Adaptive filter design for cyclostationary processes associated with sine-wave extraction operation

Abstract

Cyclostationary signals have been generally used in condition monitoring and fault diagnosis of bearings, for which the adaptive filter is important in obtaining the signal-of-interest cyclostationary process from a disturbed signal. However, due to the nonstationarity of cyclostationary signals, their adaptive filters are usually time-varying and difficult to be implemented. In this study, a new optimum filter is obtained by converting the time-varying optimization function to be time-invariant using a sine-wave extraction (SE) operator, for which the adaptive implementation is designed and analyzed. The time-varying linear minimum mean-square error is converted to be time invariant using the SE operator, and its solution is time invariant but without explicit expression. Subsequently, an adaptive implementation, called the SE least mean square (SE-LMS), is introduced, for which the mean and weighted mean-square-deviation (MSD) are explicitly expressed. The scope of step size is determined to obtain a mean-convergent and mean-square-stable SE-LMS. The steady-state SE-operated excess mean-square-error is analyzed, and its explicit expression is obtained. The SE-LMS algorithm uses only a single cyclic frequency, based on which, the combined SE-LMS algorithm is introduced for a cyclostationary signal with multiple cyclic frequencies. Finally, several simulations are designed to verify the superiority of the proposed estimators over the time-average operator-based estimator.