20082ee102_1_HW_3_solution1

20082ee102_1_HW_3_solution1 - HW3 solution, SPRING 2008, EE...

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Unformatted text preview: HW3 solution, SPRING 2008, EE 102 1. Reformulate y(t) into y ( t ) = Z - U ( t- ) x ( ) d + Z - U ( - t ) e t e- x ( ) d, = Z - [ U ( t- ) + U ( - t ) e t e- ] x ( ) d, Therefore by BT h ( t, ) = U ( t- ) + U (- ( t- )) e t- 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- ) U ( ) d = Z - [ U ( t- ) + U (- ( t- )) e t- ] U ( ) d, = Z - U ( t- ) U ( ) d + Z - U (- t + ) e t- U ( ) d, = U ( t ) Z t 1 d + e t Z max ( t, 0) e- d, = 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, ) = Z t- e- ( t- ) ( - ) d, = U ( t- ) e- ( t- ) , h ( t ) = U ( t ) e- t 1 Let x 1 ( t ) = e- ( t- 1) U ( t- 1) , and x 2 ( t ) =...
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20082ee102_1_HW_3_solution1 - HW3 solution, SPRING 2008, EE...

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