Suppose is a complex-valued signal that repeats every samples. (We are continuing to use complex-valued signals rather than real-valued ones to simplify the mathematics.) Because of the period , the values of for determine for all integer values of .
Suppose further that can be written as a sum of complex sinusoids of frequency , , , , . These are the partials, starting with the zeroth, for a signal of period . We stop at the th term because the next one would have frequency , equivalent to frequency , which is already on the list.
Given the values of , we wish to find the complex amplitudes of the partials. Suppose we want the th partial, where . The frequency of this partial is . We can find its complex amplitude by modulating downward radians per sample in frequency, so that the th partial is modulated to frequency zero. Then we pass the signal through a low-pass filter with such a low cutoff frequency that nothing but the zero-frequency partial remains. We can do this in effect by averaging over a huge number of samples; but since the signal repeats every samples, this huge average is the same as the average of the first samples. In short, to measure a sinusoidal component of a periodic signal, modulate it down to DC and then average over one period.
Let be the fundamental frequency for the period , and
let be the unit-magnitude complex number with argument :