{"references": ["B. Widrow and S.D. Stearns, Adaptive Signal Processing. Englewood Cliffs, Precentile -Hall, 1985.", "S. Haykin, Adaptive Filter Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall, 2002.", "J. Nagumo and A. Noda, \"A learning method ofr system identification\", IEEE Trans. Automat. Control, 12 (1967)282-287.", "S.J. Orfanidis, Optimum Signal Processing, An Introduction. Mac Milan, New York, 1985.", "K. Ozeki and T. Umeda, \"An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties,\" Electron. Commun. Jpn., vol 67-A, no. 5, pp.19-27, 1984", "F. Yu and M. Bouchyard, \"Recursive least-squares algorithms with good numerical stability for multichannel active noise control\" vol. 5, pp. 3221C3224. ICASSP 2001", "S. L. GaY AND s. Tavathia, \"The fast affine prjection algorithm.\" in Proc. ICASSP, PP. 3023C3026, May 1995.", "H. C. Shin and A. Y. Sayed, \"Transistent behavior of affine projection algorithms\", ICASSP, VI 353-356, 2003", "s. G. Sankaran and A. A. Beex, \"Convergence behavior of affine projection algorithms, \" IEEE Trans. Signal Processing, vol. 48, no. 4, pp.1086-1096, 2000\n[10] R. W. Harris, D. Chabries, and F. Bishop, \"A variable step (VS) adaptive filter algorithm\". IEEE Trans. Acoust. Speech Signal Process. vol. 34, i2. 309-316, 1986\n[11] A. Mader, H. Puder, and G. U. Schmidt, Step-size control for acoustic echo cancellation filters- An overview\", Signal Process., vol 80., pp. 1697-1719, Sept 2000.\n[12] H. C. Shin A. H. Sayed and W. J. Song, \"Variable step-size NLMS and affine projection algorithms, IEEE Signal Proc. Letters, Vol. 11, No. 2, pp. 132-135, Feb. 2004.\n[13] K. Ikeda, \"Convergence analysis of block orthogonal projection and affine projection algorithms\", Signal Processing, 82, 491-496, 2002.\n[14] S. L. Gay, \"Affine projection algorithm\", in Least-Mean-Square Adaptive Filters, ISBN 0-471-21570-8, Ed. S. Haykin and B. Widrow, Wiley interscience, 2003."]}

This paper presents a forgetting factor scheme for variable step-size affine projection algorithms (APA). The proposed scheme uses a forgetting processed input matrix as the projection matrix of pseudo-inverse to estimate system deviation. This method introduces temporal weights into the projection matrix, which is typically a better model of the real error's behavior than homogeneous temporal weights. The regularization overcomes the ill-conditioning introduced by both the forgetting process and the increasing size of the input matrix. This algorithm is tested by independent trials with coloured input signals and various parameter combinations. Results show that the proposed algorithm is superior in terms of convergence rate and misadjustment compared to existing algorithms. As a special case, a variable step size NLMS with forgetting factor is also presented in this paper.