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Imagine we are sampling of a gaussian signal along time and need to know the power/variance associated with the first N spectral components. So we take our favorite fft algorithm to get the PSD.

The error associated with a given estimated spectral component f(w) (w is the frequency) of a Gaussian signal follows a Chi-squared distribution with ν=2 degrees of freedom (we just have a single spectrum, no averaging, no overlapping). For instance the 95% confidence interval is given by:

.[νχ2(ν,α/2), ν/χ2(ν,1−α/2)] with ν=2 and α=0.05.

That is, we have 95% of chance to find the true F(w) in the range

[νχ2(ν,α/2)f(w), νχ2(ν,1−α/2)f(w)].

NB: f(w) is the estimated frequency component, F(w) is the true frequency component.

My question is the following: what is the error in the power/variance estimate which equals to the sum of f(w) over the N spectral components (N positive frequencies) of the spectra?

- Is it given by a χ2 law with ν=2N degrees of freedom?
- Is it given by the summation of the error in each independent frequency ?
- Something else?