Cramér-Rao Bounds: The Adaptive Array Story

Christ D. Richmond

MIT Lincoln Laboratory

Lexington, MA

Email: christ@ll.mit.edu

**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.