	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 - How the Haar Transform Is a 45-Degree Rotation

A basis is a projection onto a coordinate axis. If we have a point on a graph, we can talk of its distance from the origin, another point used for reference. But this only narrows it down to a circle, so we need more information: the angle between the vector made by the origin to the point of interest, and another line for reference (i.e., the x-axis). For a point in 2D space, these polar coordinates are sufficient information to unambiguously locate the point with respect to the origin. But we often use the Cartesian basis {1, 0} and {0, 1} to represent this point, which tells about the point in terms of a unit vector along the x-axis and another unit vector along the y-axis (they are normalized). We know that the y-axis intersects the x-axis at the origin, and that there is a 90-degree angle between them (they are orthogonal). Though, generally speaking, we could use a different set of axes (i.e., a different basis).

The Haar basis is { $1/\sqrt{2}$ , $1/\sqrt{2}$ } and { $1/\sqrt{2}$ , $-1/\sqrt{2}$ }, instead of the normal Cartesian basis. This section answers the question of what the points look like in the Haar-domain instead of the Cartesian-domain. What we will see is that the points have the same relationship to one another, but the coordinate system moves by a rotation of 45 degrees.

First, consider the points $(x_1, y_1), (x_2, y_2), etc.$ as the array $\{ x_1, y_1, x_2, y_2, etc. \}$ . The Cartesian system simply isolates the $x$ 's and $y$ 's, as in:

$\begin{displaymath}\left[ \begin{array}{ccc} \left[ 1 \; 0 \right] & 0 \; 0 \\... ...gin{array}{c} x_1 \\ y_1 \\ x_2 \\ y_2 \end{array} \right] . \end{displaymath}$

Whereas the Haar system does the following:

$\begin{displaymath}\frac{1}{\sqrt{2}} \left[ \begin{array}{cccc} \left[1 \; 1\... ...gin{array}{c} x_1 \\ y_1 \\ x_2 \\ y_2 \end{array} \right] . \end{displaymath}$

For example, the Cartesian coordinates $(1, 10)$ would be mapped to $\left(\frac{1 + 10}{\sqrt{2}}, \frac{1 - 10}{\sqrt{2}}\right)$ , or $\left(\frac{11}{\sqrt{2}}, \frac{-9}{\sqrt{2}}\right)$ in the Haar-domain.

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Chapter 9.3 - How the Haar Transform Is a 45-Degree Rotation

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