The modified Cramer-Rao bound in vector parameter estimation

TLDR

The paper extends the scalar modified Cramer‑Rao bound to vector parameter estimation in the presence of nuisance parameters. The authors derive a vector modified Cramer‑Rao bound and apply it to jointly estimate carrier frequency, phase, and symbol epoch of a linearly modulated signal in correlated impulsive noise, treating data symbols and noise power as nuisance parameters. The resulting MCRVB is generally looser than the conventional CRVB yet coincides in some practical cases; it provides tractable expressions where the conventional bound is infeasible and reproduces known results under white Gaussian noise.

Abstract

In this paper we extend the scalar modified Cramer-Rao bound (MCRB) to the estimation of a vector of nonrandom parameters in the presence of nuisance parameters. The resulting bound is denoted with the acronym MCRVB, where "V" stands for "vector". As with the scalar bound, the MCRVB is generally looser than the conventional CRVB, but the two bounds are shown to coincide in some situations of practical interest. The MCRVB is applied to the joint estimation of carrier frequency, phase, and symbol epoch of a linearly modulated waveform corrupted by correlated impulsive noise (encompassing white Gaussian noise as a particular case), wherein data symbols and noise power are regarded as nuisance parameters. In this situation, calculation of the conventional CRVB is infeasible, while application of the MCRVB leads to simple useful expressions with moderate analytical effort. When specialized to the case of white Gaussian noise, the MCRVB yields results already available in the literature in fragmentary form and simplified contexts.