Error Convergence of the Filtered-X LMS Algorithm for Multiple Harmonic Excitation

This paper presents a thorough analysis regarding the error signal convergence of the filtered-X LMS algorithm based on the approach of considering a diagonalized system describing the adaptive filter misalignment. For applications where the number of adaptive filter coefficients is chosen to be greater than or equal to the ratio of the fundamental disturbance period to the sampling period, a simple mathematical description of the error signal is derived that approximates the actual convergence process with numerically given error bounds. The analysis consists of two parts: a formal description of the error convergence using suitable expressions for eigenvalues, eigenvectors, and initial state of the filter misalignment; and a rearrangement to a form that is no longer exactly diagonal, but depends only on known values and allows for an approximation of the ideal convergence process. Finally, the exact and approximate convergence behavior are compared in simulations.