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Rose-Hulman ECE 300 - Exam 2

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Name __________________________________________________ CM____________ ECE 300 Signals and Systems Exam 2 18 October 2007 NAME ________________________________________ This exam is closed-book in nature. You may use a calculator for simple calculations, but not for things like integrals. Credit will not be given if your work is not shown! Problem 1 ________ / 25 Problem 2 ________ / 25 Problem 3 ________ / 25 Problem 4 ________ / 25 Exam 2 Total Score: _______ / 100 1Name __________________________________________________ CM____________ 1) Short Answer Questions (5 points each): a) Is the system with impulse response BIBO stable? Why or why not? ( ) ( )tht eut= b) Is the system 1() cos()ytxt⎛=⎜⎝⎠⎞⎟ BIBO stable? Why or why not? c) What is the impulse response for the system 1() ( 2)ttyte ex dλλλ−−−∞=−∫? Be sure to include appropriate unit step functions. d) Consider the two LTI systems shown below, with impulse responses shown. What is the impulse response between ()xtand ? ()yt 1() ()tht eut−= 2() 2 ( 1)ht tδ=− ()xt ()vt ()yt e) Is the function ( ) cos(4 ) sin(6 )2xtt tπππ=++ periodic? If yes, determine the fundamental period. 2Name __________________________________________________ CM____________ 2) Assume periodic signal ()xt has the spectrum shown below and a fundamental frequency3 rad/secoω= . Assume all angles are multiples of 45 degrees. a) Determine the average value and average power in()xt. b) Write and expression for ()xt in terms of cosines (and/or sines). c) Sketch the single sided power spectrum for ()xt on the graph below. Be sure to accurately label all axes. 3Name __________________________________________________ CM____________ 3) Assume periodic signal ()xt has Fourier series representation ()1kjktkjkxtejk=∞=−∞=+∑ ()xt is the input to an LTI system with transfer function given by 11.5 | | 2.51()0jHjotherwiseωωω⎧<<⎪+=⎨⎪⎩ Determine the steady state output of the system . For full credit your answer must be written in terms of cosines (and/or sines) . Clearly indicate whether you are writing your phase in degrees or in radians. ,()yt 4Name __________________________________________________ CM____________ 4) Assume ()xt is a periodic signal with period 03T=. ()xt is defined over one period as 011()00 2txtt−<≤⎧=⎨<≤⎩ a) Determine the fundamental frequency 0ω. b) Determine the average value of ()xt . c) Determine the average power in the DC component of ()xt. d) Determine an expression for the expansion coefficients,.kX, where ()ojktkxtXeω=∑ must write your expression in terms of the sinc function, and possibly a leading nential term. . Youexpo 5Name __________________________________________________ CM____________ 6Name __________________________________________________ CM____________ 7Name __________________________________________________ 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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Rose-Hulman ECE 300 - Exam 2

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