An alternate view of transmit diversity for the collocated MIMO system

We present an alternate view of transmit diversity for the Multiple Input Multiple Output (MIMO) system with collocated antennas in terms of certain subspaces determined by the signal set used. For a MIMO system with n_t antennas at the transmitter, with spatial multiplexing, deep fades are shown to occur when all the rows of the channel fade coefficient matrix fall in a finite number of n_t-1 dimensional vector subspaces of C^nt referred as the singular fade subspaces, which depend on the signal set used. It is the presence of these harmful singular fade subspaces which results in a transmit diversity order 1 for a MIMO system with spatial multiplexing. For a Space Time Block Code (STBC), the singular fade subspaces depend on the design matrix of the STBC as well as the signal set used. Full transmit diversity STBCs ensure a transmit diversity order of n_t for a signal set, by totally removing all the n_t-1 dimensional singular fade subspaces which were present for the MIMO spatial multiplexing system. In general, for an STBC over a signal set, if there exists a singular fade subspace of dimension d, where 0 ≤ d <; n_t, there would exist a pair-wise error event whose transmit diversity order is n_t-d. The notion of singular fade subspaces is extremely helpful for the design of network coding schemes in a wireless two-way relaying scenario, in which two end nodes exchange bi-directional information with the help of a relay.