know the precise value of a variable, but may have an idea of the relative likelihood that it will have one of a number of possible values. Let us call the unknown quantity X. This quantity is referred to as a random variable. If X is the value that we will get when we roll a six-sided die, then the possible values of X are 1, 2, …, 6. We describe the likelihood that X will have one of these values, say 3, by a number between 0 and 1, called the probability. If the probability that X = 3 is unity, then this means we will always get 3 when we roll the die. If this probability is zero, then we will never get the value 3.

Introduction

You are probably asking yourself why we need a second chapter on probability theory. The reason is that the modern formalism used by mathematicians to describe probability involves a number of concepts, predefined structures, and jargon that are not included in the simple approach to probability used by the majority of natural scientists, and the approach we have adopted here. This modern formalism is not required to understand probability theory. Further, in the author's experience, the vast majority of physically relevant questions can be answered, albeit nonrigorously, without the use of modern probability theory. Nevertheless, research work that is written in this modern language is not accessible unless you know the jargon. The modern formalism is used by mathematicians, some mathematical physicists, control theorists, and researchers who work in mathematical finance. Since work that is published in these fields is sometimes useful to physicists and other natural scientists, it can be worthwhile to know the jargon and the concepts that underly it.

Unfortunately a considerable investment of effort is required to learn modern probability theory in its technical detail: significant groundwork is required to define the concepts with the precision demanded by mathematicians. Here we present the concepts and jargon of modern probability theory without the rigorous mathematical technicalities. Knowing this jargon allows one to understand research articles that apply this formalism to problems in the natural sciences, control theory, and mathematical finance.