A New Derivation of Constrained Cramér–Rao Bound Via Norm Minimization

The constrained Cramér-Rao bound (CCRB) is widely used to evaluate the estimation performance for deterministic parameters with parametric constraints. An alternative derivation of the CCRB is presented from the perspective of solving a norm minimization problem under a set of linear constraints. The parametric constraints are embedded in the linear constraints by constructing a proper basis to represent the estimation error. This derivation avoids sophisticated matrix manipulations and has a clear physical meaning that the parametric constraints cut down the bases used to represent the estimation error and hence reduce the minimum norm of the error, and the bound can be achieved if and only if the error is in the subspace spanned by the bases. The result is applicable to biased estimators and singular Fisher information matrices.