# Periodicity and Harmonicity

A periodic signal has a harmonic spectrum. In the extreme case, a click train signal has a comb spectrum:

But why? After all, a solitary click has a uniform spectrum. So why should the sum of multiple clicks have a non-uniform spectrum?

The answer exposes a lot of detail of how spectra work, and gives us a glimpse into the inner workings of spectral phases. Let's start with that click: In that spectrum, we see that we can decompose a click into a sum of sine waves:

As we have seen in the spectrum of the click earlier, it is composed of all frequencies (uniform spectrum). But we see here that the individual sine waves are delayed *just so*, that they all add constructively at exactly one time, and form the click. At all other times, they cancel each other out. This per-sine delay is called the phase of that frequency.

But what happens if we have more than one click? How does that change things?

In a click train, odd frequencies in the Fourier series of the individual clicks cancel each other out (red/blue), and only harmonic frequencies (brown) remain. And this is exactly what our first spectrum showed: A periodic click train results in a harmonic spectrum.

Even though each click has a uniform spectrum, adding multiple clicks together cancels out all non-harmonic parts, and only a harmonic comb spectrum remains.