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Question: the answers for the questions are given below please explain...

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The answers for the questions are given below. Please explain how the answers to questions 4.1,4.2,4.5,and 4.6 were obtained. The correct solutions are given below.Thank you.

PROBLEMS 4.1 What is the fundamental frequency and fundamental period of the signal x(r) 3+sin 61-2 cos 6t + sin 9r- cos 12 Express it in complex Fourier series and plot the magnitudes and phases of its frequency com ponents 4.2 Consider the full-wave rectifier in Figure 2.21(a). What is its output y) if the input is u) sin 2/? What are the fundamental period and fundamental frequency of y()? Express the output in Fourier series. Does the output contains frequency other than 2 rad/s? 4.3 Use (1.7) and (1.8) to show that the periodic signal in Figure 4.2 has infinite average power 4.4 Consider a periodic signal with period 1 and x) for O S1. Show that the signal 4.5 What is the total energy of the signal in Problem 4.1? What is its average powers in one period? 4.6 What are the average powers of the input and output in Example 4.2.5? How many percentage is absolutely integrable in one period but has infinite average power How many percentage of the average power lies inside the frequency range [-7,71 of the input power is transmitted to the output?
figures for problems:

ui 0 Figure 2.21 (a) Full-wave rectifier. (b) Its characteristic. (c) Inputui (t)=sinont (dot vo(t)sin wot (solid line)
CHAPTER 4 FREQUENCY SPECTRA OF CT SIGNALS EXAMPLE 4.2.5 Conider the half-wave rectifier shown in Figure 2.20. The diode can be considesed a a switch. If the applied voltage v, is positive, the diode acts as a short circuit or the switch is closed. If w< 0, the diode acts as an open circuit or the switch is open. If )cosothen the output e,) is as shown in Figure 4.3 and can be expeessed as where pi is given in (4.10) and plotted in Figure 4.1(a. The switch is closed whem pu)-I and open when pr)-0. Using (4.10%, we have which, using cos e co‘φ-lose, + φ1+cos迴ㄧ·)l/2.can be writen as 3m 5 20%3 This output contains frequencies O. os, 20.4% … een hugh the ing contain only froquency . Thas the switching, a noalincar operation, can pencratie new fre- quencies. This fact will be used in modulation and demodulation in Chapter& Figure43 Oupa of i-ve i Not every CT periodic signal can be expeessed as a Fourier series. The conditioes for the exisience of Fourier series and the questions of comergence are complicated. Soe, for example Reference 13. We mention only the conditions oftem cited in engineering tests. A signal is said to be of bonded varianion if it has a finite number of finite discontinuities and a finite numberof maxima and minima in any finite time interval Figure 4.4a) shows the function xir)sin / for rin 10, 11 It has innitely many minima and maxima Fipue 4.3b) shows a function in O.11 it equals1in 10,0.5).05 in 10.5,0.751.025 in, and so forth It has infinitely many discontinuities These functions are mathematically contrived and cannot arise in practice. Thus we assume that all signals in ahis next are of bounded variation 2 FOURIER SERIES OF PERIODIC SIGNALS-FREQUENCY COMPONENTS 123 Figure 4.4 (ai Signal that has ininiely many maxima and minima, (b) Sipmal that has infinitely many Next we discuss a sufficiet condition If xir) is absolutely inteprable over one period, that is, then its Fourier series exists. Furthermore, all its cocfficients are finite, Indeed, because e1.(4.7) imp
CHAPTER 4 4.1 Fundamental frequency 3 rad/s. Fundamental period 2T/3. x(t) = 3e® + 1.12e-j268e® + 1.12e, 2.68e-j6e ANSWERS TO SELECTED PROBLEMS 4.2 Fundamental period π/2. Fundamental frequency 4 rad/s. 2 ej4mt 4.3 4.5 4.6 Inna-0(1/Ta)=00 Its total energy is infinity. Its average power is 16.92, 68%. 0.5, 0.25, 50%
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