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 :

The th partial of the signal is of the form:

where is the complex amplitude of the partial, and the frequency of the partial is:

We're assuming for the moment that the signal can actually be written as a sum of the partials, or in other words:

By the heterodyne-filtering argument above, we expect to be able to measure each by multiplying by the sinusoid of frequency and averaging over a period:

This is such a useful formula that it gets its own notation. The

where . The Fourier transform is a function of the variable , equal to times the amplitude of the input's th partial. So far has taken integer values but the formula makes sense for any value of if we define more generally as:

where, as before, is the (angular) fundamental frequency associated with the period .