20081ee102_1_Re-HW-2-SOLS - WINTER 2008 Put First Letter of...

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WINTER 2008: Put First Letter * of LAST Name in the corner →→ (* Otherwise Your HW will be LOST) PRINT: (LAST, Middle, First): Nhan N.M.I. LEVAN EE102: SYSTEMS & SIGNALS HW: # 2 SOLUTIONS LATE HW Solutions Are STILL HW Solutions! Posted: 01/24/2008 1. Given the following IPOP relations of two L systems: (i) y ( t ) = x (4 t - 5) , t ( -∞ , ) S 1 (ii) y ( t ) = -∞ e ( τ - t ) U ( τ - t ) x ( τ ) d τ , t ( -∞ , ) S 2 Which system is TI, which one is C? ———————————————————————————————— (i) S 1 : x ( t ) -→ [ S 1 ] -→ y ( t ) = x (4 t - 5) Note: The action of S 1 is to “multiply” t (of x ( t )) by 4 then to “substract” 5 from 4 t . x ( t - A ) -→ [ S 1 ] -→ z ( t ) = x (4 t - A - 5) y ( t - A ) = x ( 4( t - A ) - 5 ) z ( t ) = y ( t - A ) S 1 : TV (ii) S 2 : x ( t ) -→ [ S 2 ] -→ y ( t ) = -∞ e ( τ - t ) U ( τ - t ) x ( τ ) d τ , t ( -∞ , ) y ( t ) = -∞ [ e - ( t - τ ) U ( - ( t - τ ))] x ( τ ) d τ , t ( -∞ , ) S 2 : L: BT: h 2 ( t - τ ) = e - ( t - τ ) U ( - ( t - τ )) TI 1
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S 2 : C, the lower limit t : y ( t ) = t e - ( t - τ ) 1 x ( τ ) d τ , t ( -∞ , ) 2. (i) Write down the IRF h ( t, τ ) of the system of Problem 1(i). (ii) Given the IRF h ( t ) of a L,TI,C system: h ( t ) = t e - t U ( t ) , t ( -∞ , ) , and the input x ( t ) = δ ( t - 4) - ( t - 4) U ( t - 4) , t ( -∞ , ) . Find the corresponding output y ( t ). —————————————————————————————————– (i) y ( t ) := x (4 t - 5) = -∞ δ (4 t - 5 - τ
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