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About this book

This book offers a user friendly, hands-on, and systematic introduction to applied and computational harmonic analysis: to Fourier analysis, signal processing and wavelets; and to their interplay and applications. The approach is novel, and the book can be used in undergraduate courses, for example, following a first course in linear algebra, but is also suitable for use in graduate level courses. The book will benefit anyone with a basic background in linear algebra. It defines fundamental concepts in signal processing and wavelet theory, assuming only a familiarity with elementary linear algebra. No background in signal processing is needed. Additionally, the book demonstrates in detail why linear algebra is often the best way to go. Those with only a signal processing background are also introduced to the world of linear algebra, although a full course is recommended.

The book comes in two versions: one based on MATLAB, and one on Python, demonstrating the feasibility and applications of both approaches. Most of the MATLAB code is available interactively. The applications mainly involve sound and images. The book also includes a rich set of exercises, many of which are of a computational nature.

Table of Contents


Chapter 1. Sound and Fourier Series

A major part of the information we receive and perceive every day is in the form of audio. Most sounds are transferred directly from the source to our ears, like when we have a face to face conversation with someone or listen to the sounds in a forest or a street. However, a considerable part of the sounds are generated by loudspeakers in various kinds of audio machines like cell phones, digital audio players, home cinemas, radios, television sets and so on. The sounds produced by these machines are either generated from information stored inside, or electromagnetic waves are picked up by an antenna, processed, and then converted to sound.
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Chapter 2. Digital Sound and Discrete Fourier Analysis

In Chap. 1 we saw how a periodic function can be decomposed into a linear combination of sines and cosines, or equivalently, a linear combination of complex exponential functions. This kind of decomposition is, however, not very convenient from a computational point of view. The coefficients are given by integrals that in most cases cannot be evaluated exactly, so some kind of numerical integration technique needs to be applied. Transformation to the frequency domain, where meaningful operations on sound easily can be constructed, amounts to a linear transformation called the Discrete Fourier transform. We will start by defining this, and see how it can be implemented efficiently.
Øyvind Ryan

Chapter 3. Discrete Time Filters

In Sect. 1.​5 we defined filters as operations in continuous time which preserved frequencies. Such operations are important since they can change the frequency content in many ways. They are difficult to use computationally, however, since they are defined for all instances in time. This will now be addressed by changing focus to discrete-time. Filters will now be required to operate on a possibly infinite vector \(\mathbf {x}=(x_n)_{n=-\infty }^{\infty }\) of values, corresponding to concrete instances in time. Such filters will be called discrete time filters, and they too will be required to be frequency preserving. We will see that discrete time filters make analog filters computable, similarly to how the DFT made Fourier series computable in Chap. 2. The DFT will be a central ingredient also now.
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Chapter 4. Motivation for Wavelets and Some Simple Examples

In the first part of the book the focus was on approximating functions or vectors with trigonometric counterparts. We saw that Fourier series and the Discrete Fourier transform could be used to obtain such approximations, and that the FFT provided an efficient algorithm. This approach was useful for analyzing and filtering data, but had some limitations. Firstly, the frequency content is fixed over time in a trigonometric representation. This is in contrast to most sound, where the characteristics change over time. Secondly, we have seen that even if a sound has a simple trigonometric representation on two different time intervals, the representation as a whole may not be simple. In particular this is the case if the function is nonzero only on a very small time interval.
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Chapter 5. The Filter Representation of Wavelets

We saw in the previous chapter that the wavelet kernels G and H had some repeating structure in the rows and columns, similar to the circular Toeplitz structure in filters. Clearly the matrices are not filters, however. Nevertheless, we will now prove that wavelet kernels can be implemented easily in terms of filters. but that several filters are needed in the computation. Each of these filters will have an interpretation in terms of how the wavelet transform treats different frequencies. Much has been done in establishing efficient implementations of filters, and by expressing a wavelet transform in terms of filters we can take advantage of this.
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Chapter 6. Constructing Interesting Wavelets

Previously we have associated several MRA’s with filter bank transforms. Since filter bank transforms also can be constructed outside the setting of wavelets, it would be interesting to see when we can make the association the opposite way: Which filter bank transforms appear in the context of a multiresolution analysis? An answer to this question certainly could transfer much theory between wavelets and filters. Also, it may be easier to construct good filter bank transforms than good wavelet bases.
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Chapter 7. The Polyphase Representation of Filter Bank Transforms

In Chap. 5 we expressed wavelet transforms, and more generally filter bank transforms, in terms of filters. Through this one obtains intuition on a how a wavelet transform splits the input using low-pass and high-pass filters, and how the filters in the MP3 standard split the input into frequency bands.
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Chapter 8. Digital Images

Up to now all signals have been one-dimensional. Images, however, are two-dimensional by nature. In this chapter we will extend our theory so that it also applies in higher dimensions. The key mathematical concept in this extension is called the tensor product, which also will help in obtaining efficient implementations of operations on images.
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Chapter 9. Using Tensor Products to Apply Wavelets to Images

Previously we have used wavelets to analyze sound. We would also like to use them in a similar way to analyze images. In Chap. 8 we used the tensor product to construct two dimensional objects (i.e. matrices) from one-dimensional objects (i.e. vectors). Since the spaces in wavelet contexts are function spaces, we need to extend the strategy from Chap. 8 to such spaces. In this chapter we will start with this extension, then specialize to the resolution spaces V m, and extend the DWT to images. Finally we will look at several examples.
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