Extract same in phase signal from 3 different sources with shared delayed noise?

Question

If I have 3 signals coming from 3 microphones with noise $n$ delayed like this:

$S_1(t) = x(t) + n(t+\phi_1)$

$S_2(t) = x(t) + n(t+\phi_2)$

$S_3(t) = x(t) + n(t+\phi_3)$

Their shared feature $x(t)$ is in phase and exactly the same. $x(t)$ is a voice and the noise can be anything different from $x(t)$.

EDIT 1: To be more precise, the different noises also share similar features but with different phases:

$n_1(t) = \sum_{0 \to n} m_1(t+\tau_1) + m_2(t+\tau_2) + ... +m_n(t+\tau_n)$ $n_2(t) = \sum_{0 \to n} m_1(t+\tau'_1) + m_2(t+\tau'_2) + ... +m_n(t+\tau'_n)$

$etc$

I approximately know what $x(t)$ is, a voice call like "Hello Robot".

How can I extract and detect $x(t)$ from these three signals?

Answer 1

Depending on how good your approximation of x(t) is, you might want to try this:

1. Subtract your approx. of x(t) from all $S$
2. Cross correlate the signals to get $\phi_2$ and $\phi_3$ (choose $\phi_1$ = 0)
3. You now should have n(t),
4. Subtract $n(t)$ from orig $S_1$.
5. You now have x(t)

edit: Please note, that this will only work if your approximation of x(t) is quite good, and the noise you have really is equal in all mics (and not introduced due to electronics etc.)