Riemannian coding for covariance interpolation in massive MIMO frequency division duplex systems

In the context of multi-user Massive MIMO frequency division duplex (FDD) systems, the acquisition of channel state information cannot benefit from channel reciprocity. However, it is generally expected that covariance information about the downlink channel must be estimated and fed back by the user equipment (UE). As an alternative, it was also proposed to infer the downlink covariance based on the observed uplink covariance and a stored dictionary of uplink/downlink covariance matrices. This inference was performed through an interpolation in the Riemannian space of Hermitian positive definite matrices. We propose to rewrite the interpolation step as a Riemannian coding problematic. In this framework, we estimate the decomposition of the observed uplink matrix in the dictionary of uplink matrices and recover the corresponding downlink matrix assuming that its decomposition in the dictionary of downlink matrices is the same. Moreover, since this space is of large dimension in the Massive MIMO setting, it is expected that these decompositions will be sparse. We then propose new criteria based on this further constraint.