  We consider the joint constellation design problem for the noncoherent
multiple-input multiple-output multiple-access channel (MAC). By analyzing the
noncoherent maximum-likelihood detection error, we propose novel design
criteria so as to minimize the error probability. As a baseline approach, we
adapt several existing design criteria for the point-to-point channel to the
MAC. Furthermore, we propose new design criteria. Our first proposed design
metric is the dominating term in nonasymptotic lower and upper bounds on the
pairwise error probability exponent. We give a geometric interpretation of the
bound using Riemannian distance in the manifold of Hermitian positive definite
matrices. From an analysis of this metric at high signal-to-noise ratio, we
obtain further simplified metrics. For any given set of constellation sizes,
the proposed metrics can be optimized over the set of constellation symbols.
Motivated by the simplified metric, we propose a simple constellation
construction consisting in partitioning a single-user constellation. We also
provide a generalization of our previously proposed construction based on
precoding individual constellations of lower dimensions. For a fixed joint
constellation, the design metrics can be further optimized over the per-user
transmit power, especially when the users transmit at different rates.
Considering unitary space-time modulation, we investigate the option of
building each individual constellation as a set of truncated unitary matrices
scaled by the respective transmit power. Numerical results show that our
proposed metrics are meaningful, and can be used as objectives to generate
constellations through numerical optimization that perform better, for the same
transmission rate and power constraint, than a common pilot-based scheme and
the constellations optimized with existing metrics.
