Exact LTP representation of the generalized periodic-reference FxLMS algorithm

The analysis of the Filtered-x Least Mean Square (FxLMS) algorithm can be based either on a stochastic or deterministic approach. One of the drawbacks of the stochastic analysis is that it relies on several assumptions which are not justified if the reference signal is time periodic. To overcome these limitations, several deterministic methods have been suggested. However, useful results have been reported only for setups with certain filter length, secondary path model, or reference signal. In this paper, an exact Linear Time-Periodic (LTP) representation is proposed assuming only that the reference is synchronously-sampled. The representation is derived for a general FxLMS system, which also covers the multichannel topology of several parallel filters and its common narrowband modification. The representation is found to have a Linear Time-Invariant (LTI) form if the filter length is suitably chosen. The usability of the method is demonstrated in a numerically computed example by comparing it to an approximate LTI model.