7_Fourier - Chapter 7 Fourier Series and Fourier Transforms...

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133 Chapter 7 Fourier Series and Fourier Transforms Chapter Outline 7.1 INTRODUCTION TO FOURIER SERIES. ............................................. 135 7.1.1 Introduction . ................................................................................................................................. 135 7.1.2 Periodic Signals. ........................................................................................................................ 135 7.1.3 Fourier Series of a Pulse Train. .......................................................................................... 137 7.1.4 Interpretation of a Fourier Series. ...................................................................................... 140 7.2 THREE REPRESENTATIONS OF A FOURIER SERIES. ....................... 142 7.2.1 Introduction . 142 7.2.2 Cosine Representation. ........................................................................................................... 143 7.2.3 Trigonometric Representation. ............................................................................................ 143 7.2.4 Exponential Representation . ................................................................................................ 146 7.2.5 Existence Theorem. .................................................................................................................. 148 7.3 COMPUTATIONAL FORMULAS FOR THE FOURIER SERIES COEFFICIENTS . .................................................................................. 149 7.3.1 Introduction . 149 7.3.2 Trigonometric Representation. 150 7.3.3 Exponential Representation . 154 7.3.4 Use of Symmetry. ...................................................................................................................... 156 7.3.5 Summary. ...................................................................................................................................... 159 7.4 DEFINITION OF THE FOURIER TRANSFORM. .................................. 159 7.4.1 Introduction . 159 7.4.2 The Definition. ............................................................................................................................ 162 7.4.3 Existence of the Fourier Transform. ................................................................................. 163 7.5 PROPERTIES OF THE FOURIER TRANSFORM AND THE GENERALIZED FOURIER TRANSFORM. ............................................ 165 7.5.1 Introduction . 165 7.5.2 Properties of the Fourier Transform. 165 7.5.3 Generalized Fourier Transform. 171 7.6 CHAPTER SUMMARY. ........................................................................ 174 7.6.1 Fourier Series. ............................................................................................................................. 174 7.6.2 Fourier Transform. ..................................................................................................................... 175 7.7 HOMEWORK FOR CHAPTER 7. .......................................................... 178 In this chapter we introduce two very important mathematical tools for analyzing signals and systems: the Fourier series and the Fourier transform. Both of these concepts come from classical mathematics, and this chapter is devoted to developing the required mathematical background to use these concepts in this text.
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134 Chapter 7 Fourier Series and Fourier Transforms In keeping with the philosophy of this text, this chapter concentrates on the mathematics of these two concepts. Subsequent chapters apply these ideas to the analysis of signals and systems. In the first three sections of this chapter we discuss Fourier series. A Fourier series is the representation of a periodic signal by an infinite sum of sinusoids. In this chapter we define the Fourier series and discuss how to compute the Fourier series for a given periodic signal. We also give three alternative representations of the Fourier series, each of which is frequently used, and develop their interrelationship. The focus of this chapter is on the analytical interpretations of the Fourier series. For most applications of the Fourier series in signals and systems, the Fourier series of a signal is represented graphically. This graphical representation and its interpretation is developed in Chapter 8. The Fourier series is also very useful in explaining how a signal propagates through a system. This discussion is presented in Chapter 15. The Fourier transform can be viewed as an extension of the Fourier series to aperiodic signals. In the last two sections of this chapter we define the Fourier transform and develop several of its properties. The Fourier transform (and its extension to discrete-time) is one of the most useful tools for doing signal and system analysis. It will play a fundamental role throughout the rest of the text including the representation of signals in Chapter 8, the representation of systems in Chapter 12, and the analysis of a signal propagating through a system in Chapters 14 and 15.
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This note was uploaded on 09/10/2009 for the course ECE 60367 taught by Professor Meehan during the Spring '09 term at Virginia Tech.

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7_Fourier - Chapter 7 Fourier Series and Fourier Transforms...

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