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Unformatted text preview: MAE107 Homework #3 Prof. MCloskey Due Date The homework is due by Thursday, 5PM, January 29, 2009 to Mr. David Shatto in the 38-137 foyer (3rd oor Engineering 4). Problem 1 Show, Z h ( t ) u ( ) d = Z h ( ) u ( t ) d, 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 ) u 1 ( ) d 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 ) u 1 ( ) d. 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 ) y 1 ( ) d = Z h 2 ( t ) Z h 1 ( s ) u 1 ( s ) ds d. (1) Define the convolution of the impulse responses h ( t ) = Z h 2 ( t ) h 1 ( ) d. (2) Show that (1) is equivalent to y 2 ( t ) = Z h ( t ) u 1 ( ) d. 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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