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

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?

• What do you mean by "collerated"?
– MBaz
Sep 25, 2017 at 22:07
• Sorry I translated the word from French and thought it had the same meaning. I mean that these signals are exactly the same, with the same phase and the same amplitude. Sep 26, 2017 at 9:06

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)
4. Subtract $n(t)$ from orig $S_1$.