	Although DSP has long been considered an EE topic, recent developments have also generated signifi cant interest from the computer science community. DSP applications in the consumer market, such as bioinformatics, the MP3 audio format, and MPEG-based cable/satellite television have fueled a desire to understand this technology outside of hardware circles. Designed for upper division engineering and computer science students as well as practicing engineers, Digital Signal Processing Using MATLAB and Wavelets emphasizes the practical applications of signal processing. Over 100 MATLAB examples and wavelet techniques provide the latest applications of DSP, including image processing, games, fi lters, transforms, networking, parallel processing, and sound. The book also provides the mathematical processes and techniques needed to ensure an understanding of DSP theory. Designed to be incremental in diffi culty, the book will benefi t readers who are unfamiliar with complex mathematical topics or those limited in programming experience. Beginning with an introduction to MATLAB programming, it moves through filters, sinusoids, sampling, the Fourier transform, the z-transform and other key topics. An entire chapter is dedicated to the discussion of wavelets and their applications. A CD-ROM (platform independent) accompanies the book and contains source code, projects for each chapter, and the fi gures contained in the book. FEATURES: Contains over 100 short examples in MATLAB used throughout the book Includes an entire chapter on the wavelet transform Designed for the reader who does not have extensive math and programming experience Accompanied by a CD-ROM containing MATLAB examples, source code, projects, and fi gures from the book Contains modern applications of DSP and MATLAB project ideas

Digital Signal Processing Using MATLAB and Wavelets

By Michael Weeks

Chapter 9.3.1 - How The Haar Transform Affects a Point's Radius

Consider these numbers to be vectors (i.e., in polar coordinates). The magnitude and angle can be found with a little geometry. In fact, the magnitude can be found using Pythagoras' theorem. The magnitude of the Cartesian coordinates

is $\sqrt{1^2 + 10^2} = \sqrt{101}$ . The magnitude of the Haar coordinates is $\left(\frac{11}{\sqrt{2}}, \frac{-9}{\sqrt{2}}\right)$

$\begin{displaymath}\sqrt{\left(\frac{11}{\sqrt{2}}\right)^2 + \left(\frac{-9}{\s... ... + \frac{81}{2}} = \sqrt{\frac{202}{2}} = \sqrt{101} \mathrm{.}\end{displaymath}$

This example shows that the magnitudes are the same. In other words, when we consider the point $(1, 10)$

as a distance from the origin and a rotation from the positive x-axis, the Haar transform does not affect the distance. We can show that this is the case for ANY arbitrary point by replacing $(1, 10)$

above with $(x, y)$

. In the Haar-domain, we would have points $\left(\frac{x + y}{\sqrt{2}}, \frac{x - y}{\sqrt{2}}\right)$ . Now, considering these points as polar coordinates, we see that the magnitude of the Cartesian coordinates is $\sqrt{x^2 + y^2}$ . The magnitude of the Haar coordinates is

$\begin{displaymath}\sqrt{\left(\frac{x+y}{\sqrt{2}}\right)^2 + \left(\frac{x-y}{... ...2xy + y^2}{2}\right) + \left(\frac{x^2 - 2xy + y^2}{2}\right)} \end{displaymath}$

$\begin{displaymath}= \sqrt{\frac{2x^2 + 2y^2}{2} } = \sqrt{x^2 + y^2} \mathrm{.} \end{displaymath}$

Therefore, the Haar transform does not affect the distance of a point from the origin. It does, however, affect the angle of the point in relation to the x-axis.

TABLE OF CONTENTS

Chapter 9.3.1 - How The Haar Transform Affects a Point's Radius

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