First off, I apologize for possibly incorrect use of jargon, as I am new in this area.
I have a type of signal that I am assuming can be described by an arbitrary number (let's say $n$) of basis vectors, which are linearly independent. I am measuring a signal through one sensor, which gives me a vector of amplitudes. I am able to measure an arbitrary number (let's say $m$) of mixtures of the basis signals, which gives me $m$ measured signals.
The length of my vectors is 150000, sampled at a frequency of 2.5 MHz.
So here is what I think my system of equations will look like.
$$ Y = A X $$
where $Y$ is the matrix containing the mixtures (measurements) and has dimensions of $ m \times 150000 $, $X$ is the matrix containing the basis signals, and has dimensions of $ n \times 150000 $, and $A$ is a coefficient matrix that has dimensions of $ m \times n $.
I need to solve for $A$. However, since $X$ is non-square ($ n \ll 150000$), it does not have an inverse. I guess I could calculate the pseudo-inverse. Does this make sense?
I have also looked into Principal Component Analysis and how it is used in various types of spectroscopy. It seems to me that PCA assumes a set of basis signals, and I don't know whether that has a physical meaning in my case, since my signals are stochastic (but stationary). I used the PCA routine on matrix $Y$ in MATLAB, and it gave me some signals very similar to the input ones, but I did not know what to do with them.
I think I do have a valid physical model for my signals. Should I perhaps use model-based blind source separation (BSS) ?
Thanks in advance for taking the time.