102_1_102-MT-SOLS

102_1_102-MT-SOLS - UCLA DEPARTMENT OF ELECTRICAL...

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UCLA DEPARTMENT OF ELECTRICAL ENGINEERING Winter 2008 Your Name (LAST, Middle, First): LEVAN n.m.i. Nhan (Fill in the following lines — if applicable) NAME OF BEST FRIEND ON YOUR LEFT : NAME OF BEST FRIEND ON YOUR RIGHT : EE102: SYSTEMS & SIGNALS February 6, 2008 MIDTERM EXAMINATION SOLUTIONS Instructions: (i) Closed Book, Calculators Are NOT Allowed (ii) Please bring your own LOOSE papers. Write On ONE Side Only (-5 points for each two-sided page) (iii) Staple These Examination Papers With Your Papers — using your own stapler (iv) Questions are equally weighted 1
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QUESTION 1 t-Domain Analysis (Do NOT use Laplace Transforms ) (i) The IPOP relation of a SISO system S is: x ( t ) -→ [ S ] -→ y ( t ) y ( t ) = t -∞ x ( τ ) d τ + t e t e - τ x ( τ ) d τ , t ( -∞ , ) Find the IRF h ( t , τ ) of S , then state all properties of the system (no proof is required). Find the USR (Unit Step Response) g ( t ) of S . —————————————- SOLS (i) y ( t ) = t -∞ x ( τ ) d τ + t e t e - τ x ( τ ) d τ , = -∞ [ U ( t - τ ) + e ( t - τ ) U ( - ( t - τ ) ) ] x ( τ ) d τ h ( t - τ ) = U ( t - τ ) + e ( t - τ ) U ( - ( t - τ ) ) (by BT) h ( t ) = U ( t ) + e t U ( - t ) S : L, TI, NC By definition: g ( t ) = t -∞ U ( τ ) d τ + t e t e - τ U ( τ ) d τ , = t 0 1 d τ + e t t e - τ U ( τ ) d τ t 0 1 d τ = t U ( t ) e t t e - τ U ( τ ) d τ = e t 0 e - τ 1 d τ = e t , for t < 0 = e t t e - τ 1 d τ = 1 , for t 0 2
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Finally g ( t ) = ( 1 + t ) U ( t ) + e t U ( - t ) —————————————– (ii) Compute the OP y ( t ) of S given that its input x ( t ) is x ( t ) = U ( t - 4 ) , t ( -∞ , ) —————————————- SOLS (ii) Since S is TI we can have x ( t ) = U ( t - 4 ) -→ [ S ] -→ g ( t - 4
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