# What is the physical meaning of the cross-spectrum of two signals measured at two locations and different moment?

I am working with steady state pressure field measurement where sound pressure at multiple locations in the field needs to be measured. In my case, more than one microphone existing in the field at the same time will intrude the field itself. As a result, the pressure of each location needs to be measured at different moments.

My question is is it still physically meaningful to calculate the cross-spectrum of signals of two microphones? If so, what is the physical meaning of the phase of the cross-spectrum at one given frequency?

I understand that for two signals that are measured at the same time, the phase of the cross-spectrum is the phase shift due to the time lag. But when both time and space is involved, I am confused on the meaning of the phase from cross-spectrum calculation.

## 1 Answer

I figured this out myself. The answer to this is that for the steady state, phase from the cross-spectrum is a contribution of both time and space. And if two signals are measured at the same time or location, the contribution of the other factor to phase will be eliminated. The conclusion is made based on formula deduction, which I will not discuss further here.