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14 Freq Resp

# 14 Freq Resp - Chapter 14 The Frequency Response Theorem...

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485 Chapter 14 The Frequency Response Theorem Chapter Outline 14.1 THE FREQUENCY RESPONSE THEOREM USING LAPLACE TRANSFORMS. .................................................................................. 487 14.2 THE FREQUENCY RESPONSE THEOREM USING FOURIER TRANSFORMS. 492 14.3 THE FREQUENCY RESPONSE FUNCTION. ....................................... 496 14.3.1 Introduction. ............................................................................................................................... 496 14.3.2 System Representations and the Frequency Response Theorem. ................... 497 14.3.3 Definition of the Frequency Response Function . .................................................... 498 14.3.4 Power Considerations . .......................................................................................................... 500 14.4 GRAPHICAL INTERPRETATIONS OF THE FREQUENCY RESPONSE FUNCTION. ........................................................................................ 502 14.4.1 Introduction. 502 14.4.2 Magnitude and Phase Graphs . .......................................................................................... 502 14.4.3 Graphical Interpretations. .................................................................................................... 505 14.4.4 MATLAB Experiments. ....................................................................................................... 509 14.5 THE BANDWIDTH OF A SYSTEM. 511 14.5.1 Introduction. 511 14.5.2 Bandwidth of a System. 513 14.5.3 Half-Power Bandwidth . ........................................................................................................ 516 14.6 IDEAL FILTERS. ................................................................................ 519 14.6.1 Introduction. 519 14.6.2 Distortionless Filters . ............................................................................................................ 519 14.6.3 Ideal Filters . .............................................................................................................................. 520 14.7 INTRODUCTION TO FILTERING. ..................................................... 524 14.7.1 Introduction. 524 14.7.2 An Example. 525 14.7.3 Determination of the Frequency Response Function . ........................................... 528 14.7.4 Frequency Transformations. ............................................................................................... 532 14.7.5 MATLAB Experiments. 534 14.8 CHAPTER SUMMARY. ....................................................................... 535 14.9 HOMEWORK FOR CHAPTER 14. ...................................................... 537 This text is devoted to the study of signals and systems. By definition, an input signal to a system produces an output signal. In this chapter we begin to study how the characteristics of the input signal and the characteristics of the system interact to produce the output signal. To frame this discussion, we will adopt the following

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486 Chapter 14 The Frequency Response Theorem terminology. We say the input signal propagates through the system to generate the output signal. This terminology is consistent with the definition of a system, which defines the system in terms of the input and output signal. This terminology suggests that the relationship between the input and output signals and the system can be studied by comparing the input signal to the output signal. We can attribute change between these two signals to the system. We shall pursue this idea in the next two chapters. This analysis will require the tools of signal analysis and system analysis. The signal analysis will use the signal representations developed in Chapter 8. The system analysis will use the system representations and their properties developed in Chapters 10, 11, 12, and 13. Through an analysis that combines these powerful results, we will obtain deep insight into the propagation of a signal through a system. Our method of analysis will start with simple input signals, sinusoids, to establish the basic link between the input signal, the system, and the output signal. From these simple signals we will proceed to periodic signals, infinite sums of sinusoids, and then to aperiodic signals in Chapter 15. In this chapter we study the response of a system to a sinusoidal input signal. The output signal is characterized in terms of a complex quantity calculated from the transfer function of the system. So the output signal of a system is directly related to the system representation. This result is precisely stated in the frequency response theorem which is discussed in this chapter. The linkage between the input
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14 Freq Resp - Chapter 14 The Frequency Response Theorem...

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