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The Newton adaptive filtering algorithm in its original form is computationally very complex as it requires inversion of the input-signal autocorrelation matrix at every time step. Also, it may suffer from stability problems due to the inversion of the input-signal autocorrelation matrix. In this paper, we propose to replace the inverse of the input-signal autocorrelation matrix by an approximate one, assuming that the input-signal autocorrelation matrix is Toeplitz. This assumption would help us in replacing the update of the inverse of the autocorrelation matrix by the update of the autocorrelation matrix itself, and performing the multiplication of R<sup>-1</sup>x in the update equation by using the Fourier transform. This would increase the stability of the algorithm, in one hand, and decrease its computational complexity, on the other hand. Since the objective of the paper is to enhance the stability of the Newton algorithm, the performance of the proposed algorithm is compared to those of the Newton and the improved quasi-Newton (QN) algorithms in noise cancellation and system identification settings.

A Fast Quasi-Newton Adaptive Algorithm Based on Approximate Inversion of the Autocorrelation Matrix