Why is directional noise correlated noise?
This is not a homework question but I question I have from reading a signals processing paper on acoustics.
Suppose there is a sound source in a room [tex]S(t)[/tex] and two microphones [tex]X_1(t)[/tex] and [tex]X_2(t)[/tex]. Then the standard acoustic propagation model has that
[tex] X_1(t) = a_1S(t-\tau_1)+n_1(t) [/tex]
[tex] X_2(t) = a_2S(t-\tau_1)+n_2(t) [/tex]
where [tex]a_i, \tau_i, n_i [/tex] account for signal attenuation due to distance, time delay due to distance and noise, respectively.
But the paper says that if we have directional noise in the room (like a ceiling fan), then the noise at the two microphones is correlated, that is [tex] Corr(n_1(t),n_2(t)) \neq 0 [/itex].
But it seems to me the directionality isn't what's causing the correlation, but more the fact that the noise comes from a fan. That is, if we had an "omnidirectional" fan in the center of the room, the noise between the two microphones would still be correlated.
Also, how does one mathematically represent noise that is directional?