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Spacetime block codes from orthogonal designs
 IEEE Trans. Inform. Theory
, 1999
"... Abstract — We introduce space–time block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space–time block code and the encoded data is split into � streams which are simultaneously transmitted using � transmit antennas. ..."
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Cited by 996 (24 self)
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Abstract — We introduce space–time block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space–time block code and the encoded data is split into � streams which are simultaneously transmitted using � transmit antennas. The received signal at each receive antenna is a linear superposition of the � transmitted signals perturbed by noise. Maximumlikelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space–time block code and gives a maximumlikelihood decoding algorithm which is based only on linear processing at the receiver. Space–time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm. The classical mathematical framework of orthogonal designs is applied to construct space–time block codes. It is shown that space–time block codes constructed in this way only exist for few sporadic values of �. Subsequently, a generalization of orthogonal designs is shown to provide space–time block codes for both real and complex constellations for any number of transmit antennas. These codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as PAM. For an arbitrary complex constellation such as PSK and QAM, space–time block codes are designed that achieve IaP of the maximum possible transmission rate for any number of transmit antennas. For the specific cases of two, three, and four transmit antennas, space–time block codes are designed that achieve, respectively, all, QaR, and QaR of maximum possible transmission rate using arbitrary complex constellations. The best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well. Index Terms — Codes, diversity, multipath channels, multiple antennas, wireless communication.
Upper bounds of rates of complex orthogonal spacetime block codes
 IEEE Trans. Inform. Theory
, 2003
"... Abstract—In this correspondence, we derive some upper bounds of the rates of (generalized) complex orthogonal space–time block codes. We first present some new properties of complex orthogonal designs and then show that the rates of complex orthogonal space–time block codes for more than two transmi ..."
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Cited by 33 (7 self)
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Abstract—In this correspondence, we derive some upper bounds of the rates of (generalized) complex orthogonal space–time block codes. We first present some new properties of complex orthogonal designs and then show that the rates of complex orthogonal space–time block codes for more than two transmit antennas are upperbounded by Q R. We show that the rates of generalized complex orthogonal space–time block codes for more than two transmit antennas are upperbounded by R S, where the norms of column vectors may not be necessarily the same. We also present another upper bound under a certain condition. For a (generalized) complex orthogonal design, its variables are not restricted to any alphabet sets but are on the whole complex plane. In this correspondence, a (generalized) complex orthogonal design with variables over some alphabet sets on the complex plane is also considered. We obtain a condition on the alphabet sets such that a (generalized) complex orthogonal design with variables over these alphabet sets is also a conventional (generalized) complex orthogonal design and, therefore, the above upper bounds on its rate also hold. We show that commonly used quadrature amplitude modulation (QAM) constellations of sizes above R satisfy this condition. Index Terms—Complex orthogonal designs, complex orthogonal space– time block codes, Hermitian compositions of quadratic forms, Hurwitz family, Hurwitz–Radon theory. I.
On SpaceTime Block Codes from Complex Orthogonal Designs
, 2003
"... Spacetime block codes from orthogonal designs recently proposed by Alamouti, and TarokhJafarkhaniCalderbank have attracted considerable attention due to the fast maximumlikelihood (ML) decoding and the full diversity. There are two classes of spacetime block codes from orthogonal designs. One c ..."
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Cited by 25 (7 self)
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Spacetime block codes from orthogonal designs recently proposed by Alamouti, and TarokhJafarkhaniCalderbank have attracted considerable attention due to the fast maximumlikelihood (ML) decoding and the full diversity. There are two classes of spacetime block codes from orthogonal designs. One class consists of those from real orthogonal designs for real signal constellations which have been well developed in the mathematics literature. The other class consists of those from complex orthogonal designs for complex constellations for high data rates, which are not well developed as the real orthogonal designs. Since orthogonal designs can be traced back to decades, if not centuries, ago and have recently invoked considerable interests in multiantenna wireless communications, one of the goals of this paper is to provide a tutorial on both historical and most recent results on complex orthogonal designs. For spacetime block codes from both real and (generalized) complex orthogonal designs (GCODs) with or without linear processing, Tarokh, Jafarkhani and Calderbank showed that their rates cannot be greater than 1. While the maximum rate 1 can be reached for real orthogonal designs for any number of transmit antennas from the Hurwitz–Radon constructive theory, Liang and Xia recently showed that rate 1 for the GCODs (square or nonsquare size) with linear processing is not reachable for more than two transmit antennas. For GCODs of square size, the designs with the maximum rates have been known, which are related to the Hurwitz theorem. In this paper, We briefly review these results and give a simple and intuitive interpretation of the realization. For GCODs without linear processing (square or nonsquare size), we prove that the rates cannot be greater than 3/4 for more than two transmit antennas.
A treatise on quantum Clifford Algebras
"... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very e ..."
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Cited by 13 (10 self)
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on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A ‘spinorial ’ generalization of the antipode is given. The nonexistence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially generated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordization is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory. MSC2000: 16W30 Coalgebras, bialgebras, Hopf algebras; 1502 Research exposition (monographs, survey articles);
ON SPACES OF MATRICES CONTAINING A NONZERO MATRIX OF BOUNDED RANK
, 2002
"... Let Mn(R) and Sn(R) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n, n − 2, R): The minimal number ℓ such that every ℓdimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2, R) = 5 and obtain some upp ..."
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Cited by 4 (0 self)
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Let Mn(R) and Sn(R) be the spaces of n × n real matrices and real symmetric matrices respectively. We continue to study d(n, n − 2, R): The minimal number ℓ such that every ℓdimensional subspace of Sn(R) contains a nonzero matrix of rank n−2 or less. We show that d(4, 2, R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n − 2, R). We give upper bounds for the dimensions of n − 1 subspaces (subspaces where every nonzero matrix has rank n − 1) of Mn(R) and Sn(R), which are sharp in many cases. We study the subspaces of Mn(R) and Sn(R) where each nonzero matrix has rank n or n − 1. For a fixed integer q>1wefind an infinite sequence of n such that any ( q+1) dimensional sub2 space of Sn(R) contains a nonzero matrix with an eigenvalue of multiplicity at least q.
On orthogonal designs and spacetime codes
 In Proc. IEEE ISIT–02
, 2002
"... codes of size n × n offer maximum diversity gain advantage and a simple yet optimal decoding algorithm under an arbitrary signal alphabet or constellation A. However, these designs only exist for n = 2, 4, 8 when A is real and for n = 2 when A is complex. In this correspondence, we address the quest ..."
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Cited by 3 (0 self)
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codes of size n × n offer maximum diversity gain advantage and a simple yet optimal decoding algorithm under an arbitrary signal alphabet or constellation A. However, these designs only exist for n = 2, 4, 8 when A is real and for n = 2 when A is complex. In this correspondence, we address the question of the existence of ODST codes of other sizes when A is restricted to be a proper subset of either real or complex numbers. We refer to these as restrictedalphabet ODST (RAODST) codes. We show that real RAODST codes of size greater than 8 that also guarantee maximum diversity advantage do not exist. Without the maximumdiversityadvantage requirement, RAODST codes exist only when the signal constellation is binary and of the form {a, −a}, 0 < a ∈ R. Example of such binaryalphabet codes are provided for every n for which the existence of a Hadamard matrix is known. In the complex case, under the added requirement of maximum diversity advantage, we prove the nonexistence of complex RAODST codes under fairly simple assumptions regarding the signal alphabet. Keywords — Spacetime codes, orthogonal design, space time codes, MIMO, transmit diversity. I.
COMPARISON OF VOLUMES OF CONVEX BODIES IN REAL, COMPLEX, AND QUATERNIONIC SPACES
, 812
"... Abstract. The classical BusemannPetty problem (1956) asks, whether originsymmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ≤ 4 and negative if n> 4. The same question can be asked when volumes ..."
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Abstract. The classical BusemannPetty problem (1956) asks, whether originsymmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ≤ 4 and negative if n> 4. The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified exposition of this circle of problems in real, complex, and quaternionic ndimensional spaces. All cases are treated simultaneously. In particular, we show that the BusemannPetty problem in the quaternionic ndimensional space has an affirmative answer if and only if n = 2. The method relies on the properties of cosine transforms on the unit sphere. Possible generalizations are discussed. 1.
[3] R. E. Blahut, Theory and Practice of Error Control Codes. Reading,
, 1984
"... [8] D. T. Chi, “A new algorithm for correcting single burst errors with Reed ..."
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[8] D. T. Chi, “A new algorithm for correcting single burst errors with Reed
MATRIX INTERSECTION PROBLEMS FOR CONDITIONING
"... Abstract. Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. Computationally the problem is seemingly challenging. By associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nea ..."
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Abstract. Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. Computationally the problem is seemingly challenging. By associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nearness problem can be viewed to generalize the socalled Löwdin problem in quantum chemistry. For critical points in the Frobenius norm, a differential equation on the manifold of unitary matrices is derived. Another resulting matrix nearness problem allows locating points of optimality more directly, once formulated as a problem in computational algebraic geometry.
Contemporary Mathematics A Journey of Discovery: Orthogonal Matrices and Wireless Communications
"... Dedicated to Joseph Gallian on his 65th birthday and the 30th anniversary of his Duluth REU Abstract. Real orthogonal designs were first introduced in the 1970’s, followed shortly by the introduction of complex orthogonal designs. These designs can be described simply as square matrices whose column ..."
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Dedicated to Joseph Gallian on his 65th birthday and the 30th anniversary of his Duluth REU Abstract. Real orthogonal designs were first introduced in the 1970’s, followed shortly by the introduction of complex orthogonal designs. These designs can be described simply as square matrices whose columns are formally orthogonal, and their existence criteria depend on number theoretic results from the turn of the century. In 1999, generalizations of these designs were applied in the development of successful wireless communication systems, renewing interest in the theory of these orthogonal designs. This area of study represents a beautiful marriage of classical mathematics and modern engineering. This paper has two main goals. First, we provide a brief and accessible introduction to orthogonal design theory and related wireless communication systems. We include neither the mathematical proofs of the relevant results nor the technical implementation details, rather hoping that this gentle introduction will whet the reader’s appetite for further study of the relevant mathematics, the relevant engineering implementations, or, in the best case scenario, both. Second, in light of the dedication of this paper to Joe Gallian, who was an extraordinary undergraduate research advisor to so many of us and who inspired so many of us to become undergraduate research advisors ourselves, we describe the involvement of undergraduates in research involving these orthogonal designs and related communications systems. 1.