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8_Signal_Analysis

8_Signal_Analysis - Chapter 8 Spectral Content of a Signal...

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187 Chapter 8 Spectral Content of a Signal Chapter Outline 8.1 AMPLITUDE AND PHASE SPECTRA ................................................... 189 8.1.1 Introduction .................................................................................................................................. 189 8.1.2 Periodic Signals ......................................................................................................................... 190 8.1.3 Aperiodic Signals ...................................................................................................................... 197 8.1.4 Fourier Series and the Generalized Fourier Transform ........................................... 199 8.2 ENERGY AND POWER SIGNALS ......................................................... 200 8.2.1 Introduction .................................................................................................................................. 200 8.2.2 Definitions .................................................................................................................................... 200 8.3 ENERGY SPECTRAL DENSITY ........................................................... 204 8.3.1 Parseval’s Theorem .................................................................................................................. 204 8.3.2 Signal Bandwidth ...................................................................................................................... 207 8.4 POWER SPECTRAL DENSITY ............................................................. 213 8.4.1 Definition ...................................................................................................................................... 213 8.4.2 Examples ....................................................................................................................................... 214 8.5 POWER CALCULATIONS FOR PERIODIC SIGNALS .......................... 215 8.5.1 Introduction .................................................................................................................................. 215 8.5.2 Power of Periodic Signals ..................................................................................................... 216 8.5.3 Parseval’s Theorem .................................................................................................................. 217 8.5.4 Power Spectral Density of a Fourier Series ................................................................. 220 8.6 SPECTRAL CONTENT OF A SIGNAL: AN EXAMPLE ........................ 223 8.6.1 Introduction .................................................................................................................................. 223 8.6.2 Approximation of Periodic Signals Using Fourier Series ...................................... 224 8.6.3 Examples ....................................................................................................................................... 226 8.6.4 Summary ....................................................................................................................................... 231 8.7 STATIC NONLINEARITIES ................................................................. 232 8.7.1 A Nonlinear System ................................................................................................................. 232 8.7.2 Harmonic Distortion ................................................................................................................. 233 8.7.3 Total Harmonic Distortion .................................................................................................... 237 8.8 MATLAB EXPERIMENTS .................................................................... 239 8.8.1 Introduction .................................................................................................................................. 239 8.8.2 Square Wave ............................................................................................................................... 239 8.8.3 Modulated Square Wave ....................................................................................................... 241 8.9 CHAPTER SUMMARY ......................................................................... 243 8.9.1 Spectral Content of a Signal ............................................................................................... 243 8.9.2 Classification of Signals ........................................................................................................ 244 8.10 HOMEWORK FOR CHAPTER 8 ......................................................... 245

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188 Chapter 8 Spectral Content of a Signal Signals, as defined in Chapter 5, are functions of time which represent a physical variable. To understand the physical variable, we analyze the function of time, that is, the signal. Signal analysis is the systematic investigation of the properties of a mathematical model of a physical variable and the underlying physical process. The purpose of this investigation is to uncover properties of the physical process from the signal that are not immediately apparent from the physical process itself. In this chapter we begin the investigation of the properties of these representations of signals. The tools we have to analyze a signal depend on how we represent the signal. For example, a signal could be represented by an oscilloscope trace, or an analytical expression. Taking the derivative of the analytical expression may be easy, but taking the derivative of the oscilloscope trace may be nearly impossible. The central idea in signal analysis is to introduce a signal representation that gives insight into the characteristics of the signal. Generally, that means the signal representation should lend itself to numerical computation and graphical representation in such a way that the essential characteristics of the signal are exposed. This is a rather tall order, but several very successful signal representations have been introduced in the last chapter. These two representations are the Fourier series and the Fourier transform. We will study these two representations in this chapter. The Fourier series represents the signal as a sum of cosines with given frequencies. The Fourier transform of a signal is a function of a frequency variable ϖ . It is the purpose of this chapter to develop the idea that these signal representations depend on frequency. This concept is known as the spectral content of a signal. This concept plays two important roles in signal analysis. First, the
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