# QAM constellation slicer with unknown carrier phase

As shown in the image, I have a 16QAM constellation that is misaligned due to a phase rotation. In this case you can see that rotation amount is approximately $\theta = \pi/4$, but this won't be the case in general. For real-world data the phase could be a slowly varying function of time, $\theta(t)$, so that it is not enough to apply some fixed correction factor.

I am aware of differential mapping schemes that solve the phase ambiguity problem due to the constellation having $\pi/2$ symmetry, but it seems $\theta$ must still be known to perform the slicing.

One suggested solution was to try to map the received constellation point to the nearest QAM constellation point and feed a phase-locked-loop with the result, but it is not clear how this would perform when $\theta$ varies over time.

What techniques exist to recover the symbols? I have already tried various carrier recovery schemes based on feedback loops, with no success, and am interested in decision directed approaches that may avoid having to find the phase.

What you need is carrier phase synchronization. This is a complicated topic with many different approaches. The approach that you'll choose could depend on things like:

• Data-aided versus blind: Does the underlying sequence contain any known data (e.g. a training or sync sequence of some kind) that you can use to divine the phase offset? Or, do you have to synchronize with no knowledge of the modulating symbols?

Blind approaches are more general, but you can get better phase estimation performance if you're aided by the data. Also, blind approaches often introduce ambiguities into the recovered phase (i.e. there are multiple phase offset solutions that equally satisfy the blind optimization criterion).

I haven't come across any blind solutions that work equally well across all modulation types; for instance, most blind phase estimators are best applied to PSK signals. They can be made to work, suboptimally, on QAM. Various blind carrier phase estimators for QAM are documented in the literature, but I don't have any great recommendations for QAM-specific estimators. The most commonly-used blind estimator for PSK-type signals is the power-law detector. Here is an example of a paper that talks about its application to QAM.

• Coherent versus differential: As you noted, one way to get around having to synchronize to the carrier phase directly is to use differential modulation. In that case, information is carried by the difference in phase between consecutive symbols. Since the carrier phase is likely to be approximately constant across a two-symbol time period, the carrier component cancels out. This makes synchronization easier, but there is a symbol error rate performance loss on the order of 1-2 dB for differentially-coherent modulations versus fully-coherent operation.

• Feedforward versus feedback: Are you processing a continuous, indefinitely-long stream of symbols, or do you have a finitely-sized batch? A feedback approach like a phase-locked loop might be appropriate for the former, while feedforward techniques that estimate the bulk phase offset of a block of symbols at a time are best for the latter (feedback techniques have some acquisition period during which you won't get good output; if you only have short blocks of data at a time, this can be a problem).

If you're looking for a book reference, my go-to is Synchronization Techniques for Digital Receivers by Mengali. It's expensive and hard to find a copy, but I find it very thorough.

Jason wrote an excellent answer, so I'll just supplement that:

decision directed approaches that may avoid having to find the phase.

That can't work with 16QAM, because you can't do a meaningful decision without knowing how your decision boundary grid has been rotated.

So, from my point of view, the only feasible approaches here always must either

• Method 1
• directly estimate the right phase and correct that
• Method 2
• derotate the constellation to an ambiguity of $0$, $\frac \pi2$, $\pi$ or $\frac 32 \pi$, and then
• resolve the remaining ambiguity with data.

Methods of type 1 would be, for example, correlation with the shape of a known preamble (without decision!).

Methods of type 2 would be, for example, doing statistical analysis of your observed complex numbers to find a square bounding box of received constellation points, and derotate this to be parallel to the I and Q axis. Note that this would require coherency first!

In any case, 16QAM is typically used in high-rate systems, and after you've recovered the correct phase once, you'll have to track the phase for these – be it through pilot symbols, or through a continuously running PLL that gets phase error info from the decider.