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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22773–22790
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Polarization-time coding for PDL mitigation in long-haul PolMux OFDM systems

Elie Awwad, Yves Jaouën, and Ghaya Rekaya-Ben Othman  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22773-22790 (2013)
http://dx.doi.org/10.1364/OE.21.022773


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Abstract

In this paper, we present a numerical, theoretical and experimental study on the mitigation of Polarization Dependent Loss (PDL) with Polarization-Time (PT) codes in long-haul coherent optical fiber transmissions using Orthogonal Frequency Division Multiplexing (OFDM). First, we review the scheme of a polarization-multiplexed (PolMux) optical transmission and the 2 × 2 MIMO model of the optical channel with PDL. Second, we introduce the Space-Time (ST) codes originally designed for wireless Rayleigh fading channels, and evaluate their performance, as PT codes, in mitigating PDL through numerical simulations. The obtained behaviors and coding gains are different from those observed on the wireless channel. In particular, the Silver code performs better than the Golden code and the coding gains offered by PT codes and forward-error-correction (FEC) codes aggregate. We investigate the numerical results through a theoretical analysis based on the computation of an upper bound of the error probability of the optical channel with PDL. The derived upper bound yields a design criterion for optimal PDL-mitigating codes. Furthermore, a transmission experiment of PDL-mitigation in a 1000km optical fiber link with inline PDL validates the numerical and theoretical findings. The results are shown in terms of Q-factor distributions. The mean Q-factor is improved with PT coding and the variance is also narrowed.

© 2013 OSA

1. Introduction

In this paper, we will study the simplest multiplexed scheme which consists in a 2 × 2 polarization-multiplexed (PolMux) MIMO system. The implementation of these systems was made possible with the use of DSP algorithms and the development of high-speed electronics to recover the transmitted data at the receiver side in the electrical domain. The impairments undergone by the high bitrate transmitted signal can be categorized into two classes: linear and nonlinear. The major linear impairments are chromatic dispersion (CD), polarization-mode dispersion (PMD), and polarization-dependent loss (PDL) that can all be modeled using 2 × 2 transfer matrices (Jones matrices) straightforwardly leading to a 2 × 2 MIMO system representation. As for nonlinear effects, they will not be considered in the numerical and theoretical studies where we assess the penalties induced by the linear impairments. However, in the experimental study, we will observe the limitations caused by non-linear effects and determine the optimal operating point of the transmission (in terms of optimal launched signal power).

2. PT-coded coherent OFDM system model

In this section, we describe the general transmission scheme in which PT codes will be implemented. We also define the channel model of an optical transmission with PDL that will be considered in the simulations and the theoretical study.

2.1. PolMux coherent OFDM transmission scheme

2.1.1. Transmitter side

In a conventional PolMux OFDM transmission [7

7. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15(16), 9936–9947 (2007). [CrossRef] [PubMed]

], two independently modulated OFDM signals where each OFDM subcarrier carries an M-QAM symbol, are sent on two orthogonal polarization states; while in a PT-coded OFDM system, the modulated polarization states are correlated and carry a combination of M-QAM symbols. Indeed, PT coding consists in sending a combination of different modulated symbols on two polarization states (Pol1, Pol2) during several time slots. Hence, if we consider two consecutive time slots (T1, T2), the PT codeword matrix X will be:
X=[XPol1,T1XPol1,T2XPol2,T1XPol2,T2]
(1)

Fig. 1 General scheme of a PolMux OFDM transmission with Polarization-Time coding.

2.1.2. Receiver side

At the receiver side, a polarization beam splitter splits the incident optical signal on two orthogonal states and a dual polarization coherent receiver down-converts the signal to the electrical domain. Next, OFDM processing is carried including CP removal and an FFT. Each OFDM subcarrier sees a non-dispersive channel and the received symbol is given by [7

7. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15(16), 9936–9947 (2007). [CrossRef] [PubMed]

]:
Yk,i=Hk(ωk)Xk,i+Nk,i
(2)
where Xk,i is the 2 × 2 matrix of transmitted symbols on the kth subcarrier (k = 1 ... n) of the ith OFDM symbol, Yk,i is the 2 × 2 matrix of the received symbols. Hk(ωk) is the 2 × 2 channel matrix including laser phase noise, CD, PMD and PDL. Nk,i is a 2 × 2 matrix representing the additive noise. A training sequence known at the receiver is used to estimate the channel matrix for each subcarrier [3

3. S. L. Jansen, I. Morita, T. C. W. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM Transmission With 2-b/s/Hz Spectral Efficiency Over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009). [CrossRef]

]. Then, the common phase error induced by the laser phase noise is corrected and finally the transmitted data symbols can be detected and demodulated. It is important to note that ST codes were designed to bring coding and diversity gains to the MIMO scheme when they are optimally decoded according to the maximum likelihood (ML) criterion instead of using sub-optimal linear decoders. Considering the channel model in Eq. (2) and assuming all codeword matrices X equiprobable, an ML decoder estimates the codeword X with X′ according to the following criterion:
X=argminX𝒞YHX2
(3)
where 𝒞 is the set of all possible codewords. The indices k and i were dropped for clarity. The PT decoding can only be performed under the assumption of a constant H during the codeword duration (2 time slots in this case) which is the case of the optical channel [14

14. H. Bulow, W. Baumert, H. Schmuck, F. Mohr, T. Schulz, F. Kuppers, and W. Weiershausen, “Measurement of the maximum speed of PMD fluctuation in installed field fiber,” OFC/IOOC’99 2, 83–85.

]. ML decoding is performed by an exhaustive research with reasonable complexity when the considered modulation symbols come from a 4-QAM constellation. We point out that the decoding complexity is quite reduced with the use of OFDM at the cost of an increased overhead. PT coding could have been implemented in a single-carrier context with time domain equalization [15

15. S. Ben Rayana, H. Besbes, G. Rekaya-Ben Othman, and Y. Jaouën, “Joint equalization and polarization-time coding detection to mitigate PMD and PDL impairments,” in proc. of SPPCom’12, paper SpW2B.3.

, 16

16. A. Andrusier, E. Meron, M. Feder, and M. Shtaif, “An optical implementation of a space-time-trellis code for enhancing the tolerance of systems to polarization-dependent loss,” Opt. Letters 38(2), 118–120 (2013). [CrossRef]

]. However, multi-taps channel matrices would be required increasing the decoding complexity.

2.2. Long-haul optical link

Fig. 2 Structure of the considered long-haul optical link.

Each fiber span is modeled with a concatenation of M random PMD elements where each element has a transfer matrix HPMD(ωk) defined as the product of a delay matrix and a rotation matrix as in [17

17. S. R. Desbruslais and P. R. Morkel, “Simulation of polarisation mode dispersion and its effects in long-haul optically amplified lightwave systems,” IEE Colloquium on International Transmission Systems , 6/1–6/6 (1994).

]. These unitary effects cause no loss of energy to the transmitted signal unlike PDL that induces OSNR fluctuations. The transfer matrix of a PDL element is given by:
HPDL=Rα[100ε]Rα1=11+γRα[1+γ001γ]Rα1
(4)
The diagonal matrix gives the imbalanced attenuation values of the least and most attenuated polarization states and Rα is a random rotation matrix. α is uniformly drawn in [0 : 2π]. ε and γ are defined through ΓdB=10log101+γ1γ=10log10(ε) where ΓdB ≥ 0 is the PDL coefficient in dB (or simply referred to as PDL) and consists of the ratio between the highest and the lowest losses. The normalization factor in the model using the variable γ is commonly dropped in literature [5

5. T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarization dependent loss on coherent POLMUX-NRZ-DQPSK,” in proc. of OFC/NFOEC’08 , 1–3.

, 18

18. O. Vassilieva, I. Kim, Y. Akasaka, M. Bouda, and M. Sekiya, “Interplay between PDL and nonlinear effects in coherent polarization multiplexed systems,” Opt. Express 19(26), B357–B362 (2011). [CrossRef]

] because the diagonal matrix and the rotation matrices are enough to take into account the OSNR inequality between the polarization states as well as their crosstalk.

3. Numerical investigation of PDL mitigation

Many ST codes were designed for wireless MIMO channels. We will consider the most famous ones that have proven to be the best-performing codes on 2 × 2 and 2 × 1 Rayleigh fading channels: the Golden code, the Silver code and the Alamouti code. In the following, the codeword matrix X and the rate of each coding scheme is given. The rate of a code is defined as the number of transmitted symbols per time slot (ts). In an uncoded case, we simply fill the matrix with 4 different M-QAM symbols and the rate is equal to 2 M-QAM symbols/ts.

3.1. PT codeword matrices

3.1.1. The Golden code

The Golden code [13

13. J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2×2 full-rate space-time code with nonvanishing determinants,” IEEE Transactions on Information Theory 51(4), 1432–1436 (2005). [CrossRef]

] has the best performance on 2 × 2 MIMO Rayleigh fading channels. The codeword matrix of the Golden code is:
X𝒢=15[α(S1+θS2)α(S3+θS4)iα¯(S3+θ¯S4)α¯(S1+θ¯S2)]
(9)
where θ=1+52, θ¯=152, α = 1 +i, = 1 +iiθ̄ and S1, S2, S3, S4 are M-QAM symbols. The Golden code achieves a full rate of 2 symbols/ts because 4 symbols are transmitted during 2 symbol times. Hence, this code introduces no redundancy. Moreover, the determinant of the codeword matrix (corresponding to a coding gain on a Rayleigh fading channel [12

12. V. Tarokh, A. Naguib, N. Seshadri, and A.R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications 47(2), 199–207 (1999). [CrossRef]

]) is proportional to 15 which is the highest obtained value for a 2 × 2 ST code.

3.1.2. The Silver code

The Silver code has a slightly weaker performance than the Golden code but it has the advantage of having a reduced decoding complexity due to its particular structure. The codeword matrix of the Silver code is:
X𝒮=[S1+Z3S2*Z4*S2Z4S1*Z3*][Z3Z4]=17[1+i1+2i1+2i1i][S3S4]
(10)
where S1, S2, S3, S4 are M-QAM symbols. The determinant of this code is proportional to 17. The Silver code also achieves a full rate of 2 symbols/ts because 4 symbols are transmitted during 2 time slots making it also a redundancy-free code.

3.1.3. The Alamouti code

The Alamouti code has the optimal performance on a 2 × 1 MIMO Rayleigh fading channel and can be also used for 2 × 2 MIMO channels. The codeword matrix of the Alamouti code is defined by:
X𝒜=[S1S2*S2S1*]
(11)
where S1 and S2 are M-QAM symbols. Note that the codeword matrix has an orthogonal structure which makes the decoding straightforward. However, the Alamouti code introduces a redundancy as it has a rate of only 1 symbol/ts (half-rate code).

3.2. Performance analysis of PT codes

The performance of a PT-coded OFDM transmission on an optical channel with PDL will be investigated using the following frequency non-selective channel model with H defined in Eq. (8):
Yk=HXk+Nk
(12)

Figure 3(a) shows the BER evolution versus the SNRbit for a PDL of 6dB. Without PT coding, the SNR degradation induced by PDL is about 2.3dB at BER = 10−3. The SNR penalty -defined as the SNR gap to the PDL-free case at a given BER - is only 0.6dB with the Golden code and 0.3dB with the Silver code corresponding to a coding gain of 2dB. Apart from the important coding gains, these codes do not introduce any spectral efficiency penalties compared to the uncoded case as they are by construction redundancy-free. On the other hand, looking at the performance of the Alamouti code, we notice an SNR penalty of 3.6dB at BER = 10−3. This is due to the use of 16-QAM symbols. In Fig. 3(b), we observe the performance of the Alamouti code for 3 different levels of PDL. We notice that the code performs the same independently of the amount of PDL. The SNR penalties at BER = 10−3 for different PDL values had been evaluated for the first time in [8

8. S. Mumtaz, G. Rekaya, and Y. Jaouën, “Space-Time codes for optical fiber communication with polarization multiplexing,” in proc. of ICC’10 , 1–5.

] and the results were confirmed in [11

11. E. Meron, A. Andrusier, M. Feder, and M. Shtaif, “Use of space-time coding in coherent polarization-multiplexed systems suffering from polarization-dependent loss,” Opt. Letters 35(21), 3547–3549 (2010). [CrossRef]

]. For PDL values less than 4dB, the Silver code can mitigate almost all PDL effects and always performs better than the Golden code. In [10

10. S. Mumtaz, G. Rekaya-Ben Othman, Y. Jaouën, J. Li, S. Koenig, R. Schmogrow, and J. Leuthold, “Alamouti code against PDL in polarization multiplexed systems,” in proc. of SPPCom’11, paper SPTuA2.

], the performance of the Alamouti code was also found to be independent of PDL. Moreover, experimental demonstrations of PDL mitigation with PT coding were also realized in a back-to-back (no transmission) scenario [9

9. S. Mumtaz, J. Li, S. Koenig, Y. Jaouën, R. Schmogrow, G. Rekaya-Ben Othman, and J. Leuthold, “Experimental demonstration of PDL mitigation using Polarization-Time coding in PDM-OFDM systems,” in proc. of SPPCom’11, paper SPWB6.

]. However, all these observations were still analytically unexplained.

Fig. 3 Performance of PT codes obtained through Monte Carlo simulations.

3.3. Concatenation of FEC and PT coding

While PT coding technique uses the modulated symbols to form a codeword matrix and mitigate some channel effects, other coding techniques, known as forward error-correcting (FEC) or channel codes, operate on the information bits and add some redundancy in order to enhance the performance over the noisy channel. The FEC block at the transmitter side precedes the M-QAM modulation block and a corresponding FEC decoding unit at the receiver side follows the demodulation block. Linear block codes are a major class of channel codes where a binary information sequence of length k is mapped to a binary sequence of length n called a codeword. The code rate of a block code is denoted by Rc:
Rc=kn<1
(13)
The optimal decoding strategy of FEC codes is soft decision decoding (SDD) [26, Chap.7] that receives at its input a real value indicating the reliability of each coded bit (e.g. log-likelihood ratios) and computes the original information bits. A sub-optimal, yet computationally simple, decoding strategy is hard decision decoding (HDD) [26, Chap.7] that consists in quantizing the samples at the output of the PT demodulator before sending them to the FEC decoder.

We are interested, in this section, in evaluating the total gain provided by FEC coding and PT coding through Monte-Carlo simulations. The performance on an optical channel with ΓdB = 6dB of an uncoded 4-QAM scheme (no FEC, no PT) and a Silver-coded scheme is compared with a Bose-Chaudhuri-Hocquenghem (BCH) coded binary sequence where n = 63, k = 45 and an error correction capability t = 3, followed by 4-QAM modulation and PT coding (Silver code). The obtained BER curves are plotted in Fig. 4. At a BER = 10−4, the gain provided by the Silver code alone is 2.4dB. When HDD is considered, the gain provided by the BCH code alone is 1.8dB. The concatenation of both codes offer a total coding gain of 3.9dB, approximately equal to the sum of the separate coding gains 2.4 + 1.8 = 4.2dB. The summation of the FEC and PT coding gains is also observed when using SDD. In this case, the gain offered by the BCH code alone at a BER = 10−4 is 2.8dB. The concatenation with the Silver code provides a total gain of 4.9dB and the sum of the separate coding gains is 2.4 + 2.8 = 5.2dB.

Fig. 4 Bit Error Rate as a function of the SNRbit for the uncoded scheme and the Silver code, with or without FEC: (a) Hard decision decoding (HDD), (b) Soft decision decoding (SDD). The simulated FEC is a BCH(63,45) code.

4. Theoretical analysis of PDL mitigation

4.1. Upper bound of the error probability

The error probability is defined as:
Perror=Pr{XX}=X𝒞Pr{X}Pr{XX|X}
(14)
For equiprobable codewords, using the union bound, the error probability is upper-bounded by:
Perror1card(𝒞)X,X𝒞,XXPr(XX)
(15)
where card(𝒞) is the cardinality of 𝒞 and Pr(X → X′) is the pairwise error probability (PEP) supposing that X and X′ are the only possible codewords in the codeword space. To compute the PEP, we define the conditional PEP that we average over all the possible channel realizations. For an ML decoder, the conditional PEP is defined as:
Pr(XX)𝔼H[exp(H(XX)28σ2)]
(16)
Using the Gaussian properties of the noise N and applying Chernoff’s bound, the PEP can be upper-bounded by [26, Chap.4]:
Pr(XX)𝔼H[exp(H(XX)28σ2)]
(17)
where 𝔼H [] is the averaging operation over all possible channel realizations.

Averaging over H given in Eq. (8) where we consider constant values of ΓdB and a random rotation angle that varies uniformly in [0 : 2π], we get:
Pr(XX)exp(XΔ28σ2)I0(γeq8σ2a2+b2)
(18)
where I0(z) is the 0th order modified Bessel function of the first kind. XΔ=XX=(x1x2), x⃗1,2 being line vectors and:
a=x22x12
(19)
b=2Re(x1,x2)
(20)

We can approximate the error probability expression in Eq. (18) for high SNR values by using a first order approximation of I0(z) when z and get:
Pr(XX)exp(XΔ2γeqa2+b28σ2)
(21)

4.2. Design criterion

I0(z) being monotonously increasing for z ≥ 0, its minimum is at z = 0. This corresponds to null ‘a’ and ‘b’ and the obtained error probability will be independent of PDL. Consequently:

Proposition 1 A Polarization-Time code completely mitigates PDL if and only if all codeword differences satisfy:
  1. a = ||x⃗2||2 − ||x⃗1||2 = 0 and
  2. b = 2Re(〈x⃗1, x⃗2〉) = 0.

When this design criterion is met, we recover the performance over two parallel AWGN channels which is the best achievable performance:
perror,AWGNexp(XΔ28σ2)
(22)

The resulting criterion is completely different from the rank and the minimum determinant criteria derived for a Rayleigh fading channel [12

12. V. Tarokh, A. Naguib, N. Seshadri, and A.R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications 47(2), 199–207 (1999). [CrossRef]

] and defining respectively a diversity gain and a coding gain on the BER curves. If we compare the approximation of the error probability expression at high SNR in Eq. (21) to the one obtained in the case of a 2 × 2 MIMO Rayleigh fading channel, we notice different behaviors: the error probability of the Rayleigh fading channel decays as SNR−2r [12

12. V. Tarokh, A. Naguib, N. Seshadri, and A.R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications 47(2), 199–207 (1999). [CrossRef]

] where r is the minimum rank of the matrix XΔ. The diversity of the system is defined as the power of SNR−1 (2r in this case) and can be graphically discerned as the slope of the BER curves at high SNR values. While the error probability of the PDL channel decays exponentially as a function of the SNR as in the case of an additive white Gaussian channel in Eq. (22). Hence, Space-Time codes, used as Polarization-Time codes, bring no diversity gain to the optical channel with PDL. A coding gain that will be evaluated in the following, is only brought, reducing the penalty induced by PDL.

4.3. Performance analysis of PT codes

The performance of both coded and uncoded schemes will be examined using the derived upper bound of the pairwise error probability expression in Eq. (21). To compare the different schemes, we will compute the squared distance d2=XΔ2γeqa2+b2 of all combinations of codeword differences and compare the minimum values of d2, denoted dmin2, of the codes. The best code is the one that maximizes this minimum value. In Table 1, we report the minimum values of XΔ2γeqa2+b2 analytically computed for each investigated PT code at four different PDL values. To fill the table, we set a spectral efficiency of 4 bits per time slot for all coding schemes and consider an average symbol energy Es = 1 for all constellations.

Table 1. dmin2 for different coding schemes at different PDL values

table-icon
View This Table

First, we notice that the Alamouti code has the same minimal distance for all PDL values which explains why it performs the same independently of PDL in Fig. 3(b). This is due to the orthogonality of its codeword matrix (Eq. (11)) that induces a = b = 0 for all possible codeword differences. Hence, this code satisfies the criterion of Proposition 1. However, its performance is affected by the use of 16-QAM symbols giving a squared minimal distance of 0.8.

Second, we note that the Silver code is not optimal in mitigating PDL since it does not satisfy the derived design criterion. Unlike the Alamouti code, the Silver code has only some codeword differences having a = b = 0. In Fig. 5, we plot the performance of the Silver code for different PDL values. We see that the code mitigates almost all PDL when the PDL coefficient is equal to 3dB and 6dB whereas for a PDL of 10dB, the code is not able to completely palliate PDL. The computed dmin2 in Table 1 explain the behavior of the Silver code. dmin2 is given by the same codeword difference with a = 0 and b = 0 at a PDL of 3dB and 6dB, and is equal to 2. Whereas at a PDL of 10dB, it falls to 1.23 given by another codeword difference where a ≠ 0 and b ≠ 0.

Fig. 5 Bit Error Rate as a function of the SNRbit for the Silver code, obtained through Monte Carlo simulations.

Third, in Fig. 3(a), we saw that the Silver code outperforms the Golden code for ΓdB = 6dB, and both reduce the penalty that PDL causes to the uncoded scheme. Again, this result can be explained by looking at Table 1. Indeed, dmin2 is the greatest for the Silver code followed by the Golden code and then the uncoded scheme.

In conclusion, we were able to explain, in terms of error probability bounds, the performance of the Alamouti, the Silver and the Golden codes on an optical channel with PDL. These codes were designed to satisfy the rank and the minimum determinant criteria for a wireless channel that are no more relevant for the optical channel.

4.4. Concatenation of FEC and PT coding

The numerical investigation of the performance of a PolMux scheme using both a FEC code and a PT code showed that the total coding gain is equal to the sum of the separate coding gains brought by each. This result can be also theoretically explained and is mainly due to the exponential decrease of the error probability as a function of SNR.

The bit error probability of a linear block code over an AWGN channel, and after hard decision decoding, is determined by the minimum distance of the considered FEC code dmin,FEC and the crossover probability p of the equivalent binary symmetric channel (BSC) [26, Chap.7]:
Pe,HDDm=t+1n(nm)pm(1p)nm
(23)
where t=dmin,FEC12 is the error correction capability of the linear block code and dmin,FEC is the minimum Hamming distance between distinct codewords of this code. At high SNR, p tends towards zero and Eq. (23) is dominated by the first term where m = t + 1:
Pe,HDD(nt+1)pt+1=(ndFEC,HDD)pdFEC,HDD
(24)
where dFEC,HDD=dmin,FEC2.

Had we considered soft decision decoding, the bit error probability would have been [26, Chap.7]:
Pe,SDD(2k1)pdFEC,SDD
(25)
where dFEC,SDD = dmin,FEC.

p can be upper-bounded using Eq. (21) where the error probability is a codeword error probability. Given that a full-rate 2 × 2 PT codeword has 4 log2M bits, an error in decoding one PT codeword implies that one bit is erroneous in the best case or all 4 log2M bits are erroneous in the worst case, hence:
14log2MPr(XX)pPr(XX)
(26)
At high SNR, the pairwise error probability of the closest neighbors predominates the other terms in the union bound (Eq. (15)) and p is upper-bounded by:
pAdPTexp(dPT28σ2)
(27)
where dPT2=argminX,X𝒞(XΔ2γeqa2+b2) and AdPT being the average number of codewords located at the distance dPT of a given codeword. AdPT is usually called the kissing number.

Replacing p by its upper bound and substituting the average symbol energy ES for the average energy per information bit Eb using ES = 1 = rPTRcEb log2M, we get:
PeK(AdPT)dFECexp(dFECdPT2SNRbitrPTRclog2M4)
(28)
where SNRbit=Eb2σ2. K and dFEC are the constant and the power of the bit error probability of p in Eq. (23) or (25) depending on the chosen FEC decoding strategy. rPT is equal to 1 for a full-rate PT code and 0.5 for a half-rate code.

In order to evaluate the asymptotic gains provided by the concatenation of a FEC code and a PT code, we compare the following two schemes: a first scheme NC without FEC (dFEC = 1 and Rc = 1) and without PT coding (independent M-QAM symbols) and a second scheme FEC + PT using a linear block code and a full-rate PT code. At the same achieved error probability, the coding gain G of the coded scheme is given by:
G=SNRbit,FEC+PTSNRbit,NC=RcdFECdPT2dNC2
(29)
In decibels, we obtain GdB:
GdB=10log10(RcdFEC)+10log10(dPT2dNC2)=GdB,FEC+GdB,PT
(30)
The first term denote the coding gain of the FEC code and the third term denotes the coding gain provided by PT coding. Equation (30) shows that the total asymptotic gain obtained when concatenating a FEC code and a PT code is the sum of the gains provided by each code separately, validating the third result of our numerical investigation.

5. Experimental validation of PDL mitigation

The numerical and theoretical validations of PDL mitigation using PT coding are limited to a linear channel with a single-element or lumped PDL and AWGN noise at the receiver side. Initial experimental results limited to the linear regime also showed the efficiency of PT codes [9

9. S. Mumtaz, J. Li, S. Koenig, Y. Jaouën, R. Schmogrow, G. Rekaya-Ben Othman, and J. Leuthold, “Experimental demonstration of PDL mitigation using Polarization-Time coding in PDM-OFDM systems,” in proc. of SPPCom’11, paper SPWB6.

]. These simplifications allowed us to analyze the potential of MIMO codes for PolMux optical transmission using a handy and well defined channel model. The final remaining step is to test the ability of PT codes to mitigate inline PDL through a transmission experiment with distributed PDL taking into account the interactions of PDL with distributed ASE noise [28

28. C. Xie, “Polarization-dependent loss induced penalties in PDM-QPSK coherent optical communication systems,” in proc. of OFC/NFOEC’10 , 1–3.

] and non-linear effects [18

18. O. Vassilieva, I. Kim, Y. Akasaka, M. Bouda, and M. Sekiya, “Interplay between PDL and nonlinear effects in coherent polarization multiplexed systems,” Opt. Express 19(26), B357–B362 (2011). [CrossRef]

].

5.1. Experimental setup

Fig. 6 Experimental setup. (ECL: External Cavity Laser, AWG: Arbitrary Waveform Generator, MUX: Multiplexer, AO: Acousto-Optical Modulator, PS: Polarization Scrambler, OBPF: Optical Band-Pass Filter, ASE: Accumulated Spontaneous Emission source, LO: Local Oscillator, OSA: Optical Spectrum Analyzer, OSC: Tektronix 50GS/s Oscilloscope).

The transmission line consists in a recirculating loop which contains 2 spans of 100 km of SMF, a PDL element of 2dB, a polarization scrambler to randomize the polarization state between successive loops and an optical band-pass filter. After additional ASE noise loading at the receiver, the desired wavelength is filtered and the signal is detected with a dual-polarization coherent receiver. The received signal is then acquired with a real-time Tektronix oscilloscope and offline-processing [3

3. S. L. Jansen, I. Morita, T. C. W. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM Transmission With 2-b/s/Hz Spectral Efficiency Over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009). [CrossRef]

, 7

7. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15(16), 9936–9947 (2007). [CrossRef] [PubMed]

] ending with ML decoding is carried to measure the BER and the corresponding Q-factor in dB, commonly used in optical communications:
BER=0.5erfc(Q/2)andQdB=20log10(Q)
(31)

5.2. Experimental Results

5.2.1. PT coding in non-linear propagation regime

Fig. 7 BER evolution versus launched input power after 5 × 200km for the Silver-coded and 4-QAM schemes, at three different PDL values at the transmitter: 0, 3 and 6dB.

5.2.2. Mitigation of distributed in-line PDL

We remove the emulated PDL at the transmitter side and insert the PDL element of 2dB into the loop. 2000 Q-factor measurements are then carried out for the different coding schemes at the optimum operating point. The measured OSNR at the receiver is 12dB after additional ASE noise loading in order to evaluate the complete Q-factor distributions. The OSNR is proportional to SNRbit that we used along this paper:
OSNR=Rb2BrefSNRbit
(32)
where Rb is the total transmitted bitrate and Bref is the reference spectral bandwidth of 0.1nm.

The obtained Q-factor distributions are shown in Fig. 8. We also show, in the inset, the measured probability distribution of PDL and the fitted theoretical distribution. PDL is measured using the estimated 2 × 2 channel matrices and a Maxwellian probability distribution function of PDL is obtained with a 4.2dB mean (the expected theoretical mean being 4dB [19

19. A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photonics Technol. Lett. 14(3), 313–315 (2002). [CrossRef]

]).

Fig. 8 Q-factor distribution after 5 × 200km at Pin = −3dBm (OSNR0.1nm = 12dB). Inset: Experimental and theoretical probability distributions of PDL.

The Q-factor distribution for the 4-QAM-coded subcarriers is asymmetric with a large tail towards the worst Q-factors and has a mean of 11.2dB and a standard deviation (std) of 0.55dB whereas the distribution is symmetric and the mean Q-factor of the Silver- and Golden- coded subcarriers is 11.8dB (the same observed value for the PDL-free case when OSNR = 12dB). The Q-factor distributions are also narrower when PT codes are used. The Silver code gives a distribution slightly narrower than the one obtained with the Golden code: std of 0.35dB and 0.37dB respectively. For the Alamouti code, we observe a mean Q-factor of 8.3dB due to the use of 16-QAM symbols to guarantee the same spectral efficiency of the other schemes. However, its Q factor distribution is the narrowest of all with a std of 0.24dB. This is due to the powerful orthogonal structure of the Alamouti codeword matrix that makes its performance independent of the amount of accumulated PDL in the link, as found in the theoretical analysis in section 4. The observed small variance of its Q-factor distribution can be ascribed to the increased sensitivity of 16-QAM modulation to phase noise and non-linear effects.

6. Conclusion

Acknowledgments

The experimental work has been partially supported by the Celtic-Plus SASER-SIEGFRIED project.

References and links

1.

P. J. Winzer, “High-Spectral-Efficiency Optical Modulation Formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012). [CrossRef]

2.

S. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804–817 (2008). [CrossRef] [PubMed]

3.

S. L. Jansen, I. Morita, T. C. W. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM Transmission With 2-b/s/Hz Spectral Efficiency Over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009). [CrossRef]

4.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” in Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000). [CrossRef] [PubMed]

5.

T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarization dependent loss on coherent POLMUX-NRZ-DQPSK,” in proc. of OFC/NFOEC’08 , 1–3.

6.

A. Juarez, C. Bunge, S. Warm, and K. Petermann, “Perspectives of principal mode transmission in mode-division-multiplex operation,” Opt. Express 20(13), 13810–13824 (2012). [CrossRef] [PubMed]

7.

W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15(16), 9936–9947 (2007). [CrossRef] [PubMed]

8.

S. Mumtaz, G. Rekaya, and Y. Jaouën, “Space-Time codes for optical fiber communication with polarization multiplexing,” in proc. of ICC’10 , 1–5.

9.

S. Mumtaz, J. Li, S. Koenig, Y. Jaouën, R. Schmogrow, G. Rekaya-Ben Othman, and J. Leuthold, “Experimental demonstration of PDL mitigation using Polarization-Time coding in PDM-OFDM systems,” in proc. of SPPCom’11, paper SPWB6.

10.

S. Mumtaz, G. Rekaya-Ben Othman, Y. Jaouën, J. Li, S. Koenig, R. Schmogrow, and J. Leuthold, “Alamouti code against PDL in polarization multiplexed systems,” in proc. of SPPCom’11, paper SPTuA2.

11.

E. Meron, A. Andrusier, M. Feder, and M. Shtaif, “Use of space-time coding in coherent polarization-multiplexed systems suffering from polarization-dependent loss,” Opt. Letters 35(21), 3547–3549 (2010). [CrossRef]

12.

V. Tarokh, A. Naguib, N. Seshadri, and A.R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications 47(2), 199–207 (1999). [CrossRef]

13.

J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2×2 full-rate space-time code with nonvanishing determinants,” IEEE Transactions on Information Theory 51(4), 1432–1436 (2005). [CrossRef]

14.

H. Bulow, W. Baumert, H. Schmuck, F. Mohr, T. Schulz, F. Kuppers, and W. Weiershausen, “Measurement of the maximum speed of PMD fluctuation in installed field fiber,” OFC/IOOC’99 2, 83–85.

15.

S. Ben Rayana, H. Besbes, G. Rekaya-Ben Othman, and Y. Jaouën, “Joint equalization and polarization-time coding detection to mitigate PMD and PDL impairments,” in proc. of SPPCom’12, paper SpW2B.3.

16.

A. Andrusier, E. Meron, M. Feder, and M. Shtaif, “An optical implementation of a space-time-trellis code for enhancing the tolerance of systems to polarization-dependent loss,” Opt. Letters 38(2), 118–120 (2013). [CrossRef]

17.

S. R. Desbruslais and P. R. Morkel, “Simulation of polarisation mode dispersion and its effects in long-haul optically amplified lightwave systems,” IEE Colloquium on International Transmission Systems , 6/1–6/6 (1994).

18.

O. Vassilieva, I. Kim, Y. Akasaka, M. Bouda, and M. Sekiya, “Interplay between PDL and nonlinear effects in coherent polarization multiplexed systems,” Opt. Express 19(26), B357–B362 (2011). [CrossRef]

19.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photonics Technol. Lett. 14(3), 313–315 (2002). [CrossRef]

20.

N. Gisin, “Statistics of polarization dependent loss,” Optics Communications114, Elsevier (1995). [CrossRef]

21.

L. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19(7), 6790–6796 (2011). [CrossRef] [PubMed]

22.

A. Lima, I. Lima Jr., C. Menyuk, and T. Adali, “Comparison of penalties resulting from first-order and all-order polarization mode dispersion distortions in optical fiber transmission systems,” Opt. Letters 28(5), 310–312 (2003). [CrossRef]

23.

W. Shieh, “PMD-Supported Coherent Optical OFDM Systems,” IEEE Photonics Technol. Lett. 19(3), 134–136 (2007). [CrossRef]

24.

X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]

25.

M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16(18), 13918–13932 (2008). [CrossRef] [PubMed]

26.

J. Proakis and M. Salehi, Digital Communications, Fifth Edition. Mc Graw - Hill International Edition (2008).

27.

P. Delesques, E. Awwad, S. Mumtaz, G. Froc, P. Ciblat, Y. Jaouën, G. Rekaya, and C. Ware, “Mitigation of PDL in coherent optical communications: How close to the fundamental limit?,” in proc. of ECOC’12, paper P4.13.

28.

C. Xie, “Polarization-dependent loss induced penalties in PDM-QPSK coherent optical communication systems,” in proc. of OFC/NFOEC’10 , 1–3.

29.

E. Awwad, Y. Jaouën, G. Rekaya-Ben Othman, and E. Pincemin, “Polarization-Time Coded OFDM for PDL Mitigation in Long-Haul Optical Transmission Systems,” in proc. of ECOC’13, paper P3.4 (to be published).

30.

E. Awwad, Y. Jaouën, and G. Rekaya-Ben Othman, “Improving PDL Tolerance of Long-Haul PDM-OFDM Systems Using Polarization-Time Coding,” in proc. of SPPCom’12, paper SpTu2A.5.

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.1660) Fiber optics and optical communications : Coherent communications
(060.4080) Fiber optics and optical communications : Modulation
(060.4230) Fiber optics and optical communications : Multiplexing

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 31, 2013
Revised Manuscript: August 29, 2013
Manuscript Accepted: August 30, 2013
Published: September 20, 2013

Citation
Elie Awwad, Yves Jaouën, and Ghaya Rekaya-Ben Othman, "Polarization-time coding for PDL mitigation in long-haul PolMux OFDM systems," Opt. Express 21, 22773-22790 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22773


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References

  1. P. J. Winzer, “High-Spectral-Efficiency Optical Modulation Formats,” J. Lightwave Technol.30(24), 3824–3835 (2012). [CrossRef]
  2. S. Savory, “Digital filters for coherent optical receivers,” Opt. Express16, 804–817 (2008). [CrossRef] [PubMed]
  3. S. L. Jansen, I. Morita, T. C. W. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM Transmission With 2-b/s/Hz Spectral Efficiency Over 1000 km of SSMF,” J. Lightwave Technol.27(3), 177–188 (2009). [CrossRef]
  4. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” in Proc. Natl. Acad. Sci. U.S.A.97(9), 4541–4550 (2000). [CrossRef] [PubMed]
  5. T. Duthel, C. R. S. Fludger, J. Geyer, and C. Schulien, “Impact of polarization dependent loss on coherent POLMUX-NRZ-DQPSK,” in proc. of OFC/NFOEC’08, 1–3.
  6. A. Juarez, C. Bunge, S. Warm, and K. Petermann, “Perspectives of principal mode transmission in mode-division-multiplex operation,” Opt. Express20(13), 13810–13824 (2012). [CrossRef] [PubMed]
  7. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express15(16), 9936–9947 (2007). [CrossRef] [PubMed]
  8. S. Mumtaz, G. Rekaya, and Y. Jaouën, “Space-Time codes for optical fiber communication with polarization multiplexing,” in proc. of ICC’10, 1–5.
  9. S. Mumtaz, J. Li, S. Koenig, Y. Jaouën, R. Schmogrow, G. Rekaya-Ben Othman, and J. Leuthold, “Experimental demonstration of PDL mitigation using Polarization-Time coding in PDM-OFDM systems,” in proc. of SPPCom’11, paper SPWB6.
  10. S. Mumtaz, G. Rekaya-Ben Othman, Y. Jaouën, J. Li, S. Koenig, R. Schmogrow, and J. Leuthold, “Alamouti code against PDL in polarization multiplexed systems,” in proc. of SPPCom’11, paper SPTuA2.
  11. E. Meron, A. Andrusier, M. Feder, and M. Shtaif, “Use of space-time coding in coherent polarization-multiplexed systems suffering from polarization-dependent loss,” Opt. Letters35(21), 3547–3549 (2010). [CrossRef]
  12. V. Tarokh, A. Naguib, N. Seshadri, and A.R. Calderbank, “Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Transactions on Communications47(2), 199–207 (1999). [CrossRef]
  13. J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2×2 full-rate space-time code with nonvanishing determinants,” IEEE Transactions on Information Theory51(4), 1432–1436 (2005). [CrossRef]
  14. H. Bulow, W. Baumert, H. Schmuck, F. Mohr, T. Schulz, F. Kuppers, and W. Weiershausen, “Measurement of the maximum speed of PMD fluctuation in installed field fiber,” OFC/IOOC’992, 83–85.
  15. S. Ben Rayana, H. Besbes, G. Rekaya-Ben Othman, and Y. Jaouën, “Joint equalization and polarization-time coding detection to mitigate PMD and PDL impairments,” in proc. of SPPCom’12, paper SpW2B.3.
  16. A. Andrusier, E. Meron, M. Feder, and M. Shtaif, “An optical implementation of a space-time-trellis code for enhancing the tolerance of systems to polarization-dependent loss,” Opt. Letters38(2), 118–120 (2013). [CrossRef]
  17. S. R. Desbruslais and P. R. Morkel, “Simulation of polarisation mode dispersion and its effects in long-haul optically amplified lightwave systems,” IEE Colloquium on International Transmission Systems, 6/1–6/6 (1994).
  18. O. Vassilieva, I. Kim, Y. Akasaka, M. Bouda, and M. Sekiya, “Interplay between PDL and nonlinear effects in coherent polarization multiplexed systems,” Opt. Express19(26), B357–B362 (2011). [CrossRef]
  19. A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photonics Technol. Lett.14(3), 313–315 (2002). [CrossRef]
  20. N. Gisin, “Statistics of polarization dependent loss,” Optics Communications114, Elsevier (1995). [CrossRef]
  21. L. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express19(7), 6790–6796 (2011). [CrossRef] [PubMed]
  22. A. Lima, I. Lima, C. Menyuk, and T. Adali, “Comparison of penalties resulting from first-order and all-order polarization mode dispersion distortions in optical fiber transmission systems,” Opt. Letters28(5), 310–312 (2003). [CrossRef]
  23. W. Shieh, “PMD-Supported Coherent Optical OFDM Systems,” IEEE Photonics Technol. Lett.19(3), 134–136 (2007). [CrossRef]
  24. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express16(26), 21944–21957 (2008). [CrossRef] [PubMed]
  25. M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express16(18), 13918–13932 (2008). [CrossRef] [PubMed]
  26. J. Proakis and M. Salehi, Digital Communications, Fifth Edition. Mc Graw - Hill International Edition (2008).
  27. P. Delesques, E. Awwad, S. Mumtaz, G. Froc, P. Ciblat, Y. Jaouën, G. Rekaya, and C. Ware, “Mitigation of PDL in coherent optical communications: How close to the fundamental limit?,” in proc. of ECOC’12, paper P4.13.
  28. C. Xie, “Polarization-dependent loss induced penalties in PDM-QPSK coherent optical communication systems,” in proc. of OFC/NFOEC’10, 1–3.
  29. E. Awwad, Y. Jaouën, G. Rekaya-Ben Othman, and E. Pincemin, “Polarization-Time Coded OFDM for PDL Mitigation in Long-Haul Optical Transmission Systems,” in proc. of ECOC’13, paper P3.4 (to be published).
  30. E. Awwad, Y. Jaouën, and G. Rekaya-Ben Othman, “Improving PDL Tolerance of Long-Haul PDM-OFDM Systems Using Polarization-Time Coding,” in proc. of SPPCom’12, paper SpTu2A.5.

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