This paper explores the properties of Bearing Persistently Exciting (BPE) formations defined in d(d≥2)-dimensional space, a class of time-varying bearing formations whose configuration can be uniquely determined up to a translational Euclidean vector using only bearing and velocity measurements. By analyzing the bearing Laplacian matrix in relation to PE conditions on the bearings, we provide results on necessary conditions, and necessary & sufficient conditions to guarantee that a formation is BPE. We also explore the concept of Relaxed Bearing Rigidity (RBR), a particular case of BPE formation, which relaxes the classical conditions on interaction topologies required by Bearing Rigidity to maintain a rigid shape. Distributed estimators are also proposed to estimate the configuration of the BPE formation using bearing and velocity measurements. Simulation results are provided to validate the proposed estimators.