	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 8 - The z-Transform

This chapter will address questions about the z-transform, an analytical tool for systems. What is ? How does this transform work? How can it be used to combine filters? Why do delay units sometimes have a $z^{-1}$ symbol? How does this transform relate to other transforms we have seen? We will answer these questions and more in the following sections.

The z-transform is a generalized version of the Fourier transform. Like the Fourier transform, it allows us to represent a time-domain signal in terms of its frequency components. Instead of accessing signal values in term of $n$ , a discrete index related to time, we can know the response of the signal for a given frequency (as we did with the Fourier transform). The difference is that we can also specify a magnitude with the z-transform.

The z-transform serves two purposes. First, it provides a convenient way to notate the effects of filters. So far, we have used the coefficients, $h[n] = \{a, b, c, d\}$ , to describe how the output $y[n]$ relates to the input $x[n]$ . In z-transform notation, we say $ Y(z) = H(z) X(z)$ , where $H(z)$ is the z-transform of $h[n]$ . We can think of $H(z)$ as something that operates on $X(z)$ to produce the output $Y(z)$ . For this reason, $H(z)$ is also called the transfer function. Rather than use $a x[n-k]$ in the equation, we can put the coefficient and the delay ( $k$ ) together, and remove $x$ . So the filter with coefficients $h[n] = \{a, b, c, d\}$ can be described instead by the z-transform of $h[n]$ , $H(z) = a z^0 + b z^{-1} + c z^{-2} + d z^{-3}$ A second purpose of the z-transform is to tell us about the stability of the filter, but this will be explained a bit later.