I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, in frequency domain it's a Gaussian spectrum.


I want to explore more on the frequency spectrum estimation of this kind of signal when there are not enough samples or there are samples with gaps in between sets of observation or random sampling cases. In these cases, the existing DFT or correlation techniques don't work very well.


I have come across many frequency estimation techniques such as [1]. The techniques like this can estimate the discrete frequencies present in a signal when limited data is available. However, for a continuous broad band spectrum, these techniques can fail as the computational complexity is extremely high.


I have 2 questions.


1. How many coherent samples in observation is enough to characterize a Gaussian distribution in frequency? Should it be estimator dependent (I suppose not). If it's not estimator dependent, how to find this number?
2. Is it useful to get a broader perspective on harmonic analysis based on some functional analysis? I'm very curious to get a broader perspective to such estimators. Will it be helpful for my current problem?


[1] C. Andrieu and A. Doucet, “Joint bayesian model selection and estimation of noisy sinusoids via reversible jump MCMC,” IEEE Transactions on Signal Processing, vol. 61, no. 14, pp. 3653–3655, 2013.



Harmonic analysis for a beginner
***Q1.*** Your question 1 does not have a definite answer, this will very much depend on the uniformity of your data and on the noise level. To characterize your spectrum, the method of spectral estimation with a Gaussian process prior seems well suited. The paper [Bayesian Nonparametric Spectral Estimation](https://arxiv.org/abs/1809.02196) contains both a comprehensive description of the method and a computer code you can use.


***Q2.*** For question 2, [harmonic analysis](https://en.wikipedia.org/wiki/Harmonic_analysis) and [spectral analysis](https://en.wikipedia.org/wiki/Spectral_density_estimation) refer to very different fields of research, the former is not particularly relevant for the data analysis problem that you describe. Some texts on spectral analysis that you could look into: [Spectral Analysis for Physical Applications](https://books.google.nl/books/about/Spectral_Analysis_for_Physical_Applicati.html?id=FubniGJ0ECQC) and [Spectral Analysis of Signals.](https://www.maths.lu.se/fileadmin/maths/personal_staff/Andreas_Jakobsson/StoicaM05.pdf)


