16QAM coherent optical WDM systems

Optics Communications 358 (2016) 108–119

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Decision-aided maximum likelihood phase estimation with optimum block length in hybrid QPSK/16QAM coherent optical WDM systems Yong Zhang, Yulong Wang n School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China

art ic l e i nf o

a b s t r a c t

Article history: Received 16 July 2015 Received in revised form 2 September 2015 Accepted 4 September 2015

We propose a general model to entirely describe XPM effects induced by 16QAM channels in hybrid QPSK/16QAM wavelength division multiplexed (WDM) systems. A power spectral density (PSD) formula is presented to predict the statistical properties of XPM effects at the end of dispersion management (DM) fiber links. We derive the analytical expression of phase error variance for optimizing block length of QPSK channel coherent receiver with decision-aided (DA) maximum-likelihood (ML) phase estimation (PE). With our theoretical analysis, the optimum block length can be employed to improve the performance of coherent receiver. Bit error rate (BER) performance in QPSK channel is evaluated and compared through both theoretical derivation and Monte Carlo simulation. The results show that by using the DAML with optimum block length, bit signal-to-noise ratio (SNR) improvement over DA-ML with fixed block length of 10, 20 and 40 at BER of 10
3 is 0.18 dB, 0.46 dB and 0.65 dB, respectively, when in-line residual dispersion is 0 ps/nm. & 2015 Elsevier B.V. All rights reserved.

Keywords: Wavelength division multiplexed Cross phase modulation Hybrid QPSK/16QAM coherent system Decision-aided Maximum likelihood Phase estimation algorithm Optimum block length

1. Introduction Demand for large transmission capacity has been increasing exponentially due to the rapid expansion of LTE, IPTV, and Big Data. The single channel data rate of coherent optical communication systems has been increased to 100 Gb/s and even beyond by employing advanced modulation formats, such as M-ary PSK and M-ary QAM. To make the upgrade cost-effective, only a few channels of WDM grid may be upgraded depending on capacity demand. QPSK/OOK hybrid systems have been studied both experimentally [1] and numerically [2] showing that XPM is the limiting impairment on the phase modulated signals in the nonlinear regime. Nowadays, 200G 16QAM optical network is commercially available and deployed by some operators. Hybrid configurations based on 16QAM and QPSK will be demanded in the future. However, there are few researches on the performance of QPSK channel impaired by XPM, which is induced by 16QAM channels. Due to the Kerr effects in optical fibers, XPM causes two kinds of distortions: XPM induced phase noise and XPM induced intensity noise by group velocity dispersion (GVD). Although the model of XPM effects in fiber-optic systems has been investigated intensively, the previous XPM models have two primary problems. The first, XPM models ignore waveform distortions of each n

Corresponding author. E-mail address: [email protected] (Y. Wang).

http://dx.doi.org/10.1016/j.optcom.2015.09.016 0030-4018/& 2015 Elsevier B.V. All rights reserved.

channel as they propagate in fiber [1,3–6]. The second, we should employ two models to describe XPM induced intensity noise [7,8] and XPM induced phase noise [2], respectively. To overcome the problems, we derive XPM model from the nonlinear Schrodinger equation (NLSE). The Volterra series transfer function (VSTF) method expresses the NLSE as a polynomial expansion in the frequency domain and retains the most significant terms in the resulting transfer function [9]. By employing VSTF method, we present a novel general XPM model to estimate XPM effects induced by 16QAM channels. The general model can entirely estimate XPM effects in a dispersion management fiber links, and contains the waveform distortions of each channel. Furthermore, we derive a PSD formula to predict the statistical properties of XPM effects at the end of DM fiber links. Generally, in coherent optical communication, there are two mainstream carrier phase estimation algorithm: Viterbi & Viterbi (V–V) and DA-ML. Based on a nonlinear transformation of received M-ary PSK signals, the V–V algorithm is capable of accurately tracking the unknown carrier phase [10]. However, it relies heavily on nonlinear computations. Recently, DA-ML PE algorithm has been proposed due to its computational linear and efficiency in coherent receiver [11,12]. Previous research about DA-ML PE mainly concentrated on combating with the laser phase noise [13]. In this work we extend the theoretical analysis of DA-ML to investigate the impacts of XPM effects induced by 16QAM channels in QPSK channel coherent receiver. By employing our XPM model, comprehensive mathematical description regarding of phase error induced by XPM effects and additive white Gaussian noise has

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

been derived. The result can be used to obtain optimum block length. Such theoretical analysis will be helpful to optimize the performance of coherent receiver with DA-ML PE algorithm. In this paper, we first present a general XPM model based on VSTF method, analyze the statistical properties of XPM effects induced by 16QAM channels, and then achieve a general expression of received signal. After that, we derive the analytical expression of phase error in consideration of XPM effects using DAML PE algorithm, and obtain optimum block length to improve the performance of QPSK channel coherent receiver. Finally, we perform Monte Carlo simulation to verify the theoretical analysis and evaluate the BER performance.

(SMF), β2, smf is the GVD parameter of SMF, α is the fiber loss, and γ is the nonlinearity coefficient of SMF, N is the number of fiber spans, βpre and βres are the GVD cumulated in pre-compensation fiber and in-line residual GVD cumulated, respectively. The output (0) (0) (1) of optical signal consists of two parts: ALin (ω) and ANL (ω) is (ω). ALin (1) the output of optical signal, and ANL (ω) is the nonlinear perturbation at the end of DM fiber links. In WDM systems, the input consists of multiple channels. We k=M define input optical signal A (ω) = ∑k =−M Ak (ω − kΔω), where Ak (ω) is the baseband input of the kth channel and Δω is the channel spacing. After substituting A (ω) into Volterra series [4], we obtain (2M + 1)3 terms containing factors of the form * (ω2 ) An (ω − ω1 + ω2 ). The XPM term can be described as Al (ω1) Am

⎧ ⎪ XPM on Channel n if ( l = m ≠ n) * An ⎨ Al A m . ⎪ ⎩ XPM on Channel l if ( l ≠ m = n)

2. Theoretical modeling of XPM effects 2.1. XPM model based on VSTF method The VSTF is an infinite series expansion of the solution to the NLSE that can be truncated to third-order [14,15]. We describe our analytical model of multi-span DM fiber links. Fig. 1 shows the schematic of dispersion management optical fiber links. We use VSTF method to express the output of optical signal after optical fiber links. We retain only the first-order and the third-order Volterra kernels. Under the assumption of identical spans, the parameters of transmission fibers are the same in each span. After

(1) ANL ( ω) = − jγ P0 P˜k ( ω) Hxpm, k ( ω) − 2jγN P0 Pk ( 0) L eff , ,k

(3)

with

)

some calculations (for convenience a detailed derivation is given in Appendix A), the total output optical signal in the end of optical fiber links can be approximated in the frequency domain as

(2)

Our analysis assumes that consider two optical waves in DM fiber links. Following the mathematical derivation in Appendix B, our results can be generalized to the pseudorandom signal waveforms of the kth channel. The derived frequency domain expression of nonlinear perturbation induced by channel k for arbitrary modulation can be written as

⎡ − α − jβ2, smf ω 2 − jβ2, smf ( kΔω) ω) Lsmf 2 ⎢1 −e ( e j ( βpre + βres ( n − 1) )( ω + ( kΔω) ω) N ⎢ α − jβ2 ω2 − jβ2 ( kΔω) ω Hxpm, k ( ω) = ∑ ⎢ ⎢ n= 1 1 − e−( α − jβ2, smf ( kΔω)ω m ) Lsmf j ( βpre + βres ( n − 1) )( (kΔω) ω) ⎢ e + ⎢ α − jβ2, smf kΔω ω ⎣

(

109

⎤ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎦

(4)

where L eff = [1 − exp ( − αLsmf )] /α , P˜k (ω) is the Fourier transform of the kth channel power modulation, Pk (0) = 2πP¯k δ (ω), P¯k is the average power of the kth channel. The nonlinear perturbation

(0) ALin (ω) = A(ω, 0),

(1) ANL (ω)

−jγ = 4π 2

N

∑∫ n= 1

0

Lsmf

∬

⎤ ⎡ e (−α − jβ2, smf (ω1− ω)(ω1− ω 2))z ⎥ ⎢ − ( β +( − ) β )( ω − ω )( ω − ω ) j n 1 pre res 1 1 2 ⎥ dω1 dω2 dz, ⎢ ·e ⎢ ·A(ω , 0)A⁎(ω , 0)A(ω − ω + ω , 0)⎥ ⎦ ⎣ 1 2 1 2

where A = A (t , z ) is the slowly varying complex envelope of the optical field at time t and position z along the fiber, A (ω, 0) is the Fourier transform of A (t, 0) , Lsmf is length of single mode fiber

(1)

induced by channel k in the time domain can be written as (1) ⎡ ⎤ ˜ ¯ ANL , k ( t ) = IFT ⎣ − jγ P0 Pk ω Hxpm, k ω ⎦ − 2jγN P0 Pk L eff .

( )

( )

(5)

We assume that the channel 0 as M channels to its right and M (1) to its left. The overall ANL (ω) can be written as a superposition of the stemming from the individual channels. The total nonlinear perturbation can be written as

Fig. 1. The schematic of optical fiber links.

110

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

⎡ (1) ANL (t ) = IFT ⎢ ⎢⎣

⎤ − jγ P0 P˜k (ω) Hxpm, k (ω) ⎥ ⎥⎦

M

∑ k =−M, k ≠ 0 M

− 2jγN P0 P¯k L eff .

∑

+

k =−M, k ≠ 0

(6)

To solve the energy divergence problem of the VSTF method, we describe the output signal of channel 0 by using modified VSTF method [16]

⎡ A1 t ⎤ NL 0 A out ( t ) = ALin ( t )·exp ⎢⎢ 0 ⎥⎥ ⎣ ALin t ⎦

() ()

=

M ⎡ ⎛ ⎞ − jγP˜k (ω) Hxpm, k (ω) ⎟⎟ P0 ·exp ⎢ IFT ⎜⎜ ∑ ⎢⎣ ⎝ k =−M, k ≠ 0 ⎠ M ⎤ + ∑ − 2jγNP¯k Leff ⎥, ⎥⎦ k =−M, k ≠ 0

(7)

where IFT is the inverse Fourier transform. As can be seen from Eq. (7), we can describe the output signal of channel 0 at the end of DM fiber links. So we can use one general model to predict XPM effects in frequency domain as M

− jγP˜k ( ω) Hxpm, k ( ω).

∑

Expm ( ω) =

k =−M, k ≠ 0

(8)

Note that in Eq. (8), Expm (ω) is a complex, so we can easily obtain the XPM induced phase noise as Im [Expm (ω)]. Meanwhile, the XPM induced amplitude noise by GVD can be written as Re [Expm (ω)]. 2.2. XPM effects induced by 16QAM channels According Eq. (8), XPM induced phase noise in the frequency domain can be written as

θ˜xpm ( ω) = Im ⎡⎣ Expm ( ω) ⎤⎦ M

∑

=

−

k =−M, k ≠ 0

γ⎡ * , k (−ω) ⎤⎦ P˜k ( ω), ⎣ Hxpm, k (ω) + Hxpm 2

Fig. 2. The autocorrelation function of XPM induced phase noise (a) and XPM induced amplitude noise (b) by GVD after 10-span DM fiber links with SMF fiber. Dres = [0, 80] ps/nm and average input power Pave = 0 dBm/ch . Symbols: simulation; Solid line: theoretical results.

(9)

and XPM induced amplitude noise by GVD in the frequency domain can be written as

Rpm ( n) =

I˜xpm ( ω) = Re ⎡⎣ Expm ( ω) ⎤⎦ M

∑

=

k =−M, k ≠ 0

−jγ ⎡ * , k (−ω) ⎤⎦ P˜k ( ω). ⎣ Hxpm, k (ω) − Hxpm 2

M

∑ k =−M

⎡ Hxpm, k ( ω) ⎤ 2 γ ⎥ S ( 0, ω)( k ≠ 0), − ⎢ * , k ( − ω)⎥⎦ k 2 ⎢⎣ + Hxpm

(10)

(11)

and the PSD of I˜xpm (ω) is M

Cim − xpm − am ( ω) =

∑ k =−M

⎡ ⎤2 −jγ ⎢ Hxpm, k ( ω) ⎥ S ( 0, ω)( k ≠ 0), * , k ( − ω)⎥⎦ k 2 ⎢⎣ − Hxpm

(12)

where Sk (0, ω) is the PSD of the kth channel power. The power values of 16QAM transmitter can be described as the sum of P˜k (t ) and P¯k . The possible values of P˜k are +8P¯k/10 , −8P¯k/10, 0, and 0. The PSD of P˜k (t ) can be obtained as

⎛ 8P¯ ⎞2 1 ⎛ sin πfTs ⎞2 Sk ( 0, f ) = ⎜ k ⎟ Ts ⎜ ⎟ , ⎝ 10 ⎠ 2 ⎝ πfTs ⎠

∞

∫−∞ Cim − xpm ( f ) e j2πfnT

s

df

(14)

and

The PSD of θ˜xpm (ω) is

Cim − xpm ( ω) =

where Ts is symbol interval. The autocorrelation function of θ˜xpm (ω) and I˜xpm (ω) can be obtained as

(13)

R am ( n) =

∞

∫−∞ Cim − xpm − am ( f ) e j2πfnT

s

df ,

(15)

respectively. The variance of θ˜xpm (ω) and I˜xpm (ω) can be obtained as Rpm (0) and Ram (0), respectively. Fig. 2 shows the autocorrelation function of both XPM induced phase noise and XPM induced amplitude noise by GVD. From Fig. 2 (a), we should note that XPM induced phase noise has correlation in time. In the presence of in-line residual dispersion in each fiber link, the correlation of XPM induced phase noise is increased, while the variance is decreased. From Fig. 2(b), XPM induced amplitude noise has a small correlation in time. It means the statistical property of this noise is close to Gaussian white noise. However, the correlation of XPM induced amplitude will be increased with increasing in-line residual dispersion. In Fig. 2, the analytical results agree well with Monte Carlo simulation, which verifies our derivation. It is worth to note that XPM induced phase noise and amplitude noise both approximatively obey Gaussian distribution with zero mean.

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

111

2.3. Signal modeling

The phase estimate is first considered in the absence of the ASE noise, so we rewrite Eq. (21) as

Based on the analysis above, the received QPSK signal in digital signal procession (DSP) units can be described as

⎡ θ^ ( k ) = arg ⎢ ⎢⎣

r ( k) = ≅

=

P0 m ( k ) e Ixpm ( k) + j ( θ xpm+ θ xpm ( k) ) + n ( k ) ˜

¯

P0 m ( k ) e j ( θ xpm+ θ xpm ( k) ) + ˜

¯

P0 Ixpm ( k ) m ( k ) e j ( θ xpm+ θ xpm ( k) ) + n ( k ) ¯

˜

˜ ¯ P0 m ( k ) e j ( θ xpm+ θ xpm ( k) ) + ṅ ( k ),

(16)

where m (k ) is the kth data symbol, P0 is the energy of the symbol, N the average phase shift θ¯xpm = − 2P¯k ∑k = 1 γL eff , n (k ) is a complex additive white Gaussian noise. Ixpm (k ) tends to has Gaussian distribution, so ṅ (k ) can be approximated a complex additive white Gaussian noise. According Eqs. (10) and (15), the signal-to-noise (SNR) per symbol can be described as

P γs = 2 0 2 , σxpm + σase

⎤ ⎡¯ ⎤ ˜ e j ⎣ θ xpm+ θ xpm ( l) ⎦ ⎥ ⎥⎦ l= k − 1 k − Lbl

∑

⎡ = θ˜xpm ( k − 1) + θ¯xpm + arg ⎢ 1 + ⎢⎣

⎤ ˜ ˜ e j ( θ xpm ( l) − θ xpm ( k − 1) ) ⎥. ⎥⎦ l= k − 2 k − Lbl

∑

(22)

∼ We define a function of Z (Lbl ), and the function can be approximated as

∼ Z ( Lbl ) = 1 +

k − Lbl

⎡

⎤

e⎣ j ( θ xpm ( l) − θ xpm ( k − 1) ) ⎦

∑

˜

˜

l= k − 2 k − Lbl

=1+

⎡ 1 + j θ˜ ( l) − θ˜ ( k − 1) ⎤ xpm xpm ⎣ ⎦

(

∑

)

l= k − 2

⎡ j ≅ Lbl exp ⎢ ⎢⎣ Lbl

(17)

k − Lbl

∑ l= k − 2

⎛ ⎞⎤ ⎜ θ˜xpm ( l) − θ˜xpm ( k − 1) ⎟ ⎥. ⎜ ⎟⎥ ⎝ ⎠⎦

(23)

By using the approximation 1 + jx ≅ exp [jx], we rewrite Eq. (22) as

with 2 σxpm = P0 R am ( 0).

(18)

In our calculation and simulation, we assume perfect analog to digital converter (ADC), timing synchronization, polarization mode dispersion (PMD) compensation, self-phase modulation (SPM) compensation, and carrier frequency estimation in the DSP unit. PE is carried out independently for each polarization, and only one polarization is considered for simplicity [1].

3.1. Basic principle of DA-ML PE In DA-ML PE algorithm [11,12], the ML phase estimate θ^ (k ) at time t = kTs is computed using the immediate past Lbl received signals, where Lbl is called the block length. A complex phase reference (PR) V (k ) is defined as [11] k − Lbl

∑ r ( l)

^* ( l), P0 m

(19)

l= k − 1

^ (l ) is the where the superscript * denotes complex conjugate, m receiver decision on the lth received symbol, and k−L ^ (l ) 2 is the factor used to normalize the RP. In U (k ) = ∑l = k −bl1 P0 m an M-ary constellation, the decision statistics are given by [11]

qi ( k ) = arg max Re ⎡⎣ r ( k ) V * ( k ) Ci* ⎤⎦ , i

(

)

(20)

where Ci is the symbol datum.

To analyze the phase error Δθ = θ (k ) − θ^ (k ) of the DA-ML receiver, we take the argument of the phase reference V (k ) to evaluate its variance. Here θ^ (k ) is the ML phase estimation. Substituting Eq. (16) to Eq. (19), the argument of phase reference can be written as

⎤⎤ ⎡ ⎢ e j ⎡⎣ θ¯xpm+ θ˜xpm ( l) ⎤⎦ + n′̇ (l) ⎥ ⎥, ⎥⎦ ⎥⎦ ⎢ l= k − 1 ⎣

k − Lbl

∑

⎡⎣ θ˜xpm ( l) − θ˜xpm ( k − 1) ⎤⎦

∑ l= k − 2

⎡⎣ θ˜xpm ( l) ⎤⎦.

(24)

l= k − 1

Substituting Eq. (24) in Eq. (21), incorporating the ASE noise, L and assuming ∑l =bl1 n′̇ (l )⪡1, the phase estimation error can be shown as

⎡ 1 ⎢ = θ˜xpm ( k ) − Lbl ⎢⎣

k − Lbl

∑

(21)

where n′̇ (l ) is statistically identical to ṅ (l ). Here, n′̇ (l ) = ṅ (l ) / P0 m (l ) with mean is 0, and the variance is 1/γs .

⎤ 1 θ˜xpm (l) ⎥ − ⎥⎦ Lbl l= k − 1 k − Lbl

k − Lbl

∑

∑

Im {n″̇ ( l) } ,

l= k − 1

(25)

with

⎤ ⎡ ⎡ k − Lbl ⎡ ⎤⎤ j ⎢ n″̇ ( l) = n′̇ ( l) exp ⎢ − ∑ ⎢ θ˜xpm ( l) ⎥ ⎥⎥ − θ¯xpm ⎥⎥. ⎢⎣ ⎢ ⎥⎦ ⎦ Lbl ⎣ l = k − 1 ⎢⎣ ⎦

(26)

n″̇ (l ) is statistically identical to n′̇ (l ) since n′̇ (l ) is circularly symmetric, and Im(x ) is the imaginary part of x. According Eq. (14), after some algebra (the detailed derivation is given in Appendix C), the corresponding variance of phase error is then given by

E ⎡⎣ Δθ 2⎤⎦ =

1 2 L +1 Rpm ( 0) − σn′̇2 + bl 2Lbl Lbl Lbl 2 + 2 Lbl

3.2. Optimum block length

⎡ θ^ ( k ) ≡ arg ⎡⎣ V ( k ) ⎤⎦ = arg ⎢ ⎢⎣

1 = θ¯xpm + Lbl

k − Lbl

Δθ = θ˜xpm ( k ) + θ¯xpm − θ^ ( k )

3. DA-ML PE with optimum block length

V ( k ) = U−1 ( k )

1 θ^ ( k ) = θ˜xpm ( k − 1) + θ¯xpm + Lbl

Lbl

∑ Rpm ( k) k=1

Lbl − 1

∑ ( Lbl − k) Rpm ( k), k=1

(27)

where Rpm (k ) represents auto-correlation function of XPM induced phase noise at sampling time t = kTs . The contribution of the XPM induced phase noise to the variance of Δθ increases with increasing block length, while the impact of ASE noise is reduced. When the phase estimation error of ASE noise and phase noise have a same value, an optimum block length can be obtained. Fig. 3 plots phase error variance versus block length in QPSK channel coherent receiver with DA-ML algorithm. Monte Carlo simulation agree well with our analytical derivation Eq. (27). Therefore, we can use our theoretical analysis to predict optimum block length, which gives the minimum

112

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

Fig. 3. Phase error variance versus block length in QPSK channel coherent receiver with DA-ML algorithm. Dres = [0, 80] ps/nm and average input power Pave = 0 dBm/ch . Symbols: simulation; Solid line: theoretical results.

Fig. 4. Optimum block length versus SNR and block length in QPSK channel coherent receiver with DA-ML algorithm. Dres = 0 ps/nm and average input power Pave = 0 dBm/ch . Black line: optimum block length.

variance of phase estimation error and the smallest BER. The result is important since it allows an accurate determination of the block length when such a scheme is to be implemented without the need to perform lengthy simulations. 3.3. BER analysis As we can see from Eqs. (16) and (17), we can obtain that the SNR in QPSK channel coherent receiver. Therefore, the BER of QPSK signal affected by both additive complex Gaussian noise and XPM effects can be expressed as [2] 3 e −(γs /2) − BER = 8 2

γs π

∞

∑ m=1

⎤ ⎡⎛ ⎛γ ⎞ ⎞ ⎛ mπ ⎞ ⎟ ⎥ ⎢ ⎜ I(m − 1) /2 ⎜ s ⎟ ⎟ sin ⎜ ⎝ 2⎠ ⎟ ⎝ 4 ⎠ −(m2 /2) σ 2 ⎥ ⎢⎜ · ·e Δθ . ⎥ ⎢⎜ ⎛ γs ⎞ ⎟ m ⎥ ⎢ ⎜ + I(m + 1/) 2 ⎜ ⎟ ⎟ ⎝ 2 ⎠⎠ ⎦ ⎣⎝

(28)

4. Simulation and discussions Simulations in VPItransmissionMaker are performed to evaluate the accuracy of theoretical analysis. The hybrid system has 5 channels with 40 Gb/s QPSK central channel, and all the others are 80 Gb/s 16QAM.

Fig. 5. Optimum block length versus SNR and block length in QPSK channel coherent receiver with DA-ML algorithm. Dres = 80 ps/nm and average input power Pave = 0 dBm/ch . Black line: optimum block length.

Fig. 6. Performance comparison in QPSK channel coherent receiver between DAML with optimum block length and DA-ML with fixed block length ( Lbl = 10, 20, 40 ). Dres = 0 ps/nm and average input power Pave = 0 dBm/ch . Symbols: simulation; Solid line: theoretical results.

QPSK channel is demodulated by using a coherent receiver. The DM fiber links composed of a linear pre-compensating fiber, N¼ 10 identical spans with linear dispersion compensation at the end of each span, and finally a linear post-compensating fiber. The dispersion map is chosen according to “straight line rule”. The total dispersion is compensated to zero. The transmission fiber is SMF with length L¼100 km, attenuation α ¼0.2 dB/km, dispersion D = 16 ps/(km nm), nonlinear coefficient γ = 1.315 W−1 km−1. Channel spacing Δf is 50 GHz. The average input power is 0 dBm/ch in the neighborhood of optimum input power, so we have to consider the impacts of XPM effects and ASE noise on the performance of our system. From Eq. (27), we have discussed above, it can be observed that an optimum value of block length for each pair of XPM induced phase noise and ASE noise, which gives the minimal variance of the phase estimation error. Fig. 4 plots optimum block length versus the SNR and block length in QPSK coherent receiver with DA-ML algorithm, when in-line residual dispersion Dres = 0 ps/nm . Under the large variance of ASE noise, longer block length is preferred to average out the ASE noise. Because the variance of ASE noise decreases with increasing SNR, the optimum block length becomes smaller. From the Fig. 4 , the results agree with our theoretical analysis. Fig. 5 plots optimum block length as in-line residual dispersion Dres = 80 ps/nm . We can see a similar case. In the presence of

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

Fig. 7. Performance comparison in QPSK channel coherent receiver between DAML with optimum block length and DA-ML with fixed block length ( Lbl = 10, 20, 40 ). Dres = 80 ps/nm and average input power Pave = 0 dBm/ch . Symbols: simulation; Solid line: theoretical results.

113

the theoretical derivation become more accurate with higher bit SNR, it is expected that the analytical approximation for BER and Monte Carlo simulation will have even better agreement at high bit SNR values. In Fig. 6, it is obviously observed that DA-ML with optimum block length can improve the performance of QPSK channel coherent receiver. As we can see from Fig. 6, the BER performances of the DA-ML with optimum block length at BER¼ 10−3 have 0.18 dB, 0.46 dB and 0.65 dB bit SNR improvement over the DA-ML with fixed block length of 10, 20 and 40, respectively, when in-line residual dispersion Dres = 0 ps/nm . In Fig. 7, in-line residual dispersion Dres = 80 ps/nm , the BER performance of the DA-ML with optimum block length is slightly better than the DA-ML with fixed block length of 10. However, 0.21 dB and 0.42 dB bit SNR improvement at BER = 10−3 can be seen than the DA-ML with fixed block length of 20 and 40, respectively. From Figs. 6 and 7, the DA-ML with optimum block length has significant improvement in performance compared with the DA-ML with fixed block length, especially in high SNR. The Reed-Solomon (RS)(255,239) of forward error correction (FEC) scheme, in particular, has been used in a broad range of long-haul communication systems. In our simulation, the RS(255,239) encoder is employed in the transmitter of QPSK channel. At the receiver, RS (255,239) decoder is operated after DA-ML carrier PE algorithm to obtain higher coding gain. We present the BER performance results for three cases: the DA-ML with optimum block length and RS (255,239); the DA-ML with fixed block length and RS(255,239); the DA-ML with optimum block length only. Fig. 8 shows the BER performance results when SNR varies from 5 dB to 10 dB when in-line residual dispersion is 0 ps/nm and 80 ps/nm, respectively. We can see that the performance of the DA-ML(optimum Lbl)þFEC concatenation scheme is much better than the corresponding uncoded case. Because the DA-ML with optimum block length is more effective in the suppression of system impairments than the DA-ML with fixed block length, the performance of the DA-ML(optimum Lbl)þFEC concatenation scheme is better than the DA-ML(fixed Lbl)þ FEC concatenation scheme over the entire range of SNR, particularly for high SNR. From Fig. 8, we find that FEC cannot eliminate the block length effect [17] of DA-ML PE algorithm, so DA-ML(optimum Lbl)þFEC concatenation scheme can obtain higher coding gain.

5. Conclusion

Fig. 8. Performance evaluation of the DA-ML(optimum Lbl) þ FEC concatenation scheme, the DA-ML( Lbl = 10, 20, 40 ) þFEC concatenation scheme, and the DA-ML (optimum Lbl) only scheme. Average input power Pave = 0 dBm/ch .

residue dispersion in each fiber link, the variance of XPM induced phase noise will be decreased. From the comparison of Figs. 4 and 5, we note that the optimum block length becomes larger with decreasing the variance of XPM phase noise. From the above results, we can say that the larger the variance of XPM induced phase noise, the shorter the block length to be used. Monte Carlo simulations have been performed to verify the validity of optimum block length. The BER curves were compared to the approximate BER calculated using Eq. (28). The Monte Carlo simulations were performed using a series of 217 samples. The results of simulations with at least 100 errors are included in the BER comparison. Excellent agreement can be observed between the simulation and the obtained analytical approximation. Since the approximations used in

In this paper, we investigate XPM effects induced by 16QAM channels in hybrid QPSK/16QAM WDM systems. A general model is proposed to entirely describe XPM effects at the end of DM fiber links. Based on this model, we present a general expression of received signal in QPSK channel coherent receiver. DA-ML algorithm is employed to estimate the carrier phase in DSP units. The analytical expression of phase error variance for predicting optimum block length is derived. We perform Monte Carlo simulations to verify the theoretical analysis and compare the BER performance of DA-ML with optimum and fixed block length. The results show that optimum block length can be used to improve the performance of coherent receiver. The BER performance of DA-ML with optimum block length has 0.18 dB, 0.46 dB, and 0.65 dB bit SNR improvement at BER¼ 10
3 over the DA-ML with fixed block length of 10, 20 and 40, respectively, when in-line residual dispersion is 0 ps/nm. The RS(255,239) of FEC is employed in our system, the results show that the combination of DA-ML with optimum block length and RS(255,239) can obtain higher coding gain, particularly for high SNR. Our theoretical analysis will be very useful for optimizing carrier phase estimation algorithm in hybrid coherent optical WDM systems. The optimum block length of V–V phase estimation algorithm can be obtained by using the same theoretical analysis. Furthermore, we can obtain the weight factors of DA-ML PE to improve the performance of coherent receiver. These are the subject of continuing research.

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Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

Appendix A. The nonlinear perturbation after DM fiber links The generalized NLSE is commonly used to describe the slowly varying complex envelop of the optical field in the fiber. It can be expressed in the time domain as

∂A α j ∂ 2A = − A + β2 2 − jγ A 2 A, ∂z 2 2 ∂t

(29)

where A = A (t , z ) is the slowly varying complex envelope of the optical field at time t and position z along the fiber, β2 is the GVD parameter, α is the fiber loss, and γ is the nonlinearity coefficient of the fiber. The VSTF is an infinite series expansion of the solution to the NLSE that can be truncated to third-order to give

A ( ω, L smf ) ≈ H1 ( ω, L smf ) A ( ω, 0)

+

⎤ ⎥ dω1 dω2, ⎣ · A ( ω1, 0) A* ( ω2, 0) A ( ω − ω1 + ω2, 0)⎥⎦ ⎡H ω , ω , ω − ω + ω , L 3( 1 2 1 2 smf )

∫ ∫ ⎢⎢

(30)

where the first, third-order Volterra kernel transforms are

H1 ( ω, L smf ) = e−(αLsmf /2)+ j (β2 ω

2 /2) L

smf

(31)

and

H3 ( ω1, ω2, ω − ω1 + ω2, z ) = − j

γ H1 ( ω, z )· 4π 2

∫0

Lsmf

e( −αz − jβ2, smf z ( ω1− ω)( ω1− ω 2 )) dz,

(32)

where A (ω, 0) is the Fourier transform of A (t, 0), Lsmf is the length of single mode fiber (SMF), β2, smf is the GVD parameter of SMF. Our analytical model includes the effect of pre-, post-, and in-line compensating fibers. Fig. 1 shows the schematic of dispersion management optical fiber links. We use VSTF method to express the output of optical signal at the end of optical fiber links. We retain only the first-order and the third-order Volterra kernels. Under the assumption of identical spans, the parameters of transmission fibers are the same in each span. The N-span dispersion management fiber links is shown in Fig. 1. The dispersion of the pre-compensating fiber is selected according to “straight line rule” as

Dpre = −

Dsmf N−1 Dres , − 2 α

(33)

where Dsmf is the dispersion of SMF, Dres is the in-line residual dispersion per span. The nonlinearity effect of pre-, post-, and in-line compensating fibers are ignored for simplicity. The optical signal at point A after linear pre-compensating fiber in Fig. 1 can be expressed as

Apre ( ω, L pre ) = A ( ω, 0)·e j (ω

2 /2) β

2, pre Lpre ,

(34)

where L pre is the length of pre-compensating fiber. Using truncated third-order VSTF method, the optical signal at point B after SMF in Fig. 1 as (0) Asmf ( ω, Lsmf ) = A ( ω, 0)·e−j (ω

2 /2) β

2 2, pre Lpre · e−(α /2) Lsmf − j (ω /2) β 2, smf Lsmf .

(35)

The nonlinear perturbation can be expressed through a third convolution as

(1) Asmf ( ω, Lsmf )

⎡ ⎢ ⎢ ⎢ ⎢ =⎢ ⎢· ⎢ ⎢ ⎢ ⎢⎣

⎤ ⎥ ⎥ ⎞ ⎥ e( −α − jβ2, smf ( ω − ω1)( ω1− ω 2 )) z ⎟ ⎥ ⎤⎟ ⎡ ⎤⎦·e j (ω12 /2) β2, pre Lpre ⎥. , 0 A ω ( 1 ⎟ ⎥ ⎢ ⎥ dz d d ω ω ⎟ 2 1 2⎥ ⎥ ⎢ * ( ω2, 0)·e−j (ω 2 /2) β2, pre Lpre A · ⎟ ⎥ ⎢ ⎥ ⎥⎟ ⎢ ( ω − ω1+ ω 2 )2 ⎥ β2, pre Lpre ⎥⎟ ⎢⎣ ·A ( ω − ω1 + ω2, 0)·e j 2 ⎦⎠ ⎥⎦ −j

∫0

Lsmf

⎡ ⎢ ⎢ ⎢ ⎢ ⎢· ⎢ ⎢⎣

∫∫

γ −(α /2) Lsmf + j (ω 2 /2) β2, smf Lsmf e 4π 2

(36)

After some algebra, we can obtain

(1) Asmf ( ω, Lsmf )

⎡ ⎢ ⎢ ⎢ =⎢ ⎢· ⎢ ⎢ ⎣

−j

∫0

Lsmf

⎡ ⎢ ⎢ ⎢ ⎢· ⎣

γ j (ω 2 /2) β2, pre Lpre −(α /2) Lsmf + j (ω 2 /2) β2, smf Lsmf ⎤ ·e e ⎥ 4π 2 ⎥ ⎞ ⎥ e( −α − jβ2, smf ( ω − ω1)( ω1− ω 2 )) z ⎟ ⎥. ⎤ ⎟ j L − ω − ω ω − ω β 1)( 1 2 ⎦ 2, pre pre ⎥ ·e ( ⎟ dz dω1 dω2 ⎥ ⎡⎣ A ( ω1, 0)·A* ( ω2, 0)·A ( ω − ω1 + ω2, 0) ⎤⎦⎟ ⎥ ⎠ ⎦

∫∫

The optical signal at point C after linear in-line compensating fiber in Fig. 1 is

(37)

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

(0) Adcf ( ω, Ldcf ) = A ( ω, 0)·e j (ω

2 /2) β

2 2 2, pre Lpre + j (ω /2) β 2, smf Lsmf + j (ω /2) β 2, dcf L dcf ,

115

(38)

where L dcf is the length of in-line compensating fiber, β2, dcf is the GVD parameter of dispersion compensation fiber (DCF). The nonlinear perturbation after linear in-line compensating fiber can be expressed as

(1) Adcf ( ω, Ldcf )

γ j (ω 2 /2) β2, pre Lpre+ j (ω 2 /2) β2, smf Lsmf + j (ω 2 /2) β2, dcf L dcf ⎡ ⎤ ⎢ − j 4π 2 e ⎥ ⎢ ⎥ ⎡ ⎞ ⎢ ⎥ e( −α − jβsmf ( ω − ω1)( ω1− ω 2 )) z ⎟ ⎢ =⎢ ⎥. Lsmf ⎤ βpre Lpre ⎟ ⎢ j − ω − ω ω − ω ⎦ ( )( 1 1 2 ⎢· ·e dz dω1 dω2 ⎥ ⎟ ⎢ 0 ⎢ ⎥ ⎡⎣ A ( ω1, 0)·A* ( ω2, 0)·A ( ω − ω1 + ω2, 0) ⎤⎦⎟ ⎢· ⎢⎣ ⎥⎦ ⎣ ⎠

∫

∬

(39)

Under the assumption of identical spans, the length, attenuation, group-velocity dispersion, and nonlinear coefficients of the transmission fibers are the same in each span. It should be emphasized that the total cumulated dispersion Dtot = Dpre + NDres + Dpost is zero. The optical signal at point D after linear post-compensating fiber in Fig. 1 is (0) ALin ( ω ) = A ( ω , 0) .

(40)

The total nonlinear perturbation after post-compensating fiber can be expressed as

⎡ ⎢ γ (1) ANL ( ω) = − j 2 ⎢⎢ 4π ⎢ ⎣

N

∑∫ n= 1

Lsmf

0

⎤ ⎡ ( −α − jβ2, smf ( ω − ω1)( ω1− ω 2 )⎤ z ⎞ ⎦ ⎥ ⎟ ⎢e ⎟ ⎥ ⎢ −j ( βpre + ( n − 1) βres )( ω − ω1)( ω1− ω 2 ) dz d d ω ω e · 1 2 ⎥, ⎟ ⎢ ⎥ ⎢· ⎡⎣ A ( ω1, 0)·A* ( ω2, 0)·A ( ω − ω1 + ω2, 0) ⎤⎦⎟ ⎦ ⎣ ⎠

∬

(41)

where the GVD cumulated in pre-compensation fiber βpre = β2, pre L pre and the in-line residual GVD cumulated βres = β2, smf Lsmf − β2, dcf L dcf . In dispersion unmanaged fiber links, there are no pre-, post-, and in-line compensating fibers in Fig. 1. The GVD cumulated in precompensation fiber βpre = 0, the in-line residual GVD cumulated βres = β2, smf Lsmf , and the total GVD cumulated in fiber links is compensated in coherent receiver DSP units. Therefore, the total nonlinear perturbation in coherent receiver can be obtained as

⎡ ⎢ γ (1) ( ω) = − j 2 ⎢ ANL 4π ⎢ ⎢ ⎣

N

∑∫ n= 1

0

Lsmf

⎡ ⎢ ⎢ ⎢ ⎢· ⎣

⎤ ⎞ ⎟ ⎥ ⎟ dz dω dω ⎥. 1 2 ⎟ ⎥ ⎥ ⎡⎣ A ( ω1, 0)·A* ( ω2, 0)·A ( ω − ω1 + ω2, 0) ⎤⎦⎟ ⎦ ⎠ ⎤

e( −α − jβ2, smf ( ω − ω1)( ω1− ω 2 )⎦ z · e−j ( ( n − 1) βres )( ω − ω1)( ω1− ω 2 )

∫∫

(42)

From Eq. (42), we can see that our theoretical analysis can be easily extended to dispersion unmanaged fiber links.

Appendix B. XPM model for arbitrary modulation Our analysis assumes two optical waves system propagating over DM fiber links. Channel 0 is a CW with input A0 (t ) = channel k is modulated by a single sine wave

P0 , and

Pk (1 + m cos (ωm t )). With the CW of channel 0, the XPM term from the kth channel to the

channel 0 is found to be from (l, m, n) = (0, k , k ) or (k , k , 0) in Eq. (2), so Eq. (41) can be written as

116

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

(1) ANL ( ωm )

⎡ ⎢ ⎢ ⎢ ⎢ 2γ = − j 2⎢ 4π ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ 2γ = − j 2 2π P0 ⎢ ⎢ 4π ⎢⎣ ⎡ ⎢ 2γ = − j 2 2π P0 ⎢ ⎢ 4π ⎢⎣

⎡⎡ −α − jβ2, smf ( ω1− ω)( ω1− ω 2 ) ) Lsmf ⎢ ⎢ 1 − e( ⎢ ⎢ α + jβ ⎤ 2, smf ω1 − ω ω1 − ω2 ⎦ ⎢⎢ ⎢ −j βpre + ( n − 1) βres )( ω1− ω)( ω1− ω 2 ) ∞ ⎢ ⎢ ⎣ ·e ( ⎢ −∞ ⎤ ⎡ ⎢ ⎢ Ak ( ω1 − kΔω) ⎥ ⎢ · ⎢ ·A * ω − kΔω ⎥ ) ⎢ ⎢ k( 2 ⎥ ⎢ ⎢ ·2π P0 δ ( ω − ω1 + ω2 )⎥ ⎦ ⎣ ⎣

(

N

∞

)(

∑∫ ∫ −∞

n= 1

N

⎡ −α − jβ2, smf ( ω1− ω) ω) Lsmf ⎢ 1 − e( ⎤ ⎢ α + jβ 2, smf ω1 − ω⎦ ω −∞ ⎢ ⎢⎣ −j ( βpre + ( n − 1) βres )( ω1− ω) ω ·e

∑∫ n= 1

N

(

⎤ ⎤ ⎞⎡ ⎥ ⎥ ⎟ ⎢ A ( ω − kΔω) ⎟· ⎢ k 1 ⎥ dω ⎥ ⎟ ⎢ · A * ( ω1 − ω − kΔω)⎥ 1⎥ ⎟⎢ k ⎥⎦ ⎥⎦ ⎠⎣

⎡ ( −α − jβ2,smf ( ω1′− ω + kΔω)ω)Lsmf ⎢1 − e ⎢ α + jβ 2, smf ω1′ − ω + kΔω ω −∞ ⎢ ⎢⎣ −j ( βpre + ( n − 1⎤⎦ βres )( ω1′− ω + kΔω) ω ·e

∑∫ n= 1

∞

∞

(

⎤ ⎤⎞ ⎥ ⎥⎟ ⎥ ⎥⎟ ⎥ ⎥⎟ ⎥ ⎥⎦⎟ ⎟ ⎥ d d ω ω 1 2 ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ⎟ ⎥ ⎦ ⎠

)

⎤ ⎤ ⎞⎡ ⎟ ⎢ A ( ω ′) ⎥ ⎥ k 1 ⎟· ⎢ ⎥ dω ′ ⎥ . 1 ⎟ ⎢ · Ak* ( ω1′ − ω)⎥ ⎥ ⎟⎢ ⎥⎦ ⎥⎦ ⎠⎣

(43)

The kth channel Ak (t ) is modulated by a single sine wave. The frequency domain of Ak (t ) can be described as

⎡ ⎛ ⎞ m ⎛ ⎞ m ⎛ ⎞⎤ Ak ( ω) = 2π Pk ⎢ δ ⎜ ω⎟ + δ ⎜ ω + ω m ⎟ + δ ⎜ ω − ω m ⎟ ⎥. ⎣ ⎝ ⎠ ⎠ ⎠⎦ 2 ⎝ 2 ⎝

(44)

We can obtain Ak (ω1′)·Ak* (ω1′ − ω) as

⎤ ⎡ δ ( ω ′ ) δ * ( ω ′ − ω) 1 1 ⎥ ⎢ ⎥ ⎢ m * ′ ′ + − + δ ω δ ω ω ω ( ) ( ) m ⎥ 1 1 ⎢ 2 ⎥ ⎢ m ⎡ A ( ω ′) ⎤ ⎛ 2⎞ k 1 ⎥ = 4π 2Pk ⎢ + δ ( ω1′ + ω m ) δ* ( ω1′ − ω) ⎥ + o ⎜ m ⎟. ⎢ 2 ⎥ ⎢ ⎢⎣ · Ak* ( ω1′ − ω)⎦⎥ ⎝ 4 ⎠ ⎥ ⎢ m ⎢ + 2 δ ( ω1′ − ω m ) δ* ( ω1′ − ω) ⎥ ⎥ ⎢ ⎢ + m δ ( ω1′ ) δ* ( ω1′ − ω − ω m )⎥ ⎦ ⎣ 2

(45)

When modulation index is small, we keep only linear terms in m, and substitute Eq. (45) into Eq. (43)

⎧ ⎡ ⎞ 2 − jβ − α − jβ2, smf ωm 2, smf ( kΔω) ω m Lsmf ⎪ ⎟ ⎢1 − e ⎪ ⎟ ⎢ 2 α j β ω j β k ω ω − − Δ m m 2 2 ⎪ ⎟ ⎢ ⎪ N ⎢ ⎟ ⎤ 2 + kΔω ω j βpre + βres ( n − 1) ⎦⎥ ωm m ( ) ⎪ ⎟ ⎢ ·e ⎪∑ ⎢ ⎟ ⎪ n = 1 ⎢ 1 − e−( α − jβ2, smf ( kΔω)ω m ) Lsmf ⎟ + ⎪ ⎟ ⎢ α − jβ2, smf kΔω ω m ⎪ ⎟ ⎢ ⎪ ⎟ ⎢⎣ j ( βpre + βres ( n − 1) )( ( kΔω) ω m ) ⎠ ·e ⎪ ⎪ ⎪ · δ ( ω − ωm ) = − jγ 2π P0 Pk m ⎨ ⎤ ⎡ 2 + jβ ⎪ − α − jβ2, smf ωm 2, smf ( kΔω) ω m ⎥ ⎦ Lsmf ⎢1 − e ⎪ ⎢ ⎪ α − jβ2 ωm2 + jβ2, smf kΔω ω m ⎢ ⎪ ⎢ N ⎪ 2 − kΔω ω j β + β ( n − 1) ωm ( )m ⎪ + ∑ ⎢ ·e pre res ⎪ n= 1 ⎢ − α + jβ2, smf ( kΔω) ω m ) Lsmf ⎢+1 − e ( ⎪ ⎢ ⎪ α + jβ2, smf kΔω ω m ⎢ ⎪ ⎪ ⎢⎣ −j ( βpre + βres ( n − 1) )( ( kΔω) ω m ) ·e ⎪ ⎪· δ(ω + ω ) m ⎩

(

)

(

(

(

)

(

(1) Atot , k ( ωm )

)

)

(

(

(

)(

(

− 2jγN 2π P0 Pk We define a function as

1 − e−αLsmf δ ( ω). α

)

)

)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎞⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎟⎪ ⎠⎪ ⎪ ⎭

(46)

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

⎡ ⎢1 − N ⎢ Hxpm, k ( ω m ) = ∑ ⎢ ⎢ n= 1 ⎢ 1 + ⎢ ⎣

e−

( α − jβ

( ) )

2 2, smf ωm − jβ 2, smf kΔω ω m Lsmf

(

)

α − jβ2 ωm2 − jβ2 kΔω ω m − e−( α − jβ2, smf ( kΔω)ω m ) Lsmf

(

)

α − jβ2, smf kΔω ω m

ej

(β

(

117

⎤

))( ωm2 + ( kΔω)ωm )⎥

pre + β res n − 1

e j ( βpre + βres ( n − 1) )( ( kΔω) ω m )

⎥ ⎥. ⎥ ⎥ ⎥ ⎦

(47)

Hence we rewrite Eq. (46) as (1) ω γ π P0 Pk ANL ,k( m ) = − j 2

−2jγN 2π P0 Pk

m ⎧ 2Hxpm, k ( − ω m )·δ ( ω + ω m )⎫ ⎨ ⎬ 2 ⎩ + 2Hxpm, k ( ω m )·δ ( ω − ω m ) ⎭ ⎪

⎪

⎪

⎪

(48)

1 − e−αLsmf δ ( ω). α

Due to Kerr nonlinearity, XPM effects on channel 0 originate from the intensity modulation of the kth channel in the same fiber. For small modulation index m, the intensity modulation of the kth channel is Ak (t ) 2 ≈ Pk (1 + 2m cos (ωm t )). Because the total nonlinear perturbation is unchanged, we can obtain the frequency response of XPM effects:

⎧ ⎫ ⎪ Hxpm, k ( − ω m )· δ ( ω + ω m )⎪ (1) ⎬ ANL j P0 Pk m ⎨ , k ( ω m ) = − γ 2π ⎪ ⎪ ⎩ + Hxpm, k ( ω m )·δ ( ω − ω m ) ⎭

− 2jγN 2π P0 Pk

1 − e−αLsmf δ ( ω). α

(49)

When the intensity of channel k is sinusoidally modulated with angular frequency ωm , the expression for optical power is

Pk ( t ) = Ak ( t )

2

≈ Pk ( 1 + 2m cos ( ω m t ))

= P¯k + P˜k ( t ),

(50)

where P¯k is the DC component of Pk (t ), and P˜k (t ) is the AC component of Pk (t ). Evaluating the Fourier transform of Eq. (50), the frequency domain of Pk (t ) can be described as

⎧ P˜ ( ω ) = 2πP mδ ( ω − ω ) k m k m ⎪ ⎪ P k ( ω ) = ⎨ P k ( 0 ) = 2π P k δ ( ω ) . ⎪ ⎪ ⎩ P˜k ( − ω m ) = 2πPk mδ ( ω + ω m )

(51)

(1) Comparing with the frequency response of ANL , we can see that DM fiber links can be as a transfer function , k (ω m )

HDM ( ω m ) = − jγ P0 Hxpm, k ( ω m ), HDM ( − ω m ) = − jγ P0 Hxpm, k ( − ω m ), HDM ( 0) = − 2jγN P0 L eff .

(52)

From Eq. (52), we can see that the transfer function has two parts: HDM (0) for DC component and HDM (ω) for AC component. For the intensity of channel k, which is arbitrary modulation signal with period T0 , the two-sided spectrum can be written as ∞

P k ( ω ) = 2π

∑

cn δ ( ω − nω 0 ),

(53)

n =−∞

where ω0 = 2π /T0 , cn is phasor Fourier coefficients of Pk (ω). According to Eqs. (52) and (53), the nonlinear perturbation induced by channel k can be written as ∞ (1) ANL ω = − jγ P0 2π ,k( )

∑

cn δ ( ω − nω 0 ) Hxpm, k ( nω 0 ) − 2jγN 2π P0 P¯k L eff δ ( ω)

n =−∞ , n ≠ 0

= − jγ P0 P˜k ( ω) Hxpm, k ( ω) − 2jγN P0 Pk ( 0) L eff ,

(54)

where Pk (0) = 2πP¯k δ (ω). The nonlinear perturbation induced by channel k in the time domain can be written as (1) ⎡ ⎤ ˜ ¯ ANL , k ( t ) = IFT ⎣ − jγ P0 Pk ω Hxpm, k ω ⎦ − 2jγN P0 Pk L eff .

( )

( )

(55)

118

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

Appendix C. Phase error variance after DA-ML According Eq. (25), the phase estimation error in the absence of the additive noise can be shown to be

⎡ 1 ⎢ Δθ = θ˜xpm ( k ) − Lbl ⎢⎣

⎤ θ˜xpm (l) ⎥. ⎥⎦ l= k − 1 k − Lbl

∑

(56)

When Lbl = 1, the phase estimation error is

Δθ = θ˜xpm ( k ) − θ˜xpm ( k − 1).

(57)

The variance of phase estimation error can be shown as 2⎤ ⎡ E ⎡⎣ Δθ 2⎤⎦ = E ⎢⎣ θ˜xpm ( k ) − θ˜xpm ( k − 1) ⎥⎦

(

)

2 2 = E ⎡⎣ θxpm k + θxpm k − 1 − 2θ˜xpm k θ˜xpm k − 1 ⎤⎦

( )

(

)

( )

(

)

= Rpm ( 0) + Rpm ( 0) − 2Rpm ( 1) = 2 ⎡⎣ Rpm ( 0) − Rpm ( 1) ⎤⎦.

(58)

When Lbl = 2, the phase estimation error is 1 1 Δθ = θ˜xpm ( k ) − 2 θ˜xpm ( k − 1) − 2 θ˜xpm ( k − 2).

(59)

The variance of phase estimation error can be written as

⎡⎛ ⎛ ⎞ 1 ⎛ ⎞ 1 ⎛ ⎞ ⎞2⎤ E ⎡⎣ Δθ 2⎤⎦ = E ⎢ ⎜ θ˜xpm ⎜ k⎟ − θ˜xpm ⎜ k − 1⎟ − θ˜xpm ⎜ k − 2⎟ ⎟ ⎥ ⎝ ⎠ 2 ⎝ ⎠ 2 ⎝ ⎠ ⎠ ⎥⎦ ⎢⎣ ⎝ ⎤ ⎡ 2 1 2 1 2 ⎢ θ˜xpm ( k ) + θ˜xpm ( k − 1) + θ˜xpm ( k − 2) − θ˜xpm ( k ) θ˜xpm ( k − 1)⎥ 4 4 ⎥ = E⎢ 1˜ ⎥ ⎢ ˜ ˜ ˜ ⎥⎦ ⎢⎣ − θxpm ( k ) θxpm ( k − 2) + 2 θxpm ( k − 1) θxpm ( k − 2)

=

3 1 Rpm ( 0) − Rpm ( 1) − Rpm ( 2). 2 2

(60)

When Lbl = L , the phase estimation error is 2 ⎡⎛ ⎡ k−L ⎛ ⎞ ⎛ ⎞⎤⎞ ⎤ ⎥ ⎢⎜ ˜ ⎜ ⎟ 1⎢ ⎟ 2 ⎥ ⎡ ⎤ ⎜ ⎟ ˜ E ⎣ Δθ ⎦ = E ⎢ θxpm ⎜ k⎟ − ∑ θxpm ⎜ l⎟ ⎥ ⎟ ⎥ ⎜ L ⎢⎣ l = k − 1 ⎝ ⎠ ⎦ ⎠ ⎥⎦ ⎝ ⎠ ⎢⎣ ⎝

⎧ L ⎫ ⎪ ⎪ 1  ⎪ Rpm ( 0) + 2 Rpm ( 0) + ⋯ + Rpm ( 0) ⎪ L ⎪ ⎪ ⎪ ⎞ ⎪ L ⎪ ⎟ ⎪ ⎤ ⎟ ⎪ ⎡⎣ R ⎪ −2 pm ( 1) + Rpm ( 2) + Rpm ( 3) + ⋯ + Rpm ( L} ⎦ ⎪ L ⎟ ⎪ ⎪ ⎟ ⎪ ⎞ L− 1 ⎪ ⎟ ⎪ ⎟ ⎤ ⎟ ⎪+2 ⎟ ⎪ ⎡⎣ R + + ⋯ + − 1 R 2 R L 1 ( ) ( ) ( ) ⎦ pm pm pm 2 ⎪ ⎟ ⎪ ⎟ ⎪ L ⎪ ⎟ ⎬. =⎨ ⎟ ⋮ ⎟ ⎪ ⎪ ⎟ ⋮ ⎟ ⎪ ⎪ ⎟ ⎟ ⎪ 2 ⎫ ⎪ ⎟ ⎟ ⎪ ⎪ ⎟ 2 ⎤ ⎪ ⎡ ⎟ ⎪ ⎪ + 2 ⎣ Rpm ( 1) + Rpm ( 2} ⎦ ⎪ ⎟ ⎟ ⎪ L ⎪ ⎪ ⎟ ⎪ ⎟ ⎪ ⎪ ⎟ 1 ⎬ ⋯⋯ ⎫ ⎟ ⎪ ⎪ ⎟ ⎪ ⎪ 2      ⎟ ⎪ ⎡ ⎤ ⎪ ⎟ ⎪ + 2 ⎣ Rpm ( 1) ⎦ ⎬ ⎟ ⎪ ⎪ L ⎟ ⎪ ⎪ ⎟ ⎪ ⎟ ⎪ ⎭1 × 1 ⎪ ⎭ ⎠(L − 1)×(L − 1) ⎟⎠ ⎪ 2×2 ⎪ ⎩ L×L ⎭ After some algebra, Eq. (61) can be written as

(61)

Y. Zhang, Y. Wang / Optics Communications 358 (2016) 108–119

E ⎡⎣ Δθ 2⎤⎦ =

L+1 2 Rpm ( 0) − L L

L

∑ Rpm ( k) + k=1

2 L2

119

L− 1

∑ ( L − k) Rpm ( k)

(62)

k=1

Knowing that

σ2 D ⎡⎣ Im {n ( l) } ⎤⎦ = n . 2

(63)

According Eqs. (25) and (26), the contribution of additive noise to the variance of Δθ is

⎡ 1 D⎢ ⎢⎣ Lbl

k − Lbl

∑ l= k − 1

⎤ 1 Im {n″̇ (l) } ⎥ = σn′̇2 ⎥⎦ 2Lbl

(64)

When block length is Lbl , incorporating the additive noise, the phase estimation can be shown to be

E ⎡⎣ Δθ 2⎤⎦ =

1 2 L +1 Rpm ( 0) − σn′̇2 + bl 2Lbl Lbl Lbl

Lbl

∑ Rpm ( k) + k=1

2 Lbl2

Lbl − 1

∑ ( Lbl − k) Rpm ( k). k=1

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