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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