# How to obtain mean Power Spectrum from two audio channels?

I have a source of noise in a duct, on either side of the noise source I have a microphone, which register the pressure, so that I get two pressure signals: $p_1$ and $p_2$. I want to get the acoustic Sound Pressure Level (SPL) spectrum which can be estimated using a periodogram approach, more specifically using the welch procedure. This function is readily available in Matlab as

pwelch


or in python using

scipy.signal.welch


## The Question

The question is, since I have two channels of audio, how do I combine them to get a representation of the total SPL spectrum? Or is the mean SPL more appropriate?

Some ideas:

I know that for two sound sources giving rise to $p_1$ and $p_2$ then we have two cases:

Correlated sound: $<p_{tot}^2> = \frac{1}{2}(p_1^2+p_2^2)$
Uncorrelated sound: $<p_{tot}^2> = \frac{1}{2}(p_1+p_2)^2$

In my case I'm guessing I should consider the time series $p_1$ and $p_2$ to be correlated since they come from the same source but are measured on separate sides, and this would imply that the total SPL spectrum would be given by averaging the resulting spectra given by welch on the individual channels. Thoughts?