# Frequency identification as a convex problem

Suppose I have two time series, $x(t)$ and $y(t)$: $x$ is the output of a sine wave generator, and $y$ is the generator's frequency setting. Given a training set of $(x, y)$ pairs, Is it possible to formulate a convex/quasi-convex optimization problem to build a predictor of some new $y$ given $x$?

The system is certainly non-linear, since you can double the signal amplitude without changing the output (the frequency). It seems like you need to perform an argmax in the frequency domain, which I believe is a quasi-convex operation. This limits what functions you can compose, so something like computing residuals may no longer be convex. My hunch though is there may be a clever recasting of the problem, or a duality argument, etc.

I'm sure neural nets are capable of it, but for sake of discussion assume I don't have the data/processing power.

• See Part III section 1.1 – Peter K. Apr 6 '17 at 19:39
• @PeterK. ha, that will make a fine addition to my references library – Marcus Müller Apr 6 '17 at 20:38
• @PeterK. this looks great - thanks for posting! I don't have enough rep to upvote your comment but consider this a +1 – RedPanda Apr 6 '17 at 23:32
• @MarcusMüller it's a little old. Like 21+ years... 🙀 – Peter K. Apr 7 '17 at 12:46
• @RedPanda no worries. I would do a longer actual answer, but I'm traveling and don't have easy access to a laptop... and hate composing actual answers on mobile. – Peter K. Apr 7 '17 at 12:47