In Digital Communications the task of the receiver is to observe the channel outputs and to use these observations to accurately guess the data bits that were sent by the transmitter, i.e., the data bits that were fed to the modulator. Ideally, the guessing would be perfect, i.e., the receiver would make no errors. This, alas, is typically impossible because of the distortions and noise that the channel introduces. Indeed, while one can usually recover the data bits from the transmitted waveform (provided that the modulator is a one-to-one mapping), the receiver has no access to the transmitted waveform but only to the received waveform. And since the latter is typically a noisy version of the former, some errors are usually unavoidable.

In this chapter we shall begin our study of how to guess intelligently, i.e., how, given the channel output, one should guess the data bits with as low a probability of error as possible. This study will help us not only in the design of receivers but also in the design of modulators that allow for reliable decoding from the channel's output.

In the engineering literature the process of guessing the data bits based on the channel output is called “decoding.” In the statistics literature this process is called “hypothesis testing.” We like “guessing” because it demystifies the process.

In most applications the channel output is a continuous-time waveform and we seek to decode a large number of bits. Nevertheless, for pedagogical reasons, we shall begin our study with the simpler case where we wish to decode only a single data bit.

Introduction

The multivariate Gaussian distribution is arguably the most important multivariate distribution in Digital Communications. It is the extension of the univariate Gaussian distribution from scalars to vectors. A random vector of this distribution is said to be a Gaussian vector, and its components are said to be jointly Gaussian. In this chapter we shall define this distribution, provide some useful characterizations, and study some of its key properties. To emphasize its connection to the univariate distribution, we shall derive it along the same lines we followed in deriving the univariate Gaussian distribution in Chapter 19.

There are a number of equivalent ways to define the multivariate Gaussian distribution, and authors typically pick one definition and then proceed over the course of numerous pages to derive alternate characterizations. We shall also proceed in this way, but to satisfy the impatient reader's curiosity we shall state the various equivalent definitions in this section. The proof of their equivalence will be spread over the whole chapter.

In the following definition we use the notation introduced in Section 17.2. In particular, all vectors are column vectors, and we denote the components of the vector a ∈ ℝn by a(1), …, a(n).

Introduction and Setup

In Chapter 26 we addressed the problem of detecting one of M bandwidth-W signals corrupted by additive Gaussian noise that is white with respect to the bandwidth W. Except for assuming that the mean signals are integrable signals that are bandlimited to W Hz, we made no assumptions about their structure. In this chapter we study the implication of the results of Chapter 26 for Pulse Amplitude Modulation, where the mean signals correspond to different possible outputs of a PAM modulator. The conclusions we shall draw are extremely important to the design of receivers for systems employing PAM.

The most important result of this chapter is that, loosely speaking, for PAM signals contaminated by additive white Gaussian noise, the inner products between the received waveform and the time shifts of the pulse shape by integer multiples of the baud period Ts form a sufficient statistic. Thus, if we feed the received waveform to a matched filter that is matched to the pulse shape defining the PAM signals, then the matched filter's outputs sampled at integer multiples of the baud period Ts form a sufficient statistic (Theorem 5.8.2). Using this result we can reduce the guessing problem from one with an observation consisting of a continuous-time stochastic process to one with an observation consisting of a discrete-time SP. In fact, since we shall only consider the problem of detecting a finite number of data bits, the reduction will be to a finite number of random variables.

Introduction

The Power Spectral Density of a stochastic process tells us more about the SP than just its power. It tells us something about how this power is distributed among the different frequencies that the SP occupies. The purpose of this chapter is to clarify this statement and to derive the PSD of PAM signals. Most of this chapter is written informally with an emphasis on ideas and intuition as opposed to mathematical rigor. The mathematically-inclined readers will find precise statements of the key results of this chapter in Section 15.5. We emphasize that this chapter only deals with real continuous-time stochastic processes.

The classical definition of the PSD of continuous-time stochastic processes (Definition 25.7.2 ahead) is only applicable to wide-sense stationary stochastic processes, and PAM signals are not WSS. Consequently, we shall have to introduce a new concept, which we call the operational power spectral density, or the operational PSD for short. This new concept is applicable to a large family of stochastic processes that includes most WSS processes and most PAM signals. For WSS stochastic processes, the operational PSD and the classical PSD coincide (Section 25.14). In addition to being more general, the operational PSD is more intuitive in that it clarifies the origin of the words “power spectral density.” Moreover, it gives an operational meaning to the concept.

Introduction

The signals encountered in wireless communications are typically real passband signals. In this chapter we shall define such signals and define their bandwidth around a carrier frequency. We shall then explain how such signals can be represented using their complex baseband representation. We shall emphasize two relationships: that between the energy in the passband signal and in its baseband representation, and that between the bandwidth of the passband signal around the carrier frequency and the bandwidth of its baseband representation. We ask the reader to pay special attention to the fact that only real passband signals have a baseband representation.

Most of the chapter deals with the family of integrable passband signals. As we shall see in Corollary 7.2.4, an integrable passband signal must have finite energy, and this family is thus a subset of the family of energy-limited passband signals. Restricting ourselves to integrable signals—while reducing the generality of some of the results—simplifies the exposition because we can discuss the Fourier Transform without having to resort to the L2-Fourier Transform, which requires all statements to be phrased in terms of equivalence classes. But most of the derived results will also hold for the more general family of energy-limited passband signals with only slight modifications. The required modifications are discussed in Section 7.7.