Robust Multiple Signal Classification via Probability Measure Transformation

Abstract

In this paper, we introduce a new framework for robust multiple signal classification (MUSIC). The proposed framework, called robust measure-transformed (MT) MUSIC, is based on applying a transform to the probability distribution of the received signals, i.e., transformation of the probability measure defined on the observation space. In robust MT-MUSIC, the sample covariance is replaced by the empirical MT-covariance. By judicious choice of the transform we show that: (1) the resulting empirical MT-covariance is B-robust, with bounded influence function that takes negligible values for large norm outliers, and (2) under the assumption of spherically contoured noise distribution, the noise subspace can be determined from the eigendecomposition of the MT-covariance. Furthermore, we derive a new robust measure-transformed minimum description length (MDL) criterion for estimating the number of signals, and extend the MT-MUSIC framework to the case of coherent signals. The proposed approach is illustrated in simulation examples that show its advantages as compared to other robust MUSIC and MDL generalizations.

Binary Hypothesis Testing via Measure Transformed Quasi-Likelihood Ratio Test

“…Similarly to the proof of Proposition 3 in [17] it can be shown that if there exists a finite positive constant M ∈ R, such that for all y ∈ C p :…”

“…Proof. By (17), (19), the non-singularity of Σ (u) 0 and Σ (u) 1 , inequality (1) in [44], and the triangle inequality:…”

“…By (14), (17), (18), (19) and assumption A-1, the test statistic is a non-degenerate random variable that can be written as:…”

Measure-Transformed Quasi-Maximum Likelihood Estimation

“…The following definition of a transformation on the parametric probability measure P X;θ parallels the transformation on a non-parametric distribution stated as Definition 1 in [29].…”

“…Under the proposed gener-alization we show that new estimators can be obtained that can gain sensitivity to higher-order statistical moments, resilience to outliers, and yet have the computational advantages of the first and second-order methods of moments. This generalization, called measure-transformed GQMLE (MT-GQMLE), is based on the probability measure transformation framework that was recently applied to canonical correlation analysis [27], [28], and multiple signal classification [29], [30]. It is worthwhile noting that knowledge of these MT-mean vectors and MT-covariance matrices, which establishes partial information about the underlying distribution, is analogous to the side information in the minimax estimation problem considered in [31].…”