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Linear Algebra, Signal Processing, and Wavelets - A Unified Approach

Øyvind Ryan
Publication Date: 
Number of Pages: 
Springer Undergraduates Texts in Mathematics and Technology
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Jeff Ibbotson
, on
This book contains a nice approach to the kinds of Linear Algebra often used by engineers. It markets itself as useful for a second course in Linear Algebra and showcases some very nice material involving signal processing, sampling and filtering, scaling and multiresolution. It is available in two versions depending on which code is preferred –MATLAB or Python.
The materials and approach for this book come from a course created by the author at the University of Oslo entitled “Applications of Linear Algebra”. As such the book does indeed showcase some interesting material that involves orthonormal bases for vector spaces and projection into subspaces. To a mathematician, this is only a slim part of the vast subject matter of Linear Algebra and if one is looking for material on normal forms or factorization one should look elsewhere. On the other hand, if you are looking for a well-motivated approach to a major part of modern applied mathematics, then this book might be the right choice. The motivation is the analysis of sound and vision.
The beginning in Chapter 1 “ Sound and Fourier Series” is really quite a nice introduction and sets the stage for much of the rest of the book. We are introduced to periodic functions and the physical nature of sound with samples that range from a whisper to the sound made by jet engines to the enormous output of the volcanic eruption on Krakatoa. Right away the author introduces the reader to the idea of scale in looking at waveforms. This gives an intuitive basis to the more complicated mathematical background of harmonic analysis involving issues of convergence, continuity and differentiability of Fourier series representations. Nor does the author hide these concepts (although the typical convergence theorems are not hidden, they are not given formal proofs either). The Dirichlet and Fejer kernels make early appearances and the approach builds slowly to their place in the convolution representation of the Fourier series. The discrete Fourier representation appears early and the canonical matrix representation involving roots of unity is also given a central role. The first two chapters do a nice job of leading the student up to an understanding of the Fast Fourier Transform, windowing via filters, and the matrix computation of output for the processing of signals. Everything is presented very clearly and both computation and insight are cultivated through the series of exercises embedded in the text. The author assumes the reader will get her hands “dirty” with using MATLAB/Python to process audio samples utilizing a repository of sound files on github. Chapter 3 executes a nice “deep dive” into discrete-time filters via convolutions. Toeplitz and circulant matrices abound in these pages and low-pass and high-pass filters are introduced in a way that really explains the frequency response phenomenon for signal processing. Again, things are very computational – no real theorems about eigenvalues of various matrices are proven even though their significance is obvious on every page. The Z-transform makes a brief appearance towards the end of the chapter.
The remaining six chapters are devoted to an in-depth presentation of all the major ideas behind wavelets and filter banks. As before, the author does not follow the typical formal presentation of the mathematics behind wavelets but carefully motivates the underlying ideas first. Google Earth views of the Earth at various scales introduce the notion of frequency analysis at different scales and concomitant issues involving data storage. Resolution spaces based on piecewise constant functions are used to show the details of projection on various subspaces and kernel transformations (and their matrix representatives) appear right away. The author’s approach builds nicely on all the basic ideas presented for Fourier series in the first three chapters. The transition continues in the following chapters using piecewise linear bases before later chapters introduce the most general presentation of wavelets based on smoother mother functions. Throughout nice examples of standards (such as JPEG2000 and MP3 )are contrasted with each other with regard to lossy and lossless compression and the book includes some nice examples of visual reconstruction of data using DFT and DCT algorithms. Tensor products (really Kronecker products) of series and matrices are introduced in the last chapter alongside Haar wavelets. A beginning student who is unfamiliar with the mathematics behind signal processing will find much here that explains the techniques and the issues associated with their use.  Altogether the book presents a beautiful introduction to the uses of linear algebra in signal processing. 


Jeffrey Ibbotson, Ph.D., is the Smith Teaching Chair in Mathematics at Phillips Exeter Academy.