New expression for the functional transformation of the vector Cramér-Rao lower bound

Assume that it is desired to estimate α = f(θ), where f(·) is an r-dimensional function. This paper derives the general expression for the functional transformation of the vector Cramér-Rao lower bound (CRLB). The derived bound is a tight lower bound on the estimation of uncoupled parameters, i.e., parameters that can be estimated separately. Unlike previous results in the literature, this new expression is not dependent on the inverse of the Fisher´s information matrix (FIM) of the untransformed parameters, θ. Thus, it can be applied to scenarios where the FIM for θ is ill-conditioned or singular. Finally, as an application, the derived transformation is applied to determine the exact CRLB for estimation of channel parameters in amplify-and-forward relaying networks.