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The communications channel between a QPSK transmitter and receiver can introduce an arbitrary phase change. For this reason, differential QPSK is used.

Differential QPSK seeks to solve the phase ambiguity problem by encoding data using changes in phase between data points, as opposed to their absolute phase. However, now I have a question:

What kind of circuit can measure phase between two signals at different points in time?

On page 2 of this pdf a demodulator is shown, using a delay element. But in the analog world, how do you build such a delay element?

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Coax cables are delay lines. Tiny sections of solid-shield coax were used in 150MHz adaptive equalizers, prior to dataslicing and phase-locking and data recovery.

For your purposes, cut the coax to exceed the switching-time between symbols, so you have some clean regions to feed into a mixer. You'll need to trim the delays rather tightly, to N cycles + 90/180/270 degrees, for best noise rejections.

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One option is to start with a phase locked loop that tracks the phase and produces a control signal that follows the phase. That signal is the input to the tracking VCO within the loop. It represents the phase angle in its amplitude.

If your data rate is known, you can sample the phase value and store it for later comparison with the current phase value. All you need to do is ensure that you sample the phase value and store it at a rate greater than your data rate (the more the better). Thus you build up a past picture of recent phase values inside a microprocessor for instance. You use the current value and a "tap" to a past (stored) value to make the phase comparison.

You could even use a low pass filter that has a time constant about one quarter of the data symbol period. If you make a differential voltage measurement between current and filtered signals, your output is the data but somewhat differentiated. This can be restored to full data using a comparator with hysteresis or, with more sophistication, an upper and lower comparator that sets and resets an RS latch. The sophistication arises in that you can track the amplitude of the differentiated signal and modify comparator thresholds on the fly.

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