Exact convergence analysis of adaptive filter algorithms without persistently exciting condition

Exact convergence analysis of the RLS and 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 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.