My objective is to demodulate the signals with different modulation orders and schemes. In particular, I want to recover the phase first.
Let us say, my signal looks like in the constellation below
The idea is to position the constellation in such a way that the largest excursions are at 45 degrees angle. In other words, IQ diagram must be shifted by some angle. This angle is determined by the average shift of all points from the reference axis (real or imaginary). The idea itself is coming from this post, where BPSK phase recovery is described.
My problem is that I still do not understand, how do we perform the phase recovery for the higher order modulation schemes. The idea of simply mirroring the constellation to one quadrant (as it works in the case of BPSK) does not seem correct to me and also does not work well. So how we can do it?