I would like to know the algorithm for how centre channel is derived from rejecting left and right channel in audio upmixing process. Can someone explain intuitively rather than hard hitting mathematical equations to describe the derivation process ...
Here's a simplified overview:
Start with L and R signals. Run an fft on each to extract Mag and Phase at every frequency bin.
Compare the Mag and Phase values of the L and R signals. If they're "the same" within a predefined range (for example within +/- 3dB and +/- 30 degrees), assign this as "center" content.
For content that is specifically anti-correlated (i.e. Phase difference of 180 degree, again within a predefined range), assign this as "rear" content.
For all other content of varying amounts of Mag and/or Phase correlation, assign these as Front L/R or Side L/R content. For example, content that is entirely uncorrelated (Mag and Phase values seem random when we compare L and R, ie. there's no similarity between L and R), we can assign these as Side L/R, since they appear to be hard-panned.
As you can imagine, the fft window size will affect the time or "content" (i.e. frequency) resolution of the upmixer, as well as its latency. Other elements of the upmixer design will affect the "cleanness" of the extraction.
Audio up-mixers are fairly complicated and the "better" ones are either protected IP or trade secrets.
To extract a center channel you would look at the correlation between the left and right input channels and dynamically steer the correlated content towards the center and the uncorrelated towards left and right. The devil is in the details though: it's tricky to do this without creating undesirable steering artifacts. There are lot of implementation issues tow work out: time constants, energy detectors details, correlation metrics, steering curves, frequency band splicing, etc.
Here is on old patent you can plow through if you are interested: https://www.freepatentsonline.com/7016501.pdf