Blind separation of sound sources from the principle of least spatial entropy

The aim of the paper is to offer a method for separating incoherent and compact sound sources which may overlap in both the space and frequency domains. This is found of interest in acoustical applications involving the identification and ranking of sound sources stemming from different physical origins. The principle proceeds in two steps, the first one being reminiscent to source reconstruction (e.g. as in near-field acoustical holography) and the second one to blind source separation. Specifically, the source mixture is first expanded into a linear combination of spatial basis functions whose coefficients are set by backpropagating the pressures measured by an array of microphones to the source domain. This leads to a formulation similar, but no identical, to blind source separation. In the second step, these coefficients are blindly separated into uncorrelated latent variables, assigned to incoherent “virtual sources”. These are shown to be defined up to an arbitrary rotation. A unique set of sound sources is finally recovered by searching for that rotation (by conjugate gradient descent in the Stiefel manifold of unitary matrices) which maximizes their spatial compactness, as measured either by their spatial variance or their spatial entropy. This results in the proposal of two separation criteria coined “least spatial variance” and “least spatial entropy”, respectively. The same concept of spatial entropy, which is central to the paper, is also exploited in defining a new criterion, the entropic L-curve, dedicated to determining the number of active sound sources. The idea consists in considering the number of sources that achieves the best compromise between a low spatial entropy (as expected from compact sources) and a low statistical entropy (as expected from a low residual error). The proposed methodology is validated on both laboratory experiments and numerical data, and illustrated on an industrial example concerned with the ranking of sound sources on a Diesel engine. At the same time, its robustness to the estimated number of active sources is demonstrated.