TITAN Login

Events Calendar

« < May 2020 > »
S M T W T F S
26 27 28 29 30 1 2
3 4 5 6 7 8 9
10 11 12 13 14 15 16
17 18 19 20 21 22 23
24 25 26 27 28 29 30
31 1 2 3 4 5 6
« < June 2020 > »
S M T W T F S
31 1 2 3 4 5 6
7 8 9 10 11 12 13
14 15 16 17 18 19 20
21 22 23 24 25 26 27
28 29 30 1 2 3 4
No events
mod_vvisit_countermod_vvisit_countermod_vvisit_countermod_vvisit_countermod_vvisit_countermod_vvisit_countermod_vvisit_countermod_vvisit_counter
mod_vvisit_counterThis week8702
mod_vvisit_counterThis month100000
You are connecting to this web site from: 3.230.154.129

PollXT-List

Emanuele Viterbo Profile Page
Emanuele Viterbo
Emanuele Viterbo
Full Professor
DEIS- 42D 1° Floor
University of Calabria
via P. Bucci, 42
Rende (CS)
87036
ITALY
+39 0984 494748
+39 0984 494041
viterbo(at)deis.unical.it

Short Bio

Emanuele Viterbo was born in Torino, Italy, in 1966. He received his degree (Laurea) in Electrical Engineering in 1989 and his Ph.D. in 1995 in Electrical Engineering, both from the Politecnico di Torino, Torino, Italy.


From 1990 to 1992 he was with the European Patent Office, The Hague, The Netherlands, as a patent examiner in the field of dynamic recording and error-control coding. Between 1995 and 1997 he held a post-doctoral position in the Dipartimento di Elettronica of the Politecnico di Torino in Communications Techniques over Fading Channels. He became Associate Professor at Politecnico di Torino, Dipartimento di Elettronica in 2005 and since November 2006 he is Full Professor in Dipartimento di Elettronica, Informatica e Sistemistica (DEIS) at Universita' della Calabria, Italy.
In 1993 he was visiting researcher in the Communications Department of DLR, Oberpfaffenhofen, Germany. In 1994 and 1995 he was visiting the �cole Nationale Sup�rieure des T�l�communications (E.N.S.T.), Paris. In 1998 he was visiting researcher in the Information Sciences Research Center of AT&T Research, Florham Park, NJ. In 2003 he was visiting researcher at the Maths Department of EPFL, Lausanne, Switzerland. In 2004 he was visiting researcher at the Telecommunications Department of UNICAMP, Campinas, Brazil. In 2005 he was visiting researcher at the ITR of UniSA, Adelaide, Australia.

Dr. Emanuele Viterbo was awarded a NATO Advanced Fellowship in 1997 from the Italian National Research Council. His main research interests are in lattice codes for the Gaussian and fading channels, algebraic coding theory, algebraic space-time coding, digital terrestrial television broadcasting, and digital magnetic recording.

He is Associate Editor of
IEEE Transactions on Information Theory, European Transactions on Telecommunications and Journal of Communications and Networks.

Research interests


alt
alt alt alt alt alt alt alt alt

Frrique� Oggier, Jean-Claude Belfiore and Emanuele Viterbo (2007)
"Cyclic Division Algebras: A Tool for Space-Time Coding",
Foundations and Trends� in Communications and Information Theory: Vol. 4: No 1, pp 1-95.
A printed and bound version of this article is available at a 50% discount from Now Publishers.
This can be obtained by entering the promotional code MC01016 on the
order form at now publishers. You will then pay $40/Euro 40 plus postage.

alt
alt alt alt alt alt alt alt alt

Publications

Publications of Emanuele Viterbo

(last update May 2009)


Journals:

  1. Y. Hong and E. Viterbo: "Algebraic multiuser space-time block codes for 2x2 MIMO", IEEE Transactions on Vehicular Technology , to appear 2009.
  2. E. Viterbo and Y. Hong: "On the performance of Golden space-time trellis coded modulation over MIMO block fading channels", IEEE Transactions on Wireless Communications , June, 2009.
  3. A. Nordio, C.-F. Chiasserini, E.Viterbo, The Impact of Quasi-equally Spaced Sensor Topologies on Signal Reconstruction, ACM Transactions on Sensor Networks, 2009.
  4. L. Luzzi, G. Rekaya-Ben Othman, J.-C. Belfiore and E. Viterbo, "Golden Space-Time Block Coded Modulation", IEEE Transactions on Information Theory, pp. 584-597, vol. 55, n. 2, Feb. 2009.
  5. E. Biglieri, Yi Hong, E. Viterbo: "On Fast Decodable Space-Time Block Codes", IEEE Transactions on Information Theory, pp. 524-530, vol. 55, n. 2, Feb. 2009.
  6. B. Cerato, G. Masera, and E. Viterbo: "Decoding the Golden Code: A VLSI Design", IEEE Transactions on VLSI, vol. 17, n. 1, pp. 156-160, Jan. 2009, doi: 10.1109/TVLSI.2008.2003163.
  7. A. Nordio, C.-F. Chiasserini, E. Viterbo: "Performance of Field Reconstruction Techniques with Noise and Uncertain Sensor Locations", IEEE Transactions on Signal Processing , vol. 56, n. 9, pp. 4274-4285, Sept. 2008.
  8. A. Nordio, C.-F. Chiasserini, E. Viterbo: "Reconstruction of Multidimensional Signals from Irregular Noisy Samples", IEEE Transactions on Signal Processing , vol. 56, n. 8, pp. 3535-3547, Aug. 2008.
  9. B. Cerato, G. Masera, and E. Viterbo: "Decoding the Golden space-time trellis coded modulation", IEEE Communication Letters, vol. 12, n. 8, pp. 569-571, Aug. 2008.
  10. P. Pace and E. Viterbo: "Fast and accuate video PQoS estimation over wireless networks", EURASIP Journal on Advances in Signal Processing - Special Issue on Wireless Video, vol. 2008, Article ID 548741, 10 pages, 2008. doi:10.1155/2008/548741
  11. B. Cerato, G. Masera, and E. Viterbo: "Enabling VLSI Processing Blocks for MIMO-OFDM Communications", VLSI Design, vol. 2008, Article ID 351962, 10 pages, 2008. doi:10.1155/2008/351962
  12. A. Guillen i Fabregas and E. Viterbo: "Sphere Lower Bound for Rotated Lattice Constellations in Fading Channels", IEEE Transactions on Wireless Communications , vol. 7, no. 3, pp. 825-830, May 2008.
  13. E. Natalizio, V. Loscri, and E. Viterbo: "Optimal Placement of Wireless Nodes for Maximizing Path Lifetime", IEEE Communication Letters , vol. 12, no. 5, pp. 362-364, May 2008.
  14. Y. Hong, E. Viterbo, and J.-C. Belifiore: "Golden space-time trellis coded modulation", IEEE Transactions on Information Theory, vol. 53, no. 5, pp. 1689 - 1705, May 2007.
  15. F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo: "Perfect Space-Time Block Codes", IEEE Transactions on Information Theory, vol. 52, n. 9, pp. 3885-3902, Sept. 2006.
  16. E. Bayer-Fluckiger, F. Oggier, E. Viterbo: "Algebraic Lattice Constellations: bounds on performance", IEEE Transactions on Information Theory, vol. 52, n. 1, pp. 319-327, Jan. 2006.
  17. J.-C. Belfiore, G. Rekaya, E. Viterbo: "The Golden Code: A 2x2 Full-Rate Space-Time Code With Nonvanishing Determinants", IEEE Transactions on Information Theory, vol. 51, n. 4, pp. 1432-1436, Apr. 2005.
  18. R. Gaudino and E. Viterbo: "Transmitter optimization and theoretical bounds for dispersion-limited optical fiber links", IEEE Transactions on Communications, vol. 52, n. 9, pp. 1558-1565, Sept. 2004.
  19. E. Bayer-Fluckiger, F. Oggier, E. Viterbo: "New Algebraic Constructions of Rotated Z^n-Lattice Constellations for the Rayleigh Fading Channel", IEEE Transactions on Information Theory, vol. 50, n. 4, pp. 702-714, Apr. 2004.
  20. R. Gaudino and E. Viterbo: "Pulse Shape Optimization in Dispersion-Limited Direct Detection Optical Fiber Links", IEEE Communication Letters, vol. 7, n. 11, pp. 552-554, Nov. 2003.
  21. M. Meo and E. Viterbo: "Performance of Wideband CDMA Systems Supporting Multimedia Traffic", IEEE Communication Letters, vol. 5, n. 6, pp. 251-253, June 2001.
  22. E. Biglieri, G. Caire, G. Taricco, and E. Viterbo: "How fading Affects CDMA: An Asymptotic Analysis with Linear Receivers", IEEE Journal on Selected areas in Communications, vol. 19, n. 2, pp. 191-201, Feb. 2001.
  23. G. Caire, E. Leonardi, and E. Viterbo: "Modulation and Coding for the Gaussian Collision Channel", IEEE Transactions on Information Theory, vol. 46, n. 6, pp. 2007-2026, Sept. 2000.
  24. M. Greferath and E. Viterbo: "On Z4- and Z9-linear lifts of the Golay codes", IEEE Transactions on Information Theory, vol. 45, n. 7, pp. 2524-2527, Nov. 1999.
  25. M. Elia, E. Viterbo, and G. Bertinetti: "Decoding of binary separable Goppa codes using Berlekamp-Massey algorithm", Electronics Letters, Vol. 35, No. 20, pp. 1720-1721, 30th September 1999.
  26. M. Elia, G. Taricco, and E. Viterbo: "Optimal Energy Transfer over Bandlimited Communication Channels", IEEE Transactions on Information Theory, vol. 45, n. 6, pp. 2020-2029, Sept. 1999.
  27. E. Biglieri, J. K. Karlof, and E. Viterbo: "Representing Group Codes as Permutation Codes", IEEE Transactions on Information Theory, vol. 45, n. 6, pp. 2204-2207, Sept. 1999.
  28. E. Viterbo and J. Boutros: "A Universal Lattice Code Decoder for Fading Channels", IEEE Transactions on Information Theory, vol. 45, n. 5, pp. 1639-1642, July 1999.
  29. J. Boutros and E. Viterbo: "Signal Space Diversity: a power and bandwidth efficient diversity technique for the Rayleigh fading channel", IEEE Transactions on Information Theory, vol. 44, n. 4, pp. 1453-1467, July 1998.
  30. G. Taricco and E. Viterbo: "Performance of High Diversity Multidimensional Constellations", IEEE Transactions on Information Theory, vol. 44, n. 4, pp. 1539-1543, July 1998.
  31. E. Viterbo: "The Ciphered Autobiography of an 19th Century Egyptologist", CRYPTOLOGIA, vol. XXII, n. 3, pp. 231-243, July 1998.
  32. G. Caire and E. Viterbo: "Upper Bound to the Frame Error Probability of Terminated Trellis Codes", IEEE Communication Letters, vol. 2, n. 1, Jan. 1998.
  33. G. Taricco and E. Viterbo: "Performance of component-interleaved signal sets for fading channels", Electronics Letters, Vol. 32, No. 13, pp. 1170-1172, 20th June 1996.
  34. J. Boutros, E. Viterbo, C. Rastello, and J.C. Belfiore: "Good Lattice Constellations for both Rayleigh Fading and Gaussian Channels", IEEE Transactions on Information Theory, vol. 42, n. 2, pp. 502-518, March 1996.
  35. E. Viterbo and E. Biglieri: "Computing the Voronoi cell of a lattice: The diamond-cutting algorithm", IEEE Transactions on Information Theory, vol. 42, n. 1, pp. 161-171, Jan. 1996.
  36. E. Biglieri, E. Chiaberto, G. P. Maccone, and E. Viterbo: "Compensation of Nonlinearities in High-Density Magnetic Recording Channels", IEEE Transactions on Magnetics, vol. 30, n. 6, pp. 5079-5086, Nov. 1994.
  37. M. Elia and E. Viterbo: "Algebraic decoding of the ternary (11,6,5) Golay Code", Electronics Letters, Vol. 28, No. 21, pp. 2021-2022, 8th October 1992.
  38. V. Daniele, M. Gilli, and E. Viterbo: "Diffraction of a plane wave by a strip grating", Electromagnetics, n. 10, pp. 245-269, 1990.

Conference papers:

  1. A. Nordio, C.-F. Chiasserini, and E. Viterbo, Linear Field Reconstruction from Jittered Sampling, 8th international conference on Sampling Theory and Applications, (SampTA), Marseille, France, May 18-22, 2009
  2. E. Viterbo: "Irregular Sampling and Random Matrix Theory", JWCC 2008, Napa Valley, CA, Oct. 2008.
  3. Y. Hong and E. Viterbo: "Algebraic multiuser space-time block codes for 2x2 MIMO", invited paper at PIMRC 2008, Cannes, France, Sept. 2008.
  4. A. Nordio, C.-F. Chiasserini, and E. Viterbo: "On quasi-equally spaced sampling in wireless sensor networks", invited paper at PIMRC 2008, Cannes, France, Sept. 2008.
  5. V. Loscri', E. Natalizio, E. Viterbo, D. Mauro, G. D'Aquila, G. Brasili: "Carrier Independent Localization Techniques for GSM Terminals", PIMRC 2008, Cannes, France, Sept. 2008.
  6. Y. Hong and E. Viterbo: "Algebraic multiuser space-frequency block codes", ISSSTA 2008, Bologna, Italy, Aug. 2008.
  7. E. Viterbo and A. Hottinen: "Optimal user pairing for multiuser MIMO", ISSSTA 2008, Bologna, Italy, Aug. 2008.
  8. E. Biglieri, Y. Hong and E. Viterbo: "Silver space-time trellis coded modulation", EUSIPCO 2008, Lausanne, Switzerland, Aug 2008.
  9. Y. Hong, S. Shamai (Shitz), and E. Viterbo: "Algebraic-phase scrambling sequences for Code spread code division multiple access", ISIT 2008, Toronto, Canada, July 2008.
  10. J. Boutros, E. Viterbo, G.Cohen: "Convolutional Tanner Structures for Non-Ergodic Wireless Channels", ISIT 2008, Toronto, Canada, July 2008.
  11. P. Pace, M. Belcastro, E. Viterbo: "Fast and accurate PQoS estimation over 802.11g wireless networks", Proceedings of ICC 2008, Beijing China, May 2008.
  12. L. Luzzi, G. Rekaya, J.-C. Belfiore, and E. Viterbo: "Golden space-time block coded modulation", IEEE Information Theory Workshop 2008, Porto, May 2008.
  13. C. Hollanti, J. Lahtonen, K. Ranto, R. Vehkalahti, and E. Viterbo: "On the algebraic structure of a Silver code: A 2x2 perfect space-time block code", IEEE Information Theory Workshop 2008, Porto, May 2008.
  14. E. Biglieri, Y. Hong, and E. Viterbo : "On fast-decodable space-time block codes", International Zurich Seminar 2008, Zurich, Switzerland, pp. 116-119, March 2008.
  15. A. Nordio, C.-F. Chiasserini, and E. Viterbo: "Signal reconstruction in multidimensional sensor fields", International Zurich Seminar 2008, Zurich, Switzerland, pp. 56-59, March 2008.
  16. E. Viterbo: Timing Errors on Distributed Space-Time Communications, JWCC 2007, Duernstein, Austria, Oct. 2007.
  17. E. Biglieri, Y. Hong, E. Viterbo: "A Fast-Decodable, Quasi-Orthogonal Space-Time Block Code for 4x2 MIMO", Allerton Conference (2007), Monticello, IL, pp. 921-926, September 2007
  18. E. Viterbo, Y. Hong: "Robust codes for 2x2 MIMO Block Fading Channels", International Symposium on Information Theory (ISIT 2007), Nice, France, pp. 2621-2625, June 2007
  19. A. Nordio, C.-F. Chiasserini, and E. Viterbo: "The impact of quasi-equally spaced sensor layouts on field reconstruction", International Symposium on Information Processing in Sensor Networks 2007, Cambridge, Massachusset, pp. 274-282, April 25-27, 2007.
  20. A. Nordio, C.-F. Chiasserini, and E.Viterbo "Quality of field reconstruction in sensor networks", IEEE INFOCOM Mini-Symposium, Anchorage, AK, May 2007.
  21. A. Hottinen, Y. Hong, E. Viterbo, C. Mecklenbr�uker, C. Mehlf�hrer: "A comparison of high rate algebraic and non-orthogonal STBCs", WSA 2007, pp.1-5, Vienna, Austria, Feb. 2007.
  22. E. Viterbo, Y. Hong and Alex Grant: "Timing Errors on Distributed Space-Time Communications", AusCTW 2007 , Adelaide, Australia, Feb. 2007.
  23. E. Viterbo, Y. Hong: "Applications of the Golden Code ", Invited paper at Information Theory and Application (ITA 2007), UCSD, San Diego, USA, Jan. 2007
  24. B. Cerato, G. Masera, E. Viterbo: "A VLSI decoder for the Golden code", 13th IEEE International Conference on Electronics, Circuits and Systems ICECS 2006 Conference, Nice, Dec. 10-13, 2006.
  25. Y. Hong, E. Viterbo, J.-C. Belfiore: "Golden Space-Time Trellis Coded Modulation" , ISITA 2006 , Seoul, Korea, pp. 848-853, Nov. 2006.
  26. Y. Hong, E. Viterbo, and J.-C. Belfiore: "High rate Golden space-time trellis coded modulation for MIMO fading channels", Invited paper at Asilomar 2006 Conference, USA, 2006.
  27. Y. Hong, E. Viterbo, J.-C. Belfiore: "High Data Rate Trellis Coded Modulation" , NEWCOM-ACoRN Joint Workshop 2006 , Vienna, Austria, 20-22 Sept. 2006.
  28. Y. Hong, E. Viterbo, J.-C. Belfiore: "A Space-Time Block Coded Multiuser MIMO Downlink Transmission Scheme", IEEE International Symposium on Information Theory , Seattle, USA, pp. 257-261, July 2006.
  29. Albert Guillen i Fabregas, E. Viterbo: "Performance of Rotated Lattice Constellations in Fading Channels", IEEE International Symposium on Information Theory , Seattle, USA, pp. 1046-1050, July 2006.
  30. J.-C. Belfiore, E. Viterbo: "Approximating the error probability for the independent Rayleigh fading channels" , IEEE International Symposium on Information Theory , p. 362, Adelaide, Australia 2005.
  31. Carla-Fabiana Chiasserini, Alessandro Nordio, Emanuele Viterbo, On Data Acquisition and Field Reconstruction in Wireless Sensor Networks, Proc. Tyrrhenian International Workshop on Digital Communications, Sorrento, Italy, July 4-6, 2005.
  32. D. Champion, J.-C. Belfiore, G. Rekaya and E. Viterbo: "Partitionning the Golden Code: A framework to the design of Space-Time coded modulation", Canadian Workshop on Information Theory , Montreal, 2005
  33. G. Rekaya, J.-C. Belfiore and E. Viterbo: "Rectangular Algebraic Space-Time Block Codes", Canadian Workshop on Information Theory , Montreal, 2005
  34. E. Viterbo: Algebraic lattices and channel coding for digital transmission", Invited talk at Workshop on Gitter und Andwendungen, Mathematiches Forschungsinstitut, Oberwolfach, Germany, Jan. 2-8, 2005.
  35. G. Rekaya, J.-C. Belfiore, E. Viterbo: "Algebraic 3x3, 4x4, 6x6 Space-Time Codes with non-vanishing Determinants", IEEE International Symposium on Information Theory and its Applications, pp. 325-329, Parma, Italy, Oct. 10-13, 2004.
  36. G. Rekaya, J.-C. Belfiore, E. Viterbo: "A Very Efficient Lattice Reduction Tool on Fast Fading Channels", IEEE International Symposium on Information Theory and its Applications, Parma, Italy, October 10-13, 2004, p. 714-717.
  37. E. Viterbo: �Perfect Space-Time Block Codes�, JWCC 2004, Donnini, Firenze, Italy, Oct. 2004.
  38. J.-C. Belfiore, G. Rekaya, E. Viterbo: "The Golden Code: A 2x2 Full-Rate Space-Time Code with Non-Vanishing Determinants", IEEE International Symposium on Information Theory, Chicago, June 27 - July 2, 2004, pp. 310.
  39. E. Bayer-Fluckiger, F. Oggier, E. Viterbo: "Bounds on the performance of rotated lattice constellations", IEEE International Symposium on Information Theory, Chicago, June 27 - July 2, 2004, pp. 412.
  40. A. Nordio and E. Viterbo: "Trellis decoding of permutation modulations", IEEE International Symposium on Information Theory, Yokohama, June 29 - July 4, 2004, pp. 393.
  41. F. Oggier, E. Bayer-Fluckiger, E. Viterbo: "New algebraic constructions of rotated cubic lattice constellations for the Rayleigh fading channel", ITW 2003, Paris , France, March 31 - April 4, 2003, pp. 263-266.
  42. A. Nordio and E. Viterbo: "Permutation Modulation for Fading Channels", ICT 2003, Tahiti , French Polynesia, February 23 - March 1, 2003.
  43. E. Viterbo: "Permutation Modulation for Fast Fading Channels", JWCC 2002, Barolo, Italy, November 3-6, 2002.
  44. J. Boutros, G. Caire, E. Viterbo, H. Sawaya, and S. Vialle: "Turbo code at 0.03 dB from capacity limit", ISIT 2002, Lausanne, Switzerland, June 2002, p. 56.
  45. F. Alesiani, E. Biglieri, G. Taricco and E. Viterbo: "Performance of Adaptive Modulation Techniques in the UMTS System", Globecom 2001, San Antonio, Texas, USA, November, 2001, pp. CTS 16-5.
  46. E. Viterbo and Carla F. Chiasserini: "Dynamic Pricing for Connection-Oriented Services in Wireless networks", 12th International Symposium on Personal, Indoor and Mobile Radio Communications, San Diego, California, USA, 30 September - 3 October, 2001, pp. A68-A72.
  47. E. Viterbo and Carla F. Chiasserini: "Dynamic Pricing in Wireless networks", International Symposium on Telecommunications, Tehran, Iran, 1-3 September, 2001, pp. 385-388.
  48. M. Meo and E. Viterbo: "Dynamic Capacity of Wideband CDMA Systems Supporting Multimedia Traffic", International Symposium on Information Theory and Its Applications, Honolulu, Hawaii, USA, 5-8th November, 2000, pp. 152-155.
  49. E. Biglieri, G. Taricco, and E. Viterbo: "Information Theoretic Analysis of Bit-Interleaved Time-Space Codes for Fading Channels", International Symposium on Information Theory and Its Applications, Honolulu, Hawaii, USA, 5-8th November, 2000, pp. 473-476.
  50. M. Elia, G. Taricco, and E. Viterbo: "On the Classification of Binary Goppa Codes", International Symposium on Information Theory and Its Applications, Honolulu, Hawaii, USA, 5-8th November, 2000, pp. 461-464.
  51. M. Elia and E. Viterbo: "Linear sequences and Punctured linear codes", Proceedings SIC' 2000, 23-28 July 2000, Orlando, Florida, pp.660-665
  52. E. Biglieri, G. Caire, G. Taricco, E. Viterbo: "CDMA with fading: Effective bandwidth and spreading-coding tradeoff", Procedings of ISIT 2000, p. 438, Sorrento, Italy, 25-30th June, 2000.
  53. E. Biglieri, G. Taricco, E. Viterbo: "Bit-Interleaved Time-Space Codes for Fading Channels", 2000 Conference on Information Sciences and Systems, pp. WA4/1-6, Princeton University, Princeton, March 15-17th, 2000.
  54. E. Biglieri, G. Caire, G. Taricco, E. Viterbo: "CDMA design through asymptotic analysis: Fading channels", 3rd IEEE/ITG Conference on Source and Channel Coding, pp. 307-312, Muenchen, Germany, Jan. 17-19th, 2000.
  55. E. Biglieri, G. Caire, G. Taricco, E. Viterbo: "CDMA design through asymptotic analysis: Fading channels", 37th Annual Allerton Conference on Communications, Control and Computing, pp. 851-860, Allerton, USA, Sept. 22-24th, 1999.
  56. M. Greferath and E. Viterbo: "Algebraic Construction of Good Collision Resistant Signal Sets", Workshop on Coding and Cryptography, Paris, France, pp. 213-224, Jan. 11-14th, 1999.
  57. M. Greferath and E. Viterbo: On Z4- and Z9-linear lifts of the Golay codes, 1999 Information Theory and Communications Workshop, Kruger Park, South Africa, p. 117, June 1999.
  58. E. Biglieri, J. K. Karlof, and E. Viterbo: "Representing group codes as permutation codes", 1999 Information Theory and Networking Workshop, Metsovo, Greece, p. 53, June 1999.
  59. G. Caire, E. Leonardi, and E. Viterbo: Improving performance of Wireless Networks using Collision Resistant Modulations, Proceedings of Globecom '98, pp. 2186-2191, Sydney, Australia, Nov. 1998.
  60. G. Caire, E. Leonardi, and E. Viterbo: "Concatenated Coding for Packet Transmission over the Collision Channel", Proceedings of ISIT '98, p. 288, Cambridge, Massachusetts, Aug. 1998.
  61. G. Caire, E. Leonardi, and E. Viterbo: "Collision Resistant Modulation", Proceedings of ICT'98, pp. 438-444, Porto Carras, Grecia, Jun. 1998.
  62. G. Caire, E. Leonardi, and E. Viterbo: "Signal-Space Coding for the collision channel: Collision Resistant Modulation", Ninth Joint Conference on Coding and Communication, Courmayeur, Italy, Jan. 1998.
  63. M. Elia, G. Taricco, and E. Viterbo: "Multidimensional Signal Sets Generalizing PSK Modulations", Third Mediterranean Workshop on Coding and Information Integrity, Israele, Oct. 1997.
  64. M. Elia, G. Taricco, and E. Viterbo: "Optimal Energy Transfer over Bandlimited Communication Channels", Seizieme Colloque Gretsi sur le Traitement du Signal et des Images, Grenoble, France, Sept. 1997, pp. 705-708.
  65. G. Taricco and E. Viterbo: "Performance of High Diversity Multidimensional Constellations", IEEE International Symposium on Information Theory, Ulm, Germany, Jun. 1997, p. 167.
  66. J. Boutros and E. Viterbo: "Number fields and Modulations for the Fading Channel", Reseaux Euclidiens et Formes Modulaires, Colloque CIRM, Luminy, Marseille, France, Sept. 1996.
  67. J. Boutros and E. Viterbo: "New approach for transmission over fading channel", Proceedings of ICUPC '96, pp. 66-70, Boston, USA, Sept. 1996.
  68. J. Boutros and E. Viterbo: "Rotated Trellis Coded Lattices", Proceedings of the XXVth General Assembly of the International Union of Radio Science, URSI, p. 153, Lille, Francia, Aug. 1996.
  69. J. Boutros and E. Viterbo: "Rotated multidimensional QAM constellations for Rayleigh fading channels", Proceedings of the 1996 IEEE Information Theory Workshop, p. 23, Haifa, Israel, Jun. 1996.
  70. E. Viterbo and K. Fazel: "How to combat long echoes in OFDM transmission schemes: Sub-channel equalization or more powerful channel coding", Procedings of the conference Globecom '95, pp. 2069-2074, Singapore, Nov. 1995.
  71. E. Viterbo and E. Biglieri: "Computing the Voronoi cell of a lattice: The diamond-cutting algorithm", Procedings of the Int. Symp. on Information Theory, p. 182, Whistler, Canada, Sept. 1995.
  72. J. Boutros and E. Viterbo: "High diversity lattices for fading channels", Procedings of the Int. Symp. on Information Theory, p. 157, Whistler, Canada, Sept. 1995.
  73. E. Viterbo: "High-speed high-density digital magnetic recording", Sixth Joint Conference on Coding and Communication, Selva di Val Gardena, Italy, Feb. 1994.
  74. E. Viterbo and E. Biglieri: "A universal decoding algorithm for lattice codes", Quatorzieme colloque GRETSI, pp. 611-614, Juan-les-Pins, Sept. 1993.
  75. E. Biglieri and E. Viterbo: "Nonlinear Compensation for Magnetic Recording Channels", Quatorzieme colloque GRETSI, pp. 391-394, Juan-les-Pins, Sept. 1993.

Books, chapters and other reports:

  1. E. Viterbo and Y. Hong "Algebraic Coding for Fast Fading Channels", invited book chapter, in Wireless Communications over Rapidly Time-Varying Channels Editor: F. Hlawatsch and G. Matz, Publisher: Academic Press, to appear 2009
  2. F. Oggier, J.-C. Belfiore and E. Viterbo: "Cyclic Division Algebras: A Tool for Space-Time Coding", Foundations and Trends in Communications and Information Theory, Now Publishers, 2007.
  3. A. Nordio, C.-F. Chiasserini, E. Viterbo, On Data Acquisition And Field Reconstruction In Wireless Sensor Networks, in F.Davoli, S.Palazzo and S.Zappatore (Editor), Distributed Cooperative Laboratories: Networking, Instrumentation, and Measurements, Springer Berlin Heidelberg, July 2006.
  4. F. Oggier, E. Viterbo: "Algebraic number theory and code design for Rayleigh fading channels", Foundations and Trends in Communications and Information Theory, Now Publishers, 2004.
  5. E. Viterbo, "Permutation Codes" in Encyclopedia of Telecommunications, Ed. J.Proakis, John Wiley & Sons, Inc., 2002
  6. G. Taricco, E. Viterbo, and G. Caire, "Comunicazioni Elettriche - Esercizi svolti", CLUT Editrice, Torino, ISBN 88-7992-143-6, Feb. 1999.
  7. G. Arian Levi, E. Viterbo: "Simeone Levi - La storia sconosciuta di un noto egittologo", pp. 135, ISBN 88-86626-40-1, Editrice Ananke, Torino, Italy, 1999.
  8. E. Viterbo: "Tecniche matematiche computazionali per l'analisi ed il progetto di costellazioni a reticolo", Ph.D. thesis, Politecnico di Torino, Italy, 1995. (Uni-Bielefeld)
  9. F. Muratore and E. Viterbo: "A universal lattice decoder: Applications and Results", Documento Tecnico CSELT (DTR 95.0365), Torino, Italy, May 1995.
  10. E. Viterbo and K. Fazel: "Guard interval versus sub-channel equalization in OFDM system for HDTV", Internal report DLR, Institut f�r Nachrichtentechnik, Oberpfaffenhofen, Germany, June 1993.


alt

The following statement applies to all publications on this site that are published in IEEE Conferences or Journals.

Copyright IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works, must be obtained from the IEEE. Contact: Manager, Copyrights and Permissions / IEEE Service Center / 445 Hoes Lane / P.O. Box 1331 / Piscataway, NJ 08855-1331, USA. Telephone: + Intl. 908-562-3966.

See IEEE Policy & Procedures Manual (Electronic Information Dissemination) for more information.

The following statement applies, more generally, to all publications downloaded from this site.

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.



Teaching

  • Fondamenti di telecomunicazioni
  • Laboratorio di Reti di Telecomunicazioni
  • Elementi di Sicurezza nelle Reti
  • Complementi di Sicurezza nelle Reti

Miscellaneous



Invited Talks Service Slides
The ciphered autobiography of a 19th century Egyptologist
Tables of algebraic rotations
alt
The Golden Code
alt
The Perfect Space-Time Block Codes
alt
The Silver Code




Invited Talks


  • E. Viterbo: Golden Space-Time Coded Modulation, Plenary talk at the ACC Workshop Emerging Wireless Broadband Technologies, Tel Aviv, Israel, April 19th, 2007.
  • E. Viterbo: Golden Space-Time Coded Modulation, ETH, Zurich, Switzerland, March 29th, 2007.
  • E. Viterbo: Golden Space-Time Coded Modulation, Nokia Research Center, Helsinki, Finland, March 26th, 2007.
  • E. Viterbo: Algebraic tools for code design in MIMO systems, Invited lecture at the Multiuser MIMO Tutorial jointly organised by the FP6-IST projects Mascot and Surface, Vienna, Austria, March 1st, 2007.
  • E. Viterbo: Diversity code design criterion, Invited lecture at the ACE/NEWCOM Autumn School on Space-Time Coding, Politecnico di Torino, Turin, Italy, October 9, 2006.
  • E. Viterbo: Alamouti code, BLAST, orthogonal codes and associate decoders, Invited lecture at the ACE/NEWCOM Autumn School on Space-Time Coding, Politecnico di Torino, Turin, Italy, October 9, 2006.
  • E. Viterbo: Cyclic division algebra based codes (Golden codes and perfect codes), Invited lecture at the ACE/NEWCOM Autumn School on Space-Time Coding, Politecnico di Torino, Turin, Italy, October 10, 2006.
  • E. Viterbo: Golden Space-Time Coded Modulation, UCSD, San Diego, California, USA, August 4th, 2006.
  • E. Viterbo: Golden Space-Time Coded Modulation, CalTech, Pasadena, California, USA, July, 2006.
  • E. Viterbo: Lattice Coding and Decoding for Fading Channels, Invited lecture at the ACoRN Spring School 2005 Coding and Information Theory, Adelaide, Australia, September 14th, 2005.




Service


Conferences

  • Publications Chair, ISIT 2000, Sorrento, Italy, June 2000.
  • Member of Technical Program Committee, ISITA 2000, Honolulu, Hawaii, Dec. 2000.
  • Member of Technical Program Committee, ISITA 2002, Xi'an, China, 2002.
  • Member of Technical Program Committee, ITW 2003, Paris, France, April 2003.
  • Publications Co-Chair, ISITA 2004, Parma, Italy, October 2004.
  • Member of Technical Program Committee, ISIT 2005, Adelaide, Australia, Sept. 2005.


PhD Committees
  • Mohamed Oussama Damen, ``Joint Coding/Decoding in a Multiple Access System, Application to Mobile Communications'', ENST Paris, France, September 1999. (EXAMINATEUR)
  • Loic Brunel, ``Algorithmes de decodage de canal pour l'acces multiple a etalement de spectre,'' ENST Paris, France, December 1999. (RAPPORTEUR)
  • Catherine Lamy, ``Communications a grande efficacite spectrale sur le canal a evanuissements,'' ENST Paris, France, April 2000. (RAPPORTEUR)
  • Ines Kammoun, ``Non-coherent design and detection for space-time coding,'' ENST Paris, France, April 2004. (EXAMINATEUR)
  • Ghaya Rekaya, ``Nouvelles constructions algebriques de codes spatio-temporels atteignant le compromis multiplexage-diversite,'' ENST Paris, France, December 2004. (EXAMINATEUR)
  • Georgia Feideropoulou, ``Codajge Conjoint Source-Canal des Sources Video,'' ENST Paris, France, April 2005. (RAPPORTEUR)
  • Chadi Abou Rjeily, ``Construction et Analyse de Nouveaux Codes Statio-Temporels pour les Systemes Ultra Large Bande par Impulsions,'' ENST Paris, France, October 2006. (RAPPORTEUR)
  • Elie Jandot dit Danjou, ``Applications du codage spatio-temporel � des r�seaux sans fils,'' ENST Paris, France, December 2006. (RAPPORTEUR)
  • Laura Luzzi, ``Continued fractions, coding and wireless channels,'' Scuola Normale Superiore, Pisa, Italy, May 2007. (CORRELATORE)


Reviewing Assignments
  • European Transactions on Telecommunications (Associate Editor 2007-)
  • IEEE Transactions on Information Theory
  • IEEE Communications Letters
  • IEEE Transactions on Communications
  • IEEE Transactions on Vehicular Technology
  • IEEE Transactions on Signal Processing
  • Electronics Letters


Visitors at Unical 2006-2007
  • Ezio Biglieri - UPF Barcelona
  • Joseph Boutros - ENST Paris
  • Yi Hong - ITR Univesrity of South Australia


Slides

  1. 17/1/2003 Workshop on Algebra and Communications
  2. 21/11/2002 Seminaire de Theorie des Nombres, Algorithmique et Cryptographie
  3. 22/11/2002 Seminaire de Theorie des Nombres, Algorithmique et Cryptographie
  4. Dynamic Pricing in Wireless Networks
  5. Decoding Reed-Solomon Codes Beyond d/2 using Sudan's Algorithm
  6. Information Theoretic Analysis of Bit-Iterleaved Time-Space Codes for Fading Channels
  7. Signal-Space coding for the Collision Channel: Collision Resistant Modulation
  8. Algebraic Construction of Good Collision Resistant Signal Sets
  9. Improving Performance of Wireless Networks Using Collision Resistant Modulations
  10. On Z4- and Z9-linear lifts of the Golay codes
  11. Introduction to Algebraic Number Theory
  12. Rotated Multidimensional QAM Constellations for the Rayleigh Fading Channel
  13. Number Fields and Modulations for the Rayleigh Fading Channels
  14. Performance of High diversity Multidimensional Constellations for the Fading Channel


The Ciphered Autobiography of an 19th-Century Egyptologist

Simone Levi was an Simone Levi was an Italian Egyptologist who lived in Turin during the second half of the 19th century. His major work is the eight volume hieroglyphic dictionary for which, in 1886, he was awarded the prize of the Royal Academy of Lincei. He was the brother of the mother of my great-grandfather, and I first heard about him when my grandmother gave her copy of the dictionary to my father. The dictionary is a lithographic copy of his handwritten manuscript and its aim was to compare the hieroglyphic words with the corresponding Coptic and Hebrew words, in an attempt to demonstrate their derivation from hieroglyphic.
Born in 1843 in the Jewish ghetto of Carmagnola near Turin to a poor family, Simeone Levi was the seventh of the 10 children of a goldsmith. Struck by a paralysis at two, he remained disabled all his life, sufferering the limitations imposed by his handicap. After getting a degree in mathematics, he earned his living at first by teaching mathematics. His interest in Egyptology started only at 33, after he attended a series of lectures by Professor Francesco Rossi, vicedirector of the Egyptian Museum of Turin. His only other classmate was Ernesto Schiaparelli, who was to become famous for discovering Queen Nefertari's tomb. From that moment he entirely devoted himself to papyrology, having Professor Rossi as his guide and maintaining a competitive attitude towards Schiaparelli.
Last year my grandaunt Giorgina Levi (Simeone's grandgrandniece) decided to find out more about the life of her famous ancestor and initiated an historical research. Through the documents, she got in touch with the lineal descendants of Simeone, Ettora and Massimo Levi. They had several papers, books and letters of their grandfather and among them a manuscript written in an unknown alphabet. They also reported that Simeone had imposed upon his sons the duty of interpreting and reading the 355-page manuscript. But all attempts to decode the mysterious text failed, even though it was given to fairly expert, but possibly not very motivated, people to examine.


  • E. Viterbo: "The Ciphered Autobiography of an 19th Century Egyptologist", CRYPTOLOGIA, vol. XXII, n. 3, pp. 231-243, July 1998.
  • G. Arian Levi, E. Viterbo: "Simeone Levi - La Storia sconosciuta di un noto egittologo", Editrice Ananke, Torino, 1999, pp. 135, ISBN 88-86626-40-1.

The Metafont source files

sim.mf
gosim.mf
simbase.mf

The font files created by Metafont

sim.tfm
sim.300gf
sim.300pk

Sample .tex files

simdefs2.tex
simpap.tex

Sample .dvi and .ps files

simpap.dvi
simpap.ps




Full diversity rotations

Algebraic number theory provides effective means to construct rotated Zn lattices with full diversity and large minimum product distance. These two properties enable to design good signal constellations for the independent Rayleigh fading channel. The following tables provide the best known constructions for these lattices in terms of highest minimum product distance. The corresponding rotated Zn lattice generator matrices are orthogonal matrices (i.e., M*Mt=In) and can be downloaded in text form.

The authors would be happy to hear about any contribution improving over the best known rotations reported here.


Table of best known full diversity algebraic rotations



n
?
I
?
dp,min
dp,min1/n
Bound
Matrix
Notes&Refs
2
(1+sqrt(5))/2
OK
3-?
1/sqrt(5)
0.668740
0.66995
cyclo_2
2
sqrt(2)
OK
1/(2*sqrt(2)+4)
1/(2*sqrt(2))
0.59460
0.66995
ideal_2
[BOV04]
3
2cos(2?/7)
OK
2-?
1/7
0.522757
0.52461
cyclo_3
as krus_3
3 exp(2?i/13)
AK
1 1/13 0.425290 0.52461
cyclic_3a
r=2, lambda=1
3
exp(2?i/9)
AK
1
1/9
0.480749
0.52461
cyclic_3b
r=2
4
x4-x3-3x2+x+1
1/sqrt(52*29)
0.438993
0.44163
krus_4
4
sqrt(2)
sqrt(5)
1/40
0.397635
0.44163
mixed_2x2
cyclo_2Xideal_2
5
2cos(2?/11)
OK
2-?
1/112
0.383215
0.38794
cyclo_5
as krus_5
5
exp(2?i/11)
AK
1
1/112
0.383215
0.38794
cyclic_5a
as cyclo_5
r=2, lambda=1
5
exp(2?i/31)
AK
1
1/54
0.275946
0.38794
cyclic_5b
r=2
5
exp(2?i/25)
AK
1
1/312
0.253195
0.38794
cyclic_5c
r=3, lambda=16
6
2cos(2?/5)
2cos(2?/7)
1/sqrt(53*74)
0.349589
0.35032
mixed_2x3
as krus_6
6
2cos(2?/13)
OK
2-?
1/13(5/2)
0.343444
0.35032
cyclo_6
7
x7-x6-6x5+4x4
+10x3-4x2-4x+1
1/sqrt(20134393)
0.300809
0.32245
krus_7
7
exp(2?i/29)
AK
1
1/293
0.236188
0.32245
cyclic_7a
r=2, lambda=1
7
exp(2?i/49)
AK
1
1/76
0.188638
0.32245
cyclic_7b
r=3
8
2cos(2?/17)
OK
2-?
1/17(7/2)
0.289520
0.30093
cyclo_8
8 2cos(2?/32) 1/2(31/2) 0.261068 0.30093 dast_8 [DAB02]
9
2cos(2?/19)
OK
2-?
1/194
0.270187
0.28377
cyclo_9
10
2cos(2?/5)
2cos(2?/11)
1/sqrt(55*118)
0.256271
0.26973
mixed_2x5
cyclo_2Xcyclo_5
11
2cos(2?/23)
OK
2-?
1/235
0.240454
0.25801
cyclo_11
11
exp(2?i/23)
AK
1
1/235
0.240454
0.25801
cyclic_11a
as cyclo_11
r=5, lambda=6
11
exp(2?i/121)
AK
1
1/1110
0.113052
0.25801
cyclic_11b
r=2
12
2cos(2?/5)
2cos(2?/13)
1/(53*135)
0.229675
0.24807
mixed_2x6
cyclo_2Xcyclo_6
12 1/sqrt(494*56*293) 0.229487 0.24807
mixed_3x4
cyclo_3Xkrus_4
13
exp(2?i/53)
AK
1
1/536
0.160022
0.23951
cyclic_13a
r=2, lambda=1
13
exp(2?i/169)
AK
1
1/1312
0.093701
0.23951
cyclic_13b
r=3
14
2cos(2?/29)
OK
2-?
1/29(13/2)
0.209425
0.23205
cyclo_14
15
2cos(2?/31)
OK
2-?
1/317
0.201386
0.22548
cyclo_15
15
1/(75*116)
0.200328
0.22548
mixed_3x5
cyclo_3Xcyclo_5
16
1/(54*177)
0.193613
0.21965
mixed_2x8
cyclo_2Xcyclo_8
16 2cos(2?/64) 1/2(79/2) 0.180648 0.21965 dast_16 [DAB02]
17
exp(2?i/103)
AK
1
1/1038
0.112923
0.21444
cyclic_17a
r=5, lambda=26
17
exp(2?i/289)
AK
1
1/1716
0.069491
0.21444
cyclic_17b
r=3
18
2cos(2?/37)
OK
2-?
1/37(17/2)
0.181744
0.20973
cyclo_18
18
1/sqrt(59*1916)
0.180685
0.20973
mixed_3x6
cyclo_3Xcyclo_6
19
exp(2?i/191)
AK
1
1/1919
0.083083
0.20547
cyclic_19a
r=19, lambda=138
20
2cos(2?/41)
OK
2-?
1/41(19/2)
0.171367
0.20159
cyclo_20
21
2cos(2?/43)
OK
2-?
1/4310
0.166785
0.19803
cyclo_21
21 1/sqrt(497*201343933) 0.157250 0.19803 mixed_3x7 krus_3Xkrus_7
22
1/sqrt(511*2320)
0.160801
0.19475
mixed_2x11
cyclo_2Xcyclo_11
23
2cos(2?/47)
OK
2-?
1/4711
0.158599
0.19173
cyclo_23
23
exp(2?i/47)
AK
1
1/4711
0.158599
0.19173
cyclic_23a
as cyclo_23
r=5, lambda=12
24
1/sqrt(716*1721)
0.151348
0.18892
mixed_3x8
cyclo_3Xcyclic_8
25
1/(520*1110)
0.105747
0.18631
mixed_5x5
cyclo_5Xcyclic_5a
26
2cos(2?/53)
OK
2-?
1/53(25/2)
0.148259
0.18388
cyclo_26
27
1/(79*1912)
0.141243
0.18160
cyclo_3Xcyclo_9
28
1/(57*2913)
0.140051
0.17947
mixed_4x7
krus_4Xkrus_7
29
2cos(2?/59)
OK
2-?
1/5914
0.139670
0.17746
cyclo_29
29
exp(2?i/59)
AK
1
1/5914
0.139670
0.17746
cyclic_29a
as cyclo_29
r=2, lambda=1
30
2cos(2?/61)
OK
2-?
1/61(29/2)
0.137116
0.17556
cyclo_30
30
1/sqrt(1124*1325)
0.131613
0.17556
mixed_5x6
cyclo_5Xcyclo_6




Legend

n
Dimension
?
Primitive element of the algebraic number field Q(?):
- for cyclotomic constructions K=Q(?)
- for cyclic constructions K is a subfield of Q(?)
I Ideal of the ring of integers OK
?
Twisting element in the ideal lattice construction
dp,min
Minimum product distance
dp,min1/n
Normalized minimum product distance
(in red best known value, in boldface red optimal)
Bound
Upper bound on dp,min1/n based on
Odlyzko's bound for the root discriminant
Matrix
Generator matrix in the row vector convention
Refs
Bibliographic reference
OK
Ring of integers of K
AK
Ideal of OK such that AK2 is the codifferent of K




References

[O05] F. Oggier, Algebraic methods for channel coding, Ph.D. Thesis, EPFL, 2005.
[OV04] F. Oggier and E. Viterbo, "Algebraic number theory and code design for Rayleigh fading channels," in Foundations and Trends in Communications and Information Theory, vol. 1, pp. 333-415, 2004.
[BOV04] E. Bayer-Fluckiger, F. Oggier, E. Viterbo: "New Algebraic Constructions of Rotated Z^n-Lattice Constellations for the Rayleigh Fading Channel," IEEE Transactions on Information Theory, vol. 50, n. 4, pp. 702-714, Apr. 2004.
[OB03] F. Oggier, E. Bayer-Fluckiger, "Best rotated cubic lattice constellations for the Rayleigh fading channel,"
Proceedings of the IEEE International Symposium on Information Theory, Yokohama, Japan, 2003.
[DAB02] M.O. Damen, K. Abed-Meriam, J.C. Belfiore:"Diagonal algebraic space-time block codes," IEEE Transactions on Information Theory, vol. 48, pp. 628-636, Mar. 2002.
[BV98] J. Boutros and E. Viterbo: "Signal Space Diversity: a power and bandwidth efficient diversity technique for the Rayleigh fading channel", IEEE Transactions on Information Theory, vol. 44, n. 4, pp. 1453-1467, July 1998.
[GBB97] X. Giraud, E. Boutillon, and J.C. Belfiore, "Algebraic tools to build modulation schemes for fading channels" IEEE Transactions on Information Theory, vol. 43, n. 3, pp. 938 - 952, May 1997.




The Golden Code

Definition

The Golden Code is a Space-Time code for 2 transmit and 2 receive antennas, for the coherent MIMO channel.
It has been found independently by [BRV05],[YW03],[DV03].
The channel model considered is Y = H X + N, where H ={hij} is the 2x2 channel matrix with complex fading coefficients
and N the 2x2 complex Gaussian noise matrix.
The codewords X of the Golden Code are 2x2 complex matrices of the following form :

X = 1/sqrt(5) *
? [a+b?] ? [c+d?]
i ?(?) [c+d?(?)] ?(?) [a+b?(?)]

where
  • a,b,c,d are the information symbols which can be taken from any M-QAM constellation carved from Z[i]
  • i = sqrt(-1)
  • ? = (1+sqrt(5))/2 = 1.618... (Golden number)
  • ?(?) = (1-sqrt(5))/2 = 1-?
  • ? = 1 + i - i ? = 1 + i ?(?)
  • ?(?) = 1 + i - i ?(?) = 1 + i ?
Using the relations ? ?(?) = -1 and ? + ?(?) = 1 we can rewrite the codeword matrices as:
X = 1/sqrt(5) *
[1 + i ?(?)]a + [?-i]b [1 + i ?(?)]c + [?-i]d
[i - ?]c + [1 + i?(?)]d [1 + i ?]a + [?(?)-i]b

Properties

  1. Full-rank : the determinant of the difference of 2 codewords is always different from 0.
  2. Full-rate : the four degrees of freedom of the system are used, which allows to send 4 information symbols.
  3. Non-vanishing determinant for increasing rate : the minimum determinant of the Golden Code is 1/5.
  4. Cubic shaping : each layer is carved from a rotated version of Z[i]^2.
  5. It achieves the Diversity Multiplexing Frontier [YW03].
  6. The spectral efficiency is 2log2(M) bits/s/Hz.

ML Decoding with the SphereDecoder

In order to decode the Golden Code, the matrix has to be vectorized, furthermore real and imaginary part are separated,
so as to obtain an 8x8 matrix R, as shown below.

R = 1/sqrt(5) *
1 -?(?)
?(?) 1
? 1
-1 ?
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
-? -1
1 -?
1 -?(?)
?(?) 1
0 0
0 0
0 0
0 0
1 -?(?)
?(?) 1
? 1
-1 ?
1 -?
? 1
?(?) 1
-1 ?(?)
0 0
0 0
0 0
0 0

The Golden Code can be seen as a rotated Z 8 algebraic lattice [OV04], with an orthogonal generator matrix R, and sent over a channel described by an 8x8 matrix H'

H' =
Re(h11) -Im(h11)
Im(h11) Re(h11)
Re(h12) -Im(h12)
Im(h12) Re(h12)
0 0
0 0
0 0
0 0
Re(h21) -Im(h21)
Im(h21) Re(h21)
Re(h22) -Im(h22)
Im(h22) Re(h22)
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
Re(h11) -Im(h11)
Im(h11) Re(h11)
Re(h12) -Im(h12)
Im(h12) Re(h12)
0 0
0 0
0 0
0 0
Re(h21) -Im(h21)
Im(h21) Re(h21)
Re(h22) -Im(h22)
Im(h22) Re(h22)


The received vector becomes y' = H'Rx'+ n, where n is the real Gaussian noise vector and x'


x' =
Re(a)
Im(a)
Re(b)
Im(b)
Re(c)
Im(c)
Re(d)
Im(d)

The decoding of the 8-dimensional lattice with generator matrix M=H'R can be performed using the Sphere Decoder [VB98].

Performance of the Golden Code (*)

Codeword Error Rate
Bit Error Rate
(*) We thank Barbara Cerato for generating these simulation curves.

References

[BRV05] J.-C. Belfiore, G. Rekaya, E. Viterbo: "The Golden Code: A 2 x 2 Full-Rate Space-Time Code with Non-Vanishing Determinants," IEEE Transactions on Information Theory, vol. 51, n. 4, pp. 1432-1436, Apr. 2005.
[DV03] P. Dayal, M.K. Varanasi: "An Optimal Two Transmit Antenna Space-Time Code and its Stacked Extensions," Proceedings of Asilomar Conf. on Signals, Systems and Computers, Monterey, CA , November 2003.
[ORBV05] F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo: "Perfect Space Time Block Codes," submitted to IEEE Transactions on Information Theory, Sep. 2004.
[OV04] F. Oggier and E. Viterbo, "Algebraic number theory and code design for Rayleigh fading channels,"
Foundations and Trends in Communications and Information Theory, vol. 1, pp. 333-415, 2004.
[RBV05] G. Rekaya, J.-C. Belfiore, E. Viterbo: "Algebraic 3x3, 4x4 and 6x6 Space-Time Codes with Non-Vanishing Determinants," Proceedings of Intern. Symp. on Inform. Theory and its Applications, ISITA , October 2004, pp. 325-329.
[VB98] E. Viterbo and J. Boutros: "A Universal Lattice Code Decoder for Fading Channels," IEEE Transactions on Information Theory, vol. 45, n. 5, pp. 1639-1642, July 1999.
[YW03] H. Yao, G.W. Wornell: "Achieving the Full MIMO Diversity-Multiplexing Frontier with Rotation-Based Space-Time Codes," Proceedings of Allerton Conf. on Communication, Control and Computing , October 2003.

Last modified 29/5/2006 by Frederique Oggier and Emanuele Viterbo








The Silver Code

Definition

The Silver Code is a fast-decodable space-time block code for 2 transmit and 2 receive antennas, for the coherent MIMO channel. It has been found by [HT04] [HTWBook] [TH02] [TK02]. Recently it was re-found by [PGA07][SF07], which pointed out its fast-decoding properties and it was also summarized by [BHV07].


The channel model considered is Y = H X + N, where H ={h
ij} is the 2x2 channel matrix with complex fading coefficientsand N the 2x2 complex Gaussian noise matrix. The codewords X of the Golden Code are 2x2 complex matrices of the following form :

X = Xa(s1,s2) + TXb(z1,z2)


where

X
a and Xb take Alamouti structur

Xa(s1,s2) =

s1

-s2*

Xb(z1,z2) =

z1

-z2*

[z1, z2]T = U*

s3

s2

s1*

z2

z1*

s4

si , i=1,...,4, are the information QAM symbols

  • U is an unitary matrix, and the optimum U matrix is given by the following in order to maximize the minimum determiant of the codeword matrix X

U = 1/sqrt(7) *

1 + j

-1 + 2j

1 + 2j

1 -

  • T is chosen as the following matrix to gaurantee the cubic shaping property [BHV07]

T =

1

0

0

-1

Porperties

  1. Full-rank : the determinant of the difference of 2 codewords is always different from 0.
  2. Full-rate : the four degrees of freedom of the system are used, which allows to send 4 information symbols.
  3. Non-vanishing determinant for increasing rate [HLRVV08]: the minimum determinant is 1/7,slightly smaller than that of the Golden code.
  4. Cubic shaping : each layer is carved from a rotated version of Z[i]^2.
  5. The spectral efficiency is 2log2(Q) bits/s/Hz for Q-QAM.
  6. It achieves the Diversity Multiplexing Frontier.
  7. Fast-decoding property: the worst-case maximum likelihood decoding (MLD) complexity is 2M3, as compared to a standard MLD complexity M4, where M is the cardinality of the signal constellation.

Reduced-Complexity MLD with the SphereDecoder

  • Consider a linear space-time block coded MIMO, where STBC carries 4 independent QAM information symboles. In a complex vector form, we rephrase the received signal equation as vec(Y) = Fs + vec(N), where s = {si}, i=1,...,4, and F = diag(H,...,H) x G, where G is the generator matrix of the silver code given in [PGA07,BHV07], vec(?) operator stacks the m column vectors of a n x m complex matrix into a mn complex column vector.
  • Let F = [f1 | f2 | f3 | f4], where fi is a 4 dimensional column vector. Sphere decoding (SD) can be used to conduct the MLD based on QR decomposition to minimize ||Q vec(Y) Rs||, where () denotes matrix Hermitian, and F = QR, where Q is a 4 x 4 unitary matrix, and R is a 4 x 4 upper triangular matrix with the following special structure, where <a,b> denotes the inner product of a and b,


R
=

||d1||

0

<f3, e1>

<f4, e1>

0

|| d2||

<f3, e2>

<f4, e2>

0

0

|| d3||

0

0

0

0

|| d4||

where di = fi� sum( Projej fi, j = 1, �, i-1), where Projuv = <v,u>/<u,u> and ei = di/||di||.

  • Note that there are two zeros in the matrix R which lead to a reduced-complexity MLD [PGA07][SF07][BHV07].

1) <f2, e1> = 0 provides a saving of 2-dimensional complex SD tree search, i.e., we employ 2-dimensional complex SD tree search to find s3, s4, with complexity of M2. For the remaining pair (s1,s2), an Alamouti decoding is used with decoding complexity 2M. In summary, the worst-case decoding complexity is 2M3.

2) <f4, e3> = 0 leads to a faster metric computation in the relevant SD computation.

Performance of the Silver Code, compared to Golden Code (*)

References

[HT04]

A. Hottinen and O. Tirkkonen, ``Precoder designs for high rate space-time block codes,'' in Proc. Conference on Information Sciences and Systems, Princeton, NJ, March 17--19, 2004.

[HTWBook]

A. Hottinen, O. Tirkkonen and R. Wichman, ``Multi-antenna Transceiver Techniques for 3G and Beyound,'' WILEY publisher, UK.

[TH02]

O. Tirkkonen and A. Hottinen, ``Square-matrix embeddable space-time block codes for complex signal constellations,'' in IEEE Trans. Inform. Theory, vol. 48, no. 2, , pp. 384-395, February 2002.

[TK02]

O. Tirkkonen and R. Kashaev, ``Combined information and performance optimization of linear MIMO modulations,'' in Proc IEEE Int. Symp. Inform. Theory (ISIT 2002), Lausanne, Switzerland, p. 76, June 2002.

[PGA07]

J. Paredes, A.B. Gershman, and M. G. Alkhanari, ``A2x2 space--time code with non-vanishing determinants and fast maximum likelihood decoding,'' in Proc IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP2007), Honolulu, Hawaii, USA, pp. 877-880, April 2007.

[SF07]

M. Samuel and M. P. Fitz, ``Reducing the detection complexity by using 2x2 multi-strata space--time codes,'' in Proc IEEE Int. Symp. Inform. Theory (ISIT 2007), pp. 1946-1950, Nice, France, June 2007.

[BHV07]

E. Biglieri, Y. Hong and E. Viterbo, "On fast-decodable space-time block codes,"
submitted to IEEE Trans.
On Information Theory, also available on arXiv: 0708.2804, Aug. 2007

[HLRVV08]

C. Hollanti, J. Lahtonen, K. Ranto, R. Vehkalahti, and E. Viterbo, ``On the Algebraic Structure of the Silver Code,'' appear in IEEE Information Theory Workshop, Porto, Portugal, May 2008.


Last modified 2/3/2008 by Yi Hong




The Perfect Space Time Block Codes


Definition

Perfect Codes are Space-Time codes for the coherent MIMO channel.
They were defined in [ORBV06]. They are algebraic codes, built on non-commutative fields (or division algebras ).
The channel model considered is the following: if M is the number of transmit and receive antennas,

Y = H X + N (1)
where H ={hij} is the MxM channel matrix with complex fading coefficients and N the MxM complex Gaussian noise matrix.

A perfect space-time code satisfies by definition the following properties:
  1. Full-rank : the determinant of the difference of any two distinct codewords is different from 0.
  2. Full-rate : all the M2 degrees of freedom of the system are used, which allows to send M2 information symbols, either QAM or HEX.
  3. Non-vanishing determinant for increasing rate : the minimum determinant of a perfect code is lower bounded away from zero by a constant. This constant, prior to SNR normalization, does not depend on the spectral efficiency.
  4. Efficient shaping : The energy required to send the linear combination of the information symbols on each layer is similar to the energy used for sending the symbols themselves. This can be interpreted by saying that each layer is carved from a rotated version of the lattice Z[i]M or A2M, where A2 is the hexagonal lattice.
  5. It achieves the Diversity Multiplexing Gain Trade-off [ERPVL06].
  6. Uniform energy : It induces uniform average transmitted energy per antenna in all T=M time slots, i.e., all the coded symbols in the code matrix have the same average energy.

Code Construction

Perfect codes only exist in dimension 2, 3, 4, and 6 [BO06]. Codewords of a Perfect code have the form:

SUM [diag(Muj) Ej-1, j=1...M] (2)

where uj = [uj,1, .... , uj,M] , M is a MxM unitary matrix defined below, and
0 1 ... 0
0 0 1
E = ... 1
0 ... 1
g 0 ... 0 0
  • For M=2 antennas, QAM symbols are sent. There are infinitely many of them, but the most famous is the Golden Code
  • For M=3 antennas, HEX symbols are sent. (g = exp(2?i/3) and M). The minimum determinant is 1/49.
  • For M=4 antennas, QAM symbols are sent. (g = i and M). The minimum determinant is 1/1125.
  • For M=6 antennas, HEX symbols are sent. (g = -exp(2?i/3) and M). The minimum determinant is between 1/(26 74) and 1/(26 75) .

Decoder

The Sphere Decoder can be applied to decode the Perfect codes by vectorizing (1) and using (2) similarly to the Golden code.

References

[ORBV06] F. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo: "Perfect Space-Time Blocks Codes," IEEE Transactions on Information Theory, Sep. 2006.
[ERPVL06] P. Elia, K. Raj Kumar, S. A. Pawar, P. Vijay Kumar and H.-F. Lu :"Explicit, Minimum-Delay Space-Time Codes Achieving the Diversity-Multiplexing Gain Tradeoff," IEEE Transactioins on Information Theory , to appear, 2006.
[BO06]
G. Berhuy, F. Oggier: "On the Existence of Perfect Codes," submitted to IEEE Transactioins on Information Theory, 2006.


Last modified 15/7/2006 by Frederique Oggier and Emanuele Viterbo








© 2020 Titan Lab
Joomla! un software libero rilasciato sotto licenza GNU/GPL.