In a nutshell:
Consider the space of functions from $I \mapsto \mathbb{R}$ where $I \subseteq \mathbb{R}$ with the $L2$ norm. I am looking for a function that maps those functions to probability distributions over $\mathbb{R}$ such that, in the limit of an infinite interval, a periodic function is sent to a Dirac delta around its fundamental frequency.
The motivation is pitch detection, but prior to even considering the problem of pitch detection, we must define pitch and doing so in an interesting way is not obvious.
One definition is that pitch is the fundamental, or the reciprocal of the period. This is well defined for an infinite, periodic function, but things are less clear when the signal is observed over a finite window with some noise.t
Suppose now that we are given a real-world signal such as the sound from a note played on an instrument or held by a singer. That signal typically won't be exactly periodic, even an infinitesimal amount of noise prevents that. We would like a notion of pitch that is at least continuous with respect to perturbations of the signal.
Perhaps we can do a Fourier transform of the signal and extract the maximum value? Very well, but that peak could be an harmonic, for instance $s(t) = \sin(t) + (1 + 10^{-10}) \sin(2t)$ has period $2\pi$ not $\pi$. Similarly, what pitch should we attribute to $s(t) = 10^{-10} \sin(t) + \sin(2t)$? Mathematically it's $2\pi$ but $\pi$ is the more reasonable.
To have any hope of continuity, we probably need to express the pitch not a single value but as distribution. We would like this distribution
- to be continuous in the underlying signal
- for its limit to be a Dirac delta around the pitch when the signal is perfectly periodic.
- perhaps to capture a missing fundamental in the presence of telltale harmonics
What's a reasonable definition that would have those properties and, ideally, land itself well to computation?