| مستخلص: |
Abstract: This paper is motivated by the following observation. Take a 3×3 random (Haar distributed) orthogonal matrix , and use it to “rotate” the north pole, say, on the unit sphere in . This then gives a point that is uniformly distributed on the unit sphere. Now use the same orthogonal matrix to transform , giving . Simulations reported in Marzetta et al. [Marzetta, T.L., Hassibi, B., Hochwald, B.M., 2002. Structured unitary space-time autocoding constellations. IEEE Transactions on Information Theory 48 (4) 942–950] suggest that is more likely to be in the northern hemisphere than in the southern hemisphere, and, moreover, that has higher probability of being closer to the poles than the uniformly distributed point . In this paper we prove these results, in the general setting of dimension , by deriving the exact distributions of the relevant components of and . The essential questions answered are the following. Let be any fixed point on the unit sphere in , where . What are the distributions of and ? It is clear by orthogonal invariance that these distributions do not depend on , so that we can, without loss of generality, take to be . Call this the “north pole”. Then is the first component of the vector . We derive stochastic representations for the exact distributions of and in terms of random variables with known distributions. [Copyright &y& Elsevier] |
