How to extract noise when the noise on three data sets are near identical?

Question

I have a microphone array with three microphones placed in a small equilateral triangle. During recording, there is a lot of noise introduced by an ADC. However, I am not interested in reducing this noise before sampling - for learning, I want to remove this noise digitally as best as possible, given that the noise on each channel is near identical.

Given that their noise is near identical, I imagine it would be possible to correlate the signals in some way, even in the presence of a stronger signal with slight time delays. If possible, how can I remove this noise?

Answer 1

Given the three measurements $y1$, $y2$ and $y3$ coming from three unknown signals $x1$, $x2$ and $x3$, which we could assume additive to some unknown common noise $x4$, we can write: $$ y1=x1+x4\\ y2=x2+x4\\ y3=x3+x4 $$ or equivalently: $$ y=Ax\\ A=[1,0,0,1;0,1,0,1;0,0,1,1] $$ Again, assuming a enough noisy signal, we can have an standard Recursive Least Squares algorithm for solving the standard LS projection recursively: $$ \hat x=(A^TA)^{-1}A^Ty $$ This way, and if and only if the additive noise and the noisy signal suficciencies are met, you will solve for your three signals, and your coupled noise.

Because of the former, you even can express this solution using the Kalman Filter