A generalized family of saturation composition cost function based robust adaptive filters

A substantial body of literature has been devoted to the development of novel cost functions for robust adaptive filtering algorithms. These algorithms differ in computational complexity, the number of hyperparameters, convergence speed, and steady-state misalignment. Popular algorithms such as the least-mean fourth (LMF), logarithmic least mean square (LMLS), and generalized maximum correntropy criterion (GMCC) primarily rely on statistical and information-theoretic measures of the error signal. However, a significant portion of recent literature does not provide an explicit theoretical foundation for the proposed cost functions. In this work, we present a simple yet powerful approach for designing robust cost functions for adaptive algorithms, wherein the cost function is expressed as a composition of two functions that satisfy specific properties of monotonicity and convexity. Using this method, many standard robust cost functions can be represented within this framework. Additionally, we propose two new families of robust cost functions based on the hyperbolic tangent and hyperbolic secant functions. Theoretical closed-form expressions for the bounds on the adaptation parameter rate and the steady-state misalignment of the adaptive filtering algorithms based on the proposed cost function families have been derived and validated through simulations. Extensive simulations, involving channel estimation and direction-of-arrival (DOA) estimation tasks, demonstrate that the proposed algorithm family outperforms state-of-the-art cost functions.