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finals02 - Sample Final Exam Covering Chapters 1-9(finals02...

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Sample Final Exam Covering Chapters 1-9 (finals02) 1 Sample Final Exam ( finals00) Covering Chapters 1-9 of Fundamentals of Signals & Systems Problem 1 (20 marks) The unit step response of an LTI system was measured to be 3 ( ) 2 sin( ) ( ) ( ) ( ) 6 t s t e t u t u t tu t π = + . (a) [10 marks] Find the transfer function ( ) H s of the system. Specify its ROC. Sketch its pole-zero plot. Answer: 3 ( ) ( ) 6 6 3 3 3 ( ) ( ) 2 sin( ) ( ) ( ) ( ) 6 2 ( ) ( ) ( ) 2 3 1 3 1 ( ) ( ) 2 2 2 2 2 ( ) ( ) ( ) 2 3 1 ( ) 2 2 t j t j t t jt jt t t H s sS s s e t u t u t tu t e e s e u t u t tu t j j e j e s e u t u t tu t j j e s π π π + = = + = + + = + = L L L L ( ) 3 3 3 3 3 3 3 2 3 1 ( ) 2 2 ( ) ( ) ( ) 3 3 1 1 ( ) 2 2 2 2 ( ) ( ) ( ) 3 sin cos ( ) ( ) ( ) 3 3 ( 3) 1 ( jt t jt t jt t jt t jt t jt t t j e u t u t tu t j e e j e j e s u t u t tu t j s e t e t u t u t tu t s s s + + + + = + = + + = + + L L 2 2 3 2 3 2 2 2 2 2 2 2 1 1 3) 1 [( 3) 1]( 1) [ 2 3 4]( 1) [( 3) 1] [( 3) 1] (4 2 3) 4 (2 3 1) (2 3 1) (2 3 1) (2 3 1) (4 2 3) 4 [( 3) 1] [( 3) 1] s s s s s s s s s s s s s s s s s s s s s s + + + + + + + + + = = + + + + + + = = + + + + ROC: Re{ } 0 s >
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Sample Final Exam Covering Chapters 1-9 (finals02) 2 Poles are 1 2 3 0 3 3 p p j p j = = − + = − , Zeros are zeros of 2 0.21748 1.62331 s s + : 1 2 3 1.3875, 1.1700 z z z = − = = ∞ (b) [4 marks] Is the system causal? Is it stable? Justify your answers. Answer: System is causal: ROC is an open RHP and transfer function is rational . This system isn't stable as ROC doesn't include the imaginary axis (or because rightmost pole 0 has a nonnegative real part.) (c) [6 marks] Give the direct form realization (block diagram) of H s ( ) . Answer: 2 3 2 (2 3 1) (4 2 3) 4 ( ) 2 3 4 s s H s s s s + = + + Re{s} Im{s} -1.7 -1.39 1.17 -1 1 + - - + + - + W s ( ) 1 s X s ( ) 1 s 2 3 sW s ( ) s W s 2 ( ) 4 Y s ( ) 1 s s W s 3 ( ) 0 4 2 3 1 4 2 3
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Sample Final Exam Covering Chapters 1-9 (finals02) 3 Problem 2 (20 marks) The following circuit has initial conditions on the capacitor v C ( ) 0 and inductor i L ( ) 0 . (a) [4 marks] Transform the circuit using the unilateral Laplace transform. Answer: (b) [8 marks] Find the unilateral Laplace transform of v t ( ) . Answer: Let's use mesh analysis. For mesh 1: V I I I I V I s c s c s Cs s s v R s s s R s R s v R Cs s ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) = = − + + + 1 1 0 0 1 1 0 1 1 1 1 1 2 2 1 1 1 1 For mesh 2: R s s R Ls s Li s R R R Ls s L R i L L 1 1 2 2 2 1 1 1 2 2 1 0 0 1 0 [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( ) I I I I I + + = = + + Substituting, we obtain I V I I V I V 2 1 1 1 1 1 2 2 1 1 2 1 1 2 2 1 1 1 1 2 1 2 1 2 2 1 1 1 1 0 1 1 1 0 1 0 1 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ( )( )] ( ) ( ) ( ) ( ) ( ) [ ( ) ] ( ) ( ) ( ) ( s R s R s v R Cs R R R Ls s L R i R Cs R Cs R R Ls s R Cs s R Cv R Cs Li LR Cs L R R C s R R s
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