Signal processing is a fundamental discipline and plays a key role in many fields including astronomy, biology, chemistry, economics, engineering, and physics, etc. Signal processing is based on fundamental mathematical tools and methods. As a result, a deeper understanding of these mathematical aspects is crucial towards successful application of signal processing in practice and to carry out advanced research. The aim of the book Mathematical Aspects of Signal Processing by Professor Pradip Sircar is to provide the reader with a comprehensive review, along with typical applications, of four specific mathematical topics: Function representation, generalized inverse, modal decomposition, and optimization in signal processing. This book is based on the extensive experience of the author as a researcher and educator in signal processing, and the presented materials have been tested with many students over a number of years at the Indian Institute of Technology, Kanpur, India. In my opinion, this book will be a very good text for an upper division and/or graduate level course on the subject. Researchers and professional engineers will also find this book very useful as a reference.

Introduction

Signal processing is a branch of applied mathematics. The primary aim of signal processing is to extract information from a given-set of data, a procedure known as signal analysis, or conversely, to use information in generating data with some desirable property, a procedure known as signal synthesis. Mathematical methods are developed based on concepts from mathematics with the purpose of signal processing as an area of application. Various mathematical concepts find use in signal processing, and a systematic introduction of those mathematical concepts to the students who will become researchers and professionals in this area, has become an essential part of the curricula worldwide.

When the mathematical ideas are introduced in a course of mathematics, they are presented as abstractions. Theories are built in structured manner, based on the present state of knowledge and the aspiration of the mathematician to extend knowledge in some desired direction. A mathematician is often not concerned of the areas of application of the developed theories. In contrast, a researcher engaged in signal processing often considers an established theory and identifies ways how it can be adopted and applied in extracting information inherent in data or for generating data with some desirable property.

It is the purpose of the present book to introduce the mathematical concepts and their interpretations related to signal processing. It is illustrated how basic concepts mould into mathematical methods, and mathematical methods perform signal processing tasks. We take up the case study of spectral estimation of a signal in the next section. The spectral estimation, by definition, refers to computing the energy spectral density of a signal with finite energy or the power spectral density of a signal with finite power. In the case study, however, we generalize our tasks by stating that primarily we are interested in finding the frequency-content of a signal, and computation of the energy-level or power-level at each frequency is our secondary purpose. This case study will illustrate various connections of signal processing with mathematics.

I have been using the material of this book for more than two decades while teaching a foundation level course on mathematical methods in signal processing being offered to the final-year undergraduate and first-year postgraduate students of Electrical Engineering at Indian Institute of Technology (IIT) Kanpur. Signal processing being an application area of mathematics, in order to motivate students to get involved in research of signal processing, it was felt that mathematics has to be presented in an informal and thought-provoking manner. In fact, the application area of signal processing is primarily concerned with the interpretation, interrelation, and applicability of mathematical concepts, whereas a formal course of mathematics is structured on treatise of existence, uniqueness, and convergence of mathematical results. Therefore, one has to relook at mathematics and find ways to make mathematical concepts go beyond abstractions and formal proofs.

My experimentation in teaching the above mentioned course has been quite rewarding. I have identified four areas of mathematics, namely, function representation, generalized inverse, modal decomposition, and optimization for our discussions. Mathematical concepts are frequently borrowed from these four areas for signal processing applications. However, the list of areas is not complete. Fundamental results of each area are presented in the class, followed by an interactive discussion of what can be their plausible implications. Finally, signal processing applications are showcased as illustrations of basic principles. Thus, the course serves the purpose of demonstrating how advanced signal processing applications are structured on basic mathematical concepts. I know a good number of students who credited the course, and subsequently decided to work on a research problem related to signal processing applications. The book will motivate students to develop a career in the broad area of signal processing as teachers, researchers or professionals.

During the year 1998–99, while I was visiting École Nationale Supérieure des Télécommunications (ENST) Paris, I started framing the idea of writing a book on the course material. I realized soon that it will not be easy to write a book on this subject. My lectures are more like discussion sessions, and the comprehensions are informal. In a few months time, I jotted down an outline of the proposed course, and drafted some sections of a chapter. However, I could not continue my endeavour after coming back to Kanpur because of other priorities. Nevertheless, the course material was constantly updated over the years.