finals00 - Sample Final Exam Covering Chapters 1-9(finals00...

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Sample Final Exam Covering Chapters 1-9 (finals00) 1 Sample Final Exam ( finals00) Covering Chapters 1-9 of Fundamentals of Signals & Systems Problem 1 (20 marks) Consider the transfer function: 2 2 2 1 ( ) (0.01 0.1 3 1)( 2) s s H s s s s + = + + + . (a) [6 marks] Find the poles and zeros of ( ) H s (specify how many there are at ). Give all possible regions of convergence of the transfer function H s ( ) . Answer: Poles are 1 2 3 2 5 3 5 8.66 5 5 3 5 8.66 5 p p j j p j j = − = − + = − + = − = − Zeros are 1 2 3 1, 1 z z z = = = ∞ There are 3 possible ROCs: ROC1: Re{ } 5 3 8.66 s < − = − ROC2: 5 3 Re{ } 2 s < < − ROC3: Re{ } 2 s > − (b) [10 marks] Give the region of convergence of ( ) H s that corresponds to the impulse response of a stable system, and sketch it on a pole-zero plot. Is the stable system causal? Explain. Compute the impulse response h t ( ) of this stable system and give its Fourier transform ( ) H j ω . Answer: System is stable for the ROC that contains the j ω -axis: ROC3. This system is also causal as ROC3 is an open RHP and the transfer function is rational. The partial fraction expansion of H s ( ) yields: ROC3 Re{s} Im{s} -8.66 -2 -5 5 1
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Sample Final Exam Covering Chapters 1-9 (finals00) 2 2 2 2 2 2 2 5 3 2 1 100( 2 1) ( ) (0.01 0.1 3 1)( 2) ( 10 3 100)( 2) 100( 1) ( 5 3 5)( 5 3 5)( 2) 1 1 1 72.58 225.2 72.58 225.2 12.976 2 5 3 5 5 3 5 100( 1) ( 5 3 5)( 2) C A B s s s s s H s s s s s s s s s j s j s j j s s j s j s A s j s =− + + = = + + + + + + = + + + + = + + + + + + + = + + + ±²³ ±´´²´´³ ±´´²´´³ 2 2 5 2 2 2 100( 5 3 1 5 ) 10( 5 3 1 5 ) 43.512 135.24 (10 )( 5 3 2 5 ) ( 5 3 2) 5 43.512 135.24 100( 1) 900 900 12.976 69.359 ( 10 3 100) (4 20 3 100) j s j j j j j j B A j s C s s + =− + + = = = + + + + = = = = = = + + + Using the table and simplifying, we find the following impulse response: ROC Re{ } s < − 1 : { } ( 5 3 5 ) ( 5 3 5 ) 2 5 3 5 2 5 3 2 5 3 ( ) (43.512 135.24) ( ) (43.512 135.24) ( ) 12.976 ( ) 2 Re (43.512 135.24) ( ) 12.976 ( ) 2(142.07) cos(5 1.2595) ( ) 12.976 ( ) 284.14 cos(5 1.25 j t j t t t j t t t t t h t j e u t j e u t e u t e j e u t e u t e t u t e u t e t + = + + + = + + = + + = + 2 5 3 2 95) ( ) 12.976 ( ) 2 [43.512cos(5 ) 135.24sin(5 )] ( ) 12.976 ( ) t t t u t e u t e t t u t e u t + = + Fourier transform of ( ) h t is ( ) H j ω : 2 2 ( ) 2 1 ( ) (0.01( ) 0.1 3 1)( 2) j j H j j j j ω ω ω ω ω ω + = + + + 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -40 -20 0 20 40 60 80 100
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Sample Final Exam Covering Chapters 1-9 (finals00) 3 (c) [4 marks] Give the direct form realization (block diagram) of H s ( ) . Answer: Problem 2 (5 marks) True or False? (a) The Fourier transform X j ( ) ω of the product of a real signal x t ( ) and an impulse ( 1) t δ is real. Answer: False. (b) The system defined by 2 ( ) ( ) y t x t = is time-invariant. Answer: False. (c) The Fourier series coefficients a k of a purely imaginary periodic signal x t ( ) have the following property: k k a a = − . Answer: True. (d) The causal linear discrete-time system defined by [ 2] 2 [ 1] [ ] [ ] y n y n y n x n + + = is stable. Answer: False. (e) The signal 2 5 [ ] j n x n e = is not periodic. Answer: True. Problem 3 (15 marks) Consider the radar system depicted below where the radar's emitting antenna emits a pulse 100 3 ( ) sin(10 ) ( ) t x t e t u t = that gets reflected by a boat and is received by the radar's receiving + - - + + + - W s ( ) 1 s X s ( ) 1 s 10 3 2 + sW s ( ) s W s 2 ( ) 100 100 200 Y s ( ) 1 s s W s 3 ( ) 200 20 3 100 +
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Sample Final Exam Covering Chapters 1-9 (finals00) 4 antenna as 0 100 100 3 3 0 0 ( ) 0.1 sin(10 10 ) ( ) t
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