## Alamouti-type polarization-time coding in coded-modulation schemes with coherent detection

Optics Express, Vol. 16, Issue 18, pp. 14163-14172 (2008)

http://dx.doi.org/10.1364/OE.16.014163

Acrobat PDF (212 KB)

### Abstract

We present the Almouti-type polarization-time (PT) coding scheme suitable for use in multilevel (M≥2) block-coded modulation schemes with coherent detection. The PT-decoder is found it to be similar to the Alamouti combiner. We also describe how to determine the symbols log-likelihood ratios in the presence of laser phase noise. We show that the proposed scheme is able to compensate even 800 ps of differential group delay, for the system operating at 10 Gb/s, with negligible penalty. The proposed scheme outperforms equal-gain combining polarization diversity OFDM scheme. However, the polarization diversity coded-OFDM and PT-coding based coded-OFDM schemes perform comparable. The proposed scheme has the potential of doubling the spectral efficiency compared to polarization diversity schemes.

© 2008 Optical Society of America

## 1. Introduction

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]

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]

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

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]

*s*x using x-polarization channel and symbol

*s*

_{y}using y-polarization channel. In the second channel use the transmitter sends symbol -

*s**

_{y}by using x-polarization channel, and symbol

*s**

_{x}by using y-polarization. With proper combining on a receiver side, the tolerance to PMD can be improved compared to the corresponding polarization diversity scheme. We discuss two possible schemes, one in which only one polarization at the receiver side is used, and the second one in which both polarizations are used. When the channel coefficients, representing the so called channel state information (CSI), are known at the receiver side, both schemes are able to compensate for DGD of 800 ps, with negligible penalty; however, in the first scheme one polarization is not used at the receiver side, although both polarizations are used on transmitter side, resulting in 3 dB penalty compared to the second scheme. Notice that Alamouti-type coding has already been considered for use in optical communications, but in different context: in [7

7. 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]

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

7. 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]

## 2. Description of Alamouti-type polarization-time coding scheme with LDPC codes as channel codes

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

*τ*denotes DGD,

*R*=

*R*(

*θ*,

*ε*) is the rotational matrix defined by

*θ*denotes the polar angle,

*ε*denotes the azimuth angle, and

*ω*is the angular frequency. For coherent detection OFDM, the received symbol vector

*r**=[*

_{i,k}*r*

_{x,i,k}*r*

*]*

_{y,i,k}^{T}at ith OFDM symbol and kth subcarrier can be represented by

*=[*

**s**_{i,k}*s*

_{x,i,k}*s*]

_{y,i,k}^{T}denotes the transmitted symbol vector,

*n*=[

_{i,k}*n*

_{x,i,k}*n*]

_{y,i,k}^{T}denotes the noise vector dominantly determined by the amplified spontaneous emission (ASE), and the Jones matrix

*H*was introduced in Eq. (1) (we use index

*k*to denote the

*k*th subcarrier frequency

*ω*).

_{k}*ϕ*

_{T}and

*ϕ*

_{LO}denote the laser phase noise processes of transmitting and local lasers that are commonly modeled as the Wiener-Lévy processes [14], which are a zero-mean Gaussian processes with corresponding variances being 2πΔν

_{T}|

*t*| and 2πΔν

_{LO}|

*t*|, where Δν

_{T}and Δν

_{LO}are the laser linewidths of transmitting and receiving laser, respectively. The transmitted/received symbols per subcarrier are complex-valued, with real part corresponding to the in-phase coordinate and imaginary part corresponding to the quadrature coordinate of corresponding constellation point. Fig. 1 shows the magnitude responses of

*h*

_{xx}and

*h*

_{xy}coefficients of Jones against normalized frequency

*fτ*(the frequency is normalized with DGD

*τ*so that the conclusions are independent on the bit 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}tends to zero for certain frequencies, while in the second case it never becomes zero; 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

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]

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

*M*-ary quadrature-amplitude modulation (QAM) and OFDM as well. This method is based on space-time coding proposed to deal with fading in wireless communication systems, with Alamouti-type scheme [5

**16**, 1451–1458 (1998). [CrossRef]

7. 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]

**26**, 478–487 (2008). [CrossRef]

*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 identical [1

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]

*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*) in two-dimensional (2D) constellation diagram such as

_{Q,i}*M*-ary PSK or

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

*M*-ary 2D constellation.)

*i*th time instance (“the first channel use”) it sends symbol

*s*

_{x}to be transmitted using x-polarization channel and symbol

*s*

_{y}to be transmitted using y-polarization channel. In the second half of ith time instance (“the second channel use”) it sends symbol -

*s**

_{y}to be transmitted using x-polarization channel, and symbol

*s**

_{x}to be transmitted using y-polarization. Therefore, the PT-coding procedure is similar to the Alamouti-scheme [5

**16**, 1451–1458 (1998). [CrossRef]

**16**, 1451–1458 (1998). [CrossRef]

**25**, 3619–3625 (2007). [CrossRef]

*N*

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

*N*

_{FFT}input samples for inverse FFT (IFFT) (the zeros are added in the middle), and

*N*

_{G}non-zero samples are inserted to create the guard interval. For efficient chromatic dispersion and PMD compensation, the length of cyclically extended guard interval should be smaller than the total spread due to chromatic dispersion and maximum value of DGD. The cyclic extension is obtained 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 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). The same DFB laser is used as CW source, with x- and y-polarization being separated by a polarization beam splitter (PBS).

**25**, 3619–3625 (2007). [CrossRef]

12. I. B. Djordjevic, L. Xu, and T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes,” Opt. Express **16**, 10269–10278 (2008). [CrossRef] [PubMed]

*H*(

*k*) is already introduced in (1) (we use again index

*k*to denote the

*k*th subcarrier frequency

*ω*)

_{k}*m*th (

*m*=1,2) channel use of ith OFDM symbol and kth subcarrier, while

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]

11. W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Netw. **7**, 234–255 (2008). [CrossRef]

*i*th time instance are orthogonal,

*ϕ*

_{PN}to denote

*ϕ*

_{T}-

*ϕ*

_{LO}. If only one polarization is to be used we can solve either Eq. (4) or (5). However, the use of only one polarization results in 3 dB penalty with respect to the case when both polarizations are used. Following the derivation similar to that performed by Alamouti, it can be shown that the estimates of transmitted symbols at the output of PT-decoder (for ASE noise dominated scenario) can be obtain as follows

*s̃*and

_{x,i}*s̃*denote the PT-decoder estimates of symbols

_{y,i}*s*and

_{x,i}*s*transmitted in ith time instance. In case that only one polarization is to be used, say x-polarization, then the last two terms in equations (6) and (7) are to be omitted. The PT-decoder estimates are forwarded to the a posteriori probability (APP) demapper, which determines the symbol log-likelihood ratios (LLRs) in a fashion similar to that we reported in [1

_{y,i}**25**, 3619–3625 (2007). [CrossRef]

**25**, 3619–3625 (2007). [CrossRef]

**25**, 3619–3625 (2007). [CrossRef]

*g*≥10) [9], 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}).

*s*

*at*

_{i,k}*i*th OFDM symbol and

*k*th subcarrier can be estimated by:

*h*

_{xx}and

*h*

_{xy}are the channel coefficients introduced by Eq. (1),

*r*and

_{x,i,k}*r*represent the corresponding samples in x- and y-polarization branches, respectively.

_{y,i,k}*k*th subcarrier in

*i*th OFDM symbol,

*S*̃

*, are forwarded to the a posteriori probability (APP) demapper, which determines the symbol log-likelihood ratios (LLRs) λ*

_{x(y)i,k}_{x(y)}(

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

*σ*

^{2}denotes the variance of an equivalent Gaussian noise process originating from ASE noise, and map(

*s*) denotes a corresponding mapping rule (Gray mapping rule is applied here). (

*n*denotes the number of bits carried by symbol.) Notice that symbol LLRs in Eq. (9) are conditioned on the laser phase noise sample

_{b}*ϕ*

_{PN}=

*ϕ*

_{T}-

*ϕ*

_{LO}, which is a zero-mean Gaussian process (the Wiener-Lévy process [14]) with variance

*σ*

^{2}

_{PN}=2π(Δν

_{T}+Δν

_{LO})|

*| (Δν*

^{t}_{T}and Δν

_{LO}are the corresponding laser linewidths introduced earlier). This come from the fact that estimated symbols

*S*̃

*are functions of*

_{x(y)i,k}*ϕ*

_{PN}. To remove the dependence on

*ϕ*

_{PN}we have to average the likelihood function (not its logarithm), over all possible values of

*ϕ*

_{PN}:

*ϕ*

_{PN}obtained by pilot-aided channel estimation as explained in [11

11. W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Netw. **7**, 234–255 (2008). [CrossRef]

*b*(

_{j,x}*y*) the

*j*th bit in an observed symbol

*s*binary representation

*b*=(

*b*

_{1},

*b*

_{2},…,

*b*) for x- (y-) polarization. The bit LLRs required for LDPC decoding are calculated from symbol LLRs by

_{nb}*j*th bit LLR in Eq. (11) is calculated as the logarithm of the ratio of a probability that

*b*=0 and probability that

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

_{j}*s*having 0 at the position

*j*. Similarly, in the denominator summation is performed over all symbols

*s*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, as explained above (see also Fig. 2(c)).

## 3. Evaluation of the proposed coded-modulation scheme

*N*

_{QAM}=512, the oversampling is two times, OFDM signal bandwidth is set to 10 GHz, and the number of samples used cyclic extension

*N*

_{G}=256. The 4 pilots were sufficient to estimate this level of laser phase noise. For the fair comparison of different

*M*-ary schemes the OSNR on x-axis is given per information

*bit*, which is also consistent with digital communication literature [5

**16**, 1451–1458 (1998). [CrossRef]

*M*ary RZ-PSK (M=2 or 4) transmission (with duty cycle of 33%). Although the results of simulations are given for 10 Gb/s transmission they are reported in terms of normalized DGD (DGD is normalized with the symbol duration), so that the conclusions are applicable for 40 Gb/s and 100 Gb/s transmissions as well. The average launch power per

*symbol*is set to 0 dBm (and similarly as in wireless communications [5

**16**, 1451–1458 (1998). [CrossRef]

^{-6}). (Notice that comparison is done for the same launch powers per constellation symbols or per OFDM symbols.) The scheme employing only one polarization on receiver side, instead of both, faces 3 dB performance degradation, as expected. The LDPC-coded case provides a significant BER performance improvement over PT-coded scheme alone. When the CSI is known at receiver side it is sufficient to implement PT-decoder and APP demapper separately, as shown in Fig. 2(c). Notice that for corresponding turbo equalization [2

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]

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]

**19**, 1852–1854 (2007). [CrossRef]

^{17}states, which is too high for practical implementation. The proposed scheme also outperforms the scheme implemented by Nortel Networks researchers [4

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]

**16**, 873–879 (2008). [CrossRef] [PubMed]

^{-3}(see Fig. 4(a)), but better at BERs above 10

^{-2}where is the BER threshold region of large-girth LDPC codes. The Alamouti-type PT coding based LDPC-coded OFDM performs comparable to the equal-gain combining polarization diversity LDPC-coded OFDM (see Fig. 4(b)).

## 4. Conclusion

**19**, 1852–1854 (2007). [CrossRef]

## Acknowledgment

## 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. |

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. |

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 |

4. | H. Sun, K. -T. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express |

5. | S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun. |

6. | Y Han and G. Li “Polarization diversity transmitter and optical nonlinearity mitigation using polarization-time coding,” in Proc. COTA 2006, Paper no. CThC7, Whistler, Canada, 2006. |

7. | 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. |

8. | D. Penninckx and V. Morenás, “Jones matrix of polarization mode dispersion,” Opt. Lett. |

9. | 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. |

10. | E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor MIMO Wireless Communications, Cambridge University Press, Cambridge 2007. |

11. | W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Netw. |

12. | I. B. Djordjevic, L. Xu, and T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes,” Opt. Express |

13. | J. G. Proakis |

14. | M. Cvijetic |

15. | I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Comm., accepted for publication. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 17, 2008

Revised Manuscript: July 19, 2008

Manuscript Accepted: August 19, 2008

Published: August 26, 2008

**Citation**

Ivan B. Djordjevic, Lei Xu, and Ting Wang, "Alamouti-type polarization-time coding in
coded-modulation schemes with coherent
detection," Opt. Express **16**, 14163-14172 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14163

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### References

- 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]
- 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]
- 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]
- 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]
- S. Alamouti, "A simple transmit diversity technique for wireless communications," IEEE J. Sel. Areas Commun. 16, 1451-1458 (1998). [CrossRef]
- Y. Han and G. Li, "Polarization diversity transmitter and optical nonlinearity mitigation using polarization-time coding," in Proc. COTA 2006, Paper no. CThC7, Whistler, Canada, 2006.
- 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]
- D. Penninckx and V. Morenás, "Jones matrix of polarization mode dispersion," Opt. Lett. 24, 875-877 (1999). [CrossRef]
- 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.
- E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications, Cambridge University Press, Cambridge 2007.
- W. Shieh, X. Yi, Y. Ma, and Q. Yang, "Coherent optical OFDM: has its time come? [Invited]," J. Opt. Netw. 7, 234-255 (2008). [CrossRef]
- I. B. Djordjevic, L. Xu, and T. Wang, "Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes," Opt. Express 16, 10269-10278 (2008). [CrossRef] [PubMed]
- J. G. Proakis, Digital Communications (McGraw-Hill, Boston, 2001).
- M. Cvijetic, Coherent and Nonlinear Lightwave Communications (Artech House, Boston, 1996). I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, "Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization," IEEE J. Sel. Areas Comm., accepted for publication.

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