Abstract

With wavelength-division multiplexing (WDM) rapidly nearing its scalability limits, space-division multiplexing (SDM) seems the only option to further scale the capacity of optical transport networks. In order for SDM systems to continue the WDM trend of reducing energy and cost per bit with system capacity, integration will be key to SDM. Since integration is likely to introduce non-negligible crosstalk between multiple parallel transmission paths, multiple-input multiple output (MIMO) signal processing techniques will have to be used. In this paper, we discuss MIMO capacities in optical SDM systems, including related outage considerations which are an important part in the design of such systems. In order to achieve the low-outage standards required for optical transport networks, SDM transponders should be capable of individually addressing, and preferably MIMO processing all modes supported by the optical SDM waveguide. We then discuss the effect of distributed optical noise in MIMO SDM systems and focus on the impact of mode-dependent loss (MDL) on system capacity and system outage. Through extensive numerical simulations, we extract scaling rules for mode-average and mode-dependent loss and show that MIMO SDM systems composed of up to 128 segments and supporting up to 128 modes can tolerate up to 1 dB of per-segment MDL at 90% of the system’s full capacity at an outage probability of 10−4.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–10 (2010).
    [CrossRef]
  2. J. Gray and P. Shenoy, “Rules of thumb in data engineering,” Microsoft Research Technical Report MS-TR-99-100 (2000).
  3. J. L. Hennessy and D. A. Patterson, Computer Architectures: A Quantitative Approach (Morgan Kaufmann, 2003).
  4. http://top500.org/lists/2010/06/performance_development
  5. F. B. Shepherd and P. J. Winzer, “Selective randomized load balancing and mesh networks with changing demands,” J. Opt. Netw. 5(5), 320–339 (2006).
    [CrossRef]
  6. P. J. Winzer, “Beyond 100G ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
    [CrossRef]
  7. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
    [CrossRef]
  8. A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication (ECOC’09), plenary talk (2009).
  9. P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” in Optical Fiber Telecommunications V B , I. Kaminow, T. Li, and A. Willner (eds.), (Academic, 2008), ch. 2, pp. 23–94.
    [CrossRef]
  10. P. J. Winzer, “Modulation and multiplexing in optical communication systems,” IEEE-LEOS Newsletter , Feb.2009, http://photonicssociety.org/newsletters/feb09/modulation.pdf .
  11. P. J. Winzer, “Energy-efficient optical transport capacity scaling through spatial multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
    [CrossRef]
  12. R. J. Essiambre, “Impact of fiber parameters on nonlinear fiber capacity,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), OTuJ1 (2011).
  13. T. Morioka, “New generation optical infrastructure technologies: EXAT initiative towards 2020 and beyond,” Proc. Optoelectronics and Communications Conference (OECC’09), FT4 (2009).
  14. Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
    [CrossRef]
  15. C. R. Doerr and T. F. Taunay, “Silicon photonics core-, wavelength-, and polarization-diversity receiver,” IEEE Photon. Technol. Lett. 23(9), 597–599 (2011).
    [CrossRef]
  16. P. J. Winzer, A. H. Gnauck, A. Konczykowska, F. Jorge, and J.-Y. Dupuy, “Penalties from in-band crosstalk for advanced optical modulation formats,” Proc. European Conference on Optical Communication (ECOC’11), Tu.5.B.7 (2011).
  17. G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996).
    [CrossRef]
  18. J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7x97x172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB6 (2011).
  19. B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, F. V. Dimarcello, K. Abedin, P. W. Wisk, D. W. Peckham, and P. Dziedzic, “Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB7 (2011).
  20. R. Ryf, R.-J. Essiambre, S. Randel, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki, “MIMO-based crosstalk suppression in spatially multiplexed 56-Gb/s PDM-QPSK signals in strongly-coupled 3-core fiber,” accepted for publication in IEEE Photon. Technol. Lett. (2011).
  21. R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6x6 MIMO processing,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB10 (2011).
  22. M. Salsi, C. Koebele, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Transmission at 2x100Gb/s, over two modes of 40-km-long prototype few-mode fiber, using LCOS-based mode multiplexer and demultiplexer,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB9 (2011).
  23. A. Li, A. Al Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB8 (2011).
  24. S. Berdague and P. Facq, “Mode division multiplexing in optical fibers,” Appl. Opt. 21(11), 1950–1955 (1982).
    [CrossRef] [PubMed]
  25. H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
    [CrossRef] [PubMed]
  26. S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40(8), 1030–1036 (2008).
    [CrossRef]
  27. A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
    [CrossRef]
  28. S. Schoellmann, N. Schrammar, and W. Rosenkranz, “Experimental realisation of 3x3 MIMO system with mode group diversity multiplexing limited by modal noise,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’08), JWA68 (2008).
  29. M. Nazarathy and A. Agmon, “Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks,” J. Lightwave Technol. 26, 2037–2045 (2008).
    [CrossRef]
  30. B. Franz, D. Suikat, R. Dischler, F. Buchali, and H. Buelow, “High speed OFDM data transmission over 5 km GI-multimode fiber using spatial multiplexing with 2 × 4 MIMO processing,” Proc. European Conference on Optical Communication (ECOC’10), Tu.3.C.4 (2010).
  31. P. J. Winzer and G. J. Foschini, “Outage calculations for spatially multiplexed fiber links,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), OThO5 (2011).
  32. C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in long-haul transmission over bimodal optical fibre,” Proc. European Conference on Optical Communication (ECOC’10), Mo.2.C.6 (2010).
  33. H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun. 50(2), 225–234 (2002).
    [CrossRef]
  34. A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
    [CrossRef]
  35. H. Kogelnik, L. E. Nelson, and R. M. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IV B , I. P. Kaminow and T. Li (eds.), San Diego: Academic, ch. 15, 725–861 (2002).
  36. M. Brodsky, N. J. Frigo, and M. Tur, “Polarization mode dispersion,” in Optical Fiber Telecommunications V A , I. P. Kaminow, T. Li, and A. E. Willner (eds.), (Academic, 2008), ch. 17, pp. 605–670.
    [CrossRef]
  37. C. Xie, “Polarization-mode-dispersion impairments in 112-Gb/s PDM-QPSK coherent systems,” Proc. European Conference on Optical Communication (ECOC’10), Th.10.E.6 (2010).
  38. F. Mezzadri, “How to generate random matrices from the classical compact groups,” Notices of the AMS 54, 592–604 (2007).
  39. B. Wedding and C. N. Haslach, “Enhanced PMD mitigation by polarization scrambling and forward error correction,” Proc. Optical Fiber Communication Conference (OFC’01), WAA1 (2001).
  40. X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
    [CrossRef]
  41. M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16(18), 13918–13932 (2008).
    [CrossRef] [PubMed]
  42. A. Nafta, E. Meron, and M. Shtaif, “Capacity limitations in fiber-optic communications systems as a result of polarization dependent loss,” Opt. Lett. 34(23), 3613–3615 (2009).
    [CrossRef] [PubMed]
  43. E. Meron, A. Andrusier, M. Feder, and M. Shtaif, “Use of space-time coding in coherent polarization-multiplexed systems suffering from polarization dependent loss,” Opt. Lett. 35(21), 3547–3549 (2010).
    [CrossRef] [PubMed]
  44. A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
    [CrossRef]
  45. Y. Fukada, “Probability density function of polarization dependent loss (PDL) in optical transmission system composed of passive devices and connecting fibers,” J. Lightwave Technol. 20(6), 953–964 (2002).
    [CrossRef]
  46. M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
    [CrossRef]
  47. A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” J. Lightwave Technol. 22(8), 1856–1871 (2004).
    [CrossRef]
  48. L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19(7), 6790–6796 (2011)
    [CrossRef] [PubMed]
  49. A. Steinkamp, S. Vorbeck, and E. I. Voges, “Polarization mode dispersion and polarization dependent loss in optical fiber systems,” Proc. SPIE 5596, 243–254 (2004).
    [CrossRef]
  50. A. El Amari, N. Gisin, B. Perny, H. Zbinden, and C. W. Zimmer, “Statistical prediction and experimental verification of concatenations of fiber optic components with polarization dependent loss,” J. Lightwave Technol. 16(3), 332–339 (1998).
    [CrossRef]
  51. P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L.L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
    [CrossRef]
  52. A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic, 1979).
  53. J. B. Lasserre, “A trace inequality for matrix product,” IEEE Trans. Autom. Control 40(8), 1500–1501 (1995).
    [CrossRef]

2011

P. J. Winzer, “Energy-efficient optical transport capacity scaling through spatial multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
[CrossRef]

C. R. Doerr and T. F. Taunay, “Silicon photonics core-, wavelength-, and polarization-diversity receiver,” IEEE Photon. Technol. Lett. 23(9), 597–599 (2011).
[CrossRef]

L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19(7), 6790–6796 (2011)
[CrossRef] [PubMed]

2010

2009

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

A. Nafta, E. Meron, and M. Shtaif, “Capacity limitations in fiber-optic communications systems as a result of polarization dependent loss,” Opt. Lett. 34(23), 3613–3615 (2009).
[CrossRef] [PubMed]

2008

M. Shtaif, “Performance degradation in coherent polarization multiplexed systems as a result of polarization dependent loss,” Opt. Express 16(18), 13918–13932 (2008).
[CrossRef] [PubMed]

M. Brodsky, N. J. Frigo, and M. Tur, “Polarization mode dispersion,” in Optical Fiber Telecommunications V A , I. P. Kaminow, T. Li, and A. E. Willner (eds.), (Academic, 2008), ch. 17, pp. 605–670.
[CrossRef]

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” in Optical Fiber Telecommunications V B , I. Kaminow, T. Li, and A. Willner (eds.), (Academic, 2008), ch. 2, pp. 23–94.
[CrossRef]

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40(8), 1030–1036 (2008).
[CrossRef]

M. Nazarathy and A. Agmon, “Coherent transmission direct detection MIMO over short-range optical interconnects and passive optical networks,” J. Lightwave Technol. 26, 2037–2045 (2008).
[CrossRef]

2007

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

F. Mezzadri, “How to generate random matrices from the classical compact groups,” Notices of the AMS 54, 592–604 (2007).

2006

2005

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

2004

A. Steinkamp, S. Vorbeck, and E. I. Voges, “Polarization mode dispersion and polarization dependent loss in optical fiber systems,” Proc. SPIE 5596, 243–254 (2004).
[CrossRef]

A. Mecozzi and M. Shtaif, “Signal-to-noise-ratio degradation caused by polarization-dependent loss and the effect of dynamic gain equalization,” J. Lightwave Technol. 22(8), 1856–1871 (2004).
[CrossRef]

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
[CrossRef]

2003

J. L. Hennessy and D. A. Patterson, Computer Architectures: A Quantitative Approach (Morgan Kaufmann, 2003).

2002

H. Kogelnik, L. E. Nelson, and R. M. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IV B , I. P. Kaminow and T. Li (eds.), San Diego: Academic, ch. 15, 725–861 (2002).

H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun. 50(2), 225–234 (2002).
[CrossRef]

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
[CrossRef]

Y. Fukada, “Probability density function of polarization dependent loss (PDL) in optical transmission system composed of passive devices and connecting fibers,” J. Lightwave Technol. 20(6), 953–964 (2002).
[CrossRef]

M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
[CrossRef]

2000

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[CrossRef] [PubMed]

1998

1996

G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996).
[CrossRef]

1995

J. B. Lasserre, “A trace inequality for matrix product,” IEEE Trans. Autom. Control 40(8), 1500–1501 (1995).
[CrossRef]

1982

1979

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic, 1979).

Agmon, A.

Andrusier, A.

Antonelli, C.

Berdague, S.

Birk, M.

Bölcskei, H.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
[CrossRef]

H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun. 50(2), 225–234 (2002).
[CrossRef]

Brodsky, M.

M. Brodsky, N. J. Frigo, and M. Tur, “Polarization mode dispersion,” in Optical Fiber Telecommunications V A , I. P. Kaminow, T. Li, and A. E. Willner (eds.), (Academic, 2008), ch. 17, pp. 605–670.
[CrossRef]

Buhl, L.L.

Doerr, C. R.

C. R. Doerr and T. F. Taunay, “Silicon photonics core-, wavelength-, and polarization-diversity receiver,” IEEE Photon. Technol. Lett. 23(9), 597–599 (2011).
[CrossRef]

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L.L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[CrossRef]

El Amari, A.

Essiambre, R.-J.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[CrossRef]

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” in Optical Fiber Telecommunications V B , I. Kaminow, T. Li, and A. Willner (eds.), (Academic, 2008), ch. 2, pp. 23–94.
[CrossRef]

Facq, P.

Feder, M.

Foschini, G. J.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[CrossRef]

G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996).
[CrossRef]

Frigo, N. J.

M. Brodsky, N. J. Frigo, and M. Tur, “Polarization mode dispersion,” in Optical Fiber Telecommunications V A , I. P. Kaminow, T. Li, and A. E. Willner (eds.), (Academic, 2008), ch. 17, pp. 605–670.
[CrossRef]

Fukada, Y.

Gesbert, D.

H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun. 50(2), 225–234 (2002).
[CrossRef]

Giles, C. R.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Gisin, N.

Gnauck, A. H.

Goebel, B.

Gore, D. A.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
[CrossRef]

Grossman, B.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40(8), 1030–1036 (2008).
[CrossRef]

Hennessy, J. L.

J. L. Hennessy and D. A. Patterson, Computer Architectures: A Quantitative Approach (Morgan Kaufmann, 2003).

Hsu, R. C. J.

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

Jalali, B.

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

Jopson, R. M.

H. Kogelnik, L. E. Nelson, and R. M. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IV B , I. P. Kaminow and T. Li (eds.), San Diego: Academic, ch. 15, 725–861 (2002).

Kan, C.

M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
[CrossRef]

Kang, I.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Kao, Y.-H.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Kogelnik, H.

H. Kogelnik, L. E. Nelson, and R. M. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IV B , I. P. Kaminow and T. Li (eds.), San Diego: Academic, ch. 15, 725–861 (2002).

Kokubun, Y.

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

Koshiba, M.

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

Kramer, G.

Lasserre, J. B.

J. B. Lasserre, “A trace inequality for matrix product,” IEEE Trans. Autom. Control 40(8), 1500–1501 (1995).
[CrossRef]

Lewis, M.

M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
[CrossRef]

Liu, X.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Magarini, M.

Magill, P.

Marshall, A. W.

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic, 1979).

Mecozzi, A.

Meron, E.

Mezzadri, F.

F. Mezzadri, “How to generate random matrices from the classical compact groups,” Notices of the AMS 54, 592–604 (2007).

Moller, L.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Murshid, S.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40(8), 1030–1036 (2008).
[CrossRef]

Nabar, R. U.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
[CrossRef]

Nafta, A.

Narakorn, P.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40(8), 1030–1036 (2008).
[CrossRef]

Nazarathy, M.

Nelson, L. E.

L. E. Nelson, C. Antonelli, A. Mecozzi, M. Birk, P. Magill, A. Schex, and L. Rapp, “Statistics of polarization dependent loss in an installed long-haul WDM system,” Opt. Express 19(7), 6790–6796 (2011)
[CrossRef] [PubMed]

H. Kogelnik, L. E. Nelson, and R. M. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IV B , I. P. Kaminow and T. Li (eds.), San Diego: Academic, ch. 15, 725–861 (2002).

Olkin, I.

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic, 1979).

Patterson, D. A.

J. L. Hennessy and D. A. Patterson, Computer Architectures: A Quantitative Approach (Morgan Kaufmann, 2003).

Paulraj, A. J.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
[CrossRef]

H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun. 50(2), 225–234 (2002).
[CrossRef]

Perny, B.

Rapp, L.

Sayed, A. H.

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

Schex, A.

Shah, A. R.

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

Shepherd, F. B.

Shtaif, M.

Sizmann, A.

M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
[CrossRef]

Steinkamp, A.

A. Steinkamp, S. Vorbeck, and E. I. Voges, “Polarization mode dispersion and polarization dependent loss in optical fiber systems,” Proc. SPIE 5596, 243–254 (2004).
[CrossRef]

Stuart, H. R.

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[CrossRef] [PubMed]

Tarighat, A.

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

Taunay, T. F.

C. R. Doerr and T. F. Taunay, “Silicon photonics core-, wavelength-, and polarization-diversity receiver,” IEEE Photon. Technol. Lett. 23(9), 597–599 (2011).
[CrossRef]

Tkach, R. W.

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–10 (2010).
[CrossRef]

Tur, M.

M. Brodsky, N. J. Frigo, and M. Tur, “Polarization mode dispersion,” in Optical Fiber Telecommunications V A , I. P. Kaminow, T. Li, and A. E. Willner (eds.), (Academic, 2008), ch. 17, pp. 605–670.
[CrossRef]

van Wijngaarden, A. J.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Voges, E. I.

A. Steinkamp, S. Vorbeck, and E. I. Voges, “Polarization mode dispersion and polarization dependent loss in optical fiber systems,” Proc. SPIE 5596, 243–254 (2004).
[CrossRef]

Vorbeck, S.

A. Steinkamp, S. Vorbeck, and E. I. Voges, “Polarization mode dispersion and polarization dependent loss in optical fiber systems,” Proc. SPIE 5596, 243–254 (2004).
[CrossRef]

Wei, X.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Winzer, P. J.

P. J. Winzer, “Energy-efficient optical transport capacity scaling through spatial multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
[CrossRef]

P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L.L. Buhl, “Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM,” J. Lightwave Technol. 28(4), 547–556 (2010).
[CrossRef]

P. J. Winzer, “Beyond 100G ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
[CrossRef]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. 28(4), 662–701 (2010).
[CrossRef]

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” in Optical Fiber Telecommunications V B , I. Kaminow, T. Li, and A. Willner (eds.), (Academic, 2008), ch. 2, pp. 23–94.
[CrossRef]

F. B. Shepherd and P. J. Winzer, “Selective randomized load balancing and mesh networks with changing demands,” J. Opt. Netw. 5(5), 320–339 (2006).
[CrossRef]

Xie, C.

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

Yu, M.

M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
[CrossRef]

Zbinden, H.

Zimmer, C. W.

Appl. Opt.

Bell Labs Tech. J.

G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J. 1(2), 41–59 (1996).
[CrossRef]

R. W. Tkach, “Scaling optical communications for the next decade and beyond,” Bell Labs Tech. J. 14(4), 3–10 (2010).
[CrossRef]

IEEE Commun. Mag.

A. Tarighat, R. C. J. Hsu, A. R. Shah, A. H. Sayed, and B. Jalali,“Fundamentals and challenges of optical multiple-input-multiple-output multimode fiber links,” IEEE Commun. Mag. , 57–63 (2007).
[CrossRef]

P. J. Winzer, “Beyond 100G ethernet,” IEEE Commun. Mag. 48(7), 26–30 (2010).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Mecozzi and M. Shtaif, “The statistics of polarization-dependent loss in optical communication systems,” IEEE Photon. Technol. Lett. 14(3), 313–315 (2002).
[CrossRef]

M. Yu, C. Kan, M. Lewis, and A. Sizmann, “Statistics of polarization-dependent loss, insertion loss, and signal power in optical communication systems,” IEEE Photon. Technol. Lett. 14(12), 1695–1697 (2002).
[CrossRef]

X. Liu, C. R. Giles, X. Wei, A. J. van Wijngaarden, Y.-H. Kao, C. Xie, L. Moller, and I. Kang, “Demonstration of broad-band PMD mitigation in the presence of PDL through distributed fast polarization scrambling and forward-error correction,” IEEE Photon. Technol. Lett. 17(5), 1109–1111 (2005).
[CrossRef]

C. R. Doerr and T. F. Taunay, “Silicon photonics core-, wavelength-, and polarization-diversity receiver,” IEEE Photon. Technol. Lett. 23(9), 597–599 (2011).
[CrossRef]

P. J. Winzer, “Energy-efficient optical transport capacity scaling through spatial multiplexing,” IEEE Photon. Technol. Lett. 23(13), 851–853 (2011).
[CrossRef]

IEEE Trans. Autom. Control

J. B. Lasserre, “A trace inequality for matrix product,” IEEE Trans. Autom. Control 40(8), 1500–1501 (1995).
[CrossRef]

IEEE Trans. Commun.

H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDM-based spatial multiplexing systems,” IEEE Trans. Commun. 50(2), 225–234 (2002).
[CrossRef]

IEICE Electron. Express

Y. Kokubun and M. Koshiba, “Novel multi-core fibers for mode division multiplexing: proposal and design principle,” IEICE Electron. Express 6(8), 522–528 (2009).
[CrossRef]

J. Lightwave Technol.

J. Opt. Netw.

Notices of the AMS

F. Mezzadri, “How to generate random matrices from the classical compact groups,” Notices of the AMS 54, 592–604 (2007).

Opt. Express

Opt. Laser Technol.

S. Murshid, B. Grossman, and P. Narakorn, “Spatial domain multiplexing: a new dimension in fiber optic multiplexing,” Opt. Laser Technol. 40(8), 1030–1036 (2008).
[CrossRef]

Opt. Lett.

Proc. IEEE

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bölcskei, “An overview of MIMO communications—a key to Gigabit wireless,” Proc. IEEE 92(2), 198 –218 (2004).
[CrossRef]

Proc. SPIE

A. Steinkamp, S. Vorbeck, and E. I. Voges, “Polarization mode dispersion and polarization dependent loss in optical fiber systems,” Proc. SPIE 5596, 243–254 (2004).
[CrossRef]

Science

H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000).
[CrossRef] [PubMed]

Other

P. J. Winzer, A. H. Gnauck, A. Konczykowska, F. Jorge, and J.-Y. Dupuy, “Penalties from in-band crosstalk for advanced optical modulation formats,” Proc. European Conference on Optical Communication (ECOC’11), Tu.5.B.7 (2011).

J. Gray and P. Shenoy, “Rules of thumb in data engineering,” Microsoft Research Technical Report MS-TR-99-100 (2000).

J. L. Hennessy and D. A. Patterson, Computer Architectures: A Quantitative Approach (Morgan Kaufmann, 2003).

http://top500.org/lists/2010/06/performance_development

R. J. Essiambre, “Impact of fiber parameters on nonlinear fiber capacity,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), OTuJ1 (2011).

T. Morioka, “New generation optical infrastructure technologies: EXAT initiative towards 2020 and beyond,” Proc. Optoelectronics and Communications Conference (OECC’09), FT4 (2009).

A. R. Chraplyvy, “The coming capacity crunch,” European Conference on Optical Communication (ECOC’09), plenary talk (2009).

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” in Optical Fiber Telecommunications V B , I. Kaminow, T. Li, and A. Willner (eds.), (Academic, 2008), ch. 2, pp. 23–94.
[CrossRef]

P. J. Winzer, “Modulation and multiplexing in optical communication systems,” IEEE-LEOS Newsletter , Feb.2009, http://photonicssociety.org/newsletters/feb09/modulation.pdf .

J. Sakaguchi, Y. Awaji, N. Wada, A. Kanno, T. Kawanishi, T. Hayashi, T. Taru, T. Kobayashi, and M. Watanabe, “109-Tb/s (7x97x172-Gb/s SDM/WDM/PDM) QPSK transmission through 16.8-km homogeneous multi-core fiber,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB6 (2011).

B. Zhu, T. F. Taunay, M. Fishteyn, X. Liu, S. Chandrasekhar, M. F. Yan, J. M. Fini, E. M. Monberg, F. V. Dimarcello, K. Abedin, P. W. Wisk, D. W. Peckham, and P. Dziedzic, “Space-, wavelength-, polarization-division multiplexed transmission of 56-Tb/s over a 76.8-km seven-core fiber,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB7 (2011).

R. Ryf, R.-J. Essiambre, S. Randel, A. H. Gnauck, P. J. Winzer, T. Hayashi, T. Taru, and T. Sasaki, “MIMO-based crosstalk suppression in spatially multiplexed 56-Gb/s PDM-QPSK signals in strongly-coupled 3-core fiber,” accepted for publication in IEEE Photon. Technol. Lett. (2011).

R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, R.-J. Essiambre, P. J. Winzer, D. W. Peckham, A. McCurdy, and R. Lingle, “Space-division multiplexing over 10 km of three-mode fiber using coherent 6x6 MIMO processing,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB10 (2011).

M. Salsi, C. Koebele, D. Sperti, P. Tran, P. Brindel, H. Mardoyan, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Astruc, L. Provost, F. Cerou, and G. Charlet, “Transmission at 2x100Gb/s, over two modes of 40-km-long prototype few-mode fiber, using LCOS-based mode multiplexer and demultiplexer,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB9 (2011).

A. Li, A. Al Amin, X. Chen, and W. Shieh, “Reception of mode and polarization multiplexed 107-Gb/s CO-OFDM signal over a two-mode fiber,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), PDPB8 (2011).

A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and its Applications (Academic, 1979).

B. Wedding and C. N. Haslach, “Enhanced PMD mitigation by polarization scrambling and forward error correction,” Proc. Optical Fiber Communication Conference (OFC’01), WAA1 (2001).

H. Kogelnik, L. E. Nelson, and R. M. Jopson, “Polarization mode dispersion,” in Optical Fiber Telecommunications IV B , I. P. Kaminow and T. Li (eds.), San Diego: Academic, ch. 15, 725–861 (2002).

M. Brodsky, N. J. Frigo, and M. Tur, “Polarization mode dispersion,” in Optical Fiber Telecommunications V A , I. P. Kaminow, T. Li, and A. E. Willner (eds.), (Academic, 2008), ch. 17, pp. 605–670.
[CrossRef]

C. Xie, “Polarization-mode-dispersion impairments in 112-Gb/s PDM-QPSK coherent systems,” Proc. European Conference on Optical Communication (ECOC’10), Th.10.E.6 (2010).

S. Schoellmann, N. Schrammar, and W. Rosenkranz, “Experimental realisation of 3x3 MIMO system with mode group diversity multiplexing limited by modal noise,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’08), JWA68 (2008).

B. Franz, D. Suikat, R. Dischler, F. Buchali, and H. Buelow, “High speed OFDM data transmission over 5 km GI-multimode fiber using spatial multiplexing with 2 × 4 MIMO processing,” Proc. European Conference on Optical Communication (ECOC’10), Tu.3.C.4 (2010).

P. J. Winzer and G. J. Foschini, “Outage calculations for spatially multiplexed fiber links,” Proc. Optical Fiber Communications Conference (OFC/NFOEC’11), OThO5 (2011).

C. Koebele, M. Salsi, G. Charlet, and S. Bigo, “Nonlinear effects in long-haul transmission over bimodal optical fibre,” Proc. European Conference on Optical Communication (ECOC’10), Mo.2.C.6 (2010).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

(a) Traffic growth in US data networks [1] and evolution of supercomputer processing power [4], dictating machine-to-machine traffic growth. (b) Experimentally achieved WDM spectral efficiencies (blue) have approached the fundamental Shannon limit (red) to within a factor of 2 (Fig. after [11]).

Fig. 2
Fig. 2

Spatial multiplexing exploits the only known physical dimension that has not yet been used in optical transport systems. Implementations include fiber bundles, multi-core, and multi-mode fiber (Fig. after [10]).

Fig. 3
Fig. 3

Spatial multiplexing uses M parallel, integrated transmission paths to increase system capacity by a factor of M. Integration of system components is key for sustainable scalability.

Fig. 4
Fig. 4

(a) Basic MIMO system model for MT = 3, MR = 2, and M = 4. (b) Histogram of MIMO capacities C , normalized to the single-mode capacity CS . The shaded area represents the probability that C is smaller than CT , leading to system outage if the system is designed to code for capacity CT .

Fig. 5
Fig. 5

(a) Achievable SDM capacities CT normalized to the single-mode capacity CS at a given outage probability for different combinations of transmit × receive modes on a waveguide supporting M = 4 modes. (b) Impact of mode-average SNR on the outage performance of an under-addressed 4-mode waveguide (dashed). Also shown are the step-like outage curves for a highly frequency-selective channel (dotted). (c) Outage performance of systems using more than one fixed (dashed) combination of mode sets (shaded areas); solid curves: dynamic switching among all possible mode set combinations; dotted lines: highly frequency-selective channel.

Fig. 6
Fig. 6

(a) Achievable SDM capacity CT relative to the single-mode capacity CS at 10−4 outage and 20 dB SNR, as a function of the number of modes MTR processed by a fixed mode set transponder. The number of modes M supported by the waveguide parameterizes the curves. (b) Normalized version of (a). (c) Capacity gain of dynamically mode set switching transponders on a waveguide supporting M = 8 modes.

Fig. 7
Fig. 7

System models for a MIMO channel with distributed noise.

Fig. 8
Fig. 8

Achievable SDM capacities CT normalized to M times the single-mode capacity CS at a given outage probability for K = 64 segments and M = 16 modes, with the per-segment MDL as a parameter. The SNR is varied from 10 to 40 dB among the four subplots. Solid red curves pertain to noise loading at the receiver, while dashed blue curves represent distributed noise loading. Dotted curves for MDL S of 5 and 2 dB are the capacities of the highly frequency-selective channel.

Fig. 9
Fig. 9

System capacity at 10−4 outage probability as a function of per-segment MDL and number of segments K. (a) Systems with M = 32 modes and 20 dB SNR. Red circles and blue squares are numerically exact results for receive-side and distributed noise loading, respectively. Solid red and dashed blue lines represent the Gaussian approximation with numerically exact moments. Dotted black lines use approximate moments. (b) Systems with M = {8, 16, 32, 128} modes and K = {2, 16, 64, 128} segments, represented by the Gaussian approximations with exact moments for 20 dB SNR and noise loading at the receiver.

Fig. 10
Fig. 10

System capacity at 10−4 outage probability as a function of the aggregate system MDL with the number of segments K and modes M as parameters.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

y = E 0 L H x + n ,
L = 1 M tr { H ˜ H ˜ } = 1 M i = 1 M λ ˜ i ;
C = i = 1 r log 2 ( 1 + λ i E 0 L N 0 ) ,
SNR = E 0 L N 0
C r C S = r log 2 ( 1 + SNR ) ,
P out = 0 C T p C ( C ) d C ,
H 1 = ( 4 / 3 0 0 2 / 3 ) and H 2 = ( 2 / 3 0 0 4 / 3 ) .
R n = n n = ( N 1 H K H 3 H 2 H 2 H 3 H K + N 2 H K H 3 H 3 H K + + N K 1 H K H K + N K I M ) ,
R n = i = 1 K N i I M = N 0 I M .
y = E 0 L H 0 x + G n 0 ,
y = E 0 L G 1 H 0 x + n 0 ,
MDL S = max { v k k } min { v k k } .
L Σ 1 ,
σ L Σ α K M MDL S 2 ,
MDL Σ [ dB ] K MDL S [ dB ] .
C n β K MDL S [ dB ] 2 ,
σ C n γ K M MDL S [ dB ] 2 ,
P out ( C T ) = 0.5 erfc ( C C T 2 σ C ) ,
C r log 2 ( 1 + SNR M / r ) .
C r log 2 ( 1 + SNR ) = r C S .
tr ( HH ) = i = 1 r λ i = μ M .
C = r log 2 ( 1 + SNR μ / r ) .
tr { HH } = tr { S R U S T S T U S R } = tr { S R U S T U } = tr { S R D } i = 1 M λ i ( S R ) λ i ( D ) = min { M T , M R } .

Metrics