We consider the adaptive detection of a signal of interest embedded in colored noise, when the environment is nonhomogeneous, i.e., when the training samples used for adaptation do not share the same covariance matrix as the vector under test. A Bayesian framework is proposed where the covariance matrices of the primary and the secondary data are assumed to be random, with some appropriate joint distribution. The prior distributions of these matrices require a rough knowledge about the environment. This provides a flexible, yet simple, knowledge-aided model where the degree of nonhomogeneity can be tuned through some scalar variables. Within this framework, an approximate generalized likelihood ratio test is formulated. Accordingly, two Bayesian versions of the adaptive matched filter are presented, where the conventional maximum likelihood estimate of the primary data covariance matrix is replaced either by its minimum mean-square error estimate or by its maximum a posteriori estimate. Two detectors require generating samples distributed according to the joint posterior distribution of primary and secondary data covariance matrices. This is achieved through the use of a Gibbs sampling strategy. Numerical simulations illustrate the performances of these detectors, and compare them with those of the conventional adaptive matched filter.
We address the problem of detecting a signal of interest in the presence of noise with unknown covariance matrix, using a set of training samples. We consider a situation where the environment is not homogeneous, i.e., when the covariance matrices of the primary and the secondary data are different. A knowledge-aided Bayesian framework is proposed, where these covariance matrices are considered as random, and some information about the covariance matrix of the training samples is available. Within this framework, the maximum a priori (MAP) estimate of the primary data covariance matrix is derived. It is shown that it amounts to colored loading of the sample covariance matrix of the secondary data. The MAP estimate is in turn used to yield a Bayesian version of the adaptive matched filter. Numerical simulations illustrate the performance of this detector, and compare it with the conventional adaptive matched filter.
We consider the problem of estimating the covariance matrix of an observation vector, using heterogeneous training samples, i.e., samples whose covariance matrices are not exactly . More precisely, we assume that the training samples can be clustered into groups, each one containing snapshots sharing the same covariance matrix . Furthermore, a Bayesian approach is proposed in which the matrices are assumed to be random with some prior distribution. We consider two different assumptions for . In a fully Bayesian framework, is assumed to be random with a given prior distribution. Under this assumption, we derive the minimum mean-square error (MMSE) estimator of which is implemented using a Gibbs-sampling strategy. Moreover, a simpler scheme based on a weighted sample covariance matrix (SCM) is also considered. The weights minimizing the mean square error (MSE) of the estimated covariance matrix are derived. Furthermore, we consider estimators based on colored or diagonal loading of the weighted SCM, and we determine theoretically the optimal level of loading. Finally, in order to relax the a priori assumptions about the covariance matrix , the second part of the paper assumes that this matrix is deterministic and derives its maximum-likelihood estimator. Numerical simulations are presented to illustrate the performance of the different estimation schemes.
The problem of estimating the covariance matrix of a primary vector from heterogeneous samples and some prior knowledge is addressed, under the framework of knowledge-aided space-time adaptive processing (KA-STAP). More precisely, a Gaussian scenario is considered where the covariance matrix of the secondary data may differ from the one of interest. Additionally, some knowledge on the primary data is supposed to be available and summarized in a prior matrix. Two KA-estimation schemes are presented in a Bayesian framework whereby the minimum mean square error (MMSE) estimates are derived. The first scheme is an extension of a previous work and takes into account the nonhomogeneity via an original relation. In search of simplicity and to reduce the computational load, a second estimation scheme, less complex, is proposed and omits the fact that the environment may be heterogeneous. Along the estimation process, not only the covariance matrix is estimated but also some parameters representing the degree of a priori and/or the degree of heterogeneity. Performance of the two approaches are then compared using STAP synthetic data. STAP filter shapes are analyzed and also compared with a colored loading technique.
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