05 0052 recall that 1 log 0 2 log log005

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Unformatted text preview: − (�2 ) −1 2 1 − �2 2 − 2�2N +2 1 − �2 2 − 2�2N +2 1 − �2 2 − 2�2N +2 2 − 2�2N +2 �2N +2 (2N + 2) log(�) → → → → → ← ← N +1 → N→ � � 2 R −1 1 − �2 � � 2R −R+1 1 − �2 2R + (1 − R)(1 − �2 ) 1 − �2 R + 1 − (1 − R)�2 (� 1 − �2 > 0) 1.9 − 0.1�2 (plugging in R=0.9) 2 0.05 + 0.05� (simplifying a bit) 2 log(0.05 + 0.05� ) log(0.05 + 0.05�2 ) (recall that � < 1 � log(�) < 0) 2 log(�) log(0.05 + 0.05�2 ) −1 2 log(�) After choosing an integer N that satisfies the inequality above, W can be chosen such that W → N �0 . 2 Problem 1 Consider the LTI system with impulse response given in O&W 3.34. Find the Fourier series representation of the output y (t) for the following input. x(t) 2 ··· −2 −5 −4 −3 −1 ··· 4 1 2 3 5 6 t −1 From O & W 3.34, the impulse response of the LTI system is: h(t) = e−4|t| . From the figure above, we can see that x(t) has a period T = 3 � �0 = 2� . 3 First, we calculate...
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This document was uploaded on 02/09/2014.

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