This appendix is intended to help the reader by summarizing some known formulas and results from matrix algebra and probability theory. It is absolutely not a selfcontained treatment of linear algebra or probability theory, nor is it a replacement for a systematic treatment or a course on these topics. However, it may be helpful as a refresher for students who have not studied these subjects for some time.

All the results presented here are well-known and their proofs can be found in many textbooks on linear algebra, probability theory and statistics. Since our discussion is somewhat scattered, and since we believe that the proof of some of the results can constitute appropriate problems for students when this book is used in a classroom setting, we present most of the results in the form of exercises.

Complex Baseband Representation of Bandpass Signals

Most signals in a wireless communication system have a narrowband character, and such narrowband (real-valued) signals can be represented with complex variables in a neat way. The general theory for doing so is treated in most digital communication textbooks (see, e.g., [Proakis, 2001, Sec. 4.1]), but since the complex baseband representation is a foundation for all signal models used in this book we offer a brief treatment of the topic.

The book in hand is the result of merging an original plan of writing a monograph on orthogonal space-time block coding with the ambition of authoring a more textbook-like treatment of modern MIMO communication theory, and it may therefore have the attributes of both these categories of literature. A major part of the book has resulted from the development of graduate courses at the universities where the authors are active, whereas a smaller part is the outcome of the authors' own research.

Our text covers many aspects of MIMO communication theory, although not all topics are treated in equal depth. The discussion starts with a review of MIMO channel modeling along with error probability and information theoretical properties of such channels, from which we go on to analyze the concepts of receive and transmit diversity. We discuss linear space-time block coding for flat and frequency selective fading channels, along with its associated receiver structures, both in general and in particular with emphasis on orthogonal space-time block coding. We also treat several special topics, including space-time coding for informed transmitters and space-time coding in a multiuser environment. Furthermore we include material about the existence and design of amicable orthogonal designs, a topic which is relevant in the context of orthogonal space-time block coding.

The text focuses on principles rather than on specific practical applications, although we have tried to illustrate some results in the context of contemporary wireless standards.

The demand for capacity in cellular and wireless local area networks has grown in a literally explosive manner during the last decade. In particular, the need for wireless Internet access and multimedia applications require an increase in information throughput with orders of magnitude compared to the data rates made available by today's technology. One major technological breakthrough that will make this increase in data rate possible is the use of multiple antennas at the transmitters and receivers in the system. A system with multiple transmit and receive antennas is often called a multiple-input multiple-output (MIMO) system. The feasibility of implementing MIMO systems and the associated signal processing algorithms is enabled by the corresponding increase of computational power of integrated circuits, which is generally believed to grow with time in an exponential fashion.

Why Space-Time Diversity?

Depending on the surrounding environment, a transmitted radio signal usually propagates through several different paths before it reaches the receiver antenna. This phenomenon is often referred to as multipath propagation. The radio signal received by the receiver antenna consists of the superposition of the various multipaths. If there is no line-of-sight between the transmitter and the receiver, the attenuation coefficients corresponding to different paths are often assumed to be independent and identically distributed, in which case the central limit theorem [papoulis, 2002, ch. 7] applies and the resulting path gain can be modelled as a complex Gaussian random variable (which has a uniformly distributed phase and a Rayleigh distributed magnitude).

This chapter will take the first step towards exposing the reader to the concept of transmit diversity. We begin by studying a system where the channel is known to the transmitter. Next we assume that the channel is unknown at the transmitter site and show how transmit diversity can be achieved via repeated transmission of a single symbol, and how such transmit diversity is related to the receive diversity discussed in the previous chapter. Finally, we provide a more systematic discussion of space-time coding and also set the framework for the rest of this book. In most parts of this chapter, we assume a frequency flat channel (frequency selective channels will be discussed in Chapter 8).

Optimal Beamforming with Channel Known at Transmitter

As a preparation we shall begin by studying a system with nr ≥ 1 receive and nt > 1 transmit antennas, where both the transmitter and receiver know the propagation channel. This knowledge about the channel can be used to adapt the weights for each transmit antenna in such a way that the SNR at the receiver is maximized. Doing so is sometimes called “beamforming,” although it (like the beamforming in Section 5.1) may not have the physical interpretation of forming a beam. All the analysis presented in this section is straightforward; some of it can also be found in [ganesan and stoica, 2001b].

All the discussion so far in this book (with the exception of Sections 3.3 and 6.1) has focused on the case when the channel is unknown to the transmitter. In this chapter, we will briefly study how partial channel knowledge at the transmitter can be used to improve the system performance. In particular we will study linearly precoded STBC.

Introduction

If the transmitter knows the channel, then it is optimal from an error probability point of view to use what we referred to in Section 6.1 as beamforming. In the case of one receive antenna (nr = 1), beamforming amounts to transmitting a symbol weighted by the conjugate transpose of the channel, h* (for nr = 1, H becomes a row vector hT). Although doing so might lack the interpretation of beamforming in a strict physical sense (depending on the antenna configuration), it shares the main signal processing attributes thereof.

For a given transmit power, the performance obtained via transmit diversity using STBC is in general inferior to that of beamforming, or receive diversity, by a factor that is sometimes called the “array gain.” Loosely speaking, this is so since space-time coding methods spread power uniformly in all directions in space, while beamforming uses information about the channel to steer energy in the particular direction of the receiver.

This short chapter is aimed at highlighting some issues related to the use of space-time coding in a multiuser environment. We will see that in general, the use of a space-time code changes the statistical properties of the transmitted signal compared with conventional transmission. We will also describe some simple techniques for multiuser interference suppression in a system that uses orthogonal STBC.

Introduction

Since the radio spectrum is a finite resource, the radio frequencies will always be shared. For this reason, all users in a system will suffer from co-channel interference, that is, disturbing radio signals from other users who use the same carrier frequency. The capacity of a system is related to how often, or how densely, the carrier frequencies are reused in the system. The more densely the frequencies are used, the higher the system capacity, but also the higher the level of co-channel interference. Therefore a signal processing algorithm that can suppress co-channel interference at the receiver, or maintain a functional communication link at a higher interference level, can also increase the system capacity. In a cellular system, the sharing of frequencies is usually coordinated by the network, whereas for some indoor local area networks, and so-called ad-hoc networks, it may be harder to assign radio resources in a coordinated manner; consequently the problem of mitigating co-channel interference may be even more important for such networks.