practice mdt solution - UCLA DEPARTMENT OF ELECTRICAL...

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UCLA DEPARTMENT OF ELECTRICAL ENGINEERING EE102: SYSTEMS & SIGNALS Practice Midterm Examination II- Solution Write Your Discussion Session in the Corner !!%% (* Otherwise Your Midterm might be LOST) Your name :——————————————————– Instructions: Closed Book except one double-sided cheat sheet, Calculators are NOT Allowed Good Luck! 1
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Question 1 The figure below is the “poles-zeros” plot of a system transfer function H ( s ). (i) Find H ( s ) given H (0) = 1. (ii) Find Laplace transform of the output signal Y ( s ) if input is x ( t ) = e - 2 t u ( t ). (iii) Find the time domain output signal y ( t ) from Y ( s ). Im(s) Re(s) 1 2j -1 -2j Solution: (i) The transform function is H ( s ) = A ( s - 1) ( s + 1 + 2 j )( s + 1 - 2 j ) = A s - 1 ( s + 1) 2 + 4 = A s - 1 s 2 + 2 s + 5 Since H (0) = - A/ 5 = 1, therefore, A = - 5. H ( s ) = - 5 s - 1 s 2 + 2 s + 5 (ii) X ( s ) = 1 s + 2 thus Y ( s ) = - 5( s - 1) ( s + 2)( s 2 + 2 s + 5) = - 5 B s + 2 + Cs + D s 2 + 2 s + 5 2
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Multiply both sides by ( s +2)( s 2 +2 s +5) and compare coe ffi cients of powers of s on both sides to get B = - 3 / 5, C = 3 / 5, and D = 1. Y ( s ) = A B s + 2 + C ( s + 1) + ( D - C ) ( s + 1) 2 + (2) 2 Y ( s ) = A B s + 2 + C ( s + 1) ( s + 1) 2 + (2) 2 + D - C 2 2 ( s + 1) 2 + (2) 2 Take inverse Laplace transform to obtain y ( t ) = A Be - 2 t + Ce - t cos(2 t ) + D - C 2 e - t sin(2 t ) u ( t ) Substitute values of A , B , C and D to get y ( t ) = - 5 - 3 5 e - 2 t + 3 5 e - t cos(2 t ) + 1 5 e - t sin(2 t ) u ( t ) y ( t ) = 3 e - 2 t - 3 e - t cos(2 t ) - e - t sin(2 t ) u ( t ) 3
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Question 2 Consider the following di erential equation d 2 y ( t ) dt 2 + 6 dy ( t ) dt + 25 y ( t ) = dx ( t ) dt + 3 x ( t ) , t > 0 with initial condition y 0 (0) = 0 , y (0) = 0 , x (0) = 0 .
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