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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.

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    $\begingroup$ 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. $\endgroup$ Nov 14 '15 at 14:25
  • $\begingroup$ Hello and i'm wondering have you found any solution to generate subharmonics? $\endgroup$
    – ZR Han
    Jun 9 '21 at 2:10
  • $\begingroup$ @ZRHan just posted some stuff in an answer below $\endgroup$ Jun 9 '21 at 20:25
  • $\begingroup$ @MikeBattaglia Thanks a lot! $\endgroup$
    – ZR Han
    Jun 11 '21 at 1:32
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Check out this paper. I would have made a comment but not high enough rep.

http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1211087&url=http%3A%2F%2Fieeexplore.ieee.org%2Fiel5%2F81%2F27258%2F01211087

Looks like you need multiple in to get subharmonics in Volterra series

The abstract states "Subharmonic generation is a complex nonlinear phenomenon which can arise from nonlinear oscillations, bifurcation and chaos. It is well known that single-input-single-output Volterra series cannot currently be applied to model systems which exhibit subharmonics. A new modeling alternative is introduced in this paper which overcomes these restrictions by using local multiple input single output Volterra models. The generalized frequency-response functions can then be applied to interpret systems with subharmonics in the frequency domain."

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  • $\begingroup$ Thanks, although I'm a bit confused. When we talk about a MISO system, which input are we generating subharmonics of? Both? I can't see the paper, so I'm a bit confused here. $\endgroup$ Nov 19 '15 at 9:26
  • $\begingroup$ full disclosure - I have never done MISO Volterra as I can limit my analysis that I do to SISO but I would imagine that you could look at analygous to a subharmonic mixer if you have ever came across one. In this case, a low frequency tone is mixed with a high frequency one, say something like LO = 10 GHz and RF = 100.1 GHz, the mixer product, IF would be at multiple subharmonics but if you wanted the 10 harmonic of the LO, IF would be 100 MHz ( this is done a lot with millimeter wave systems) - but really, I would read the paper I linked here. $\endgroup$
    – johnnymopo
    Nov 20 '15 at 3:05
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Many years later I was asked to write some of what I've learned on this. The short answer is that it depends on what the term "subharmonics" means - things are very different if we're looking at bilateral signals or unilateral ones. So this is a partial answer

In a very strict sense it's not possible to generate subharmonics using any time-invariant system, linear or nonlinear. Since Volterra series represent nonlinear time-invariant systems, they can't be used to generate subharmonics either.

The proof is pretty straightforward. Let's say we have a time-invariant system $F[x(t)](t)$ which maps $\sin(t)$ to $\sin(t/2)$. Then because it's time-invariant, we have

$$F[\sin(t+2\pi)](t) = F[\sin(t)](t+2\pi)$$

Which means that, if it maps $\sin(t)$ to $\sin(t/2)$, then with time-invariance we get

$$F[\sin(t+2\pi)](t) = \sin((t + 2\pi)/2) = \sin(t/2 + \pi)$$

But, we have that $\sin(t+2\pi)$ = $sin(t)$ to begin with, so we get

$$F[\sin(t+2\pi)](t) = F[\sin(t)](t) = \sin(t/2)$$

So we'd have $\sin(t/2) = \sin(t/2 + \pi)$, which isn't possible. So no, in general, you can't get subharmonics with a nonlinear time-invariant system.

However, of course, in the "real world" we typically don't use bilaterally infinite signals, but rather one-sided causal signals which are identically $0$ prior to $t=0$, e.g. we care about signals like $\sin(t) u(t)$ where $u(t)$ is a Heaviside step function. Things are much subtler in this situation, so this is just a partial answer.

There does theoretically exist a nonlinear time-invariant system which sends $\sin(t) u(t)$ to $\sin(t/2) u(t/2)$ - a trivial example is to wait for the first nonzero value and then play everything after that at half speed. Such a thing would be difficult to represent as a Volterra series, but it's a simple example of a nonlinear time-invariant system that can generate "one-sided subharmonics." Such a system doesn't even make sense for bilateral signals (as there is no first nonzero value), but if you only care about causal signals, then this kind of thing becomes possible.

For something less contrived, any of the available time/pitch stretching algorithms, such as a phase vocoder, would seem to be able to generate subharmonics (simply by transposing the pitch down an octave without doing any timescale stretching). If the hop size is one sample (or if we're looking at an analog system with a sliding windowed short-time Fourier transform), such things are also technically time-invariant. Again, such systems are typically thought of in terms of one-sided signals - it's very interesting to think of how something like a phase vocoder would even work with something like a bilaterally infinite sinusoid. What would the phase be at $t=0$?

The point of all that is that there is some subtlety regarding the math; for bilateral signals the answer is clear, but for unilateral ones there's some interesting stuff going on that would make it possible to generate "one-sided subharmonics" for "one-sided signals." For analytic functionals, I'm not sure what the corresponding Volterra series would look like, though.

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