Structured Unitary Space-Time Autocoding Constellations

We recently showed that arbitrarily reliable communication is possible within a single coherence interval in Rayleigh flat fading, where no one has knowledge of the propagation matrix, as the symbol-duration of the coherence interval and the number of transmit antennas grow simultaneously. This effect, where the space-time signals act as their own channel codes, is called autocoding. For relatively short (e.g., 16-symbol) coherence intervals, a codebook of isotropically random unitary space-time signals theoretically supports transmission rates that are a significant fraction of autocapacity with an extremely low probability of error. However a constellation of the required size (typically L = 2 sup (80)) is impossible to generate and store, and due to lack of structure there is little hope of finding a fast decoding scheme. In this paper we propose a random, but highly structured, constellation that is completely specified by log sub 2 L independent isotropically distributed unitary matrices. The distinguishing property of this construction is that any two signals in the constellation are pairwise statistically independent and isotropically distributed. Thus, the pairwise probability of error, and hence the union bound on the block probability of error, of the structured constellation is identical to that of a fully random constellation of independent signals. As part of this work we have established a subsidiary result that is interesting in its own right: the square (or for that matter, any integer power greater than one) of an isotropically random unitary matrix is not isotropically random, with two exceptions: 1) a one-by-one complex unitary matrix, and 2) a two-ty-two real orthogonal matrix.