Wave Theory of Information [Book Review

Abstract

Information theory textbooks and courses largely limit themselves to a purely mathematical treatment of the subject. However, the real world is not just limited by laws of mathematics, but also by laws of physics. This book attempts to fill this gap at least in the context of wave theory. The book weaves beautiful concepts in physics and information theory in a manner that is equally engaging to a student and an expert in information theory. The author is a renowned researcher from UC San Diego, who is responsible for one of the central results characterizing the degrees of freedom of electromagnetic (EM) waves in a scattering environment. Indeed, central to the book is the concept — and the quantification — of the number of degrees of freedom of spatio-temporal signals. A rigorous treatment of degrees of freedom of bandlimited signals is provided in Chapters 2 and 3. Chapters 4 and 5 provide the background on EM waves and their propagation. Fundamental results from information theory are discussed on the way, including those on representation, communication and sensing of signals in space and time. Chapter 7 discusses fundamentals of wireless communication, including an overview of current technologies and strategies for point-to-point and network communication, with an emphasis on attainable degrees of freedom. This builds up to Chapters 8 and 9, where it is shown how, in an arbitrary scattering environment, the radiated field filtered by the Green’s operator becomes spatially bandlimited. Finally, the number of degrees of freedom of bandlimited waves propagating in physical media is characterized through the concept of surface cut-set integral. This establishes limits on amount of information that can radiate from a scattering system, an aspect of communication theory that has been of interest for several decades before significant advances in the 2000s. Ensuing chapters discuss additional in-depth results. At the end of every chapter there is a brief history of how the concepts presented in that chapter developed. I believe that this book is a great success for the following reason: it enables information theorists with interests in physics to understand and contribute to the solution of informational puzzles and paradoxes in physics, which can (and will) enrich both areas. The book is timely: information theory is expanding its horizons and now, more than ever, there is a need to closely tie the information- theoretic understanding with fundamental physics. The book is also timeless: it serves as a great illustration of how, in our attempt to expand these horizons, we still need to embed our understanding in intellectual depth and mathematical rigor and avoid the “scientific bandwagon” that Shannon warned us of in his 1956 article.