Abstract:
A group of unitary matrices is called fixed-point-free (fpf) if all
non-identity elements of the group have no eigenvalues at unity. Such groups
are useful in multiple-antenna communications, especially in multiple-antenna
differential modulation, since they constitute a fully-diverse constellation.
In a previous work, all finite fpf groups were classified.
In this note we consider infinite groups and, in particular, their most
interesting case, Lie groups. Two such fpf Lie groups are
currently widely used in communications: the group of unit modulus scalars,
from which various phase modulation schemes, such as QPSK, are derived,
and the 2x2 orthogonal designs of Alamouti, on which many
two-transmit-antenna schemes are based. In Lie-group-theoretic jargon these
are referred to as U(1) and SU(2). A natural question is whether there
exist other fpf Lie groups. We answer this question in the negative:
U(1) and SU(2) are all there are.
Status:
Technical Memorandum, Bell Laboratories, Lucent Technologies.
Submitted to IEEE Trans. Info. Theory.
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