OOK coherent optical WDM systems

Optics Communications 365 (2016) 237–247

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

Invited Paper

A flexible decision-aided maximum likelihood phase estimation in hybrid QPSK/OOK 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 28 September 2015 Received in revised form 4 November 2015 Accepted 4 December 2015 Available online 12 January 2016

Although decision-aided (DA) maximum likelihood (ML) phase estimation (PE) algorithm has been investigated intensively, block length effect impacts system performance and leads to the increasing of hardware complexity. In this paper, a flexible DA-ML algorithm is proposed in hybrid QPSK/OOK coherent optical wavelength division multiplexed (WDM) systems. We present a general cross phase modulation (XPM) model based on Volterra series transfer function (VSTF) method to describe XPM effects induced by OOK channels at the end of dispersion management (DM) fiber links. Based on our model, the weighted factors obtained from maximum likelihood method are introduced to eliminate the block length effect. We derive the analytical expression of phase error variance for the performance prediction of coherent receiver with the flexible DA-ML algorithm. Bit error ratio (BER) performance is evaluated and compared through both theoretical derivation and Monte Carlo (MC) simulation. The results show that our flexible DA-ML algorithm has significant improvement in performance compared with the conventional DA-ML algorithm as block length is a fixed value. Compared with the conventional DA-ML with optimum block length, our flexible DA-ML can obtain better system performance. It means our flexible DA-ML algorithm is more effective for mitigating phase noise than conventional DA-ML algorithm. & 2015 Elsevier B.V. All rights reserved.

Keywords: Wavelength division multiplexed Cross phase modulation Hybrid QPSK/OOK coherent system Decision-aided Maximum likelihood Phase estimation algorithm

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 [1]. Generally, in coherent optical communication, there are two mainstream carrier phase estimation algorithms: Viterbi & Viterbi (V–V) [2] and DA-ML [3]. Based on a nonlinear transformation of received M-ary PSK signals, the V–V algorithm is capable of accurately tracking the unknown carrier phase [4]. However, it relies heavily on nonlinear computations. Recently, DAML PE algorithm has been proposed due to its computational linear and efficiency in coherent receiver [5,6]. Previous research about DA-ML PE mainly concentrated on combating with the laser phase noise [3,5–8]. In this work we extend the theoretical analysis of DA-ML to investigate the impacts of XPM effects induced by OOK channels in QPSK channel coherent receiver. In QPSK/OOK hybrid optical coherent systems, XPM is the major n

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

http://dx.doi.org/10.1016/j.optcom.2015.12.005 0030-4018/& 2015 Elsevier B.V. All rights reserved.

impairment on the phase modulated signals in the nonlinear regime [9,10]. Due to Kerr nonlinearity, XPM causes two kinds of distortions: XPM induced phase noise and XPM induced intensity noise by group velocity dispersion (GVD). The previous XPM models have two primary problems: first, XPM models ignore waveform distortions as they propagate in fiber [9,11–14]; second, we should employ two independent models to describe XPM induced phase noise and intensity noise [10,15,16]. 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 [17]. By employing VSTF method, we propose a general model to entirely estimate XPM effects in a dispersion management fiber links. Based on our XPM model, we derive a power spectral density (PSD) formula to predict the statistical properties of XPM effects, and achieve a general expression of QPSK channel received signal at the end of DM fiber links. Although DA-ML algorithm has been investigated intensively, block length effect is the most primary problem in system performance optimization. In conventional DA-ML algorithm, optimum block length results in optimum performance, which has been studied both experimentally [18] and numerically [3,5,19,20]. To obtain optimum block length, the computational complexity of

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Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

Fig. 1. The schematic of optical fiber links.

DA-ML algorithm will be increased [19,20]. Due to block length effect, conventional DA-ML with optimum block length needs to add hardware module to adjust the length of pilot symbols, so that it will increase hardware complexity in coherent receiver DSP units. To overcome block length effect, we propose a flexible DAML algorithm [7,8]. We take into account the time-varying phase noise, instead of the assumption of constant phase noise over the entire observation interval. Based on our XPM model, maximum likelihood is used to obtain weighted factors of the observation interval. Compared with conventional DA-ML algorithm, the contribution of the phase noise with longer block length is significantly reduced, while the impact of additive noise slight increases. There will be a system BER floor with increasing block length, so our proposed flexible DA-ML algorithm can eliminate block length effect. Compared with the conventional DA-ML, our flexible DA-ML can obtain better performance in QPSK channel coherent receiver. In this paper, we first present a general XPM model based on VSTF method, analyze the statistical properties of XPM effects induced by OOK channels, and then achieve a general expression at the end of DM fiber links. After that, we explain the principle of the flexible DA-ML scheme and analyze the weighted factors. Furthermore, we derive the analytical expression of phase error variance for the performance prediction of QPSK channel coherent receiver with the flexible DA-ML scheme. Finally, we perform MC simulation to verify the theoretical analysis and evaluate the BER performance.

∬

A( ω, L ) ≈ H1( ω, L )A( ω, 0) +

⎡ ⎤ ⎢ H3( ω1, ω 2, ω − ω1 + ω 2, L ) ⎥ ⎢ ⎥dω1 dω 2, ⎢ ⎥ ⎢⎣ ·A( ω1, 0)A⁎ ( ω 2, 0)A( ω − ω1 + ω 2, 0)⎥⎦ (2)

where the first, third-order Volterra kernel transforms are

(

)

H1 ω, L smf = e−αL /2 + j(β2ω

2

/2)L

and

H3( ω1, ω2, ω − ω1 + ω2, z ) =−j

γ H1( ω, z )· 4π 2

∫0

L

e( − αz − jβ2z ( ω1− ω)( ω1− ω 2)) dz,

In this section, we first present the expression of output optical signal at the end of DM fiber links based on VSTF method. A general XPM model is proposed to entirely describe XPM effects induced by OOK channels. Next, based on our XPM model, we provide a PSD formula to predict the statistical properties of XPM effects, and discuss the major characteristics of XPM effects. In the third part, we present the general received complex signal model in QPSK channel coherent receiver. 2.1. XPM model based on VSTF method 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

(1)

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 [21]

(4)

where A(ω, 0) is the Fourier transform of A(t, 0), L is the length of fiber. Our analytical model includes the effect of pre-, post-, and inline 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 total output optical signal in the end of optical fiber links can be approximated in the frequency domain as [22] ( 0) ω = A( ω , 0), A ( 1)( ω ) A NL Lin ( ) ⎡ ⎤ ⎢ − α − jβ 2, smf ( ω1− ω )( ω1− ω 2) z ⎥ ⎢e ⎥ N ⎢ −j β + n − 1 β ⎥ L smf ω − ω ω − ω − jγ = ∑ ∫ ∬ ⎢⎢ ·e pre ( ) res ( 1 )( 1 2)⎥⎥dz, 2 0 4π n = 1 ⁎ ⎢ · A( ω1, 0)A ( ω 2, 0) ⎥ ⎢ ⎥ ⎢⎣ · A( ω − ω1 + ω 2, 0)dω1dω 2 ⎥⎦

(

)

(

2. Theoretical modeling of XPM effects

(3)

)

(5)

where A(ω) is the Fourier transform of A(t ), Lsmf is length of single mode fiber (SMF), β2, smf is the GVD parameter of SMF, N is the number of fiber spans, βpre and βres are the GVD cumulated in precompensation fiber and in-line residual GVD cumulated, respec(0) tively. The output of optical signal consists of two parts: ALin (ω) and (0) (1) (1) ANL (ω). ALin (ω) is 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 expression 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 [12], we obtain (2M + 1)3 terms containing factors of the form ⁎ Al (ω1)Am (ω2)An (ω − ω1 + ω2). The XPM term can be described as

⎧ ⎪ XPM on Channel n ⁎ Al A m An⎨ ⎪ ⎩ XPM on Channel l

( l = m ≠ n) , ( l ≠ m = n)

(6)

Our analysis assumes that consider two optical waves in DM fiber links. Following the mathematical derivation in [22], 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

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

( 1) ( ω) = − jγ P P˜ ( ω)H ANL 0 k xpm, k( ω) − 2jγN P0 Pk( 0)Leff ,k

(7)

I˜xpm( ω) = Re⎡⎣ Expm( ω)⎤⎦ M

with

∑

=

k =−M, k ≠ 0

Hxpm, k( ω) ⎡ − α − jβ2, smf ω2 − jβ2, smf ( kΔω)ω)Lsmf ⎢1−e ( · ⎢ α − jβ2ω2 − jβ2( kΔω)ω N ⎢ = ∑ ⎢ e j( βpre + βres ( n − 1))( ω2 + ( kΔω)ω) n= 1 ⎢ ⎢ 1 − e−( α − jβ2, smf ( kΔω)ωm)Lsmf ⎢+ ·e j( βpre + βres ( n − 1))( ( kΔω)ω) α − jβ2, smf ( kΔω)ω ⎢⎣

⎤ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥⎦

M

M

∑ k =−M, k ≠ 0

k =−M

(8)

(9)

⎤ − jγ P0 P˜k( ω)Hxpm, k( ω)⎥ ⎥⎦

To solve the energy divergence problem of the VSTF method, we describe the output signal of channel 0 by using modified VSTF method [23]

⎤ ⎥ ⎥. ⎥⎦

(11)

M

∑

− jγP˜k( ω)Hxpm, k( ω).

k =−M, k ≠ 0

Cim − xpm − am( ω) M

∑

=

k =−M

2

−jγ ⎡ ⎤ ⁎ Hxpm, k( ω) − Hxpm , k( − ω)⎦ Sk( 0, ω)( k ≠ 0), 2 ⎣

(16)

where Sk(0, ω) is the PSD of the kth channel power. The power values of OOK transmitter can be described as the sum of P˜k(t ) and P¯k . Average power P¯k is Ppeak/2. The possible values of P˜ are +P /2 and −P /2. The PSD of P˜ (t ) can be obtained as k

k

peak

peak

2 2 Sk( 0, f ) = Ppeak/2 Ts⎡⎣ sin πfTs / πfTs ⎤⎦ ,

(

R pm( n) =

)

(

)(

)

(17)

(12)

Note that in Eq. (12), Expm is a complex, so we can easily obtain the XPM induced phase noise in the frequency domain as Im[Expm(ω)]. Meanwhile, the XPM induced amplitude noise by GVD can be written as Re[Expm(ω)].

∞

∫−∞ Cim − xpm( f )e j2πfnT

s

df

(18)

and

Ram( n) =

where IFT is the inverse Fourier transform. As can be seen from Eq. (11), 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

Expm( ω) =

and the PSD of I˜xpm(ω) is

(10)

k =−M, k ≠ 0

⎡ 1 ⎤ ⎢ A ( t) ⎥ Aout ( t ) = A 0 ( t )· exp⎢ NL ⎥ Lin 0 ⎣ ALin ( t ) ⎦ ⎡ ⎛ ⎞ M M ⎢ ⎜ ⎟ = P 0 · exp⎢ IFT ⎜ − jγP˜ k ( ω )Hxpm, k ( ω )⎟ + − 2jγNP¯ kL eff ∑ ∑ ⎜ ⎟ ⎢⎣ ⎝ k =−M , k ≠ 0 ⎠ k =−M , k ≠ 0

(15)

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

− 2jγN P0 P¯kLeff .

∑

(14)

2 γ ⁎ − ⎡⎣ Hxpm, k( ω) + Hxpm, k( − ω)⎤⎦ Sk( 0, ω) 2

( k ≠ 0) ,

M

+

∑

Cim − xpm( ω) =

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

⎡ ( 1) t = IFT ⎢ ANL () ⎢⎣

−jγ ⎡ ⎤˜ ⁎ ⎣ Hxpm, k( ω) − Hxpm, k( − ω)⎦Pk( ω). 2

The PSD of θ˜xpm(ω) is

where Leff = [1 − exp( − αLsmf )] /α , P˜k(ω) is the Fourier transform of the kth channel power modulation, Pk(0) = 2πP¯kδ (ω), P¯k is the average power of the kth channel. The nonlinear perturbation induced by channel k in the time domain can be written as

( 1) t = IFT ⎡ − jγ P P˜ ( ω)H ⎤ ¯ ANL ⎣ 0 k xpm, k( ω)⎦ − 2jγN P0 PkLeff . ,k( )

239

∞

∫−∞ Cim − xpm − am( f )e j2πfnT

s

df ,

(19)

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 that 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 MC simulation, which verifies our derivation. It is worth to note that XPM induced phase noise and amplitude noise both approximately obey Gaussian distribution with zero mean.

2.2. XPM effects induced by OOK channels According to Eq. (11), XPM induced phase noise in the frequency domain can be written as

θ˜xpm( ω) = Im⎡⎣ Expm( ω)⎤⎦ M

=

∑ k =−M, k ≠ 0

−

γ⎡ ⎤˜ ⁎ ⎣ Hxpm, k( ω) + Hxpm, k( − ω)⎦Pk( ω), 2

(13)

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

2.3. Signal modeling XPM effects originate from the intensity modulation of other copropagating optical signal in the same fiber [16], while laser phase noise (LPN) resulting from laser is a phase fluctuation, therefore laser phase noise and XPM effects are independent. Based on the analysis above, the received QPSK signal in DSP units in the presence of ASE noise, LPN together with XPM effects, can be described as

240

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

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 [9].

3. Flexible DA-ML PE algorithm In current coherent optical communication system, instead of optical phase-locked loop, the DSP-based PE algorithms are introduced to estimate carrier phase. A computationally efficient DAML PE has been proposed in [5]. However, block length effect is the most primary problem of conventional DA-ML PE algorithm, because it impacts system performance and leads to the increasing of hardware complexity. In this section, we propose a flexible DA-ML PE algorithm to eliminate block length effect. We derive the mathematical expression of phase error variance induced by XPM effects, LPN and ASE noise using our flexible DA-ML PE. The BER performance of QPSK channel coherent receiver is analyzed by theoretical derivation. 3.1. The principle of flexible DA-ML According to Eq. (20), the received signal over the entire observation interval {r (k − l ) , 1 ≤ l ≤ Lbl} can be written as r ( k − l) =

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 , peak input power Ppeak = 0 dBm/ch . Symbols: simulation; solid line: theoretical results.

r( k) =

˜

⎡ j( θ¯xpm+ θ˜xpm( k) + θlpn( k)) + n( k ) ⎢ P0 m( k )e ≅⎢ ˜ ¯ ⎢⎣ + P0 Ixpm( k )m( k )e j( θxpm+ θxpm( k) + θlpn( k)) =

P0 m( k )e (

) + ṅ( k),

j θ¯xpm+ θ˜xpm( k) + θlpn( k)

⎤ ⎥ ⎥ ⎥⎦ (20)

k=1

P0 2 + σase

, (21)

with

σn¨2

P0σn2̇ .

= where Var{n¨ (k − l )} = In order to retrieve the kth information m(k), the carrier phase noise is estimated based on the received signals over the immediate past Lbl symbols. The observed vector can be written as T r ̇ = ⎡⎣ r (̇ k − Lbl ), …, r (̇ k − 2), r (̇ k − 1)⎤⎦ .

(22)

LPN is typically modeled as random-walk process, which is expressed as [3] k

∑ u =−∞

v( u)

(23)

where v(u)′s are independent identically distributed Gaussian 2 random variables with zero-mean and variance σlpn = 2π (Δv )T . Here, Δv is the beat linewidth of signal and local oscillator (LO) and T is the symbol period. In our calculation and simulation, we assume perfect analog to digital converter (ADC), timing synchronization, polarization mode dispersion (PMD) compensation, self-phase modulation (SPM)

(26) l−1 ∑u = 0 v(u)

Assuming that the variable [θ˜xpm(k ) − θ˜xpm(k − l )] and are two small values compared to one, Eq. (25) can be written as ⎡ l−1 ⎢ ⎡ ⎤ j θ¯ + θ˜ k +θ k ⎢ 1 + j ∑ v ( u) r (̇ k − l ) ≈ P0· e ⎣ xpm xpm( ) lpn( )⎦ ⎢ u =0 ⎢ ˜ ˜ ⎣ + j θ xpm( k − l ) − θ xpm( k )

(

2 σxpm = P0Ram( 0).

θlpn( k ) =

(25)

n¨ ( k − l )

additive white Gaussian noise. Ixpm(k ) tends to be Gaussian distribution, so ṅ(k ) can be approximated a complex additive white Gaussian noise. According to Eqs. (19) and (20), the signal-to-noise (SNR) per symbol can be described as 2 σxpm

where Lbl is the block length. r (k − l ) is first multiplied by P0 m⁎(k − l ) to remove the data dependency. Thus,

+ n(̇ k − l )· P0 m⁎( k − l ) , 

where m(k ) is the kth data symbol, P0 is the energy of the symbol, N the average phase shift θ¯xpm = − 2P¯k ∑ γLeff , n(k ) is a complex

γs =

(24)

+ n(̇ k − l ),

⎡ ⎛ ⎞⎤ l−1 ⎥ ⎢ r (̇ k − l ) = P0· exp⎢ j⎜ θ¯xpm + θ˜xpm( k − l ) + θ lpn ( k ) + ∑ v ( u) ⎟⎥ ⎟ ⎜ ⎠⎦ ⎣ ⎝ u =0

P0 m( k )e Ixpm( k) + j( θxpm+ θxpm( k) + θlpn( k)) + n( k ) ¯

⎡ ⎛ ⎞⎤ l−1 ⎥ ⎢ P0 m( k − l )exp⎢ j⎜ θ¯xpm + θ˜xpm( k − l ) + θ lpn ( k ) + ∑ v ( u) ⎟⎥ ⎟ ⎜ ⎠⎦ ⎣ ⎝ u =0

)

⎤ ⎥ ⎥ ¨ ⎥ + n( k − l ). ⎥ ⎦

(27)

Because θ˜xpm(k ) and v(u) approximately obey Gaussian distribution with zero mean, r (̇ k − l ) can be as a complex Gaussian random variable. Thus, the observed vector ṙ given by Eq. (26) has a Nvariate complex Gaussian distribution

fr ̇ θ ( r ̇ θ ) = where

⎧ ⎫ H exp⎨ −( r ̇ − m r )̇ C−1( r ̇ − m r )̇ ⎬ , ( π ) det C ⎩ ⎭ 1

N

⎪

⎪

⎪

⎪

(28)

θ is (θ¯xpm + θ˜xpm + θlpn). The mean vector mr ̇ is defined by

⎡¯ ⎤ T ˜ m r ̇ = E{r}̇ = P0·e j⎣ θxpm+ θxpm( k) + θlpn( k)⎦ ⎡⎣ 1, 1, …, 1⎤⎦ ⎡¯

˜

⎤

= P0·e j⎣ θxpm+ θxpm( k) + θlpn( k)⎦1. Each element of the covariance matrix C has a value

(29)

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

C xy = E{(r (̇ k − x ) − E{r (̇ k − x )})⁎(r (̇ k − y ) − E{r (̇ k − y )})} ⎧⎛ ⎞⁎ ⎫ ⎡ ⎤ ⎪ ⎜ P · e j⎣ θ¯ xpm+ θ˜ xpm( k ) + θlpn( k )⎦ ⎟ ⎪ ⎪⎜ 0 ⎟ ⎪ ⎡ ⎤⎟ ⎪ x−1 ⎪⎜ ⎪ ⎜ · j⎢ θ˜xpm( k − x ) − θ˜xpm( k ) + ∑ v ( u)⎥ ⎟ ⎪ ⎢ ⎥⎟ ⎪ ⎪⎜ ⎣ ⎦ u =0 ⎪⎜ ⎟⎟ ⎪ ⎜ ⎪ ⎝ + n¨ ( k − x ) ⎪ ⎠ ⎪ ⎪ ⎬. = E⎨ ⎞⎪ ⎡ ⎤ ⎪ ⎛ j⎣ θ¯ xpm+ θ˜ xpm( k ) + θlpn( k )⎦ ⎟⎪ ⎪ ⎜ P0· e ⎟ ⎪ ⎜ ⎡ ⎤⎟⎪ y−1 ⎪ ·⎜ ⎢ θ˜ ⎥⎟⎪ ˜ ⎜ · − − + j k y k v w θ ∑ ) xpm( ) ( )⎥ ⎪ ⎪ ⎢ xpm( ⎪ ⎜ ⎣ ⎦⎟⎪ w =0 ⎟⎟ ⎪ ⎪ ⎜⎜ ⎪ ⎝ + n¨ ( k − y ) ⎠⎪ ⎩ ⎭

(

)

(

)

(30)

241

^ (k − l ) is the receivers decision on m(k − l ). PR, m Fig. 3 presents the coefficient vectors for different SNR and Dres. Beat linewidth per laser is 2 MHz. It is evident that a large number of block length Lbl is needed to average out the additive noise as SNR is low. When XPM effects are high, only a few received signals are useful in the estimation process, because XPM induced phase noise becomes less correlated over the observation interval when Dres = 0 ps/nm . From the weighted coefficients, we can see that the phase estimator gives decreasing emphasis to the earlier symbols that far away symbol k whose phase is estimated in Eq. (39). The decision statistics are given by

Because laser phase noise, XPM effects and ASE noise are independent, after some algebra, we can obtain each element of the covariance matrix C as

^ ( k ) = arg max Re⎡⎣ r ( k )V ⁎( k )C ⁎⎤⎦ m i

⎡ 2 2 2 ⎢ P0σ ṅ δ ( x − y ) + P0 min( x , y )σ lpn C xy = ⎢ 2 ⎢⎣ + P0 { R pm( 0} + R pm( x − y ) − R pm( x ) − R pm( y )

for PSK signal, where Ci is the symbol datum. The receiver com^ (k ) = C if putes r (k )V⁎(k )C ⁎ for each Ci, and decides that m

)

⎤ ⎥ ⎥( x , y ∈ ( 1, 2, … , Lbl )), ⎥⎦

(31)

where δ is the Kronecker delta function, Rpm is shown in Eq. (18). The estimate of θ(k ) is found by maximizing the probability distribution function (pdf) according to Eq. (28). That is minimizing the log-likelihood function

Λln = ln{

1

( π)

N

} − ( r ̇ − m r )̇ C−1( r ̇ − m r )̇

(

)

= ln{·} − r ̇ HC−1r ̇ + m rḢ C−1m r ̇ − m rḢ C−1r ̇ − r ̇ HC−1m r ̇ . ∂Λln ∂θ (k )

The solution to

(32)

= 0 is found to be

(33)

Thus,

1T C−1r ̇ 1T C−1r ̇ 1 1 arg{ H −1 } = arg{ ⁎ }. 2 2 ṙ C 1 1T C−1r ̇

(

)

(34)

Knowing that

⎧ X⎫ 1 Im{X} arg⎨ ⁎ ⎬ = tan−1 . ⎩X ⎭ 2 Re{X}

(35)

Eq. (34) can be written as

) ⎫⎪⎬. ) ⎪⎭

⎧ Im 1T C−1r ̇ ⎪ θML = tan−1⎨ T −1 ⎪ ⎩ Re 1 C r ̇

( (

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 Eqs. (24) (in 39), the argument of phase reference can be written as

θ (k )= θML

⎡ ⎤ ⎡ ⎤ ∂Λln ⎛ ⎞ = 0 − ⎜ 0 + 0 + jP0 · e−j⎣ θML ( k )⎦1T C −1ṙ − jP0 · e j⎣ θML ( k )⎦ṙ HC −11⎟ = 0. ⎝ ⎠ ∂θ (k ) θ (k)= θ ML

θML =

i

i

Re[r (k )V⁎(k )Ci⁎] is maximize. The flexible DA-ML structure is shown in Fig. 4. Consisting of phase estimator in Eq. (39) and the data decision in Eq. (40).

3.2. Phase error variance

H

det C

(40)

i

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

⎡ ⎞⎤ ⎡¯ ⎤ ˜ ^ ⁎( k − l⎤⎦n(̇ k − l ) ⎟⎥ ⎢ αlP0e j⎣ θxpm+ θxpm( k − l ) + θlpn( k − l )⎦ + αl P0 m ⎢ ∑ ⎟⎥ 2 ⎟⎥ ^ ( k − l) l=1 ⎢ P0 m ⎢⎣ ⎠⎥⎦

⎡ = arg⎢ ⎢ ⎣

∑ ⎢ αle j⎣ θxpm+ θxpm( k − l⎦ + θlpn( k − l )⎦ + αln′̇ ( k − l )⎟⎥⎥,

Lbl

Lbl

⎡

l=1 ⎣

⎡¯

˜

⎤

⎤

⎞⎤ ⎠⎦

^ (k − l ) is statistically identical to where n′̇ (k − l ) = αlṅ(k − l ) / P0 m ṅ(k − l ) with the mean being 0, and the variance is 1/γs . The phase estimate is first considered in the absence of the ASE noise, so we rewrite Eq. (41) as

(36)

Eq. (36) can be reformulated using the coefficient vector

α=

1T C−1 1T C−11

(37)

as

θ^( k ) = θML = tan−1

{ Re{ ∑

⁎

L

}, }

^ ( k − l)r ( k − l) Im ∑l =bl1 αl P0 m Lbl α l= 1 l

^ ⁎( k − l)r ( k − l) P0 m

(38) T

where αl is the (k − l ) th weighted coefficient, 1 = [1, 1, …1] is a L vector of Lbl ones, α is the normalized vector and ∑l =bl1 αl = 1. In order to totally eliminate the nonlinear operation of Eq. (38), we introduce a phase reference (PR) as

Lbl

^ ⁎( k − l)r ( k − l). V ( k ) = U −1( k ) ∑ αl P0 m l= 1 L

Here, U (k ) = ∑l =bl1

(39)

2

^ (k − l ) is the factor used to normalize the P0 m

(41)

Fig. 3. The weighted coefficient vectors.

242

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247 ⎡ Lbl ⎢˜ ⎡ ⎤ ⎢ θ xpm( k − 1) + ∑ αl⎣ θ˜ xpm( k − l ) − θ˜ xpm( k − 1) ⎦ + θlpn( k − 1) ⎢ l=2 θ^( k ) = θ¯ xpm + ⎢ ⎢ Lbl ⎡ ⎤ ⎢ + ∑ αl⎣⎢ θlpn( k − l ) − θlpn( k − 1) ⎦⎥ ⎢⎣ l = 2 ⎞⎤ Lbl ⎡ ⎛ Lbl l−1 ⎢⎜ ⎟⎥ ⎡ ⎤ = θ¯ xpm + ∑ αl⎣ θ˜ xpm( k − l ) ⎦ + ∑ αl⎢ ⎜ θlpn( k ) + ∑ v( u)⎟⎥ ⎜ ⎟ u =0 l = 1 ⎣⎢ ⎝ l=1 ⎠⎥⎦

(

)

(

)

(

= θ¯ xpm +

Lbl

∑

l=1

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

)

⎡ ⎤ αl⎣ θ˜ xpm( k − l ) ⎦ + θlpn( k ) +

(

)

⎛ l−1 ⎞ ⎜ ⎟ αl⎜ ∑ v( u)⎟ . ⎜ ⎟ l=1 ⎝ u =0 ⎠ Lbl

∑

(44)

Substituting Eq. (44) in Eq. (41), incorporating the ASE noise, and L assuming that ∑l =bl1 αln′̇ (k − l )⪡1, the phase estimation error can be shown as Fig. 4. The flexible DA-ML PE receiver structure.

Δθ = θ¯xpm + θ˜xpm( k ) + θ lpn( k ) − θ^( k ) ⎡ Lbl ⎛ ⎡ ⎤ ⎤ j θ¯ k−l +θ k−l ⎞ + θ˜ ^ θ ( k ) = arg⎢ ∑ ⎜ αle ⎣ xpm xpm( ) lpn( )⎦⎟⎥ ⎢ ⎠⎥⎦ ⎣ l=1 ⎝

⎡ Lbl ⎢ θ˜ ⎡ ˜ ⎤ ⎢ xpm( k ) − ∑ αl⎣ j θxpm( k − l ) ⎦ − l=1 ⎢ = ⎢ Lbl ⎢ ⎡ ⎤ ̇ ⎢ − ∑ ⎣ Im αln″( k − l ) ⎦ ⎣ l=1

(

⎡ ⎤ ⎡¯ ⎤ ˜ ⎢ α1e j⎣ θ xpm+ θ xpm( k − 1) + θlpn( k − 1)⎦ ⎥ ⎢ ⎥ Lbl ⎛ = arg⎢ ⎡¯ ⎤⎞ ⎥ ˜ +θ j θ ( k − l )+ θlpn( k − l )⎦⎟⎥ ⎢ + ∑ ⎜ αle ⎣ xpm xpm ⎝ ⎠ ⎥⎦ ⎢⎣ l=2

(

)

Lbl

⎛

l−1

⎞

l=1

⎝

u =0

⎠

∑ αl⎜⎜ ∑ v( u)⎟⎟

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(45)

with ⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

⎡ θ¯ + θ˜xpm( k − 1) + θ lpn( k − 1) ⎢ xpm ⎢ ⎡ ⎡ θ˜ ⎛ k − l ) − θ˜ xpm( k − 1)⎤⎞⎞ ⎢ ⎢ xpm( ⎥⎟⎟ ⎢ j⎢ L ⎜ =⎢ ⎤⎥ ⎟ ⎢ 1 bl ⎜ + θ lpn( k − l ) − θlpn( k − 1⎦ ⎦⎟ ⎣ + + arg 1 α e ∑ ⎢ ⎢ ⎜ l ⎟⎟ α1 ⎢ ⎢ ⎟⎟ l=2 ⎜ ⎟⎟ ⎜ ⎢ ⎠⎠ ⎝ ⎣⎢ ⎣

⎧ ⎡ ⎞⎞⎫ Lbl ⎛ l − 1 Lbl ⎪ ⎢ ⎟⎪ ṅ″( k − l ) = ṅ′( k − l )exp⎨ − j⎢ θ¯xpm + ∑ αl⎡⎣ j θ˜xpm( k − l ) ⎤⎦ + θlpn( k ⎤⎦ + ∑ αl⎜⎜ ∑ v( u)⎟⎟⎟⎬ , ⎟ ⎪ ⎢ ⎠⎠⎪ l=1 ⎝ u =0 l=1 ⎭ ⎩ ⎣

(

(42)

We define a function of Z˜ (Lbl ), and the function can be approximated as L ⎡˜ ⎞ ⎤ ˜ 1 bl ⎛ Z˜ ( Lbl ) = 1 + ∑ ⎜ α le j⎣ θxpm( k − l )− θxpm( k − 1⎦ + θlpn( k − l )− θlpn( k − 1) ⎟ α1 l = 2 ⎝ ⎠

)

⎡ ⎤ Lbl ⎡ ⎛ ˜ θ xpm( k − l] − θ˜xpm( k − 1) ⎞⎞⎟⎥ 1⎢ ⎟ 1 + ∑ α l⎢⎢ j⎜ ⎜ +θ ⎟⎟⎥ α1 ⎢⎢ − − − k l k θ 1 ) lpn ( ) ⎠⎠⎥⎦ lpn ( l=2 ⎣ ⎝ ⎣ ⎤ ⎡ Lbl ⎡ ⎛ ˜ ˜ θ − − θ − k l k 1 ( ( ) ⎞⎟⎞⎟⎥ ] xpm xpm 1 ⎢ ≅ exp⎢ j ∑ α l⎢⎢ ⎜ ⎥. ⎟ ⎟ ⎜ α1 +θ k − l ) − θ lpn ( k − 1) ⎠⎠⎥ ⎦ ⎣⎢ l = 2 ⎣ ⎝ lpn ( ≅

(43)

)

(46)

where n″̇ (l ) is statistically identical to n′̇ (l ) since n′̇ (l ) is circularly symmetric, and Im(x ) is the imaginary part of x. The approximation: arg[1 + jx] ≈ Im[x] is made in deriving Eq. (45). According Eqs. (42) and (18), after some algebra (the detailed derivation is given in Appendix A), Eq. (45) leads to a Gaussian approximation for Δθ with moments E[Δθ ] = 0, ⎡ L bl − 1 ⎛ L bl L bl ⎞2 ⎢ 1 ⎡ L bl ⎤ ⎢ ∑ α 2⎥σ 2 + ∑ ⎜ ⎟ σ 2 + R (0) − 2 ∑ α R ⎢ α ∑ l pm l pm (l ) ⎟ lpn ⎜ 2 ⎢⎣l = 1 l ⎥⎦ n˙ ′ ⎢ u = 0 ⎝l = u + 1 ⎠ l=1 E⎡⎣Δθ 2⎤⎦ = ⎢ L bl L bl ⎢ ⎢ + ∑ ∑ ααR i j pm ( i − j ) ⎢ i =1 j =1 ⎣

⎤ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

(47)

By using the approximation 1 + jx ≅ exp[jx], we rewrite Eq. (42) as Here, letting the weighted coefficients are equal, which is α = (1/Lbl )·[1, 1, …1]1 × Lbl . Thus, Eq. (47) can be written as

Fig. 5. Phase error variance versus block length in QPSK channel coherent receiver with conventional DA-ML and flexible DA-ML. SNR is 7 dB, Dres = 80 ps/nm , Beat linewidth ¼ 2 MHz, peak input power Ppeak = 0 dBm/ch . Symbols: simulation; solid line: theoretical results.

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

243

Fig. 8. Performance comparison in QPSK channel coherent receiver between conventional DA-ML and flexible DA-ML algorithm. block length Lbl = [10, 20, 40]. Dres = 80 ps/nm , Beat linewidth ¼ 2 MHz, peak input power Ppeak = 0 dBm/ch . Symbols: simulation; solid line: theoretical results.

Fig. 6. Phase error variance versus SNR and block length in QPSK channel coherent receiver with conventional DA-ML and flexible DA-ML algorithm. Dres = 80 ps/nm , Beat linewidth ¼ 2 MHz, peak input power Ppeak = 0 dBm/ch .

Fig. 9. Performance comparison in QPSK channel coherent receiver between conventional DA-ML (optimum Lbl) and flexible DA-ML ( Lbl = 60 ) algorithm. Dres = [0, 80] ps/nm , Beat linewidth ¼2 MHz, peak input power Ppeak = 0 dBm/ch . Symbols: simulation; solid line: theoretical results.

Fig. 7. Performance comparison in QPSK channel coherent receiver between conventional DA-ML and flexible DA-ML algorithm. Block length Lbl = [10, 20, 40]. Dres = 0 ps/nm , Beat linewidth ¼ 2 MHz, peak input power Ppeak = 0 dBm/ch . Symbols: simulation; Solid line: theoretical results.

E ⎡⎣ Δθ ⎤⎦ = 0, ⎡ ⎢ 1 σ 2 + ( Lbl + 1)( 2Lbl + 1) σ 2 + Lbl + 1 R ( 0) lpn pm ⎢ 2Lbl ṅ′ 6Lbl Lbl 2⎤ ⎡ E ⎣ Δθ ⎦ = ⎢ L Lbl − 1 ⎢ 2 bl ∑ R pm( l ) + 22 ∑ ( Lbl − l )R pm( l ) ⎢− L Lbl l = 1 ⎢⎣ bl l = 1

⎤ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥⎦

(48)

We can see that the result of Eq. (48) is consistent with the phase error variance of the conventional DA-ML PE in [22,24,5], which proves that the conventional DA-ML PE is a special case of our

proposed flexible DA-ML PE algorithm. Fig. 5 plots phase error variance versus block length in QPSK channel coherent receiver with conventional DA-ML and flexible DA-ML. We compare MC simulation results with the theoretical results of Eqs. (47) and (48). In Fig. 5, it is noteworthy that the variance floor occurs with increasing block length Lbl in the flexible DA-ML algorithm, while there is an optimum block length to minimum variance in the conventional DA-ML algorithm. Compared with conventional DA-ML, the contribution of the phase noise to the phase error variance with longer block length is significantly reduced, while the impact of additive noise slight increases. This observation indicates the flexible DA-ML receiver does not suffer from the block length effect. In other words, block length need not be optimized to achieve the optimum performance in flexible DA-ML. Therefore the flexible DA-ML algorithm can reduce the hardware complexity of conventional DA-ML. 3.3. BER analysis As we can see from Eqs. (20) and (21), we can obtain the symbol SNR of received signal in QPSK channel coherent receiver DSP

244

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

Fig. 10. Hardware complexity comparison for three schemes: conventional DA-ML with fixed block length, conventional DA-ML with optimum block length, and flexible DAML in DSP units.

units. Meanwhile, we can achieve the variance of phase error by using Eqs. (47) and (48). Therefore, the BER of QPSK signal affected by additive complex Gaussian noise, LPN and XPM effects can be expressed as [10] BER =

3 e−γs /2 − 8 2

γs π

⎤ ⎡ mπ 2 ⎢⎛ ⎛ γs ⎞ ⎛ γs ⎞ ⎞ sin 4 2 ⎥ · e−(m /2)σΔθ ⎥. ⎢ ⎜ I(m − 1) /2⎜⎝ ⎟⎠ + I(m + 1) /2⎜⎝ ⎟⎠ ⎟· ⎝ 2 2 ⎠ m ⎥ m=1 ⎢ ⎦ ⎣ ∞

∑

( )

(49)

4. Simulation and discussions Simulations in VPItransmissionMaker are performed to verify the validity of our theoretical analysis. The hybrid system has 5 channels with 40 Gb/s QPSK central channel, and all the others are 10 Gb/s OOK. QPSK channel is demodulated by using a coherent receiver. The DM fiber links composed of a linear pre-compensating fiber, N ¼10 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 L¼ 100 km, attenuation α ¼0.2 dB/km, dispersion D¼ 16 ps/(km nm), nonlinear coefficient γ ¼1.315 W  1 km  1. Channel spacing Δf is 50 GHz. The 3 dB linewith of transmitter laser and local oscillator (LO) is 2 MHz. 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, laser phase noise and ASE noise on the performance of our system. From Eq. (48), we have discussed above, the contribution of the XPM induced phase noise and the laser phase noise to the variance

of Δθ increases with increasing block length Lbl, while the impact of ASE noise is reduced. Fig. 6(a) plots phase error variance versus the SNR and block length in QPSK coherent receiver with conventional DA-ML algorithm, when in-line residual dispersion Dres = 80 ps/nm and Beat linewidth = 2 MHz . It can be observed that an optimum value of block length for each pair phase noise and ASE noise. The optimum block length corresponds to the minimal variance of the phase estimation error. The variance of ASE noise decreases with increasing SNR, so the optimum block length becomes smaller in high SNR. According to Eq. (47), Fig. 6(b) plots phase error variance versus block length Lbl and SNR in flexible DA-ML algorithm. We can see that the variance floor occurs with increasing block length Lbl. It means that our flexible DA-ML algorithm can eliminate the block length effect. Monte Carlo simulations have been performed to verify the validity of our flexible DA-ML algorithm. Figs. 7 and 8 show that the BER curves were compared to the approximate BER calculated using Eq. (49). The Monte Carlo simulations were performed using a series of 217 samples. The results of simulations with at least 50 errors are included in the BER comparison. Excellent agreement can be observed between the simulation and the obtained analytical approximation in low SNR. Since the approximations used in the theoretical derivation become more accurate with higher 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. 7, it is obviously observed that the flexible DA-ML with fixed block length can significantly improve the performance of QPSK channel coherent receiver, when in-line residual dispersion Dres = 0 ps/nm and the linewidth per laser is 2 MHz. Compared

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

245

Fig. 11. Computational complexity comparison of conventional DA-ML with fixed block length, conventional DA-ML with optimum blocks length and flexible DA-ML with fixed block length.

with the conventional DA-ML, 0.92 dB, and 2.41 dB SNR improvement at BER ¼ 10  3 can be seen with a fixed block length of 10 and 20, respectively. Moreover, when block length is 40, there is significantly more than 3 dB system performance improvement. It is worth to note that the BER of flexible DA-ML are almost completely overlapped at the different block length. This observation indicates that the flexible DA-ML does not suffer from the block length effect. In Fig. 8, in-line residual dispersion Dres = 80 ps/nm , we can see a similar case. The BER performance of the flexible DA-ML at BER ¼10  3 has 0.21 dB, 1.12 dB, and 2.83 dB SNR improvement over the conventional DA-ML with a fixed block length of 10, 20 and 40, respectively, when in-line residual dispersion Dres = 80 ps/nm and the linewidth per laser is 2 MHz. From Figs. 7 and 8, we can obviously see that the performance of flexible DA-ML is better than conventional DA-ML, especially in high SNR. In Fig. 9, optimum block length is used to obtain optimal system performance in conventional DA-ML algorithm, while flexible DAML uses the fixed block length ( Lbl = 60). As shown in Fig. 9, the SNR improvement of the flexible DA-ML is about 0.31 dB and 0.11 dB over the conventional DA-ML at the BER level of 10  3 with the in-line residual dispersion of 0 ps/nm and 80 ps/nm, respectively. This result shows that the flexible DA-ML is more effective for mitigating phase noise than conventional DA-ML algorithm. In Fig. 10, we present hardware complexity comparison for three schemes: conventional DA-ML with fixed block length, conventional DA-ML with optimum block length, and flexible DA-ML with fixed block length in DSP units. The received frame of QPSK

channel consists of pilot symbols and consecutive data symbols. The received symbols are distorted by XPM effects, laser phase noise and ASE noise. In DA-ML PE algorithm, known pilot symbols are employed to start the DA-ML receiver of QPSK channel up. After DA-ML PE and data detection, we can obtain estimated data symbols. From Fig. 10, we can find that conventional DA-ML with fixed block length is the most efficient. However, coherent receiver performance will be influenced by block length effect. As can be seen from Figs. 7 and 8, the system performance of conventional DA-ML with fixed block length is the worst. To improve the performance of coherent receiver, optimum block length is proposed in hybrid WDM system [22]. From Fig. 6(a), we can find that optimum block length can give the minimum variance of phase estimation error in conventional DA-ML algorithm. When block length is above or below optimum block length, the variance of phase error will be increased. Therefore, the DA-ML with optimum block length can achieve significant improvement in performance compared with DA-ML with fixed block length. However, in conventional DA-ML with optimum block length, we must add hardware module to adjust the block length of known pilot symbols and received pilot symbols, respectively. Compared with conventional DA-ML with fixed block length, it means that hardware complexity is increased in DSP units. To eliminate block length effect, a flexible DA-ML algorithm is proposed in this paper. From Fig. 6(b), we can see that the variance floor occurs with increasing block length. It means that block length need not be optimized to achieve optimum performance in our flexible DA-ML algorithm.

246

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

Compared with conventional DA-ML with optimum block length, our flexible DA-ML can decrease hardware complexity in DSP units and achieve better system performance. Meanwhile, compared with conventional DA-ML with fixed block length, our flexible DAML has significant improvement in system performance. In Fig. 11, we present computational complexity comparison for three schemes: conventional DA-ML with fixed block length, conventional DA-ML with optimum block length and flexible DA-ML with fixed block length. In the “main function” of the three schemes, the computation of reference phase is a one-shot process and the data decision of the three schemes is the same. Therefore, the “main function” of the three schemes has the same computation complexity. From Fig. 11, we can find the conventional DA-ML with fixed block length has the smallest computation complexity. In conventional DA-ML with optimum block length, we need to obtain optimum block length, so this process increases algorithm computational complexity. The computational complexity of this process is O(4Lbl − 1). First, we need to obtain different block lengths corresponding to the phase error variance of ASE and phase noise, and the computational complexity is (2Lbl ). Second, we can obtain the absolute of difference between ASE phase error variance and phase noise phase error variance, and the computational complexity is (Lbl ). Finally, we can find the least absolute value corresponding to optimum block length, and the computational complexity is (Lbl − 1). In flexible DA-ML with fixed block length, the weighted factors are introduced to eliminate block length effect, so computational complexity is increased to obtain weighted factors. The computational complexity of weighted factors is O(Lbl2 + 1). First, we need to calculate each element of the covariance matrix C in Eq. (30), so the computational complexity is (Lbl2 ). Second, we can obtain weighted factors by once calculation. The computational complexity of conventional DA-ML with fixed block length is O(main function), however the system performance of conventional DA-ML with fixed block length is the worst. The computational complexity of conventional DA-ML with optimum block length is O(main function) + O(4Lbl − 1). The computational complexity of flexible DA-ML with optimum block length is O(main function) + O(Lbl2 + 1). Although the flexible DA-ML algorithm has higher computational complexity, our flexible DA-ML can reduce the hardware complexity in DSP units and obtain better system performance. In addition, the computational complexity of the three PE schemes mainly depends on O(main function), so our flexible DA-ML receiver is feasible in the real systems.

5. Conclusion In this paper, we investigate XPM effects induced by OOK channels in hybrid QPSK/OOK WDM systems. A general model based on VSTF method is proposed to entirely describe XPM effects at the end of DM fiber links. Based on our model, the flexible DAML algorithm is proposed to eliminate the block length effect of conventional DA-ML algorithm in QPSK channel coherent receiver DSP units. We derive the analytical expression of phase error variance for the performance prediction of QPSK channel coherent receiver with the flexible DA-ML PE algorithm. BER performance is evaluated and compared through both theoretical derivation and MC simulation. The results show that our flexible DA-ML can eliminate the block length effect of conventional DA-ML algorithm. The flexible DA-ML with fixed block length has significant improvement in performance compared with the conventional DAML with fixed block length, especially in high SNR. Compared with the conventional DA-ML with optimum block length, our flexible DA-ML can obtain better system performance and reduce the hardware complexity in coherent receiver DSP units. Our theoretical analysis will be very useful to improve the performance of

coherent receiver. The flexible V–V PE algorithm can be obtained by using the same theoretical analysis. This is the subject of continuing research.

Acknowledgments This work is supported by the Innovation Project of Jiangsu, China (No.CXZZ13 0481).

Appendix A. Phase error variance after flexible DA-ML First of all, we analyse the contribution of XPM induced phase noise to the phase estimation error. According Eq. (45), the phase estimation error in the absence of the additive noise and laser phase noise can be shown to be Lbl

Δθ = θ˜xpm( k ) −

∑ αl⎡⎣ j( θ˜xpm( k − l))⎤⎦.

(50)

l= 1

When Lbl = 1, the phase estimation error is

Δθ = θ˜xpm( k ) − α1θ˜xpm( k − 1).

(51)

The variance of phase estimation error can be shown as 2⎤ ⎡ E⎡⎣ Δθ 2⎤⎦ = E⎢ θ˜xpm( k ) − α1θ˜xpm( k − 1) ⎥ ⎣ ⎦

(

)

2 ⎡ 2 ⎤ = E⎣ θ˜xpm( k ) + α12θ˜xpm( k − 1) − 2α1θ˜xpm( k )θ˜xpm( k − 1)⎦

= R pm( 0) + α12R pm( 0) − 2α1R pm( 1).

(52)

When Lbl = 2, the phase estimation error is

Δθ = θ˜xpm( k ) − α1θ˜xpm( k − 1) − α2θ˜xpm( k − 2).

(53)

The variance of phase estimation error can be written as 2⎤ ⎡ E⎡⎣ Δθ 2⎤⎦ = E⎢ θ˜xpm( k ) − α1θ˜xpm( k − 1) − α2θ˜xpm( k − 2) ⎥ ⎣ ⎦

(

)

⎡ 2 2 2 2 2 ⎢ θ˜xpm( k ) + α1 θ˜xpm( k − 1) + α2 θ˜xpm( k − 2) ⎢ = E⎢ − 2α1θ˜xpm( k )θ˜xpm( k − 1) − 2α2θ˜xpm( k )θ˜xpm( k − 2) ⎢ ⎢⎣ + 2α1α2θ˜xpm( k − 1)θ˜xpm( k − 2) ⎡ R ( 0) − 2α R ( 1) − 2α R ( 2) ⎤ 1 pm 2 pm pm ⎥. =⎢ 2 ⎢ +α 2R ⎥ 1 pm( 0) + α2 R pm( 0) + 2α1α2R pm( 1) ⎦ ⎣

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

(54)

When Lbl = L , the phase estimation error is ⎡⎛ ⎡ ⎢ E ⎡⎣ Δθ 2⎤⎦ = E ⎢ ⎜ θ˜xpm( k ) − ⎢ ⎜ ⎢⎣ ⎢⎝ ⎣

2⎤ ⎤⎞ ⎥ ⎥ ⎦⎠ ⎥ ⎦

L

∑ αlθ˜xpm( k − l )⎥⎥⎟⎟ l=1

L

 = R pm( 0) + α12R pm( 0) + ⋯ + αL2R pm( 0) L

 −2 ⎡⎣ α1R pm( 1) + α2R pm( 2) + ⋯ + αLR pm( L )⎤⎦ ⎤ ⎥ ⎥ ⎥ ⎥ ⋮ ⎥ ⋮ ⎥ ⎞ 2 ⎥ ⎟ ⎥  ⎥ +2 ⎡⎣ αL − 2αL − 1R pm( 1) + αL − 2αLR pm( 2}⎤⎦ ⎟ ⎟ ⋯⎥ 1 ⎥ ⎟ ⎥ ⎟  ⎟ ⎥ +2 ⎡⎣ αL − 1αLR pm( 1)⎤⎦ }1 × 1 ⎠2 × 2 ⎥⎦ L −1

 +2 ⎡⎣ α1α2R pm( 1) + ⋯ + α1αLR pm( L − 1)}

( L − 1)·( L − 1)

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ L·L

(55)

Y. Zhang, Y. Wang / Optics Communications 365 (2016) 237–247

After some algebra, Eq. (55) can be written as Lbl

E⎡⎣ Δθ 2⎤⎦ = R pm( 0) − 2 ∑ αlR pm( l) + l= 1

Lbl Lbl

∑ ∑ αiαjR pm( i − j ). i=1 j=1

(56)

Noting that the laser phase noise, additive noise and XPM induced phase noise are independent, so we can obtain the contribution of laser phase noise to the phase estimation error [8]

E⎡⎣Δθ 2⎤⎦ =

⎞2 2 ⎜ ∑ αl⎟ σlpn . ⎜ ⎟ ⎝l = u + 1 ⎠

Lbl − 1 ⎛ Lbl

∑

u= 0

(57)

Knowing that

σ2 D⎡⎣ Im{n( l)}⎤⎦ = n , 2

(58)

According Eqs. (45) and (46), the contribution of additive noise to the variance of Δθ is

⎡ 1 D⎢ ⎢⎣ Lbl

Lbl

⎤

l= 1

⎥⎦

∑ Im( αln″̇ ( k − l))⎥ =

⎡ 1⎢ 2 ⎢⎣

Lbl

⎤

l= 1

⎥⎦

∑ αl2⎥σn2′̇ .

(59)

When block length is Lbl , meanwhile, additive noise, laser phase noise and XPM induced phase noise are considered, the phase estimation can be shown to be

⎤ ⎡ ⎡ Lbl ⎤ Lbl − 1 ⎛ Lbl ⎞2 ⎥ ⎢ 1⎢ 2⎥ 2 2 ⎜ ⎟ ∑ ∑ α σ + α σ + R 0 ∑ ( ) pm l⎟ lpn l ⎥ ⎜ ⎢ 2⎢ n˙ ′ ⎥ ⎦ ⎣ ⎝ ⎠ 1 0 u l = = l= u+ 1 ⎥. E⎡⎣Δθ 2⎤⎦ = ⎢ ⎥ ⎢ Lbl Lbl Lbl ⎢ − 2 ∑ α R (l) + ∑ ∑ α α R ( i − j ) ⎥ i j pm l pm ⎥ ⎢ i=1 j=1 l= 1 ⎦ ⎣

(60)

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