Error in summation of spectral components

[SpecGuest]

HI everyone,

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?

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

Thanks for your help

 

[Dr. Courtney]

Independent errors would add in quadrature.

But if you are summing to find the area under the curve (integrating over a frequency range), then the errors are not independent.

I'd figure out a way to take multiple trials and use the standard error of the mean for my error estimate for random errors. Then I'd add that in quadrature with estimates for instrumental and other possible systematic errors. I'd also use our more accurate Fourier transform code (see below) which allows greater accuracy and precision at every step.

As a double check, one can always start your analysis method with a known perfect signal (sum of sine waves of known frequency, amplitude, and phase) and then add random errors of known mean and distribution to them and see how they impact the errors that result from your Fourier transform methods. This is a powerful double check that your methods for estimating errors in your frequency analysis process are reasonable.

 

[SpecGuest]

Thank you for this answer. Actually, I did the double check, that was my first step and also the reason why I'm asking here since I was told that the uncertainty was option 1 in my previous post. But it turns out, unless I did a mistake somewhere in my test program, that option 1 is not the right answer! It is not clear to me whether the uncertainty in each of the spectral components are independent or not. There are some good reasons to think they are not but I guess this also depends on how you evaluate your PSD.
Anyway, thank you for the reference about the EI method. I'll check that.