Name CM ECE 300 Signals and Systems Exam 3 17 February 2009 This exam is closed book in nature Credit will not be given for work not shown You may not use calculators Problem 1 Problem 2 Problem 3 Problem 4 30 30 30 10 Exam 3 Total Score 100 1 Name CM 1 Fourier Series 30 points The plot of x t can be used to answer parts a and b below x t 1 1 1 t 1 a Determine the Fourier Series coefficients for x t Do NOT express your answer as a sinc function but do simplify your answer as much as possible b You found the Fourier Coefficients of a waveform that is different from the one above to be as shown below Express these coefficients as a sinc function with a possible gain phase term 3 ck j 2 k 2 jk jk 43 3 e e 2 Name CM 2 30 points An LTI system has the following transfer function 6e j 2 H 0 4 6 otherwise The input to the system is given by 5t x t sinc a What is X b What is y t c What is the energy in x t 3 Name CM 3 30 points Determine X or x t You must show your work You may leave solutions in terms of integrals derivatives or convolutions i e you do not have to explicitly solve them but you must go further than the basic Fourier Inverse Fourier Transform Definitions a For x t e 4t determine the corresponding Fourier transform X 2 b For X c For x t 2 determine the corresponding inverse Fourier transform x t 1 j 2 2 2 2 t 2 3 2 determine the corresponding Fourier transform X 4 Name CM 4 10 points Starting from the Fourier transform or inverse transform integrals X x t e j t dt x t 1 2 X e j t d If x t X then derive the corresponding Fourier Transform of x t a 5 Name CM 6 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 7
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