hw5_solution

# hw5_solution - MAE107 Homework#5 Solution Prof M’Closkey...

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Unformatted text preview: MAE107 Homework #5 Solution Prof. M’Closkey 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 αe- α ( t- τ ) ce- cτ dτ = αc c- α ( e- αt- e- ct ) . Thus, V out ( t ) = αc c- α ( e- αt- e- ct ) μ ( 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 + τ ) u ( τ ) dτ = Z ∞- t ce- c ( t + τ ) ce- cτ dτ = c 2 e- ct Z ∞- t e- 2 cτ dτ = 1 2 ce ct . • Case 2: t ≥ 0. In this case, R V in V in ( t ) = Z ∞ ce- c ( t + τ ) ce- cτ dτ = 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 + τ ) V in ( τ ) dτ = Z ∞- t αc c- α e- α ( t + τ )- e- c ( t + τ ) ce- cτ dτ = αc 2 c- α e- αt Z ∞- t e- ( α + c ) τ dτ- e- ct Z ∞- t e- 2 cτ dτ = αc 2 c- α 1 α + c e- ct- 1 2 c e ct = αc 2( α + c ) e ct . • Case 2. t < 0. In this case, R V out V in ( t ) = Z ∞-∞ V out ( t + τ ) V in ( τ ) dτ = αc 2 c- α 1 α + c e- αt- 1 2 c e- ct . Summarizing, R V out V in ( t ) = αc 2( α + c ) e ct t < αc 2 c- α 1 α + c e- αt- 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- τ ) R V in V in ( τ ) dτ = Z t-∞ αe- α ( t- τ ) 1 2 ce cτ dτ = αc 2 e- αt Z t-∞ e ( α + c ) τ dτ = αc 2( α + c ) e ct . • Case 2. t ≥ 0. Z ∞-∞ h ( t- τ ) R V in V in ( τ ) dτ = Z-∞ h ( t- τ ) R V in V in ( τ ) dτ + Z t h ( t- τ ) R V in V in ( τ ) = Z-∞ αe- α ( t- τ ) 1 2 ce cτ dτ + Z t αe- α ( t- τ ) 1 2 ce- cτ dτ = αc 2 e- αt Z-∞ e ( α + c )...
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hw5_solution - MAE107 Homework#5 Solution Prof M’Closkey...

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