Color Shift Keying Demodulation

In Visible Light Communication, Color Shift Keying Modulation is done as shown in the block diagram

The relation or equations to convert (x,y) into (Pr,Pg,Pb) is also shown in the picture

Now, for the demodulation part how can i get the xp,yp(or call it recieved x and y for easy undertanding) values back. It seems that if i write these equations again at demodulation side, it returns the values calculated at the modulation side. It should be different because there will be channel and noise effect. Xr, Yr, Xg, Yb etc are all known.

Answer 1

[mile amm]

the demodulation of csk