Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system

Zibo Zheng; Nan Cui; Hengying Xu; Xiaoguang Zhang; Wenbo Zhang; Lixia Xi; Yuanyuan Fang; Liangchuan Li

doi:10.1364/OE.26.007211

1. Introduction

With the explosive increasing traffic brought from media, terminal and Internet, demand of higher capacity of communication system and network is increasing rapidly in recent years [1]. High modulation format, polarization division multiplexing (PDM), wavelength division multiplexing (WDM), spatial division multiplexing (SDM) are most popular multiplexing dimensions for high capacity transmission in optical system [1]. PDM coherent optical system has become effective and crucial system nowadays, which uses two orthogonal states of polarization (SOP) to double the system capacity. Naturally, polarization effects caused by PDM technique have drawn people’s attention because they will induce the impairments on polarizing signals. Polarization mode dispersion (PMD), rotation of SOP (RSOP) and polarization dependent loss (PDL) are three main effects existed in optical links [2,3]. PDL attenuates the signal unequally in different SOPs. PMD and RSOP are generated in fiber due to distributed birefringence which are time-varying according to practical variation of the environment. Finite-impulse-response (FIR) filter based on constant modulus algorithm (CMA) or multiple modulus algorithm (MMA) in multiple-input-multiple-output (MIMO) structure is classical solution for polarization demultiplexing (PolDeMux) [4]. However, it has been found that much pressures are produced on the ability of tracking and convergence of MIMO if RSOP increases to several hundred or mega radian per second. Actually, some extreme environments, like lightning strike, will not only make direct striking vibration on the optical ground wire cable (OPGW) which will generate a polarization rotation in the fiber in OPGW, but also induce the Kerr effect and Faraday effect which induce same polarization rotation in the OPGW a distance of several hundred kilometers away of the location of lightning events [5,6]. It has been reported that this kind of fast polarization rotation of SOP will be as fast as 400krad/s or even 2Mrad/s [5]. In that case, performance of CMA/MMA degrades observably, even leads to failure.

Besides, large PMD may accumulate during long haul transmission and induce strong birefringence. The combination of RSOP and PMD will make more heavy burden to MIMO equalization. Note that parameter of PMD is not just differential group delay (DGD) which is the magnitude of PMD vector. Principal states of polarization (PSPs) are also another important parameter, the slow one of which is the direction of PMD vector. Therefore, if RSOP and PMD exist in the fiber simultaneously, a fast RSOP means also a fast rotation of PSP, which implies time-varying PMD even when DGD of PMD is unchanged. Some reports showed that the appropriate arrangement of CMA/MMA can be competent for large DGD with slow variation of PSP [7,8]. However, the existence of both fast RSOP and large PMD will make MIMO algorithm out of work. It is most important to find a fast response method to cope with the extreme polarization environments [6].

Kalman filter is a kind of optimum adaptive filter algorithm and has won a big success in other fields like engineering control [9–11]. Some previous works on impairments mitigation or recovery using Kalman filter in optical fiber communications have been reported [12–18], some of them solve the problems of polarization effect in Stokes space [12,13]. But most of works on polarization effect equalization utilizing Stokes space only involved fast RSOP tracking, few of them take PMD into account, because large PMD will make the constellation points in Stokes space scattered around with the result that we cannot get the information of RSOP in Stokes space any more. A few works on joint equalization of fast RSOP and PMD only achieved DGD compensation no more than a single symbol period because the PMD compensation reported was implemented in time domain [13,15] while the distortion induced by PMD is actually in frequency domain.

In this paper, a novel frequency domain Kalman scheme for solving large range PMD and fast RSOP is proposed. First of all, we analyze the fiber channel including both RSOP and PMD, and obtain a simplified version of generally accepted model for combination of RSOP and PMD for the further study. With this simplified model for fiber channel and a special design for the proposed scheme, we carry out equalizations with Kalman filter both in frequency and time domain, the compensation for large PMD is implemented in frequency domain and the fast RSOP tracking is implemented in time domain. The special design and arrangement for proposed frequency domain Kalman structure are introduced. In order to make fast and stable convergence of Kalman filter, the process and measurement noise covariance parameters q and r should be initially optimized. In this paper, a half analytic and half empirical relation of q-r is obtained. The proposed scheme is tested on 28Gbaud PDM-QPSK coherent optical communication system both by simulation and experiment. Simulation results show that using the proposed Kalman scheme we can get the OSNR requirements at BER of 3.8e-3 (7% FEC threshold) from 12.8 to 13.3dB (around 13dB within 0.5dB) under the RSOP variation from 200krad/s to 2 Mrad/s with DGD as large as 100 ps. With OSNR of 14dB which is about 1dB OSNR penalty, the proposed Kalman scheme performs well under the all combinations of RSOP from 200krad/s to 2Mrad/s with PMD from 20ps to 190ps, keeping BERs below 3.8e-3 (7% FEC threshold). However, CMA is barely competent for RSOP of 200krad/s. The experiment verifies the good performance of proposed Kalman scheme, with the Q-factor above 10.4dB with PMD up to 215ps (about 6 times of symbol period) combined with RSOP from 200krad/s to 2Mrad/s. Besides, lower computational complexity of proposed scheme is hold compared with CMA (about 1/5).

2. Principle

2.1 Polarization effects modelling in fiber

Different impairments are existed in optical transmission systems such as attenuation, noise, chromatic dispersion (CD), frequency offset (FO) and phase noise (PN) of laser, etc. When PDM technique is applied, distortion raised by polarization effects which mainly consist of RSOP, PMD and PDL will be counted as the very crucial part of overall impairments. In this paper we only focus on the solutions for large PMD combined with ultra-fast RSOP because these two polarization effects are time-varying and statistically in nature, while PDL is relatively static effect whose equalization method based on Kalman filter is left for our next work. In order to study the equalization solution for severe PMD and RSOP, an effective polarization model need to be built to help us cope with these phenomena. Commonly, a generalized model for combination of PMD and RSOP is RSOP1 + PMD + RSOP2 as shown in Fig. 1.

Fig. 1 Generalized polarization model in fiber span.

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According to fundamental theory [19], the first order PMD is a vector $\vec{τ} = Δ τ \hat{p}$ in Stokes space which includes its magnitude DGD $Δ τ$ and its direction $\hat{p}$ corresponding to its slow principal state of polarization (PSP). In Jones space the first order PMD is represented by a transformation matrix as in Eq. (1)

\begin{matrix} U_{P M D} (ω) = (\begin{matrix} \cos θ & - e^{- j γ} \sin θ \\ e^{j γ} \sin θ & \cos θ \end{matrix}) (\begin{matrix} \exp (j ω Δ τ / 2) & 0 \\ 0 & \exp (- j ω Δ τ / 2) \end{matrix}) {(\begin{matrix} \cos θ & - e^{- j γ} \sin θ \\ e^{j γ} \sin θ & \cos θ \end{matrix})}^{- 1} \\ = R_{p s p} Λ (ω) R_{p s p}^{- 1} \end{matrix}

where

U_{P M D} (ω)

described in Eq. (1) means that PMD induces a DGD

Δ τ

between two orthogonal PSPs (fast PSP and slow PSP). Actually in the matrix

R_{p s p} = (\begin{matrix} \cos θ & - e^{- j γ} \sin θ \\ e^{j γ} \sin θ & \cos θ \end{matrix})

, the two column vectors

{(c o s θ, e^{j γ} s i n θ)}^{T}

and

{(- e^{- j γ} \sin θ, \cos θ)}^{T}

, which are two orthogonal elliptical polarization states, represent the fast and slow output PSPs of PMD, while matrix

Λ (ω) = (\begin{matrix} \exp (j ω Δ τ / 2) & 0 \\ 0 & \exp (- j ω Δ τ / 2) \end{matrix})

means the phase retardation of

ω Δ τ

in frequency domain between two orthogonal PSPs. It is worth to notice some works in the literature [16] regarded the fast and slow PSPs as only linear polarized states not in general the elliptical polarized which was equivalent to

γ = 0

in Eq. (1) and is the special case of PMD. For the reason that PMD is the time delay between two orthogonal elliptical polarization states we call it elliptical PMD (EPSP-PMD). In fact, among previous PolDeMux works using Kalman filter in Stokes space, those equalizations just involving RSOP without PMD are more successful [12–14] because PMD will scatter the constellation points to destroy information of RSOP in Stokes space. When we face the joint effect of PMD and RSOP as indicated in Fig. 1, we have the transform matrix as

U (ω) = R_{2} U_{P M D} (ω) R_{1}

where

R_{i} = (\begin{matrix} \cos θ_{i} & - e^{- j γ_{i}} \sin θ_{i} \\ e^{j γ_{i}} \sin θ_{i} & \cos θ_{i} \end{matrix}), i = 1, 2

. Actually

R_{i}

can take the similar form of unitary matrix as

R_{p s p}

[3,19]. Now we will change Eq. (2) into another form and other meaning can be read out

\begin{array}{l} U (ω) = R_{2} R_{p s p} Λ (ω) R_{p s p}^{- 1} R_{1} = R_{2} R_{p s p} Λ (ω) R_{p s p}^{- 1} \cdot (R_{2}^{- 1} R_{2}) \cdot R_{1} \\ = (R_{2} R_{p s p}) Λ (ω) {(R_{2} R_{p s p})}^{- 1} \cdot (R_{2} R_{1}) \\ = R_{n e w - p s p} Λ (ω) R_{n e w - p s p}^{- 1} \cdot R^{'} = U_{n e w - P M D} (ω) \cdot R^{'} \end{array}

where

R_{2} R_{p s p}

constitutes the new PSPs for PMD, and

R_{2} R_{1}

constitutes the new RSOP

R^{'}

. Apparently the polarization effect model in fiber RSOP1 + PMD + RSOP2 is simplified as RSOP′ + newPMD, as such RSOP and PMD are separated as shown in Fig. 2. We can solve PMD at first and then RSOP. In the model described in Eq. (3), we can find that a fast changing RSOP will definitely induce the new fast changing PSPs of PMD on which CMA cannot work. Since Kalman filter has fast convergence and less complexity, we can use Kalman filter to compensate PMD with fast changing PSPs in frequency domain and then to track fast changing RSOP in time domain as in following sections.

Fig. 2 Simplified polarization impairment model.

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2.2 The principle and implementation process of proposed Kalman scheme

As analyzed in Section 2.1, we simplify a general model of RSOP1 + PMD + RSOP2 in Fig. 1 to a model of RSOP′ + newPMD in Fig. 2. For this simplified model, we can at first solve PMD and then track RSOP. As mentioned above, in a few works to compensate PMD using Kalman filter, the compensated DGD was only within a single symbol period because the PMD compensations were merely implemented in time domain while the PMD impairment is actually induced in frequency domain. In this paper we propose a novel window-split Kalman scheme for PMD compensation in frequency domain. By careful and smart design of Kalman structure, the proposed scheme can easily equalize large DGD over several symbol periods with fast PSP and RSOP variation up to 2Mrad/s. In order to realize the best performance of the Kalman scheme, following three design issues should be considered: 1) state and measurement vector selection; 2) compensation architecture; 3) noise optimization. We will describe above issues one by one in detail as follows.

For the selection of state vector for Kalman filter, PMD vector $\vec{τ} = Δ τ \hat{p}$ can also be represented as $\overset{⇀}{τ} = {(τ_{1}, τ_{2}, τ_{3})}^{T}$ where $τ_{1}, τ_{2}, τ_{3}$ are three components of vector $\overset{⇀}{τ}$ with DGD $Δ τ = \sqrt{τ_{1}^{2} + τ_{2}^{2} + τ_{3}^{2}}$ and PSP $\hat{p} = {(τ_{1}, τ_{2}, τ_{3})}^{T} / Δ τ$ . Therefore we select $τ_{1}, τ_{2}, τ_{3}$ as the part of Kalman state parameters to monitor PMD variation including both DGD and PSP. The RSOP in fiber channel generally has three parameters $κ, α, β$ which appear in following RSOP equalization matrix Eq. (7). So the final state vector for Kalman filter is

x = {(τ_{1}, τ_{2}, τ_{3}, κ, α, β)}^{T}

We will make the measurement in Stokes space, because after compensation of PMD and equalization of RSOP, the constellation points for PDM-QPSK signals will lie in S₂-S₃ plane in Stokes space with $S_{1}^{i d e a l} = 0$ and normalized intensity $S_{0}^{i d e a l} = Constant$ . So we select measurement vector $z$ and hence innovation vector $e$ as

z = (\begin{matrix} S_{0}^{i d e a l} \\ S_{1}^{i d e a l} \end{matrix}) and e (x) = z (x) - h (x) = (\begin{matrix} Constant \\ 0 \end{matrix}) - (\begin{matrix} u_{x} u_{x}^{*} + u_{y} u_{y}^{*} \\ u_{x} u_{x}^{*} - u_{y} u_{y}^{*} \end{matrix})

where

u_{x}, u_{y}

are x and y components of equalized PDM-QPSK signals, h is measurement function. For compensation architecture, we chose PMD compensation and RSOP equalization in Jones space whose transform matrices are [3,19]

U_{c o m p} (ω) = \cos (\frac{ω Δ τ}{2}) I - \frac{j (\overset{⇀}{τ} \cdot \overset{⇀}{σ})}{Δ τ} \sin (\frac{ω Δ τ}{2})

and

R_{e q} = (\begin{matrix} e^{j α} \cos κ & - e^{j β} \sin κ \\ e^{- j β} \sin κ & e^{- j α} \cos κ \end{matrix})

where

κ

is azimuth rotation angle, and

α, β

are phase rotation angles,

\overset{⇀}{σ} = {(σ_{1}, σ_{2}, σ_{3})}^{T}

and

σ_{1}, σ_{2}, σ_{3}

are Pauli Matrices [3,19].

U_{c o m p} (ω)

will be conducted in frequency domain, and

R_{e q}

will be carried out in time domain. We can show that

U_{c o m p} (ω)

is exactly equal to

U_{P M D}^{†} (ω)

in Eq. (1), and then can be used as the compensation matrix for PMD. Ref [3]. showed RSOP in fiber channel was three-parameter dependent. Finally, noise optimization helps us to provide stable and excellent convergence performance of the Kalman scheme which will be discussed in detail in Section 2.3.

In order for PMD compensation to be conducted in frequency domain, we design a window-split structure. Its specific structure is demonstrated in Fig. 3. Symbol sequence is cut off from data stream by a fixed-length window as the signal queue before each Kalman iteration. The signal queue is at first transformed into the form in frequency domain by fast Fourier transformation (FFT) and then is sent to compensation process. For next iteration, the window slides forward by several symbols which is denote as slide step $Δ S$ to cover new piece of signal queue. All the symbols of frequency domain signal queue are multiplied by $U_{c o m p} (ω)$ to realize PMD compensation, and then are inversely transformed into the form in time domain by IFFT. In time domain, the RSOP equalization using matrix $R_{e q}$ is implemented on the symbols in the middle of each window whose length is equal to step length $Δ S$ . Then the next iteration will repeat again.

Fig. 3 Window-split structure and slide operation.

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After problem analysis and vectors selection, next procedure is to construct the whole Kalman equalization scheme. According to the Kalman filter theory [9–11], the mathematical model is described as Eq. (8) which includes process equation and measurement equation.

{\begin{cases} x_{k} = f (x_{k - 1}) + w_{k - 1} \\ z_{k} = h (x_{k}) + v_{k} \end{cases}

In Eq. (8),

x, z

are state vector and measurement vector, whose dimensions are 6 and 2 in this paper respectively.

f (\cdot), h (\cdot)

are transition function and measurement function separately.

w_{k - 1}, v_{k}

are Gaussian noise vectors with zero expectation and covariance matrices

Q_{k - 1}, R_{k}

. Subscript k and k-1 denote the sequence order. In this paper, the process equation is linear with

f (x_{k - 1}) = x_{k - 1}

, while the measurement equation is nonlinear according to Eq. (5). Also, the compensation or equalization in the proposed Kalman scheme follows the physical mechanism which is independent of prediction and correction processes as described in Eqs. (6)-(7). Therefore, extended Kalman filter (EKF) is utilized in this paper. The estimation formulas of EKF is given as following:

Initialization: {\hat{x}}_{0} = E [x_{0}], P_{0} = E [(x_{0} - {\hat{x}}_{0}) {(x_{0} - {\hat{x}}_{0})}^{T}]

Prediction: {\hat{x}}_{k | k - 1} = F_{k - 1} {\hat{x}}_{k - 1}, P_{k | k - 1} = F_{k - 1} P_{k - 1} F_{k - 1}^{T} + Q_{k - 1}

Correction : G_{k} = P_{k | k - 1} H_{k}^{T} {(H_{k} P_{k | k - 1} H_{k}^{T} + R_{k})}^{- 1}

{\hat{x}}_{k} = {\hat{x}}_{k | k - 1} + G_{k} (z - H_{k} {\hat{x}}_{k | k - 1}), P_{k} = (I - G_{k} H_{k}) P_{k | k - 1}

Subscript k|k-1 and k (or k-1) here mean prior estimation and posterior estimation.

P

represents the covariance matrix of state vector,

H = \partial h (x) / \partial x

is the Jacobi matrix of measurement equation expanded at prior estimation point

{\hat{x}}_{k | k - 1}

. In Eq. (5), we have set

z

as standard values to play the role of criterion of correction, it will remain unchanged along with sequence order k.

F

is given as unit matrix in our scheme.

The first step of Kalman filter processing is initialization, in which initial expectation and covariance need to be assigned. Moreover, the noise covariance matrices $Q_{k - 1}, R_{k}$ are also required. Appropriate covariance will make great performance improvement compared to a bad one. The detail of initialization will be discussed in Section 2.3. After initialization, iteration of prediction Eq. (10) and correction Eqs. (11)-(12) go on repeating. Prediction process provides the prior estimation using previous posterior estimation. $F$ is taken as unit matrix due to the ideal of this application. Since described parameters of PMD $(τ_{1}, τ_{2}, τ_{3})$ and RSOP $(κ, α, β)$ tracked in state vector are varied randomly in real optical fiber environment, we cannot figure out how they change. So our basic idea is to do nothing on prior estimation for state vector in prediction process Eq. (10). Updating of the state vector comes from the correction process described in Eq. (12) according to the Kalman gain $G_{k}$ and the innovation $e_{k} = z_{k} - h ({\hat{x}}_{k | k - 1})$ .

The overall diagram of the proposed Kalman scheme is shown in Fig. 4. We can see in the diagram that the compensation procedure (shown in right frame) is independent of prediction and correction procedures (shown in left frame), which offers more flexibility of the proposed Kalman scheme. Input signal goes through the window-split process and is transformed to frequency domain for new-PMD compensation. After IFFT the symbols in the middle of the window (with the length of $Δ S$ ) will be selected for RSOP recovery in time domain. Equalized signal is sent to the left frame for proceeding correction and prediction. When the proposed Kalman filter come to convergent, the output signal will satisfy the criterion in measurement vector, so impairment caused by PMD and RSOP are eliminated.

Fig. 4 The diagram of Window-split structured frequency domain Kalman filter.

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2.3 Initialization analysis

First of all we have a look at the state vector $x = {(τ_{1}, τ_{2}, τ_{3}, κ, α, β)}^{T}$ in Eq. (4) which can be divided into two parts as PMD parameters (τ₁, τ₂, τ₃) and RSOP parameters (κ, α, β). The orders of magnitude of these two parts of parameters are noteworthy. The magnitude of PMD could be vary from dozens to hundreds of picosecond (10⁻¹² s) whereas the rotation of angles for RSOP would be several times of π. The enormous gap in order of magnitudes between these two parts of parameters will make the elements in 6 × 6 matrices $Q_{k}, R_{k}$ so different in order of magnitude and hence make the different relative calculation errors to different elements or even singularity problem of matrices. So numerical normalization is necessary due to the precision limitation of DSP computation. We set symbol interval $T_{s}$ (for a 28Gbaud system $T_{s} \approx 35 ps$ ) and constant $2 π$ as the numerical reference to normalize state vector which given as follows:

x_{n o r m} = {(\frac{τ_{1}}{T_{s}}, \frac{τ_{2}}{T_{s}}, \frac{τ_{3}}{T_{s}}, \frac{κ}{2 π}, \frac{α}{2 π}, \frac{β}{2 π})}^{T}

These reference values constraint the numerical floating of state vector components and fill the order gap. Considering dynamic channel impairment, for example, where DGD ranges from tens picosecond up to two hundred and RSOP from 100 krad/s to 2 Mrad/s, which is quite wide range of variation and able to cover most scenarios in optical link, numerical values of components in state vector can be unified to 10⁰-10¹ scope after normalization This scaling is helpful to uniformed state vector components and maintain stability of filter algorithm.

Process noise and measurement noise are another inevitable issue of Kalman filter. They bring the uncertainty to corresponding process and are assumed as Gaussian distribution with zero expectation and covariance $Q_{k}, R_{k}$ . These statistical parameters of noise will make great influence on filter’s effectiveness. Mismatch of covariance values would cause a severe performance degradation, even lead to divergence. Considering the application of Kalman filter is rather flexible and system dependent, it is nearly impossible to have a common method for optimum values decision. Therefore, faced to the severe polarization impairment as modelled in this paper, we utilize following trace analysis to build a guidance for setting the covariance matrices of $Q_{k}, R_{k}$ .

We consider that covariance matrices of process noise and measurement noise can maintain constant diagonal matrices for the polarization environment, and diagonal elements are all the same. This idea comes from the assumption that different variable noises are independent one another, and from simplicity for the analytical work. Based on these assumptions, we rewrite posterior covariance of state vector and Kalman gain of Eqs. (10)-(12) as:

\begin{array}{l} P_{k} = (I - G_{k} H_{k}) (P_{k - 1} + q I_{6}) \\ G_{k} = (P_{k - 1} + q I_{6}) H_{k}^{T} {[H_{k} (P_{k - 1} + q I_{6}) H_{k}^{T} + r I_{2}]}^{- 1} \end{array}

where scalar q and r correspond to diagonal elements in

Q_{k}

and

R_{k}

matrices,

I_{m}

means unit matrix with m^th order. Considering the requirement of stable convergence, present posterior covariance

P_{k}

should be required not much different compared with previous one, otherwise this difference will accumulate by iteration steps and finally cause divergence of the algorithm. This requirement implies the underlying relation between q and r. The straightforward and rigorous solution for explicit expression for q-r relation is tedious and complicated. But we can make a rough estimation as follows. Trace operation Tr (.) is taken for describing the matrices property during the iteration. So requirement for stable convergence implies:

Tr (P_{k}) = T r (P_{k - 1})

Still, the further calculation is inconvenient due to complex form of Kalman filter. For the purpose of simplification, two conditions are assumed: 1) non-diagonal elements in $P_{k - 1}$ are far less than diagonal ones and the matrix is treated as diagonal matrix, 2) scalar value r is far larger than the element values in matrix $H_{k} (P_{k - 1} + q I) H_{k}^{T}$ . In this case, taking trace operation on both sides of Eq. (14) and applying Eq. (15), we obtain an equation:

\sum_{i, j}^{m, n} h_{i j}^{2} p_{i}^{2} + 2 q \sum_{i, j}^{m, n} h_{i j}^{2} p_{i} + q^{2} \sum_{i, j}^{m, n} h_{i j}^{2} - m q r = 0

where m = 6, n = 2 mean dimension order of state and measurement vector,

h_{i j}

is real number element in matrix

H_{k}

.

p_{i}

is diagonal element in

P_{k - 1}

. These quadratic equations about

p_{i}

are drawn from Eq. (16) and the discriminant is always greater than zero for positive real numbers of q, r. Two rational roots for each

p_{i}

are:

p_{i} = - q \pm \frac{\sqrt{q r}}{\sum_{j}^{n} h_{i j}^{2}}

Since $p_{i}$ represents the meaning of covariance, so positive value should be given. More than that, it is reasonable to be larger than introduced noise for covariance in system process. That is $p_{i} > q$ , then

r > 4 {(\sum_{j}^{n} h_{i j}^{2})}^{2} q, \frac{q}{r} < a upper boundary

is derived. In fact, all the discussions above are built on the premise that the filter has already been in convergent status. However, ultra-fast dynamic tracing is asked for enough update coefficients which are contained in Kalman gain. Elements in

G_{k}

can be computed by Eq. (14) and (17) under previous assumptions and finally expressed in Eq. (19), where we can find that excessively large r will reduce coefficients too dramatically to catch up with actual variation. Ratio q/r has to be limited above certain threshold for normal tracking function.

g_{_{i j}} = \sqrt{\frac{q}{r}} \frac{h_{j i}}{\sum_{j}^{n} h_{i j}^{2}}, \frac{q}{r} > a lower boundary

So the values of q/r are confined in a region between two boundaries. The analysis process above may not be that rigorous but it indeed provides helpfulness for initialization. At last, specific values for different environments will be testified and decided in Section 3.2.

3. Simulation and experiment

3.1 Simulation platform

To verify that the proposed Kalman scheme is effective for equalization of signal distortion due to co-existed large PMD and ultra-fast RSOP in coherent optical communications, we construct a simulation platform of a 28Gbaud PDM-QPSK Nyquist coherent optical communication system as shown in Fig. 5. AWG provides raised cosine pulse shaping with 0.1 roll-off factor and optical signal generated form transmitter experiences ASE noise, CD distortion, polarization effects successively. PMD and RSOP is contained in fiber channel as the same as the model in Fig. 1. Before receiver, an optical Gaussian filter is applied with bandwidth 33GHz. 300kHz linewidth is introduced in transmitter continuous wave (CW) laser and local oscillator. CMA algorithm and the proposed Kalman scheme are used separately to treat the in-phase/quadrature data stream for polarization demultiplexing to recover the received signal from PMD and RSOP impairment. In order to focus on the equalization of PMD and RSOP we assume all other impairments such as CD are already eliminated in previous equalization algorithms. Viterbi-Viterbi phase equalizer (VVPE) is used to provide phase noise recovery for the reason of presence of linewidth. Besides, 16-symbol window-length and 4-symbol slide step are adopted in window-split operation, which decide FFT/IFFT are 32 points since Kalman filter runs based on 2 samples per symbol. The performance of proposed scheme including the proposed Kalman + VVPE will be compared with CMA + VVPE in the following sections.

Fig. 5 Simulation platform diagram.

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3.2 Optimization of $Q$ and $R$

Before impairment compensation simulation, specific noise covariance parameters q-r should be optimized for best recovery performance. In Section 2.3, noise analysis has been carried out theoretically and the boundaries of ratio q/r are shown. A large amount of simulation tests are proceeded in this section to find the optimum range of q and r to check verification of noise analysis in Section 2.3.

Retrospectively, we regard the covariance matrices as the form of $Q = q I_{6}, R = r I_{2}$ , $q, r$ are constant scalar values. A large amount of simulation tests are in most severe and most slight polarization impairment to find out if any combination of q and r can satisfy compensation requirement under different circumstances. Figure 6 shows results in colored contour maps with bit-error-rate (BER) as the function of q and r values.

Fig. 6 q, r vs. log₁₀(BER) with DGD-RSOP-ONSR: (a) 20ps-200krad/s-13dB (b) 200ps-2Mrad/s-13dB (c) 20ps-200krad/s-17dB (d) 200ps-2Mrad/s-17dB.

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q, r are tested under four different scenarios by the severe/slight impairment with two kinds of OSNR 13dB and17dB. 20ps DGD is around 0.8 times of symbol interval whereas 200ps is around 5.8 times of it. For RSOP the rotation speeds of 200krad/s and 2Mrad/s are considered. By carefully reviewing Figs. 6(a)-6(d), we indeed see the upper boundary and lower boundary for ratio q/r, and approximately linear relationship between q and r for different scenarios Considering trade-off of BER performances among four scenarios above, the mean optimum point $(q, r) = (10^{- 4}, 10)$ is determined for the proposed Kalman scheme. The approximate linear relationship $r = 10^{5} q$ is obtained. The q and r relation as shown in Fig. 6 proves that the analysis in Section 2.3 is reasonable and can be regarded as the guidance for getting optimum $Q$ and $R$ .

3.3 Simulation verification

Comprehensive verification on our proposed scheme is carried out on the 28 Gbaud PDM-QPSK simulation platform. We assume that the variation speed of PSPs holds the same as the rotation speed of RSOP whose azimuth and phase angles are linearly increasing with time. In the rest of this paper we assume the speed of PSPs is equal to the speed of RSOP without mention again. Offline DSP is applied for QPSK symbol equalization. Adopted CMA owns 17-tap MIMO structure with update coefficient of $10^{- 4}$ . Besides, each symbol goes through the CMA number of repetition times CNT = 5 for better PolDeMux operation. For proposed Kalman scheme, window length, slide step and covariance maintain unchanged which are given by 16-symbol, 4-symbol and $q = 10^{- 4}, r = 10$ respectively. Overall performance evaluation and comparison are illustrated in Fig. 7.

Fig. 7 Performance evaluation (a) BER vs. OSNR (with 100ps DGD) (b) BER vs. RSOP (14dB OSNR, and different DGD) (c) PMD vector tracing curve (200ps-2Mrad/s, 14dB OSNR) (d) SOP rotation angle tracing (200ps-2Mrad/s, 17dB OSNR), Convergence curves of posterior covariance of state vector (2Mrad/s RSOP, 14dB OSNR, q = 10⁻⁴, r = 10) (e) with different DGD (f) with different q-r values (200ps DGD).

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Figure 7(a) illustrates the BER performance along with OSNR with DGD of 100ps. The speed of RSOP (also PSPs) changes 200krad/s to 2Mrad/s. We can see that CMA is barely competent for signal recovery under 200krad/s RSOP with OSNR requirement 15dB at BER = 3.8e-3 (7% FEC threshold). But it is unqualified when RSOP speed is above 400krad/s, no matter how large the OSNR is. The Kalman scheme provides excellent performance under the different situations of RSOP from 200krad/s to 2Mrad/s with DGD 100ps. It is worth to note that the OSNR requirements at BER of 3.8e-3 is only around 13dB, from 12.8dB to 13.3dB, within 0.5dB for all the situations of RSOP and PMD impairments mentioned above. However, we can see a BER rise when OSNR is lower than 11dB. This phenomenon mainly comes from the mistaken recovery of phase noise by VVPE algorithm (300kHz laser linewidth assumed). High level of noise changes the location of symbols on constellation plane and influences decision after 4th-power operation. Figure 7(b) depicts the BER performance vs. RSOP speed with different amount of DGD from 20ps to 190ps when OSNR is set to 14dB. It was reported that CMA can be used to cope with large DGD with slow variation of PSP which is confirmed in Fig. 8(b) that BER values at speed of PSP (also RSOP) variation of 200krad/s locate nearly around 3.8e-3 for different DGDs from 20ps to 190ps. But when PSP variation speed is higher than 200krad/s, BER increases rapidly for all case of DGD, which proves that CMA has not fast response time enough for fast variation of polarization effects. However the proposed Kalman scheme shows much more stable lower BER curves and nearly no degradation along with RSOP variation even at DGD up to 190ps. Figure 7(c) exhibit the dynamic tracing results of components of PMD vector $τ_{1}, τ_{2}, τ_{3}$ using the Kalman scheme. The polarization impairment in Fig. 7(c) is assumed to be 190ps DGD, 2Mrad/s RSOP (also PSP) under 14dB OSNR. We can clearly conclude that the Kalman scheme can catch up with fast PMD variation just within 310 symbols period (about 11μs) which implies low latency of the scheme. At the same time RSOP tracing performance is also shown in Fig. 7(d), in which we can see that the tracing results of rotation angles θ and γ agree well with the original linearly increased ones under the situation of 190ps DGD, 2Mrad/s RSOP (also PSP), 17 dB OSNR. Figure 7(e) shows the trace of matrix P vs. iteration. As in Eq. (15) we assume that trace of P will keep invariant for every iteration in order to maintain stable convergence. Figure 7(e) indeed verifies invariant behaviors of trace(P) which means stableness and effectiveness of the Kalman filter even for the extreme case of large DGD 200ps and ultra-fast RSOP 2Mrad/s. It should be mentioned here the level of trace(P) is determined by Jacobi matrix H as shown in Eq. (17). Figure 7(f) partly conforms Eq. (17) and hence the initialization theory in Sec. 2.3 are correct. We can find in Eq. (17) that if q and r all increase N times then the trace(P) will increase the same times. Three cases are put forward that if we take the red curve as a base (q = 10⁻⁴, r = 10). The q and r for blue and yellow curves are 2 and 5 times of the base curve respectively, trace(P) are obtained same times accordingly.

Fig. 8 Experiment set up of 28GBaud PDM-QPSK system.

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Computational complexity is another important aspect of algorithm evaluation. Since FFT/IFFT are applied in the proposed scheme, window length will definitely determine performance of the scheme. Wider window length makes FFT/IFFT more accurate, but increase the complexity. Slide step is also factor of scheme performance. Larger step will speed the iteration and reduce the complexity per symbol (in window length) on average, but would not ensure stably convergence. So the trade-off must be considered for optimum performance. Denote window length as L_w, slide step as $Δ S$ , CMA tap number and repetition times number as N and CNT, respectively. The complexity comparison is given in Table 1.

Table 1. Computational complexity comparison (per symbol)

View Table

The statistic of computational complexity is based on optimum implementation. Complex multiplication is counted for 4 real multiplications and 2 real additions, and 2 real additions for complex addition. Complex exponential operation exp(j•) is composed of 2 looking up table (LUT) operations. In that case, if we take L_w = 16, $Δ S$ = 4 as used in above simulation, the total number of real multiplication/addition per symbol are calculated as 406.5/364 for the Kalman scheme. For CMA, we take N = 17 and CNT = 5, 3015/2040 is obtained which is more than 5 times to the Kalman scheme. In addition, if we take CNT = 1 for CMA, the total number of real multiplication/addition per symbol 595/408 is still more than the Kalman scheme. The advantage of less complexity for Kalman filter is clearly shown.

Actually, the innovation vector $e$ represents the convergence criterion for Kalman filter. Kalman filter possesses flexibility so that we can choose appropriate measurement space for measurement. For PDM-QPSK, we find all the constellation points should lie in S₂-S₃ plane in Stokes space after PolDemux. So we choose Stokes space as measurement space. For PDM-16QAM (quadrature amplitude modulation), after PolDemux all the equalized signal samples will locate at the circumferences of three circles in x or y branch constellation planes. Therefore, we can choose x and y branch constellation planes as measurement space for PDM-16QAM. Hence, the proposed Kalman scheme is also applicable to PDM-16QAM format provided we change the innovation vector $e$ in Eq. (5) into [15]

e (x) = (\begin{matrix} 0 \\ 0 \end{matrix}) - (\begin{matrix} (u_{x} u_{x}^{*} - r_{1}) (u_{x} u_{x}^{*} - r_{2}) (u_{x} u_{x}^{*} - r_{3}) \\ (u_{y} u_{y}^{*} - r_{1}) (u_{y} u_{y}^{*} - r_{2}) (u_{y} u_{y}^{*} - r_{3}) \end{matrix})

where

r_{1}, r_{2}

and

r_{3}

represent three radii of three circles.

3.4 Experiment verification

We also build a 28Gbaud PDM-QPSK coherent experiment platform to verify the effectiveness of the proposed Kalman scheme. The experiment setup is shown in Fig. 8.

The 28Gbaud optical PDM-QPSK signal is generated by a 65GS/s arbitrary wave generator (AWG) modulated with 0.75 roll-off factor root raised cosine pulse. A commercially available software (Keysight 81195A) which is integrated in AWG is used to simulate endless RSOP rotation and PMD variation. The software provides cascaded 7-segment PMD emulators each two of which are connected by polarization controller. All PMD segments and polarization controllers are emulated by their corresponding 2 × 2 Jones matrices. It can generate emulated PMD with DGD up to 220ps, and RSOP with rotation speed from 2.4rad/s to 175Mrad/s. At the receiver side, Gaussian BPF with around 33GHz bandwidth is configured and optical signals are received by 80GS/s coherent receiver and saved for offline treatment. The central frequencies of transmitter and local lasers are around 193.414THz. The linewidths for transmitter and local lasers are all approximately 100kHz. The loss of fiber (0.2 dB/km) is compensated by pre-amplification. The parameter settings of the proposed Kalman scheme are as $q = 10^{- 5}$ , $r = 1$ , 16-symbol window length and 2-symbol slide step. On the other hand, 35-tap CMA operating five iterations for each symbol is applied for comparison reference. Besides, VVPE is also used for inevitable linewidth of lasers at transmitter end and receiver end. The Kalman process flow is as same as in Fig. 6. Q-factor is calculated [20] for the performance evaluation of equalization effectiveness. OSNR is set around 17-dB.

Figure 9 illustrates the experiment performance by evaluate Q-factor in the different combinations of RSOP and PMD. Apparently Fig. 9 has the same performance trend as in Fig. 7(b) that the proposed Kalman scheme achieves better performance with Q-factors nearly 4 dB higher than 7% FEC (BER = 3.8e-3) threshold. On the other hand, CMA is not competent to recover the distortion with Q-factor beneath the 7% FEC (BER = 3.8e-3) threshold when speed of RSOP is larger than about 500krad/s. Especially when DGD value is as high as 153.85ps and 215.39ps, Q-factors are generally 2-4 dB lower than the FEC threshold. The inset in Fig. 9 gives an example of constellation map of QPSK signals (after VVPE recovery) which processed by Kalman and CMA respectively with 215.39ps DGD and 2Mrad/s RSOP. In the figure we can find out that CMA fails to recover the QPSK symbols to the single constant radius which is definitely out of work, whereas the Kalman scheme gives clear correction constellation map. Therefore, the experimental results confirm that the proposed Kalman scheme is indeed a good PolDemux technique for solving the impairment due to the polarization effect of combination of large PMD and ultra-fast RSOP, and also prove that above analytical theory and simulations are reasonable.

Fig. 9 Q-factor vs. RSOP with different DGDs (OSNR = 17dB). (inset) Examples of recovery constellations by Kalman and CMA under the situation of 2Mra/s RSOP and 215.39ps DGD.

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4. Conclusion

In this paper, we at first give a simplified and equivalent model for general model of polarization effect of combination of PMD and RSOP, which allows the equalizations simple, focused on solving large PMD and ultra-fast RSOP, a frequency domain Kalman scheme is proposed with a window-split structure. Using an extended Kalman filter the window-split structure is designed for large DGD recovery in frequency domain and fast RSOP tracking in time domain. Specific half analytical and half empirical theory for the initialization of process and measurement noise covariance is presented. With this theory we find that the ratio of q/r has the upper and lower boundaries which is verified by the numerical simulation. We also find a nearly linear relation $r = 10^{5} q$ which is a condition for stable implementation of proposed Kalman scheme. The simulation and experiment were conducted to check the PolDemaux performance of the proposed Kalman scheme in 28 Gbaud PDM-QPSK system under the situations of different DGD and RSOP speed, and the performance comparison with CMA is also made. The simulation and experiment results exhibit much better performance compared with CMA for solving large PMD and ultra-fast RSOP, even under the extreme environment of 200ps DGD and 2Mrad/s RSOP. Besides, less computational complexity of the proposed Kalman scheme is found compared with CMA.

Funding

National Natural Science Foundation of China (61571057, 61527820, 61575082); Huawei Technology Project (YBN2017030025).

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	Real Multiplication	Real Addition	LUT
CMA	(36N + 8)*CNT	24N*CNT	0
Kalman scheme	[666 + 12L_wlog₂(2L_w)]/ $Δ S$	[496 + 12L_wlog₂(2L_w)]/ $Δ S$	27

Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system

Abstract

1. Introduction