HW07_Ch09

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Electrical Engineering

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Nov 24, 2024

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HW07 Problem1: For the following impulse response ℎ ( 𝑡𝑡 ) = [1 + 2 sin( 𝑡𝑡 )] 𝑒𝑒 −3𝑡𝑡 𝑢𝑢 ( 𝑡𝑡 ) a) Find its Laplace transform, 𝐻𝐻 ( 𝑠𝑠 ). b) Plot the pole-zero diagram in s-domain, and mark ROC in it. c) Provide approximate magnitude response, | 𝐻𝐻 ( 𝑗𝑗𝑗𝑗 )| , and specify what type of filter it corresponds to. Justify your answer. d) Is this system stable? Justify your answer. Problem2: A causal LTI system with input signal 𝑥𝑥 ( 𝑡𝑡 ) output 𝑦𝑦 ( 𝑡𝑡 ) , is described by the following linear differential equation, 𝑦𝑦 ′′ ( 𝑡𝑡 ) − 3 𝑦𝑦 ′ ( 𝑡𝑡 ) + 2 𝑦𝑦 ( 𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) a) Find the system’s transfer function 𝐻𝐻 ( 𝑠𝑠 ) . b) Plot the pole-zero diagram in s-domain, and mark ROC in it. c) Is this system stable? Justify your answer. d) If it exists, find 𝐻𝐻 ( 𝑗𝑗𝑗𝑗 ) , provide the approximate magnitude response, | 𝐻𝐻 ( 𝑗𝑗𝑗𝑗 )| , and specify what type of filter it corresponds to. Justify your answer. e) Using the one-sided LT, find the output, 𝑦𝑦 ( 𝑡𝑡 ) , when 𝑥𝑥 ( 𝑡𝑡 ) = 𝑢𝑢 ( 𝑡𝑡 ) and all initial conditions are zero. f) Using the one-sided LT, find the output, 𝑦𝑦 ( 𝑡𝑡 ) , when 𝑥𝑥 ( 𝑡𝑡 ) = 0 and the initial conditions on 𝑦𝑦 ( 𝑡𝑡 ) are 𝑦𝑦 (0) = 1 and 𝑦𝑦 ′ (0) = 0 . Note that ℒ � 𝑑𝑑 2 𝑑𝑑𝑡𝑡 𝑥𝑥 ( 𝑡𝑡 ) � = 𝑠𝑠 2 𝑋𝑋 ( 𝑠𝑠 ) − 𝑠𝑠𝑥𝑥 (0 − ) − 𝑥𝑥 ′ (0 − ) . g) Using the one-sided LT, find the output, 𝑦𝑦 ( 𝑡𝑡 ) , when 𝑥𝑥 ( 𝑡𝑡 ) = 𝑢𝑢 ( 𝑡𝑡 ) and the initial 1conditions on 𝑦𝑦 ( 𝑡𝑡 ) are 𝑦𝑦 (0) = 1 and 𝑦𝑦 ′ (0) = 0 . Problem3: Repeat the question in Problem 2 for and LTI system described by the following linear differential equation: 𝑦𝑦 ′′ ( 𝑡𝑡 ) + 3 𝑦𝑦 ′ ( 𝑡𝑡 ) + 2 𝑦𝑦 ( 𝑡𝑡 ) = 𝑥𝑥 ( 𝑡𝑡 ) . Problem4: Given the following pole-zero diagram with ROC in the s-domain of 𝑋𝑋 ( 𝑠𝑠 ) , find 𝑥𝑥 ( 𝑡𝑡 ) . Note that the scaling factors are not required to be specified; for example, you can write 𝑋𝑋 ( 𝑠𝑠 ) = 𝐴𝐴 𝑠𝑠+1 + 𝐵𝐵 𝑠𝑠+2 resulting 𝑥𝑥 ( 𝑡𝑡 ) = [ 𝐴𝐴𝑒𝑒 −𝑡𝑡 + 𝐵𝐵𝑒𝑒 −2𝑡𝑡 ] 𝑢𝑢 ( 𝑡𝑡 ) . a) ROC 𝜎𝜎 𝑗𝑗 × × -1 1 1

b) ROC 𝜎𝜎 𝑗𝑗 × × -3 -2 -1 1 o o

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