A Signal Theoretic Introduction to Random Processes

Presenting a rigorous introduction to the modelling and characterization of random phenomena, this book stands out from the existing texts in this field by characterizing random processes in signal theory. Instead of approaching randomness directly from random processes theory, the author places an emphasis on statistical signal theory, mathematical rigor, and using finite-time interval to establish results of random processes. One advantage to this theoretical approach is the unique framework provided to augment existing theory in the context of many unsolved problems. The use of a signal theory basis provides a general framework for defining functions used for characterizing random phenomena including the autocorrelation function and the power spectral density. The signal basis set approach for defining the power spectral density, which is the most widely used measure for characterizing random phenomena, provides a simple and natural interpretation of this function for the general case and for the usual case where a sinusoidal basis set is assumed. The usual approach to the power spectral density is through an autocorrelation function where an indirect interpretation can be provided for the restricted sinusoidal basis set case. Also, results for the important random phenomena encountered in the electronic and communications engineering field are given and include: the random walk, Brownian motion, the random telegraph signal, the Poisson point process, the Poisson counting process, shot noise, white noise, 1/"f" noise, signaling random processes, jittered random processes, random clustering, and birth-death random processes. Finally, the mathematical rigor underpins all aspects throughout the book, which demonstrates clarity and precision in the statement of results. The first five chapters provides the necessary background on the mathematical, signal theory, random variable theory, and random process theory to facilitate further development of random processes and random phenomena in the next chapters. Chapter 6 provides details the prototypical random processes that are fundamental to electrical, electronic, and communication engineering. Consequently, chapter 7-10 coverage includes details of the characterization of random phenomena from an engineering perspective: probability mass function/probability density function evolution, autocorrelation function, and power spectral density. Chapter 11 features an introduction to order statistics, which provides the background for a discussion of the Poisson point random process in Chapter 12. Chapter 13 introduces birth-death random processes and then Chapter 14 provides an introduction to first passage time theory.