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?