Discrete Fourier series

Given `2N` samples of real data (where j `= 0;1;2;...;2N-1`) taken at equally spaced intervals , where T is the period, the corresponding Fourier series is:

where and is the mean value of .

From the original `2N` data points, we have exactly `2N` calculated coefficients, that is, we have `N+1` 's and `N-1` 's with "real" information.

If t`=0` at , then for each we have , for j `= 0;1;...;2N-1`, and so

The amplitude, A, and the phase, P, are calculated as follows:

Fourier coefficients
Restrictions