20092ee102_1_S09-102-HW-1-Sols

20092ee102_1_S09-102-HW-1-Sols - Spring 2009 Put First...

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Spring 2009: Put First Letter of LAST Name in the corner →→%% ( Otherwise Your HW Will 1 Be LOST! ) PRINT: (LAST , Middle, First) Nhan, NMI, LEVAN HW: # 1 A LATE HW IS INDEED A NON-HW! Posted: April 2 Hand In To NNMIL April 9 Attach This Sheet To Your HW Notations := means “equal be de±nition” x [ a, b ) means a x < b cm:= check me 1. Calculate: I := Z −∞ e t U ( t + 1) t U ( t 1) dt I ( t ) = Z −∞ [ δ ( t τ ) e ( t τ ) ] U ( t τ ) e τ U ( τ ) SOLS. This Q is not that hard. One needs to get rid of the USF (Unit Step Function) in the integrand. This will result in, possibly, ±nite limits for the integrals. To do this it is helpful to sketch the USF ±rst! You ±nd: I := Z −∞ e t U ( t + 1) t U ( t 1) dt = Z 1 e t 1 t 1 dt = 2 e 1 I ( t ) := Z −∞ [ δ ( t τ ) e ( t τ ) ] U ( t τ ) e τ U ( τ ) = e t U (0) U ( t ) Z −∞ e ( t τ ) U ( t τ ) e τ U ( τ ) 1 intentionally 1
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Now if you plot U ( t τ ) U ( τ ) as a function of τ you will see that For t 0: Z −∞ e ( t τ ) U ( t τ ) e τ U ( τ
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This note was uploaded on 03/29/2011 for the course ELECTRICAL EE102 taught by Professor Levan during the Spring '09 term at UCLA.

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20092ee102_1_S09-102-HW-1-Sols - Spring 2009 Put First...

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