Joint blind equalization of CD and RSOP using a time-frequency domain Kalman filter structure in Stokes vector direct detection system

Nan Cui; Zibo Zheng; Xiaoguang Zhang; Wei Yi; Ruipu Guo; Wenbo Zhang; Xianfeng Tang; Hengying Xu; Lixia Xi

doi:10.1364/OE.27.011557

1. Introduction

The success of emerging applications such as big data, 5G wireless networks, and virtual reality has resulted in increasing demands for data centers (DCs), which covers distances of several hundred meters to tens of kilometers [1]. In such short-reach applications, both transceiver costs and spectral efficiency (SE) need to be considered. Compared with coherent approaches, direct detection (DD) has proved to be more economical for short-reach links due to low device cost and complexity [2–4]. Nevertheless, the conventional DD method is intensity modulation and direct detection (IM-DD), in which only amplitude is used, leading to low SE. Recently, D. Che et al. proposed a “Stokes vector direct detection (SV-DD)” system, which can achieve both amplitude and phase modulation of the transmitting signals due to the self-coherent detection structure, which possesses the same SE as single-polarization coherent detection [5]. Therefore, SV-DD may be one of the promising DD architectures for short-reach optical links. Chromatic dispersion (CD) and rotation of state of polarization (RSOP) are two main impairments in the systems with high data speed over distances of several tens to one hundred kilometers. The accumulated CD can be as high as 1700 ps/nm (corresponding to a 100km G.652 single-mode fiber). In addition, RSOP is produced due to time-varying disturbance events in optical fibers. In some extreme weather scenarios such as lightning strikes, the transient electric and magnetic fields covering the neighboring area for several kilometers will be introduced. This will induce Kerr and Faraday effects in the fiber cable in this area. These transient events will generate ultrafast RSOP with speeds exceeding 300 krad/s or even 2 Mrad/s [6–8].

For an SV-DD system, the CD can be compensated by the pre-compensation at the transmitter or by the post-compensation at the receiver. With respect to CD pre-compensation, we should know the prior information of the actual transmission distance and the dispersion coefficient of the fiber. Moreover, there is always a residual CD (rCD) because of the mismatch between the evaluated CD and the real value induced in the optical fiber channel. On the other hand, at the receiver, the CD post-compensation can be correctly performed on the condition that the RSOP should be tracked in advance [9–12]. However, the CD (or rCD) will cause the constellation clouds to scatter about in Stokes space resulting in the failure of the blind RSOP equalization [13,14]. In addition, we will see in the analysis in Section 2 that for SV-DD systems the received signals are processed in Stokes space and the distortions induced by CD and RSOP have a nonlinear form. As a result, the CD and RSOP cannot be equalized sequentially using the post blind algorithms. Therefore, the naturally straightforward solution adopted in the literature is to use the training symbols (TS) to extract information on the RSOP prior to implementing the CD compensation [9,10,12]. TS based algorithms solved RSOP equalization problem in the expense of a bit overhead for training symbols (although it may be a low overhead), and the performance degradation due to the rCD.

In recently years, Kalman filter algorithm has been found to be used in the equalization in coherent optical communication. There are a number of significant publications which showed good impairment equalization abilities, such as polarization-state tracking [15–18], linear or non-linear phase noise tracking [19–21] and nonlinear channel impairments mitigation [22,23]. In addition, Kalamn filter has been shown to have advantages when dealing with RSOP in SV-DD systems [14]. Inspired by the previous works, we introduce a special Kalman filter structure to establish a blind joint compensation algorithm for both CD and RSOP in an SV-DD system.

In this article, we propose a joint blind post-equalization method to cope with the combined effects of CD and RSOP in an SV-DD system using a new time-frequency domain Kalman filter structure in which CD and RSOP can be equalized simultaneously rather than sequentially. The Kalman filter (KF) was originally presented for implementation in the time domain, while CD is induced in the frequency domain. For this reason, we designed a window structure to cover several received data symbols to allow the proposed Kalman filter to perform RSOP compensation in the time domain, followed by an transformation into the frequency domain to perform CD equalization.

The article is organized as follows. At first, we briefly introduce the concept of the SV-DD system and perform a simple analysis of the nonlinear form of the combined effect of CD and RSOP in the SV-DD system. Secondly, based on this distortion and on the Stokes vector receiver (SVR) model, we propose the equalization method based on the Kalman filter both in the time and frequency domain. The tracking of RSOP is performed in the time domain and the compensation for CD is conducted in the frequency domain. The proposed method was evaluated in a 28 Gbaud 16-QAM SV-DD system and the results showed this approach is very effective in the performance of joint blind equalization of CD and RSOP.

2. Problems associated with the nonlinear form of the combined effect of CD and RSOP in SV-DD system

The transmitter and receiver of the SV-DD system are illustrated in Fig. 1. At the transmitter, the continuous wave (CW) is divided into two orthogonal polarization light beam ( $X$ and $Y$ ) using a polarization beam splitter (PBS). One polarized component is introduced to an inphase quadrature modulator (IQM) that is driven by electrical signals for the modulated signal $E_{x}^{t}$ . The other component acts as a carrier $C$ . The two components are combined using a polarization beam combiner (PBC) and propagate through an optical fiber. At the receiver, the received signal $E_{x}^{r}$ and $E_{y}^{r}$ is obtained by a PBS. One part originating from of $E_{x}^{r}$ and $E_{y}^{r}$ gets into balanced photodetector 1 (B-PD1), producing an intensity given by $S_{1}^{r} = | E_{x}^{r} |^{2} - | E_{y}^{r} |^{2}$ . The other part which also originates from $E_{x}^{r}$ and $E_{y}^{r}$ enters into a 90° optical hybrid and prior to detection by B-PD2 and B-PD3, resulting in the signals $S_{2}^{r} = 2 Re (E_{x}^{r} \cdot E_{y}^{r}^{*})$ and $S_{3}^{r} = 2Im (E_{x}^{r} \cdot E_{y}^{r}^{*})$ . Combing the three outputs of B-PD1, B-PD2, and B-PD3, we obtain the received Stokes vector $S^{r} = {[\begin{matrix} S_{1}^{r} & S_{2}^{r} & S_{3}^{r} \end{matrix}]}^{T}$ .

Fig. 1 (a) SV-DD system transmitter. (b) SV-DD system receiver.

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When the optical waves propagate in the fiber, the signal suffers from both RSOP and CD distortion. Now, we will introduce a simple mathematical analysis to highlight the difficulties associated with compensating for CD and RSOP in the SV-DD system. At the transmitter, the signal can be described using the Jones vector as follows:

J^{t} = [\begin{matrix} E_{x}^{t} \\ C \end{matrix}]

According to the transformation relationship between a Jones vector and a Stokes vector, we obtain the corresponding Stokes vector at the transmitter end as follows:

S^{t} = [\begin{matrix} S_{1}^{t} \\ S_{2}^{t} \\ S_{3}^{t} \end{matrix}] = [\begin{matrix} | E_{x}^{t} |^{2} - | C |^{2} \\ 2 Re (E_{x}^{t} \cdot C^{*}) \\ 2Im (E_{x}^{t} \cdot C^{*}) \end{matrix}]

where

Re {\cdot}

and

Im {\cdot}

denote the real and imaginary parts, superscript

^{*}

denotes the complex conjugate. From the second and third rows of Eq. (2), we obtain

2 \cdot E_{x}^{t} \cdot C^{*} = S_{2}^{t} + j S_{3}^{t}

, which implies that

S_{2}^{t} + j S_{3}^{t}

contains the full amplitude and phase information of the transmitted signal

E_{x}^{t}

since the carrier C represents CW light whose amplitude can be considered as a constant with respect to time. Therefore, the desired recovered information at the receiver is

S_{2}^{t} + j S_{3}^{t}

.

As optical waves propagate in the fiber, the signal suffers from both RSOP and CD distortions and the received Jones vector $J^{r}$ can be expressed as [9]:

J^{r} = [\begin{matrix} E_{x}^{r} \\ E_{y}^{r} \end{matrix}] = \hat{D} \hat{U} [\begin{matrix} E_{x}^{t} \\ C \end{matrix}] = \exp [- j (\frac{β_{2} L}{2}) \frac{\partial^{2}}{\partial t^{2}}] [\begin{matrix} a & b \\ - b^{*} & a^{*} \end{matrix}] [\begin{matrix} E_{x}^{t} \\ C \end{matrix}]

where

\hat{D} = - j (β_{2} L / 2) \partial^{2} / \partial t^{2}

denotes the CD operator in which L and

β_{2}

are the fiber length and dispersion parameters. The unitary matrix constructed by the parameters

a

and

b

describes the RSOP in the fiber. Firstly, we will examine the situation whereby only CD distortion exists in the fiber. The carrier C is CW light that is not impacted by CD. After the receiver,

S_{2}^{r} + j S_{3}^{r}

can be written as follows:

S_{2}^{r} + j S_{3}^{r} = \overset{⌢}{D} {2 E_{x}^{t} \cdot C^{*}} = \overset{⌢}{D} {S_{2}^{t} + j S_{3}^{t}}

which takes a linear form of the CD operator. In this case,

S_{2}^{r} + j S_{3}^{r}

can be directly recovered subsequent to CD post-compensation by applying

{\hat{D}}^{- 1}

to Eq. (4). However, if the RSOP and the CD co-exist in the fiber, the result is quite different. According to Eq. (3), after the receiver,

S_{2}^{r} + j S_{3}^{r}

can be written as

S_{2}^{r} + j S_{3}^{r} = 2 E_{x}^{r} E_{y}^{r *} = a^{2} \overset{⌢}{D} {S_{2}^{t} + j S_{3}^{t}} - b^{2} (\overset{⌢}{D} {S_{2}^{t} + j S_{3}^{t}}) - 2 a b ({| \overset{⌢}{D} E_{x}^{t} |}^{2} - {| C |}^{2})

Here, the received $S_{2}^{r} + j S_{3}^{r}$ shows a nonlinear form of $\hat{D}$ and $\hat{U}$ . If we then separately perform CD and RSOP equalizations, e.g., first apply ${\hat{D}}^{- 1}$ to Eq. (5), it is not possible to completely recover the signal affected by CD distortion because of the nonlinear factor $| \hat{D} |^{2}$ in the third term in Eq. (5). Another option is to first compensate for RSOP. However, in the presence of CD, the “clouds” of samples in Stokes space will scatter about, as shown in Fig. 2, and thus the RSOP information becomes inaccessible.

Fig. 2 Stokes space representations of the 16QAM signal (a) ideal signal. (b) signal suffering from the accumulated CD of 300 ps/nm.

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The above analysis reveals that the CD and the RSOP cannot be equalized individually or sequentially using ordinary blind post algorithms. As such, other solutions should be considered. There were some examples of works in the literature based on training symbol assisted RSOP equalization prior to the application of ${\hat{D}}^{- 1}$ [9,10,12], which implies the absence of the 3rd term by making b = 0. However, as previously mentioned, the training symbol assisted RSOP equalization sacrifices a bit overhead for training symbols (although it may be a low overhead), and the performance degradation due to the rCD.

We are of the opinion that the aforementioned problem can be solved if an equalization scheme could be designed in which CD and RSOP are simultaneously equalized. We have determined that by designing a specially organized Kalman filter structure, joint blind post-equalization can be achieved for both CD and time-varying RSOP in the SV-DD system. Kalman filters have been successfully applied in optical fiber communications [14–28]. Moreover, there has been much progress in the application of the Kalman filter, and numerous structures of this filter have been designed to solve a wide range of problems in the different scenarios for optical communication systems [14,24–28]. By exploiting these advances, CD and the RSOP joint compensation in an SV-DD system may be possible. The detailed structural design of the Kalman filter used in this investigation will be described in the next section.

3. Solution for joint equalization of CD and RSOP in SV-DD system

In Section 2, we concluded that the CD and the RSOP cannot be equalized individually using the ordinary blind post algorithms in an SV-DD system because the CD operator $\hat{D}$ and the RSOP operator $\hat{U}$ act as nonlinear forms. We will design a special Kalman filter structure which can jointly equalize CD and time-varying RSOP simultaneously. In this section, we will describe in detail the special structured of the Kalman filter and its operation.

It is widely known that the RSOP distortion occurs in the time domain while CD is induced in the frequency domain and leads to pulse broadening. This pulse broaden causes inter-symbol interference in the time domain. However, the Kalman filter was originally designed to be implemented in the time domain. Therefore, we have to construct a Kalman filter structure for implementation in both the time and frequency domains. For this reason, we constructed a sliding window structure with a length L that can cover a number of received symbols [25]. At first, the time-varying RSOP can be tracked and equalized in the time domain. These symbols are then transformed into the frequency domain so that CD compensation can be performed. The sliding window then shifts forward with a step size of $Δ l$ to cover the next queue of symbols, and another iteration is repeated. The sliding window structure is illustrated in Fig. 3.

Fig. 3 The diagram of the sliding window structure.

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We designed different kinds of Kalman filter structures to equalize linear distortions in optical coherent communication systems [14,24–28]. It was determined that the Kalman filter is flexibility and is easily modified. The appropriate design for the practical problem under consideration is important. In this report, apart from the sliding window structure, the proposed Kalman filter design focuses on the following three important issues: 1) the selection of the appropriate state vector whose components will be monitored by the Kalman filter; 2) the appropriate compensation architecture to completely compensate for the distortions induced by CD and RSOP; 3) the appropriate design of the measurement innovation or residual. These issues will be examined individually in the following sections, although not in the listed order (because the performing and reasoning sequences are different).

We will begin by examining issue 2). For the compensation architecture, the appropriate compensation matrices for the RSOP ( ${\hat{H}}_{S}^{- 1}$ ) and CD ( ${\hat{D}}^{- 1}$ ) should be determined. ${\hat{H}}_{S}^{- 1}$ is performed in the time domain and ${\hat{D}}^{- 1}$ is performed in the frequency domain. In our previous work, we highlighted that the RSOP equalization operator should include 3 independent parameters and took a 2 × 2 matrix ${\hat{H}}_{J}^{- 1}$ in Jones space and a 3 × 3 matrix ${\hat{H}}_{S}^{- 1}$ in Stokes space as in Eqs. (6) and (7) [28]:

{\hat{H}}_{J}^{- 1} = (\begin{matrix} \cos κ e^{- j ς} & \sin κ e^{j η} \\ - \sin κ e^{- j η} & \cos κ e^{j ς} \end{matrix})

and

{\hat{H}}_{S}^{- 1} = (\begin{matrix} \cos 2 κ & \sin 2 κ \cos (η + ζ) & - \sin 2 κ \sin (η + ζ) \\ - \sin 2 κ \cos (η - ζ) & \cos^{2} κ \cos 2 ζ - \sin^{2} κ \cos 2 η & \sin^{2} κ \sin 2 η - \cos^{2} κ \sin 2 ζ \\ \sin 2 κ \sin (η - ζ) & \cos^{2} κ \sin 2 ζ + \sin^{2} κ \sin 2 η & \cos^{2} κ \cos 2 ζ + \sin^{2} κ \cos 2 η \end{matrix})

where

κ

,

η

and

ζ

represent three rotation angles.

The CD compensation operator in the frequency domain can be described as:

{\hat{D}}^{- 1} (Φ, ω) = \exp (- j \frac{Φ λ^{2} ω^{2}}{4 π c})

where

Φ

is the accumulated CD in units of ps/nm.

Next, we examine issue 1). According to Eqs. (7) and (8), the RSOP in the fiber can be completely characterized by the parameters $κ$ , $η$ and $ζ$ , and the CD by $Φ$ . Therefore, we choose the state vector form as shown in Eq. (9). The first 3 parameters reflect the RSOP distortion as in Eq. (7) and the 4th parameter represents the level of distortion induced by CD as shown in Eq. (8).

x = {(κ, η, ζ, Φ)}^{T}

Finally, we consider issue 3). For the Kalman filter, the measurement and measurement innovation are most important and help to define the filter converge. The constellation space was chosen as the measurement space. After equalization of RSOP and CD, the constellation of 16 QAM signals should converge to three circles with radii of $r_{1}, r_{2}$ and $r_{3}$ . Therefore, the measurement vector $z$ and the residual $e$ should be:

e (x) = z (x) - h (x) = 0 - (u_{x} u_{x}^{*} - r_{1}^{2}) (u_{x} u_{x}^{*} - r_{2}^{2}) (u_{x} u_{x}^{*} - r_{3}^{2})

where

u

is the equalized signal. It will be seen that in each iteration, it is this procedure serves to examine the quality of the RSOP and CD compensations and ensures that simultaneous compensation of RSOP and CD is possible.

After the aforementioned three important design issues have been considered, the entire structure of the proposed Kalman filter should be considered. The recursion equations of the Kalman algorithm are given as Eqs. (11)–(15). In Fig. 4, it is seen that the filter mainly consists of three stages: the prediction stage, the compensation stage and the correction stage as shown.

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

P_{k | k - 1} = P_{k - 1} + Q_{k - 1}

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_{k} - h_{k} ({\hat{x}}_{k | k - 1}))

P_{k} = (I - G_{k} H_{k}) P_{k | k - 1}

where the subscript

k / k - 1

and

k

(or

k - 1

) denote the a priori estimate and a posteriori state estimate.

h

is the measurement function, and

H

is the Jacobian matrix of partial derivatives of

h (x_{k})

with respect to

x_{k}

.

G k

is the Kalman gain.

P

is the error covariance matrix.

Q_{k - 1}

and

R_{k}

denote the noise covariance matrices.

Fig. 4 The flow chart of the time-frequency domain KF.

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All the matrices of KF have to be defined and some parameters must be initialized. The initial state vector and error covariance matrix are chosen as ${\hat{x}}_{0} = {(0, 0, 0, 0)}^{T}$ and $P_{0} = d i a g [10^{- 8}, 10^{- 8}, 10^{- 8}, 0.8]$ . Moreover, by optimizing the noise covariance matrices, $Q_{k - 1}$ and $R_{k}$ are given as $Q = d i a g (5 \times 10^{- 8}, 5 \times 10^{- 8}, 5 \times 10^{- 8}, 10^{- 4})$ and $R = 12000$ . After initialization, the recursive algorithm can now be applied. As shown in Fig. 4, in the prediction stage, the priori estimation of ${\hat{x}}_{k | k - 1}$ and $P_{k | k - 1}$ are obtained from the posteriori estimations of the previous iteration. In the compensation stage, based on the priori estimation of the state vector ${\hat{x}}_{k | k - 1}$ and the extra sliding window operation, RSOP and CD are compensated in the time and frequency domains respectively. At the correction stage, the state vector is updated according to the priori estimation of ${\hat{x}}_{k | k - 1}$ and the innovation $e_{k} = z_{k} - h ({\hat{x}}_{k | k - 1})$ which is scaled by the Kalman gain $G k$ , and is then fed to the prediction stage for the next iteration. It can be seen that using the proposed specially designed Kalman structure, this innovation checks the performance after completion of both RSOP and CD compensation, and facilitates the simultaneous compensation of these two factors.

4. Simulation and analysis

4.1 Simulation configuration

The performance of the time-frequency domain Kalman method was evaluated using numerical simulations in a 28-Gbaud 16-QAM SV-DD system. In the simulation, we only consider the impairments of ASE (amplified spontaneous emission) noise, CD and RSOP. The simulation platform is shown in Fig. 5. At the transmitter, the CW is divided into two beams. The upper beam is introduced to an IQM that is driven by electrical signals for the modulated signal $E_{x}^{t}$ . The lower beam acts as the optical carrier $C$ . The two beams are then combined using a PBC. Along the fiber, the optical signal suffers from ASE noise, CD and RSOP distortion successively. The CD is assumed to be static, and the RSOP is dynamic with a linear increase of the azimuth angle and the phase angle with time. At the receiver, the Stokes vector is directly received by the SVR, which contains a 90° hybrid and three B-PDs. In the DSP module, the proposed Kalman method is employed to compensate for RSOP and CD. Finally, the BER (bit error rate) is calculated.

Fig. 5 Numerical simulation system framework.

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4.2 Performance and discussions

For the proposed KF method, the sliding window step is set as 4-symbol. In addition, the window length undoubtedly determines the final performance of the joint equalization method because the FFT is used. A large window length will enlarge the CD tolerance of the SV-DD receiver and limit the tracking speed response to the time-varying RSOP, and also increases the complexity. However, a narrow window length will obviously limit the CD equalization performance and increase the response speed. Figure 6 shows the BER performances for different window lengths N with the value of 2e4, 2e6, 2e7, 2e8, 2e9 and 2e11. It is evident that when N is 2e4, the proposed Kalman filter is sensitive to the increase of CD. When N is greater than 2^7, the performance of the proposed scheme cannot work well at the ultra-fast speed of RSOP. Considering the trade-off between the complexity and compensation performance, we set the window length as 2e7 in the following discussions.

Fig. 6 Performance evaluation (a) BER vs. the accumulate CD for different window lengths of the Kalman filter (24 dB OSNR, 1 Mrad/s RSOP). (b) BER vs. RSOP for different window lengths of the Kalman filter (24 dB OSNR, 510ps/nm accumulate CD).

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Then we comprehensively verified the performance of the proposed KF method. Figure 7(a) represents the BER as a function of the OSNR with different accumulated CD and speed of variation of RSOP. In order to verify the effectiveness of the proposed KF method, CD pre-compensation (CD-pre) is implemented at the transmitter (assume CD is completely compensated), the training symbols (TS) assisted RSOP equalization method [29] and least mean square (LMS) [9] are utilized at the receiver. For the TS assisted algorithm 9 training symbols were added in front of every 320 symbols, which implies 2.7% overhead. When using LMS, the step sizes is optimized as 1e-5 and the filter lengths is set as 25. We can see that, for weak impairment cases (CD = 1020ps/nm + RSOP = 300krad/s, and CD = 1530ps/nm + RSOP = 600krad/s), the BER performance for CD-pre + LMS is comparable to Kalman method (the slight performance degradation compare to Kalman can be seen). But for severe impairment case of RSOP = 1Mkrad/s and the accumulated CD = 2040 ps/nm, the LMS method fails to work, while Kalman works well, and CD-pre + TS shows a slight worse performance. In summary, the proposed Kalman scheme provides the excellent performance under the different situations of RSOP from 300krad/s to 1Mrad/s with the accumulated CD from 1020ps/nm to 2040ps/nm. Figure 7(b) depicts the OSNR penalty vs. RSOP speed with different accumulated CD from 510ps/nm to 2040ps/nm. The simulation results show that under the severe distortion of RSOP = 1 Mrad/s and the accumulated CD = 2040 ps/nm, the proposed KF scheme has lower OSNR penalty (around 0.8dB) than CD-pre + TS method, which mean that at the extreme scenario of severe impairment induced by CD and RSOP the proposed KF scheme has lower OSNR requirement compare to CD-pre + TS method. This advantage is important for a direct detection system since direct detection systems commonly require higher OSNR compared with the coherent communication system. We also discuss the performance under lower RSOP speeds. Figures 7(c) depict BER values with different speed of RSOP at OSNR of 23 dB and the accumulated CD of 1530ps/nm. The simulation results show that the proposed Kalman scheme exhibits good performance under the different speed of RSOP from 100krad/s to 200krad/s. According to the above analysis, we can clearly conclude the proposed Kalman scheme has better performance compared to the CD-pre + TS method in both ordinary environment and extreme scenario. Therefore, the Kalman method is one of good promising algorithms for SV-DD system to cope with both CD and RSOP. The above discussions for CD-pre + TS are assumed that CD is completely compensated at the transmitter. Now we investigate the rCD tolerance of the CD-pre + TS method as shown Fig. 7(d). We can see that, with the error range about 3% (120 km G. 652 fiber), the TS method cannot work normally. For KF, which is a totally blind equalization method, the problem is the ability how many total CD can be compensated rather than the tolerance to rCD. We make a star mark to indicate the BER performance for the KF to blindly compensate the total CD.

Fig. 7 Performance evaluation (a) BER vs. OSNR for different accumulated CD and RSOP. (b) OSNR penalty as a function of different accumulated CD and RSOP. (c) BER vs. RSOP. (d) BER vs. the residual CD.

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Then the performance of the proposed Kalman method is investigated with different channel parameters. Figure 8(a) depicts the accumulated CD tolerance of the proposed Kalman method with different amount of RSOP and OSNR of 24 dB. The results show that the method has a good performance for a relatively large CD (more than 2550 ps/nm, corresponding to a 150 km G.652 fiber) combined with ultra-fast RSOP (up to 2 Mrad/s). This implies, that the proposed method is indeed appropriate for joint blind equalization both for CD and RSOP in an SV-DD system. Then we discuss the CD compensation accuracy of the proposed method at OSNR of 24 dB, RSOP of 1 Mrad/s and the accumulated CD of 2040 ps/nm. The mean CD estimation errors from 50 independent tests were calculated and the results are shown in Fig. 8(b). It is evident that the mean error is smaller than 5 ps/nm when OSNR is higher than 21 dB, which means that the proposed method has an accuracy better than 99.7%. At the same time CD tracing performance is also shown in Fig. 8(c). We can see that within 512 iterations the CD tracing is completed. At this moment, we can switch our Kalman scheme to a smart mode of the proposed scheme, in which CD monitoring and updating cease.

Fig. 8 Performance evaluation (a) BER vs. different accumulated CD (OSNR = 24 dB). (b) the accumulated CD estimation mean error vs. OSNR (1 Mrad/s RSOP, 2040 ps/nm accumulated CD). (c) the accumulated CD vector tracing curve.

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Now we make an explanation about the normal mode and the smart mode of the proposed scheme. For the normal mode, section 3 has made the detailed discussions in which both RSOP and CD are tracked and updating is performed forever. Since CD is static impairment, for complexity saving, we can construct a smart mode of the proposed scheme, in which after the blind CD compensation is achieved by the normal mode with a correct CD estimation, the CD monitoring and updating are stopped, only RSOP tracking continues with the CD compensation just by multiplying a factor in Eq. (8).

4.3 Computational complexity

In this section, we compare complexities of the Kalman filter and the CD-pre + TS. All the computations are based on the optimum implementation. A complex multiplier is counted for 4 real multipliers and 2 real adders, and a complex addition is equivalent to 2 real adders. The results are shown in Table1.

Table 1. Computational complexity comparison

View Table

It can be clearly seen that the CD-pre + TS method is significantly less complexity demanding, with the ratio of total number of real multipliers / adders as 81/46. For the Kalman method, the window length N undoubtedly determines its complexity and the CD tolerance. So the trade-off must be considered. In practical application, we can choose the appropriate N value based on the required transmission distance of the fiber link. As shown in Figs. 6(a), when the window lengths N is set to be 2^7, the accumulated CD tolerances can achieve 2550ps/nm (equivalent to a 150 km G. 652 fiber), corresponding to a high complexity. When the transmission distance is less than 90 km, we can set N as 2^6. In this case, the ratio of total number of real multipliers / adders is calculated as 545/305 (for smart mode), which is less than half complexity of that using 2^7 window length (1185/625 for smart mode). Nevertheless, the Kalman based method still has relative larger computational overhead. However, we believe we can make further simplification of the Kalman structure to reduce the complexity in our next works.

4.4 Comprehensively assessment

For an SV-DD short reach system, one of the problems is the equalization algorithm both responsible for CD and RSOP. TS based algorithm and the Kalman based scheme proposed in this paper are compared in detail. We take the CD-pre + TS as the comparison counterpart of the proposed Kalman scheme. The two algorithms show their pros and cons in above discussions (sections 4.2 and 4.3).

Now we hope to make a comprehensively assessment for these two kinds of algorithms. TS based algorithm exhibits extreme lower complexity, relative strong ability for tracking RSOP with a degradation for ultra-fast RSOP, strong CD compensation ability, but a weaker rCD tolerance. In contrast, proposed Kalman scheme is a blind equalization algorithm both for CD and RSOP. It possesses strong RSOP tracking ability (up to 2 Mrad/s, which belongs to an extreme environment), relative large accumulated CD tolerance (more than 2550 ps/nm, corresponding to a 150 km G.652 fiber), but relative high complexity.

Therefore, in the ordinary environments such as ground fiber cables without outside strong vibration impact, TS based algorithms may be a good choice. However, the proposed Kalman scheme provides a universal equalization algorithm to cover both ordinary environment and the extreme scenarios, such as lightning strikes in a rainy day, especially for the systems with aerial fiber cables. But the high complexity is a problem to be further solved.

5. Conclusions

In this report, the difficulties associated with compensating for the combined distortions of CD and RSOP in an SV-DD system were discussed in detail. A joint blind equalization scheme was proposed for equalization of CD and RSOP based on a new time-frequency domain Kalman filter structure. This specially designed Kalman structure facilitates the joint CD and RSOP compensation in SV-DD systems. The proposed scheme was evaluated in a 28-Gbaud 16-QAM SV-DD system. The results show that the proposed KF method has better performance, higher spectral efficiency but higher complexity compared with CD-pre + TS methods. With a OSNR of 24 dB, the scheme is strongly immune to CD (more than 2550 ps/nm) combined with ultra-fast RSOP (up to 2 Mrad/s). Moreover, the mean CD estimation errors are smaller than 5 ps/nm when the OSNR is higher than 21 dB and the RSOP is as high as 1 Mrad/s. Therefore, the proposed Kalman scheme is indeed appropriate for SV-DD systems for joint blind equalization for CD and RSOP.

Funding

National Natural Science Foundation of China (61571057, 61527820,61575082); BUPT Excellent Ph.D. Students Foundation (CX2019317).

References

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Joint blind equalization of CD and RSOP using a time-frequency domain Kalman filter structure in Stokes vector direct detection system

Abstract

1. Introduction

2. Problems associated with the nonlinear form of the combined effect of CD and RSOP in SV-DD system

3. Solution for joint equalization of CD and RSOP in SV-DD system

4. Simulation and analysis

4.1 Simulation configuration

4.2 Performance and discussions

4.3 Computational complexity

4.4 Comprehensively assessment

5. Conclusions

Funding

References

Cited By

Figures (8)

Tables (1)

Equations (15)

Optics Express

	Window Lengths	Mode	Real Multiplication	Real Addition
KF	2^7	Normal	1435	763
	2^7	Smart	1185	625
	2^6	Normal	667	380
	2^6	Smart	545	305
CD-pre + TS			81	46