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Unformatted text preview: complex. Thus, the Fourier representation will need only 3 complex sinusoids with 3 complex coefficients X , 1 X and 2 X . Essentially, we can always choose these coefficients such that the Fourier sum ( 29 n j n j n j e X e X e X n x 3 2 2 2 3 1 2 1 3 2 π + + = matches ( 29 x , ( 29 1 x and ( 29 2 x at = n , 1 and 2 exactly. This is possible because the exercise is like solving 3 linear equations with 3 unknowns. One and only one solution exists. In fact, having another sinusoid, n j e X 3 3 2 3 to the series is useless and does not help at all. This is because n j n j j n j n j e e e e e 3 2 3 3 3 2 2 3 3 2 3 3 2 = = =and ( 29 n j n j n j n j n j n j n j e X e X e X X e X e X e X e X 3 2 2 2 3 1 2 1 3 2 3 3 3 2 3 3 2 2 2 3 1 2 1 3 2 + + + = + + + Thus, the effect of n j e X 3 3 2 3 π can be absorbed in the sinusoid n j e X 3 3 2 ....
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This note was uploaded on 08/02/2009 for the course ECE EE2009 taught by Professor Prof.c.c.ko during the Fall '07 term at National University of Singapore.
 Fall '07
 Prof.C.C.Ko

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