On posterior distributions for signals in Gaussian noise with unknown covariance matrix

Abstract

Abstract-A Bayesian approach to estimate parameters of signals embedded in complex Gaussian noise with unknown color is presented. The study specifically focuses on a Bayesian treatment of the unknown noise covariance matrix making up a nuisance parameter in such problems. By integrating out uncertainties regarding the noise color, an enhanced ability to estimate both the signal parameters as well as properties of the error is exploited. Several noninformative priors for the covariance matrix, such as the reference prior, the Jeffreys prior, and modifications to this, are considered. Some of the priors result in analytical solutions, whereas others demand numerical approximations. In the linear signal model, connections are made between the standard Adaptive Maximum Likelihood (AML) estimate and a Bayesian solution using the Jeffreys prior. With adjustments to the Jeffreys prior, correspondence to the regularized solution is also established. This in turn enables a formal treatment of the regularization parameter. Simulations indicate that significant improvements, compared to the AML estimator, can be obtained by considering both the derived regularized solutions as well as the one obtained using the reference prior. The simulations also indicate the possibility of enhancing the predictions of properties of the error as uncertainties in the noise color are acknowledged.

Knowledge-Aided Adaptive Coherence Estimator in Stochastic Partially Homogeneous Environments

“…Recently, a Bayesian approach to the detection problem emerged [3], [4], where the covariance matrix is assumed to be randomly distributed with some prior distribution. The resulting detectors are often referred to as knowledgeaided (KA) detectors for the stochastic homogeneous environment.…”

“…Furthermore, we assume the covariance matrix R to be random and has a complex inverse Wishart distribution, i.e., R ∼ CW −1 ((µ − N )R, µ) [3], [4], [9]:…”

“…If λ = 1, the stochastic partially homogeneous model reduces to the stochastic homogeneous model [3], [4]. According to the stochastic partially homogeneous model, the scale-invariant generalized likelihood ratio test (GLRT) is developed within a Bayesian framework.…”

Bayesian Complex Amplitude Estimation and Adaptive Matched Filter Detection in Low-Rank Interference

“…Intrinsic Cramér-Rao bounds (CRBs) for low-rank subspace estimation are developed in [1] and low-rank subspace tracking is discussed in [11]. Bayesian and non-Bayesian approaches for complex amplitude estimation have been developed and analyzed in [12]- [13] and [14]- [17] (see also references therein) assuming unstructured covariance matrix of interference and noise. A Bayesian approach for estimating interference-plus-noise covariance matrices in knowledge-aided radar is outlined in [18], where numerous examples of available a priori information are given for the radar problem.…”

“…A Bayesian approach for estimating interference-plus-noise covariance matrices in knowledge-aided radar is outlined in [18], where numerous examples of available a priori information are given for the radar problem. The methods in [12]- [18] ignore the low-rank structure of the interference. In this paper, we develop an iterated conditional modes (ICM) algorithm for Bayesian estimation of complex signal amplitudes in low-rank interference and propose a (non-Bayesian) adaptive-matched-filter (AMF) detector that utilizes the ICM estimates of the unknown parameters.…”