20075ee102_1_SolutionsMidTermI

# 20075ee102_1_SolutionsMidTermI - EE 102 – Solutions for...

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Unformatted text preview: EE 102 – Solutions for Midterm I Problem 1: 1. The answer is yes. Recall that the system S 1 is memoryless if there exists a function f : Reals → Reals + satisfying S 1 ( x )( t ) = f ( x ( t ) ) . In this case we have f ( a ) = a 2 + a +1 for any a ∈ Reals . 2. The answer is no. The system S defined the diagram in Figure 1 is given by S = S 2 ◦ S 1 . Thus for any x ∈ [ Reals → Reals ] and for any t ∈ Reals we have: S ( x )( t ) = p x 2 (- 2 t ) + x (- 2 t ) + 1 Let us take, for example, t = 1. Then S ( x )(1) = p x 2 (- 2) + x (- 2) + 1. This shows that for t = 1 we cannot write S ( x )(1) in the form f ( x (1) ) , for some function f : Reals → Reals , since S ( x )(1) depends on the value of x for t =- 2. 3. The answer is no. The system S defined the diagram in Figure 2 is given by S = S 2 ◦ S 3 . But this composition cannot be formed since the image of S 3 is Reals and thus not contained in the domain of S 2 which is Reals + . Problem 2: 1. States = { q 1 ,q 2 ,q 3 } , Inputs = { , 1 } and Outputs = {* , , 1 } ....
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20075ee102_1_SolutionsMidTermI - EE 102 – Solutions for...

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