Why does the sum of n·sin[4πf₀n+2φ] approximate zero?

[sauravrt]

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

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

$$f_o$$ is not near 0 or 1/2

Saurav

 

[mikeph]

Perhaps it's because you are averaging over a sin function, which has average 0?

 

[CRGreathouse]

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).