A direct derivation of transforms for wave-domain adaptive filtering based on circular harmonics

Wave-domain adaptive filtering (WDAF) is advantageous for modeling loudspeaker-enclosure-microphone systems with many loudspeakers and microphones (MIMO LEMS). The necessary transforms may be derived by considering free field propagation between loudspeaker and microphones first, followed by wave field analysis of the true and simulated microphone signals. However, transforms obtained following this strategy are not invertible when there are more loudspeakers than microphones. As invertibility of the transforms is required for many applications, this restricts the applicability of WDAF. In this paper we propose a strategy to describe the free field propagation directly in the wave domain, so that invertible transforms can be defined independently from the number of actually used microphones. With this strategy, transforms for a MIMO LEMS comprising a uniform circular microphone array and an arbitrary loudspeaker array are derived.