# differential phase tracking loop for the receiver

For a DBPSK modulation, how can we track the differential phase? I thought about using a phase tracking loop but I did not find articles for this.

• Can you clarify why you want to track differential phase? I was starting to answer on the traditional DBPSK demodulator (delay and multiply) but not sure if you still need to track differential phase for other reasons? Perhaps you want to have a coherent receiver for 3 dB better performance and do not have control over what the transmit waveform is? – Dan Boschen Dec 1 '18 at 12:39

There is no need to track the phase difference at all. What is needed is to compare the "average" RF phase during the current bit interval to the "average" RF phase during the previous bit interval. If the two phases are the same, or near enough the same, the receiver decides that the phase did not change during the transition from the previous bit interval to the current bit interval and so the differential encoding at the transmitter is telling us that a $$0$$ data bit is being transmitted. If the two phases are vastly different, then we are being told that a $$1$$ data bit is being transmitted.

How is all this done? Well, use the part of a QPSK receiver that separates the RF signal into I and Q signals, and then match-filter (or otherwise process) the I and Q signals to produce complex-valued outputs $$Z_1 = (X_1,Y_1)$$ and $$Z_2 =(X_2,Y_2)$$ where $$X$$ is the output of the I branch and $$Y$$ the output of the Q branch of the receiver. Note that the $$X$$'s and $$Y$$'s are individual real numbers: they are the sample values obtained from the I and Q branch filters/processors at the chosen sampling instants, not continuous-time signals or vectors of sample values that DSP processors or computers use to represent continuous-time signals.

The hard-decision DBPSK decision device considers the question:

Is the new symbol $$Z_2$$ closer to the old symbol $$Z_1$$ or closer to the negative $$-Z_1$$ of the old symbol?

and thus compares

$$(X_2-X_1)^2 + (Y_2-Y_1)^2 \gtrless (X_2+X_1)^2 + (Y_2+Y_1)^2$$

which can be simplified to a sign comparison on $$\langle Z_1,Z_2\rangle = X_1X_2+Y_1Y_2$$. Note that this is essentially asking

Are the two vectors $$Z_1$$ and $$Z_2$$ are pointing in roughly the same direction (in which case the inner product or dot product is positive) or in roughly opposite direction (in which case the dot product is negative)?

A third viewpoint thinks of $$Z_1$$ and $$Z_2$$ as complex numbers and asks

Is $$\text{Re}(Z_1Z_2^*) = X_1X_2+Y_1Y_2$$ positive or negative?

But, regardless of the viewpoint, the end result is the same.

Note that there is no explicit phase comparison in all this: those boxes marked $$\arctan\left(\frac yx\right)$$ on block diagrams of communications receivers are more for conceptual understanding than for implementation guidance.

It is assumed that the receiver has a good estimate of the received carrier frequency but not of the carrier phase. Frequency acquisition can be done by squaring the RF signal (to eliminate the phase modulation), high-pass filtering (to remove the DC in the squared signal which has DC plus an unmodulated double-frequency signal in it), and frequency-halving, to recover the carrier frequency. In fact, the output of the frequency halver can be used as the local oscillator in the receiver front end. It is not necessary to track the RF phase. All that we are interested in is whether the phase in the current interval was the same as the phase in the previous interval or was it different? Where exactly $$Z_1$$ and $$Z_2$$ individually lie in the complex plane is not of importance; their relative position is. This is different from phase-coherent demodulation of plain vanilla BPSK in which there is a single (real-number) output that must be compared to $$0$$ to determine the transmitted bit and woe betide the designer who ignores the possibility of phase ambiguity caused by the frequency halver. Note also that we are assuming that the bit interval timing is known with a fair degree of accuracy so that the DBPSK receiver knows when it is the correct time to sample the I and Q branch outputs to get the $$Z$$'s.