Bias of feedback cancellation algorithms in hearing aids based on direct closed loop identification

The undesired effects of acoustic feedback of hearing aids can be reduced with an internal feedback path that is an estimate of the external feedback path. This paper analyzes the limiting estimate of the feedback for feedback cancellation schemes that apply some recursive prediction error method with a quadratic norm, e.g., least mean square (LMS) and recursive least squares (RLS), to the output and input signals of the hearing aid to identify the feedback path. The data used for identification is then collected in closed loop and the estimate used in one recursion will affect the data used in succeeding recursions. These properties have to be considered in the analysis. The analysis shows that the limiting estimate may be biased if there is an error in the used model of the input signal to the hearing aid, and that the system is not identifiable unless a second input signal to the system is added to the output of the hearing aid or the signal processing of the hearing aid used to modify the signal to the impaired ear is nonlinear. The limiting estimate is presented as the solution to an optimization problem in the frequency domain. An analytical expression of the limiting estimate is presented for a special case. For other cases an algorithm is presented that can be used to find a numerical solution. The results can be useful when the model structure used with the recursive identification is chosen.