How is this summation approx valid?

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SUMMARY

The approximation \(\frac{1}{N}\sum_{n=0}^{N-1}n\sin[4\pi f_o n + 2\phi] \approx 0\) is valid when \(f_o\) is not near 0 or 1/2. This is primarily due to the averaging effect of the sine function, which oscillates between positive and negative values, leading to cancellation. Saurav confirms that unless specific conditions are met, the result will be small, likely not exceeding \(O(N^{-1/2})\), but certainly tending towards \(o(1)\) for randomly chosen \(f_o\) and \(\phi\).

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I came across this approximation in a book. I am not sure why this approximation is valid..

[tex]\frac{1}{N}\sum_{n=0}^{N-1}n.sin[4\pi f_o n + 2\phi] \approx 0[/tex]

[tex]f_o[/tex] is not near 0 or 1/2

Saurav
 
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Perhaps it's because you are averaging over a sin function, which has average 0?
 
Pretty much. Unless something funny is going on, the positives and the negatives should pretty well cancel out.

If it is -- if you manage to get sin = 1 at each point -- you can get as high as (N - 1)/2. But for randomly-chosen f_0 and phi, I'd expect to get something pretty small. Maybe not O(N^(-1/2)), but certainly o(1).
 

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