As the integer k increases by one the complex vectors

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Unformatted text preview: in the difference equation: 1 y [n] − y [n − 1] = x[n] + 2x[n − 4] 4 H (ej� ) ej�n − 1 · H (ej � ) ej�(n−1) = ej�n + 2 · ej �(n−4) 4 1 · H (ej � ) ej�n ej�(−1) = ej�n + 2 · ej �n ej�(−4) 4 � � � � 1 −j� j� j�n = ej �n 1 + 2 e−j�4 H (e ) e 1− e 4 H (ej� ) ej�n − � H (ej� ) = 1 + 2 e−j�4 . 1 − 1 e−j� 4 Then, we find the Fourier series coefficients, ak , of the given input, possibly by dissecting the input expression into a summation of complex exponentials: x[n] = 2 + sin(ωn/4) − 2 cos(ωn/2) = 2 + sin(�0 n) − 2 cos(2�0 n), ω where �0 = is the Greatest Common Factor for the sinusoids frequencies 4 ej�0 n − e−j�0 n ej 2�0 n + e−j 2�0 n = 2 ej (0)�0 n + −2· 2j 2 1 j (1)�0 n 1 j (−1)�0 n = 2 ej (0)�0 n + e − e − ej (2)�0 n − ej (−2)�0 n 2j 2j � N = 8 � ak has only eight distinct values and is periodic with a period of N = 8. 8 � ⎧−1, ⎧ ⎧ ⎧−...
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This document was uploaded on 02/09/2014.

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