# Fisher information of spectrum estimator

Perhaps someone could point me to a reference

I would like to calculate the Fisher information of a spectrum estimator. The spectrum estimator is a windowed Fourier transform of the data.

More precisely, the data is a discrete time series $x(t_i)$, and each of the $x(t_{i})$ are independent and Gaussian distributed with mean zero and an unknown variance. The windowed finite Fourier transform of $x(t_0)$ through $x(t_{N-1})$ is given by:

$$y(f) = \sum_{t=0}^{N-1}exp(-i 2 \pi f t) x(t) a(t)$$

a(t) is a window function that decays to 0 at $t=0$ and $t=N-1$. The estimate of the spectrum is then given by:

$$\hat{S}(f) = |y(f)|^2$$

I know that y(f) should be Gaussian distributed and that the spectrum estimator is chi-squared distributed, but going through the math looks pretty hairy. I was wondering if there was any reference that might go through a derivation of the Fisher information for this particular problem.

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