On the Cramér-Rao bound for the constrained and unconstrained complex parameters

We derive a complex form of the unconstrained and constrained Cramer-Rao lower bound (CRB) of composite real parameters formed by stacking the real and imaginary part of the complex parameters. The derived complex constrained and unconstrained CRB is easy to calculate and possesses similar structure as in the real parameter case but with the real covariance, Jacobian and the Fisher information matrix replaced by complex matrices with analogous interpretations. The advantage of the complex CRB is that it is oftentimes easier to calculate than its real form. It is highlighted that a statistic that attains a bound on the complex covariance matrix alone do not necessarily attain the CRB since complex covariance matrix does not provide a full second-order description of a complex statistic since also the pseudo-covariance matrix is needed. Our derivations also lead to some new insights and theory that are similar to real CRB theory.