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