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Capacity Bounds under Imperfect Polarization Tracking

Mohammad Farsi, Magnus Karlsson, Erik Agrell

TL;DR

This work addresses capacity limits for polarization-drift channels in optical links under imperfect channel knowledge by modeling SOP fluctuations as a random unitary channel. The authors derive an achievable information rate (AIR) under mismatched decoding and show that a unitary channel estimate tightens the bound; they then introduce a data-aided Kabsch estimator to enforce unitary estimates, showing it outperforms conventional LS, especially with small pilot overhead. Through Monte Carlo simulations for dual-polarization channels (n=2), Kabsch yields AIR gains of roughly 0.2–0.35 bits/symbol over LS across a range of SNRs, validating the practical relevance of unitary-channel estimation for polarization tracking. The results provide capacity insights for DP optical channels and suggest actionable improvements for polarization maintenance in SDM systems.

Abstract

In optical fiber communication, due to the random variation of the environment, the state of polarization (SOP) fluctuates randomly with time leading to distortion and performance degradation. The memory-less SOP fluctuations can be regarded as a two-by-two random unitary matrix. In this paper, for what we believe to be the first time, the capacity of the polarization drift channel under an average power constraint with imperfect channel knowledge is characterized. An achievable information rate (AIR) is derived when imperfect channel knowledge is available and is shown to be highly dependent on the channel estimation technique. It is also shown that a tighter lower bound can be achieved when a unitary estimation of the channel is available. However, the conventional estimation algorithms do not guarantee a unitary channel estimation. Therefore, by considering the unitary constraint of the channel, a data-aided channel estimator based on the Kabsch algorithm is proposed, and its performance is numerically evaluated in terms of AIR. Monte Carlo simulations show that Kabsch outperforms the least-square error algorithm. In particular, with complex, Gaussian inputs and eight pilot symbols per block, Kabsch improves the AIR by 0:2 to 0:35 bits/symbol throughout the range of studied signal-to-noise ratios.

Capacity Bounds under Imperfect Polarization Tracking

TL;DR

This work addresses capacity limits for polarization-drift channels in optical links under imperfect channel knowledge by modeling SOP fluctuations as a random unitary channel. The authors derive an achievable information rate (AIR) under mismatched decoding and show that a unitary channel estimate tightens the bound; they then introduce a data-aided Kabsch estimator to enforce unitary estimates, showing it outperforms conventional LS, especially with small pilot overhead. Through Monte Carlo simulations for dual-polarization channels (n=2), Kabsch yields AIR gains of roughly 0.2–0.35 bits/symbol over LS across a range of SNRs, validating the practical relevance of unitary-channel estimation for polarization tracking. The results provide capacity insights for DP optical channels and suggest actionable improvements for polarization maintenance in SDM systems.

Abstract

In optical fiber communication, due to the random variation of the environment, the state of polarization (SOP) fluctuates randomly with time leading to distortion and performance degradation. The memory-less SOP fluctuations can be regarded as a two-by-two random unitary matrix. In this paper, for what we believe to be the first time, the capacity of the polarization drift channel under an average power constraint with imperfect channel knowledge is characterized. An achievable information rate (AIR) is derived when imperfect channel knowledge is available and is shown to be highly dependent on the channel estimation technique. It is also shown that a tighter lower bound can be achieved when a unitary estimation of the channel is available. However, the conventional estimation algorithms do not guarantee a unitary channel estimation. Therefore, by considering the unitary constraint of the channel, a data-aided channel estimator based on the Kabsch algorithm is proposed, and its performance is numerically evaluated in terms of AIR. Monte Carlo simulations show that Kabsch outperforms the least-square error algorithm. In particular, with complex, Gaussian inputs and eight pilot symbols per block, Kabsch improves the AIR by 0:2 to 0:35 bits/symbol throughout the range of studied signal-to-noise ratios.
Paper Structure (10 sections, 5 theorems, 35 equations, 4 figures)

This paper contains 10 sections, 5 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Channel Model and Mutual Information
  3. Channel Model
  4. Mutual Information with Perfect Knowledge of Channel at the Receiver
  5. the Mutual information in presence of Channel Estimation Error
  6. Channel Estimation
  7. LS Algorithm
  8. Kabsch Algorithm
  9. Numerical Results
  10. Conclusion

Key Result

Theorem 1

Consider an arbitrary complex MIMO channel matrix $\mathrm{H}$ and a fixed estimated channel matrix $\mathrm{\hat{H}}$. Defining ${\mathrm{E} = \mathrm{H}-\mathrm{\hat{H}}}$, the AIR is where the definitions of $\Lambda_{\underline{x}}$, $Q$, and $\hat{\Lambda}_{\underline{x}}$ can be found in eq:Lambda, eq:Q, and eq:Lambda_Hat, respectively.

Figures (4)

  • Figure 1: Transmission block model
  • Figure 2: The average AIR $\bar{I}_q$ of the DP channel as a function of SNR $\eta$ and estimation error per DOF $\mathcal{E}^2$.
  • Figure 3: The average AIR $\bar{I}_q$ when ${L = 8}$DP pilot symbols are used. (a) Using $\mathcal{CN}$ inputs where for LS and Kabsch, $\bar{I}_q$ is according to \ref{['eq:AIR_Rand_Est']} and \ref{['eq:AIR_Unitary_Est']}, respectively. (b) Using uniformly distributed DP-$16$-QAM inputs where $\bar{I}_q$ for both LS and Kabsch is according to \ref{['eq:AIR_Discrete']}.
  • Figure 4: The average information gap $I-\bar{I}_q$ for a range of pilot lengths. (a) $\mathcal{CN}$ inputs where $\bar{I}_q$ for LS and Kabsch algorithms is according to \ref{['eq:AIR_Rand_Est']} and \ref{['eq:AIR_Unitary_Est']}, respectively. (b) Uniformly distributed DP-$16$-QAM inputs where $\bar{I}_q$ for both LS and Kabsch is according to \ref{['eq:AIR_Discrete']}.

Theorems & Definitions (5)

  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4