## High-dimensional modulation for coherent optical communications systems |

Optics Express, Vol. 22, Issue 7, pp. 8798-8812 (2014)

http://dx.doi.org/10.1364/OE.22.008798

Acrobat PDF (2045 KB)

### Abstract

In this paper, we examine the performance of several modulation formats in more than four dimensions for coherent optical communications systems. We compare two high-dimensional modulation design methodologies based on spherical cutting of lattices and block coding of a ‘base constellation’ of binary phase shift keying (BPSK) on each dimension. The performances of modulation formats generated with these methodologies is analyzed in the asymptotic signal-to-noise ratio regime and for an additive white Gaussian noise (AWGN) channel. We then study the application of both types of high-dimensional modulation formats to standard single-mode fiber (SSMF) transmission systems. For modulation with spectral efficiencies comparable to dual-polarization (DP-) BPSK, polarization-switched quaternary phase shift keying (PS-QPSK) and DP-QPSK, we demonstrate SNR gains of up to 3 dB, 0.9 dB and 1 dB respectively, at a BER of 10^{−3}.

© 2014 Optical Society of America

## 1. Introduction

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

2. “Implementation Agreement for Integrated Dual Polarization Intradyne Coherent Receivers,” Optical Internetworking Forum (2010). http://www.oiforum.com/public/documents/OIF_DPC_RX-01.0.pdf.

3. S. Betti, F. Curti, G. De Marchis, and E. Iannone, “Exploiting fibre optics transmission capacity: 4-quadrature multilevel signalling,” Electron. Lett. **26**, 992–993 (1990). [CrossRef]

4. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express **17**, 10814–10819 (2009). [CrossRef] [PubMed]

5. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. **27**, 5115–5126 (2009). [CrossRef]

6. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PSQPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express **18**, 11360–11371 (2010). [CrossRef] [PubMed]

8. D. S. Millar, D. Lavery, S. Makovejs, C. Behrens, B. C. Thomsen, P. Bayvel, and S. J. Savory, “Generation and long-haul transmission of polarization-switched QPSK at 42.9 Gb/s,” Opt. Express **19**, 9296–9302 (2011). [CrossRef] [PubMed]

11. M. Sjödin, P. Johannisson, J. Li, E. Agrell, P. A. Andrekson, and M. Karlsson, “Comparison of 128-SP-QAM with PM-16-QAM,” Opt. Express20, 8356–8366 (2012). [CrossRef] [PubMed]

22. J. A. Anguita, J. Herreros, and I. B. Djordjevic, “Coherent Multimode OAM Superpositions for Multidimensional Modulation,” IEEE Photonics J. **6**, 1–11 (2014). [CrossRef]

23. I. B. Djordjevic, M. Cvijetic, and C. Lin, “Multidimensional Signaling and Coding Enabling Multi-Tb/s Optical Transport and Networking: Multidimensional aspects of coded modulation,” IEEE Sig. Proc. Mag. **31**22, 104–117 (2014). [CrossRef]

25. M. Nakazawa, T. Hirooka, M. Yoshida, and K. Kasai, “Extremely Higher-Order Modulation Formats,” in *Optical Fiber Telecommunications VIB*, I. Kaminow, T. Li, and A. E. Willner, eds. (Elsevier, 2013), pp. 297–336. [CrossRef]

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

28. G.-H. Gho and J. M. Kahn, “Rate-Adaptive Modulation and Coding for Optical Fiber Transmission Systems,” J. Lightwave Technol. **30**, 1812–1818 (2012). [CrossRef]

*N*-dimensional require all

*N*components of the signal field to be jointly considered when performing symbol estimation. For highly dispersive channels, we may consider the combined effects of ASE and fiber nonlinearity as AWGN [29

29. P. Poggiolini, “The GN Model of Non-Linear Propagation in Uncompensated Coherent Optical Systems,” J. Lightwave Technol. **30**, 3857–3879 (2012). [CrossRef]

*p*-bit words to be modulated. These

*p*-bit words are then encoded with a block encoder to form

*N*-bit codewords, which are then mapped to an

*N*-dimensional field. The field is then partially serialized to form a 4-D signal where a single high-dimensional symbol is transmitted over several time-slots. Modulation, transmission and coherent detection of these 4-D signals are carried out in the conventional manner. After coherent detection and equalization (not shown), the 4-D signal is then deserialized into an

*N*-dimensional field. This field is then used for minimum Euclidean distance symbol detection (this may be considered as joint demapping and decoding), resulting in a

*p*-bit hard decision. This

*p*-bit hard decision is then serialized, and sent to the sink.

## 2. High-dimensional modulation formats

*p*denote the number of bits per high-dimensional symbol, and

*N*denote the number of modulation dimensions. A high-dimensional modulation format maps each

*p*-bit symbol to one of

*M*= 2

*constellation points in the*

^{p}*N*dimensional signal space (see Fig. 1). High-dimensional modulation designs must therefore optimize (i) locations of

*M*constellation points and (ii) corresponding

*p*-bit labelings for each of the constellation points. Throughout the paper, we use the notation ‘

*p*b-

*N*D’ for modulation formats of constellations generated by sphere-cutting of lattices, where

*p*is the number of bits per symbol and

*N*is the dimensionality of the constellation space. For block-coded modulation, we use the notation [

*N*,

*p*,

*h*] - consistent with the coding literature - where

*h*is the minimum Hamming distance of the code. Although this is not consistent with the notation used in our previous work [16–20], the large variety of codes used in this work necessitates this more generalized notation. In this section, we describe the design of such high-dimensional modulation formats using two approaches: spherical cutting of optimal lattices, and linear block codes over a base constellation of BPSK.

### 2.1. Sphere cutting of densest lattices

*N*-dimensional hyperspheres which are optimized by finding the densest possible non-overlapping arrangement in

*N*-dimensional space [5

5. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. **27**, 5115–5126 (2009). [CrossRef]

31. N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy clusters of hard spheres,” Discrete & Computational Geometry **14**, 237–259 (1995). [CrossRef]

*N*dimensional hypersphere of points from the lattice – optimization of this cutting procedure is described in detail in [30]. We then have a symbol alphabet which is optimized in terms of minimum separation between constellation points, but has no bit labeling.

*N*= 2, the densest lattice is the hexagonal lattice [32]. The optimized constellation by cutting the 2-D hexagonal lattice reduces to BPSK constellations for

*M*= 2 (

*p*= 1) and rhombus constellation for

*M*= 4 (

*p*= 2) which has the same minimum Euclidean distance with the QPSK constellation. For higher

*p*, comparing the optimized constellations cut from 2-D hexagonal lattice to the constellations cut from 2-D rectangular lattice, 0.6, 0.76 and 0.81 dB increase in asymptotic power efficiency [5

5. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. Lightwave Technol. **27**, 5115–5126 (2009). [CrossRef]

*M*= 16, 64 and 256 (

*p*= 4, 6 and 8) [33]. The asymptotic power efficiency is defined as

*d*

_{min}is the minimum Euclidean distance between constellation points, and

*E*

_{b}is the energy per bit.

*M*because of the regular lattice structure. An interesting feature of this constellation is that the maximum and minimum distance points both have the same distance from the reference point. We may consider the reference point as a high-dimensional sphere, with additional high-dimensional spheres (representing the other constellation points) of the same size touching it. As long as the total number of neighboring constellation points is lower than the kissing number (240 for the

*E*

_{8}lattice), each neighboring sphere can be touching the reference sphere. The centers of all neighboring spheres may therefore be the same distance from the center of the reference sphere.

*M*−1)! possibilities. As will be discussed in Section 3, the increased number of near neighbors for high dimensional constellations means that although sphere-cut high-dimensional formats can maximize the minimum Euclidean distance between constellation points and provide significant improvements in symbol error ratio (SER), the gain for uncoded bit error ratio (BER) may not be so significant in the low SNR regimes. In this paper, we have employed a random search algorithm for bit-labeling. For a fixed SNR, we tested the BER of many randomly selected labelings. The best labeling was selected when either the improvement expected from testing additional labelings approached zero, or the maximum number of test labelings was reached. Although this approach does not determine the optimal labeling, it was considered to be computationally tractable, unlike an exhaustive search of all possible labelings.

*a-priori*and extrinsic mutual information and also higher total extrinsic mutual information. Iterative decoding of high-dimensional formats may therefore provide additional benefits in performance not revealed by the investigation of uncoded performance presented in this paper. We also note that this technique may be prohibitively computationally intensive for very large numbers of dimensions.

### 2.2. Linear binary block codes

34. G. D. Forney Jr, R. G. Gallager, G. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Sel. Areas Commun. **2**, 632–647 (1984). [CrossRef]

35. G. Forney Jr, G. David, and L.-F. Wei., “Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. **7**, 877–892 (1989). [CrossRef]

36. K. Onohara, T. Sugihara, Y. Konishi, Y. Miyata, T. Inoue, S. Kametani, K. Sugihara, K. Kubo, H. Yoshida, and T. Mizuochi, “Soft-Decision-Based Forward Error Correction for 100 Gb/s Transport Systems,” IEEE J. Sel. Top. Quantum Electron. **16**, 1258–1267 (2010). [CrossRef]

37. J. H. van Lint, “A survey of perfect codes,” Rocky Mountain Journal of Mathematics **5**, 199–224 (1975). [CrossRef]

*N*, message length

*p*and minimum Hamming distance between codewords

*h*, with BSPK mapping for each of the

*N*binary outputs, yields

*M*= 2

*constellation points in*

^{p}*N*dimensional space and corresponding binary labelings. For example, for

*N*= 4, DP-QPSK modulation can be considered as two independent uncoded modulation dimensions with

*h*= 1. Using an inner [4,3,2] single-parity check code results in PS-QPSK (also equivalent to 3b-4D sphere-cut modulation) [4

4. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Opt. Express **17**, 10814–10819 (2009). [CrossRef] [PubMed]

*N*is a multiple of 4 while it is not necessary condition):

*N*,

*p*,

*h*] with the corresponding block code name for block-coded modulation formats with block length

*N*, message length

*p*, and the minimum Hamming distance

*h*. As an illustration, the extended Hamming [8,4,4] code constellation is shown in Fig. 3, which has the same spectral efficiency and minimum Euclidean distance with the 4b-8D constellation in Fig. 2. Again, we have plotted both the full constellation, and the minimum and maximum Euclidean distance points relative to a reference point. For this constellation, we can clearly see that the minimum distance constellation point requires a total of 4 changes of signal level (either I or Q) for the 8 signal dimensions. Similarly, the maximum distance point is achieved when the signal level changes on all 8 dimensions (maximum distance corresponding to a change of signal on all dimensions requires the use of a self-dual code such as the extended Hamming [8,4,4] code).

31. N. J. A. Sloane, R. H. Hardin, T. D. S. Duff, and J. H. Conway, “Minimal-energy clusters of hard spheres,” Discrete & Computational Geometry **14**, 237–259 (1995). [CrossRef]

37. J. H. van Lint, “A survey of perfect codes,” Rocky Mountain Journal of Mathematics **5**, 199–224 (1975). [CrossRef]

*− 1,2*

^{r}*−*

^{r}*r*− 1,3] Hamming codes and the [23,12,7] binary Golay code are the only two (non-trivial) perfect binary codes. Indeed, the Golay code is the only binary quasi-perfect code with

*h*> 4. The quasi-perfect codes can be obtained by adding a single parity bit to the perfect codes to ensure even parity over the set of all codewords, thus increasing Hamming distance by 1. Extended Hamming [2

*, 2*

^{r}*−*

^{r}*r*− 1,4] codes and the extended Golay [24,12,8] code belong to this family.

## 3. Asymptotic power efficiency and noise sensitivity

*d*

_{min}between constellation points determines the SER as the nearest neighbor errors become dominant. One commonly used parameter to quantify the asymptotic noise performance is the sensitivity penalty 1/

*γ*where

*γ*is the asymptotic power efficiency. Note that for 1 bit/symbol/dimension case of DPQPSK,

*γ*= 1, hence sensitivity penalty can also be considered as the performance penalty with respect to DP-QPSK for asymptotically high SNR.

*N*= 2, 4, 6, 8 and 16, respectively. In [5

**27**, 5115–5126 (2009). [CrossRef]

*N*= 2 is obtained for

*M*= 3 and for

*N*= 3 for

*M*= 4, both using simplex configurations, i.e. same length vectors with same angular separation. For

*N*> 3, simplex configurations are not optimal constellations in terms of power efficiency. For

*N*= 4, the highest power efficiency is obtained for

*M*= 8, with the 3b-4D PS-QPSK constellation. Among the values we consider for sphere-cut constellation design, for

*N*= 6, 8 and 16, we observe the highest power efficiencies for

*M*= 2

^{4}, 2

^{4}and 2

^{11}(4b-6D, 4b-8D and 11b-16D modulation formats).

*d*

_{min}, the total number of nearest neighbors at that distance increases the SER proportionally. For reference, the average number of neighbors within 1.01 ·

*d*

_{min}distance is listed in Table 1. Although the maximum number of spheres which can touch an adjacent sphere (the so-called kissing number problem) is unknown for an arbitrary number of dimensions, upper and lower bounds (including lattice arrangements) indicate that the kissing number increases exponentially with number of dimensions [32]. From this we may infer that the number of nearest neighbors in the high spectral efficiency region increases exponentially for sphere-cut high-dimensional modulation. Of course, for finite constellations, the maximum number of nearest neighbors is at most the number of points in the constellation.

*E*

_{b}/

*N*

_{0}where

*N*

_{0}is the unilateral power spectral density of the noise per dimension. Monte Carlo simulations were performed for an ideal transmitter and receiver using minimum Euclidean distance decisions, and more than 10

^{7}bits for each SNR level.

^{−3}. This gain is reduced to 1 dB and 1.5 dB respectively at a BER of 10

^{−2}. For the sphere-cut modulation formats presented in Fig. 6(a) with spectral efficiency of 0.687 and 0.75 bit/symbol/dimension (3b-4D – also known as PS-QPSK – and 11b-16D), we note a gain compared with DP-QPSK at a BER of 10

^{−3}of 1 dB and 1.9 dB respectively. Increasing BER to 10

^{−2}reduces the gain to 0.6 and 0.8 dB respectively. The formats presented in Fig. 7(a) with 0.875 bit/symbol/dimension (7b-8D) and 1 bit/symbol/dimension (6b-6D and 8b-8D) spectral efficiency modulation formats to DP-QPSK, result in gains of 1.1, 0.5 and 0.9 dB respectively when compared with DP-QPSK at a BER of 10

^{−3}. These gains are reduced to 0.3, −0.1 and 0.1 dB at a BER of 10

^{−2}.

^{−3}, we observe improvements in sensitivity of 1.5, 1.7, 2.9 and 3 dB. For a BER of 10

^{−2}, we note improvements of 1, 1.1, 1.8 and 1.9 dB. Modulation formats presented in Fig. 6(b) are [4,3,2] (identical to PS-QPSK), [15,11,3] and 16,11,4], with spectral efficiency of 0.75, 0.733 and 0.688 bit/symbol/dimension. At a BER of 10

^{−3}, we can see improvements in sensitivity of 1, 1.9 and 2.2 dB, respectively. These gains are reduced to 0.6, 1.2 and 1.3 dB at a BER of 10

^{−2}. At higher spectral efficiency, we are asymptotically limited to 1 bit/symbol/dimension. Therefore in Fig. 7(b), we examine two single party check coded formats: [8,7,2] and [16,15,2], with respective spectral efficiencies of 0.875 and 0.938 bit/symbol/dimension. These formats offer a gain of 1.1 and 1 dB over DP-QPSK at a BER of 10

^{−3}, and a gain of 0.6 and 0.4 dB at a BER of 10

^{−2}.

## 4. Transmission over single-mode fiber

^{−3}; (ii) 125 Gb/s with 20% FEC overhead and a BER threshold of 10

^{−2}. For each case, we consider three groups of modulation formats: low, intermediate and high spectral efficiencies, that are listed in Table 2. The wavelength spacings are chosen to be wide enough to account for the different spectral widths of the modulated signals, and are also listed in Table 2. The total transmission link of 6000 km consists of 75 spans of 80 km SMF, with optical loss compensated by EDFA, each with 5 dB of noise figure. To quantify the transmission performance in terms of span loss budget for a target BER, variable optical attenuators are used [6

6. P. Poggiolini, G. Bosco, A. Carena, V. Curri, and F. Forghieri, “Performance evaluation of coherent WDM PSQPSK (HEXA) accounting for non-linear fiber propagation effects,” Opt. Express **18**, 11360–11371 (2010). [CrossRef] [PubMed]

*order Bessel filter with (single sided) −3 dB cut-off frequency of 0.35 times the (double sided) Nyquist bandwidth. Optical to electrical conversion is performed by ideal, linear dual-polarization I/Q modulators for each wavelength.*

^{th}*order Bessel filter with −3 dB cut-off frequency of 0.35 times the Nyquist bandwidth to emulate limited receiver bandwidth, and then sampled at the rate of twice the symbol rate. Digital signal processing (DSP) is then performed on the received signals in the following order prior to demodulation: normalization, chromatic dispersion compensation by frequency-domain equalization, time-domain data-aided equalization by least-mean-square (LMS) algorithm. Demodulation is then performed by selecting the minimum Euclidean distance between the received signal and each possible transmitted symbol in the signal alphabet. The transmitted bits were then compared to the received bits to calculate the BER.*

^{th}^{−2}and 10

^{−3}in Fig. 5 as expected. As the launch power increases and nonlinearity becomes significant, relative performances of the modulation formats alter differently. For both cases of 7% FEC and 20% FEC, block-coded modulation formats outperform sphere-cut modulation formats in the highly nonlinear regime, which we attribute to the increased peak-to-average power ratio in sphere-cut constellations. For BER of 10

^{−3}, extended Golay [24,12,8] coded modulation yields span loss budget improvement around 3.2 dB gain with respect to DP-BPSK in the linear propagation regime, whereas the difference between the peak span loss budget values is around 4.8 dB. Although 12b-24D sphere-cut modulation demonstrates similar performance to extended Golay [24,12,8] coded modulation in the linear propagation regime, the peak span loss budget difference is around 1.2 dB between these two 24-D modulation formats for BER of 10

^{−3}. The 8-D modulation formats have more similar performances. In the linear propagation regime, 4b-8D sphere-cut modulation yields 1.6 dB improvement and extended Hamming [8,4,4] coded modulation yields 1.8 dB improvement with respect to DP-BPSK. Between the peak points, this improvement increases to 2.7 dB and 2.8 dB for 4b-8D sphere-cut modulation and extended Hamming [8,4,4] coded modulation. The SNR improvements are lower for BER of 10

^{−2}in all cases. The difference between peak points with respect to DPBPSK ares 1.5 dB for 4b-8D sphere-cut modulation, 1.7 dB for extended Hamming [8,4,4] coded modulation, 1.7 dB for 12b-24D sphere-cut modulation and 2.7 dB for extended Golay [24,12,8] coded modulation.

^{−2}and 10

^{−3}in Fig. 6 for the AWGN channel. Moreover, robustness against nonlinearity is again considerably higher for the block-coded modulation format comparing to the sphere-cut modulation format. In the linear propagation regime, span loss budget improvement with respect to PS-QPSK is 0.9 dB for 11b-16D sphere-cut modulation and 1.2 dB for extended Hamming [16,11,4] coded modulation for BER of 10

^{−3}. The improvement between the peak points become 1 dB for 11b-16D sphere-cut modulation and 2.2 dB for extended Hamming [16,11,4] coded modulation for the same BER of 10

^{−3}. For BER of 10

^{−2}, performances of 11b–16D sphere-cut modulation and PSQPSK become very similar, and extended Hamming [16,11,4] coded modulation results in span loss budget improvement up to 1.5 dB with respect to other two.

^{−3}, 8b-8D sphere-cut modulation provides 0.9 dB improvement with respect to DP-QPSK in the linear region, while performance degrades above the optimum launch power. The performance deterioration of sphere-cut modulation formats in highly nonlinear regime can also be clearly observed for 8b-8D modulation format. For BER of 10

^{−2}, as expected from the additive white Gaussian channel characteristics, span loss budgets of 8b-8D sphere-cut modulation and DP-QPSK are almost same in the linear propagation regime and 8b-8D modulation underperforms when nonlinearity becomes significant.

## 5. Conclusions

^{−3}, and 1.9 dB for a BER of 10

^{−2}. At a spectral efficiency of 0.73 bit/symbol/dimension, the block coded [15,11,3] format provides gains over PS-QPSK of 0.9 dB and 0.6 dB at BERs of 10

^{−3}and 10

^{−2}respectively, with a reduction in spectral efficiency of just 2.2%. Using a [16,15,2] single parity check code, gains over DP-QPSK of 1 dB and 0.4 dB were demonstrated at BERs of 10

^{−3}and 10

^{−2}respectively.

^{−3}∼ 10

^{−2}, overall performance comparisons for the joint use of such high-dimensional modulation and strong modern FEC codes remains a topic for further investigation.

## Acknowledgements

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32. | J. H. Conway and N. J. A. Sloane, |

33. | C. D. Murphy, “High-order optimum hexagonal constellations,” IEEE PIMRC |

34. | G. D. Forney Jr, R. G. Gallager, G. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Sel. Areas Commun. |

35. | G. Forney Jr, G. David, and L.-F. Wei., “Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. |

36. | K. Onohara, T. Sugihara, Y. Konishi, Y. Miyata, T. Inoue, S. Kametani, K. Sugihara, K. Kubo, H. Yoshida, and T. Mizuochi, “Soft-Decision-Based Forward Error Correction for 100 Gb/s Transport Systems,” IEEE J. Sel. Top. Quantum Electron. |

37. | J. H. van Lint, “A survey of perfect codes,” Rocky Mountain Journal of Mathematics |

**OCIS Codes**

(060.4080) Fiber optics and optical communications : Modulation

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Optical Communications

**History**

Original Manuscript: February 20, 2014

Revised Manuscript: March 28, 2014

Manuscript Accepted: March 29, 2014

Published: April 4, 2014

**Citation**

David S. Millar, Toshiaki Koike-Akino, Sercan Ö. Arık, Keisuke Kojima, Kieran Parsons, Tsuyoshi Yoshida, and Takashi Sugihara, "High-dimensional modulation for coherent optical communications systems," Opt. Express **22**, 8798-8812 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8798

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