An evaluation of the Bayesian CRLB for time-varying MIMO channel estimation using complex-valued differentials

In this paper, we derive the Cramer-Rao Lower Bound (CRLB) within the Bayesian setup for time-varying multiple-input/multiple-output channel estimation. We consider the transmitting signal to be composed of linearly precoded information symbols and training symbols such that they share the same channel resource. This framework consisting of linearly precoded data and super-imposed training sequences is referred to as an affine-precoding scheme. In order to facilitate the characterization of time-varying channels parsimoniously, basis-expansion models (BEMs) have been widely applied. We consider a complex-exponential BEM to describe the time-varying channel affecting the transmitted blocks. Following this system model, we utilize the theory of complex-valued differentials to perform complex-valued differentiation. Specifically, we use a generalized chain rule for differentiating a complex scalar quantity with respect to a complex vector to derive the Fisher Information Matrix (FIM). Finally, simulation results are presented to explain the trends in the CRLBs for two time-varying channels.