Unitary space-time codes and the Cayley transform

A recently proposed method for communicating with multiple antennas over block fading channels is unitary space-time modulation (USTM), so-called because the transmitted signals form a matrix with orthonormal columns. Since channel knowledge is not required at the receiver, USTM schemes are suitable for use on wireless links where channel tracking is undesirable or infeasible. Recent results have shown that, if suitably designed, USTM schemes can achieve full channel capacity at high SNR. While all this is well recognized, what is not clear is how to generate good performing constellations of (non-square) unitary matrices, that lend themselves to efficient encoding/decoding. The schemes proposed so far either exhibit poor performance, especially at high rates, or have no efficient decoding algorithms. In this paper, we propose to use the Cayley transform to design USTM constellations. This work is a generalization, to the non-square case, of the Cayley codes that have been proposed for differential USTM. The codes are designed based on an information-theoretic criterion, and lend themselves to polynomial-time (often cubic) near-maximum-likelihood decoding using a sphere decoding algorithm.