Name CM ECE 300 Signals and Systems Exam 2 25 April 2006 This exam is closed book in nature You are not to use a computer during the exam but you can use a calculator for simple calculations Problem 1 Problem 2 Problem 3 Problem 4 20 30 35 15 Exam 2 Total Score 100 1 Name CM 1 20 points Consider the following system composed of two subsystems Assume both systems are initially at rest no initial enery and c 0 x t v t v t x t c v t y t a Determine the impulse response for each subsystem b Determine the impulse response between input x t and output y t 2 Name CM 2 30 points Consider a causal linear time invariant system with impulse response given by h t e t 1 u t 1 The input to the system is given by x t u t u t 1 u t 3 Using graphical convolution determine the output y t for 2 t 5 Note the limited range of t we are interested in Specifically you must a Flip and slide h t b Show graphs displaying both h t l and x l 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 e Evaluate the integrals 3 Name CM 3 35 points The spectrum of a periodic signal x t which has period T 2 seconds is shown below All angles are multiples of 45 degrees Amplitude 2 1 5 1 0 5 0 3 2 1 0 Harmonic 1 2 3 2 1 0 Harmonic 1 2 3 Phase degrees 100 50 0 50 100 3 a Determine an expression for x t in terms of cosines and constants Your expression must be purely real b Determine the average value of x t c Determine the average power in x t jw Now assume x t is the input to a LTI system with transfer function H w 1 jw d Determine the average power in the output y t e Determine an expression for y t in terms of cosines and constants Your expression must be purely real 4 Name CM 4 15 points Assume periodic signal x t has the Fourier series representation x t ckx e jkwot x t is the input to an LTI system with the system model y t 3 d x t 2 dt y jkw t Determine the Fourier series representation of y t where y t ck e o Specifically y x show how ck is related to ck 5 Name CM 6 Name CM 7 Name CM 8 Name CM Some Potentially Useful Relationships T E lim x t T T 2 2 dt x t dt T 2 1 x t dt T 2T T P lim e jx cos x jsin x j 1 1 jx jx e e 2j 1 jx jx cos x e e 2 sin x 1 1 cos 2 x cos 2x 2 2 1 1 sin 2 x cos 2x 2 2 T T t t rect 0 u t t 0 u t t0 2 2 T 9
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