# Phase ambiguity in QAM modulation

I'm currently studying a decision directed carrier recovery scheme. Basically a PLL try to estimate the phase error in the current received symbol, the phase detector compensate the incoming symbol's phase with the estimated one and then the result is multiplied my the complex conjugate of the decision taken. My notes say:

Another phenomenon that should be taken into account is the phase ambiguity of $n \pi/2$ that affects the decisions with conventional QAM constellations.

What is he talking about? I don't understand why he tells that there is phase ambiguity in the decision of a QAM symbol. Also citing Wikipedia:

Most QAM constellations also have $\pi/2$ phase symmetry

To me there is no phase ambiguity, in fact if I have a 4-QAM with 1 symbol in each quadrant I can just sample the I and Q branch after a matched filter and than perform a simple unambiguous atan like this:

Typically, phase ambiguity arises in the demodulation process when the carrier recovery is done via a loop that tracks the fourth harmonic of the received signal. So why can't the loop track the carrier frequency directly? Well, the received signal doesn't have a carrier signal -- it has a modulated carrier signal whose phase is going to change at every symbol interval which makes the phase untrackable by a typical loop -- and it is necessary to take the fourth power to eliminate the modulation and produce a signal with unchanging phase. As a result, the local oscillator (LO) in the receiver might be in any of four different phases differing by $\pi/2$, and the local reference signals to the mixers in the I and Q branches of the receiver might end up being not the expected \begin{align}\big(\cos(\omega t), -\sin(\omega t)\big)& & \text{LO and received signal are in phase}\end{align} as reference signals (note that the Q branch reference leads the I branch reference by $\pi/2$) but instead \begin{align}\big(\sin(\omega t),\cos(\omega t)\big)& & \text{LO lags received signal by }\pi/2\end{align} or \begin{align}\big(-\cos(\omega t), \sin(\omega t)\big)& & \text{LO lags received signal by }\pi\end{align} or \begin{align}\big( -\sin(\omega t),-\cos(\omega t)\big)& & \text{LO lags received signal by }3\pi/2.\end{align} The result is that instead of the I and Q bits appearing at their respective demodulator outputs (the receiver produces $(I,Q)$), the receiver might be producing $(\bar{Q},I)$, or $(\bar{I},Q)$, or $(\bar{Q},I)$ where the overbars denote bit complements. Differential demodulation is commonly used to eliminate the phase ambiguity.