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 Problem 2 Problem 3 Problem 4 Problem 5 15 15 20 25 25 Exam 2 Total Score 100 1 Name CM 1 15 points Assume x t and y t are periodic signal with Fourier series representations and x t X k e jk 0t k y t Yk e jk 0t k Assume also that x t and y t are related by the differential equation y t a 2 y t x t a Write the Yk in terms of the X k b If x t is the input to an LTI system with transfer function H j with output y t what is the transfer function H j 2 Name CM 2 15 points Assume we are computing the Fourier series coefficients and after evaluating the integrals we end up with Xk e jk e j 3k jk Write X k in terms of the sinc function 3 Name CM 3 20 points Assume x t is a periodic signal with a Fourier series representation and the following graph displays the spectrum of x t Assume the fundamental frequency is 0 4 rad sec Note that the phase is in radians and all phases are multiples of 1 radian a What is the average value of x t b What is the average power in x t c What is the average power in the second harmonic of x t c Write x t in terms of sines and cosines 4 Name CM 4 25 points Assume x t is a periodic signal with Fourier series representation x t 2 k 2 1 jk e jk 4 t k Assume x t is the input to an LTI system with transfer function 3 j H j 4e 10 0 3 3 11 11 Determine the steady state output of the system y t 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 5 Name CM 5 25 points Graphical Convolution and System Properties Consider a linear time invariant system with impulse response given by h t sin t u t 1 u t 2 and input x t u t 1 u t 2 2u t 3 2u t 5 shown below a Is this system causal Why or why not b Is this system BIBO stable Why or why not 6 Name CM c Using graphical convolution determine the output y t h t x t Specifically you must a Flip and slide h t NOT x t b Show graphs displaying both h t and x for 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 y t Your integrals must be complete in that they cannot contain the symbols x or h t 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 7 Name CM 8 Name CM 9 Name CM 10 Name CM Some Potentially Useful Relationships T E lim T x t 2 dt T x t 2 dt T 2 1 x t dt T 2T T P lim e jx cos x jsin x j 1 cos x 1 jx e e jx 2 sin x 1 jx jx e e 2j cos 2 x 1 1 cos 2x 2 2 sin 2 x 1 1 cos 2x 2 2 T T t t0 rect u t t0 u t t0 2 2 T 11
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