On the CramrRan Bound Under Parametric Constraints

This letter presents a simple expression for the Cramér-Rao bound (CRB) for parametric estimation under differentiable, deterministic constraints on the parameters. In contrast to previous works, the constrained CRB presented here does not require that the Fisher information matrix (FIM) for the unconstrained problem be of full rank. This is a useful extension because, for several signal processing problems (such as blind channel identification), the unconstrained problem is unidentifiable. Our expression for the constrained CRB depends only on the unconstrained FIM and a basis of the nullspace of the constraint's gradient matrix. We show that our constrained CRB formula reduces to the known expression when the FIM for the unconstrained problem is nonsingular. A necessary and sufficient condition for the existence of the constrained CRB is also derived.