Why is directional noise correlated noise?

[CantorSet]

Hi everyone,

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 $$S(t)$$ and two microphones $$X_1(t)$$ and $$X_2(t)$$. Then the standard acoustic propagation model has that

$$X_1(t) = a_1S(t-\tau_1)+n_1(t)$$

and

$$X_2(t) = a_2S(t-\tau_1)+n_2(t)$$

where $$a_i, \tau_i, n_i$$ 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?

 

[fleem]

In this context, I'd have to assume that by "directional", they simply mean the sound has one point source, as you say (and not that it is anisotropic!). On a side note, practically speaking there will be notable multipath, making things more difficult.

By the way, that Tau in the second equation should be tau sub 2, not Tau sub 1 (unless the microphones are the same distance from the sound source)

To mathematically describe noise that comes from a point source (described here as "directional"), treat it as if it were just another signal source term, albeit an undesirable one.

 

[CantorSet]

In this context, I'd have to assume that by "directional", they simply mean the sound has one point source, as you say (and not that it is anisotropic!). On a side note, practically speaking there will be notable multipath, making things more difficult.

By the way, that Tau in the second equation should be tau sub 2, not Tau sub 1 (unless the microphones are the same distance from the sound source)

To mathematically describe noise that comes from a point source (described here as "directional"), treat it as if it were just another signal source term, albeit an undesirable one.

Thanks for your help, Fleem.