_Signal_FAQ20_Number_of_Sinusoids_in_DFS

_Signal_FAQ20_Number_of_Sinusoids_in_DFS - complex. Thus,...

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Why is it that we just need to have 3 sinusoids to represent a periodic digital signal with period 3? Consider an analog periodic signal ( 29 t x a with a period of, say, 4 seconds. To represent this analog signal using Fourier series, we need an infinite number of complex sinusoids. Fundamentally, each complex sinusoid is characterized by only a complex Fourier coefficient which gives the amplitude and phase of the sinusoid. However, over a period of ( 29 t x a , say, from 0 to 4 seconds, there are an infinite number of signal values for ( 29 t x a . Essentially, ( 29 t x a is a continuous function of time and specifying ( 29 t x a uniquely over any finite interval will require an infinite number of values. Thus, it is to be expected that an infinite number of sinusoids is needed in the Fourier series. Now consider a periodic digital signal ( 29 n x , sampled at 1 Hz, that has a period of only 3 samples. The signal has 3 only unique values, ( 29 0 x , ( 29 1 x and ( 29 2 x , each of which is in general
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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.

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_Signal_FAQ20_Number_of_Sinusoids_in_DFS - complex. Thus,...

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