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20092ee102_1_S09-102-HW-2-Sols

20092ee102_1_S09-102-HW-2-Sols - Spring 2009 Put First...

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Unformatted text preview: Spring 2009: Put First Letter of LAST Name in the corner →→%% L (Otherwise Your HW Will Be LOST!) PRINT: (LAST , Middle,First): LEVAN, NMI, Nhan EE102: SYSTEMS & SIGNALS HW: # 2 A LATE HW IS A NON-HW! Posted: April 09 Hand In To Me 1 : April 16 Attach This Sheet To Your HW (Otherwise It Will Be Lost!) 1. Let x ( · ) and y ( · ) denote input and corresponding output of a system S . Given the following IPOP relations: (i) y ( t ) = x ( α t ) , t ∈ ( −∞ , ∞ ) , where α is a real constant. (ii) y ( t ) = R ∞ −∞ sin( τ − t ) U ( τ − t ) x ( τ ) dτ, t ∈ ( −∞ , ∞ ) . Show whether: in case (i) the system is TV while in case (ii) the system is NC. SOLS. We are going to use BT throughout this HW — as much as we can. So the first task is to put any given IPOP relation into the so-called BT-Form, i.e., t ∈ ( −∞ , ∞ ) : x ( t ) −→ [ S : L, h ( t, τ )] −→ y ( t ) y ( t ) = Z ∞ −∞ h ( t, τ ) x ( τ ) dτ (i) First, given the IPOP relation of S : y ( t ) = T [ x ( t )] = x ( αt ) Then S is clearly L (if you do not believe this then do the L test). Secondly, You do not need to consider the trivial case α = 1. Then You have, for non-zero α 6 = 1: y ( t ) = x ( αt ) = Z ∞ −∞ δ ( αt − τ ) x ( τ ) dτ = Z ∞ −∞ δ ( α [ t − τ α ]) x ( τ ) dτ 1 during the break of the Thursday Lecture 1 Therefore, by BT: α 6 = 1 : h ( t, τ ) := δ ( α [ t − τ α ]) 6 = h ( t − τ ) ⇒ S is TV (ii) For the system described by: y ( t ) = Z ∞ −∞ sin( τ − t ) U ( τ − t ) x ( τ ) dτ, t ∈ ( −∞ , ∞ ) = Z ∞ t ← sin( τ − t ) [ U ( τ − t ) = 1] x ( τ ) dτ, t ∈ ( −∞ , ∞ ) Therefore, by BT, S is NC....
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20092ee102_1_S09-102-HW-2-Sols - Spring 2009 Put First...

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