Phase and Frequency Recovery Algorithms for Probabilistically Shaped Transmission

In additive white Gaussian noise (AWGN) channels, probabilistic shaping (PS) approaches the information-theoretical capacity using typical square $M$-ary quadrature amplitude modulation ($M$-QAM) constellations with nonuniform a priori probabilities. However, shaped constellations may have a detrimental impact on the chain of digital signal processing algorithms of typical coherent receivers, reducing or even eliminating the expected shaping gains. In this article, we evaluate carrier phase recovery (CPR) and carrier frequency recovery (CFR) algorithms for probabilistically shaped transmission. The results are obtained by offline processing of experimental data with laser imperfections and additive noise loading. We investigate an existing two-stage CPR, where the first stage is based on pilots with 3.1% overhead, and the second stage uses the blind phase search (BPS) algorithm. We show that the BPS stage is severely impaired by PS at moderate and low SNRs. These impairments can be partially compensated using extremely long noise rejection windows, which may be undesirable from a computational complexity standpoint. Conversely, using only the pilot-based stage exhibits penalties at high SNRs, as phase noise dominates over additive noise. In order to obtain a suitable performance over the full range of SNRs, we propose to detune the PS factor (with respect to the AWGN-optimized value) to optimize the BPS stage. In scenarios with relatively low phase noise and low modulation orders, we show that an equivalent performance can be obtained without shaping factor detuning by operating with the two-stage approach at high SNRs and switching off the BPS stage at moderate and low SNRs. We also show that typical blind CFR based on the 4$^{th}$ power algorithm is severely affected by PS at moderate block lengths, but its expected performance can be recovered using very long block lengths. Alternatively, a pilot-based algorithm with 3.1% overhead (employing the same pilots used for the first stage of the two-stage CPR algorithm) achieves a suitable performance in all the investigated scenarios.