عنوان مقاله [English]
One of the most important challenges for the adaptive filtering is the slow convergence rate of adaptive algorithm against highly correlated input signals. The affine projection adaptive algorithm (APA) is an extension of the well-known normalized least mean square (NLMS) algorithm which achieves a higher convergence rate in both full-band and sub-band structures. In this paper, to further improve the convergence rate of the algorithm against high-correlation input signals in the application of sparse linear system modeling, a sub-band APA is proposed in which the number of projection vectors is determined as a function of the estimated sub-band mean square error (MSE). In addition, variable sub-band step-sizes are proposed as a function of filter weights and the estimated MSE such that at the initial convergence stage, bigger weights make an increased contribution to the adaptation process while during convergence, the contributions approach the same amount. The proposed idea improves the convergence rate and lowers the steady-state MSE. Simulation results for the sparse linear system modeling and for the application of acoustic echo cancellation, verify the superiority of the proposed algorithm in the convergence performance and estimation accuracy of the acoustic path coefficients over its counterparts.
 B. Farhang-Boroujeny, Adaptive Filters: Theory and Applications, 2nd ed., John Wiley & Sons, USA, 2013.
 حسین شریف زاده و نیما امجدی، " مروری بر انواع الگوریتمهای فراکاوشی در بهینهسازی"، نشریه مدلسازی در مهندسی، دوره 12، شماره 83 ، پاییز 1393، صفحه 28-43.
 M. Bekrani, R. Bibak, and M. Lotfizad, “Improved clipped affine projection adaptive algorithm”, IET Signal Processing, Vol. 13, No. 1, Feb. 2019, pp. 103 –111.
 K. Ozeki, and T. Umeda, “An adaptive filtering algorithm using an orthogonal projection to an affine subspace and its properties”, Electronics and Communications in Japan.,Vol. 67, No. 5, May 1984, pp 19-27.
 F. Huang, J. Zhang, and S. Zhang, “Combined-step-size affine projection sign algorithm for robust adaptive filtering in impulsive interference environments”, IEEE Transactions on Circuits and Systems, vol. 63, No. 5, May 2016, pp 493-497.
 Z. Wang, and H. Zhao, “An improved affine projection subband adaptive filter”, 35th Chinese Control Conference, China, July 2016, pp. 3162-3165.
 K. Lee, Y. Baek, and Y. Park, “Nonlinear acoustic echo cancellation using a nonlinear postprocessor with a linearly constrained affine projection algorithm”, IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 62, No. 9, September 2015, pp. 881-885.
 V. A.Niţă, R. A. Dobre, S. Ciochina, and C. Paleologu, “Improved convergence model of the affine projection algorithm for system identification”, International Symposium on Signals, Circuits and Systems, Romania, July 2017.
 S. L. Gay, “A fast converging, low complexity adaptive filtering algorithm”, IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, USA, October 1993.
 S. Douglas, “The fast affine projection algorithm for active noise control”, Asilomar Conference on Signals, Systems and Computers, USA, October 1995, pp. 1245-1249.
 H. Zhao, and Z. Zheng, “Bias-compensated affine-projection-like algorithms with noisy input”, Electronic Letters, Vol. 52, No. 9, April 2016, pp. 712-71.
 T. Zhang, H. Q. Jiao, and Z. C. Lei, “Individual-Activation-Factor Memory Proportionate Affine Projection Algorithm With Evolving Regularization”, IEEE Access, Vol. 5, March 2017, pp. 4939-4946.
 S. I. Koike, “Analysis of Affine Projection Normalized Correlation Algorithm”, International Symposium on Intelligent Signal Processing and Communication Systems, Thailand, October 2016.
 M. R. Petraglia, D. B. Haddad, and E. L. Marques, “Affine projection subband adaptive filter with low computational complexity”, IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 63, No. 10, October 2016, pp. 989-993.
 G. Pathak, B. Singh, and B. K. Panigrahi, “Control of wind-diesel microgrid using affine projection-like algorithm”, IEEE Transactions on Industrial Informatics, Vol. 12, No. 2, April 2016, pp. 524-531.
 J. H. Husøy, and M. S. E. Abadi, “On the convergence speed of the Normalized Subband Adaptive Filter: Some new insights and interpretations”, International Symposium on Signals, Circuits and Systems, Romania, July 2017.
 Y. Huang, J. Benesty, and J. Chen, Acoustic MIMO Signal Processing, Springer-Verlag New York, Inc., NJ, USA, 2006.
 J. Maheshwari, and N. V. George, “Robust modeling of acoustic paths using a sparse adaptive algorithm”, Applied Acoustics, Vol. 101, January 2016, pp. 122–126.
 M. Bekrani, A. W. H. Khong, and M. Lotfizad, “A Linear Neural Network based Approach to Stereophonic Acoustic Echo Cancellation”, IEEE Transaction on Audio, Speech, and Language Processing, Vol. 19, No. 6, August 2011, pp. 1743- 1753.
 M. Bekrani, A. W. H. Khong, and M. Lotfizad, “A Clipping-based Selective-Tap Adaptive Filtering Approach for Stereophonic Acoustic Echo Cancellation”, IEEE Transaction on Audio, Speech, and Language Processing, Vol. 19, No. 6, August 2011, pp. 1826- 1836.
 S. Pradhan, V. Patel, K. Patel, J. Maheshwari, and N. V. George, “Acoustic feedback cancellation in digital hearing aids: A sparse adaptive filtering approach”, Applied Acoustics, Vol. 122, July 2017, pp. 138–145.
 Y. R. Chen, Ch.Yuan, and M. S. Kuo, “Active Noise Control and Secondary Path Modeling Algorithms for Earphones”, American Control Conference, USA, May 2017, pp. 246-251.
 M. Bekrani, and H. Zayyani, “A Weighted Soft-Max PNLMS Algorithm for Sparse System Identification”, International Journal of Information & Communication Technology Research, Vol. 8, No. 3, Summer 2016, pp. 7-14.
 S. H. Yim, S. Lee, and W. J. Song, “A proportionate diffusion LMS algorithm for sparse distributed estimation”, IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 62, No. 10, October 2015, pp. 992-996.
 H. Deng, and M. Doroslovacki, “Proportionate adaptive algorithms for network echo cancellation”, IEEE Transactions on Signal Processing, Vol. 54, No. 5, May 2006, pp. 1794-1803.
 C. Paleologu, J. Benesty, and S. Ciochină, “An improved proportionate NLMS algorithm based on the norm”, International Conference on Acoustics Speech and Signal Processing, USA, March 2010, pp. 309-312.
 F. Albu, J. Liu, and S. L. Grant, “A fast filtering block-sparse proportionate affine projection sign algorithm”, International Conference on Communications, Romania, June 2016, pp. 29-32.
 J. Liu, and S. L. Grant, “Proportionate adaptive filtering for block-sparse system identification”, IEEE/ACM Transactions on Audio, Speech, and Language Processing, Vol. 24, No. 4, April 2016, pp. 623-630.
 A. Gonzalez, M. Ferrer, F. Albu, and M. Diego,"Affine projection algorithms: Evolution to smart and fast algorithms and applications", Proc. 20th European Signal Processing Conference, Romania, Aug. 2012, pp. 1965-1969.
 S. G. Sankaran, and A. A. Beex," Convergence Behavior of Affine Projection Algorithms", IEEE Transactions on Signal Processing, Vol. 48, No.4, April 2000, pp. 1086-1096.
 L. Liao, and A. W. H. Khong, "Sparseness-Controlled Affine Projection Algorithm for Echo Cancelation", Proceedings of APSIPA Annual Summit and Conference, Biopolis, 2010, pp. 355–361.
 V. Kılıç, M. Bamard, W. Wang, and J. Kittler, “Audio assisted robust visual tracking with adaptive particle filtering”, IEEE Transactions on Multimedia, Vol. 17, No. 2, Feb. 2015, pp. 186-200.
 K. Nishikawa, and H. Kiya, “New structure of affine projection algorithm using a novel subband adaptive system”, Third Workshop on Signal Processing Advances in Wireless Communications, China, March 2001, pp. 364-367.
 Z. Zheng, Zh. Liu, and Y. Dong, "Steady-State and Tracking Analyses of Improved Proportionate Affine Projection Algorithm", IEEE Transactions on Circuits and Systems II: Express Briefs, Vol. 65, No. 11, Nov. 2018, pp. 1793 – 1797.
 D. DasGupta, “In-Place Matrix Inversion by Modified Gauss-Jordan Algorithm”, Applied Mathematics, Vol. 4, No. 10, Oct. 2013, pp. 1392-1396.
 مهدی نیلی احمدآبادی، عباس افشاری و احمد شرفی، "شبیهسازی آکوستیکی محیط بسته با استفاده از روش عددی تفاضل محدود حوزه زمان"، نشریه مدلسازی در مهندسی، دوره 12، شماره 39 ، زمستان 1393، صفحه 89-98.
 J. Allen, D. Berkley, “Image Method for Efficiently Simulating Small-Room Acoustics”, Journal of the Acoustical Society of America, Vol. 65, No. 4, April 1979, pp. 943-950.