Variable matrix-type step-size affine projection algorithm with orthogonalized input vectors

• An APA is realized with orthogonalized input vectors to implement it using sum of NLMS-like updating equations.
• We make the matrix-type step-size APA by adopting the individual step sizes corresponding to each of NLMS-like updating equations.
• We analyze the MSD of the matrix-type step-size APA that is different from the conventional APA.
• We derive the individual step sizes by minimizing the MSD to achieve the fastest convergence on every iteration.
• The algorithm works well even for badly-excited input signals because of the regularization-absorbed step sizes.

In this paper, we propose a variable matrix-type step-size affine projection algorithm (APA) with orthogonalized input vectors. We generate orthogonalized input vectors using the Gram–Schmidt process to implement the weight update equation of the APA using the sum of normalized least mean squares (NLMS)-like updating equations. This method allows us to use individual step sizes corresponding to each NLMS-like equation, which is equivalent to adopting the step size in the form of a diagonal matrix in the APA. We adopt a variable step-size scheme, in which the individual step sizes are determined to minimize the mean square deviation of the APA in order to achieve the fastest convergence on every iteration. Furthermore, because of the weight vector updated successively only along each innovative one among the reused inputs and effect of the regularization absorbed into the derived step size, the algorithm works well even for badly excited input signals. Experimental results show that our proposed algorithm has almost optimal performance in terms of convergence rate and steady-state estimation error, and these results are remarkable especially for badly excited input signals.