Optimal signal constellation design for ultra-high-speed optical transport in the presence of nonlinear phase noise

Tao Liu; Ivan B. Djordjevic

doi:10.1364/OE.22.032188

1. Introduction

The fiber nonlinearities can be considered as one of the most important factor in coherent long-haul optical transmission system [1]. Due to fiber Kerr nonlinear effects, the refractive index of optical fiber increases with optical intensity to slightly slow down the propagation speed, inducing intensity depending nonlinear phase shift, which includes the nonlinear phase noise (NLPN) because of the noise component in optical intensity introduced by amplifier [2]. In order to improve the performance of the long-haul optical transport systems with coherent detection, the nonlinear phase noise should be considered in order to increase data rates and/or extend the transmission distance of optical transmission systems.

The optimal design of a signal constellation, i.e., placing M constellation points in the complex plane such that symbol error probability (SEP) or a specific criterion is minimized under an average or peak power constraint, is a classical problem in communication theory [3]. However, only few results are known about the constellation design for optical channel with nonlinear effects. A maximum likelihood (ML) detector for phase-shift keying modulated signals and two-stage detector for 16-QAM constellation has been proposed by Lau and Kahn [4]. Beygi has proposed 16-point ring constellation sets for combating the effect of NLPN [5]. Also, design of APSK constellations for coherent optical channels with NLPN is proposed by Hager in [6]. Although these methods can achieve better SEP performance, there is no consideration about an overall algorithm used to design the optimal signal constellation in the presence of phase noise, applicable to arbitrary signal constellation size.

In this paper, a new optimal constellation design algorithm is proposed that is applicable to coherent detection system in the presence of either linear or nonlinear phase noise. As an extension of the optimum signal constellation design (OSCD), which is introduced in [7], the proposed algorithm uses the cumulative log-likelihood function instead of the Euclidian distance, which is applicable to both linear and nonlinear phase noise-dominating scenarios. The optimum source distribution used in the algorithm is generated by maximizing the channel capacity based on well-known Arimoto-Blahut algorithm [8]. In this scheme, the proposed algorithm can interact with channel impairments and log-likelihood function calculation by using the same criterion. Also, an LDPC coded modulation scheme, representing the generalization of [9], suitable for use in combination with constellations obtained by this algorithm is proposed. Monte Carlo simulations indicate that the signal constellations obtained by the proposed algorithm outperforms corresponding QAM counterparts significantly in both linear and nonlinear phase noise dominated scenarios. Notice that in [7] the constellations obtained for ASE noise dominated scenario are evaluated in the channel dominated by ASE noise. In this paper, we design the signal constellations for the phase noise dominated channel, which is a key difference. Simulation results indicate that new constellations we obtained outperform previous constellations for ASE noise dominated scenario and significantly outperform QAM constellations in both linear or nonlinear phase noise dominated channel models.

The paper is organized as follows. In section 2, we introduce the channel model suitable for both linear and nonlinear phase noise dominated channels. In section 3, we present the proposed optimal signal constellation design algorithm for these two channels. In section 4, we introduced the cumulative log-likelihood function and our proposed coded modulation scheme employing the new signal constellation sets obtained by algorithm introduced in section 3. The results of the Monte Carlo simulation are summarized in section 5. Concluding remarks can be found in section 6.

2. Channel model

In this section, we introduce the equivalent coherent optical channel model suitable for both linear and nonlinear phase noise channels. The phase noise dominated channel modeling is a well-investigated topic in optical transmission [10], [11]. Also, the statistical description of the residual error after carrier recovery is introduced in [12], [13]. In this paper, the model we used [6] corresponds to one particular situation, when either the carrier phase estimation (CPE) is imperfect or the nonlinear phase noise is imperfectly compensated for. The proposed signal constellation design algorithm is applicable to the phase noise channel models [10]- [13] as well.

2.1 Coherent optical channel dominated by the linear phase noise

The equivalent channel model for coherent detection, upon compensation of linear and nonlinear impairments and CPE, can be represented as:

r_{k} = s (a_{k}, θ_{k}) + z_{k}, r_{k} = {[r_{k}^{(1)} \dots r_{k}^{(i)} \dots r_{k}^{(N)}]}^{T}

s (a_{k}, θ_{k}) = e^{j θ_{k}} {[a_{k}^{(1)} \dots a_{k}^{(i)} \dots a_{k}^{(N)}]}^{T}, z_{k} = {[z_{k}^{(1)} \dots z_{k}^{(i)} \dots z_{k}^{(N)}]}^{T}

where

r_{k}^{(i)}

is the ith component of observation vector at kth symbol interval,

a_{k}^{(i)}

is the ith coordinate of transmitted symbol (at kth symbol interval), and

z_{k}

is the corresponding noise vector (with Gaussian-like distribution of components). The

θ_{k}

denotes the residual phase error (at kth time instance) due to laser phase noise, nonlinear phase noise, and imperfect CPE.

N

denotes the number of components in the observation vector and

j

denotes the imaginary unit. In polarization-division multiplexing (PDM), Eqs. (1) and (2) correspond to either x- or y- polarization state. In 4-D signaling, the components above represent projections along in-phase and quadrature basis functions corresponding to x- and y- polarization. The model above is applicable to few-mode fiber (FMF) applications as well. For instance, to describe the laser phase noise and imperfect CPE, the Wiener phase noise model can be used:

θ_{k} = (θ_{k - 1} + Δ θ_{k}) m o d 2 π

where

Δ θ_{k}

is zero-mean Gaussian process of variance

δ_{Δ θ} = 2 π Δ f T_{s}

. The

T_{s}

denotes the symbol duration and

Δ f

denotes either linewidth or frequency offset. The phase slips can also be modeled by Markov-like process of certain memory. The probability density function (PDF) of the phase increment in Eq. (3) is given by

p_{Δ Θ} (Δ θ_{k}) = \sum_{n = - \infty}^{\infty} p (0, δ_{Δ θ}^{2}, Δ θ_{k} - n 2 π)

where

p (0, δ_{Δ θ}^{2}, Δ θ_{k} - n 2 π)

denotes the Gaussian PDF of zero-mean, variance

δ_{Δ θ}^{2}

, and argument

Δ θ_{k} - n 2 π

. The resulting noise process is Gaussian-like with the power spectral density of

N_{0}

so that the corresponding conditional probability density function is given by:

p_{R} (r | a_{k}, θ_{k}) = e^{- \frac{{| | r_{k} - s (a_{k}, θ_{k}) | |}^{2}}{N_{0}}} / (π N_{0})

For non-Gaussian channels, we will need to use the method of histograms to estimate the conditional probability density function

p_{R} (r | a_{k}, θ_{k})

.

2.2 Coherent optical channel dominated by the nonlinear phase noise

In this case, our channel model is nonlinear phase noise model with discrete amplification for the finite number of fiber spans. When the optical signal is periodically amplified by EDFAs, the nonlinear phase noise is unavoidably added to the optical signal and accumulated as the number spans increases. For convenience, we consider the discrete memoryless channel model, which is introduced in [1] and can be described as follows:

Y = (X + Z) e^{- j Φ_{N L}}

where

X \in X

is the channel input,

Z

is the total additive noise, and

Y

is the channel observation. In each fiber span, the overall nonlinear phase shift

Φ_{N L}

is given by

Φ_{N L} = \int_{0}^{L} γ P (z) d z = γ L_{e f f} P

where

P

is the launch power and

γ

is the nonlinear Kerr-parameter. For a fiber span length of

L

with attenuation coefficient of

α

, the power evolution is described as

P (z) = P e^{- α L}

and the effective length is defined as

L_{e f f} = \frac{1 - e^{- α L}}{α}

For a system with

N_{A}

fiber spans, the overall nonlinear phase noise is given by:

Φ_{N L} = γ L_{e f f} {{| E_{0} + n_{1} |}^{2} + {| E_{0} + n_{1} + n_{2} |}^{2} + \dots + {| E_{0} + n_{1} + \dots + n_{N_{A}} |}^{2}}

where

E_{0}

is the baseband representation of the transmitted electric field,

n_{k}

is independent identically distributed zero-mean circular Gaussian random complex variable with variance

δ_{0}^{2}

. The total additive noise at the end of all fiber segments has the variance

δ^{2} ≜ E [Z^{2}] = 2 N_{A} δ_{0}^{2}

and can be calculated as [14]

δ^{2} = 2 n_{s p} h ν α Δ ν N_{A} L

where all parameters are summarized in the Table 1 [4].

Table 1. Parameters of the System under Study

View Table

In this channel model, the variance of the phase noise is dependent on the channel input and the channel is specified by the number of spans, transmission length, and the launch power. In Fig. 1, we show the received constellation diagrams for 16-QAM constellation with $K = 25$ , $L = 2000 K m$ for different powers P.

Fig. 1 Constellation diagrams of received signal with 16-QAM constellation, L = 2000km, for different launch powers.

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3. Optimal signal constellation design for phase noise dominated channels

In the presence of phase noise, we can use an algorithm similar to OSCD algorithm [7], but now changing the optimization criterion. Instead of minimizing the mean square error, we define the cumulative log-likelihood function and get the optimal signal constellation that maximizes this function. Namely, the Euclidean distance receiver is optimum only for the AWGN channel. For the linear or nonlinear phase error dominated channel, the Arimoto-Blahut algorithm can be applied to find the optimal source distribution. The resulting OSCD constellation optimized from the training sequences generated from the optimal source distribution will make the input symbols to approximately have similar optimal source distribution, as the channel model is Gaussian plus phase noise. Our proposed algorithm, named here as a log-likelihood ratio-based OSCD (LLR-OSCD) algorithm, can be formulated as follows.

0) Initialization: Choose the signal constellation that will be used for initialization of size M.
1) Generate the training sequence from the optimum source distribution, denoted as ${x_{j}; j = 0, \dots, n - 1}$ .
2) Group the samples from this sequence into $M$ clusters. The membership to the cluster is determined based on LLR of sample point and candidate signal constellation points from previous iteration. Each sample point is assigned to the cluster with the largest LLR. Given the $m$ th subset (cluster) with $N$ candidate constellation points, denoted as ${\hat{A}}_{m} = {y_{i}; i = 1, \dots, N}$ , find the log-likelihood (LL) function of partition $P ({\hat{A}}_{m}) = {S_{i}; i = 1, \dots, N}$ , as follows $L L_{m} = L L ({{\hat{A}}_{m}, P ({\hat{A}}_{m})}) = n^{- 1} \sum_{k = 0}^{n - 1} \max_{y \in {\hat{A}}_{m}} L L (x_{k}, y)$
The function $L L (x_{j}, y)$ represents the cumulative log-likelihood function defined as
$L L (x_{k}, y) = \frac{1}{N S} \sum_{i = 1}^{N S} - \frac{{x_{k 1} - R e [(y_{1} + y_{2} j) e^{- j \times P N_{i}}]}^{2} + {x_{k 2} - I m [(y_{1} + y_{2} j) e^{- j \times P N_{i}}]}^{2}}{2 δ^{2}}$
where $N S$ denotes the number of phase noise samples and the corresponding phase noise sample is denoted as $P N_{i}$ . $x_{k 1}$ and $x_{k 2}$ denote the first and second coordinates of the point $x_{k}$ . Similarly, $y_{1}$ and $y_{2}$ denote the coordinates of the received point $y$ . The equation above is written by having in mind a two-dimensional (2D) signal constellation design, but it can straightforwardly be generalized for arbitrary dimensionality.
3) Determine the new signal constellation points as center of mass for each cluster.
Repeat the steps 2)-3) until convergence is achieved.

The $P N_{i}$ can be generated by target channel model introduced in Section 2 for both linear and nonlinear phase noise case. That is to say, if $P N_{i}$ is generated by linear phase noise model, then the resulting constellation is obtained for linear phase noise dominated scenario. Meanwhile, if $P N_{i}$ is generated by the nonlinear phase noise model, the result is obtained for nonlinear phase noise dominated scenario. The number of noise sample is chosen as 50 and the length of training sequence is $10^{4}$ for 8-ary constellation and $10^{5}$ for 16-ary constellation. We use QAM constellation as a starting constellation in initialization step.

As an illustration, in Fig. 2 we provide the result of the proposed algorithm. The proposed constellations, which can be named as LLR-OSCD constellations, for linear phase noise model are shown in Fig. 2(a), by setting the frequency offset × symbol duration product to $10^{- 3}$ . Meanwhile, the constellation, which is called NL-OSCD, for nonlinear phase noise is also shown in Fig. 2(b). The result is generated at launch power of –6 dB for the transmission length of 2000km.

Fig. 2 The optimized 2D 16-ary signal constellations for: (a) linear phase noise model, (b) nonlinear phase noise model (after 2000 km of SMF and launch power of – 6dBm).

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4. Proposed LDPC coded modulation scheme

For convenience, we can use the likelihood function defined as

L (a_{k}, θ_{k}) = \frac{p_{R} (r | a_{k}, θ_{k})}{p_{R} (r | a_{k} = 0)}

If the sequence of

L = T / T_{s}

statistically independent symbols,

a = {[a_{1} \dots a_{L}]}^{T}

, is transmitted, the corresponding likelihood function will be:

l (a, θ) = \prod_{l = 1}^{L} L (a_{k}, θ_{k})

To avoid numerical overflow problems, the log-likelihood function should be used instead, so we have that

l (a, θ) = l o g L (a, θ)

Clearly, the maximum likelihood (ML) approach leads to exponential increase in complexity as sequence length

L

increases. Here we propose a different strategy, which is inspired by Monte Carlo integration method. Namely, to calculate the log-likelihood function we will need to perform the following

L

-dimensional numerical integration:

l (a) = log {\int^{​} \dots \int^{​} \exp [l (a, θ)] p_{Θ} (θ) d θ}

Instead of numerical integration, we propose to use Monte Carlo integration. Namely, by using the Monte Carlo integration, the log-likelihood function

l (a)

can be estimated as:

l (a) = log E_{θ} {\exp [l (a, θ)]}

where the expectation averaging

E_{θ}

is performed for different phase noise realizations. This method is particularly simple for memoryless phase noise processes, Wiener phase noise process and cyclic slip phase noise process described as a Markov process of reasonable memory. It can be shown that complexity of this method is

O {(m^{2} + S) N_{r}}

, where

m

is the channel memory,

S

is the sequence length and

N_{r}

is the number of phase noise realizations. Compared to the ML method, whose complexity is

O {M^{L}}

, where

M

is the signal constellation size, complexity is significantly lower for long sequences to be detected. This method requires the knowledge of Markov phase noise process, which is quite easy to characterize by training. In particular, for the Wiener phase noise process and memoryless phase noise process, the Gaussian noise generator is only needed. Also, we can also use the nonlinear phase noise model introduced in Section 2 to generate the phase noise process.

Our proposed LDPC coded modulation scheme used in the simulations below is shown in the Fig. 3. The $b$ independent data are first encoded by an $(n, k)$ LDPC encoder and written in row-wise fashion into $b \times n$ block interleaver. Then either NL-OSCD or LLR-OSCD can be used for modulation and 2D (I/Q) modulator performs the electrical-to-optical conversion. The polarization division multiplexing is used. On receiver side, after polarization diversity receiver, which provides the projection along both polarizations and in-phase and quadrature channels, we perform compensation of linear and nonlinear impairments as well as CPE. We then calculate the log-likelihood function based on Eq. (17), by using the Monte Carlo integration approach introduced above.

Fig. 3 The LDPC coded modulation scheme with Monte Carlo integration to evaluate LLRs. PBS/C: polarization beam splitter/combiner, LPF: low-pass filter, ADC: analog-to-digital converter, APP: a posteriori probability.

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5. Numerical results

The LDPC coded modulation scheme intorduced in Section 4 is used in simulation. The symbol rate is set to $R_{s} = 32.25 G S / s$ and the quasi-cyclic LDPC (16935, 13550) code of rate 0.8 is used. All the results are obtained for 3 outer iterations and 20 inner (LDPC decoder) iterations.

5.1 Simulation for linear phase noise channel

The BER performance for LDPC-coded polarization-division multiplexed OSCD and LLR-OSCD constellation sets are summarized in the Fig. 4. We have shown the performance different LLR-OSCD constellation sets optimized with different initial constellations (put in brackets) and different number of phase noise samples, denotes as NS. The constellation size is 16 and the frequency offset × symbol duration product is set to $10^{- 3}$ . The green line denotes the BER performance of LLR-OSCD that uses the IPQ (Iterative Polar Quantization) as initial constellation, while the black line is optimized from QAM. It is evident that the 16-ary LLR-OSCD optimized from QAM performs the best for 500 phase noise samples. The proposed LLR-OSCD with (NS = 500) outperforms conventional QAM by 1.1 dB and OSCD by 0.2 dB.

Fig. 4 BER performance for proposed LLR-OSCDs for 16-ary 2D constellation.

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5.2 Simulation for the nonlinear phase noise channel

In simulations related to the nonlinear phase noise channel, we first fix the number of spans and the span length (the span length is 80km for all simulations), while vary the launch power in order to determine the optimum launch power (minimizing the BER), as illustrated in Fig. 5. The Fig. 5 clearly indicates the existence of the optimal launch power for the total transmission distance of 4000 km for both 8-ary NL-OSCD and 8-QAM. It can also be noticed that the nonlinear tolerance of 8-NL-OSCD is better. NL-OSCD (1, 2000) curve denotes that this constellation has been obtained by employing the algorithm described in Sec. 3 with the launch power of 1 dBm and for total transmission distance of 2000 km. We can see that NL-OSCDs with different parameters have similar optimal power around 1 dBm. Then we fix the span length and launch power in order to see the performance of different constellations against the total transmission length. The total transmission distance that can be achieved by employing NL-OSCDs and LDPC coding is around 6700 km, as shown in Fig. 6.

Fig. 5 BER vs. launch power for uncoded 8-ary NL-OSCD and 8-QAM.

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Fig. 6 BER vs transmission length plot for LDPC-coded 8-ary NL-OSCD and 8-star-QAM.

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In Fig. 6, we use LLR-OSCD to denote the constellation designed for linear phase noise dominated scenario. It is obvious to see that the NL-OSCD outperforms 8-ary star-QAM for almost 500 km. Notice that LLR-OSCD performs a little worse than the QAM, which indicates that LLR-OSCD constellation obtained for linear phase noise channel model is not applicable to the nonlinear phase noise channels. We then perform the similar study for 16-ary NL-OSCD and the results are summarized in Fig. 7 and Fig. 8.

Fig. 7 BER vs. launch power for uncoded 16-ary NL-OSCD and 16-QAM.

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Fig. 8 BER vs transmission length plot for LDPC-coded 16-ary NL-OSCD and 16-QAM.

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The optimal power for 16-ary NL-OSCD is 4 dBm, as shown in Fig. 7, and the total transmission distance exceeds 4400 km, as shown in Fig. 8. The transmission distance can be further increased by employing the turbo equalization principle, which is out of scope of this paper. The Fig. 7 indicates that 16-ary NL-OSCD has better nonlinear tolerance than 16-QAM. On the other hand, Fig. 8 indicates that LDPC-coded 16-ary NL-OSCD outperforms 16-QAM by almost 1000 km and LLR-OSCD by 600 km. The new signal constellations obtained by using the proposed algorithm clearly have better nonlinear tolerance and can significantly extend the total transmission distance of QAM. The main reason for this improvement is due to the fact that during the signal constellation design, the nonlinear nature of the channel is taken into account.

In Fig. 9, we provide an interesting result of our study. If we increase the span length to 100 km, the total transmission length of LDPC-coded 8 ary-NL-OSCD get increased to almost 11000 km with the launch power of −1.5 dB. Clearly, the LDPC-coded 8-ary NL-OSCD outperforms LPDC-coded 16-QM QAM for almost 3000 km. We can conclude that there exists an optimal span length for the system, with which we can achieve better performance. However, in Fig. 9, in addition to constellation optimization, launch power optimization, the span length is optimized as well.

Fig. 9 BER vs. transmission length plots for LDPC-coded 16-ary NL-OSCD and 16-QAM when span length is 100 km.

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6. Conclusion

In this paper, we proposed a new signal constellation designed algorithm, which can be used to design signal constellations suitable for coherent optical channels dominated by either linear or nonlinear phase noise. By using the log-likelihood function as the design criterion in signal constellation design instead of the Euclidean distance, we can take the nonlinear nature of the channel into account. The Monte Carlo simulations demonstrate that for 8-ary and 16-ary cases, LDPC-coded NL-OSCDs outperform the corresponding LDPC-coded QAM constellations for almost 500 km and 1000 km, respectively, for span length of 80 km. The LLR-OSCD outperforms QAM constellation by 1.1 dB for 8-ary, for phase noise dominated channel. We have found that the signal constellation is both launch power and distance dependent, as expected. Further, by optimizing the span length together with the launch power the transmission distance can be further increased. The proposed signal constellation design algorithm can be easily expanded to the multi-dimensional case, which will be the focus of the future work.

Acknowledgments

This work was supported in part by NSF under Grant CCF-0952711.

References and links

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6. C. Hager, A. Grell, A. Alvarado, and E. Agrell, “Design of APSK constellations for coherent optical channels with nonlinear phase noise,” IEEE Trans. Commun. 61(8), 3362–3373 (2013).

7. I. Djordjevic, T. Liu, L. Xu, and T. Wang, “Optimum signal constellation design for high-speed optical transmission,” in Optical Fiber Communication Conference, OSA Technical Digest (Optical Society of America, 2012), paper OW3H.2. [CrossRef]

8. T. Cover and J. Thomas, Elements of Information Theory (Wiley, 1991).

9. I. B. Djordjevic and T. Wang, “On the LDPC-coded modulation for ultra-high-speed optical transport in the presence of phase noise,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference 2013, OSA Technical Digest (online) (Optical Society of America, 2013), paper OM2B.1. [CrossRef]

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Symbol	Typical value
$γ$	$1.2 W^{- 1} k m^{- 1}$	Nonlinear Kerr-parameter
$n_{s p}$	$1.41$	Spontaneous emission factor
$h$	^{$6.626 \times 1 0^{- 44} J \cdot s$}	Planck’s constant
$ν$	$1.936 \times 1 0^{- 14} H z$	Optical carrier frequency
$α$	$0.0578 K m^{- 1}$	Fiber loss
$Δ ν$	$42.6 G H z$	Optical bandwidth

Optimal signal constellation design for ultra-high-speed optical transport in the presence of nonlinear phase noise

Abstract

1. Introduction

2. Channel model

2.1 Coherent optical channel dominated by the linear phase noise

2.2 Coherent optical channel dominated by the nonlinear phase noise

3. Optimal signal constellation design for phase noise dominated channels

4. Proposed LDPC coded modulation scheme

5. Numerical results

5.1 Simulation for linear phase noise channel

5.2 Simulation for the nonlinear phase noise channel

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (9)

Tables (1)

Equations (17)

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