Study of the asymptotic Cramér–Rao Bound for the COLD uniform linear array

In this paper, we study the Cocentered Orthogonal Loop and Dipole pairs Uniform Linear Array (COLD-ULA) which is sensitive to the source polarization in the context of the localization of time-varying narrow-band far-field sources. We derive and analyze nonmatrix expressions of the deterministic Cramér–Rao Bound (CRB(COLD)) for the direction and the polarization parameters under the assumption that all the sources are lying in the azimuthal plane. We denote this bound by ACRB(COLD), where the “A” stands for Asymptotic, meaning that the presented results are derived under the assumption that the number of sensors is sufficiently large. While, to our knowledge, closed-form (nonmatrix) expressions of the CRB(COLD) for multiple time-varying polarized sources signal do not exist in the literature, we show that the ACRB(COLD) takes a closed-form (nonmatrix) expression in this context and is a good approximation of the CRB(COLD) even if the number of sensor is moderate (about ten), if the source signals are not spatially too close. Our approach has two important advantages: (i) the computational complexity of the proposed closed-form of the bound is very low, compared to the brute force computation of a matrix-based deterministic CRB in case of time-varying model parameters and (ii) useful informations can be deduced from the closed-form expression on the behavior of the bound. In particular, we prove that the ACRB(COLD) for the direction parameter is not affected by the knowledge or the lack of it concerning the polarization parameters. Another conclusion is that with a COLD-ULA, more model parameters can be estimated than for the uniformly polarized ULA without degrading the estimation accuracy of the localization parameter. Finally, we also study the ACRB(COLD) for a priori known complex amplitudes.