Weak–Noise Modulation–Estimation of Vector Parameters

We address the problem of modulating a parameter onto a power–limited signal, transmitted over a discrete–time Gaussian channel and estimating this parameter at the receiver. Continuing an earlier work, where the optimal trade–off between the weak–noise estimation performance and the outage probability (threshold–effect breakdown) was studied for a single (scalar) parameter, here we extend the derivation of the weak–noise estimation performance to the case of a multi–dimensional vector parameter. This turns out to be a non–trivial extension, that provides a few insights, and it has some interesting implications, which are discussed in depth. Several modifications and extensions of the basic setup are also studied and discussed.