Exact Convergence Analysis of Adaptive Filter Algorithms Without the Persistently Exciting Condition

Exact convergence analysis of the recursive least square and least mean square (LMS) algorithms in adaptive filtering is presented for the case of sinusoidal signal cancellation without the persistently exciting condition. This situation occurs when the number of tap coefficients of the adaptive filter exceeds that of the complex sinusoids in the input signal. The convergent point of both algorithms is shown to be the one determined by the pseudo inverse of the deterministic covariance matrix. The convergence proof for the LMS algorithm is based on the Lyapunov function method. Finally, the validity of the obtained results is supported by simulation results