Improper Complex-Valued Bhattacharyya Distance

Motivated by application of complex-valued signal processing techniques in statistical pattern recognition, classification, and Gaussian mixture (GM) modeling, this paper derives analytical expressions for computing the Bhattacharyya coefficient/distance (BC/BD) between two improper complex-valued Gaussian distributions. The BC/BD is one of the most widely used statistical measures for evaluating class separability in classification problems, feature extraction in pattern recognition, and for GM reduction (GMR) purposes. The BC provides an upper bound on the Bayes error, which is commonly known as the best criterion to evaluate feature sets. Although the computation of the BC/BD between real-valued signals is a well-known result, it has not yet been extended to the case of improper complex-valued Gaussian densities. This paper addresses this gap. We analyze the role of the pseudocovariance matrix, which characterizes the noncircularity of the signal, and show that it carries critical second-order statistical information for computing the BC/BD. We derive upper and lower bounds on the BD in terms of the eigenvalues of the covariance and pseudocovariance matrices of the underlying densities. The theoretical bounds are then used to introduce the concept of $\beta $ -dominance in the context of statistical distance measures. The BC is pseudometric, since it fails to satisfy the triangle inequality. Using the Matusita distance (a full-metric variant of the BC), we propose an intuitively pleasing indirect distance measure for comparing two general GMs. Finally, we investigate the application of the proposed BC/BD measures for GMR purposes and develop two BC-based GMR algorithms.