Abstract

In this paper, we highlight that it is inadequate to describe the rotation of the state of polarization (RSOP) in a fiber channel with the 2-parameter description model, which was mostly used in the literature. This inadequate model may result in problems in polarization demultiplexing (PolDemux) because the RSOP in a fiber channel is actually a 3-parameter issue that will influence the state of polarization (SOP) of the optical signal propagating in the fiber and is different from the 2-parameter SOP itself. Considering three examples of the 2-parameter RSOP models typically used in the literature, we provide an in-depth analysis of the reasons why the 2-parameter RSOP model cannot represent the RSOP in the fiber channel and the problems that arise for PolDemux in the coherent optical receiver. We present a 3-parameter solution for the RSOP in the fiber channel. Based on this solution, we propose a DSP tracking and equalization scheme for the fast time-varying RSOP using the extended Kalman filter (EKF). The proposed scheme is proved to be universal and can solve all the PolDemux problems based on the 2- or 3-parameter RSOP model and exhibits good performance in the time-varying RSOP scenarios.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

In recent years, the rapid development of new services, such as cloud computing, online high–definition video and virtual reality, has produced an explosive growth of the global IP data traffic. To support the ever-growing bandwidth demands, the capacity of optical fiber communication system in single wavelength channel has increased exponentially from 1 Gb/s to 100 Gb/s, even exceeding 400 Gb/s [1–3]. As a result, polarization effects play an increasingly important role (either a positive role or a negative role) in an optical communication system. For the positive role, the polarization can be used as a physical dimension in the polarization division multiplexing (PDM) optical fiber system to double the spectral efficiency. In contrast, the negative impact, which should not be ignored, is that polarization-related impairments will lead to the degradation of the system performance. The three main polarization impairment effects in an optical fiber communication system are the rotation of state of polarization (RSOP), polarization mode dispersion (PMD) and polarization dependent loss (PDL) [4,5], all of which are required to be equalized at the receiver end. Among these effects, the RSOP is essential to a PDM system for the requirement of polarization demultiplexing (PolDemux). Therefore, in this paper, we focus on the RSOP modelling for fiber channel, and the equalization issues related to RSOP. The main trouble of RSOP for the PDM system is its high variation speed that imposes a heavier burden on the ordinary MIMO (multiple input and multiple output) equalization algorithm to keep pace. Indeed, there are some extreme scenarios such as vibration sources nearby and lightning strike, in which the speed of RSOP will be up to 5.1 Mrad/s for an optical link [6–8]. It is essential for us to make a deep analysis about the appropriate RSOP modelling for fiber channel, and hence the fast response and effective RSOP equalization schemes, in order for us to cope with the ultra-fast RSOP variation in the extreme environments.

In the literature, in Jones space the RSOP impairment is modeled as 2 × 2 matrix J = (jxx, jxy; jyx, jyy), mostly with only 2 independent parameters [9–14]. We find that it comes from the confusion or misunderstanding regarding the concept of SOP and RSOP. The SOP refers to the state of polarization of optical signals itself in a fiber and requires only 2 parameters for its normalized vector representation. Alternatively, the RSOP refers to a polarization effect that causes the rotation of a signal’s SOP in a fiber channel and requires 3 parameters for its matrix representation. This paper will answer why an RSOP modelling should be with 3 independent parameters, and show the mostly used 2-parameter RSOP models are the special cases of 3-parameter model.

As for the RSOP equalization scheme, the widely used schemes are MIMO equalization algorithms, such as constant modulus algorithm (CMA). It can equalize the RSOP impairment to some extent and have some limited tolerance to other impairments, such us residual chromatic dispersion, PMD and PDL [15]. However, the performance of CMA will decline dramatically or even be completely insufficient when the speed of RSOP exceeds several hundreds of kilo-rad per second.

On the other hand, recently, an efficient method of PolDemux in Stokes space has been proposed that has attracted widespread attention [9–14]. The operation principle of this type of PolDemux technique is based on the calculation of the Stokes parameters of the received signal samples in Stokes space to obtain a best fitting plane, whose normal vector identifies the two orthogonal polarization states at the receiver. The normal vector is forced to its original direction to achieve polarization demultiplexing [9]. The main advantage of this approach is that it does not require pre-identification of the modulation format transmitted and has a high convergence speed. In particular, Stokes space-based geometric approach proposed by [11] is robust and compatible to time-varying RSOP.

Nevertheless, among these Stokes-based PoDemux algorithms, as mentioned above, the confusion regarding the concept of SOP and RSOP will result in problems for the implementation of RSOP equalization. We can, for example, find this misunderstanding or confusion regarding this discrepancy between the SOP and the RSOP in [9–14]. The RSOP fiber channel models adopted are Eq. (9) in [9] and Eqs. (7)-(9) in [11], all of which include only 2 parameters for the RSOP matrix representation. This confusion between the 2-parameter SOP and the 3-parameter RSOP leads to problems for PolDemux in Stokes space; addressing this confusion is the motivation of this paper.

In this paper, first, we will highlight the reasons why those previous works used a 2-parameter RSOP for a fiber channel and the resulting issues in following that approach. Second, we propose a 3-parameter solution for this issue in PolDemux in Stokes space combined with the extended Kalman filter. Finally, numerical verifications are demonstrated.

2. RSOP model in a fiber channel

2.1 SOP representation with 2-parameters

In [9,11–14], nearly all the RSOP models for fiber channel are established based on the idea that, through the RSOP transformation in a fiber channel, a pair of orthogonal x and y linearly polarized states can be transformed into an arbitrary pair of orthogonal elliptically polarized states. And hence these transformations were regarded as the RSOP models for a fiber channel. The x-polarized and y-polarized states that are orthogonal to each other can be represented as

|x-polarizaed=(10)and|y-polarizaed=(01)
The arbitrary elliptical polarization state can be represented in two different ways [4,16–18], as shown in Fig. 1. One approach uses the representation in laboratory x-y coordinate system using the parameters of αandδ, by which the elliptical polarization state is denoted as|E(α,δ).αdenotes the amplitude ratio angle with the relationtanα=Ay/Ax, and δdenotes the phase difference withδ=φyφx. The arbitrary two orthogonal elliptical polarization states are denoted by |E1(α,δ) and |E2(α,δ).
|E1(α,δ)=(cosαsinαejδ)and|E2(α,δ)=(sinαejδcosα)
or|E1(α,δ)=(cosαejδ/2sinαejδ/2)and|E2(α,δ)=(sinαejδ/2cosαejδ/2)
Therefore, one would think the RSOP modelJshould satisfy |E1(α,δ)=J|x-polarizedand|E2(α,δ)=J|y-polarized.

 figure: Fig. 1

Fig. 1 An arbitrary elliptical polarization can be represented by either |E(α,δ)or|E(θ,β), which are two SOP representations, one in the laboratory x-y coordinate system and the other in the principal ξ-η coordinate system.

Download Full Size | PPT Slide | PDF

According to Fig. 1 we can also use the principal ξ-η coordinate system using another pair of parameters θ andβto describe an arbitrary elliptical polarization state, which is denoted as |E(θ,β) [18]. Here, θ denotes the orientation angle or azimuth angle, and β denotes the ellipticity angle with the relationtanβ=Bη/Bξ. Using θandβ, the arbitrary two orthogonal elliptical polarization states are

|E1(θ,β)=(cosθcosβjsinθsinβsinθcosβ+jcosθsinβ)and|E2(θ,β)=(cosθsinβ+jsinθcosβsinθsinβjcosθcosβ)
One would also think RSOP matrix model J should satisfy |E1(θ,β)=J|x-polarized and |E2(θ,β)=J|y-polarized.

|E(α,δ) or |E(θ,β) can be mapped to Stokes space either on the observable polarization sphere (usingS2-S3-S1coordinate system) or on the Poincaré sphere (using S1-S2-S3coordinate system) [16], as shown in Fig. 2, and with the relations given in Eq. (5).

 figure: Fig. 2

Fig. 2 The polarization states represented (a) on the observable polarization sphere and (b) on the Poincaré sphere corresponding to the laboratory x-y coordinate system and the principal ξ-η coordinate system.

Download Full Size | PPT Slide | PDF

S2=S0sin2αcosδS3=S0sin2αsinδS1=S0cos2αandS1=S0cos2θcos2βS2=S0sin2θcos2βS3=S0sin2β

We note that, in Stokes space, whenever we represent a state of polarization (SOP) using the observable polarization sphere or the Poincaré sphere, two parameters are adequate to describe an SOP. We can conclude that an SOP requires 2-parameter representation. Under the idea that theJmatrix of the RSOP model for a fiber channel should satisfy|E1(α,δ)=J|x-polarizedor |E1(θ,β)=J|x-polarized, can one conclude from the above reasoning that an RSOPJmatrix must also be a 2-parameter representation? Before we answer this question, we first consider some examples of the forms in which theJmatrix can take.

2.2 The RSOP model for a fiber channel adopted in previous works

2.2.1 (α,δ)-two-step RSOP model

According to Fig. 3, we can establish an RSOP J matrix through a two-step process using parameters(α,δ). The first step: an x-linearly polarized SOP for the input signal is converted into α-linearly polarized SOP by α angle rotation through a rotation matrix. The second step: the α-linearly polarized SOP (has an outer frame with lengths of 2cosα and 2sinα with zero phase difference between its x and y components) is converted into an arbitrary elliptical SOP by introducing a phase retarder, as shown in Fig. 3. For the reason that we take two steps to obtain the final representation of polarization rotation, we call this RSOP model an (α,δ)-two-step RSOP model, as shown in Eq. (6); this model can be found in [9,11–13].

 figure: Fig. 3

Fig. 3 The flow process for establishing the (α,δ)-two-step RSOP model.

Download Full Size | PPT Slide | PDF

J1(α,δ)=(ejδ/200ejδ/2)(cosαsinαsinαcosα)=(cosαejδ/2sinαejδ/2sinαejδ/2cosαejδ/2)

2.2.2 (α,δ)-one-step RSOP model

Following Fig. 4, we can also establish a RSOP J matrix through a one-step process using parameters(α,δ). This time, we find that any pair of orthogonal elliptical SOPs (cosα,sinαejδ)T and (sinαejδ,cosα)T in Eq. (2) can be the columns of transformation matrix in Eq. (7) such that the horizontal and vertical linearly SOPs (1,0)T and (0,1)T can be transformed into above arbitrary pair of orthogonal elliptical SOPs. Because the process uses a one-step process, we call this model the (α,δ)-one-step RSOP model, with the result that

 figure: Fig. 4

Fig. 4 The flow process for establishing the (α,δ)-one-step RSOP model.

Download Full Size | PPT Slide | PDF

J2(α,δ)=(cosαsinαejδsinαejδcosα)

This model can be found in [11,12,14]. Two parametersαandδin Eqs. (6)-(7) correspond to the angles of2αandδin the observable polarization sphere [16], as shown in Fig. 2(a).

2.2.3 (θ,β)-RSOP model

According to Fig. 5, we can establish an RSOPJ matrix through a two-step process using parameters(θ,β). The first step: an x-linearly polarized SOP for the input signal is converted into a right-handed zero angle orientation elliptical SOP (cosβ,jsinβ)T through a transformation matrix (cosβjsinβjsinβcosβ) where 2cosβ and 2sinβ denote the outer frame of the SOP and j denotes π/2 phase difference between its x and y components. The second step: the zero angle orientation ellipse is transformed to a θangle orientation ellipse through a rotation matrix. The RSOP J matrix of this model has the following form:

 figure: Fig. 5

Fig. 5 The flow process for establishing the(θ,β) RSOP model.

Download Full Size | PPT Slide | PDF

J3(θ,β)=(cosθsinθsinθcosθ)(cosβjsinβjsinβcosβ)=(cosθcosβjsinθsinβsinθcosβ+jcosθsinβsinθcosβ+jcosθsinβcosθcosβjsinθsinβ)

This model can be found in [18]. For this (θ,β) RSOP model, two parameters correspond to the angles of 2θand 2βin Poincaré sphere, as shown in Fig. 2(b).

2.3 Why should an RSOP model be a 3-parameter representation for a fiber channel?

As we know, for an optical fiber, if we do not take PDL into account, the Jones transformation matrix representing polarization effect should be unitary matrix U, whose Cayley-Klein form is [4]

U=(a+jbc+jdc+jdajb)witha2+b2+c2+d2=1
where only 3 parameters are independent.

From another point of view, in Stokes space, any SOP transformation through a fiber without PDL can be considered as a rotation of the SOP Stokes vector with the angle φaround an axis denoted by a unit vector r^. The corresponding rotation 3 × 3 matrix in Stokes space has the counterpart 2 × 2 Jones matrix in Jones space given in Eq. (10) [19].

U=Icos(φ/2)jr^σsin(φ/2)
where σ=(σ1,σ2,σ3)T, and σ1, σ2, σ3 are Pauli matrices [4,19]. The orientation of the rotation axis r^ has two independent orientation angles, and the rotation angle φ around r^ is another independent parameter. Therefore, an RSOP representation in the fiber channel requires three independent parameters.

In fact [20], had proved analytically and experimentally that at least 3 instead of 2 degrees of freedom for a polarization controller (PC) are needed to achieve the goal of SOP transformation from any input SOP into any other output SOP in Stokes space, which can be regarded as equivalent SOP transformation in a fiber channel. Indeed [20] proved that there are some special input SOPs that cannot be transformed into other arbitrary output SOPs on the Poincaré sphere using the PC with only 2 degrees of freedom. Therefore, we can conclude that an RSOP model matrix should be a 3-parameter representation other than 2-parameter representation.

2.4 The problem induced by the 2-parameter RSOP representation

2.4.1 RSOP effects on optical signals with different 2-parameter RSOP models

In this paper, we only focus on the RSOP for the fiber polarization effect that does not include PMD and PDL. If we let |rTx,|rRx and J be the dual-polarization signal vectors (2 × 1) at the transmitter, receiver, and the RSOP rotations matrix (2 × 2), then

|rRx=J|rTx
corresponds to the signal distortion procedure from transmitter to receiver through fiber channel.

To evaluate the effects of different 2-parameter RSOP models, as mentioned in 2.2, on the optical signals propagating in fiber channel, we let the signals suffer from the three different RSOP matrices mentioned above in Eqs. (6)-(8). The evaluations and comparisons are implemented both in Stokes space and in constellation space in Fig. 6, with the static RSOPs shown in Figs. 6(a)-6(c), and the time-varying RSOPs shown in Figs. 6(d)-6(f), respectively. In all the cases, the start SOPs at the transmitter end are dual-polarization with|x-polarized and |y-polarized QPSK signals, and in Stokes space, all the QPSK signal samples are located in the S2-S3 plane with its fitting plane’s normal vector pointing along the S1 axis [9]. We can see in Figs. 6(a)-6(c) that when a QPSK signal experience the RSOPs described by Eqs. (6)-(8) (with angles{α=pi/6;δ=pi/3}and the corresponding set {θ=arctan(3/2)/2;{β=arcsin(4/3)/2} according to Eq. (5)), although the fitting plane of the signal constellation samples in Stokes space has the same orientation, and the normal vector is n^=[0.4980.4320.751]T(left pictures in Figs. 6(a)-6(c)), but the constellation points in constellation space show different patterns (right pictures in Figs. 6(a)-6(c)). For the cases in Figs. 6(d)-6(f), one parameter is fixed (α in the (α,δ) model, and θ in the (θ,β) model), and another is linearly increased with time (δin the (α,δ) model, βin the (θ,β) model). We can see in Figs. 6(d)-6(f) that the time-varying trajectories of both signal constellation samples in Stokes space, and constellation points in the constellation plane are different one another. Therefore, from this point of view, a 2-parameter representation is insufficient for describing the RSOP in the fiber channel because different models based on Eqs. (6)-(8) have different impacts on the optical signals propagating in the fiber.

 figure: Fig. 6

Fig. 6 For each subfigure, the left column shows Stokes space representation for the PDM-QPSK signal, and the right column represents its x-polarization corresponding constellation diagrams. The signal suffers a static RSOP rotation by {α=pi/6;δ=pi/3} for (a) the J1-model and (b) the J2-model, and {θ=arctan(3/2)/2;β=arcsin(4/3)/2} for (c) the J3-model. The signal suffers a time-varying RSOP rotation by (d) the J1-model, (e) the J2-model and (f) the J3-model.

Download Full Size | PPT Slide | PDF

2.4.2 RSOP compensation based on the 2-parameter representation

The signal equalization procedure (assuming perfect equalization) can be expressed as follows:

|rEq=J1|rRx
where |rEq represents the signal after equalization. The task for equalization is to monitor or find the exact form of J matrix and make a transformation usingJ1.

The following question arises: if the 2-parameter RSOP representations mentioned in Section 2.2 is acceptable, can following equations be satisfied simultaneously?

|rRx=Ji|rTxand|rEq=Jj1|rRxwithji.
The equivalent question is: could we use one inverse matrix of the 2-parameter representation in Eqs. (6)-(8) to equalize the other RSOP represented by the different equation in one of these three representations except itself? We can see that, according to following discussion, the answer is no. In fact, for a real fiber communication system, the equalization matrix should universally cover all the RSOP models.

In the literature, the authors of [9-10] have proposed and developed a PolDemux technique based on Stokes space. The idea behind the PolDemux technique is that, for a PDM-M-QAM modulation format, if there is no RSOP impairment, then its corresponding constellation samples in Stokes space constitute a lens-like disk whose symmetrical plane (or fitting plane) lies in the S2-S3plane, with its normal vectors pointing along the +S1andS1 directions. The two ends of the normal vectors H=(1,0,0)Τand V=(1,0,0)Τrepresent the horizontal polarization state (H-state) and the vertical polarization state (V-state), respectively. In particular, for the PDM-QPSK format, its constellation samples are located at (0,±1,0)Τand (0,0,±1)Τ. An RSOP makes the orientation (also the normal vectors) of the plane deviate from the S2-S3 plane. As a result, the normal vectors become G1=(a,b,c)Τ and G2=(a,b,c)Τ. Therefore, the PolDemux technique based on Stokes space involves forcing this plane to go back to the S2-S3 plane or its normal vectors to go back to H and V to achieve polarization demultiplexing of the optical signals.

To implement the PolDemux method based on Stokes space, we must (i) find the best fitting plane and its normal vectors G1 and G2 in Stokes space and obtain the 2-parameters α and δ (or θ and β) by

cos2α=aa2+b2+c2andtanδ=cb
or
tan2θ=baandsin2β=ca2+b2+c2.
(ii) conduct the equalization in Jones space by transformation matrix J1, whose form is one of Eqs. (6)-(8); (iii) check the equalization result and go to (i) again for next iteration. If we take the form J1(α,δ) in Eq. (6) and the equalization matrix also take the formJ11(α,δ), then we will surely achieve the good equalization performance.

However, what if we conduct the equalization according to Eq. (13)? Equation (13) indicates that, if the 2-parameter representation for the RSOP in the fiber channel is acceptable, then all the RSOP model forms are equivalent, in particular, any equalization transformation matrix Jj1 can recover the RSOP impairments induced by any other RSOP modelsJi,when ji.

To evaluate the question of equalization according to Eq. (13), we performed a simulation for a PDM-QPSK coherent optical communication system. The RSOP model we chose is the (α,δ)-one-step RSOP model represented by J2(α,δ):{α=pi/6;δ=pi/3}. We used two equalization transformation matrices J11(α,δ) and J31(θ,β), and checked the results. The results are shown in Fig. 7.

 figure: Fig. 7

Fig. 7 Stokes space representations of the sample SOP: (a) back-to-back scenario; (b) signal before SOP compensation; (c) signal after SOP compensation usingJ11(α,δ); (d) signal after SOP compensation usingJ31(θ,β).

Download Full Size | PPT Slide | PDF

Figure 7(a) shows the constellation samples in Stokes space under the situation of no RSOP impairment. We can see that the 4 constellation samples are located at (0,±1,0)Τand (0,0,±1)Τ, constituting a plane lying in the S2-S3plane. Figure 7(b) shows that the RSOP J2(α,δ){α=pi/6;δ=pi/3} makes the plane deviate from the S2-S3 plane. Figure 7(c)-7(d) are the results when we perform the equalizations using J11(α,δ) and J31(θ,β). We can see that, after equalization, although the constellation sample planes all return to the S2-S3plane, and their normal vectors return to H and V, the 4 constellation samples shift from(0,±1,0)Τ and (0,0,±1)Τ, which means the impairment induced by the RSOP is not recovered completely.

The phenomena shown in Fig. 7 and the above discussion make us realize that the 2-parameter RSOP models are not sufficient and not universal; they are only some special cases of the 3-parameter representation of the RSOP. Therefore, the equalization matrix we choose must be a form with 3 independent parameters. We cannot obtain these parameters from the information in Stokes space because the number of independent parameters in Stokes space is only 2 ((α,δ) or (θ,β)). In fact, for a spherical coordinate system, only two angles are independent. The previous equalization schemes [9,10] by step (i) to (iii) mentioned above are incomplete.

3. A solution scheme

In this section, we propose a PolDemux scheme or a RSOP equalization scheme based on Stokes space and the Kalman filter. The important key issues using the Kalman filter are: (i) choose the appropriate state parameters to be monitored by the Kalman Filter; (ii) perform the correct equalization using the right transformation matrix; (iii) adopt the right measurement space to measure the measurement innovation or residual; (iv) make the appropriate initialization.

We first focus on the issues in (i) to (iii). In [12], which is a typical PolDemux scheme using both Stokes space and the Kalman filter, the authors choose the state vector as G1,xk=(ak,bk,ck)T defined in Stokes space, where only 2 parameters are independent because aks1,k+bks2,k+cks3,k=0 (constitute a plane) and ak2+bk2+ck2=1 (unit vector). They performed equalization with J11(α,δ) in Jones space. They adopted Stokes space as the measurement space in which the measurement innovation is

ek=(01)(aks1,k+bks2,k+cks3,kak2+bk2+ck2)
where si,k with i=1,2,3is the Stokes vector for the kth symbol.

We can see that, for issue (i), Stokes space can only offer 2 parameter information, and J11(α,δ) cannot cover the universal RSOP models.

Next, we describe our RSOP equalization scheme. We start with issue (ii). According to the discussions above, the equalization transformation matrix should be based on 3 parameters and unitary and can take the following form [4]:

UEq(ξ,η,κ)=U1(ξ,η,κ)=(cosκejξsinκejηsinκejηcosκejξ)1
such that
|rEq=UEq(ξ,η,κ)|rRx
where ξand η correspond to phase angles, and κ corresponds to the amplitude ratio angle. We can also find that the forms of U(ξ,η,κ) and UEq(ξ,η,κ) are each a unitary matrix that takes the Cayley-Klein form. In fact, when ξ=0, η=δ/2, and κ=α, U(ξ,η,κ) degenerates into the (α,δ) two-step RSOP model described in Eq. (6); when ξ=0, η=δ, and κ=α, U(ξ,η,κ)degenerates into the (α,δ) one-step RSOP model described in Eq. (7); when ξ=arctan(tanθtanβ), η=arctan(tanβcotθ)andsinκ=±sin2θcos2β+sin2βcos2θ, U(ξ,η,κ)degenerates into the (θ,β) RSOP model described in Eq. (8).

The Kalman filter possesses flexibilities in choosing the spaces adopted for the state vector, the equalization and the measurement. As mentioned above [12], chose Stokes space for state vector (α,δ)T, the Jones space for the equalization (process as in Eq. (12)) and Stokes space again for the measurement (process as in Eq. (14)). In our proposed scheme, we choose the state vector in the Jones space as

xk=(ξk,ηk,κk)T
We also choose the equalization in the Jones space as in Eq. (18), and the measurement space as Stokes space with measurement vector zk and innovation vector ek as
ek=zkh(xk)=(00)(s1,k(s2,k2s0,k2)(s3,k2s0,k2))
where hkis the measurement function. Note that the three parameters ξ, η, and κ have no counterparts in Stokes space because the SOP described in Stokes space only has two independent parameters.

In this paper, because hk is nonlinear with the state vector xk, the extended Kalman filter (EKF) is utilized. The main estimation formulas of EKF are given as follows [10,21,22]:

x^k=x^k1
Pk=Pk1+Q
Gk=PkH*T(HPkH*T+R)1
x^k=x^k+Gk(zkh(x^k))
Pk=(IGkH)Pk
where the superscript minus denotes the a priori estimate, x^kis the a posteriori state estimate, Pk denotes the error covariance matrix, and H is the Jacobi matrix of measurement equation expanded at prior estimation point x^k. Gkis the Kalman gain matrix. The matrices Qand Rdescribe the state covariance matrix and the measurement covariance matrix, respectively.

4. Simulation results

As a means to verify the theory described above, we performed numerical simulations in the PDM-QPSK coherent optical transmission system at 28 GBaud. The time-varying RSOP models were numerically emulated by four different matrices of J1(α,δ), J2(α,δ), J3(θ,β), and U(ξ,η,κ)mentioned above in Eqs. (6)-(8) and (17) under the OSNR of 15 dB. As shown in Fig. 8, in the receiver, the DSP mainly includes modules of resampling and normalization, orthogonalization, CD compensation, PolDemux, recovery of frequency offset and phase noise, decision, and BER calculation. In order to focus on the equalization of RSOP, we assume all other impairments such as CD are already compensated in previous equalization parts in DSP. PDL and PMD are also not taken into account in this paper. The frequency offset between the transmitter and receiver lasers is set as 500MHz, and the linewidths of them are all set as 100kHz. The simulation in this paper is mainly involved in the brown shaded parts of DSP. In the PolDemux module, the proposed Kalman scheme based on UEq(ξ,η,κ), CMA and the Stokes method are implemented separately for comparison. We exploit the fourth-power method for the frequency offset estimation and the Viterbi-Viterbi phase equalizer (VVPE) for the phase noise recovery. For the Kalman scheme based on UEq(ξ,η,κ), some parameters must be initialized. The state covariance matrix Q and the measurement covariance matrix R giving the optimum performance are set to beQ=diag(105,105,105)and R=diag(200,200).

 figure: Fig. 8

Fig. 8 Schematic diagram of DSP module.

Download Full Size | PPT Slide | PDF

Firstly, we have simulated the PolDemux performances respectively using the proposed 3-parameter UEq(ξ,η,κ) based EKF method, CMA, and the Stokes space based algorithms to compensate for the signal distortions induced by RSOP models J1(α,δ),J2(α,δ), J3(θ,β) and U(ξ,η,κ), respectively. The results are shown in Fig. 9(a), 9(b), 9(c), and 9(d). We take the algorithm chosen in [12] to represent the Stokes space based algorithms in [9–14]. Actually, the equalization matrix in [12] is the inverse matrix of (α,δ)-two-step RSOP model J11(α,δ). The step sizes of CMA is set as 5e-4, the filter lengths of CMA is set as 11. We can see in Fig. 9 that the proposed 3-parameter based method performs best and behaves universal to all the RSOP models mentioned above. It has the RSOP variation speed tolerance more than 130 Mrad/s. We can also find that the Stokes space based algorithm behaves better in the cases of J1(α,δ) and J2(α,δ) models, and performs worse in the cases of J3(θ,β) and U(ξ,η,κ) models. The reason is that the parameters (α,δ) chosen in Stokes space based algorithm are the same as in models of J1(α,δ) and J2(α,δ), and different from those in models of J3(θ,β) and U(ξ,η,κ).

 figure: Fig. 9

Fig. 9 BER as the functions of the speed of the RSOP under (a) the J1-model, (b) the J2-model, (c) theJ3-model and (d) the U-model.

Download Full Size | PPT Slide | PDF

Then we make the comparison of the convergence performance of the proposed method, CMA, and the Stokes space based method at the RSOP of 1Mrad/s. For the proposed method and the Stokes method, the convergence curves of the estimated parameters (α,δ)are given. For CMA, the absolute values of the first row elements |h11| and |h12| of the equalization matrix are tracked. The results are shown in Fig. 10. It can be seen that the proposed method and the Stokes method provides a quicker convergence rate. The tracked parameters (α,δ) achieve their convergence within just tens of samples, which represents an important advantage over the CMA algorithm.

 figure: Fig. 10

Fig. 10 Speed of convergence of (a) the proposed EKF method, (b) CMA, (c) the Stokes method.

Download Full Size | PPT Slide | PDF

5. Conclusions

In this paper, we found that the 2-parameter representation models are inadequate to describe the RSOP in a fiber channel. The 2-parameter SOP is quite different from the 3-parameter RSOP, and the description of the RSOP with 3 parameters is required. The mistaken use of 2-parameter RSOP instead of the 3-parameter RSOP leads to problems for PolDemux or RSOP equalization. It is proved analytically that the 2-parameter RSOP representations that used in the literature are actually the special cases of the 3-parameter RSOP representation used in this paper. Focused on the problems in RSOP equalization, we proposed a solution scheme using a 3-parameter RSOP representation and a Kalman filter. The proposed scheme was implemented and verified in the 28 Gbaud PDM-QPSK optical fiber coherent system. The numerical simulation results show that the proposed scheme based on the 3-parameter RSOP model is universal and independent of the RSOP models used in the literature. The RSOP tracking speed of the proposed scheme can be as high as approximately 180, 170, 160 and 130 Mrad/s for the four models considered in this paper.

Funding

National Natural Science Foundation of China (61571057, 61527820, 61575082); Huawei Technology Project (YBN2017030025); Open Fund of the Guangdong Provincial Key Laboratory of Optical Fiber Sensing and Communications (Jinan University); State Grid Corporation of China (5101/2017-3205A).

References

1. M. Birk, P. Gerard, R. Curto, L. Nelson, X. Zhou, P. Magill, T. Schmidt, C. Malouin, B. Zhang, E. Ibragimov, S. Khatana, M. Glavanovic, R. Lofland, R. Marcoccia, R. Saunders, G. Nicholl, M. Nowell, and F. Forghieri, “Real-time single-carrier coherent 100 Gb/s PM-QPSK field trial,” J. Lightwave Technol. 29(4), 417–425 (2011). [CrossRef]  

2. P. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012). [CrossRef]  

3. G. Raybon, “High symbol rate transmission systems for data rates from 400 Gb/s to 1Tb/s,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper M3G.1. [CrossRef]  

4. J. N. Damask, Polarization Optics in Telecommunications (Springer,2005).

5. Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016). [CrossRef]   [PubMed]  

6. M. Kuscherov and M. Herrmann, “Lightning affects coherent optical transmission in aerial fiber,” (lightwave, 2016), http://www.lightwaveonline.com/articles/2016/03/lightning-affects-coherent-optical-transmission-in-aerial-fiber.html.

7. H. Yaffe, “Are ultrafast SOP events affecting your coherent receivers?” https://newridgetech.com/are-ultrafast-sop-events-affecting-your-receivers.

8. D. Charlton, S. Clarke, D. Doucet, M. O’Sullivan, D. L. Peterson, D. Wilson, G. Wellbrock, and M. Bélanger, “Field measurements of SOP transients in OPGW, with time and location correlation to lightning strikes,” Opt. Express 25(9), 9689–9696 (2017). [CrossRef]   [PubMed]  

9. B. Szafraniec, B. Nebendahl, and T. Marshall, “Polarization demultiplexing in Stokes space,” Opt. Express 18(17), 17928–17939 (2010). [CrossRef]   [PubMed]  

10. B. Szafraniec, T. Marshall, and B. Nebendahl, “Performance monitoring and measurement techniques for coherent optical systems,” J. Lightwave Technol. 31(4), 648–663 (2013). [CrossRef]  

11. N. Muga and A. Pinto, “Adaptive 3-D stokes space-based polarization demultiplexing algorithm,” J. Lightwave Technol. 32(19), 3290–3298 (2014). [CrossRef]  

12. N. Muga and A. Pinto, “Extended Kalman filter vs. geometrical approach for Stokes space-based polarization demultiplexing,” J. Lightwave Technol. 33(23), 4826–4833 (2015). [CrossRef]  

13. Z. Yu, X. Yi, Q. Yang, M. Luo, J. Zhang, L. Chen, and K. Qiu, “Polarization demultiplexing in stokes space for coherent optical PDM-OFDM,” Opt. Express 21(3), 3885–3890 (2013). [CrossRef]   [PubMed]  

14. Z. Yu, X. Yi, J. Zhang, D. Zhao, and K. Qiu, “Experimental demonstration of polarization-dependent loss monitoring and compensation in Stokes space for coherent optical PDM-OFDM,” J. Lightwave Technol. 32(23), 3926–3931 (2014).

15. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef]   [PubMed]  

16. E. Collett, Polarized Light in Fiber Optics (SPIE,2003).

17. A. Kumar and A. Ghatak, Polarization of Light with Applications in Optical Fibers (SPIE,2011).

18. R. Noé, Essentials of Modern Optical Fiber Communication (Springer,2016).

19. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000). [CrossRef]   [PubMed]  

20. X. Zhang and Y. Zheng, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17(7), 2509–2513 (2008). [CrossRef]  

21. S. S. Haykin, Adaptive Filter Theory, 4th ed. (Pearson Education India, 2008).

22. H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017). [CrossRef]  

References

  • View by:

  1. M. Birk, P. Gerard, R. Curto, L. Nelson, X. Zhou, P. Magill, T. Schmidt, C. Malouin, B. Zhang, E. Ibragimov, S. Khatana, M. Glavanovic, R. Lofland, R. Marcoccia, R. Saunders, G. Nicholl, M. Nowell, and F. Forghieri, “Real-time single-carrier coherent 100 Gb/s PM-QPSK field trial,” J. Lightwave Technol. 29(4), 417–425 (2011).
    [Crossref]
  2. P. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012).
    [Crossref]
  3. G. Raybon, “High symbol rate transmission systems for data rates from 400 Gb/s to 1Tb/s,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper M3G.1.
    [Crossref]
  4. J. N. Damask, Polarization Optics in Telecommunications (Springer,2005).
  5. Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
    [Crossref] [PubMed]
  6. M. Kuscherov and M. Herrmann, “Lightning affects coherent optical transmission in aerial fiber,” (lightwave, 2016), http://www.lightwaveonline.com/articles/2016/03/lightning-affects-coherent-optical-transmission-in-aerial-fiber.html .
  7. H. Yaffe, “Are ultrafast SOP events affecting your coherent receivers?” https://newridgetech.com/are-ultrafast-sop-events-affecting-your-receivers .
  8. D. Charlton, S. Clarke, D. Doucet, M. O’Sullivan, D. L. Peterson, D. Wilson, G. Wellbrock, and M. Bélanger, “Field measurements of SOP transients in OPGW, with time and location correlation to lightning strikes,” Opt. Express 25(9), 9689–9696 (2017).
    [Crossref] [PubMed]
  9. B. Szafraniec, B. Nebendahl, and T. Marshall, “Polarization demultiplexing in Stokes space,” Opt. Express 18(17), 17928–17939 (2010).
    [Crossref] [PubMed]
  10. B. Szafraniec, T. Marshall, and B. Nebendahl, “Performance monitoring and measurement techniques for coherent optical systems,” J. Lightwave Technol. 31(4), 648–663 (2013).
    [Crossref]
  11. N. Muga and A. Pinto, “Adaptive 3-D stokes space-based polarization demultiplexing algorithm,” J. Lightwave Technol. 32(19), 3290–3298 (2014).
    [Crossref]
  12. N. Muga and A. Pinto, “Extended Kalman filter vs. geometrical approach for Stokes space-based polarization demultiplexing,” J. Lightwave Technol. 33(23), 4826–4833 (2015).
    [Crossref]
  13. Z. Yu, X. Yi, Q. Yang, M. Luo, J. Zhang, L. Chen, and K. Qiu, “Polarization demultiplexing in stokes space for coherent optical PDM-OFDM,” Opt. Express 21(3), 3885–3890 (2013).
    [Crossref] [PubMed]
  14. Z. Yu, X. Yi, J. Zhang, D. Zhao, and K. Qiu, “Experimental demonstration of polarization-dependent loss monitoring and compensation in Stokes space for coherent optical PDM-OFDM,” J. Lightwave Technol. 32(23), 3926–3931 (2014).
  15. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
    [Crossref] [PubMed]
  16. E. Collett, Polarized Light in Fiber Optics (SPIE,2003).
  17. A. Kumar and A. Ghatak, Polarization of Light with Applications in Optical Fibers (SPIE,2011).
  18. R. Noé, Essentials of Modern Optical Fiber Communication (Springer,2016).
  19. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
    [Crossref] [PubMed]
  20. X. Zhang and Y. Zheng, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17(7), 2509–2513 (2008).
    [Crossref]
  21. S. S. Haykin, Adaptive Filter Theory, 4th ed. (Pearson Education India, 2008).
  22. H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
    [Crossref]

2017 (2)

D. Charlton, S. Clarke, D. Doucet, M. O’Sullivan, D. L. Peterson, D. Wilson, G. Wellbrock, and M. Bélanger, “Field measurements of SOP transients in OPGW, with time and location correlation to lightning strikes,” Opt. Express 25(9), 9689–9696 (2017).
[Crossref] [PubMed]

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

2016 (1)

2015 (1)

2014 (2)

2013 (2)

2012 (1)

2011 (1)

2010 (1)

2008 (2)

X. Zhang and Y. Zheng, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17(7), 2509–2513 (2008).
[Crossref]

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008).
[Crossref] [PubMed]

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Bai, C.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Bélanger, M.

Birk, M.

Charlton, D.

Chen, L.

Clarke, S.

Curto, R.

Doucet, D.

Feng, Y.

Forghieri, F.

Gerard, P.

Glavanovic, M.

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Ibragimov, E.

Khatana, S.

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Li, L.

Lin, J.

Lofland, R.

Luo, M.

Magill, P.

Malouin, C.

Marcoccia, R.

Marshall, T.

Muga, N.

Nebendahl, B.

Nelson, L.

Nicholl, G.

Nowell, M.

O’Sullivan, M.

Peterson, D. L.

Pinto, A.

Qiu, K.

Saunders, R.

Savory, S. J.

Schmidt, T.

Szafraniec, B.

Tang, X.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Wellbrock, G.

Wilson, D.

Winzer, P.

Xi, L.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Xu, H.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Yang, Q.

Yi, X.

Yu, Z.

Zhang, B.

Zhang, J.

Zhang, W.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Zhang, X.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

X. Zhang and Y. Zheng, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17(7), 2509–2513 (2008).
[Crossref]

Zhao, D.

Zheng, H.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Zheng, Y.

X. Zhang and Y. Zheng, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17(7), 2509–2513 (2008).
[Crossref]

Zhou, X.

Chin. Phys. B (1)

X. Zhang and Y. Zheng, “The number of least degrees of freedom required for a polarization controller to transform any state of polarization to any other output covering the entire Poincaré sphere,” Chin. Phys. B 17(7), 2509–2513 (2008).
[Crossref]

IEEE Photonics J. (1)

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

J. Lightwave Technol. (6)

Opt. Express (5)

Proc. Natl. Acad. Sci. U.S.A. (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Other (8)

S. S. Haykin, Adaptive Filter Theory, 4th ed. (Pearson Education India, 2008).

M. Kuscherov and M. Herrmann, “Lightning affects coherent optical transmission in aerial fiber,” (lightwave, 2016), http://www.lightwaveonline.com/articles/2016/03/lightning-affects-coherent-optical-transmission-in-aerial-fiber.html .

H. Yaffe, “Are ultrafast SOP events affecting your coherent receivers?” https://newridgetech.com/are-ultrafast-sop-events-affecting-your-receivers .

G. Raybon, “High symbol rate transmission systems for data rates from 400 Gb/s to 1Tb/s,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper M3G.1.
[Crossref]

J. N. Damask, Polarization Optics in Telecommunications (Springer,2005).

E. Collett, Polarized Light in Fiber Optics (SPIE,2003).

A. Kumar and A. Ghatak, Polarization of Light with Applications in Optical Fibers (SPIE,2011).

R. Noé, Essentials of Modern Optical Fiber Communication (Springer,2016).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1 An arbitrary elliptical polarization can be represented by either | E( α,δ )or | E( θ,β ), which are two SOP representations, one in the laboratory x-y coordinate system and the other in the principal ξ-η coordinate system.
Fig. 2
Fig. 2 The polarization states represented (a) on the observable polarization sphere and (b) on the Poincaré sphere corresponding to the laboratory x-y coordinate system and the principal ξ-η coordinate system.
Fig. 3
Fig. 3 The flow process for establishing the (α,δ)-two-step RSOP model.
Fig. 4
Fig. 4 The flow process for establishing the (α,δ)-one-step RSOP model.
Fig. 5
Fig. 5 The flow process for establishing the (θ,β) RSOP model.
Fig. 6
Fig. 6 For each subfigure, the left column shows Stokes space representation for the PDM-QPSK signal, and the right column represents its x-polarization corresponding constellation diagrams. The signal suffers a static RSOP rotation by {α=pi/6;δ=pi/3} for (a) the J 1 -model and (b) the J 2 -model, and {θ=arctan( 3 /2)/2;β=arcsin(4/3)/2} for (c) the J 3 -model. The signal suffers a time-varying RSOP rotation by (d) the J 1 -model, (e) the J 2 -model and (f) the J 3 -model.
Fig. 7
Fig. 7 Stokes space representations of the sample SOP: (a) back-to-back scenario; (b) signal before SOP compensation; (c) signal after SOP compensation using J 1 1 ( α,δ ); (d) signal after SOP compensation using J 3 1 ( θ,β ).
Fig. 8
Fig. 8 Schematic diagram of DSP module.
Fig. 9
Fig. 9 BER as the functions of the speed of the RSOP under (a) the J 1 -model, (b) the J 2 -model, (c) the J 3 -model and (d) the U-model.
Fig. 10
Fig. 10 Speed of convergence of (a) the proposed EKF method, (b) CMA, (c) the Stokes method.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

| x-polarizaed=( 1 0 ) and | y-polarizaed=( 0 1 )
| E 1 ( α,δ )=( cosα sinα e jδ ) and | E 2 ( α,δ )=( sinα e jδ cosα )
or | E 1 ( α,δ )=( cosα e jδ/2 sinα e jδ/2 ) and | E 2 ( α,δ )=( sinα e jδ/2 cosα e jδ/2 )
| E 1 ( θ,β )=( cosθcosβjsinθsinβ sinθcosβ+jcosθsinβ ) and | E 2 ( θ,β )=( cosθsinβ+jsinθcosβ sinθsinβjcosθcosβ )
S 2 = S 0 sin2αcosδ S 3 = S 0 sin2αsinδ S 1 = S 0 cos2α and S 1 = S 0 cos2θcos2β S 2 = S 0 sin2θcos2β S 3 = S 0 sin2β
J 1 (α,δ)=( e jδ/2 0 0 e jδ/2 )( cosα sinα sinα cosα )=( cosα e jδ/2 sinα e jδ/2 sinα e jδ/2 cosα e jδ/2 )
J 2 (α,δ)=( cosα sinα e jδ sinα e jδ cosα )
J 3 ( θ,β )=( cosθ sinθ sinθ cosθ )( cosβ jsinβ jsinβ cosβ ) =( cosθcosβjsinθsinβ sinθcosβ+jcosθsinβ sinθcosβ+jcosθsinβ cosθcosβjsinθsinβ )
U=( a+jb c+jd c+jd ajb ) with a 2 + b 2 + c 2 + d 2 =1
U=Icos( φ/2 )j r ^ σ sin( φ/2 )
| r Rx =J| r Tx
| r Eq = J 1 | r Rx
| r Rx = J i | r Tx and | r Eq = J j 1 | r Rx with ji.
cos2α= a a 2 + b 2 + c 2 and tanδ= c b
tan2θ= b a and sin2β= c a 2 + b 2 + c 2 .
e k =( 0 1 )( a k s 1,k + b k s 2,k + c k s 3,k a k 2 + b k 2 + c k 2 )
U Eq ( ξ,η,κ )= U 1 ( ξ,η,κ )= ( cosκ e jξ sinκ e jη sinκ e jη cosκ e jξ ) 1
| r Eq = U Eq ( ξ,η,κ )| r Rx
x k = ( ξ k , η k , κ k ) T
e k = z k h( x k )=( 0 0 )( s 1,k ( s 2,k 2 s 0,k 2 )( s 3,k 2 s 0,k 2 ) )
x ^ k = x ^ k1
P k = P k1 +Q
Gk= P k H *T ( H P k H *T +R ) 1
x ^ k= x ^ k +Gk( z k h( x ^ k ))
P k =( I G k H ) P k

Metrics