New Results on Deterministic Cramér–Rao Bounds for Real and Complex Parameters

Misspecified Cramér-rao bounds for complex unconstrained and constrained parameters

“…Also in this case, it is trivial to verify that, when there is no mismatch, i.e. when B θ 0 = −A θ 0 , the bound in (39) is equal to the one derived [7] and [9].…”

“…f (v, v * ). To this end, we will make extensive use of the following two mappings ( [6], [9], [16] and [17]). We define as the real representation of v the mapping:…”

“…Specifically, we will not derive any new covariance inequality directly in the complex field (as done e.g. in [8] and [9] for the classical CRB) but, relying on the interrelated representations of vectors in C N , C 2N and R 2N , we will transform the real MCRB in its complex form. Although less general than the framework detailed in [9], our approach avoids technical complications implied by reproducing the proofs given in [4] and [3] for real parameters directly on the complex field.…”

“…Consequently, there exists ∆ ∈ C 2d×(2d−2k) whose columns form an orthonormal basis for the null space of F θ , i.e. F θ ∆ = 0 and ∆ H ∆ = I ( [7], [9]). The following theorem provide the complex form of the CMCRB.…”

A constrained hybrid Cramér-Rao bound for parameter estimation

“…However, in many applications, the vector of unknown parameters is constrained to lie in a proper non-open subset of the original parameter space. Since then, numerous works [8] have been devoted to extend the results introduced in [7]: 1) by providing useful technical results such as a general reparameterization inequality and the equivalence between parameterization change and equality constraints; 2) by studying the CRB modified by constraints either required by the model or required to solve identifiability issues; 3) by investigating the use of parameters constraints from a different perspective: the value of side (a priori) information on estimation performance. All these works have shown the versatility of deterministic constrained Cramér-Rao bound (CCRB) for estimation performance analysis and design of a system of measurement.…”