EE 350 EXAM III 12 November 2009 Last Name Print First Name Print ID number Last 4 digits Section DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total 100 Test Form A INSTRUCTIONS 1 You have 2 hours to complete this exam 2 This is a closed book exam You may use one 8 5 11 note sheet 3 Calculators are not allowed 4 Solve each part of the problem in the space following the question If you need more space continue your solution on the reverse side labeling the page with the question number for example Problem 1 2 Continued NO credit will be given to solutions that do not meet this requirement 5 DO NOT REMOVE ANY PAGES FROM THIS EXAM Loose papers will not be accepted and a grade of ZERO will be assigned 6 The quality of your analysis and evaluation is as important as your answers Your reasoning must be precise and clear your complete English sentences should convey what you are doing To receive credit you must show your work 1 Problem 1 25 Points 1 12 points A periodic signal g t has the representation cos t cos2 t cos 4 t sin 4 t g t cos 2 t 3 4 6 Specify the compact trigonometric Fourier series representation by specifying the numeric value of a 2 points the average value C0 of g t b 2 points the fundamental frequency 0 of g t c 8 points and all nonzero values of the parameters Cn and n 2 2 13 points Figure 1 shows a periodic signal f t Figure 1 Periodic signal f t a 8 points Determine the trigonometric Fourier series representation of f t by specifying the parameters a0 an bn b 5 points Using the results from part a determine the exponential Fourier series representation of f t by specifying the coefficients Dn for all values of n 3 Problem 2 25 points 1 10 points Figure 2 shows two periodic signals f t and g t that both have the same fundamental period To The complex exponential Fourier series representation of the signals f t and g t have the coefficients Dnf and Dng respectively Figure 2 Periodic signals f t and g t a 2 points Without explicitly calculating the Fourier series coefficients Dnf what conclusions can be drawn about the coefficients Dnf b 2 points Without explicitly calculating the Fourier series coefficients Dng what conclusions can be drawn about the coefficients Dng c 2 points For a given value of n 6 0 do you expect Dnf to be larger or smaller than Dng Justify your answer in one or two short sentences d 4 points In order to verify your answer in part c use the observation that g t to derive an expression for Dng in terms of Dnf 4 df dt 2 6 points A periodic signal f t has a fundamental frequency 5 rad sec and the complex exponential Fourier series coefficients 30 n when n is odd Dnf 0 when n is zero or even The periodic signal f t is applied to a filter whose frequency response function is 0 30 H 30 0 0 otherwise Determine the Fourier series coefficients Dny of the periodic output signal y t of the filter In order to receive full credit your solution must specify the range of values of n for which Dny 6 0 5 3 9 points A periodic signal f t with fundamental period To 0 1 sec has the trigonometric Fourier series representation a0 1 an 1 n2 for n 1 2 3 4 5 an 0 for n 6 7 8 bn 0 for all n The periodic signal f t is passed through a filter whose frequency response function is H 10 2 1 2 Let y t denote the output of the filter a 2 points Find an expression for the magnitude H and the phase angle 6 H in terms of b 7 points Without using the MatLab command lsim compose a MatLab script that determines and plots the steady state filter response y t over the time interval 1 t 1 using 1000 time points 6 Problem 3 25 points 1 12 points Signals f t and g t have the Fourier transforms F 2 e G 2 e respectively a 2 points Without determining the inverse Fourier transform what conclusions can be drawn about the signal f t b 2 points Without determining the inverse Fourier transform what conclusions can be drawn about the signal g t c 2 points Using an appropriate Fourier transform property what is the relationship between the signals f t and g t d 6 points By direct integration determine the signal g t 7 2 6 points Consider the real valued signal s t e t cos o t a 2 points Without determining the Fourier transform S of the signal s t what conclusions can be drawn about S b 4 points Using the Fourier transform pair e a t 2a a 2 2 and the appropriate Fourier transform property determine the Fourier transform S 8 3 7 points A signal y t has the Fourier transform Y 2 rect What is the energy Ey of the signal y t 9 3 Problem 4 25 points 1 17 points Figure 3 shows a partial block diagram of a signal processing system while Figure 4 shows the Fourier spectra of the input signals m1 t and m2 t Figure 3 Signal processing system Figure 4 Fourier spectra of the input signals a 5 points As shown in Figure 3 the signal w t is the multiplier output that connects to the lower input of the summer Determine an expression for the Fourier transform W in terms of M2 and sketch W in Figure 5 To receive full credit for your sketch you must correctly label both the frequency and amplitude axes Figure 5 Fourier spectra of W 10 b 5 points As shown in Figure 3 the signal x t is the output of the summer Sketch X in Figure 6 To receive full credit for your sketch you must correctly label both the frequency and amplitude axes Figure 6 Fourier spectra of X c 7 points As shown in Figure 3 the signal y t is the output of the second multiplier Sketch Y in Figure 7 To receive full credit for your sketch you must correctly label both the frequency and amplitude axes Figure 7 Fourier spectra of Y 11 2 8 points Suppose that the impulse response function for the LTI system shown in Figure 8 is chosen as h t f t where f t is the input to the system and let Y denote the Fourier transform of the zero state response of the system Figure 8 Linear time invariant system with input f t and output y t Show that the energy of the signal f t is given by Ef 1 2 Z 12 Y d
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