hw5_solution

hw5_solution - MAE107 Homework #5 Solution Prof. MCloskey...

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Unformatted text preview: MAE107 Homework #5 Solution Prof. MCloskey Problem 1 1. Since V in = 0 for t < 0 and the system is causal, V out = 0 for t < 0. For t 0, V out ( t ) = Z t e- ( t- ) ce- c d = c c- ( e- t- e- ct ) . Thus, V out ( t ) = c c- ( e- t- e- ct ) ( t ) . 2. The auto-correlation of V in is computed for two intervals of time: Case 1: t < 0. In this case, R V in V in ( t ) = Z - u ( t + ) u ( ) d = Z - t ce- c ( t + ) ce- c d = c 2 e- ct Z - t e- 2 c d = 1 2 ce ct . Case 2: t 0. In this case, R V in V in ( t ) = Z ce- c ( t + ) ce- c d = c 2 e- ct Z 1 2 ce- ct . Summarizing, R V in V in ( t ) = 1 2 ce- c | t | , t (- , ) . 3. The cross-correlation R V out V in is also computed over two time intervals. 1 Case 1. t < 0. R V out V in ( t ) = Z - V out ( t + ) V in ( ) d = Z - t c c- e- ( t + )- e- c ( t + ) ce- c d = c 2 c- e- t Z - t e- ( + c ) d- e- ct Z - t e- 2 c d = c 2 c- 1 + c e- ct- 1 2 c e ct = c 2( + c ) e ct . Case 2. t < 0. In this case, R V out V in ( t ) = Z - V out ( t + ) V in ( ) d = c 2 c- 1 + c e- t- 1 2 c e- ct . Summarizing, R V out V in ( t ) = c 2( + c ) e ct t < c 2 c- 1 + c e- t- 1 2 c e- ct t . 4. Circuit output when the input is R V in V in is also calculated over two time intervals: Case 1. t < 0. Z - h ( t- ) R V in V in ( ) d = Z t- e- ( t- ) 1 2 ce c d = c 2 e- t Z t- e ( + c ) d = c 2( + c ) e ct . Case 2. t 0. Z - h ( t- ) R V in V in ( ) d = Z- h ( t- ) R V in V in ( ) d + Z t h ( t- ) R V in V in ( ) = Z- e- ( t- ) 1 2 ce c d + Z t e- ( t- ) 1 2 ce- c d = c 2 e- t Z- e ( + c )...
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hw5_solution - MAE107 Homework #5 Solution Prof. MCloskey...

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