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 1 Name CM Multiple Choice Questions 20 points 4 points each 1 Consider a system with sinusoidal input and output shown below 2 Input Output 1 5 1 0 5 0 0 5 1 1 5 2 0 1 2 3 Time sec 4 5 6 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 x t ce j t is a 0 b c 2 c c 2 d c 2 2 e none of these 3 The average power in the signal x t A cos t is a A 2 b A c A2 d j A2 2 e none of these t 4 The signal x t e j t 1 e 4 is a not periodic b periodic with fundamental period 2 seconds c periodic with fundamental period 4 seconds d periodic with fundamental period 8 seconds e none of the above t 5 Is the system y t e t x 1 d causal a yes b no 2 Name 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 t 1 y t e t x 3 d 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 y t sin t y t t 2 x t 3 Name CM 7 30 points Determining Impulse Responses Be sure to include all necessary unit step functions in your answers t a Determine the impulse response for the system y t x t x d t 1 b Determine the impulse response for the system y t e t x 3 d c Determine the impulse response for the system y t 3 y t 2 x t 1 d Determine the impulse response for the system below x t h1 t 2u t 3 v t h2 t 2 t 1 y t 4 Name CM 8 30 points Graphical Convolution Consider a linear time invariant system with impulse response given by h t t u t 1 u t 2 and input x t 2u t 2 3u t 3 u t 4 shown below h t x t 2 2 1 2 t 2 3 4 t 1 1 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 Made careful sketches 5 Name CM 6 Name CM 7 Name CM 8 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 9
View Full Document
Unlocking...