Analysis of adaptive feedback and echo cancelation algorithms in a general multiple-microphone and single-loudspeaker system

In this paper, we analyze a general multiple-microphone and single-loudspeaker system, where an adaptive algorithm is used to cancel acoustic feedback/echo and a beamformer processes the feedback/echo canceled signals. This system can be viewed as part of a typical hearing aid system and/or a traditional acoustic echo cancellation system. We introduce and derive an approximation of a useful frequency domain measure - the power transfer function - and show how to predict the system stability bound, convergence rate and the steady-state behavior across time and frequency. Furthermore, we show how the derived expressions can be used to determine e.g. the step size parameter in the adaptive algorithms to achieve a desired system property e.g. convergence rate at a specific frequency.