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Chapter 12
The Convolution Representation and the
Fourier Transfer Function
Chapter Outline
12.1 THE CONVOLUTION REPRESENTATION.
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406
12.1.1 Introduction.
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406
12.1.2 Examples .
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408
12.1.3 Forms of the Convolution Integral.
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410
12.2 GRAPHICAL CONVOLUTION.
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412
12.3 THE RELATIONSHIP BETWEEN THE CONVOLUTION INTEGRAL
AND OTHER SYSTEM REPRESENTATIONS.
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417
12.3.1 Introduction.
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12.3.2 Relationship Between the Transfer Function and Convolution Integral .
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12.3.3 Relationship with the State Space Representation.
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418
12.3.4 Nonzero Initial Conditions .
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421
12.4 THE FOURIER TRANSFER FUNCTION.
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423
12.4.1 Introduction.
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12.4.2 Definition.
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12.4.3 Relationship to the Convolution Representation.
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424
12.4.4 Examples .
425
12.4.5 Relationship to the Laplace Transfer Function.
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426
12.5 CHAPTER SUMMARY.
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427
12.6 HOMEWORK FOR CHAPTER 12.
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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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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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 Spring '09
 MEEHAN
 Laplace, ........., Impulse response

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