## Abstract

We propose a training symbol based channel estimation (TS-EST) algorithm that estimates the 2 × 2 Jones channel matrix. The estimated matrix entries are then used as the initial center taps of the 2 × 2 butterfly equalizer. Employing very few training symbols for TS-EST, ultrafast polarization tracking is achieved and tap update can be initially pursued using the decision-directed least mean squares (DD-LMS) algorithm to mitigate residual intersymbol interference (ISI). We experimentally verify the proposed TS-EST algorithm for 112 Gbps PDM-QPSK and 224 Gbps PDM-16QAM systems using 10 and 40 training symbols for TS-EST, respectively. Steady-state and transient bit error rates (BERs) achieved using the TS-EST algorithm are compared to those obtained using the constant modulus algorithm (CMA) and the training symbol least mean squares (TS-LMS) algorithm and results show that the proposed TS-EST algorithm provides the same steady-state BER with a superior convergence speed. Also, the tolerance of the proposed TS-EST algorithm to laser phase noise and fiber nonlinearity is experimentally verified. Finally, we show by simulation that the superior tracking speed of the TS-EST algorithm allows not only for initial polarization tracking but also for tracking fast polarization transients if four training symbols are periodically sent during steady-state operation with an overhead as low as 0.57%.

© 2012 OSA

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### Equations (10)

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(1)
$$R=\left[\begin{array}{cc}a& b\\ -{b}^{*}& {a}^{*}\end{array}\right]$$
(2)
$$a={e}^{j\delta}\mathrm{cos}\theta ,b={e}^{j\phi}\mathrm{sin}\theta $$
(3)
$${S}_{rx}[n]={e}^{j\psi [n]}R{S}_{tx}[n]={e}^{j\psi [n]}\left[\begin{array}{c}a{s}_{tx}^{x}[n]+b{s}_{tx}^{y}[n]\\ -{b}^{*}{s}_{tx}^{x}[n]+{a}^{*}{s}_{tx}^{y}[n]\end{array}\right]$$
(4)
$$\psi \left[n\right]=2\pi \Delta fT+{\psi}_{pn}[n]$$
(5)
$${T}_{rx}[k]=c{e}^{j\left(\xi +\psi [k]\right)}\left[\begin{array}{c}a+b\\ -{b}^{*}+{a}^{*}\end{array}\right],{T}_{rx}[k+1]=c{e}^{j\left(\xi +\psi [k+1]\right)}\left[\begin{array}{c}a-b\\ -{b}^{*}-{a}^{*}\end{array}\right]$$
(6)
$$\left|a\right|\approx \sqrt{0.5\left(1+\frac{1}{N}\mathrm{Re}\left\{{\displaystyle \sum _{i=0}^{N/2-1}\left(\begin{array}{l}{T}_{rx}^{x}[2i]{T}_{rx}^{x}{}^{*}[2i+1]-\mathrm{...}\\ {T}_{rx}^{y}[2i]{T}_{rx}^{y}{}^{*}[2i+1]\end{array}\right)}\right\}\right)}$$
(7)
$$\left|b\right|\approx \sqrt{1-{\left|a\right|}^{2}}$$
(8)
$$\mathrm{arg}\left\{a\right\}+\mathrm{arg}\left\{b\right\}=\mathrm{arg}\left\{-{\displaystyle \sum _{i=0}^{N/2-1}\left(\begin{array}{l}{T}_{rx}^{x}[2i]{T}_{rx}^{y}{}^{*}[2i+1]+\mathrm{...}\\ {T}_{rx}^{y}{}^{*}[2i]{T}_{rx}^{y}[2i+1]\end{array}\right)}\right\}$$
(9)
$${R}_{1}=\left[\begin{array}{cc}\left|a\right|& \left|b\right|{e}^{j(\mathrm{arg}\left\{a\right\}+\mathrm{arg}\left\{b\right\})}\\ -\left|b\right|{e}^{-j(\mathrm{arg}\left\{a\right\}+\mathrm{arg}\left\{b\right\})}& \left|a\right|\end{array}\right]$$
(10)
$$R=\left[\begin{array}{cc}\mathrm{cos}\omega t& \mathrm{sin}\omega t\\ -\mathrm{sin}\omega t& \mathrm{cos}\omega t\end{array}\right]$$