Tracking analysis of the LMF and LMMN adaptive algorithms

Although the least mean fourth (LMF) and the least mean mixed norm (LMMN) adaptive algorithms are recommended for highly nonstationary environments, their tracking capabilities are not yet fully understood. This is mainly due to the fact that both algorithms involve nonlinear update equations for the weight error vector. We present a new approach to the tracking analysis of the LMF and LMMN algorithms, which bypasses the need for working directly with the weight error vector, and is based on a fundamental energy-preserving relation. By studying the energy flow through the system in steady-state, we derive expressions for the steady-state excess mean square error (EMSE) for both algorithms. We also derive optimal parameter values that minimize the EMSE in each case, and support our conclusions by simulations.