Name __________________________________________________ CM____________ ECE 300 Signals and Systems Exam 2 24 April, 2008 NAME ________________________________________ This exam is closed-book in nature. You may use a calculator for simple calculations, but not for things like integrals. You must show all of your work. Credit will not be given for work not shown. Problem 1 ________ / 15 Problem 2 ________ / 15 Problem 3 ________ / 20 Problem 4 ________ / 25 Problem 5 ________ / 25 Exam 2 Total Score: _______ / 100 1Name __________________________________________________ CM____________ 1. (15 points) Assume ()xt and are periodic signal with Fourier series representations, and ()yt 0()jkktkext Xω=∑ 0()jkktkeyt Yω=∑ Assume also that ()xt and are related by the differential equation ()yt ()2()(yt a yt xt)−+= a) Write the in terms of the kYkX b) If ()xt is the input to an LTI system with transfer function )(Hjω with output , what is the transfer function ()yt)(Hjω ? 2Name __________________________________________________ CM____________ 2. (15 points) Assume we are computing the Fourier series coefficients, and after evaluating the integrals we end up with 3jk j kkejkXe−= WritekX in terms of the sinc function. 3Name __________________________________________________ CM____________ 3. (20 points) Assume()xt is a periodic signal with a Fourier series representation, and the following graph displays the spectrum of ()xt. Assume the fundamental frequency is 04ω= rad/sec. Note that the phase is in radians, and all phases are multiples of 1 radian. a) What is the average value of ()xt? b) What is the average power in ()xt? c) What is the average power in the second harmonic of ()xt? c) Write()xt in terms of sines and cosines. 4Name __________________________________________________ CM____________ 4. (25 points) Assume ()xt is a periodic signal with Fourier series representation 42() 21kkjk txtejk==−∞∞=++∑ Assume ()xt is the input to an LTI system with transfer function 103||3() 4 3||10||11jHj eωωωωω−<⎧⎪⎪1=<<⎨⎪>⎪⎩ Determine the steady state output of the system, . Your answer must be written in terms of sines and cosines, not complex exponentials. Your answer must also be in either degrees or radians, but not a mixture. ()yt 5Name __________________________________________________ CM____________ 5. (25 points) Graphical Convolution and System Properties Consider a linear time invariant system with impulse response given by () sin( [ ( 1) ( ))2ht ut utt ]π=− + − − and input () ( 1) ( 2) 2( 3) 2( 5)xt ut ut ut ut=−−−− −+ − shown below a) Is this system causal? Why or why not? b) Is this system BIBO stable? Why or why not? 6Name __________________________________________________ CM____________ c) Using graphical convolution, determine the output() () ()yt ht xt=∗ Specifically, you must a) Flip and slide , ()htNOT ()xt b) Show graphs displaying both (ht )λ− and ()xλfor each region of interest c) Determine the range of t for which each part of your solution is valid d) Set up any necessary integrals to compute . Your integrals must be complete, in that they cannot contain the symbols ()yt)(xλ or(ht )λ−but must contain the actual functions. e) DO NOT EVALUATE THE INTEGRALS!! Hints: (1) Pay attention to the width of h(t) (2) It is the endpoints of h(t) that matter the most 7Name __________________________________________________ CM____________ 8Name __________________________________________________ CM____________ 9Name __________________________________________________ CM____________ 10Name __________________________________________________ CM____________ Some Potentially Useful Relationships () ()T22TTE lim xt dt xt dt∞∞→∞−−∞==∫∫ ()T2TT1Plim xtd2T∞→∞−=∫t ()()jxecosxjsinx=+ j1=− ()jx jx1cos x e e2−⎡⎤=+⎣⎦ ()jx jx1sin x e e2j−⎡⎤=−⎣⎦ () ()211cos x cos 2x22=+ () ()211sin x cos 2x22=− 000ttTTrect u t t u t tT2−⎛⎞⎛⎞⎛=−+−−−⎜⎟⎜⎜⎟⎝⎠⎝⎝⎠2⎞⎟⎠
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