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Covering the fundamentals of detection and estimation theory, this systematic guide describes statistical tools that can be used to analyze, design, implement and optimize real-world systems. Detailed derivations of the various statistical methods are provided, ensuring a deeper understanding of the basics. Packed with practical insights, it uses extensive examples from communication, telecommunication and radar engineering to illustrate how theoretical results are derived and applied in practice. A unique blend of theory and applications and over 80 analytical and computational end-of-chapter problems make this an ideal resource for both graduate students and professional engineers.
The second half of the twentieth century experienced an explosive growth in information technology, including data transmission, processing, and computation. This trend will continue at an even faster pace in the twenty-first century. Radios and televisions started in the 1920s and 1940s respectively, and involved transmission from a single transmitter to multiple receivers using AM and FM modulations. Baseband analog telephony, starting in the 1900s, was originally suited only for local area person-to-person communication. It became possible to have long-distance communication after using cascades of regeneration repeaters based on digital PCM modulation. Various digital modulations with and without coding, across microwave, satellite, and optical fiber links, allowed the explosive transmissions of data around the world starting in the 1950s–1960s. The emergence of Ethernet, local area net, and, finally, the World Wide Web in the 1980s–1990s allowed almost unlimited communication from any computer to another computer. In the first decade of the twenty-first century, by using wireless communication technology, we have achieved cellular telephony and instant/personal data services for humans, and ubiquitous data collection and transmission using ad hoc and sensor networks. By using cable, optical fibers, and direct satellite communications, real-time on-demand wideband data services in offices and homes are feasible.
This publication was conceived as a textbook for a first-year graduate course in the Signals and Systems Area of the Electrical Engineering Department at UCLA to introduce basic statistical concepts of detection and estimation and their applications to engineering problems to students in communication, telecommunication, control, and signal processing. Students majoring in electromagnetics and antenna design often take this course as well. It is not the intention of this book to cover as many topics as possible, but to treat each topic with enough detail so a motivated student can duplicate independently some of the thinking processes of the originators of these concepts. Whenever possible, examples with some numerical values are provided to help the reader understand the theories and concepts. For most engineering students, overly formal and rigorous mathematical methods are probably neither appreciated nor desirable. However, in recent years, more advanced analytical tools have proved useful even in practical applications. For example, tools involving eigenvalue–eigenvector expansions for colored noise communication and radar detection; non-convex optimization methods for signal classification; non-quadratic estimation criteria for robust estimation; non-Gaussian statistics for fading channel modeling; and compressive sensing methodology for signal representation, are all introduced in the book.
Hypothesis testing is a concept originated in statistics by Fisher  and Neyman–Pearson  and forms the basis of detection of signals in noises in communication and radar systems.
Simple hypothesis testing
Suppose we measure the outcome of a real-valued r.v. X. This r.v. can come from two pdf's associated with the hypotheses, H0 or H1. Under H0, the conditional probability of X is denoted by p0(x) = p(x∣H0), −∞ < x < ∞, and under H1, the conditional probability of X is denoted by p1(x) = p(x|H1), − ∞ < x < ∞. This hypothesis is called “simple” if the two conditional pdf's are fully known (i.e., there are no unknown parameters in these two functions). From the observed x value (which is a realization of the r.v. X), we want to find a strategy to decide on H0 or H1 in some optimum statistical manner.
Example 3.1 The binary hypothesis problem in deciding between H0 or H1 is ideally suited to model the radar problem in which the hypothesis H0 is associated with the absence of a target and the hypothesis H1 is associated with the presence of a target. In a binary hypothesis problem, there are four possible states, whether H0 or H1 is true and whether the decision is to declare H0 or to declare H1. Table 3.1 summarizes these four states and the associated names and probabilities.
In this chapter we consider various analytical and simulation tools for system performance analysis of communication and radar receiver problems. In Section 8.1, we treat the analysis of receiver performance with Gaussian noise, first using the closure property of Gaussian vectors under linear operations. We then address this issue without using this closure property. Section 8.2 deals with the analysis of receiver performance with Gaussian noise and other random interferences caused by intersymbol interferences (ISI) due to bandlimitation of the transmission channel. Section 8.2.1 introduces the evaluation of the average probability of error based on the moment bounding method. Section 8.3 considers the analysis of receiver performance with non-Gaussian noises including the spherically invariant random processes (SIRP). By exploiting some basic properties of SIRP, Section 8.3.1 obtains a closed form expression for the receiver. We determine the average probability of error for the binary detection problem with additive multivariate t-distributed noise (which is a member of SIRP). Section 8.3.2 again uses some properties of SIRP to model wireless fading channels with various fading envelope statistics. By using Fox H-function representations of these pdfs, novel average probability of error expressions under fading conditions can be obtained. Section 8.3.3 treats the probabilities of a false alarm and detection of a radar problem with a robustness constraint. Section 8.4 first shows a generic practical communication/radar system, which may have various complex operations, making analytical evaluation of system performance in many cases difficult.
In Chapter 4, we considered the detection of known binary deterministic signals in Gaussian noises. In this chapter, we consider the detection and classification of M-ary deterministic signals. In Section 5.1, we introduce the problem of detecting M given signal waveforms in AWGN. Section 5.2 introduces the Gram–Schmidt orthonormalization method to obtain a set of N orthonormal signal vectors or waveforms from a set of N linearly independent signal vectors or waveforms. These orthonormal vectors or signal waveforms are used as a basis for representing M-ary signal vectors or waveforms in their detection. Section 5.3 treats the detection of M-ary given signals in AWGN. Optimum decisions under the Bayes criterion, the minimum probability of error criterion, the maximum a posteriori criterion, and the minimum distance decision rule are considered. Simple minimum distance signal vector geometry concepts are used to evaluate symbol error probabilities of various commonly encountered M-ary modulations including binary frequency-shifted-keying (BFSK), binary phase-shifted-keying (BPSK), quadra phase-shifted-keying (QPSK), and quadra-amplitude-modulation (QAM) communication systems. Section 5.4 considers optimum signal design for M-ary systems. Section 5.5 introduces linearly and non-linearly separable and support vector machine (SVM) concepts used in classification of M deterministic pattern vectors. A brief conclusion is given in Section 5.6. Some general comments are given in Section 5.7. References and homework problems are given at the end of this chapter.
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