This work is concerned with the analysis and development of a class of robust adaptive filtering algorithms which are capable of operating effectively in an environment of impulsive noise. Adaptive signal processing and particularly adaptive filtering involving linear filters provides a powerful approach to many signal processing problems. The capacity of adaptive algorithms to operate when limited a priori information is available makes them ideally matched to many practical applications. The performance of linear adaptive filters is, however, severely degraded when impulsive noise is present in the filter inputs. Such impulsive interference occurs frequently in many applications. The work here is concerned with the development of a class of nonlinear adaptive filters which are insensitive to the presence of such interference. This is achieved through a proposed Median Least-Mean- Squares (MLMS) algorithm. This algorithm operates by replacing the instantaneous estimate of the gradient of the mean-squared error performance surface (as used by the well-known LMS adaptive filter) by the sample median of that quantity. In this work, the feasibility of the MLMS is established and a theory for the MLMS, both for random and deterministic inputs, is developed. A general class of robust adaptive filtering procedures based on order statistics (of which the MLMS is a single member) is developed. The properties of key order statistical adaptive filters including the kth rank and trimmed mean are established, and algorithms designed to facilitate adaptation to optimize the order statistical filter in relation to the statistics of the input are developed.
