# Detecting periodic components in discrete-time signals [closed]

I am looking for real data examples where it is of interest to detect the presence of a periodic component of some discrete-time signal when the period of the periodic component is not known.

Suppose that $$\{X_t:t\in\mathbb Z\}$$ is a discrete-time signal (univariate or possibly multivariate) and consider the situation where this signal can be modeled as $$X_t=s_t+\varepsilon_t$$ for each $$t\in\mathbb Z$$, where $$\{s_t:t\in\mathbb Z\}$$ is a deterministic periodic component such that $$s_t=s_{t+p}$$ for all $$t\in\mathbb Z$$ and $$\sum_{t=1}^ps_t=0$$ with some period $$p\ge2$$ while $$\{\varepsilon_t:t\in\mathbb Z\}$$ is some stationary random noise sequence. We observe $$X_1,\ldots,X_n$$ and we want to detect the presence of the deterministic periodic component $$\{s_t:t\in\mathbb Z\}$$, i.e. we want to know whether $$X_t=\varepsilon_t$$ for $$t\in\mathbb Z$$ or $$X_t=s_t+\varepsilon_t$$ for $$t\in\mathbb Z$$ with some period $$p\ge2$$.

In many situations, we expect the presence of certain periodic components and hence it is not reasonable to assume that $$p$$ is unknown. For example, if we record temperature, then it is clear that we should expect to see daily and yearly periodic components. It does not make a lot of sense to check, say, for a three day periodic component. In many situations, some theory suggests that there should be some periodic components present and then it makes sense to try to detect these particular periodic components instead of assuming that the period is unknown.

However, I suspect that there are examples (maybe in engineering or astronomy) where it is not clear whether the periodic component should be present and what period of the periodic component we should expect to detect.

I am looking for real data examples (preferably multivariate but univariate examples are also of interest) where it is not clear whether the periodic component should be present and where it is not clear what period we should expect to detect. References to textbooks, papers or websites are very welcome.

I have no background in signal processing and my question or phrasing might seem strange. Please let me know if I should clarify some parts of my question.

Any help is much appreciated!

• The phrasing seems fine, the question is surprising / not overly sensible: Anything can be useful in signal processing, and detection of periodicities is practically the same as spectral estimation, and has myriads of applications and algorithms. Does that actually answer your question? Could you maybe add a bit of explanation how you come to this question? – Marcus Müller Dec 30 '20 at 18:12
• @MarcusMüller I am playing around with some statistical methodologies that detect periodicities in multivariate discrete-time signals (or multivariate time series). We observe $\{x_t:t\in\mathbb Z\}$ and we want to know if there any periodic components present in the signal. I am looking for examples of situations where such questions are of interest so that I could test different methodologies and see how they work. I hope this makes my question a bit clearer. Would you have any particular data examples in mind? – Cm7F7Bb Dec 30 '20 at 18:24
• ah, so you're actually asking for methods to detect periodicities to compare to your own? – Marcus Müller Dec 30 '20 at 18:37
• hm, yeah, this is the spectral estimation problem, yes. You're asking for the presence of an unknown periodic signal. This happens in thousand and one place, really. I could write something about detection probabilities in Q-tone multicarrier systems, about pulse-position modulation, about at least four kinds of radar, about hearbeat detection in search and rescue, about pulsar observations, about… Again, shelves of books on spectral estimation in libraries. Could you maybe tell us a bit about your actual your field or background? Are you a particle physicist, a geologist, an astronomer, a – Marcus Müller Jan 5 at 19:36
• Ah! Statistics! OK, then Steven Kay's Fundamentals of Statistical Signal Processing: Detection theory is possibly a book you've seen! I like it. What might be relevant to your problem might also be Detection of Signals in Noise by McDonough; I don't know whether that is well-available, though. From statistics' origin, communications theory takes a few cool parametric estimators. I can only imagine you might like the MUSIC algorithm (in its spectral, not its Direction-of-Arrival version); also, try ROOT-MUSIC. – Marcus Müller Jan 5 at 21:23