The Foundations of Digital Signal Processing: Theory, Algorithms and Hardware Design

Chapter 8: Introduction to Laplace Space and the Laplace Transform

8.1 Introduction

Both the Laplace transform and the z-transform are closely related to, respectively, the continuous Fourier transform and the discrete time Fourier transform. However, because they employ a complex frequency variable ( s or z) rather than a purely imaginary one ( j ?), they are more general in scope. The Laplace transform is for example, ubiquitously employed for the analysis and design of electrical circuits such as filters and networks, and is ideally suited for the analysis of transient response phenomena (Hickmann, 1999). Similarly the z-transform is an indispensable tool for the design and analysis of digital filters, especially infinite impulse response (IIR) filters, of which we will have much to say in this and later chapters.

We will commence our treatment of this subject with an investigation into the definitions, properties and uses of the Laplace transform. However, a word of caution on this matter is appropriate before we proceed further. Some texts on DSP include very detailed descriptions of the Laplace transform, presumably because it is considered that discrete domain processing cannot be adequately understood without a thorough grounding in continuous analytical methods. In contrast, other books on DSP ignore it all together, possibly because it is believed that discrete tools alone need to be applied in the design of discrete processing algorithms. In this chapter, we will steer a middle course. Sufficient information into the background and uses of Laplace will be provided to enable, you, the reader, to understand...

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