Calculating standard deviation from power spectral density

Question

I have set of data (x) for power spectral density (Gaussian white noise), and I am trying to find the standard deviation from it. The result looks wrong because I know that the standard deviation must be with a range of 5.

I used below code

sd=std(x)

Data

Thanks

Answer 1

Take the inverse fourier transform, and look at the element at time zero. Note that this is actually just the sum of all the values ( \\int{X(w)e^{iwt}}dw | t=0 = \\int X(w) dw ), with some scaling constant applied which depends on how you defined the psd in a discrete frequency domain

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The power spectral density, S_xx(w) , is equal to F{R_xx(tau)} , the fourier transform of the autocorrelation, R_xx(tau) = E[x(t)x(t+tau)] .

Since you want the standard deviation, you can get R_xx(0) = E[x(t)^2] , and then std^2 = E[x(t)^2] - E[x(t)]^2 .

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Unfortunately, it seems you have no way to recover E[x(t)] . Perhaps you already know this is 0?