20085ee102_1_HW-2-SOLS

20085ee102_1_HW-2-SOLS - 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): LEVAN, nmi, Nhan EE102: SYSTEMS & SIGNALS HW: # 2 A LATE HW CANNOT BE A HW! Posted: W, October 8 Hand In To Me 1 : M, October 20 Attach This Sheet To Your HW (Otherwise It May Be Lost!) 1. Let x ( · ) and y ( · ) denote input and corresponding output of a system S , respectively. Given the following IPOP relations: (i) y ( t ) = x (2 t - 1) , t ( -∞ , ) , (ii) y ( t ) = ± -∞ e - τ ( t - τ ) U ( t - τ ) x ( τ ) d τ , t ( -∞ , ) . In each case verify whether S is: L, TI, Causal . —————————————————— SOLS. (i) S is clealry L. To see this, first you know that by definition of y ( t ): y ( t ) = T [ x ( t )] = x (2 t - 1) — i.e., to obtain y ( t ) you replace t of x(t) by (2 t - 1). Therefore, you can write T [ ² x ( t ) := ax 1 ( t ) + bx 2 ( t )] = ² x (2 t - 1) = a x 1 (2 t - 1) + b x 2 (2 t - 1) = a T [ x 1 ( t )] + b T [ x 2 ( t )] , t R . This proves Linearity! S is also TV. Let us do the TI test: y ( t ) = T [ x ( t )] = x (2 t - 1) y ( t - A ) = x (2[ t - A ] - 1) = x (2 t - 2 A - 1) . 1 during the break of Monday Lecture 1
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z ( t ) = T [ x ( t - A )] = x ( 2t - 1 - A ) Therefore z ( t ) ± = y ( t - A ) S is TV . How about C or NC ? It follows from the IPOP relation y ( t ) = x (2 t - 1) , t ( -∞ , ) that the OP at time t, i.e., y ( t ) is the value of the IP at time 2 t - 1, i.e., x (2 t - 1). Hence if t < 2 t - 1 t > 1 then S is NC. However, if 2 t - 1 t t 1 then S is C. (ii) y ( t ) = ± -∞ [ e - τ ( t - τ ) U ( t - τ )] x ( τ ) d τ , t ( -∞ , ) . This is the place to take advantage of the “Power of BT”! The given IPOP relation is already in the BT form. Therefore, since the system is L (do you believe this?) we can “lift” the IRF h ( t, τ ) right out of the “box”, i.e., the IPOP relation. We find h ( t, τ ) = e - τ ( t - τ ) U ( t - τ ) , t, τ ( -∞ , ) which is NOT a function of ( t - τ ), therefore S is TV. S is clearly C because of the term U ( t - τ ) we can rewrite its IPOP relation as y ( t ) = ± t -∞ [ e - τ ( t - τ ) 1 ] x ( τ ) d τ , t > -∞ .
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This note was uploaded on 04/03/2009 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

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20085ee102_1_HW-2-SOLS - FALL 2008: Put First Letter of...

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