A Riemannian Metric for Noncoherent Constellation Design and Its Application to Multiple Access Channel

We extend the study of the joint constellation design problem for noncoherent multiple-input multiple-output multipleaccess channels in Rayleigh block fading. First, we derive the pairwise error probability (PEP) exponent of the noncoherent maximum-likelihood detector to within a multiplicative factor of two. In particular, the lower bound of this exponent is obtained from a Chernoff upper bound of the error probability. Then, we show that the PEP exponent scales linearly with the Riemannian distance between the shifted Gram matrices of the symbols. This gives a geometric interpretation for our proposed metrics: a pair of joint symbols achieves a low PEP if the corresponding shifted Gram matrices are well separated, i.e., joined by a long geodesic, in the manifold of Hermitian positive definite matrices. Finally, we run numerical results to show that our metrics are meaningful for joint constellation design and evaluation, and result in constellations that outperform the constellations optimized with existing metrics and a pilot-based scheme.