# Local Standard Deviation characterization

Given the power spectrum of a correlated discrete random signal (e.g. non-white Gaussian noise), how is it possible to obtain an expression for its local standard deviation, i.e. the average standard deviation value which would be measured on a limited number $N$ of consecutive samples?

I am currently facing this problem the context of Image Processing, and for the moment I am experimentally obtaining rather close estimates of the local standard deviation on colored noise with power: $S_n(f)=$ $1\over\alpha^2+f^2$using:

$\sigma_{loc}\simeq (R_n(0)-R_n((N+1)/2))^{0.5}$

where $R_n(r)$ is the auticorrelation of the signal at distance $r$.

The rough idea behind this rule of thumb is that I do not have to compute standard deviation against the mean of the signal but instead against the biased average of its $N$-samples realizations.

I was looking for a stricter mathematical treatment of the problem.

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