A fast affine projection algorithm based on a modified Toeplitz matrix

This paper presents a stable approach to solve the linear system of equations associated with the affine projection (AP) algorithm. Previously, the correlation matrix was approximated as a Toeplitz matrix and computationally efficient methods can be used to solve the inverse of the matrix. However, it was also found that a relatively large regularization parameter should be employed to ensure the algorithm's stability even for a float-point implementation, which also slows down the algorithm's convergence rate. We point out that the instability of the algorithm is mainly attributed to that the condition number of the Toeplitz matrix is much larger than that of the original correlation matrix. Motivated by the schemes used in speech coders, three approaches are proposed to improve the stability of the inverse of the Toeplitz matrix. Simulation results confirm that the time-windowing method achieves the best performance.