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Is it possible to accurately predict the noise level if we add two or more noisy signals. For the sake of simplicity, let us say, the noise is independent and Gaussian.

Suppose we wish to add signals $S_{total}$= $S_A$+$S_B$+$S_C$+... and each has an associated noise , expressed as standard deviation of the baseline signal $\sigma_A$, $\sigma_B$ and $\sigma_C$

If the standard deviation in each signal =1 we would have noise standard deviation as $\sqrt{(1)^2+(1)^2+(1)^2}=\sqrt{3}$. In chemical literature, this is how the noise is estimated if we add noisy signals for signal averaging.

Nevertheless, having a unit standard deviation is rarely the case. So what is the rigorous way to estimate noise in added noisy signals, assuming they all have uncorrelated noise and it is Gaussian, with the general case of standard deviations not being equal?

Context The context of question is related to instrumental analysis as used in chemistry. For example, sometimes one has a mixture of several compounds, A, B, C, .... One can use a separation technique and couple it to a detector that can detect those compounds one by one as they come out of a chromatography column as a function of time. These detectors are universal, which means they can detect any compound which flows through them, and they generate a signal like this:

Chromatogram

Some detectors are not universal, and they can detect only one compound under certain conditions. So one has to repeat the experiment to detect A, then repeat it to detect B, and so on. However, one can add the signal for A, B, and C to see how it would look like if the detector were able to detect all of them in the same experiment. So in that case, one may sum up the signals of A, B and C to show how the total signal would look like.

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If the noises signal are mutually uncorrelated the power of the sum is the sum of the powers.

For three signals you get $$\sigma_{total} = \sqrt{\sigma_A^2+\sigma_B^2+\sigma_C^2}$$

Works any type of uncorrelated noise, Gaussian or not.

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