# HW3 - MAE107 Homework#3 Prof M’Closkey Due Date The...

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Unformatted text preview: MAE107 Homework #3 Prof. M’Closkey Due Date The homework is due by Thursday, 5PM, January 29, 2009 to Mr. David Shatto in the 38-137 foyer (3rd ﬂoor Engineering 4). Problem 1 Show, Z ∞ −∞ h ( t − τ ) u ( τ ) dτ = Z ∞ −∞ h ( τ ) u ( t − τ ) dτ, by changing the variable of integration. Problem 2 Consider the series connection of two subsystems, u 1- System 1- y 1 ( t ) = R ∞ −∞ h 1 ( t − τ ) u 1 ( τ ) dτ System 2- y 2 The impulse response of System 1 is denoted h 1 and the impulse response of System 2 is denoted h 2 . The output of System 1 is given as the convolution y 1 ( t ) = Z ∞ −∞ h 1 ( t − τ ) u 1 ( τ ) dτ. The input to System 2 is the output of System 1 so the output of System 2 is computed to be y 2 ( t ) = Z ∞ −∞ h 2 ( t − τ ) y 1 ( τ ) dτ = Z ∞ −∞ h 2 ( t − τ ) Z ∞ −∞ h 1 ( τ − s ) u 1 ( s ) ds dτ. (1) Define the convolution of the impulse responses ˜ h ( t ) = Z ∞ −∞ h 2 ( t − τ ) h 1 ( τ ) dτ. (2) Show that (1) is equivalent to y 2 ( t ) = Z ∞ −∞ ˜ h ( t − τ ) u 1 ( τ ) dτ. 1 In other words, the impulse response of the series connection (i.e., from input u 1 to output y 2 ) is the convolution of the impulse responses of its subsystems.the convolution of the impulse responses of its subsystems....
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HW3 - MAE107 Homework#3 Prof M’Closkey Due Date The...

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