## Polarization-time coding for PDL mitigation in long-haul PolMux OFDM systems |

Optics Express, Vol. 21, Issue 19, pp. 22773-22790 (2013)

http://dx.doi.org/10.1364/OE.21.022773

Acrobat PDF (1667 KB)

### 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

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]

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

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]

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

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]

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

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]

## 2. PT-coded coherent OFDM system model

### 2.1. PolMux coherent OFDM transmission scheme

#### 2.1.1. Transmitter side

**X**modulates the first polarization state and the second modulates the second orthogonal polarization state. The columns of

**X**will be carried by the same subcarrier of two consecutive OFDM symbols. After the assignment of symbols to the different subcarriers, conventional OFDM processing is realized including an inverse Fast-Fourier Transform (iFFT) and the insertion of a well-dimensioned cyclic prefix (CP) to absorb all CD and PMD [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]

#### 2.1.2. Receiver side

**15**(16), 9936–9947 (2007). [CrossRef] [PubMed]

**X**

*is the 2 × 2 matrix of transmitted symbols on the*

_{k,i}*k*subcarrier (

^{th}*k*= 1 ...

*n*) of the

*i*OFDM symbol,

^{th}**Y**

*is the 2 × 2 matrix of the received symbols.*

_{k,i}**H**

*(*

_{k}*ω*) is the 2 × 2 channel matrix including laser phase noise, CD, PMD and PDL.

_{k}**N**

*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*

_{k,i}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]

**X**equiprobable, an ML decoder estimates the codeword

**X**with

**X′**according to the following criterion: 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]. 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, 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]

### 2.2. Long-haul optical link

*N*spans where each span contains a fiber modeled as a concatenation of random PMD elements and inline components with PDL as seen in Fig. 2. Each span is followed by an Erbium doped fiber amplifier (EDFA) operating in constant power mode to raise the signal power to the initially injected power at the transmitter. These amplifiers also add noise to the signal caused by amplified spontaneous emission (ASE). The accumulated ASE noise at the end of the link is the dominant noise source in the whole transmission scheme. In the following, the channel model of the link will be described and simplified to carry the numerical and theoretical studies of PDL mitigation with PT codes. We will, in particular, look at the statistics of PDL, its frequency dependence and the noise properties at the receiver side. We do not consider CD and phase noise because these two effects are polarization independent.

_{S}*M*random PMD elements where each element has a transfer matrix

**H**

*(*

_{PMD}*ω*) defined as the product of a delay matrix and a rotation matrix as in [17]. 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: The diagonal matrix gives the imbalanced attenuation values of the least and most attenuated polarization states and

_{k}**R**

*is a random rotation matrix.*

_{α}*α*is uniformly drawn in [0 : 2

*π*].

*ε*and

*γ*are defined through

*dB*≥ 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, 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]

*γ*≪ 1. However, many of these components are interspersed in long-haul optical links leading to a significant accumulated PDL. An overview of previous works on PDL, especially by Mecozzi et al. [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]

20. N. Gisin, “Statistics of polarization dependent loss,” *Optics Communications*114, Elsevier (1995). [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]

*of an optical link is a Maxwellian distribution when we consider many low-PDL components. Yet, field measurements of PDL in [21*

_{dB}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]

*was truncated because of the presence of a limited number of elements in the link having an appreciable PDL. In all cases, the global Γ*

_{dB}*is a random variable depending on the number of PDL elements in the link and their individual PDL.*

_{dB}*γ*is drawn from identical independently distributed (iid) normal distributions with mean and variance determined by the desired mean global Γ

*and the number of PDL elements in the link as in [19*

_{dB}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]

**14**(3), 313–315 (2002). [CrossRef]

*N*> 10 spans and

_{S}*M*= 20 and the obtained distributions perfectly fitted the expected theoretical Maxwellian distributions.

*𝒴*at the output of the link can be written in function of the input signal

_{k}**X**

*and the in-line injected noise: with*

_{k}*ℋ*

_{j→NS}=

*ℋ*

_{Ns}ℋ_{Ns−1}...

*ℋ*

_{j+1}

*ℋ*and

_{j}*ℋ*being the transfer matrix of the

_{j}*j*span. Each component of the noise

^{th}**N**

*added after the*

_{j}*j*span is white and Gaussian distributed with a zero mean and a variance

^{th}*N*

_{0}per real dimension. Because of PDL,

*𝒩*will be polarized and its coherency matrix

_{k}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]

**N**

*is a zero-mean white Gaussian noise of variance*

_{k}*σ*

^{2}=

*N*

_{S}N_{0}per real dimension. The effective PDL experienced by the signal and given by the ratio of the eigenvalues of

**H**

*can be written as a product of two unitary matrices*

_{k}**U**and

**V**and a diagonal matrix with

*γ*= (

_{eq}*λ*−

_{max}*λ*)/(

_{min}*λ*+

_{max}*λ*) and

_{min}*a*= (

*λ*+

_{max}*λ*)/2 where

_{min}*λ*and

_{max}*λ*are the eigenvalues of

_{min}*a*is an inevitable loss coefficient induced by PDL. Being particularly interested in the differential attenuation and crosstalk effects due to PDL and polarization rotations, we will consider unitary rotation matrices. Moreover, knowing that the dynamics of the optical channel present rather slow variations [14], we will only regard constant values of

**H**to model an optical link with PDL:

## 3. Numerical investigation of PDL mitigation

**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

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]

*α*= 1 +

*i*−

*iθ*,

*ᾱ*= 1 +

*i*−

*iθ̄*and

*S*

_{1},

*S*

_{2},

*S*

_{3},

*S*

_{4}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]

#### 3.1.2. The Silver code

*S*

_{1},

*S*

_{2},

*S*

_{3},

*S*

_{4}are M-QAM symbols. The determinant of this code is proportional to

#### 3.1.3. The Alamouti code

*S*

_{1}and

*S*

_{2}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

**H**defined in Eq. (8):

**N**

*is modeled as additive white with iid circularly-symmetric complex Gaussian components*

_{k}*𝒩*(0, 2

*σ*

^{2}). At the transmitter, 4-QAM symbols will be used for the uncoded case as well as to fill the codeword matrices of the Silver and the Golden code (full-rate codes). On the other hand, 16-QAM symbols will be used for the Alamouti code (half-rate code) in order to compare the schemes at the same spectral efficiency of 4 information bits/ts. At the receiver, ML decoding is performed by an exhaustive search with a reasonable complexity. In the case of an uncoded scheme, there are 16 possible codewords in

*𝒞*. In case of the Silver or the Golden code,

*𝒞*has 4

^{4}= 256 codewords. The Alamouti code benefits from its orthogonal structure and achieves the ML criterion with a single decoding operation. Monte Carlo simulations are carried in order to evaluate the performance of the different coding schemes in terms of BERs.

*for a PDL of 6dB. Without PT coding, the SNR degradation induced by PDL is about 2.3dB at BER = 10*

_{bit}^{−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] 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]

### 3.3. Concatenation of FEC and PT coding

*k*is mapped to a binary sequence of length

*n*called a codeword. The code rate of a block code is denoted by

*R*: 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.

_{c}*= 6*

_{dB}*dB*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 2.4

*dB*. When HDD is considered, the gain provided by the BCH code alone is 1.8

*dB*. The concatenation of both codes offer a total coding gain of 3.9

*dB*, approximately equal to the sum of the separate coding gains 2.4 + 1.8 = 4.2

*dB*. 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 2.8

*dB*. The concatenation with the Silver code provides a total gain of 4.9

*dB*and the sum of the separate coding gains is 2.4 + 2.8 = 5.2

*dB*.

## 4. Theoretical analysis of PDL mitigation

- the Silver code performs better than the Golden code, unlike the case of the wireless Rayleigh fading channel.
- the performance of the Alamouti code is independent of the amount of PDL in the link.
- the total coding gain obtained when using both a FEC code and a PT code is equal to the sum of the separate coding gains.

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]

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]

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]

### 4.1. Upper bound of the error probability

*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: Using the Gaussian properties of the noise

*N*and applying Chernoff’s bound, the PEP can be upper-bounded by [26, Chap.4]: where 𝔼

*[] is the averaging operation over all possible channel realizations.*

_{H}**H**given in Eq. (8) where we consider constant values of Γ

*and a random rotation angle that varies uniformly in [0 : 2*

_{dB}*π*], we get: where

*I*

_{0}(

*z*) is the 0

*order modified Bessel function of the first kind.*

^{th}*x⃗*

_{1,2}being line vectors and:

*I*

_{0}(

*z*) when

*z*→

**∞**and get:

### 4.2. Design criterion

*I*

_{0}(

*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:*

**47**(2), 199–207 (1999). [CrossRef]

^{−2}

*[12*

^{r}**47**(2), 199–207 (1999). [CrossRef]

*r*is the minimum rank of the matrix

**X**

_{Δ}. The diversity of the system is defined as the power of SNR

^{−1}(2

*r*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

*d*

^{2}, denoted

*E*= 1 for all constellations.

_{s}*Proposition 1*. However, its performance is affected by the use of 16-QAM symbols giving a squared minimal distance of 0.8.

*= 6*

_{dB}*dB*, and both reduce the penalty that PDL causes to the uncoded scheme. Again, this result can be explained by looking at Table 1. Indeed,

### 4.4. Concatenation of FEC and PT coding

*d*and the crossover probability

_{min,FEC}*p*of the equivalent binary symmetric channel (BSC) [26, Chap.7]: where

*d*is the minimum Hamming distance between distinct codewords of this code. At high SNR,

_{min,FEC}*p*tends towards zero and Eq. (23) is dominated by the first term where

*m*=

*t*+ 1: where

*d*=

_{FEC,SDD}*d*.

_{min,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 log

_{2}

*M*bits, an error in decoding one PT codeword implies that one bit is erroneous in the best case or all 4 log

_{2}

*M*bits are erroneous in the worst case, hence: 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: where

*A*

_{dPT}being the average number of codewords located at the distance

*d*of a given codeword.

_{PT}*A*

_{dPT}is usually called the kissing number.

*p*by its upper bound and substituting the average symbol energy

*E*for the average energy per information bit

_{S}*E*using

_{b}*E*= 1 =

_{S}*r*log

_{PT}R_{c}E_{b}_{2}

*M*, we get: where

*K*and

*d*are the constant and the power of the bit error probability of

_{FEC}*p*in Eq. (23) or (25) depending on the chosen FEC decoding strategy.

*r*is equal to 1 for a full-rate PT code and 0.5 for a half-rate code.

_{PT}*NC*without FEC (

*d*= 1 and

_{FEC}*R*= 1) and without PT coding (independent M-QAM symbols) and a second scheme

_{c}*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: In decibels, we obtain

*G*: 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.

_{dB}## 5. Experimental validation of PDL mitigation

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

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]

**27**(3), 177–188 (2009). [CrossRef]

**15**(16), 9936–9947 (2007). [CrossRef] [PubMed]

### 5.2. Experimental Results

#### 5.2.1. PT coding in non-linear propagation regime

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]

*Q*= 10.5

*dB*at −3 dBm for

*PDL*= 6

*dB*) while the Silver code completely compensates the penalty at

*PDL*= 3

*dB*and nearly all penalty at

*PDL*= 6

*dB*(

*Q*= 12.3

*dB*at −3dBm). Second, the curves can be separated into three regimes: a linear regime, up to −6dBm, where non-linear effects are negligible and the coding gains match the numerical results obtained with a linear channel model; a moderate non-linear regime, between −6dBm and 0dBm, where a minimum BER is reached; and a severe non-linear regime where the performance of all schemes is deteriorated. An important result is that the PT-coded OFDM does not induce any extra penalties in presence of non-linear effects. This experimental validation confirms a previous numerical investigation of the behavior of PT codes in non-linear propagation regime [30].

#### 5.2.2. Mitigation of distributed in-line PDL

*that we used along this paper: where*

_{bit}*R*is the total transmitted bitrate and

_{b}*B*is the reference spectral bandwidth of 0.1

_{ref}*nm*.

**14**(3), 313–315 (2002). [CrossRef]

## 6. Conclusion

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]

## Acknowledgments

## References and links

1. | P. J. Winzer, “High-Spectral-Efficiency Optical Modulation Formats,” J. Lightwave Technol. |

2. | S. Savory, “Digital filters for coherent optical receivers,” Opt. Express |

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. |

4. | J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” in Proc. Natl. Acad. Sci. U.S.A. |

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 |

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 |

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 |

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 |

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 |

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 |

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 |

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. | A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photonics Technol. Lett. |

20. | N. Gisin, “Statistics of polarization dependent loss,” |

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 |

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 |

23. | W. Shieh, “PMD-Supported Coherent Optical OFDM Systems,” IEEE Photonics Technol. Lett. |

24. | X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express |

25. | M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express |

26. | J. Proakis and M. Salehi, |

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

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