Median power and median correlation theory

We show that the maximum likelihood (ML) estimate of location under the Laplacian model, which forms the basis for weighted median filters, can be generalized to correlation estimates based on weighted medians. Much like linear sample correlations, the resultant median correlation estimates have a surprisingly simple structure. Unlike linear correlations, median correlations are robust to data contamination. Notably, weights in this framework do not assume fixed values as with weighted median filters but take on random values determined by the underlying data itself. The underlying parameters associated with the sample median correlations are obtained, leading to well-defined expressions that can be used in subspace-based signal processing algorithms. The properties of median correlations are illustrated through a number of simulations where the MUltiple SIgnal Classification (MUSIC) algorithm is applied on linear and median sample correlation matrices for real-valued frequency estimation applications. This paper thus unveils new and powerful capabilities of weighted medians for use in modern signal processing applications.