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20082ee102_1_HW_3_solution1

20082ee102_1_HW_3_solution1 - HW3 solution SPRING 2008 EE...

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HW3 solution, SPRING 2008, EE 102 1. Reformulate y(t) into y ( t ) = Z -∞ U ( t - τ ) x ( τ ) + Z -∞ U ( τ - t ) e t e - τ x ( τ ) dτ, = Z -∞ [ U ( t - τ ) + U ( τ - t ) e t e - τ ] x ( τ ) dτ, Therefore by BT h ( t, τ ) = U ( t - τ ) + U ( - ( t - τ )) e t - τ We obtain h ( t ) = U ( t ) + U ( - t ) e t . By BT, and we introduce the notation max(a,b)= the larger of a or b. For example, max(2,5)=5. We have g ( t ) = Z -∞ h ( t - τ ) U ( τ ) = Z -∞ [ U ( t - τ ) + U ( - ( t - τ )) e t - τ ] U ( τ ) dτ, = Z -∞ U ( t - τ ) U ( τ ) + Z -∞ U ( - t + τ ) e t - τ U ( τ ) dτ, = U ( t ) Z t 0 1 + e t Z max ( t, 0) e - τ dτ, = tU ( t ) + e t - max ( t, 0) , If t < 0, the above formula simplifies to g ( t ) = e t . If t 0, the above simplifies to g ( t ) = t + 1 . 2. IRF: h ( t, σ ) = Z t -∞ e - ( t - τ ) δ ( τ - σ ) dτ, = U ( t - σ ) e - ( t - σ ) , h ( t ) = U ( t ) e - t 1
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Let x 1 ( t ) = e - ( t - 1) U ( t - 1) , and x 2 ( t ) = U (2 - t ). The response of the system to x 1 ( t ) is y 1 ( t ) = Z -∞ h ( t - τ ) x 1 ( τ ) = e - t +1 Z -∞ U ( t - τ ) U ( τ - 1) = e - t +1 ( t - 1) U ( t - 1) . In the following, we denote min(a,b)= the smaller value of a or b. For example, min(2,3)=2.
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