We now know that information is conveyed in a digital communication system by selecting one of a set of signals to transmit. The received signal is a distorted and noisy version of the transmitted signal. A fundamental problem in receiver design, therefore, is to decide, based on the received signal, which of the set of possible signals was actually sent. The task of the link designer is to make the probability of error in this decision as small as possible, given the system constraints. Here, we examine the problem of receiver design for a simple channel model, in which the received signal equals one of M possible deterministic signals, plus white Gaussian noise (WGN). This is called the additive white Gaussian noise (AWGN) channel model. An understanding of transceiver design principles for this channel is one of the first steps in learning digital communication theory. White Gaussian noise is an excellent model for thermal noise in receivers, whose PSD is typically flat over most signal bandwidths of interest.

In practice, when a transmitted signal goes through a channel, at the very least, it gets attenuated and delayed, and (if it is a passband signal) undergoes a change of carrier phase. Thus, the model considered here applies to a receiver that can estimate the effects of the channel, and produce a noiseless copy of the received signal corresponding to each possible transmitted signal.

Communication has been one of the deepest needs of the human race throughout recorded history. It is essential to forming social unions, to educating the young, and to expressing a myriad of emotions and needs. Good communication is central to a civilized society.

The various communication disciplines in engineering have the purpose of providing technological aids to human communication. One could view the smoke signals and drum rolls of primitive societies as being technological aids to communication, but communication technology as we view it today became important with telegraphy, then telephony, then video, then computer communication, and today the amazing mixture of all of these in inexpensive, small portable devices.

Initially these technologies were developed as separate networks and were viewed as having little in common. As these networks grew, however, the fact that all parts of a given network had to work together, coupled with the fact that different components were developed at different times using different design methodologies, caused an increased focus on the underlying principles and architectural understanding required for continued system evolution.

This need for basic principles was probably best understood at American Telephone and Telegraph (AT&T), where Bell Laboratories was created as the research and development arm of AT&T. The Math Center at Bell Labs became the predominant center for communication research in the world, and held that position until quite recently.

Introduction

Chapter 7 showed how to characterize noise as a random process. This chapter uses that characterization to retrieve the signal from the noise-corrupted received waveform. As one might guess, this is not possible without occasional errors when the noise is unusually large. The objective is to retrieve the data while minimizing the effect of these errors. This process of retrieving data from a noise-corrupted version is known as detection.

Detection, decision making, hypothesis testing, and decoding are synonyms. The word detection refers to the effort to detect whether some phenomenon is present or not on the basis of observations. For example, a radar system uses observations to detect whether or not a target is present; a quality control system attempts to detect whether a unit is defective; a medical test detects whether a given disease is present. The meaning of detection has been extended in the digital communication field from a yes/no decision to a decision at the receiver between a finite set of possible transmitted signals. Such a decision between a set of possible transmitted signals is also called decoding, but here the possible set is usually regarded as the set of codewords in a code rather than the set of signals in a signal set. Decision making is, again, the process of deciding between a number of mutually exclusive alternatives.

Introduction

A general block diagram of a point-to-point digital communication system was given in Figure 1.1. The source encoder converts the sequence of symbols from the source to a sequence of binary digits, preferably using as few binary digits per symbol as possible. The source decoder performs the inverse operation. Initially, in the spirit of source/channel separation, we ignore the possibility that errors are made in the channel decoder and assume that the source decoder operates on the source encoder output.

We first distinguish between three important classes of sources.

1. • Discrete sources The output of a discrete source is a sequence of symbols from a known discrete alphabet X. This alphabet could be the alphanumeric characters, the characters on a computer keyboard, English letters, Chinese characters, the symbols in sheet music (arranged in some systematic fashion), binary digits, etc. The discrete alphabets in this chapter are assumed to contain a finite set of symbols.

2. It is often convenient to view the sequence of symbols as occurring at some fixed rate in time, but there is no need to bring time into the picture (for example, the source sequence might reside in a computer file and the encoding can be done off-line).

3. This chapter focuses on source coding and decoding for discrete sources. Supplementary references for source coding are given in Gallager (1968, chap. 3) and Cover and Thomas (2006, chap. 5). A more elementary partial treatment is given in Proakis and Salehi (1994, sect. 4.1–4.3).

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Introduction

Chapter 6 discussed modulation and demodulation, but replaced any detailed discussion of the noise by the assumption that a minimal separation is required between each pair of signal points. This chapter develops the underlying principles needed to understand noise, and Chapter 8 shows how to use these principles in detecting signals in the presence of noise.

Noise is usually the fundamental limitation for communication over physical channels. This can be seen intuitively by accepting for the moment that different possible transmitted waveforms must have a difference of some minimum energy to overcome the noise. This difference reflects back to a required distance between signal points, which, along with a transmitted power constraint, limits the number of bits per signal that can be transmitted.

The transmission rate in bits per second is then limited by the product of the number of bits per signal times the number of signals per second, i.e. the number of degrees of freedom per second that signals can occupy. This intuitive view is substantially correct, but must be understood at a deeper level, which will come from a probabilistic model of the noise.

This chapter and the next will adopt the assumption that the channel output waveform has the form y(t) = x(t) + z(t), where x(t) is the channel input and z(t) is the noise.

In Chapter 4, we showed that any ℒ2 function u(t) can be expanded in various orthogonal expansions, using such sets of orthogonal functions as the T-spaced truncated sinusoids or the sinc-weighted sinusoids. Thus u(t) may be specified (up to ℒ2-equivalence) by a countably infinite sequence such as {uk,m; −∞ < k, m < ∞} of coefficients in such an expansion.

In engineering, n-tuples of numbers are often referred to as vectors, and the use of vector notation is very helpful in manipulating these n-tuples. The collection of n-tuples of real numbers is called ℝn and that of complex numbers ℂn. It turns out that the most important properties of these n-tuples also apply to countably infinite sequences of real or complex numbers. It should not be surprising, after the results of the previous chapters, that these properties also apply to ℒ2 waveforms.

A vector space is essentially a collection of objects (such as the collection of real n-tuples) along with a set of rules for manipulating those objects. There is a set of axioms describing precisely how these objects and rules work. Any properties that follow from those axioms must then apply to any vector space, i.e. any set of objects satisfying those axioms. These axioms are satisfied by ℝn and ℂn, and we will soon see that they are also satisfied by the class of countable sequences and the class of ℒ2 waveforms.

Introduction

Digital modulation (or channel encoding) is the process of converting an input sequence of bits into a waveform suitable for transmission over a communication channel. Demodulation (channel decoding) is the corresponding process at the receiver of converting the received waveform into a (perhaps noisy) replica of the input bit sequence. Chapter 1 discussed the reasons for using a bit sequence as the interface between an arbitrary source and an arbitrary channel, and Chapters 2 and 3 discussed how to encode the source output into a bit sequence.

Chapters 4 and 5 developed the signal-space view of waveforms. As explained in those chapters, the source and channel waveforms of interest can be represented as real or complex ℒ2 vectors. Any such vector can be viewed as a conventional function of time, x(t). Given an orthonormal basis {ϕ1(t), ϕ2(t), …} of ℒ2, any such x(t) can be represented as

Each xj in (6.1) can be uniquely calculated from x(t), and the above series converges in ℒ2 to x(t). Moreover, starting from any sequence satisfying Σj|xj|2 < ∞, there is an ℒ2 function x(t) satisfying (6.1) with ℒ2-convergence. This provides a simple and generic way of going back and forth between functions of time and sequences of numbers.

Introduction

This chapter provides a brief treatment of wireless digital communication systems. More extensive treatments are found in many texts, particularly Tse and Viswanath (2005) and Goldsmith (2005). As the name suggests, wireless systems operate via transmission through space rather than through a wired connection. This has the advantage of allowing users to make and receive calls almost anywhere, including while in motion. Wireless communication is sometimes called mobile communication, since many of the new technical issues arise from motion of the transmitter or receiver.

There are two major new problems to be addressed in wireless that do not arise with wires. The first is that the communication channel often varies with time. The second is that there is often interference between multiple users. In previous chapters, modulation and coding techniques have been viewed as ways to combat the noise on communication channels. In wireless systems, these techniques must also combat time-variation and interference. This will cause major changes both in the modeling of the channel and the type of modulation and coding.

Wireless communication, despite the hype of the popular press, is a field that has been around for over 100 years, starting around 1897 with Marconi's successful demonstrations of wireless telegraphy. By 1901, radio reception across the Atlantic Ocean had been established, illustrating that rapid progress in technology has also been around for quite a while.

Digital communication is an enormous and rapidly growing industry, roughly comparable in size to the computer industry. The objective of this text is to study those aspects of digital communication systems that are unique. That is, rather than focusing on hardware and software for these systems (which is much like that in many other fields), we focus on the fundamental system aspects of modern digital communication.

Digital communication is a field in which theoretical ideas have had an unusually powerful impact on system design and practice. The basis of the theory was developed in 1948 by Claude Shannon, and is called information theory. For the first 25 years or so of its existence, information theory served as a rich source of academic research problems and as a tantalizing suggestion that communication systems could be made more efficient and more reliable by using these approaches. Other than small experiments and a few highly specialized military systems, the theory had little interaction with practice. By the mid 1970s, however, mainstream systems using information-theoretic ideas began to be widely implemented. The first reason for this was the increasing number of engineers who understood both information theory and communication system practice. The second reason was that the low cost and increasing processing power of digital hardware made it possible to implement the sophisticated algorithms suggested by information theory.

Introduction

This chapter has a dual objective. The first is to understand analog data compression, i.e. the compression of sources such as voice for which the output is an arbitrarily varying real- or complex-valued function of time; we denote such functions as waveforms. The second is to begin studying the waveforms that are typically transmitted at the input and received at the output of communication channels. The same set of mathematical tools is required for the understanding and representation of both source and channel waveforms; the development of these results is the central topic of this chapter.

These results about waveforms are standard topics in mathematical courses on analysis, real and complex variables, functional analysis, and linear algebra. They are stated here without the precision or generality of a good mathematics text, but with considerably more precision and interpretation than is found in most engineering texts.

Analog sources

The output of many analog sources (voice is the typical example) can be represented as a waveform, {u(t): ℝ → ℝ} or {u(t): ℝ → ℂ}. Often, as with voice, we are interested only in real waveforms, but the simple generalization to complex waveforms is essential for Fourier analysis and for baseband modeling of communication channels. Since a real-valued function can be viewed as a special case of a complex-valued function, the results for complex functions are also useful for real functions.

Introduction to quantization

Chapter 2 discussed coding and decoding for discrete sources. Discrete sources are a subject of interest in their own right (for text, computer files, etc.) and also serve as the inner layer for encoding analog source sequences and waveform sources (see Figure 3.1). This chapter treats coding and decoding for a sequence of analog values. Source coding for analog values is usually called quantization. Note that this is also the middle layer for waveform encoding/decoding.

The input to the quantizer will be modeled as a sequence U1, U2, …, of analog random variables (rvs). The motivation for this is much the same as that for modeling the input to a discrete source encoder as a sequence of random symbols. That is, the design of a quantizer should be responsive to the set of possible inputs rather than being designed for only a single sequence of numerical inputs. Also, it is desirable to treat very rare inputs differently from very common inputs, and a probability density is an ideal approach for this. Initially, U1, U2, … will be taken as independent identically distributed (iid) analog rvs with some given probability density function (pdf) fu(u).

A quantizer, by definition, maps the incoming sequence U1, U2, …, into a sequence of discrete rvs V1, V2, … where the objective is that Vm, for each m in the sequence, should represent Um with as little distortion as possible.

We have already seen that optical networks come in a large number of various flavors. Optical networks may have different topologies, may be transparent or opaque, and may deploy time, space, and/or wavelength division multiplexing (TDM, SDM, and/or WDM). They may comprise tunable devices, for example, tunable transmitters, tunable optical filters, and/or tunable wavelength converters (TWCs). Furthermore, to improve their flexibility optical networks may make use of reconfigurable optical add-drop multiplexers (ROADMs) and/or reconfigurable optical cross-connects. We will use the term optical switching networks to refer to all the various types of flexible and reconfigurable optical networks that use any of the aforementioned multiplexing, tuning, and switching techniques. Thus, optical switching networks are single-channel or multichannel (WDM) networks whose configuration can be changed dynamically in response to varying traffic loads and network failures by controlling the state of their tunable and/or reconfigurable network elements accordingly. Optical switching networks are widely deployed in today's wide, metropolitan, access, and local area networks and can be found at every level of the existing network infrastructure hierarchy.

End-to-end optical networks

Optical switching networks have been commonly used in backbone networks in order to cope with the ever-increasing amount of traffic originating from an increasing number of users and bandwidth-hungry applications. As shown in Fig. 2.1, optical switching networks can be found not only in wide area long-haul backbone networks but they also become increasingly the medium of choice in metro(politan), access, and local area networks (Berthelon et al., 2000). As a matter of fact, both telcos and cable providers are steadily moving the fiber-to-copper discontinuity point out toward the end users at the network periphery.

In this part, we discuss and describe in great detail various switching techniques for optical wide area networks (WANs). A number of different optical switching techniques have been proposed for backbone wavelength division multiplexing (WDM) networks over the last few years. Our overview will focus on the major optical switching techniques that can be found in today's operational long-haul WDM networks or are expected to be likely deployed in future optical WANs. In our overview we do not claim to provide a comprehensive description of all proposed switching techniques. Instead, we try to focus on the major optical switching techniques and describe their underlying principles and operation at length. We believe that our overview of carefully selected optical switching techniques fully covers the different types of switching techniques available for optical WANs and helps the reader gain sufficient knowledge to anticipate and understand any of the unmentioned optical switching techniques that in most cases might be viewed as extensions or hybrids of the optical switching techniques discussed. For instance, a so-called light-trail is a generalization of a conventional point-to-point lightpath in which data can be dropped and added at any node along the path, as opposed to a lightpath where data can be added only by the source and dropped only by the destination node, respectively (Gumaste and Zheng, 2005). Another good example is fractional lambda switching (FλS) (Baldi and Ofek, 2002). FλS uses the globally available coordinated universal time (UTC) as a common time reference to synchronize all optical switches throughout the FλS network.