Mathematics

The Mathematics of Signal Processing

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Basically, this is a book about mathematics, pitched at the advanced
undergraduate/beginning graduate level, where ideas from signal processing
are used to motivate much of the material, and applications
of the theory to signal processing are featured. It is meant for math
students who are interested in potential applications of mathematical
structures and for students from the fields of application who want to
understand the mathematical foundations of their subject. The first few
chapters cover rather standard material in Fourier analysis, functional
analysis, probability theory and linear algebra, but the topics are carefully
chosen to prepare the student for the more technical applications
to come. The mathematical core is the treatment of the linear system
y =fx in both definite-dimensional and indefinite-dimensional cases. This
breaks up naturally into three categories in which the system is determined,
over-determined or under-determined. Each has different mathematical
aspects and leads to different types of application. There are a
number of books with some overlap in coverage with this volume, e.g.,
[11, 15, 17, 19, 53, 69, 71, 72, 73, 82, 84, 95, 99, 101], and we have
proted from them. However, our text has a number of features, including
its coverage of subject matter, that together make it unique. An
important aspect of this book on the interface between fields is that it
is largely self-contained. Many such books continually refer the reader
elsewhere for essential background material. We have tried to avoid this.
We assume the reader has a basic familiarity with advanced calculus and
with linear algebra and matrix theory up through the diagonalization
of symmetric or self-adjoint matrices. Most of the remaining development
of topics is self-contained. When we do need to call on technical
results not proved in the text, we try to be specific. Little in the way
of formal knowledge about signal processing is assumed. Thus while this means that many interesting topics cannot be covered in a text of
modest size, the topics that are treated have a logical coherence, and
the reader is not continually distracted by appeals to other books and
papers. There are many exercises. In most of the volume the logic of the
mathematical topics predominates, but in a few chapters, particularly
for compressive sensing and for parsimonious representation of data, the
issues in the area of application predominate and mathematical topics
are introduced as appropriate to tackle the applied problems. Some of
the sections, designated by \Digging deeper" are more technical and
can be mostly skipped on a rst reading. We usually give a nontechnical
description of the principal results of these sections. The book is su-
ciently
exible to provide relatively easy access to new ideas for students
or instructors who wish to skip around, while lling in the background
details for others. We include a large list of references for the reader who
wants to \dig deeper." In particular, this is the case in the chapter on
the parsimonious representation of data.
This book arose from courses we have both taught and from ongoing
research. The idea of writing the book originated while the rst author
was a New Directions Professor of Imaging at the Institute for Mathematics
and its Applications, The University of Minnesota during the 05{
06 academic year. The authors acknowledge support from the National
Science Foundation; the Centre for High Performance Computing, Cape
Town; the Institute for Mathematics and its Applications, University
of Minnesota; the School of Computational and Applied Mathematics,
the University of the Witwatersrand, Johannesburg; Georgia Southern
University; and the United States Oce of Airforce Research. We are indebted
to a large number of colleagues and students who have provided
valuable feedback on this project, particularly Li Lin and Peter Mueller
who tested the compressive sensing algorithms. All gures in this book
were generated by us from open source programs such as CVX, Maple
or MATLAB, or from licensed MATLAB wavelet and signal processing
toolboxes.
In closing, we thank the sta at Cambridge University Press, especially
David Tranah and Jon Billam, for their support and cooperation during
the preparation of this volume and we look forward to working with
them on future projects.

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English

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1-107-60104-8

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