1. Discussion. In this opening chapter, we offer a review of the basic facts we need from measure theory for the rest of the book (it doubles as an introduction to our pedagogic method). For readers seeking a true introduction to the subject, we recommend first perusing, e.g. Folland (1984); experts meanwhile may safely jump to Chapter 2.

2. Comment. When an exercise is given in the middle of a proof, the end of the exercise will be signaled by a dot:

The conclusion of a proof is signaled by the box sign at the right margin, thus:

Basic definitions

In this subchapter, we discuss algebras, σ-algebras, generation of a σ-algebra by a family of subsets, completion with respect to a measure and relevant definitions.

3. Definition. Let Ω be a set. An algebra of subsets of Ω is a non-empty collection A of subsets of Ω that is closed under finite unions and complementation. A σ-algebra is a collection A of subsets of Ω that is closed under countable unions and complementation.

4. Comment. Every algebra of subsets of Ω contains the trivial algebra {∅, Ω}.

5. Exercise. Let Ω be a set and let C be a family of subsets of Ω. Show that the intersection of all σ-algebras of subsets of Ω containing C is itself a σ-algebra.

Hilbert space

The prototype qubit systems of the last chapter are very simple, but they can be generalized to more complicated versions. We can send a photon through an interferometer with three, four or more distinct beams. We can perform experiments on particles with higher intrinsic angular momentum than the spin-½ particles we have discussed. And we can analyze atomic systems in situations that involve more than two different energy levels. For these cases and others, we will need a more general version of quantum theory.

That theory will include two pieces. First, we will have a general mathematical structure that is applicable to many kinds of system. Here the qubit case will be our guide, since many of the basic concepts for other quantum systems are already present in the qubit case. Second, we will have to describe how to apply the quantum formalism to specific physical situations. Though the quantum systems we discuss will appear quite various, they share strong family resemblances that are expressed in the common mathematical framework. Keeping the framework in mind will help us understand specific examples; keeping the examples in mind will help us understand the framework.

The states of a quantum system are described by kets |ψ〉, which obey the principle of superposition. This means that the kets are elements of an abstract vector space ℋ called a Hilbert space.

Introduction

Digital communication systems have been studied for many decades, and they have become an integral part of the technological world we live in. Many excellent books in recent years have told the story of this communication revolution, and have explained in considerable depth the theory and applications. Since the late 1990s particularly, there have been a number of significant contributions to digital communications from the signal processing community. This book presents a number of these recent developments, with emphasis on the use of filter bank precoders and equalizers. Optimization of these systems will be one of the main themes in this book. Both multiple-input multiple-output (MIMO) systems and single-input single-output (SISO) systems will be considered. It is assumed that the reader has had some exposure to digital communications and signal processing at the introductory level. Many text books cover this prerequisite, and some are mentioned at the beginning of Sec. 1.5.

Before we describe the contents of the book we first give an introductory description of analog and digital communication systems in the next few sections. The scope and outline of the book will be described in Sec. 1.5.

Introduction

In Sec. 5.7 we described decision feedback equalizers (DFEs). These equalizers use past decisions on symbols to refine the decision on the present symbol. In this chapter we extend this idea to decision feedback equalization in memoryless MIMO channels. Here past decisions within a block are used to improve future decisions in that block, at a given block-time instant. The basic idea is introduced in Sec. 19.2. The decision feedback transceiver has a precoder F, feedforward equalizer G, and a feedback matrix B. In Sec. 19.3 we show how these matrices can be jointly optimized to minimize the mean square reconstruction error, under the zero-forcing constraint. We will see that in DFE systems the minimized mean square error depends on the geometric mean of 1/σh,k rather than on their arithmetic mean (where σh,k are the channel singular values). This transition from AM to GM is a fundamental property of DFE systems, and results in substantial decrease of the mean square error. In Sec. 19.4 we extend the joint optimization to the case where there is no zero forcing. DFE systems with minimum symbol error rate are presented in Sec. 19.5. Error probability plots are presented in Sec. 19.6, showing the superiority of DFE transceivers over linear transceivers. It turns out that optimal DFE systems are related to systems optimizing mutual information; this is elaborated in Sec. 19.7. A brief description of DFE receivers based on QR decomposition, and the VBLAST receiver are given in Sec. 19.8.

Introduction

In this chapter we review a number of well established ideas in digital communication. Section 5.2 revisits the matched filter introduced in Sec. 2.5 and discusses it from the viewpoint of information sufficiency and reconstructibility. We establish the generality of matched filtering as a fundamental front-end tool in receiver design. It is often necessary to make sure that the noise sequence at the input of the detector is white. The digital filter that ensures such a property, the so-called sampled noise whitening filter, is described in Sec. 5.3. A vector space interpretation of matched filtering is presented in Sec. 5.4, and offers a very useful viewpoint.

Estimation of the transmitted symbol stream from the received noisy stream is one of the basic tasks performed in any communication receiver. The foundation for this comes from optimal sequence estimation theory, which is briefly discussed in Sec. 5.5. This includes a review of maximum likelihood and maximum a posteriori methods. The Viterbi algorithm for sequence estimation is described in Sec. 5.6. While this is one of the most well known algorithms, simpler suboptimal methods, such as decision feedback equalizers (DFE), have also become popular. The motivation for DFE, which is a nonlinear equalizer, is explained in Sec. 5.7. A nonlinear precoder called the Tomlinson-Harashima precoder can be used as an alternative to the nonlinear equalizer at the receiver, and is described in Sec. 5.8.