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.

Introduction

In layman's terms, a sufficient statistic for guessing M based on the observable Y is a random variable or a collection of random variables that contains all the information in Y that is relevant for guessing M. This is a particularly useful concept when the sufficient statistic is more concise than the observables. For example, if we observe the results of a thousand coin tosses Y1, …, Y1000 and we wish to test whether the coin is fair or has a bias of ¼, then a sufficient statistic turns out to be the number of “heads”among the outcomes Y1, …, Y1000. Another example was encountered in Section 20.12. There the observable was a two-dimensional random vector, and the sufficient statistic summarized the information that was relevant for guessing H in a scalar random variable; see (20.69).

In this chapter we provide a formal definition of sufficient statistics in the multihypothesis setting and explore the concept in some detail. We shall see that our definition is compatible with Definition 20.12.2, which we gave for the binary case. We only address the case where the observations take value in the d-dimensional Euclidean space ℝd. Extensions to observations consisting of a stochastic process are discussed in Section 26.3. Also, we only treat the case of guessing among a finite number of alternatives.

Introduction

In this chapter we finally address the detection problem in continuous time. The setup is described in Section 26.2. The key result of this chapter is that—even though in this setup the observation consists of a stochastic process (i.e., a continuum of random variables)—the problem can be reduced without loss of optimality to a finite-dimensional problem where the observation consists of a random vector. Before stating this result precisely in Section 26.4, we shall take a detour in Section 26.3 to discuss the definition of sufficient statistics when the observation consists of a continuous-time SP. The proof of the main result is delayed until Section 26.8. In Section 26.5 we analyze the conditional law of the sufficient statistic vector under each of the hypotheses. This analysis enables us in Section 26.6 to derive an optimal guessing rule and in Section 26.7 to analyze its performance. Section 26.9 addresses the front-end filter, which is a critical element of any practical implementation of the decision rule. Extensions to passband detection are then described in Section 26.10, followed by some examples in Section 26.11. Section 26.12 treats the problem of detection in “colored” noise, and the chapter concludes with a discussion of the detection problem for mean signals that are not bandlimited.

Signal processing deals with the analysis, interpretation, and manipulation of signals. Signals of interest include sound, images, biological signals, radar signals, and many others. Processing of such signals includes filtering, storage and reconstruction, separation of information from noise (e.g. aircraft identification by radar), compression (e.g. image compression), and feature extraction (e.g. speech-to-text conversion). In communications systems, signal processing is mostly performed at OSI layer 1, the physical layer (modulation, equalization, multiplexing, radio transmission, etc.), as well as at OSI layer 6, the presentation layer (source coding, including analog-to-digital conversion, and data compression). In cognitive radio networks, the major task of signal processing is spectrum sensing for detecting the unused spectrum and sharing it without harmful interference to other users. One important requirement in a cognitive radio network is sensing spectrum holes reliably and efficiently. Spectrum sensing techniques can be classified into three categories. First, cognitive radios must be capable of determining if a signal from a primary transmitter is locally present in a certain spectrum. Several approaches are used for transmitter detection, such as matched filter detection, energy detection, cyclostationary feature detection, and wavelet detection. Second, collaborative detection refers to spectrum sensing methods where information from multiple cognitive radio users is exploited for primary user detection. Third, the sensing devices can be separated from the secondary users and can be deployed into the cognitive network by the cognitive radio service provider. By doing this, the cost of the secondary user devices can be reduced and the hidden terminal problem/exposed terminal problem can be mitigated.