Gradient-based adaptive algorithms for systems with external feedback paths

The effect on error surface gradient of feedback from output to input of an adaptive filter in a system identification configuration is explored. This feedback can occur in any application of adaptive filters in which the output of the filter is radiated in some way so that it may leak back into the filter input. One example where this can happen is in the active control of acoustic noise, where it is often not possible acoustically isolate the secondary source (fed by the output of the adaptive filter) from the sound field detector (which provides the input signal for the adaptive filter). It is shown that, since the error surface is altered by the presence of the feedback, the common gradient algorithms are no longer generally appropriate. The circumstances when they will or will not work are delineated