EE 350 Problem Set 7 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 November 4 in the homework slot outside 121 EE East.Problem Weight Score32 2033 2034 2035 2036 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 4, sections 4.1 through 4.4• Priemer Chapter 11Problem 32: (20 points)Consider a periodic signal f(t), with fundamental perio d To, that has the exponential Fourier series representationf(t) =∞Xn=−∞Dneωont,where ωo= 2π/ToandDn=1ToZTof(t)e−ωontdt.1. (2 points) When f(t) is a real-valued, show that D−n= D∗n. This is known as the complex conjugatesymmetry property or the Herm itian property of real signals.2. (2 points) Show that when f(t) is an even function o f time tha t Dnis an even function of n.3. (2 points) Show that when f(t) is an odd function of time that Dnis an odd function of n.4. (2 points) Using the results from parts 1 and 2, show that if f(t) is real-valued and an even function of time,then the coefficients Dnare real- valued and an even function of n.5. (2 points) Using the results from parts 1 and 3, show that if f(t) is real-valued and a n odd function of time,then the coefficients Dnare imag inary and an odd f unction of n.6. (4 points) Show that the signal f(−t) has the Fourier series co efficients D−n. This is known as the timereversal property.7. (6 points) Consider three periodic signals whose Fourier series representations aref1(t) =2π∞Xn=−∞11 + πn2e4ntf2(t) =∞Xn=−∞cos(3n)e3nt/2f3(t) =5Xn=1 sin(6n)e2πntUsing the properties of the exponential Fourier series, determine if the signals are(a) (3 points) real or complex valued,(b) (3 points) an even or odd function of time, o r neither.Problem 33: (20 points)Suppose that f(t) is a periodic sig nal with exponential Fourier series coefficients Dn. Show that the power Pfoff(t) isPf=∞Xn=−∞|Dn|2,and, if f(t) is real-valued,Pf= D2o+ 2∞Xn=1|Dn|2.This is Parseval’s theorem for the exponential Fourier series.Problem 34: (20 points)1. (10 points) Consider the periodic signalf(t) = 7 + 3 cos(2t) + 4 sin(2t) + 3 sin(3t) −5√2 cos5t +π4.(a) (2 points) Determine the fundamental frequency ωo.(b) (5 points) Determine the complex exponential Fourier series coefficients.(c) (3 poi nts) Sketch the Fourier magnitude spectrum |Dn| versus ω and the Fourier phase spectrum6Dnversus ω.2. (10 poi nts) Determine the complex exponential Fourier series coefficients and the compact trigonometric Fourierseries coefficients of the full-wave rectified waveformf(t) = A|sin(ωot)|.Problem 35: (20 points)This problem considers the response of a LTI system to a periodic input represented by a complex exponential Fourierseries.1. (10 points) A periodic signal f(t) with period Toand complex exponential Fourier series coefficients Dfnispassed through a LTI system with frequency response function H(ω). Show that the sinusoidal steady-stateresponse of the system is given byy(t) =∞Xn=−∞DfnH(nωo)enωot.2. (10 points) Apply the result f rom part 1 to determine the response of a lowpass filter.(a) (4 po ints) Determine the fundamental frequency and non-zero complex exponential Fourier series coeffi-cients of the periodic signalf(t) = −2 + 5 sin(πt) + 10 cos(3πt)and sketch the Fourier magnitude spectrum |Dfn| versus ω and the Fourier phase spectrum6Dfnversus ω.(b) (4 points) The signal f(t) from part (a) is passed through an ideal low-pass filter whose frequency responsefunction isH(ω) =2e−ω/4|ω| ≤ 2π0 |ω| > 2πto produce an output signal y(t). Find an expression for the resulting sinusoidal steady-state responsey(t) of the system and express your result as the sum of real-valued sinusoidal term(s) and if necessary, aDC offset.(c) (2 points) Find the power of the signal y(t) in part (b) using Parseval’s theorem derived in Problem 33.Problem 36: (20 points)Consider a linear time-invariant system represented by the ODE¨y + 4 ˙y + 4y = 8f.1. (4 points) Determine the zero-state unit-step response of the system.2. (2 points) Using your result in part 1, determi ne the impulse response function representation, h (t), of thesystem.3. (2 points) In Problem Set5 Problem 24 part 2, you showed that the frequency response function of a system isdetermined from its ODE representation asH(ω) =˜Y˜F=P (ω)Q(ω).Determine H(ω) for the system considered in this problem.4. (6 points) Determine the Fourier transform of the impulse response function determined in part 2, a nd expressyour answer in the standard formH(ω) =bm(ω)m+ bm−1(ω)m−1+ ···+ b1(ω) + b0(ω)n+ an−1(ω)n−1+ ···+ a1(ω) + a0,and verify that it matches the result obtained in part 2.5. (6 points) Using MATLAB:(a) (3 points) Verify your result in part 4 by determining the Fourier transform of h(t) usi ng the SymbolicMath Toolbox.(b) (3 points) Generate and plot the magnitude and phase response of the frequency response function usingthe result in part (a). Plot the magnitude in decibels and the phase in degrees from 0.01 Hz to 10 Hzusing a frequency vector of 100 l ogarithmicall y equally spaced poi nts.To receive credit:• Attach an M-file to your solution that contains the code to complete parts a and b. Include your nameand section number at the top of M-file using the comment symbol %.• Plo t the m agnitude and phase plot in a single figure, with the magnitude plot appearing in an uppersubplot and the phase pl ot appearing in a lower subplot.• 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 36 Part 5.• Use the MATLAB command gtext to place your name and section name within the figure.• Attach the figure to your
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