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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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