"This is a major book about a serious subject — the combination of engineering and mathematics that goes into modern signal processing: discrete time, continuous time, sampling, filtering, and compression. The theory is beautiful and the applications are so important and widespread."
Gil Strang, Massachusetts Institute of Technology

"This book (FSP) and its companion (FWSP) bring a refreshing new, and comprehensive approach to teaching the fundamentals of signal processing, from analysis and decompositions, to multiscale representations, approximations, and many other aspects that have a tremendous impact in modern information technology. Whereas classical texts were usually written for students in electrical or communication engineering programs, FSP and FWSP start from basic concepts in algebra and geometry, with the benefit of being easily accessible to a much broader set of readers, and also help those readers develop strong abstract reasoning and intuition about signals and processing operators.  A must read!"
Rico Malvar, Microsoft Research

"This is a wonderful book that connects together all the elements of modern signal processing. From functional analysis and probability theory, to linear algebra and computational methods, it’s all here and seamlessly integrated, along with a summary of history and developments in the field. A real tour-de-force, and a must-have on every signal processor's shelf!"
Robert D. Nowak, University of Wisconsin, Madison

"Finally a wonderful and accessible book for teaching modern signal processing to undergraduate students."
Stéphane Mallat, École Normale Supérieure

"Most introductory signal processing textbooks focus on classical transforms, and study how these can be used. Instead, Foundations of Signal Processing encourages readers to think of signals first. It develops a ‘signal-centric’ view, one that focuses on signals, their representation and approximation, through the introduction of signal spaces. Unlike most entry-level signal processing texts, this general view, which can be applied to many different signal classes, is introduced right at the beginning. From this, starting from basic concepts, and placing an emphasis on intuition, this book develops mathematical tools that give the readers gets a fresh perspective on classical results, while providing them with the tools to understand many state of the art signal representation techniques."
Antonio Ortega, University of Southern California

"Foundations of Signal Processing by Vetterli, Kovacevic, and Goyal, is a pleasure to read. Drawing on the authors’ rich experience of research and teaching of signal processing and signal representations, it provides an intellectually cohesive and modern view of the subject from the geometric point of view of vector spaces. Emphasizing Hilbert spaces, where fine technicalities can be relegated to backstage, this textbook strikes an excellent balance between intuition and mathematical rigor, that will appeal to both undergraduate and graduate engineering students. The last two chapters, on sampling and interpolation, and on localization and uncertainty, take full advantage of the machinery developed in the previous chapters to present these two very important topics of modern signal processing, that previously were only found in specialized monographs. The explanations of advanced topics are exceptionally lucid, exposing the reader to the ideas and thought processes behind the results and their derivation. Students will learn … why things work, at a deep level, which will equip them for independent further reading and research. I look forward to using this text in my own teaching."
Yoram Bresler, University of Illinois, Urbana-Champaign

"Foundations of Signal Processing by Martin Vetterli, Jelena Kovacevic, and Vivek K Goyal lives up to its title by providing a thorough tour of the subject matter based on selected tools from real analysis which allow sufficient generality to develop the foundations of the classical Fourier methods along with modern wavelet approaches. A key distinction of the book is the use of Hilbert space ideas to provide a geometric interpretation and intuition that enhances both the classic and modern approaches by providing a unified view of their similarities and relative merits in the most important special cases. Many of the specific examples of signal processing considered can be viewed as examples of projections onto subspaces, which yield immediate properties and descriptions from the underlying fundamentals. The development is both pedagogically and theoretically sound, proceeding from the underlying mathematics through discrete-time systems to the more complicated continuous time systems into a wonderfully general and enlightening treatment of sampling and interpolation operations connecting discrete and continuous time. All of the important signal classes are considered and their basic properties and interrelations developed and summarized. The book then develops important topics not ordinarily found in signal processing texts – the accuracy of approximations involving the truncation of series expansions and the quantization of series coefficients, and the localization of signals in the time-frequency plane.

The completeness of the book results in a lengthy volume of roughly 800 pages, but it is easy to navigate to extract portions of interest while saving the many byways and special topics for future reference. The chapter introductions are particularly good at setting the stage in a simple but informative context and then sketching the details to come for the remainder of the chapter. Each chapter closes with a ‘Chapter at a glance’ section highlighting the primary ideas and results. The book will be a welcome addition to the library of students, practitioners, and researchers in signal processing for learning, reviewing, and referencing the broad array of tools and properties now available to analyze, synthesize, and understand signal processing systems."
Robert M. Gray, Stanford University and Boston University