Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems

Yiqiao Feng; Linqian Li; Jiachuan Lin; Hengying Xu; Wenbo Zhang; Xianfeng Tang; Lixia Xi; Xiaoguang Zhang

doi:10.1364/OE.24.025491

1. Introduction

With the dramatic increase in demand for broadband network, the technical proposal based on advanced modulation formats, polarization division multiplexing (PDM), coherent detection and digital signal processing (DSP) is a powerful way to improve transmission rate in coherent optical communication system. PDM system, which use two orthogonal states of polarization (SOPs) to transmit two channels of optical signals at the same wavelength, can double the rate of transmission and therefore attracts much attention. PDM system is more sensitive to polarization effect. Three main polarization effects in optical fiber communication system are rotation of SOP (RSOP), polarization mode dispersion (PMD) and polarization dependent loss (PDL). SOP of optical signals change randomly during transmission due to the time-varying birefringence distributed along the fiber, whose time variation can be as fast as several hundred krad/s of the Stokes vector on the Poincare sphere [1]. Eventually, the crosstalk between the two polarizations is caused by joint effects of SOP, PMD and PDL. Hence, a joint treatment with high-speed convergence of SOP tracing, PMD and PDL equalization is significant in PDM coherent optical communication system. Due to the development of DSP techniques, a series of blind polarization demultiplexing (PolDemux) algorithms have been proposed. Constant modulus algorithm (CMA) using four adaptive finite-impulse-response (FIR) filters is widely adapted to equalize various linear impairment for PDM signals of quadrature phase-shift keying (QPSK) modulation format. However, CMA has a low speed of convergence and suffers from the problem of singularity [2]. Multi-modulus algorithm (MMA) including radius-directed algorithm (RDA), cascaded multi-modulus algorithm (CMMA) and other variants proposed for advanced modulation formats have the same disadvantages with CMA [3,4]. Stokes space (SS) method can achieve fast convergence, but it applies only to QPSK format with small value of PMD, and behaves degraded performance in a time varying SOP environment [5,6]. A natural idea is to combine SS method and CMA/MMA to improve convergence speed and solve the singularity problem. Unfortunately, it has low speed of tracing SOP [7]. The other categories of algorithms using extended Kalman filter (EKF) are proposed for SOP tracing and PolDemux [5,6,8–11]. However, these algorithms only can trace SOP with small value of PMD or equalize for PMD without fast time varying RSOP.

In this paper, we present a novel method based on extended Kalman filter (EKF), which has a capability to cope with multi-polarization effects such as SOP tracing, PolDemux, and equalization of PMD and PDL in a coherent detection system with polarization division multiplexing 16 quadrature amplitude modulation (PDM-16QAM). The mathematical model of the joint multi-polarization-effect treatment method is presented, and simulation verification is carried out. The results show that the method has better bit-error-rate (BER) performance than CMA-MMA when the OSNR is greater than 21 dB, which possesses great tracing speed of SOP, large equalization range of PDL and moderate tolerance to PMD compared with CMA-MMA. In principle, the proposed method can be applicable to other higher modulation formats, not limited to 16QAM. Finally, the table of complexity comparison between proposed scheme and CMA-MMA-BPS is given.

2. Principle

2.1 Channel model

To elaborate the principle, we consider optical transmission model in Fig. 1, in which the effects of SOP variation, chromatic dispersion (CD), PMD, and PDL are included.

Fig. 1 The model of optical transmission.

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And the model can be expressed as follow:

q (t) = K ℱ^{- 1} {M e^{\frac{1}{2} j D_{ω} ω^{2}} ℱ {R x (t) e^{j (Δ ω t + θ)}}} + η,

in which

q (t)

, _$K$, _$M$, _{$D_{ω}$}, _$ω$, _$R$,

x (t)

, _{$Δ ω$}, _$θ$, _$η$ represent the received signal in dual-polarization, PDL Jones matrix, the first-order PMD Jones matrix in frequency domain, chromatic dispersion (CD) coefficient, optical angular frequency, SOP rotation matrix, the transmitted signal, the frequency offset between the transmitter laser and the local laser, carrier phase noise and additive white Gaussian noise respectively.

ℱ {\cdot}

and

ℱ^{- 1} {\cdot}

represent the Fourier transform and inverse Fourier transform. The SOP rotation matrix can be expressed as [6]:

R = Α Φ = [\begin{matrix} \cos α & - \sin α \\ \sin α & \cos α \end{matrix}] [\begin{matrix} e^{- j \frac{φ}{2}} & 0 \\ 0 & e^{j \frac{φ}{2}} \end{matrix}],

where _$j$ is the imaginary unit, _{$2 α$}, _$φ$ represent the azimuth angle and the phase angle of SOP in Stokes space. The first-order PMD Jones matrix can be expressed as [6]:

M = \cos (ω τ) I + \frac{\sin (ω τ)}{τ} N,

in which _$I$ represents unit matrix, _$τ$ is half of differential group delay (DGD). When _{$ω τ$}is not small enough, we cannot use approximate expression in [6]. _$N$ is the sum of Pauli matrices of PMD vector components:

N = [\begin{matrix} j τ_{1} & j τ_{2} + τ_{3} \\ j τ_{2} - τ_{3} & - j τ_{1} \end{matrix}],

where _{$τ_{1}$}, _{$τ_{2}$} and _{$τ_{3}$} are the three components of PMD vector in Stokes space. We can express _$τ$as:

τ = \sqrt{τ_{1}^{2} + τ_{2}^{2} + τ_{3}^{2}} .

Hence, the Jones vector _{$Z (ω) = e^{\frac{1}{2} j D_{ω} ω^{2}} ℱ {R x (t) e^{j (Δ ω t + θ)}}$}, which describes an optical dual-polarization signal in frequency domain in Eq. (1), is transformed by PMD matrix _$M$. The Jones vector in time domain after suffering from PMD can be expressed as:

\begin{matrix} y (t) = ℱ^{- 1} {M Z (ω)} \\ = ℱ^{- 1} {\cos (ω τ) Z (ω) + \frac{\sin (ω τ)}{τ} N Z (ω)} \\ = \frac{1}{2} [z (t + τ) + z (t - τ)] - j \frac{N}{2 τ} [z (t + τ) - z (t - τ)], \end{matrix}

in which, _{$z (t)$} is the inverse Fourier transform of _{$Z (ω)$}. Equation (6) can be written as opposite form:

z (t) = ℱ^{- 1} {M^{- 1} Y (ω)} = \frac{1}{2} [y (t + τ) + y (t - τ)] + j \frac{N}{2 τ} [y (t + τ) - y (t - τ)] \cdot

The Jones matrix _$K$represents PDL described as [7,12]:

K = R_{1}^{- 1} [\begin{matrix} \sqrt{1 + ρ} & 0 \\ 0 & \sqrt{1 - ρ} \end{matrix}] R_{1},

where _$ρ$ is a real number (_{$- 1 \leq ρ \leq 1$}), and _{$R_{1}$} is a rotation matrix converting eigenstates of PDL into the x- and y-polarization. _{$R_{1}$} can be expressed as:

R_{1} = [\begin{matrix} \cos β & - \sin β \\ \sin β & \cos β \end{matrix}],

where _$β$ is the angle between the eigenstates vector of PDL and the laboratory coordinate. PDL can be also expressed in dBs as:

Γ (d B) = 10 \log_{10} \frac{1 - ρ}{1 + ρ} .

2.2 Principle of tracing and equalization

CD as a static impairment is polarization independent, hence we assume that it has been compensated by other module before the following polarization treatment module discussed in this paper. Omit CD item and substitute Eqs. (2) and (7) into Eq. (1), the SOP tracing and PolDemux process can be expressed as:

\begin{matrix} Φ x (t) e^{j (Δ ω t + θ)} + η^{'} = Α^{- 1} ℱ^{- 1} {M^{- 1} ℱ {K^{- 1} q (t)}} \\ = \frac{1}{2 τ} Α^{- 1} (τ I + j N) K^{- 1} [q (t + τ) - q (t - τ)] . \end{matrix}

Where I is identity matrix. Using time discretization with _{$t = k T_{s}$} (_{$T_{s}$} is sampling interval), Eq. (11) can be expressed approximately:

Φ_{k} x_{k} e^{j (k Δ ω T_{s} + θ_{k})} + {η^{'}}_{k} = \frac{1}{2 r T_{s}} Α_{k}^{- 1} (r T_{s} I + j N) K_{k}^{- 1} [(q_{k + r} - q_{k - r})] .

Where _{$r = r o u n d (τ / T_{s})$} represents the nearest integer of normalized _$τ$. When _$r$ is zero, Eq. (12) degenerates into the Eq. (29) in [6].

The proposed polarization effect treatment module as shown in Fig. 2 can be divided into two stages. The first stage equalizes PDL, the azimuth angle _$α$ of SOP and PMD, based on the right side of Eq. (12). The second stage compensates the phase angle _$φ$ of SOP and phase noise, based on the left side of Eq. (12).

Fig. 2 The configuration model of receiver including the proposed polarization effect treatment module.

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We note that PDL will give rise to the translation of signal points in Stokes space, and the normalized distance between the coordinate origin and the center of gravity of signal points is _$ρ$ in Eq. (8) [12]. Although time varying SOP and PMD changed the envelope of signal points, we can still estimate PDL by calculating the center of gravity of signal points. The _$ρ$ in Eq. (8) and the _$β$ in Eq. (9) can be calculated as follows:

ρ = \sqrt{x_{k}^{2} + y_{k}^{2} + z_{k}^{2}},

β = \frac{1}{2} arc \tan (\frac{y_{k}}{x_{k}}) .

where _{$(x_{k}, y_{k}, z_{k})$} represent the normalized coordinates of the center of gravity of signal points used for computation in Stokes space. After we get _$ρ$ and _$β$, we can obtain the inverse matrix of _$K$, and PDL is then equalized at first in stage 1.

For 16QAM or other higher order modulation formats, there are too many ideal discrete points in Stokes space compared with in Jones space, that is hard to equalize SOP rotation and PMD simultaneously. Therefore we propose a two stage method for advanced modulation formats to track and equalize the impairment due to SOP rotation and PMD in Jones space using extended Kalman filter.

When using Kalman filter, it is important to construct a series of measurement parameters. After equalizing PMD and a part of SOP rotation, the 16QAM signal constellation should converge to three circles, just like the output if we would use CMA/MMA as PolDemux method. However, generally, we expect above mentioned measurement parameters to converge to a constructed constant or a constant vector rather than three. Hence a natural idea is to construct a measurement vector whose component is as follow:

h_{k} = (u_{k} \cdot u_{k}^{*} - r_{1}^{2}) (u_{k} \cdot u_{k}^{*} - r_{2}^{2}) (u_{k} \cdot u_{k}^{*} - r_{3}^{2}),

in which _$u$ and _$u^{*}$ are equalized signal and its complex conjugation. _{$r_{1}$}, _{$r_{2}$} and _{$r_{3}$} represent three radii of the three circles. For dual-polarization signal, the equation above can be written as:

h_{k} = [\begin{matrix} (u_{x, k} u_{x, k}^{*} - r_{1}^{2}) (u_{x, k} u_{x, k}^{*} - r_{2}^{2}) (u_{x, k} u_{x, k}^{*} - r_{3}^{2}) \\ (u_{y, k} u_{y, k}^{*} - r_{1}^{2}) (u_{y, k} u_{y, k}^{*} - r_{2}^{2}) (u_{y, k} u_{y, k}^{*} - r_{3}^{2}) \end{matrix}],

in which _{$u_{k} = {[u_{x, k}, u_{y, k}]}^{T} = Φ_{k} x_{k} e^{j (k Δ ω T_{s} + θ_{k})} + {η^{'}}_{k}$} represents equalized dual-polarization signal. For tracing PMD and SOP azimuth angle, we define a state vector which contains four unknown variables on the right side of Eq. (12):

s_{k} = {[α_{k}, τ_{1, k}, τ_{2, k}, τ_{3, k}]}^{T} .

Hence the innovation vector for the extended Kalman filter can be written as follow:

δ (s_{k}) = [\begin{matrix} 0 \\ 0 \end{matrix}] - h (s_{k}) .

Obviously, the proposed method can be extended to other high order modulation formats, just need to increase or decrease the number of the radius rather than increase the dimension of residual in [10]. Since _{$h_{k}$} exhibits nonlinear expressions of the state vector _{$s_{k}$}, it needs to be approximated by a Jacobian matrix of partial derivatives _$H$, which can be calculated as follow [13]:

H_{i j, k} = \frac{\partial h_{i, k}}{\partial s_{j, k}}, i = 1, 2 and j = 1, 2, 3, 4.

Applying extended Kalman filter recursive algorithm, the following equations can be obtained [13]:

K_{k} = P_{k}^{-} H^{†} ({\hat{s}}_{k}^{-}) {[H ({\hat{s}}_{k}^{-}) P_{k}^{-} H^{†} ({\hat{s}}_{k}^{-}) + Q_{2}]}^{- 1},

{\hat{s}}_{k} = {\hat{s}}_{k}^{-} + K_{k} δ ({\hat{s}}_{k}^{-}),

{\hat{s}}_{k + 1}^{-} = {\hat{s}}_{k},

P_{k} = P_{k}^{-} - K_{k} H ({\hat{s}}_{k}^{-}) P_{k}^{-},

P_{k + 1}^{-} = P_{k} + Q_{1},

where superscript “_$†$” represents Hermitian conjugation, superscript “_$-$” represents a predicted estimate, _{${\hat{s}}_{k}$} is the filtered estimate of the state vector _{$s_{k}$}, the matrices _{$Q_{1}$}, _{$Q_{2}$}, _{$P_{k}$} and _{$K_{k}$} are the correlation matrix of process noise, correlation matrix of measurement noise, correlation matrix of the error in _{${\hat{s}}_{k}$} and Kalman gain, respectively.

Through the first stage extended Kalman filter above, only frequency offset _{$Δ ω$}, phase angle _$φ$ of SOP, and phase noise _$θ$ are unknown now. Frequency offset as a static impairment like CD, does not need to be compensated here using Kalman filter. Therefore, for reducing the complexity of the proposed algorithm, we assume here that frequency offset has be compensated by other more simple algorithm before carrying out the second stage. We trace the phase angle and phase noise by using another extended Kalman filter which is cascaded after the first one. Since the output of the first EKF has converged to three circles, we can partition 16QAM to two QPSKs and one 8-PSK, and then trace the variables by the two QPSKs. The ideal constellation points of QPSK should locate on the angle bisector of x-axis and y-axis, for this reason, we can construct the following state vector and measurement vector:

{s^{'}}_{k} = {[φ_{k}, θ_{k}]}^{T},

{h^{'}}_{Q P S K, k} = [\begin{matrix} Re {{u^{'}}_{Q P S K, x, k}}^{2} - Im {{u^{'}}_{Q P S K, x, k}}^{2} \\ Re {{u^{'}}_{Q P S K, y, k}}^{2} - Im {{u^{'}}_{Q P S K, y, k}}^{2} \end{matrix}],

where _{$Re {\cdot}$} and _{$Im {\cdot}$} represent real part operator and image part operator respectively. Therefore the innovation vector for the second Kalman filter can be expressed as:

{δ^{'}}_{k} ({s^{'}}_{k}) = [\begin{matrix} 0 \\ 0 \end{matrix}] - {h^{'}}_{Q P S K} ({s^{'}}_{k}) .

Applying Eqs. (25)-(27) to Eqs. (19)-(24), we can get another complete extended Kalman filter for the tracing of phase angle of SOP and phase noise.

3. Simulation and discussions

3.1 Simulation configuration

To verify the proposed method, we conduct numerical simulations for PDM-16QAM as shown in Fig. 3. At the transmitter, a 224Gbit/s PDM-16QAM signal which contains two 28Gbaud/s orthogonal polarization 16QAM signals is generated. The endless SOP rotation is emulated by sweeping azimuth angle _{$2 α$} from _$0^{\circ}$ to _{$360^{\circ}$} and the phase angle _$φ$ from _{$- 180^{\circ}$} to _{$180^{\circ}$}. CD is subsequently added. PMD is emulated by cascading 100 randomly distributed first-order PMD matrices [14,15], and the mean DGD of the PMD matrices is taken as the value of PMD. Then PDL is emulated by one PDL matrix with a fixed rotation angle. After that we add amplified spontaneous emission (ASE) noise into the signal. At the receiver, the signal is divided into two polarization components by polarization beam splitter (PBS), then sampled 4 samples per symbol by analog to digital converter (ADC). CD compensation is followed in the frequency domain. Then the first stage of the proposed algorithm is used for PolDemux. As the comparison reference, CMA-MMA is used independently. After that, frequency estimation is conducted. Then we use the second stage of the proposed algorithm to estimate phase angle of SOP and phase noise. Also, as a comparison reference blind phase search (BPS) algorithm is used independently. At last, BER is calculated.

Fig. 3 Numerical simulation system framework with two extended Kalman filters.

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3.2 Performance comparison

Firstly, we compare the performance between the proposed method and CMA-MMA-BPS carrying 50 simulations per tested parameters and setting 2¹⁶ symbols per simulation. For a Kalman filter, initialization is important [16]. In the first stage of the proposed algorithm, we set the _{$Q_{1}$} and _{$Q_{2}$} as _{$d i a g [1 e - 6, 1 e - 8, 1 e - 8, 1 e - 8]$} and _{$d i a g [500, 500]$} after multiple optimization, where _{$d i a g [a_{1}, a_{2}, ..., a_{n}]$} represents a _{$n \times n$} diagonal matrix. In the second stage of our algorithm, we set the _{${Q^{'}}_{1}$} and _{${Q^{'}}_{2}$} as _{$d i a g [1 e - 5, 1 e - 3]$} and _{$d i a g [10, 10]$}. When using CMA-MMA-BPS, the step sizes of CMA and MMA are set as 1e-4 and 1e-6, respectively, the filter lengths of CMA and MMA are both set as 35, the number of the test phase angles for BPS is set as 32 and the block length for BPS is set as 65. All the initialization parameters for two Kalman filters and convergence parameters for CMA-MMA-BPS mentioned above are chosen through the optimizations. Figure 4(a) shows the BER for CMA-MMA-BPS and the proposed method (here denoted as EKF) as a function of OSNR. The DGD, PDL, angular frequency of azimuth angle _$α$ and angular frequency of phase angle _$φ$ are 14 ps, 4 dB, 500 krad/s and 400 krad/s, respectively. It is clear that the proposed method offers an enhanced performance for OSNR values greater than 21 dB. The performance between EKF and CMA-MMA-BPS is compared at OSNR of 23 dB in Figs. 4(b)-4(e). Figures 4(b)-4(c) show that the dynamic tracing performance of the proposed method is much greater than CMA-MMA-BPS. Figure 4(d) shows the BER performances of two algorithms as a function of PDL, and it indicates the proposed method has better PDL tolerance than CMA-MMA-BPS. Figure 4(e) shows that the two algorithms have similar PMD tolerance.

Fig. 4 BER as the functions of (a) OSNR, (b) angular frequency of azimuth angle _$α$, (c) angular frequency of phase angle _$φ$, (d) PDL, and (e) PMD, respectively. (b)-(e) are at OSNR of 23 dB.

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Then the dynamic tracing performance of the proposed method is investigated with different channel parameters. The BER as functions of angular frequencies of azimuth angle _$α$ and phase angle _$φ$ with different PMD and PDL are shown in Figs. 5(a) and 5(b). It has little effect on the BER when PMD and PDL are changed. The proposed method can dynamically track the angular frequency of azimuth angle _$α$ at 110 Mrad/s and angular frequency of phase angle _$φ$ at 1200 krad/s. Figure 5(c) shows the BER curves with different PMD and RSOP while PDL changes. The performance of the proposed method is stable when PDL is less than 10 dB, and it is slightly degraded when the PDL is greater than 10 dB. Figure 5(d) shows that the proposed method can equalize PMD when DGD is smaller than half of the symbol period. The tolerance will reduce when DGD becomes larger than half of the symbol period.

Fig. 5 BER as the functions of (a)_$α$, (b)_$φ$, (c) PDL and (d) PMD, respectively, at OSNR equals to 23 dB.

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Finally, we discuss the convergence performance of the proposed method at OSNR of 22dB, DGD of 14 ps and PDL of 4 dB. The angular frequencies of azimuth angle _$α$ and phase angle _$φ$ are 500 krad/s and 500 krad/s. As shown in Fig. 6, the curves _$α$ and _$φ$ represent the convergence performance of the two Kalman filters, respectively. _$α$ can approximately converge after around 200 symbols. _$φ$ can converge after around 100 symbols.

Fig. 6 (a) Convergence procedure of_$α$ and _$φ$ at OSNR of 22dB, DGD of 14 ps and PDL of 4 dB. (b) the detail of (a) marked with the circle.

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3.3 Complexity comparison

In order to indicate clearly the superiority of the proposed method, complexity comparison between CMA-MMA-BPS and Kalman algorithm for each symbol is shown in Table 1. The computation is based on optimum implementation. For example, two order real matrix inversion is composed of 7 real multiplications and 1 real addition. Complex multiplication is composed of 4 real multiplications and 2 real additions, and complex addition is composed of 2 real additions. Complex exponential operation _{$\exp (j Δ)$} is composed of 2 looking up table (LUT) operations. In Table 1, N₁ which is the number of taps of CMA is set to be 35. N₂ which is the number of taps of MMA is set to be 35. N₃ which is the smoothing filter length for sliding average methods for BPS is set to be 65. B which is the number of test phase angles for BPS algorithm is set to be 32. P represents the polarization state of signal, P = 1 for single polarization signal and P = 2 for dual polarizations signal. Therefore, the total complexity of CMA-MMA-BPS in the simulation includes 231576 real multiplications, 226568 real additions, 66627 comparisons and 132 LUTs for each symbol, which is much higher than the proposed scheme.

Table 1. Complexity Comparison between CMA-MMA-BPS and Kalman Method for Each Symbol

View Table

4. Conclusions

In this paper, we propose a novel joint multi-polarization-effect monitoring and equalization method based on two extended Kalman filters, which can cope with SOP tracing, PolDemux, equalization of PDL and PMD in PDM-16QAM system. The first EKF can reach convergence after around 200 symbols and the second EKF can reach convergence after about 100 symbols. The maximal angular frequencies of azimuth angle and phase angle can be tracked up to about 110 Mrad/s and 1200 krad/s respectively, which means its dynamic capability of SOP tracing is much better than CMA/MMA algorithm. The PDL and PMD for which proposed method can equalize is about 10 dB and more than half of the symbol period. In addition, the complexity of our method is much lower than CMA-MMA-BPS scheme. In principle, proposed algorithm can be extended for other higher order modulation formats not limited to QPSK and 16QAM. Therefore, the proposed algorithm can be a good substitute for CMA-MMA-BPS schema in coherent PDM-M-QAM systems.

Funding

National Natural Science Foundation of China (NSFC) (61571057, 61501213, 61527820, 61575082); Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications) (IPOC2015ZZ02).

Acknowledgments

We thank the reviewers for their comments.

References and links

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16. R. L. Eubank, A Kalman Filter Primer (Chapman and Hall, 2006).

	Real Multiplication	Real Addition	Comparison	LUT
PDL Equalization	47	34	0	6
1st EKF (16QAM)	231	217	0	2
2nd EKF (16QAM)	157	108	2	13
Total	435	359	2	21
CMA	36N₁-8	24N₁	0	0
MMA (16QAM)	36N₂ + 4	24N₂ + 8	3	0
BPS (16QAM)	(55N₃B + 2N₃)P	(54N₃B-B + 2N₃ + 22)P	(16N₃B + B)P	(2B + 2)P
Total	(36N₁-8) + (36N₂ + 4) + (55N₃B + 2N₃)P	24N₁ + (24N₂ + 8) + (54N₃B-B + 2N₃ + 22)P	3 + (16N₃B + B)P	(2B + 2)P

Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems

Abstract

1. Introduction

2. Principle

2.1 Channel model

2.2 Principle of tracing and equalization

3. Simulation and discussions

3.1 Simulation configuration

3.2 Performance comparison

3.3 Complexity comparison

4. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Tables (1)

Equations (27)

Optics Express