## Which is the most power-efficient modulation format in optical links?

Optics Express, Vol. 17, Issue 13, pp. 10814-10819 (2009)

http://dx.doi.org/10.1364/OE.17.010814

Acrobat PDF (170 KB)

### Abstract

By exploiting the electromagnetic wave’s four-dimensional signal space, we find that for the additive white Gaussian noise channel, the modulation format with best sensitivity to be an 8-level format with 1.76 dB asymptotic gain over BPSK, for uncoded optical transmission with coherent detection. Low-complexity modulators are presented for the format, as well as an interpretation in terms of quantum-limited sensitivity.

© 2009 Optical Society of America

## 1. Introduction

1. S. Tsukamoto, D. Ly-Gagnon, K. Katoh, and K. Kikuchi, “Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission,” in *Proceedings of Optical Fiber Communication and National Fiber Optic Engineers Conference*, OFC/NFOEC, vol. 6, p. PDP29 (2005).

2. G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, “Performance comparison of singly-polarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments,” Electron. Lett. **43(20)**, 1109–1111 (2007).
[CrossRef]

3. H. Sun, K. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express **16(2)**, 873–879 (2008).
[CrossRef]

1. S. Tsukamoto, D. Ly-Gagnon, K. Katoh, and K. Kikuchi, “Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission,” in *Proceedings of Optical Fiber Communication and National Fiber Optic Engineers Conference*, OFC/NFOEC, vol. 6, p. PDP29 (2005).

3. H. Sun, K. Wu, and K. Roberts, “Real-time measurements of a 40 Gb/s coherent system,” Opt. Express **16(2)**, 873–879 (2008).
[CrossRef]

5. J. R. Pierce, “Optical Channels: Practical Limits with Photon Counting,” IEEE Trans. Commun. **26(12)**, 1819–1821 (1978).
[CrossRef]

## 2. Global comparison of performance

*M*-ary modulation format with the least average power requirement to reach a given BER in an AWGN environment is equivalent (in the limit of low BER) to the problem of placing

*M*points so that their minimum distance is maximized under an average energy constraint. Alternatively, the minimum distance can be kept constant and the average energy minimized, which is in turn equivalent to packing

*M*rigid spheres so that their average squared distance

*E*from the origin is minimized. Furthermore, it can be shown that the bit error rate asymptotically becomes well approximated by the union bound [6, p. 195], and that the dominating term for the

_{s}*BER*depends on the signal power

*P*as

*R*is the bit rate and

*N*

_{0}is the (single-sided) noise spectral density. The

*asymptotic power efficiency*

*γ*is defined as [6, p. 220]

*γ*=

*d*

^{2}

_{min}/(4

*E*), where

_{b}*E*=

_{b}*E*/log

_{s}_{2}

*M*is the energy per bit, and

*d*

_{min}is the sphere diameter or, equivalently, the minimum (Euclidean) distance between constellation points. Observe that

*γ*, which is usually given in dB, depends on the constellation geometry only, not on the transmitted power. It is 0 dB for BPSK and QPSK, and it can therefore be interpreted as the

*sensitivity gain*over BPSK to transmit the same data rate. In this paper we will, for what we believe is the first time, present the maximum sensitivity gain for all constellation sizes

*M*≤32 and dimensions

*N*≤4.

*N*=4), corresponding to two quadratures in two polarization states. The constellation vectors are formed from the real and imaginary part of the electric field’s x and y polarization components as [7

7. S. Betti, F. Curti, G. De Marchis, and E. Iannone, “A novel multilevel coherent optical system: 4-quadrature signaling,” J. Lightwave Technol. **9(4)**, 514–523 (1991).
[CrossRef]

*E*

_{x,r},

*E*

_{x,i},

*E*

_{y,r},

*E*

_{y,i}). As an example, the DP-QPSK format can be expressed (in normalized units) as the 16 levels

*C*

_{1}={(±1,±1,±1,±1)}, allowing for any sign selection, and it has

*d*

_{min}=2,

*E*=4, and

_{s}*γ*=0 dB just as BPSK and QPSK.

*M*nonoverlapping spheres in

*N*-dimensional space. To find the packing that minimizes the average squared distance from the origin is a geometric problem that can be solved by numerical optimization. One starts with

*M*randomly positioned nonoverlap- ping spheres, which are then made to relax into a closely-packed cluster by suitable attractive and repelling forces. Unfortunately, there exist many packings that are locally optimal in this respect. Therefore the process is repeated for a large number of random initial conditions until the best packing emerges, which can be very time consuming. The sphere-packing problem has been addressed previously in the literature, mostly via such numerical optimization. Rigorous mathematical proofs of optimality have been obtained only in a few special cases. For example, optimum constellations for dimensions

*N*=2 and

*N*=3 were discussed in [8] and [9], respectively, and results for

*N*=4 are available online [10

10. N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-Energy Clusters, Library of 3-d clusters, Library of 4-d clusters,” (1997). URL http://www.research.att.com/~njas/cluster/.

*SE*vs. the sensitivity penalty 1/

*γ*for the optimum constellations. Such a chart is the conventional way of comparing modulation formats [4

4. E. Ip, A. Lau, D. Barros, and J. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008). Erratum vol. 16, no. 26, p. 21943, 2008. [CrossRef]

*dimension pair*, as suggested in [6, p. 219]), so that

*SE*=log

_{2}(

*M*)/(

*N*/2). This definition of

*SE*will cause BPSK, QPSK, and DP-QPSK to have

*SE*=2, since BPSK has dimension

*N*=1. The leftmost points in this graph are thus the most power-efficient modulation formats, and we may note that for small

*N*this occurs for

*simplices*, i.e., the equilateral triangle (or 3-PSK format) for

*N*=2 and the tetrahedron (

*M*=4) for

*N*=3. These modulation formats have received limited practical interest, due to the difficulty of (i) generating them and (ii) mapping bits to symbols when

*M*is not a power of 2.

## 3. The PS-QPSK format

*N*=4. Instead, the overall optimum occurs for

*M*=8, showing an improved asymptotic sensitivity of 1.76 dB (or 1.5 times) over BPSK. This 8-level modulation format consists of the levels

*C*

_{2}={(±2,0,0,0), (0,±2,0,0), (0,0,±2,0), (0,0,0,±2)}. This normalization makes the amplitude the same as for the DP-QPSK format discussed above. This is the four-dimensional version of

*biorthogonal signaling*[12, pp. 198–203], [13

13. L. Xiao and X. Dong, “New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception,” IEEE Trans. Wireless Commun. **4(4)**, 1418–1424 (2005).
[CrossRef]

*cross-polytope*, or

*16-cell*, since it is bounded by 16 tetrahedrons. It has been suggested previously to be used for signal modulation, see, e.g., [7

7. S. Betti, F. Curti, G. De Marchis, and E. Iannone, “A novel multilevel coherent optical system: 4-quadrature signaling,” J. Lightwave Technol. **9(4)**, 514–523 (1991).
[CrossRef]

14. D. Saha and T. Birdsall, “Quadrature-quadrature phase-shift keying,” IEEE Trans. Commun. **37(5)**, 437–448 (1989).
[CrossRef]

*C*

_{2}in a conventional (see e.g. [2

2. G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, “Performance comparison of singly-polarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments,” Electron. Lett. **43(20)**, 1109–1111 (2007).
[CrossRef]

*C*

_{2}format, that might give rise to simpler transmitter structures. By a 45° phase rotation, the constellation may be expressed as

*C*′

_{2}=√2{(±1,±1,0,0), (0,0,±1,±1)}, which is QPSK transmission in

*either*the x

*or*the y polarization. Thus, two bits are transmitted via QPSK and the third bit determines whether the x or y polarization is used. Therefore, we will refer to this format as polarization-switched QPSK (PS-QPSK). A schematic transmitter for PS-QPSK is shown in Fig. 2 (a), showing a standard QPSK transmitter followed by a polarization modulator. Moreover, a 45° polarization rotation gives another way of expressing the PS-QPSK format:

*C*

_{2}″=±{(1,1,1,1), (1,1,-1,-1), (1,-1,1,-1), (1,-1,-1,1)}, revealing it to be a subset of the DP-QPSK (

*C*

_{1}) levels; namely, those having an even number of minus signs. This means that the PS-QPSK format can be obtained from the conventional DP-QPSK transmitter by using two XOR gates, which will force the driving bits

*b*

_{1},

*b*

_{2},

*b*

_{3},

*b*

_{4}to have even parity, as shown in Fig. 2 (b).

## 4. Bit- and symbol error rates

13. L. Xiao and X. Dong, “New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception,” IEEE Trans. Wireless Commun. **4(4)**, 1418–1424 (2005).
[CrossRef]

*BER*

_{PS-QPSK}≈

*SER*

_{PS-QPSK}/2. (An exact expression is given in [12, p. 203].) The BER for PS-QPSK and BPSK/DP-QPSK is shown in Fig. 3. The required

*E*/

_{b}*N*

_{0}at a BER of 10

^{-3}is 5.82 dB for PS-QPSK and 6.79 dB for BPSK, while at 10

^{-9}we have 11.04 dB for PS-QPSK and 12.55 dB for BPSK. As the BER decreases, the

*E*/

_{b}*N*

_{0}difference approaches 10log

_{10}(3/2)=1.76 dB.

## 5. Sensitivity limits

*n*is the average number of received photons per bit,

_{b}*N*is the number of amplifiers in the link and

_{A}*n*is the spontaneous emission noise factor from the inline amplifiers. In fact, Eq. (3) holds for both heterodyne and homodyne receivers limited by ASE noise. Since

_{sp}*N*>1, we see that in the limiting case (a single amplifier with a 3 dB noise figure), the sensitivity in terms of number of photons per bit is given directly by

_{a}n_{sp}*E*/

_{b}*N*

_{0}. For

*BER*=10

^{-9}, this translates into the well-known [4

4. E. Ip, A. Lau, D. Barros, and J. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008). Erratum vol. 16, no. 26, p. 21943, 2008. [CrossRef]

*BER*=10

^{-3}, we get 4.5 photons per bit for BPSK (which was given in dB units in [4

4. E. Ip, A. Lau, D. Barros, and J. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008). Erratum vol. 16, no. 26, p. 21943, 2008. [CrossRef]

## 6. Conclusions

^{-9}, this improves the ASE-limited sensitivity from 18 (for BPSK) to 13 (for PS-QPSK) photons per bit. We conclude that the PS-QPSK format has the best sensitivity attainable in optical systems, unless the constellation dimension is extended, e.g., by the use of error-correcting codes. Thus, the PS-QPSK format is the answer to the deceptively simple question posed in the title.

## Acknowledgements

## References and links

1. | S. Tsukamoto, D. Ly-Gagnon, K. Katoh, and K. Kikuchi, “Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission,” in |

2. | G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, “Performance comparison of singly-polarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments,” Electron. Lett. |

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

4. | E. Ip, A. Lau, D. Barros, and J. Kahn, “Coherent detection in optical fiber systems,” Opt. Express16(2), 753–791 (2008). Erratum vol. 16, no. 26, p. 21943, 2008. [CrossRef] |

5. | J. R. Pierce, “Optical Channels: Practical Limits with Photon Counting,” IEEE Trans. Commun. |

6. | S. Benedetto and E. Biglieri, |

7. | S. Betti, F. Curti, G. De Marchis, and E. Iannone, “A novel multilevel coherent optical system: 4-quadrature signaling,” J. Lightwave Technol. |

8. | R. L. Graham and N. J. A. Sloane, “Penny-packing and two-dimensional codes,” Discrete and Computational Geometry |

9. | N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-Energy Clusters of Hard Spheres,” Discrete and Computational Geometry |

10. | N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, “Minimal-Energy Clusters, Library of 3-d clusters, Library of 4-d clusters,” (1997). URL http://www.research.att.com/~njas/cluster/. |

11. | J. Kahn and K.-P. Ho, “Spectral efficiency limits and modulation/detection techniques for DWDM systems,” IEEE J. Sel. Top. Quantum Electron. |

12. | M. K. Simon, S. Hinedi, and W. C. Lindsey, |

13. | L. Xiao and X. Dong, “New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception,” IEEE Trans. Wireless Commun. |

14. | D. Saha and T. Birdsall, “Quadrature-quadrature phase-shift keying,” IEEE Trans. Commun. |

**OCIS Codes**

(060.4080) Fiber optics and optical communications : Modulation

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 3, 2009

Revised Manuscript: June 5, 2009

Manuscript Accepted: June 7, 2009

Published: June 12, 2009

**Citation**

Magnus Karlsson and Erik Agrell, "Which is the most power-efficient modulation format in optical links?," Opt. Express **17**, 10814-10819 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-13-10814

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### References

- S. Tsukamoto, D. Ly-Gagnon, K. Katoh, and K. Kikuchi, "Coherent demodulation of 40-Gbit/s polarizationmultiplexed QPSK signals with 16-GHz spacing after 200-km transmission," in Proceedings of Optical Fiber Communication and National Fiber Optic Engineers Conference, OFC/NFOEC, Vol. 6, p. PDP29 (2005).
- G. Charlet, M. Salsi, J. Renaudier, O. Pardo, H. Mardoyan, and S. Bigo, "Performance comparison of singlypolarised and polarisation-multiplexed coherent transmission at 10 Gbauds under linear impairments," Electron. Lett. 43(20), 1109-1111 (2007). [CrossRef]
- H. Sun, K. Wu, and K. Roberts, "Real-time measurements of a 40 Gb/s coherent system," Opt. Express 16(2), 873-879 (2008). [CrossRef]
- E. Ip, A. Lau, D. Barros, and J. Kahn, "Coherent detection in optical fiber systems," Opt. Express 16(2), 753-791 (2008). Erratum, Vol. 16, no. 26, p. 21943, (2008). [CrossRef]
- J. R. Pierce, "Optical Channels: Practical Limits with Photon Counting," IEEE Trans. Commun. 26(12), 1819- 1821 (1978). [CrossRef]
- S. Benedetto and E. Biglieri, Principles of Digital Transmission: With Wireless Applications (Kluwer Academic Publishers, 1999).
- S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: 4-quadrature signaling," J. Lightwave Technol. 9(4), 514-523 (1991). [CrossRef]
- R. L. Graham and N. J. A. Sloane, "Penny-packing and two-dimensional codes," Discrete and Computational Geometry 5(1), 1-11 (1990).
- N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters of Hard Spheres," Discrete and Computational Geometry 14(3), 237-259 (1995).
- N. J. A. Sloane, R. H. Hardin, T. S. Duff, and J. H. Conway, "Minimal-Energy Clusters, Library of 3-d clusters, Library of 4-d clusters," (1997). URLhttp://www.research.att.com/∼njas/cluster/.
- J. Kahn and K.-P. Ho, "Spectral efficiency limits and modulation/detection techniques for DWDM systems," IEEE J. Sel. Top. Quantum Electron. 10(2), 259-272 (2004).
- M. K. Simon, S. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection (Prentice-Hall, 1995).
- L. Xiao and X. Dong, "New analytical expressions for orthogonal, biorthogonal, and transorthogonal signaling in Nakagami fading channels with diversity reception," IEEE Trans.Wireless Commun. 4(4), 1418-1424 (2005). [CrossRef]
- D. Saha and T. Birdsall, "Quadrature-quadrature phase-shift keying," IEEE Trans. Commun. 37(5), 437-448 (1989). [CrossRef]

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