Multi-dimensional scaling (MDS), Euclidean distance matrices (EDMs), matrix completion and outlier identification in array calibration and beamforming

Abstract

The matrix of inter-element distances, a Euclidean Distance Matrix (EDM), is symmetric with zero diagonal. It thus has N(N-1)/2 distinct values, which we know are a function of 3N-6 xyz coordinates. Indeed, the EDM for the elements of a 3-D array has rank 3. As a result, it is possible to reconstruct missing elements of the EDM, using a process called matrix completion. Array calibration (aka array element localization or AEL) is determining the relative 3-D coordinates of all array elements, so that they can be incorporated into various beamforming processes. The eigenvectors of a complete and noise-free EDM immediately yield the array element xyz coordinates, using a method called Multi-Dimensional Scaling (MDS). If an EDM has missing measurements and/or is noisy, its matrix completion can recover a complete and more accurate EDM that then yields better relative xyz coordinates. Alternatively, testing a noisy EDM for its know properties can be used to identify outlier measurements (among the inter-element distances), before AEL is performed. We will present the mathematics underlying EDMs, MDS, and matrix completion, and demonstrate AEL results for incomplete and noisy EDMs, using these concepts.