Ring Modulator in software ?
I suspect this would be a tall order for a PIC at audio frequencies...
the output can't be simply calculated from instantaneous input voltages, you need to find the frequency of every harmonic in both inputs (FFT ?). This will give you two lists of sine wave frequencies, one for input 'A', one for input 'B'
You then need to take each frequency in turn from input 'A' and add it to each frequency from input 'B', then do the same again but subtract, some of these may be negative numbers - so ignore the sign to make them positive.
You now need to simultaneously generate all these sine waves.
I have ignored amplitudes as that would be rwelatively easy after programming the rest but I think the Fourier Transforms would be too much to do in real time , let alone all the other calculations. I have also ignored relative phases, I don't think they would be required.
This procedure would be quite feasable on a modern desk computer - programs like Reason can do it so it is possible.