9_Laplace_Transforms

9_Laplace_Transforms - Chapter 9 The Laplace Transform...

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255 Chapter 9 The Laplace Transform Chapter Outline 9.1 DEFINITION OF THE LAPLACE TRANSFORM. .................................. 256 9.1.1 Definitions. ................................................................................................................................... 256 9.1.2 Existence of the Laplace Transform . ............................................................................... 258 9.1.3 The Impulse Function. ............................................................................................................ 260 9.1.4 Region of Convergence. ......................................................................................................... 260 9.2 PROPERTIES OF THE LAPLACE TRANSFORM. ................................ 262 9.3 PARTIAL FRACTION EXPANSION. ..................................................... 268 9.3.1 Definition. ..................................................................................................................................... 268 9.3.2 Partial Fraction Inversion. ..................................................................................................... 271 9.3.3 MATLAB Experiments. .......................................................................................................... 274 9.4 LAPLACE TRANSFORM SOLUTION TO DIFFERENTIAL EQUATIONS . ....................................................................................... 274 9.4.1 Solving Differential Equations. ........................................................................................... 274 9.4.2 Implications of the Pole Locations. .................................................................................. 276 9.5 RELATIONSHIP TO FOURIER TRANSFORMS. ................................... 279 9.6 CHAPTER SUMMARY. ........................................................................ 282 9.7 HOMEWORK FOR CHAPTER 9. .......................................................... 286 A signal is a function that represents the time variation of a physical variable. To this point we have introduced two quite distinct representations of a signal. The first class of representations of a signal, developed in Chapter 5, is the class of functions that depend on time. This type of representation is quite obvious. The second class of signal representations, derived in Chapter 7, is based on the Fourier transform of the signal representations in the first class (when the Fourier transform exists). These signal representations depend on a real frequency variable ϖ . In this chapter we will define a third class of signal representations that depend on a complex variable s . This class of signal representations is the Laplace transform of the first class of signal representations (when the Laplace transform exists). This third class of signal representations, like the second class of signal representations, doesn’t represent the signals directly in the time domain. There are many advantages and insights to this type of signal representation, which we will develop in the coming chapters. We can make similar comments about system representations, which we will discuss in detail in Chapters 10 - 15. In this chapter we introduce the Laplace transform as a mathematical tool for analyzing signals and systems. After the definition of the Laplace transform the properties of the transform are developed. Finally, these properties are applied to the
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256 Chapter 9 The Laplace Transform solution of a differential equation. The last section of this chapter discusses the relationship between the definitions of the Laplace and Fourier transforms. In keeping with the philosophy of the text, the primary focus of this chapter is on the mathematical details of the Laplace transform. In the following chapters we will apply this transform to the analysis of signals and systems. Computer Usage : The Laplace transform has also appeared as a useful representation for signals and systems for computer packages. Hence, knowledge of the transform is required for effective use of many computer packages. Summary of Sections Section 9.1: We define the Laplace transform.
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9_Laplace_Transforms - Chapter 9 The Laplace Transform...

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