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I am doing signal analysis for a time series and the assumption of signal is

S = F + e

Where S is the original signal, F is the frequency component and e is white noise (auto-regressive time series with moving average = 0). I have a large number of samples, say, N samples; and I can do wavelet decomposition for all them to obtain N number of coefficients arrays.

Now my question is what's the proper way to combine all those coefficients to achieve an "average" of the coefficients, so that e would have smallest standard deviation? I tried to do arithmetic average or average norm times average coefficients vector by cross product; but I couldn't prove either of them is a proper way.

Thanks in advance for your help.

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It sounds like this is a numerical problem without analytic solution. So I would suggest to use a numerical optimizer to minimize the standard deviation, e.g. MATLABs FMINCON() function can minimize virtually any expression that can be calculated.

So for your problem, I recommend to calculate:

$$\text{Fmincon}(\sigma(x))$$

where $x$ being your coefficient vector.

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