12_Convolution

12_Convolution - Chapter 12 The Convolution Representation...

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405 Chapter 12 The Convolution Representation and the Fourier Transfer Function Chapter Outline 12.1 THE CONVOLUTION REPRESENTATION. ........................................ 406 12.1.1 Introduction. ............................................................................................................................... 406 12.1.2 Examples . ................................................................................................................................... 408 12.1.3 Forms of the Convolution Integral. ................................................................................. 410 12.2 GRAPHICAL CONVOLUTION. ........................................................... 412 12.3 THE RELATIONSHIP BETWEEN THE CONVOLUTION INTEGRAL AND OTHER SYSTEM REPRESENTATIONS. .................................... 417 12.3.1 Introduction. 417 12.3.2 Relationship Between the Transfer Function and Convolution Integral . ..... 417 12.3.3 Relationship with the State Space Representation. ............................................... 418 12.3.4 Nonzero Initial Conditions . ................................................................................................ 421 12.4 THE FOURIER TRANSFER FUNCTION. ............................................. 423 12.4.1 Introduction. 423 12.4.2 Definition. 423 12.4.3 Relationship to the Convolution Representation. .................................................... 424 12.4.4 Examples . 425 12.4.5 Relationship to the Laplace Transfer Function. ....................................................... 426 12.5 CHAPTER SUMMARY. ....................................................................... 427 12.6 HOMEWORK FOR CHAPTER 12. ...................................................... 429 In Chapters 10 and 11 we have studied the Laplace transfer function and state space equations as system representations. In this chapter we introduce the third and fourth system representations: the convolution integral and the Fourier transfer function. These two system representations complete the set of major system representations we will study in this text. The convolution representation is somewhat different from the previous system representations in that it expresses the relationship between the input and output signal in terms of an integral, not a differential equation or a transform of a differential equation. This representation has certain advantages in the representation of systems which are not conveniently expressed as one of the two previous representations. Hence, this representation is very useful as an analysis tool in advanced system analysis. We also present a graphical method for evaluating the convolution integral.
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406 Chapter 12 The Convolution Representation The second system representation we introduce in this chapter is the Fourier transfer function. The Fourier transfer function is very similar in concept to the Laplace transfer function except that the Fourier transform is used instead of the Laplace transform. The differences between the transforms leads to differences in the types of problems these two transfer functions are used to solve. The Fourier transfer function plays a central role in the analysis of the frequency response of systems studied in Chapters 14 and 15. The convolution integral is closely related to the Laplace transfer function and (not so obviously) to the state space representations. These relationships are developed in this chapter. In the course of this development we introduce a general formula for the solution of the state equations which is of interest in its own right. The relationship between the Fourier transfer function and the other three system representations is also developed. Summary of Sections Section 12.1: We introduce the convolution integral and develop its basic properties.
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12_Convolution - Chapter 12 The Convolution Representation...

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