Ideal focusing and optimally-conditioned systems in sound field control with loudspeaker arrays

The focusing operation inherent to sound field control based on inverse filters is a fundamental aspect of the overall sound field reproduction that completely determines the system robustness and stability. The behaviour of the system is fundamentally tied to the amount of acoustic crosstalk at each control point that manifests as a result of the focusing operation. The maximisation of the crosstalk at just one point in space leads to linear dependence in the plant matrix and system instability. 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 plant matrix is optimally-conditioned. First, the application of binaural audio reproduction using an array of loudspeakers is examined and cases of ideal focusing are presented. In the process, the Optimal Source Distribution is re-derived and shown to be a specific case of super ideal focusing. Secondly, the application of recreating multiple sound zones using a linear array is examined and the conditions to achieve super ideal focusing at control points angled arbitrarily in the far-field are derived. In all cases, the ability to maintain ideal focusing as a function of frequency requires proportional changes in the source or control point geometry.