## Four-dimensional optical multiband-OFDM for beyond 1.4 Tb/s serial optical transmission |

Optics Express, Vol. 19, Issue 2, pp. 876-882 (2011)

http://dx.doi.org/10.1364/OE.19.000876

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### Abstract

We propose a four-dimensional (4D) coded multiband-OFDM scheme suitable for beyond 1.4 Tb/s serial optical transport. The proposed scheme organizes the *N*-dimensional (*N*D) signal constellation points in the form of signal matrix; employs 2D-inverse FFT and 2D-FFT to perform modulation and demodulation, respectively; and exploits both orthogonal polarizations. This scheme can fully exploit advantages of OFDM to deal with chromatic dispersion, PMD and PDL effects; and multidimensional signal constellations to improve OSNR sensitivity of conventional optical OFDM. The improvement of 4D-OFDM over corresponding polarization-multiplexed QAM (with the same number of constellation points) ranges from 1.79 dB for 16 signal constellation point-four-dimensional-OFDM (16-4D-OFDM) up to 4.53 dB for 128-4D-OFDM.

© 2011 OSA

## 1. Introduction

4. H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express **18**(19), 20546–20551 (2010). [CrossRef] [PubMed]

*N*-dimensional signal constellation point be represented as

**= (**

*S**S*

^{(0)},

*S*

^{(1)},…,

*S*

^{(}

^{N}^{-1)}), where

*S*

^{(}

^{l}^{)}(

*l*= 0,…,

*N*-1) is the

*l*th coordinate. Let the duration of the signal frame be

*M*signal constellation points. We can represent the signal constellation points in matrix form, by placing the coordinates of signal constellation points along the columns of a signal matrix. We further apply two-dimensional inverse fast Fourier transform (2D-IFFT) to obtain 2D-IFFT array of complex numbers. The coordinates of complex numbers can be considered as in-phase (I) and quadrature (Q) channels, while even and odd rows of two-dimensional array can be mapped to x- and y-polarizations, respectively. The arbitrary

*N*-dimensional (

*N*= 2,3,4,5,…) signal constellation can be used in combination with this scheme. Because in optical channel we have four bases functions (in-phase, quadrature, x-polarization and y-polarization) available, the full advantage of this scheme can be obtained by employing the 4D signal constellations. All other steps of this 2D-OFDM scheme are similar to conventional coherent optical OFDM. On receiver side, we use the conventional polarization-diversity receiver, followed by 2D-FFT demapper. Therefore, this scheme can fully exploit the advantages of OFDM as an efficient way to deal with chromatic dispersion, polarization mode dispersion (PMD) and polarization dependent loss (PDL) effects. At the same time we can exploit the advantages of multidimensional signal constellation to improve the optical signal-to-noise ratio (OSNR) sensitivity of conventional optical OFDM dramatically. We also describe a 4D coherent optical multiband-OFDM scheme enabling 1.4 Tb/s serial optical transmission.

## 2. Description of 4D coded optical OFDM

*m*independent data streams are encoded using different LDPC (

*n*,

*k*) codes (

_{l}*l*= 1,…,

*m*), where

*n*denotes the codeword length and

*k*is the information word length of

_{l}*l*th component code. The codewords are written row-wise into

*m*×

*n*bit interleaver. The

*m*bits are taken from bit interleaver column-wise at every symbol slot

*i*and are used as input of 4D mapper, which selects one constellation point out of 2

*, depending on information content. The 4D mapper is implemented as a look-up table (LUT) with*

^{m}*m*input bits serving as a memory address that selects the four coordinates of 4D signal constellation point. The outputs of 4D mapper are written column-wise into 4 ×

*M*symbol-like interleaver. The content of symbol interleaver can be represented as a two-dimensional array (matrix) as follows:where the

*j*th column

*S**= [*

_{j}*S*

_{j}_{,0}

*S*

_{j}_{,1}

*S*

_{j}_{,2}

*S*

_{j}_{,3}]

^{T}represents the coordinates of

*j*th 4D signal constellation point (

*S*

_{j}^{(0)},

*S*

_{j}^{(1)},

*S*

_{j}^{(2)},

*S*

_{j}^{(3)}) (

*j*= 0,1,…,

*M*-1). Therefore, the rows correspond to the dimensions and columns to subcarriers. Notice that conventional PolMux OFDM requires 2 × 2

*M*signaling matrix for the same amount of data, and the bandwidth requirements are therefore identical. This two-dimensional array is used as input to the two-dimensional inverse discrete Fourier transform (2D-IDFT) block, which calculates the IDFT as followswhere

*M*/4 sub-matrix blocks of type (1), so that the duration of signal per axis is

*M*. In Eq. (2), we use

*s*= (

_{ij}*s*

_{ij}_{,I},

*s*

_{ij}_{,Q}), with subscripts I and Q corresponding to in-phase and quadrature channels, respectively. More details on multidimensional DFT can be found in [5

5. L. J. Stankovic, *Digital Signal Processing* (Naucna Knjiga, 1990). [PubMed]

6. C. Chakrabarti and J. JaJa, “VLSI architectures for multidimensional transforms,” IEEE Trans. Comput. **40**(9), 1053–1057 (1991). [CrossRef]

*s*

_{x,}

*= (*

_{i}*I*

_{x,}

_{i}Q_{x,}

*) [*

_{i}*s*

_{y,}

*= (*

_{i}*I*

_{y,}

_{i}Q_{y,}

*)], corresponding to x- (y-) polarization, are used (after digital-to-analog (D/A) conversion) as inputs to the I/Q modulator as shown in Fig. 2(b) of reference [4*

_{i}4. H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express **18**(19), 20546–20551 (2010). [CrossRef] [PubMed]

*N*-dimensional (

*N*= 2,3,4,5,…) signal constellation can be used. Because in optical channel we have four bases functions available, the full advantage of this scheme can be obtained by employing the 4D signal constellations. (The cyclic extension insertion principle and A/D conversion operation are very similar to that in conventional OFDM and as such are not discussed here.)

*M*×

*M*log

_{2}

*M*complex multiplications and additions. The complexity of this algorithm is

*M*

^{4}/[2

*M*

^{2}log

_{2}

*M*] times lower than that of direct 2D-DFT computation. Upon the deinterleaving, the symbol log-likelihood ratios (LLRs) are calculated in the

*a posteriori*probability (APP) demapper using the following equation,where

*c*denotes the

_{j}*j*th bit in the observed symbol

**binary representation**

*S***= (**

*c**c*

_{0},

*c*

_{1},…). In Eq. (8), we use

*L*(

*c*

^{(}

^{t}^{)}

*) to denote the LDPC decoder output in current iteration (iteration*

_{j}*t*). In the above equations

*R**denotes the received constellation point, and*

_{i}

*S*_{0}denotes the referent constellation point. The

*P*(

*R**|*

_{i}

*S**), from Eq. (6), denotes the conditional probability that can be estimated by collection of histograms. The bit LLRs*

_{i}*L*(

*c*) are determined from symbol LLRs byTherefore, the

_{j}*j*th bit reliability is calculated as the logarithm of the ratio of a probability that

*c*= 0 and probability that

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

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

*S**j*, while in the denominator over all symbols

**having 1 at the position**

*S**j*. With

*L*(

_{a}*c*) we denoted the prior (extrinsic) information determined from the APP demapper. The inner summation in Eq. (9) is performed over all bits of symbol

_{k}**, selected in the outer summation, for which**

*S**c*= 0,

_{k}*k*≠

*j*. The bit LLRs are forwarded to LDPC decoders, which provide extrinsic bit LLRs for demapper according to Eqs. (8) and (7), and are used as inputs to Eq. (6) as the prior information based on Eq. (7).

*k*th subcarrier in

*i*th OFDM symbol

*R*

_{i}_{,}

*= [*

_{k}*R*

^{(0)}

_{i}_{,}

_{k}R^{(1)}

_{i}_{,}

_{k}R^{(2)}

_{i}_{,}

_{k}R^{(3)}

_{i}_{,}

*]*

_{k}^{T}can be represented bywhere

*S*

_{i}_{,}

*= [*

_{k}*S*

^{(0)}

_{i}_{,}

_{k}S^{(1)}

_{i}_{,}

_{k}S^{(2)}

_{i}_{,}

_{k}S^{(3)}

_{i}_{,}

*]*

_{k}^{T}denotes the transmitted symbol vector of

*k*th subcarrier in

*i*th OFDM symbol. We use superscript (

*l*) to denote the

*l*th (

*l*= 0,1,2,3) coordinate of corresponding signal constellation point. In Eq. (10),

*N**= [*

_{i,k}*N*

^{(0)}

_{i,k}N^{(1)}

_{i,k}N^{(2}

^{)}_{i,k}N^{(3}

*]*

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

*ϕ*

_{T}and

*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, and

*H**denotes the channel matrix of*

_{k}*k*th subcarrier, which is similar to the Jones matrix described in [7,8

8. I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. **27**(16), 3518–3530 (2009). [CrossRef]

## 3. Description of 4D optical multiband-OFDM

10. Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express **17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

11. Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. **27**(16), 3511–3517 (2009). [CrossRef]

10. Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express **17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

11. Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. **27**(16), 3511–3517 (2009). [CrossRef]

11. Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. **27**(16), 3511–3517 (2009). [CrossRef]

10. Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express **17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

**27**(16), 3511–3517 (2009). [CrossRef]

*f*

_{G}=

*m*Δ

*f*

_{sc}(

*m*is a positive integer), where Δ

*f*

_{sc}is the subcarrier spacing. Because the central frequencies of neighboring OFDM bands are orthogonal to each other, we may simplify the separation of OFDM bands by anti-aliasing filters, as explained in [10

**17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

**27**(16), 3511–3517 (2009). [CrossRef]

**17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

**17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

**27**(16), 3511–3517 (2009). [CrossRef]

**17**(11), 9421–9427 (2009). [CrossRef] [PubMed]

**27**(16), 3511–3517 (2009). [CrossRef]

12. R. Nagarajan, C. H. Joyner, R. P. Schneider, J. S. Bostak, T. Butrie, A. G. Dentai, V. G. Dominic, P. W. Evans, M. Kato, M. Kauffman, D. J. H. Lambert, S. K. Mathis, A. Mathur, R. H. Miles, M. L. Mitchell, M. J. Missey, S. Murthy, A. C. Nilsson, F. H. Peters, S. C. Pennypacker, J. L. Pleumeekers, R. A. Salvatore, R. K. Schlenker, R. B. Taylor, M. F. Huan-Shang Tsai, J. Van Leeuwen, M. Webjorn, D. Ziari, J. Perkins, S. G. Singh, M. S. Grubb, D. G. Reffle, F. A. Mehuys, Kish, and D. F. Welch, “Large-scale photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. **11**(1), 50–65 (2005). [CrossRef]

## 4. Performance analysis

4. H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express **18**(19), 20546–20551 (2010). [CrossRef] [PubMed]

*g*

_{r}, ± 1/

*g*

_{r},0), and the remaining 32 points are those of the 32-point constellation. We can see that 16-4D-OFDM outperforms PolMux-8-QAM-OFDM by 1.79 dB, 32-4D-OFDM outperforms PolMux-16-QAM-OFDM by 2.14 dB and 64-4D-OFDM outperforms PolMux-32-QAM-OFDM by 2.07 dB, and finally 128-64-4D outperforms PolMux-64-QAM by 4.53dB. The comparison is, therefore, performed for the same number of constellation points.

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. Hong, and T. Schmidt, “40G and 100G modules enable next generation networks,” in Proc. SPIE, Communications and Photonics Conference and Exhibition 2009 (ACP 2009) |

2. | H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Modified hybrid subcarrier/amplitude/ phase/polarization LDPC-coded modulation for 400 Gb/s optical transmission and beyond,” Opt. Express |

3. | H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Multidimensional LDPC-coded modulation for beyond 400 Gb/s per wavelength transmission,” IEEE Photon. Technol. Lett. |

4. | H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express |

5. | L. J. Stankovic, |

6. | C. Chakrabarti and J. JaJa, “VLSI architectures for multidimensional transforms,” IEEE Trans. Comput. |

7. | W. Shieh, and I. Djordjevic, |

8. | I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. |

9. | J. McDonough, “Moving standards to 100 GbE and beyond,” IEEE Appl. Pract. |

10. | Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express |

11. | Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. |

12. | R. Nagarajan, C. H. Joyner, R. P. Schneider, J. S. Bostak, T. Butrie, A. G. Dentai, V. G. Dominic, P. W. Evans, M. Kato, M. Kauffman, D. J. H. Lambert, S. K. Mathis, A. Mathur, R. H. Miles, M. L. Mitchell, M. J. Missey, S. Murthy, A. C. Nilsson, F. H. Peters, S. C. Pennypacker, J. L. Pleumeekers, R. A. Salvatore, R. K. Schlenker, R. B. Taylor, M. F. Huan-Shang Tsai, J. Van Leeuwen, M. Webjorn, D. Ziari, J. Perkins, S. G. Singh, M. S. Grubb, D. G. Reffle, F. A. Mehuys, Kish, and D. F. Welch, “Large-scale photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. |

13. | R. van Nee, and R. Prasad, |

14. | J. G. Proakis, |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.4080) Fiber optics and optical communications : Modulation

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 3, 2010

Revised Manuscript: December 23, 2010

Manuscript Accepted: December 24, 2010

Published: January 6, 2011

**Citation**

Ivan Djordjevic, Hussam G. Batshon, Lei Xu, and Ting Wang, "Four-dimensional optical multiband-OFDM for beyond 1.4 Tb/s serial optical transmission," Opt. Express **19**, 876-882 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-876

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### References

- J. Hong and T. Schmidt, “40G and 100G modules enable next generation networks,” in Proc. SPIE, Communications and Photonics Conference and Exhibition 2009 (ACP 2009) 7631, 763115 (2009).
- H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Modified hybrid subcarrier/amplitude/ phase/polarization LDPC-coded modulation for 400 Gb/s optical transmission and beyond,” Opt. Express 18(13), 14108–14113 (2010). [CrossRef] [PubMed]
- H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Multidimensional LDPC-coded modulation for beyond 400 Gb/s per wavelength transmission,” IEEE Photon. Technol. Lett. 21(16), 1139–1141 (2009). [CrossRef]
- H. G. Batshon, I. B. Djordjevic, and T. Schmidt, “Ultra high speed optical transmission using subcarrier-multiplexed four-dimensional LDPC-coded modulation,” Opt. Express 18(19), 20546–20551 (2010). [CrossRef] [PubMed]
- L. J. Stankovic, Digital Signal Processing (Naucna Knjiga, 1990). [PubMed]
- C. Chakrabarti and J. JaJa, “VLSI architectures for multidimensional transforms,” IEEE Trans. Comput. 40(9), 1053–1057 (1991). [CrossRef]
- W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier/Academic Press, 2009).
- I. B. Djordjevic, M. Arabaci, and L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. 27(16), 3518–3530 (2009). [CrossRef]
- J. McDonough, “Moving standards to 100 GbE and beyond,” IEEE Appl. Pract. 45, 6–9 (2007).
- Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express 17(11), 9421–9427 (2009). [CrossRef] [PubMed]
- Y. Tang and W. Shieh, “Coherent optical OFDM transmission up to 1 Tb/s per channel,” J. Lightwave Technol. 27(16), 3511–3517 (2009). [CrossRef]
- R. Nagarajan, C. H. Joyner, R. P. Schneider, J. S. Bostak, T. Butrie, A. G. Dentai, V. G. Dominic, P. W. Evans, M. Kato, M. Kauffman, D. J. H. Lambert, S. K. Mathis, A. Mathur, R. H. Miles, M. L. Mitchell, M. J. Missey, S. Murthy, A. C. Nilsson, F. H. Peters, S. C. Pennypacker, J. L. Pleumeekers, R. A. Salvatore, R. K. Schlenker, R. B. Taylor, M. F. Huan-Shang Tsai, J. Van Leeuwen, M. Webjorn, D. Ziari, J. Perkins, S. G. Singh, M. S. Grubb, D. G. Reffle, F. A. Mehuys, Kish, and D. F. Welch, “Large-scale photonic integrated circuits,” IEEE J. Sel. Top. Quantum Electron. 11(1), 50–65 (2005). [CrossRef]
- R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications (Artech House, 2000).
- J. G. Proakis, Digital Communications (McGraw-Hill, 2001).

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