Name __________________________________________________ CM____________ ECE 300 Signals and Systems Exam 1 27 March, 2008 NAME ________________________________________ This exam is closed-book in nature. You are not to use a calculator or computer during the exam. Credit will not be given for work not shown. Problem 1-5 ________ / 20 Problem 6 ________ / 20 Problem 7 ________ / 30 Problems 8 ________ / 30 Exam 1 Total Score: _______ / 100 1Name __________________________________________________ CM____________ Multiple Choice Questions (20 points, 4 points each) 1. Consider a system with sinusoidal input and output shown below: 0 1 2 3 4 5 6-2-1.5-1-0.500.511.52Time(sec)InputOutput Which of the following statements is true: a) The system is linear. b) The system is not linear. c) There is not enough information to determine whether the system is linear or not linear. 2. The average power in the signal ( )jtxtceω= is a) 0 b) ||2c c) d) 2 |c| 2|c|2 e) none of these 3. The average power in the signal () cos( )xt A tωθ=+ is a) 2A b) A c) d) 2A22A e) none of these 4. The signal(1)4()tjjtxte eππ+=+ is a) not periodic b) periodic with fundamental period 2π seconds c) periodic with fundamental period 4 seconds d) periodic with fundamental period 8 seconds e) none of the above 5. Is the system ()() ( 1)ttytex dλλλ−−−∞=+∫ causal? a) yes b) no 2Name __________________________________________________ CM____________ 6. (20 points) Linearity and Time-Invariance a) Using a formal test, such as was shown in class, determine if the following system is time-invariant. Be sure to show all your work. ()1() ( 3)ttyt e x dλλλ−−−−∞−=∫ b) Using a formal test, such as was shown in class, determine if the following system is linear. Be sure to show all your work. 2( ) sin( ) ( ) ( )yttyttx+=t 3Name __________________________________________________ CM____________ 7. (30 points) Determining Impulse Responses Be sure to include all necessary unit step functions in your answers! a) Determine the impulse response for the system ( ) ( ) )(tyt xt x dλλ∞−=+∫ b) Determine the impulse response for the system1()(() 3)ttyte dxλλλ−−−−∞=+∫ c) Determine the impulse response for the system() 3 () 2 ( 1)yt yt xt−=− d) Determine the impulse response for the system below 1() 2 ( 3)ht ut=− 2() 2 ( 1)ht tδ=+ ()xt ()vt ()yt 4Name __________________________________________________ CM____________ 8. (30 points) Graphical Convolution Consider a linear time invariant system with impulse response given by () [ ( 1) ( 2)]ht t ut ut=+−− and input() 2 ( 2) 3 ( 3) ( 4)xt ut ut ut=−− −+ −, shown below -1 -1 2 2 h(t) t 3 -1 2 2 x(t) t 4 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) Made careful sketches 5Name __________________________________________________ CM____________ 6Name __________________________________________________ CM____________ 7Name __________________________________________________ CM____________ 8Name __________________________________________________ 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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