Pi/4 dqpsk carrier recovery

I think we don't have to do carrier phase recovery. Do i need to do carrier frequency recovery. My symbol rate is high around 2 MHz so i am thinking carrier freq offset is less compared the symbol rate. Or does my gardner timing recovery can take care of this carrier freq offset recovery. I have another question should i do differential decoding then timing recovery or timing recovery followed by differential decoding . Because the pi/4 dqps will have valid symbols at zero level also ie symbol with zero I data and Q non zero data. Similary zero Q data and I non zero data. So i am thinking if do timing recovery first i may get proper output as garder looks for zeros crossing between symbols. Is my explanation right

• It would be good if you could clarify the application or what is most important to you, cost, power, size etc. as that drives the architecture. Dec 18 '19 at 3:30

Yes you are correct, differential modulations such as DBPSK, DQPSK, $$pi/4$$ DQPSK do not need to perform carrier recovery as you can compare one symbol to the next to perform demodulation using a delay and multiply approach. This results in a simple and potentially low power receiver, at the expense of performance compared to coherent modulation schemes.
As far as a static carrier offset this should be no issue- the worst case is if the carrier frequency is changing from one symbol to the next (and at some level it always is, in particular phase noise)- in this regard simply consider how much your symbol rotates from one symbol to the next when you demodulate with this approach and you can directly see the impact of frequency offset (and if the SNR degradation from that rotation based on the resulting symbol distance would be acceptable). For example, a 2 degree rotation would be negligible, while with a 45 degree rotation you be on the threshold of an incorrect symbol decision with no other added noise (using the demodulation approach I describe below). The impact of this can be quantified from the phase noise, and how I approach that with a differential demodulation is to consider with regards to phase how a multiplier for small angles is a "phase adder" thus the delay and multiply demod process is a high pass filter with regards to phase noise. (The filter is given by $$1-z^{-1}$$). It has a gain of 2 at the high frequency portion (symbol rate) and a null at DC- so it passes (and doubles) the phase noise at those higher offsets). Ultimately you can integrate the high pass filtered phase noise from DC out to the symbol rate and double that to determine the rms phase error, and determine performance issues with consideration to the impact of phase offsets as I described above. What you will find in doing this is the demodulator will likely be very robust against phase noise.