# Is it possible to use a Volterra series to generate subharmonics?

Taylor series can create harmonics of the frequencies in an input signal. I'm wondering if it is likewise possible to use Volterra series to create sub-harmonics of the frequencies in that signal.

Some degree of finesse is required in posing the question:

Taylor series don't only create harmonics; there's intermodulation distortion as well. Likewise, it's ok with me if a suitable Volterra series doesn't only create sub-harmonics, but also creates something similar to intermodulation distortion, or even some degree of harmonic distortion.

However, I would like to exclude trivial cases where the input signal gets so wrecked that it ends up looking like white noise, which would obviously create sub-harmonics (along with every other frequency you can imagine!). I'm not sure how to formalize this requirement precisely, other than to say that I hope it's clear what I'm driving at.

Here's one way to formalize the behaviour I want:

1. Given an input $\cos(\omega t)$, yields an output containing $\cos(\frac{\omega}{n} t)$ for some fixed $n$, ideally a natural number
2. Given a general input, a sub-harmonic is generated corresponding to every frequency in the input
3. Other "intermodulation distortion"-type artifacts are allowed, "within reason"

The answer can be expressed for either discrete or continuous signals.

• interesting question. i don't know for sure, but i suspect that you cannot generate sub-harmonics without a time-varying component in the algorithm. Volterra is non-linear, but is also time-invariant. – robert bristow-johnson Nov 14 '15 at 14:25