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Emanuele Viterbo Profile Page
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
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.
Publications
Publications of EmanueleViterbo
(last update May 2009)
Journals:
Y. Hong and E. Viterbo: "Algebraic multiuser space-time block codes for 2x2 MIMO", IEEE Transactions on Vehicular Technology , to appear2009.
A. Nordio, C.-F. Chiasserini, E.Viterbo, The Impact of Quasi-equally Spaced Sensor Topologies on Signal Reconstruction, ACM Transactions on Sensor Networks, 2009.
L. Luzzi, G. Rekaya-Ben Othman, J.-C. Belfioreand E. Viterbo, "Golden Space-Time Block Coded Modulation", IEEE Transactions on Information Theory, pp. 584-597, vol. 55, n. 2, Feb. 2009.
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.
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.
V. Daniele, M. Gilli, and E. Viterbo: "Diffraction of a plane wave by a strip grating", Electromagnetics, n. 10, pp. 245-269, 1990.
Conferencepapers:
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
E. Viterbo: "Irregular Sampling and Random Matrix Theory", JWCC 2008, Napa Valley, CA, Oct. 2008.
E. Viterbo, Y. Hong: "Applications of the Golden Code ", Invited paper at Information Theory and Application (ITA 2007), UCSD, San Diego, USA, Jan. 2007
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.
Carla-FabianaChiasserini, Alessandro Nordio, EmanueleViterbo, On Data Acquisition and Field Reconstruction in Wireless Sensor Networks, Proc. Tyrrhenian International Workshop on Digital Communications, Sorrento, Italy, July 4-6, 2005.
E. Viterbo and Carla F. Chiasserini: "Dynamic Pricing in Wireless networks", International Symposium on Telecommunications, Tehran, Iran, 1-3 September, 2001, pp. 385-388.
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.
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.
E. Viterbo: "High-speed high-density digital magnetic recording", Sixth Joint Conference on Coding and Communication, Selvadi Val Gardena, Italy, Feb. 1994.
E. Biglieri and E. Viterbo: "Nonlinear Compensation for Magnetic Recording Channels", Quatorziemecolloque GRETSI, pp. 391-394, Juan-les-Pins, Sept. 1993.
Books, chapters and other reports:
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
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.
F. Muratore and E. Viterbo: "A universal lattice decoder: Applications and Results", Documento Tecnico CSELT (DTR 95.0365), Torino, Italy, May 1995.
E. Viterbo and K. Fazel: "Guard interval versus sub-channel equalization in OFDM system for HDTV", Internal report DLR, Institutf�rNachrichtentechnik, Oberpfaffenhofen, Germany, June 1993.
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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: 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.
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-)
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.
Algebraic number theory provides effective means to construct rotated Z^{n} 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 Z^{n} lattice generator matrices are orthogonal matrices (i.e., M*M^{t}=I_{n}) 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.
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
Full-rank : the determinant of the difference of 2 codewords is always different from 0.
Full-rate : the four degrees of freedom of the system are used, which allows to send 4 information symbols.
Non-vanishing determinant for increasing rate : the minimum determinant of the Golden Code is 1/5.
Cubic shaping : each layer is carved from a rotated version of Z[i]^2.
It achieves the Diversity Multiplexing Frontier [YW03].
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].
(*) 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.
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.
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 ={hij} 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 = X_{a}(s_{1},s_{2})+ TX_{b}(z_{1},z_{2})
�
�
where
X_{a}and X_{b} take Alamouti structur
X_{a}(s_{1},s_{2}) =
s_{1}
-s_{2}^{*}
X_{b}(z_{1},z_{2}) =
z_{1}
-z_{2}^{*}
[z_{1}, z_{2}]^{T} = U*
s_{3}
s_{2}
s_{1}^{*}
z_{2}
z_{1}^{*}
s_{4}
s_{i} , 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
Full-rank : the determinant of the difference of 2 codewords is always different from 0.
Full-rate : the four degrees of freedom of the system are used, which allows to send 4 information symbols.
Non-vanishing determinant for increasing rate [HLRVV08]: the minimum determinant is 1/7,slightly smaller than that of the Golden code.
Cubic shaping : each layer is carved from a rotated version of Z[i]^2.
The spectral efficiency is 2log2(Q) bits/s/Hz for Q-QAM.
It achieves the Diversity Multiplexing Frontier.
Fast-decoding property: the worst-case maximum likelihood decoding (MLD) complexity is 2M^{3}, as compared to a standard MLD complexity M^{4}, 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 = {s_{i}}, 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 =[f_{1}| f_{2}| f_{3}| f_{4}], where f_{i} 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, andF = 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 =
�
||d_{1}||
0
<f_{3}, e_{1}>
<f_{4}, e_{1}>
0
|| d_{2}||
<f_{3}, e_{2}>
<f_{4}, e_{2}>
0
0
|| d_{3}||
0
0
0
0
|| d_{4}||
where d_{i}= f_{i}� sum( Proj_{e}_{j }f_{i}, j = 1, �, i-1), where Proj_{u}v =<v,u>/<u,u>and e_{i}= d_{i}/||d_{i}||.
Note that there are two zeros in the matrix R which lead to a reduced-complexity MLD [PGA07][SF07][BHV07].
1) <f_{2}, e_{1}>= 0provides a saving of 2-dimensional complex SD tree search, i.e., we employ 2-dimensional complex SD tree search to find s_{3}, s_{4}, with complexity of M^{2}. For the remaining pair (s_{1},s_{2}), an Alamouti decoding is used with decoding complexity 2M. In summary, the worst-case decoding complexity is 2M^{3}.
2) <f_{4}, e_{3}>= 0leads to a faster metric computation in the relevant SD computation.
Performance of the Silver Code, compared to Golden Code (*)
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.
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:
Full-rank : the determinant of the difference of any two distinct codewords is different from 0.
Full-rate : all the M^{2} degrees of freedom of the system are used, which allows to send M^{2} information symbols, either QAM or HEX.
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.
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 A_{2}^{M}, where A_{2} is the hexagonal lattice.
It achieves the Diversity Multiplexing Gain Trade-off [ERPVL06].
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(Mu_{j}) E^{j-1}, j=1...M] (2)
where u_{j = }[u_{j,1}, .... , u_{j,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/(2^{6} 7^{4}) and 1/(2^{6} 7^{5}) .
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.