## Abstract

We propose a novel configuration of the finite-impulse-response (FIR) filter adapted by the phase-dependent decision-directed least-mean-square (DD-LMS) algorithm in digital coherent optical receivers. Since fast carrier-phase fluctuations are removed from the error signal which updates tap coefficients of the FIR filter, we can achieve stable adaptation of filter-tap coefficients for higher-order quadrature-amplitude modulation (QAM) signals. Computer simulations show that our proposed scheme is much more tolerant to the phase noise and the frequency offset than the conventional DD-LMS scheme. Such theoretical predictions are also validated experimentally by using a 10-Gsymbol/s dual-polarization 16-QAM signal.

©2012 Optical Society of America

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### Equations (28)

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(1)
$$E\left(n\right)={\left[E\left(n\right),E\left(n-1\right),\dots ,E\left(n-M\right)\right]}^{T},$$
(2)
$$p\left(n\right)={\left[p\left(n\right),p\left(n-1\right),\dots ,p\left(n-M\right)\right]}^{T}.$$
(3)
$${E}^{\prime}\left(n\right)=p{\left(n\right)}^{T}E\left(n\right).$$
(4)
$$p\left(n+1\right)=p\left(n\right)+{\mu}_{p}{e}_{p}\left(n\right)E{\left(n\right)}^{\ast},$$
(5)
$${e}_{CMA}\left(n\right)={E}^{\prime}\left(n\right)\left\{{r}^{2}-{\left|{E}^{\prime}\left(n\right)\right|}^{2}\right\},$$
(6)
$${e}_{MMA}\left(n\right)={E}^{\prime}\left(n\right)\left\{r{\left(n\right)}^{2}-{\left|{E}^{\prime}\left(n\right)\right|}^{2}\right\},$$
(7)
$${e}_{PI-LMS}\left(n\right)={E}^{\prime}\left(n\right)\left\{{\left|d\left(n\right)\right|}^{2}-{\left|{E}^{\prime}\left(n\right)\right|}^{2}\right\},$$
(8)
$${e}_{PD-LMS}\left(n\right)=d\left(\text{n}\right)-{E}^{\prime}\left(n\right).$$
(9)
$${e}_{p}\left(n\right)=d\left(n\right){\left\{f(n)/\left|f(n)\right|\right\}}^{-1}-{E}^{\prime}\left(n\right),$$
(10)
$$f\left(n+1\right)=f\left(n\right)+\frac{{\mu}_{f}}{{\left|{E}^{\prime}\left(n\right)\right|}^{2}+\epsilon}{e}_{f}\left(n\right){E}^{\prime}{\left(n\right)}^{*},$$
(11)
$${e}_{f}\left(n\right)=d\left(n\right)-f\left(n\right){E}^{\prime}\left(n\right),$$
(12)
$$p\left(n+1\right)=\left(1-{{\mu}^{\prime}}_{p}\right)p\left(n\right)+{{\mu}^{\prime}}_{p}\frac{d\left(n\right)}{E\left(n\right)}\frac{\left|f\left(n\right)\right|}{f\left(n\right)},$$
(13)
$$f\left(n+1\right)=\left(1-{\mu}_{f}\right)f\left(n\right)+{\mu}_{f}\frac{d\left(n\right)}{p\left(n\right)E\left(n\right)},$$
(14)
$$\alpha =\frac{1}{N}{\displaystyle \sum _{n=1}^{N}\frac{{\varphi}_{p}\left(n+1\right)-{\varphi}_{p}\left(n\right)}{{\varphi}_{f}\left(n+1\right)-{\varphi}_{f}\left(n\right)}},$$
(15)
$$\alpha \approx \frac{{\mu}_{p}}{{\mu}_{f}}\frac{1}{M+1}.$$
(16)
$$s\left(n+1\right)=s\left(n\right)+\frac{{\mu}_{s}}{{\left|f\left(n\right){E}^{\prime}\left(n\right)\right|}^{2}+\epsilon}{e}_{s}\left(n\right){\left\{f\left(n\right){E}^{\prime}\left(n\right)\right\}}^{\ast},$$
(17)
$${e}_{s}\left(n\right)=d\left(n\right)-s\left(n\right)f\left(n\right){E}^{\prime}\left(n\right),$$
(18)
$${e}_{p}\left(n\right)=d\left(n\right){\left\{f\left(n\right)/\left|f\left(n\right)\right|\right\}}^{-1}{\left\{s\left(n\right)/\left|s\left(n\right)\right|\right\}}^{-1}-{E}^{\prime}\left(n\right).$$
(19)
$${E}_{x,y}\left(n\right)={\left[{E}_{x,y}\left(n\right),{E}_{x,y}\left(n-1\right),\dots ,{E}_{x,y}\left(n-M\right)\right]}^{T}$$
(20)
$${p}_{k,l}\left(n\right)={\left[{p}_{k,l}\left(n\right),{p}_{k,l}\left(n-1\right),\dots ,{p}_{k,l}\left(n-M\right)\right]}^{T}.$$
(21)
$$\begin{array}{l}{p}_{xx}\left(n+1\right)={p}_{xx}\left(n\right)+{\mu}_{p}{e}_{px}\left(n\right){E}_{x}{\left(n\right)}^{*},\\ {p}_{xy}\left(n+1\right)={p}_{xy}\left(n\right)+{\mu}_{p}{e}_{px}\left(n\right){E}_{y}{\left(n\right)}^{*},\\ {p}_{yx}\left(n+1\right)={p}_{yx}\left(n\right)+{\mu}_{p}{e}_{py}\left(n\right){E}_{x}{\left(n\right)}^{*},\\ {p}_{yy}\left(n+1\right)={p}_{yy}\left(n\right)+{\mu}_{p}{e}_{py}\left(n\right){E}_{y}{\left(n\right)}^{*},\end{array}$$
(22)
$$\begin{array}{l}{e}_{px}\left(n\right)={d}_{x}\left(n\right){\left\{{f}_{x}(n)/\left|{f}_{x}(n)\right|\right\}}^{-1}{\left\{{s}_{x}(n)/\left|{s}_{x}(n)\right|\right\}}^{-1}-{{E}^{\prime}}_{x}\left(n\right),\\ {e}_{py}\left(n\right)={d}_{y}\left(n\right){\left\{{f}_{y}(n)/\left|{f}_{y}(n)\right|\right\}}^{-1}{\left\{{s}_{y}(n)/\left|{s}_{y}(n)\right|\right\}}^{-1}-{{E}^{\prime}}_{y}\left(n\right),\end{array}$$
(23)
$$\begin{array}{l}{{E}^{\prime}}_{x}\left(n\right)={p}_{xx}{\left(n\right)}^{T}{E}_{x}\left(n\right)+{p}_{xy}{\left(n\right)}^{T}{E}_{y}\left(n\right),\\ {{E}^{\prime}}_{y}\left(n\right)={p}_{yx}{\left(n\right)}^{T}{E}_{x}\left(n\right)+{p}_{yy}{\left(n\right)}^{T}{E}_{y}\left(n\right).\end{array}$$
(24)
$${f}_{x,y}\left(n+1\right)={f}_{x,y}\left(n\right)+\frac{{\mu}_{f}}{{\left|{{E}^{\prime}}_{x,y}\left(n\right)\right|}^{2}+\epsilon}{e}_{fx,y}\left(n\right){{E}^{\prime}}_{x,y}{\left(n\right)}^{\ast}\text{,}$$
(25)
$${e}_{fx,y}\left(n\right)={d}_{x,y}\left(n\right)-{f}_{x,y}\left(n\right){{E}^{\prime}}_{x,y}\left(n\right),$$
(26)
$${s}_{x,y}\left(n+1\right)={s}_{x,y}\left(n\right)+\frac{{\mu}_{s}}{{\left|{f}_{x,y}\left(n\right){{E}^{\prime}}_{x,y}\left(n\right)\right|}^{2}+\epsilon}{e}_{sx,y}\left(n\right){\left\{{f}_{x,y}\left(n\right){{E}^{\prime}}_{x,y}\left(n\right)\right\}}^{\ast}\text{,}$$
(27)
$${e}_{sx,y}\left(n\right)={d}_{x,y}\left(n\right)-{s}_{x,y}\left(n\right){f}_{x,y}\left(n\right){{E}^{\prime}}_{x,y}\left(n\right),$$
(28)
$${f}_{ave}\left(n\right)=\frac{{f}_{x}\left(n\right)+{f}_{y}\left(n\right)}{2}.$$