# 1 1 ej 4 then we nd the fourier series coecients ak of

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Unformatted text preview: oﬀ frequencies which satisfy (2.1) would be suﬃcient. Let’s say, for practicality sake, that we want to choose the cutoﬀ frequencies in the midpoints between the desired and undesired frequency components. This gives us the following speciﬁc values: �1 = (0 + 1)�0 , 2 �2 = (2 + 3)�0 , 2 � �1 = ω , 4 �2 = 7 �3 = (3 + 4)�0 , 2 5ω , 4 �3 = 7ω . 2 where �0 = ω 2 Problem 3 Consider a causal discrete-time LTI system whose input x[n] and output y [n] are related by the following diﬀerence equation: 1 y [n] − y [n − 1] = x[n] + 2x[n − 4] 4 Find the Fourier series representation of the output y [n] when the input is x[n] = 2 + sin(ωn/4) − 2 cos(ωn/2). First, let’s ﬁnd the frequency response of the system from the diﬀerence equation by injecting an input, x[n], that is an eigenfunction of the LTI system: x[n] = ej �n � y [n] = H (ej� ) ej�n H (ej� ) is the frequency response characterizing the system or the eigenvalue of the system. By substituting x[n] and y [n]...
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## This document was uploaded on 02/09/2014.

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