I have a power spectrum of random field $$P(k) = \text{const} \times \exp{(-r_1 \times |k|/2)}$$ and its normalized so $\sigma^2 = 1$, the field had been smoothed by gaussian kernel $$ g_s(k) = \exp{(-r_2 \times k^2/2)}$$ The smoothed field have been measured and then added random uncorrelated errors. I have the data points ($x_{data}$), the measured smoothed field ($f_{data}$) and the errors ($\sigma_{data}$).

I need to use Wiener Filter to reconstruct the original field, I understand the form of Wiener filter and I know I need to calculate the auto-correlation of the smoothed field ($ff$) that has been measured and also the cross-correlation of the "original" field and the measured field ($fy$) and then $$W = \frac{P_{ff}}{P_{fy}}$$ where $P_{ff}$ and $P_{fy}$ are power spectrums.

What I don't understand is how I'm supposed to estimate $fy$ cross-correlation? I don't have y, that is the whole point. If I knew it I wouldn't have any problem.

My second problem is that I don't have any idea what I'm supposed to do with the error's data, I haven't seen any reference on that anywhere.


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