How to perform a phase recovery for higher order modulation schemes

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?

Thanks!

• See this post on using a decision directed phase detector which works well for this application dsp.stackexchange.com/questions/74667/… May 8 '21 at 19:07
• And more specifically this one dsp.stackexchange.com/questions/31497/… May 8 '21 at 19:14
• And this shows the implementation usable for higher order QAM as well as QPSK dsp.stackexchange.com/questions/51856/… May 8 '21 at 19:34
• @DanBoschen thank you for the advice! The concept of PLL is new to me, so I would like to clarify one thing: my approach in demodulating QPSK would be to define the "reference points" on the constellation (+- sqrt(2) +- j sqrt(2)) and then compare all received symbols with them. Then I multiply the received symbol with complex conjugate of the nearest neighbor (reference point) and I obtain the phase that can be used to recover the phase of the symbols. Is this right? May 8 '21 at 20:54
• Yes you have that right- this gives you an error which you should accumulate as the loop filter in the PLL; this is diagrammed in the link I gave above so if something in that link isn’t clear you can ask under that question May 8 '21 at 22:59