Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part33

Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part33

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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 37 Copyright © Orchard Publications Line Spectra Figure 7.35. Line spectrum for half wave rectifier, Page 7 17 The line spectra of other waveforms can be easily constructed from the Fourier series. Example 7.4 Compute the exponential Fourier series for the waveform of Figure 7.36, and plot its line spectra. Assume . Solution: This recurrent rectangular pulse is used extensively in digital communications systems. To deter- mine how faithfully such pulses will be transmitted, it is necessary to know the frequency compo- nents. Figure 7.36. Waveform for Example 7.4 As shown in Figure 7.36, the pulse duration is . Thus, the recurrence interval (period) , is times the pulse duration. In other words, is the ratio of the pulse repetition time to the dura- tion of each pulse. For this example, the components of the exponential Fourier series are found from (7.110) The value of the average ( component) is found by letting . Then, from (7.110) we obtain or n ω t 0 1 2 4 6 8 A/ π A/ 2 DC ω 1 = 0 −π/κ 2 π T ω t A π π/κ T/ κ −π 2 π Tk T kk C n 1 2 π ------ Ae jnt t d π π A 2 π e t d π k π k == DC n 0 = C 0 A 2 π ------t π k π k A 2 π π k -- π k +  
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Chapter 7 Fourier Series 7 38 Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition Copyright © Orchard Publications (7.111) For the values for , integration of (7.110) yields or (7.112) and thus, (7.113) The relation of (7.113) has the form, and the line spectra are shown in Figures 7.37 through 7.39, for , and respectively by using the MATLAB scripts below. fplot('sin(2.*x)./(2.*x)',[ 4 4 0.4 1.2]) fplot('sin(5.*x)./(5.*x)',[ 4 4 0.4 1.2]) fplot('sin(10.*x)./(10.*x)',[ 4 4 0.4 1.2]) Figure 7.37. Line spectrum of (7.113) for C 0 A k --- = n0 C n A jn2 π --------------e jnt π k π k A n π ------ e jn π k e π k j2 ------------------------------------- A n π n π k   sin A n π k () sin n π ------------------------- == = = C n A k n π k sin n π k = ft A k n π k sin n π k n = = x sin x k2 = k5 = k1 0 = -4 -3 -2 -1 0 1 2 3 4 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 K=2 =
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Signals and Systems with MATLAB Computing and Simulink Modeling, Fourth Edition 7 39 Copyright © Orchard Publications Line Spectra Figure 7.38. Line spectrum of (7.113) for Figure 7.39. Line spectrum of (7.113) for
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This note was uploaded on 11/20/2009 for the course EE EE 102 taught by Professor Bar during the Fall '09 term at UCLA.

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Signals.and.Systems.with.MATLAB.Computing.and.Simulink.Modeling.4th_Part33

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