Tensor Formulation of the Cramer-Rao Lower Bound for Beamspace Channel Estimation in mmWave MIMO-OFDM

In this paper, we study the deterministic Cramer-Rao lower bound (CRLB) of channel estimation in the reduced-dimensional DFT beamspace for MIMO-OFDM-based systems and propose a new tensor formulation of it. We also present a compact procedure to compute the deterministic CRLB for 3-D channel estimation that can be further extended to an arbitrary number of dimensions. Moreover, we summarize the derivation of the deterministic CRLB, where we show that parts of the procedure to calculate the CRLB can be efficiently represented by tensors. For instance, the derivative of the channel tensor with respect to a scalar parameter can be written in the form of a Canonical Polyadic Decomposition (CPD), while for the calculation of the Fisher Information Matrix (FIM) the generalized tensor inner product can be used instead of operations with unfoldings. The new formulation simplifies the procedure to calculate the CRLB and requires a smaller number of operations compared to conventional approaches based only on linear algebra. Simulation results confirm the new formulation and present the behavior of the CRLB for different scenarios.