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 Problem 2 Problem 3 Problem 4 25 25 25 25 Exam 2 Total Score 100 1 Name CM 1 Short Answer Questions 5 points each a Is the system with impulse response h t et u t BIBO stable Why or why not 1 b Is the system y t cos BIBO stable Why or why not x t c What is the impulse response for the system y t e t t 1 e x 2 d 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 x t and y t x t h1 t e t u t v t h2 t 2 t 1 y t e Is the function x t cos 4 t sin 6 t periodic If yes determine the 2 fundamental period 2 Name CM 2 Assume periodic signal x t has the spectrum shown below and a fundamental frequency o 3 rad sec Assume all angles are multiples of 45 degrees a Determine the average value and average power in x t b Write and expression for x t in terms of cosines and or sines c Sketch the single sided power spectrum for x t on the graph below Be sure to accurately label all axes 3 Name CM 3 Assume periodic signal x t has Fourier series representation x t k jk 1 jk e jkt k x t is the input to an LTI system with transfer function given by 1 1 5 2 5 H j 1 j 0 otherwise Determine the steady state output of the system y t 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 4 Name CM 4 Assume x t is a periodic signal with period T0 3 x t is defined over one period as 1 1 t 0 x t 0 0 t 2 a Determine the fundamental frequency 0 b Determine the average value of x t c Determine the average power in the DC component of x t d Determine an expression for the expansion coefficients X k where x t X k e jk ot You must write your expression in terms of the sinc function and possibly a leading exponential term 5 Name CM 6 Name CM 7 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 8
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