I am currently working on narrowband beamforming and looking to compute the transfer function for a linear combiner based LMS. While doing a quick search I have found the following article which presents a direct approach for computing the Transfer Function of the LMS algorithm given a single input signal with tapped delay lines i.e. FIR filter. LMS Transfer Function LMS Transfer Function IEEE
However, in beamforming applications the input signal arrives with a temporal (phase) delay on an array of antennas hence the assumption in the presented article at estimating the correlation matrix at equations (5) and creating the correlation like in equation (7) becomes invalid. Therefore, I would like to kindly ask on how is it possible to proceed in computing a transfer function for a beamforming LMS model with multiple antennas i.e. multiple spatial inputs where the only correlation between the samples is temporal based(phase). Additionally while using the Z transform the it is unclear on how to translate the delay from temporal to spatial.
What I am currently searching in literature: Spatial temporal signal processing, Multi-dimensional Z transform, and Transfer functions.
I would be great if I can get additional help to narrow my research and jump on the correct track/algorithms that can help in completing the derivation.
Thank you Best Regards