## Abstract

We present two PMD compensation schemes suitable for use in multilevel (*M*≥2) block-coded modulation schemes with coherent detection. The first scheme is based on a BLAST-type polarization-interference cancellation scheme, and the second scheme is based on iterative polarization cancellation. Both schemes use the LDPC codes as channel codes. The proposed PMD compensations schemes are evaluated by employing coded-OFDM and coherent detection. When used in combination with girth-10 LDPC codes those schemes outperform polarization-time coding based OFDM by 1 dB at BER of 10^{-9}, and provide two times higher spectral efficiency. The proposed schemes perform comparable and are able to compensate even 1200 ps of differential group delay with negligible penalty.

©2008 Optical Society of America

## 1. Introduction

The bit-error ratio (BER) performance of fiber-optic communication systems operating at high data rates is degraded by intrachannel and interchannel fiber nonlinearities, polarization mode dispersion (PMD), and chromatic dispersion [1]. To deal with PMD a number of methods have been proposed recently, four of them seem to be able successfully to tackle the PMD effects: (i) turbo equalization [2], (ii) polarization diversity orthogonal frequency division multiplexing (OFDM) [3], (iii) the channel equalization scheme described in [4], and Alamouti-type [5] polarization-time (PT) coding scheme introduced by authors in [6].

In this paper we propose two alternative schemes suitable for PMD compensation, which do not require the increase of complexity as the differential group delay (DGD) increases. The first scheme is based on the Bell Laboratories layered space-time architecture (BLAST) [7], originally proposed to deal with spatial interference in wireless communications. We consider two versions of this scheme [8]: (a) the zero-forcing vertical-BLAST scheme (ZF V-BLAST), and the minimum-mean-square-error vertical-BLAST (MMSE V-BLAST) scheme. Because the ZF V-BLAST scheme is derived by ignoring the influence of amplified spontaneous emission (ASE) noise, we proposed the second scheme that uses the output of ZF V-BLAST scheme as starting point and removes the remaining polarization interference in an iterative fashion. This approach also leads to reducing the influence of ASE noise. We evaluate the performance of those schemes when used in combination with coherent detection based OFDM. We describe how to use those schemes together with multilevel modulation and forward error correction (FEC). The arbitrary FEC scheme can be used with proposed PMD compensation schemes, however, the use of low-density parity-check (LDPC) codes leads to near channel capacity achieving performance [9]. The proposed schemes outperform the polarization-diversity OFDM scheme [3], and PT-based OFDM [6] in terms of both BER and spectral efficiency.

## 2. Description of proposed PMD compensation schemes

For the first-order PMD study the Jones matrix, neglecting the polarization dependent loss and depolarization effects, can be represented by [10]

where *τ* denotes DGD, *ω* is the angular frequency, and *R* = *R*(*θ*,*ε*) is the rotational matrix [10]
$R=\left[\begin{array}{cc}\mathrm{cos}\left(\frac{\theta}{2}\right){e}^{\frac{j\epsilon}{2}}& \mathrm{sin}\left(\frac{\theta}{2}\right){e}^{-\frac{j\epsilon}{2}}\\ -\mathrm{sin}\left(\frac{\theta}{2}\right){e}^{\frac{j\epsilon}{2}}& \mathrm{cos}\left(\frac{\theta}{2}\right){e}^{-\frac{j\epsilon}{2}}\end{array}\right],$

with *θ* being the polar angle, and *ε* being the azimuth angle. For the OFDM with coherent detection, the received symbol vector of *k*th subcarrier in ith OFDM symbol *r** _{i}*,

*=[*

_{k}*r*

*,*

_{x}*,*

_{i}

_{k}*r*

*,*

_{y}*,*

_{i}*]*

_{k}*can be represented by*

^{T}where **s**
* _{i}*,

*=[*

_{k}*s*

*,*

_{x}*,*

_{i}

_{k}*n*

*,*

_{y}*,*

_{i}*]*

_{k}*denotes the transmitted symbol vector of*

^{T}*k*th subcarrier in

*i*th OFDM symbol, for both polarizations,

*n**,*

_{i}*=[*

_{k}*n*

*,*

_{x}*,*

_{i}

_{k}*n*

*,*

_{y}*,*

_{i}*]*

_{k}*denotes the noise vector dominantly determined by the amplified spontaneous emission (ASE) noise;*

^{T}*ϕ*

*and*

_{T}*ϕ*

_{LO}denote the laser phase noise processes of transmitting and local lasers,

*ϕ*

_{CD}(

*k*) denotes the phase distortion of

*k*th subcarrier due to chromatic dispersion (CD), and the Jones matrix of

*k*th subcarrier

*H**is already introduced in (1). The transmitted/received symbols are complex-valued, with real part corresponding to the in-phase coordinate and imaginary part corresponding to the quadrature coordinate. Figure 1 shows the magnitude responses of*

_{k}*h*

_{xx}and

*h*

_{xy}coefficients of the Jones channel matrix against a normalized frequency

*f*

*τ*(the frequency is normalized with DGD

*τ*so that the conclusions are independent on the data rate) for two different cases: (a)

*θ*=π/2 and

*ε*=0, and (b)

*θ*=π/3 and

*ε*=0. In the first case channel coefficient

*h*

_{xx}completely fades away for certain frequencies, while in the second case it never completely fades away; suggesting that the first case represents the worst case scenario. To avoid this problem, in direct detection OFDM systems someone can redistribute the transmitted power among subcarriers not being under fading, or use the polarization-diversity coherent detection OFDM [3]. We propose two alternative approaches instead, which can be used for a number of modulation formats including

*M*-ary phase-shift keying (PSK),

*M*-ary quadrature-amplitude modulation (QAM) and OFDM.

The polarization interference cancellation scheme based on V-BLAST algorithm, which uses an LDPC code as channel code, is shown in Fig. 2. The bit streams originating from *m* different information sources are encoded using different (*n*,*k*
* _{i}*) LDPC codes of code rate

*r*

*=*

_{i}*k*

*/*

_{i}*n*.

*k*

*denotes the number of information bits of*

_{i}*i*th (

*i*= 1,2,…,

*m*) component LDPC code, and

*n*denotes the codeword length, which is the same for all LDPC codes. The use of different LDPC codes allows us to optimally allocate the code rates. The bit-interleaved coded modulation (BICM) scheme can be considered as a special multilevel coding (MLC) scheme in which all of the component codes are of the same rate [1]. The outputs of

*m*LDPC encoders are written row-wise into a block-interleaver block. The mapper accepts

*m*bits at time instance

*i*from the (

*mxn*) interleaver column-wise and determines the corresponding

*M*-ary (

*M*= 2

*) signal constellation point (*

^{m}*ϕ*

*,*

_{I}*,*

_{i}*ϕ*

*,*

_{Q}*) in a two-dimensional (2D) constellation diagram such as*

_{i}*M*-ary PSK or

*M*-ary QAM. (The coordinates correspond to in-phase and quadrature components of

*M*-ary 2D constellation.) The 2D signal constellation points are split into two streams for OFDM transmitters (see Fig. 2(b)) corresponding to the x- and y-polarizations. The QAM constellation points are considered to be the values of the fast Fourier transform (FFT) of a multi-carrier OFDM signal. The OFDM symbol is generated as follows:

*N*

_{QAM}input QAM symbols are zero-padded to obtain

*N*

_{FFT}input samples for inverse FFT (IFFT),

*N*

_{G}non-zero samples are inserted to create the guard interval, and the OFDM symbol is multiplied by the window function. For efficient chromatic dispersion and PMD compensation, the length of cyclically extended guard interval should be linger than the total spread due to chromatic dispersion and DGD. The cyclic extension is accomplished by repeating the last

*N*

_{G}/2 samples of the effective OFDM symbol part (

*N*

_{FFT}samples) as a prefix, and repeating the first

*N*

_{G}/2 samples as a suffix. After D/A conversion (DAC), the RF OFDM signal is converted into the optical domain using the dual-drive Mach-Zehnder modulator (MZM). Two MZMs are needed, one for each polarization. The outputs of MZMs are combined using the polarization beam combiner (PBC). One DFB laser is used as a CW source, with x- and y-polarization separated by the polarization beam splitter (PBS).

On the receiver side, in PT-coded OFDM, we have the option to use only one polarization or to use both polarizations. The polarization diversity OFDM [3], and polarization interference cancellation schemes proposed here require the use of both polarizations. The receiver architecture employing both polarizations is shown in Fig. 2(c).

The configuration of polarization interference cancellation scheme by a BLAST-algorithm is shown in Fig. 3(a). The received symbol vector in *k*th subcarrier of *i*th OFDM symbol in both polarization (see Eq. (2)) is linearly processed, the processing is described by matrix *C** _{k}* related to channel matrix

*H**as shown below, and the estimate of polarization interference obtained from preliminary decisions*

_{k}**s**̃

*,*

_{i}*, denoted as*

_{k}

*D*

_{k}*̃*

**s***,*

_{i}*, is removed from received symbol*

_{k}

*r**,*

_{i}*. The Euclidean detector can be used to create the preliminary decisions. When the presence of ASE noise is ignored, the zero-forcing V-BLAST polarization interference cancellation scheme results. The matrices*

_{k}**C**

*and*

_{k}

**D***can be determined from QR-factorization of channel matrix*

_{k}**H**

*=*

_{k}*Q*

_{k}

*R**, as follows*

_{k}where *I* is the identity matrix, and with diag() we denoted the diagonal elements of *R** _{k}*. Notice that elements at the main diagonal in

*D**are zero in order to have only polarization interference be removed. (We use † to denote the simultaneous transposition and complex-conjugation.) In the presence of ASE noise, the matrices*

_{k}

*C**and*

_{k}

**D***can be determined by minimizing the MSE, which leads to*

_{k}where **S**
* _{k}* is the upper triangular matrix obtained by Cholesky factorization of

**H**^{†}

_{k}

*H**+*

_{k}*/*

**I***SNR*=

*S*^{†}

_{k}

*S**, where SNR denotes corresponding electrical SNR, and*

_{k}*I*is the identity matrix. The derivation of (3),(4) is equivalent to that for wireless communications [8], and as such is omitted here. Because the ZF V-BLAST is derived by ignoring the influence of ASE noise, we propose to use the ZF V-BLAST as starting point, and perform the polarization interference cancellation in an iterative fashion as shown in Fig. 3(b). If

**r**̃

^{(l)}

*,*

_{i}*denotes the processed received symbol of*

_{k}*k*th subcarrier in

*i*th OFDM symbol (for both polarizations) in

*l*th iteration, then corresponding received symbol in (

*l*+1)th iteration can be found by

where **s**̃^{(l)}
* _{i}*,

*denotes the transmitted symbol (of*

_{k}*k*th subcarrier in

*i*th OFDM symbol (for both polarizations)) estimate in

*l*th iteration. The matrices

**C**_{k}and

*D**are already introduced in (3). Notice that different matrix operations applied in (3)–(5) are trivial because the dimensionality of matrices is small, 2 × 2.*

_{k}The BLAST-detector soft estimates of symbols carried by *k*th subcarrier in *i*th OFDM symbol, *s*
* _{i}*,

*,*

_{k}*(*

_{x}_{y}), are forwarded to the a posteriori probability (APP) demapper, which determines the symbol log-likelihood ratios (LLRs) λ

_{x}(

_{y})(

_{q}) (

_{q}= 0,1,…,2

*-1) of x- (y-) polarization by*

^{b}$$\phantom{\rule{10em}{0ex}}-\frac{{\left(\mathrm{Im}\left[{\tilde{s}}_{i,k,x\left(y\right)}\right]-\mathrm{Im}\left[\mathrm{QAM}\left(\mathrm{map}\left(q\right)\right)\right]\right)}^{2}}{2{\sigma}^{2}};\phantom{\rule{.4em}{0ex}}q=0,1,\dots ,{2}^{b}-1$$

where Re[] and Im[] denote the real and imaginary part of a complex number, QAM denotes the QAM-constellation diagram, *σ*
^{2} denotes the variance of an equivalent Gaussian noise process originating from ASE noise, and map(*q*) denotes a corresponding mapping rule (Gray mapping is applied here). (*b* denotes the number of bits per constellation point.) Let us denote by *v*
* _{j}*,

*(*

_{x}*) the*

_{y}*j*th bit in an observed symbol

*q*binary representation

**=(**

*v**v*

_{1},

*v*

_{2},…,

*v*

_{b}) for x- (y-) polarization. The bit LLRs needed for LDPC decoding are calculated from symbol LLRs by

Therefore, the *j*th bit reliability is calculated as the logarithm of the ratio of a probability that *v*
* _{j}* = 0 and probability that

*v*

_{j}= 1. In the nominator, the summation is done over all symbols

*q*having 0 at the position

*j*, while in the denominator over all symbols

*q*having 1 at the position

*j*. The extrinsic LLRs are iterated backward and forward until convergence or pre-determined number of iterations has been reached. The LDPC code used in this paper belongs to the class of quasi-cyclic (array) codes of large girth (

*g*≥10) [11], so that the corresponding decoder complexity is low compared to random LDPC codes, and do not exhibit the error floor phenomena in the region of interest in fiber-optics communications (≤10

^{-15}).

## 3. Evaluation of proposed PMD compensation schemes

We are turning our attention to the BER performance evaluation of the proposed schemes. The results of simulation for uncoded OFDM for different PMD compensation schemes are shown in Fig. 4. The OFDM system parameters are chosen as follows: *N*
_{QAM} = 512, oversampling is two times, OFDM signal bandwidth is set to 10 GHz, and *N*
_{G} = 256 samples. The MMSE V-BLAST and iterative polarization cancellation schemes (with ZF V-BLAST as starting point) perform identically (only MMSE curve is shown because the curves overlap each other), while ZF V-BLAST is slightly worse. Polarization diversity OFDM outperforms MMSE V-BLAST at low BERs, but performs comparable at BERs above 10^{-2}, which is the threshold region of girth-10 LDPC codes employed here. Moreover, the spectral efficiency of MMSE V-BLAST is twice higher because in polarization diversity OFDM the same symbol is transmitted twice over both polarizations. The MMSE V-BLAST OFDM outperforms the PT-coding based OFDM at both low and high BERs, and has two times higher spectral efficiency. The MMSE V-BLAST OFDM scheme is able to compensate even 1200 ps of DGD with negligible penalty. Notice that for corresponding turbo equalization [2] or maximum-likelihood sequence estimation schemes, the detector complexity grows exponentially as DGD increases, and normalized DGD of 800 ps it would require the trellis description (see [2]) with 2^{17} states, which is too high for practical implementation. Our schemes, although of lower complexity, are able to compensate up to 1200 ps of DGD with negligible penalty. The proposed schemes also outperform the scheme implemented by Nortel Networks researchers [4], capable of compensating the rapidly varying first order PMD with peak DGD of 150 ps.

The results of simulations for LDPC-coded OFDM when MMSE V-BLAST polarization cancellation scheme is used are shown in Fig. 5. The girth-10 LDPC(16935,13550) code of rate 0.8 and column weight 3 is used in simulations. This code does not exhibit error floor phenomena for the region of interest in optical communications (see [11]). At BER of 10^{-9} the LDPC-coded OFDM with MMSE V-BLAST polarization interference cancellation scheme outperforms PT-coding based OFDM by about 1 dB, and has the spectral efficiency twice higher.

In simulations shown in Figs. 4–5 the average launch symbol power was set to -3 dBm.

## 5. Conclusion

We proposed two alternative PMD compensation schemes to turbo equalization, channel equalization scheme from [4], polarization diversity OFDM [3], and PT-coding based OFDM. The proposed schemes are suitable for use in multilevel (*M*≥2) block-coded modulation schemes with coherent detection. In contrast to the PMD turbo equalization scheme whose complexity grows exponentially as DGD increases, the complexity of the proposed schemes stays the same. The spectral efficiency of proposed schemes is two times higher than that of polarization diversity OFDM and PT-coding based OFDM. The first scheme is based on MMSE V-BLAST algorithm, used in MIMO wireless communications to deal with spatial interference. The second scheme is based on iterative polarization interference cancellation. Those two schemes perform comparable, and are able to compensate up to 1200 ps of DGD with negligible penalty. When used in combination with girth-10 LDPC codes, those schemes outperform PT-coding based OFDM by 1 dB at BER of 10^{-9}.

## Acknowledgments

This work was supported in part by the National Science Foundation (NSF) under Grant IHCS-0725405.

## References and Links

**1. **I. B. Djordjevic, M. Cvijetic, L. Xu, and T. Wang, “Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission,” J. Lightwave Technol. **25**, 3619–3625 (2007). [CrossRef]

**2. **L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, and F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett. **19**, 1852–1854 (2007). [CrossRef]

**3. **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**, 9936–9947 (2007). [CrossRef] [PubMed]

**4. **H. Sun, K. -T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express **16**, 873–879 (2008). [CrossRef] [PubMed]

**5. **S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun. **16**, 1451–1458 (1998). [CrossRef]

**6. **I. B. Djordjevic, L. Xu, and T. Wang, “PMD compensation in multilevel coded-modulation schemes with coherent detection using Alamouti-type polarization-time coding,” in Proc. IEEE LEOS Summer Topicals 2008, pp. 103–104, July 2008. [CrossRef]

**7. **G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J. **1**, 41–59 (1996). [CrossRef]

**8. **E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, *MIMO Wireless Communications* (Cambridge University Press, Cambridge2007). [CrossRef]

**9. **I. B. Djordjevic, S. Denic, J. Anguita, B. Vasic, and M. A. Neifeld, “LDPC-coded MIMO optical communication over the atmospheric turbulence channel,” J. Lightwave Technol. **26**, 478–487 (2008). [CrossRef]

**10. **D. Penninckx and V. Morenás, “Jones matrix of polarization mode dispersion,” Opt. Lett. **24**, 875–877 (1999). [CrossRef]

**11. **I. B. Djordjevic, L. Xu, T. Wang, and M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, Paper no. JWA53.