Cramér-Rao Bounds: The Adaptive Array Story
Christ D. Richmond
MIT Lincoln Laboratory
Abstract Reed, Mallett, and Brennan posed the classical problem of adaptive detection in 1974, and Kelly more formally in 1985 with the addition of target parameter estimation. Nearly thirty years later many aspects and modifications of this problem have been addressed and published, including nontrivial deviations from the original problem statement such as detection in the presence of non-Guassianity, array manifold mismatch, and inhomogeneities. In most cases it is of interest to quantify detection and estimation losses resulting from these types of deviations from the ideal case. Parameter estimation naturally follows target detection. Cramér-Rao bounds (CRBs), yielding theoretical limits on achievable accuracy, are often used in practice to ascertain the effectiveness of a proposed parameter estimation algorithm. Extensive research on CRBs exists in the literature. CRBs on target parameter estimates for the classical adaptive detection/estimation problem posed by Reed et. al. and Kelly in the constant (Swerling 0) target case are in fact immediately deducible from the results of Zeira et. al. (real Gaussian processes, 1990) and Francos et. al. (complex Gaussian processes, 1995). They demonstrate that for constant targets the colored noise (i.e. white noise plus interference) covariance is decoupled from the target parameters via use of the well-known Slepian-Bang formula. Thus, colored noise-only training losses are not reflected in the CRBs for Swerling 0 target parameters. In this paper we briefly outline the argument for this and likewise extend analysis to include targets with Swerling II fluctuations. In the case of Swerling II targets the colored noise parameters are coupled to those of the signal, and thus colored noise-only training losses are reflected in the CRBs for target parameters. We find, however, that both the Swerling II fluctuation loss and colored noise-only training losses are quite neglible away from interference, but can be pronounced near interference.
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