13 x j xt e j dt 0 et

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Unformatted text preview: 2 otherwise The question mark in the above expression indicates that we don’t have enough in­ formation to determine the system’s frequency response at those frequencies. This is simply b ecause the input that excited the system did not contain those frequencies. (II) Consider equations (1) and (2) to be the synthesis equations for x[n] and y [n], respec­ tively, and assume the fundamental frequency, �0 , to be the Greatest Common Factor for the sinusoids frequencies, i.e. �0 = � and N = 8. 4 1 j�n/2 1 −j�n/2 1 j�n 1 −j�n − e + e + ej�n/4 ej�/4 + e−j�n/4 e−j�/4 e +e 2 2j 2j 2 �� �� �� �� 1 j (−4)�0 n −1 j (2)�0 n 1 1 j (4)�0 n j (0)�0 n e + e + e + ej (−2)�0 n = (2) e + 2 2j 2j 2 ⎩ ⎩ � � + ej�/4 ej (1)�0 n + e−j�/4 ej (−1)�0 n 1 1 y [n] = 4ej 0 − ej�n + e−j�n + ej�n/4 + e−j�n/4 j j j (0)�0 n = (4) e + (j ) ej (4)�0 n + (−j ) ej (−4)�0 n + 0 + 0 + (1) ej (1)�0 n + (1) ej (−1)�0 n , x[n] = 2ej 0 + where the integer b etween parenthesis in the exponential corresponds to the index k and the number b etween parenthesis in front of the exponential is the corresponding ak and bk for...
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This document was uploaded on 02/09/2014.

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