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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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