# 20_FFTs - Chapter 20 Spectral Content of Discrete Signals...

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763 Chapter 20 Spectral Content of Discrete Signals Chapter Outline 20.1 DISCRETE-TIME ENERGY SIGNALS. ............................................... 765 20.1.1 Introduction. ............................................................................................................................... 765 20.1.2 Amplitude and Phase Spectra. ......................................................................................... 765 20.1.3 Signal Energy . .......................................................................................................................... 768 20.1.4 Classification of Signals. .................................................................................................... 770 20.1.5 MATLAB Experiments. ....................................................................................................... 772 20.2 DISCRETE-TIME POWER SIGNALS. ................................................. 773 20.2.1 Introduction. 773 20.2.2 Discrete-Time Periodic Signals. ...................................................................................... 773 20.2.3 Power Spectral Density . ...................................................................................................... 776 20.2.4 Correlation Functions . .......................................................................................................... 779 20.2.5 MATLAB Experiments. 781 20.3 COMPUTING THE FOURIER TRANSFORM: THE DFT. .................... 781 20.3.1 Summary of the DFT Development. .............................................................................. 787 20.3.2 Properties of the DFT . 787 20.4 EXAMPLES OF THE DFT. ................................................................. 788 20.4.1 Introduction. 788 20.4.2 Numerical Example. .............................................................................................................. 789 20.4.3 Frequency Axis Scaling of the DFT. ............................................................................. 791 20.4.4 An Analytical Example . 793 20.4.5 Zero Padding. ............................................................................................................................ 795 20.4.6 The FFT. ..................................................................................................................................... 795 20.4.7 MATLAB Experiments. 796 20.5 CHAPTER SUMMARY. ....................................................................... 797 20.6 HOMEWORK FOR CHAPTER 20. ...................................................... 800 One of the central themes of this text is the representation of signals and the characterization of the properties of the signal based on the representation. In Chapter 8 we showed that one of the most important representations of a continuous- time signal is the Fourier transform of that signal. The Fourier transform provides unique insight into the characteristics of the signal through the amplitude and phase spectra and the energy spectral density. These characterizations provided by the Fourier transform are extended to periodic signals through the Fourier series representation. Similar concepts also exist for aperiodic power signals. The essential role of these frequency domain signal concepts in the analysis of signals propagating through systems is explained in Chapter 15.

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764 Chapter 20 Spectral Content of Discrete Signals One purpose of this chapter is to extend these concepts to discrete-time signals. The discrete-time Fourier transform (DTFT) provides a frequency domain representation of a discrete-time signal that is analogous to the Fourier transform of a continuous-time signal. Similarly, the exponential Fourier series is defined for periodic discrete-time signals. Using these two representations we can easily extend all of the frequency domain characterizations from continuous-time signals to discrete-time signals. These extensions are accomplished in the first two sections of this chapter. The similarities and differences between the continuous-time and discrete-time concepts are highlighted. We have seen that the Fourier transform (or the discrete-time Fourier transform) of a signal lends great insight into the properties of the signal and how that signal propagates through a system. The amplitude and phase spectra along with the energy spectral density provide information about the signal that isn’t obvious from the time history. So it is obvious that we would like to determine the Fourier transform of signals that we observe in practice. In the examples presented thus far, we have based our Fourier analysis on the analytical representation of the signal.
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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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20_FFTs - Chapter 20 Spectral Content of Discrete Signals...

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