# 2 y t 1 cos 3 t 2 1 ej 3 t 2 1 2 e j 2 3 t 2 1

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Unformatted text preview: oblem 2 The p eriodic triangular wave shown below has Fourier series coeﬃcients a k . x(t) 1 ··· ··· −4 −2 0 2 4 t � ⎧2 sin(kω/2) e−jk�/2 , � j (kω )2 ak = ⎧1 �, 2 k=0 ≤ k = 0. Consider the LTI system with frequency response H (j� ) depicted below: H (j� ) A2 x(t) H (j� ) y (t) A1 A3 −�3 −�2 −�1 �1 �2 �3 � Determine values of A1 , A2 , A3 , �1 , �2 , and �3 of the LTI ﬁlter H (j� ) such that � � 3ω y (t) = 1 − cos t. 2 At the b eginning, it is worth noting that the output y (t) contains only a DC component and � a single sinusoid with a frequency of 32 . H (j� ) is a linear system so the output will only have frequency components that exit in the input. Knowing that the input x(t) has a DC component and a fundamental frequency of �0 = � , let’s dissect y (t) into a DC component 2 5 and complex exponentials with a fundamental frequency of �0 = � . 2 y (t) = 1 − cos � 3ω t 2 � =1− ej 3� t 2 1 − 2 e −j 2 3� t 2 1 j (3) � t 1 j (−3) � t 2 e 2− e 2 2...
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## This document was uploaded on 02/09/2014.

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