Fast RLS-type algorithm for unbiased equation-error adaptive IIR filtering based on approximate inverse-power iteration

In this paper, a fast approximate inverse-power (AIP) iteration is applied to compute recursively the total least squares (TLS) solution for unbiased equation-error adaptive infinite impulse response (IIR) filtering, which is established by approximating the well-known inverse-power iteration with Galerkin method. The AIP algorithm is based on an interesting modification of the inverse-power iteration in which the first entry of the parameter vector is constrained to the negative one. The distinctive feature of the proposed algorithm lies in its high computational efficiency, which is characterized by efficient computation of the fast gain vector (FGV), adaptation of the interesting modification of the inverse-power iteration, and rank-one update of the augmented correlation matrix. The shift structure of the input data vector is exploited to develop a fast algorithm for computing the gain vector. This FGV algorithm can be implemented at a numerical cost lower than the well-known fast Kalman algorithm. Since the first entry of the parameter vector has been fixed as the negative one and the weight vector is updated along the input data vector, a very efficient AIP algorithm is obtained by using the FGV algorithm. The proposed AIP algorithm is of computational complexity O(L) per iteration. Moreover, with no need to use the well-known matrix-inversion lemma, the AIP algorithm has another attractive feature of numerical stability. The proposed algorithm is shown to have global convergence. Simulation examples are included to demonstrate the effectiveness of the proposed AIP algorithm.