  Motivated by massive deployment of low data rate Internet of things (IoT) and
ehealth devices with requirement for highly reliable communications, this paper
proposes receive beamforming techniques for the uplink of a single-input
multiple-output (SIMO) multiple access channel (MAC), based on a per-user
probability of error metric and one-dimensional signalling. Although
beamforming by directly minimizing probability of error (MPE) has potential
advantages over classical beamforming methods such as zero-forcing and minimum
mean square error beamforming, MPE beamforming results in a non-convex and a
highly nonlinear optimization problem. In this paper, by adding a set of
modulation-based constraints, the MPE beamforming problem is transformed into a
convex programming problem. Then, a simplified version of the MPE beamforming
is proposed which reduces the exponential number of constraints in the MPE
beamforming problem. The simplified problem is also shown to be a convex
programming problem. The complexity of the simplified problem is further
reduced by minimizing a convex function which serves as an upper bound on the
error probability. Minimization of this upper bound results in the introduction
of a new metric, which is termed signal minus interference to noise ratio
(SMINR). It is shown that maximizing SMINR leads to a closed-form expression
for beamforming vectors as well as improved performance over existing
beamforming methods.
