Focusing Phenomena in Linear Discrete Inverse Problems in Acoustics

Abstract: The focusing operation inherent to the linear discrete inverse problem is formalised. The development is given in the context of sound-field reproduction where the source strengths are the inverse solution needed to recreate a prescribed pressure field at discrete locations. The behaviour of the system is fundamentally tied to the amount of acoustic crosstalk at each control point as a result of the focusing operation inherent to the pseudoinverse. The maximisation of the crosstalk at just one point leads to linear dependence in the system. On the other hand, its minimisation leads to the ideal focusing state wherein the sources can selectively focus at each point, while a null is created at all other points. Two theoretical case studies are presented that demonstrate ideal and super ideal focusing, wherein the latter the condition number is unitary. First, the application of binaural audio reproduction using an array of loudspeakers is examined and several cases of ideal focusing are presented. In the process, the Optimal Source Distribution is re-derived and shown to be a case of super ideal focusing. Secondly, the application of recreating multiple sound zones is examined using a uniform linear array. The conditions are derived to achieve ideal focusing at control points positioned arbitrarily in the far-field. In all cases, the ability to maintain ideal focusing as a function of frequency requires proportional changes in the source or control point geometry.