Tensor polyadic decomposition for antenna array processing

In the present framework, a tensor is understood as a multi-way array of complex numbers indexed by three (or more) indices. The decomposition of such tensors into a sum of decomposable (i.e. rank-1) terms is called ''Polyadic Decomposition'' (PD), and qualified as ''canonical'' (CPD) if it is unique up to trivial indeterminacies. The idea is to use the CPD to identify the location of radiating sources in the far-field from several sensor subarrays, deduced from each other by a translation in space. The main difficulty of this problem is that noise is present, so that the measurement tensor must be fitted by a low-rank approximate, and that the infimum of the distance between the two is not always reached. Our contribution is three-fold. We first propose to minimize the latter distance under a constraint ensuring the existence of the minimum. Next, we compute the Cram{e}r-Rao bounds related to the localization problem, in which nuisance parameters are involved (namely the translations between subarrays). Then we demonstrate that the CPD-based localization algorithm performs better than ESPRIT when more than 2 subarrays are used, performances being the same for 2 subarrays. Some inaccuracies found in the literature are also pointed out.