Linearly constrained Mean-Fourth error adaptive filtering for Deficient-Length systems: Analysis and performance characterization

The Least Mean Fourth (LMF) adaptive algorithm is recognized for its superior performance over the Least Mean Square (LMS) algorithm in non-Gaussian noise environments. However, its behavior under the practical and widespread condition of a deficient-length filter—where the adaptive filter is shorter than the unknown system—has remained largely unexplored. To our knowledge, this is the first comprehensive theoretical analysis of the deficient-length LMF algorithm. Using a Linearly Constrained Minimum Mean Fourth Error (LCMMFE) formulation, we present new and analytical models that characterize the mean as well as mean-square behavior of the algorithm. This includes closed-form expressions for the optimal weight vector, the minimum mean fourth error (MFE), and the steady-state error variance. The accuracy of these derivations is validated through extensive Monte Carlo simulations under Gaussian and non-Gaussian noise. Furthermore, a comparative analysis with the Variable Least-Mean Mixed-Norm (VLMMN) algorithm reveals a critical performance boundary: while VLMMN successfully combines the fast convergence of LMS with the low steady-state error of LMF in low-deficiency scenarios, severe filter deficiency introduces a dominant modeling error that masks these algorithmic advantages, forcing all algorithms to converge to a common performance floor. The framework is further benchmarked against recent robust non-Gaussian adaptive algorithms and validated on real electroencephalogram (EEG) data, confirming that the deficient-length LMF retains its predicted behavior under practical, non-ideal conditions.