Diversity Analysis of Group Vector Perturbation Precoding

We study the effect on diversity of using the sphere encoder-based vector perturbation (VP) technique of Peel and the lattice reduction-aided techniques (Babai approximations) of Windpassinger on a subset of the transmitted symbols. For a system with
n
diversity channels, we prove that the diversity order gap between the maximum diversity of
n
achieved by the full-size VP and the minimum diversity of 1 provided by linear precoding can be bridged by applying VP to a group of transmitted symbols. In particular, we prove that the diversity order is
n
-
l
, where
l
is the number of symbols that are not vector-perturbed. We show that this result holds true for lattice-reduction with both the rounding-off and nearest plane Babai approximations. The result shows that in terms of diversity, the group of vector-perturbed users counts as a single user. In fact, in a multiple-input and multiple-output Gaussian broadcast channel with
l
linearly precoded users, we can add up to
n
-
l
VP users and the diversity order for all users will decrease only by one, irrespective of how many VP users are added.