Minimum Phase Conditions in Kramers-Kronig Optical Receivers

Theoretically, for continuous signals, Kramers-Kronig (KK) receivers should be able to precisely reconstruct a band-limited signal's phase from its envelope if the minimum phase (MP) condition is satisfied. However, simulations have shown that even if the MP condition is satisfied, for sampled signals the reconstructed signal can contain large errors. In this article, we explore the effectiveness of DSP-based KK receivers close to the MP condition using single-tone and multi-tones. For a single-tone signal, we use a precise mathematical analysis to demonstrate large distortions in the reconstructed signal even when the MP condition is clearly satisfied and illustrate this on the complex-plane. We then study multi-tone signals spectrally, and show that there are strong intermodulation distortion tones when MP is satisfied that either directly fall in-band, or could be aliased back in-band during down-sampling. Satisfying MP does, however, eliminate signal-signal beat interference, as expected with KK processing. High up-sampling ratios reduce the spectral density of these tones, which explains the improved performance seen in experimental systems. We then explore the error-vector magnitude close to the boundary of MP for various carrier-to-signal power ratios and up-sampling ratios, to illustrate that there is a soft transition around the MP condition. Overall, this work shows that the MP condition is not sufficient to guarantee error-free performance in noiseless conditions.