102_1_MidtermSpring2008Solution

# 102_1_MidtermSpring2008Solution - HW6 solution SPRING 2008...

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HW6 solution, SPRING 2008, EE 102 1. (i)It’s Linear, and Causal. (TI or TV?) Let x ( t ) = δ ( t - τ ) y ( t ) = 1 t + 1 U ( t - τ ) It’s TV. (ii) y ( t ) = 1 t + 1 Z t 0 U ( σ - 1) = 1 t + 1 U ( t - 1)( t - 1) (iii) y ( t ) = 1 t + 1 Z t 0 δ ( σ - 2) = 1 t + 1 U ( t - 2) . 2. y ( t ) = [ e - t U ( t )] * x ( t ), Thus, h 1 ( t ) = e - t U ( t ) For S 2 , h 2 ( t ) = δ ( t ) - U ( t ). h 21 ( t ) = Z -∞ h 1 ( t - σ ) h 2 ( σ ) = [2 e - t - 1] U ( t ) . (ii) BT: y ( t ) = Z -∞ h ( t,τ ) x ( τ ) For TI system, h ( t,τ ) = h ( t - τ ) For the property given in the question: y ( t ) = Z -∞ h ( t - τ ) x ( τ ) = Z 0 -∞ h ( t - τ ) x ( τ ) + Z 1 h ( t - τ ) x ( τ ) 3. (i) Original formula= ee t - 1 δ ( t - 1) + e - 1 e - ( t - 1) U ( t - 1) e - s [ e + e - 1 1 s +1 ] (ii) 1 2 [ 1 s +2 + cos 2 π 3 ( s +2 ( s +2) 2 +4 ) + sin 2 π 3 1 ( s +2) 2 +4 ] 1

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102_1_MidtermSpring2008Solution - HW6 solution SPRING 2008...

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