EE 350 Problem Set 6 Cover Sheet Fall 2014Last Name (Print):First Name (Print):ID number (Last 4 digits):Sectio n:Submission deadl ines:• Turn in the written solutions by 4:00 pm on Tuesday October 28 in the homework slot outside 121 EE East.Problem Weight Score27 2028 2029 2030 2031 20Total 100The solution submitted for grading represents my own analysis of the problem, and not that of another student.Signature:Neatly print the name(s) of the students you colla bo rated with on this assignment.Reading assignment:• Lathi Chapter 3, Sections 3.4, 3.5, 3.7, 3.8, and 3.9• Priemer Chapter 6, Sections 6.5 through 6.10Problem 27: (20 points)A signal f(t) is said to be periodic if for some positive constant Tof(t) = f(t + To), f or all t ≥ 0.The smallest nonzero value of Tosatisfying the l ast equality is called the fundamental period of f(t). If a sig nalf(t) is not periodic, it is called aperiod ic. Determine whether or not each of the f oll owing continuous-time signalsis periodic. If the signal is periodic, determine the fundamental period.1. (4 points) f(t) = cos2(10t + 45◦) + cos(5t + 60◦)2. (4 points) f(t) = cos(t) + sin(πt)3. (4 points) φn(t) = enωotwhere n is any integer, ωo= 2π/To, a nd Tois some positive real- valued constant.4. (4 points)f(t) = ao+∞Xn=1ancos(n ωot) +∞Xn=1bnsin(n ωot),where a0, a1, a2, . . . , b1, b2. . . are real-valued constant coefficients, ωo= 2π/To, and Tois some positive real-valued constant.5. (4 points)f(t) =∞Xn=−∞Dne n ωot,where the Dnare complex-valued constant coefficients, ωo= 2π/To, and Tois some positive real-valuedconstant.Problem 28: (20 points)1. (5 points) Consider real-valued periodic functions of the formxn(t) = cos(nωot)where ωo= 2 π/Toand Tois the period. D etermine the value of < xn, xm> for two cases, n 6= m and n = m 6= 0.For what condition, if any, are the functions xn(t) and xm(t) orthogonal over the interval [−To/2, To/2]?2. (5 points) Consider real-valued periodic functions of the formxn(t) = sin(nωot)where ωo= 2 π/Toand Tois the period. D etermine the value of < xn, xm> for two cases, n 6= m and n = m 6= 0.For what condition, if any, are the functions xn(t) and xm(t) orthogonal over the interval [−To/2, To/2]?3. (5 points) Over the interval [−To/2, To/2], show that< sin(nωot), cos(mωot) > = 0 for all n and m,where ωo= 2 π/To.4. (5 points) Consider a set of periodic complex-valued signalsxn(t) = enωotn = 0, ±1, ±2, . . .where ωo= 2 π/Toand Tois the period. Show that over the interval [−To/2, To/2]< xn(t), xm(t) >=0 m 6= nTom = n.Problem 29: (20 points)When determining the trigonometric Fourier series representation of a signal f(t), the analysis can be sim plified byexploiting the symmetry properties of f(t).1. (6 points) Show that if f(t) is an even, real-valued periodic function of time with period To, thena0=2ToZTo/20f(t)dtan=4ToZTo/20f(t) cos(nωot)dtbn= 0and anis an even function of n.2. (6 points) Show that if f(t) is an odd, real-valued periodic function of time with perio d To, thena0= 0an= 0bn=4ToZTo/20f(t) si n(nωot)dtand bnis an odd function of n.3. (8 points) If a periodic signal f(t) with fundam ental period Tosatisfiesft −To2= −f(t),the signal f(t) is said to have a half-wave symmetry. In signals with half-wave symmetry, the two-halves ofone period of the signal are of identical shape except that one half is the negative of the other. If a periodicsignal f(t) with fundamental period Tosatisfies the the half-wave symmetry condition, thenft −To2= −f(t).In this case, show that all the even-numbered harmonic coefficients (a0, a2, a4, . . . , b2, b4, . . .) vanish, and thatthe odd-numbered harmonic coefficients (a1, a3, . . . , b1, b3, . . .) are given byan=4ToZTo/20f(t) cos(nωot)dtbn=4ToZTo/20f(t) si n(nωot)dt,where anis an even function of n while bnis an odd function of n .Problem 30: (20 points)Determine the trigonometric Fourier series coefficients (a0, an, bn) for each of the periodic signals in Figure 1.Whenever possible, use the results from Problem 29 to simpli fy the calculations.Figure 1: Periodic signals.Problem 31: (20 points)1. (5 p oints) A half-wave rectifier produces the periodic waveform f(t) in Figure 2. Determine the compacttrigonometric Fourier series coefficients (C0, Cn, θn) of the signal f(t).Figure 2: Output waveform of a half-wave rectifier.2. (5 points) In order to generate a constant voltage from the half-wave rectifier output, it is necessary to attenuatethe sinusoi dal components of the waveform f(t) so that the DC component Codomi nates. To this end, thesignal f(t) is passed through the passive RC low-pass filter in Figure 3 that is represented by the frequencyresponse functionH(ω) =1ωτ + 1,where the time constant τ is RC. Show that the sinusoidal steady-state output y(t) can be expressed asy(t) = Co+∞Xn=1Cnp(ωon τ)2+ 1cosn ωot + θn− tan−1(ωon τ ) .Figure 3: Passive RC filter network.3. (10 points) Use MATLAB to generate and plot the signals f(t) and y(t) in a singl e figure using the followingparameters and guidelines:• Set A = 10 V, To= 1/60 s, and τ = 30 ms.• Use a tim e vector that spans the interval [ 0, 5To] with one thousand uniformly spaced points.• G enerate an exact plot of the input waveform using the MATLAB function sin.• Approximate the output o f the filter network asy(t) ≈ Co+100Xn=1Cnp(ωon τ)2+ 1cosn ωot + θn− tan−1(ωon τ ) .• Plo t f(t) and y(t) in a single figure as a function of time in ms. Display f(t) and y(t) usi ng dashed andsolid curves, respectively.• Use the MATLAB function legend to distinguish the plots.• Appropriately label the x and y axes; no credit is given for MATLAB plots whose axes are unlabeled!• Use the title command to appropriately label the figure, for example, Problem 31 Part 3.• Use the MATLAB command gtext to place your name and section name within the figure.To receive credit:• Turn in the figure along wi th a copy of your m-file.• Include your name and section number at the top of m-file using the comment symbol
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