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20115ee102_1_hw3_soln

# 20115ee102_1_hw3_soln - 1 EE102 Systems and Signals Fall...

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Unformatted text preview: 1 EE102 Systems and Signals Fall Quarter 2011 Jin Hyung Lee Homework #3 Solutions 1. Analytically compute the convolution ( f * g )( t ) , where f ( t ) and g ( t ) are f ( t ) = u ( t ) e- t g ( t ) = rect( t ) and sketch a plot of the result. Solution There are several ways to do this. The most direct, is to write out the convolution integral ( f * g )( t ) = Z ∞-∞ e- τ u ( τ )rect( t- τ ) dτ = Z ∞ e- τ rect( t- τ ) dτ If t <- 1 / 2 , the rect() is completely to the left of the origin, and the integral is zero. If- 1 / 2 < t < 1 / 2 the rect() overlaps the origin, and the limits of the integral are from 0 to t+1/2 ( f * g )( t ) = Z t +1 / 2 e- τ dτ =- e- τ t +1 / 2 = 1- e- ( t +1 / 2) If t > 1 / 2 , then the rect() is to the right of the origin, and ( f * g )( t ) = Z t +1 / 2 t- 1 / 2 e- τ dτ =- e- τ t +1 / 2 t- 1 / 2 = e- ( t- 1 / 2)- e- ( t +1 / 2) = e- t ( e 1 / 2- e- 1 / 2 ) . This is plotted below. − 2 − 1 1 2 3 4 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time Amplitude 2. Show the somewhat surprising result that the convolution of two impulse functions, y ( t ) = Z ∞-∞ δ...
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20115ee102_1_hw3_soln - 1 EE102 Systems and Signals Fall...

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