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20085ee102_1_F08-102-HW-3

20085ee102_1_F08-102-HW-3 - FALL 2008 Put First Letter of...

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FALL 2008: Put First Letter of LAST Name in the corner →→ ( Otherwise Your HW May Be LOST! ) PRINT: (LAST , Middle, First):——————————————————– EE102: SYSTEMS & SIGNALS HW: # 3 A LATE HW CANNOT BE A HW! Posted: W, October 15 Hand In To Me 1 : M, October 27 Attach This Sheet To Your HW (Otherwise It May Be Lost!) 0. 2 The IPOP relation of a system S is y ( t ) = t -∞ x ( τ ) d τ + t e - ( t - τ ) x ( τ ) d τ , -∞ < t < . Write down the IRF h ( t ) and the USR (Unit Step Response) g ( t ) of the sys- tem. SOLS. Rewrite y ( t ) in the BT form: y ( t ) = -∞ U ( t - τ ) x ( τ ) d τ + -∞ e - ( t - τ ) U ( τ - t ) x ( τ ) d τ , t R, = -∞ [ U ( t - τ ) + e - ( t - τ ) U ( τ - t )] x ( τ ) d τ . Therefore by BT h ( t, τ ) = U ( t - τ )+ e - ( t - τ ) U ( - ( t - τ )) = h ( t - τ ) h ( t ) = U ( t ) + e - t U ( - t ) . By definition U ( t ) -→ [ S ] -→ g ( t ) . 1 during the break of Monday Lecture 2 This Problem is for your eyes only 1
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Therefore, by BT, g ( t ) = -∞ h ( t - τ ) U ( τ ) d τ , = -∞ [ U ( t - τ ) + e - ( t - τ ) U ( - ( t - τ )] U ( τ ) d τ , = UDIP — You Do It Please .
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