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Journals >Chinese Optics Letters >Volume 20 >Issue 2 >Page 020603 > Article

Chinese Optics Letters
Vol. 20, Issue 2, 020603 (2022)

Impacts of the measurement equation modification of the adaptive Kalman filter on joint polarization and laser phase noise tracking

Shaohua Hu¹, Jing Zhang^1、*, Qun Liu¹, Linchangchun Bai¹, Xingwen Yi², Bo Xu¹, and Kun Qiu¹

Author Affiliations

¹Key Laboratory of Optical Fiber Sensing and Communications, University of Electronic Science and Technology of China, Chengdu 611731, China

²School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou 510275, China

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DOI: 10.3788/COL202220.020603 Cite this Article Set citation alerts

Shaohua Hu, Jing Zhang, Qun Liu, Linchangchun Bai, Xingwen Yi, Bo Xu, Kun Qiu. Impacts of the measurement equation modification of the adaptive Kalman filter on joint polarization and laser phase noise tracking[J]. Chinese Optics Letters, 2022, 20(2): 020603 Copy Citation Text

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Abstract

Kalman filtering (KF) has good potential in fast rotation of state of polarization (RSOP) tracking. Different measurement equations cause the diverse RSOP tracking performances. We compare the conventional KF (CKF) and the modified KF (MKF), which have different measurement equations. Semi-theoretical analysis indicates the lower conditional variances of measurement residuals and process noise of MKF. Compared with CKF, the MKF has

> 3 dB

optical signal-to-noise ratio (OSNR) improvement at the 10 MHz scrambling rate in simulation. For MKF, more significant tracking speed improvement exists for lower OSNR. MKF can be smoothly combined with an adaptive algorithm, which outperforms adaptive CKF throughout the simulations.

Keywords

adaptive filtering coherent transmission optics communication polarization de-multiplexing

y (n) = H_{CD}^{- 1} {H_{CD} [H x (n) + N]},

(1)

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H = [\begin{matrix} H_{xx} & H_{xy} \\ H_{yx} & H_{yy} \end{matrix}] = [\begin{matrix} \cos κ (n) e^{j ξ (n)} - \sin κ (n) e^{j η (n)} \\ \sin κ (n) e^{- j η (n)} \cos κ (n) e^{- j ξ (n)} \end{matrix}] e^{j ϕ (n)},

(2)

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{\hat{z}}_{CKF} = [\begin{matrix} {\hat{x}}_{x} (n) \\ {\hat{x}}_{y} (n) \end{matrix}] = \hat{G} [\begin{matrix} y_{x} (n) \\ y_{y} (n) \end{matrix}] = \hat{G} y,

(3)

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{\hat{z}}_{MKF} = [\begin{matrix} {\hat{y}}_{x} (n) \\ {\hat{y}}_{y} (n) \end{matrix}] = \hat{H} [\begin{matrix} x_{x} (n) \\ x_{y} (n) \end{matrix}] = \hat{H} x,

(4)

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δ_{CKF} = z_{CKF} - {\hat{z}}_{CKF} = [\begin{matrix} x_{x} (n) \\ x_{y} (n) \end{matrix}] - \hat{G} [\begin{matrix} y_{x} (n) \\ y_{y} (n) \end{matrix}],

(5)

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δ_{MKF} = z_{MKF} - {\hat{z}}_{MKF} = [\begin{matrix} y_{x} (n) \\ y_{y} (n) \end{matrix}] - \hat{H} [\begin{matrix} x_{x} (n) \\ x_{y} (n) \end{matrix}] .

(6)

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\hat{G} = G + Δ_{G} = H^{- 1} + Δ_{G},

(7)

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\hat{H} = H + Δ_{H} .

(8)

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δ_{CKF} = x - \hat{x} = x - \hat{G} (H \cdot x + N) = x - (H^{- 1} + Δ_{G}) (H \cdot x + N) = - (G + Δ_{G}) N - Δ_{G} H x = - \hat{G} N - Δ_{G} H x .

(9)

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δ_{MKF} = y - \hat{y} = y - \hat{H} x = (H x + N) - \hat{H} x = (H x + N) - (H + Δ_{H}) x = N - Δ_{H} x .

(10)

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Δ_{G} = - (δ_{CKF} + H^{- 1} N) / y = - (δ_{CKF} + H^{- 1} N) / (H x + N),

(11)

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Δ_{H} = (N - δ_{MKF}) / x .

(12)

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D (δ_{CKF} | x) = E_{Δ_{G}} [D (\hat{G} N + Δ_{G} H x)] .

(13)

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D (δ_{MKF} | x) = E_{Δ_{H}} [D (N - Δ_{H} x)],

(14)

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D (Δ_{G} | x) = E_{δ_{CKF}} {D [(δ_{CKF} + H^{- 1} N) / (H x + N)]},

(15)

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D (Δ_{H} | x) = E_{δ_{MKF}} {D [(N - δ_{MKF}) / x]},

(16)

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D (δ_{MKF} | Δ_{H}, x) \leq D (δ_{CKF} | Δ_{G}, x),

(17)

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D (Δ_{H} | δ_{MKF}, x) \leq D (Δ_{G} | δ_{CKF}, x),

(18)

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D (Δ_{H} | x) < D (Δ_{G} | x),

(19)

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D (δ_{MKF} | x) < D (δ_{CKF} | x) .

(20)

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R_{k} = α_{R} R_{k - 1} + (1 - α_{R}) \frac{tr ({δ̆}_{MKF, k} {δ̆}_{MKF, k}^{H} + M_{n} P_{k} M_{n}^{H})}{tr (R_{k - 1})} R_{k - 1},

(21)

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Q_{k} = α_{Q} Q_{k - 1} + (1 - α_{Q}) \frac{tr (K_{k} {δ̆}_{MKF, k} {δ̆}_{MKF, k}^{H} K_{k}^{H})}{tr (Q_{k - 1})} Q_{k - 1},

(22)

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M_{n} = [\begin{matrix} x_{x} (n) & x_{y} (n) & 0 & 0 \\ 0 & 0 & x_{x} (n) & x_{y} (n) \end{matrix}],

(23)

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s_{k} = [{\hat{H}}_{x x} (k), {\hat{H}}_{x y} (k), {\hat{H}}_{y x} (k), {\hat{H}}_{y y} (k)],

(24)

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Shaohua Hu, Jing Zhang, Qun Liu, Linchangchun Bai, Xingwen Yi, Bo Xu, Kun Qiu. Impacts of the measurement equation modification of the adaptive Kalman filter on joint polarization and laser phase noise tracking[J]. Chinese Optics Letters, 2022, 20(2): 020603

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