Blind Equalization for Coherent Optical Communications Based on Variational Inference

Abstract

We investigate the potential of adaptive blind equalizers based on variational inference for carrier recovery in optical communications. These equalizers are based on a lowcomplexity approximation of maximum likelihood channel estimation. We generalize the concept of variational autoencoder (VAE) equalizers to higher order modulation formats encompassing probabilistic constellation shaping (PCS), ubiquitous in optical communications, oversampling at the receiver, and dualpolarization transmission. Besides black-box equalizers based on convolutional neural networks, we propose a model-based equalizer based on a linear butterfly filter and train the filter coefficients using the variational inference paradigm. As a byproduct, the VAE also provides a reliable channel estimation. We analyze the VAE in terms of performance and flexibility over a classical additive white Gaussian noise (AWGN) channel with inter-symbol interference (ISI) and over a dispersive linear optical dual-polarization channel. We show that it can extend the application range of blind adaptive equalizers by outperforming the state-of-the-art constant-modulus algorithm (CMA) for PCS for both fixed but also time-varying channels. The evaluation is accompanied with a hyperparameter analysis.

Blind Equalization and Channel Estimation in Coherent Optical Communications Using Variational Autoencoders

“…Since the KL divergence depends on ln 𝑝( 𝒚), it is not easily computable and thus not suitable as objective function. However, the evidence is independent of 𝑄(𝒙| 𝒚), so the last term in (1) is only an additive constant regarding the optimization. Hence, maximizing the evidence lower bound…”

“…7. Results for uniform and PCS-64-QAM through an AWGN channel with ISI and without learning rate reduction-see also [1]; the simulations for the non-blind DFE and MMSE are without pulse-shaping at 𝑁 os = 1 sps. after every frame-index 𝑘 being an integer-multiple of 20, so the given 𝜖 lr represents the initial value.…”

“…Parts of this paper have been presented at the Advanced Photonics Congress, Signal Processing in Photonic Communications (SPPCom), 2021 [1].…”

“…Fig.8. Results for uniform 64-QAM through an AWGN channel with ISI and without learning rate reduction-see also[1]: 𝒉 sim,2 = [0.055 + j0.017, −1.345 − j0.452, 1.007 + j1.152, 0.348 + j0.315]; the for the non-blind DFE and MMSE are without pulse-shaping at 𝑁 os = 1 sps.…”

Blind Polarization Demultiplexing of Probabilistically Shaped Signals

“…A similar idea of dividing constellation into two or more parts based on symbol magnitude was proposed in [6]. Very recently, a variational autoencoder-based approach for blind demultiplexing was presented [7].…”