Woyczyński W.A. A First Course in Statistics for Signal Analysis

Basel: Birkhӓuser, 2011. — 279 p.This book was designed as a text for a first, one-semester course in statistical signal analysis for students in engineering and physical sciences. It had been developed over the last few years as lecture notes used by the author in classes mainly populated by electrical, systems, computer, and biomedical engineering juniors/seniors, and graduate students in sciences and engineering who have not been previously exposed to this material. It was also used for industrial audiences as educational and training materials, and for an introductory time-series analysis class.The only prerequisite for this course is a basic two- to three-semester calculus sequence; no probability or statistics background is assumed except the usual high school elementary introduction. The emphasis is on a crisp and concise, but fairly rigorous, presentation of fundamental concepts in the statistical theory of stationary random signals and the relationships between them. The author’s goal was to write a compact but readable book of fewer than 200 p., countering the recent trend toward fatter and fatter textbooks.Since Fourier series and transforms are of fundamental importance in random signal analysis and processing, this material is developed from scratch in Chapter 2, emphasizing the time-domain vs. frequency-domain duality. Our experience showed that although harmonic analysis is normally included in the calculus syllabi, students’ practical understanding of its concepts is often hazy.Chapter 3 introduces basic concepts of probability theory, the law of large numbers and the stability of fluctuations law, and statistical parametric inference procedures based on the latter.In Chapter 4 the fundamental concept of a stationary random signal and its autocorrelation structure is introduced.This time-domain analysis is then expanded to the frequency domain by a discussion in Chapter 5 of power spectra of stationary signals.How stationary signals are affected by their transmission through linear systems is the subject of Chapter 6.This transmission analysis permits a preliminary study of the issues of designing filters with the optimal signal-to-noise ratio; this is done in Chapter 7.Chapter 8 concentrates on Gaussian signals where the autocorrelation structure completely determines all the statistical properties of the signal.The text concludes, in Chapter 9, with the description of algorithms for computer simulations of stationary random signals with a given power spectrum density. The routines are based on the general spectral representation theorem for such signals, which is also derived in this chapter.The book is essentially self-contained, assuming the necessary calculus background mentioned above. A complimentary bibliography, for readers who would like to pursue the study of random signals in greater depth, is described at the end of this volume.Some general advice to students using this book: The material is deliberately written in a compact, economical style. To achieve the understanding needed for independent solving of the problems listed at the end of each chapter in the Problems and Exercises sections, it is not sufficient to read through the text in the manner you would read through a newspaper or a novel. It is necessary to look at every single statement with a magnifying glass and decode it in your technical language so that you can use it operationally and not just be able to talk about it. The only practical way to accomplish this goal is to go through each section with pencil and paper, explicitly completing, if necessary, routine analytic intermediate steps that were omitted in the exposition for the sake of the clarity of the presentation of the bigger picture. It is the latter that the author wants you to keep at the end of the day; there is no danger in forgetting all the little details if you know that you can recover them by yourself when you need them.Description of Signals.
Spectral Representation of Deterministic Signals: Fourier Series and Transforms.
Random Quantities and Random Vectors.
Stationary Signals.
Power Spectra of Stationary Signals.
Transmission of Stationary Signals Through Linear Systems.
Optimization of Signal-to-Noise Ratio in Linear Systems.
Gaussian Signals, Covariance Matrices, and Sample Path Properties.
Spectral Representation of Discrete-Time Stationary Signals and Their Computer Simulations.
Solutions to Selected Problems and Exercises.