{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:43:06Z","timestamp":1760240586523,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2019,7,24]],"date-time":"2019-07-24T00:00:00Z","timestamp":1563926400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Adaptive algorithms with differential step-sizes (related to the filter coefficients) are well known in the literature, most frequently as \u201cproportionate\u201d algorithms. Usually, they are derived on a heuristic basis. In this paper, we introduce an algorithm resulting from an optimization criterion. Thereby, we obtain a benchmark algorithm and also another version with lower computational complexity, which is rigorously valid for less correlated input signals. Simulation results confirm the theory and outline the performance of the algorithms. Unfortunately, the good performance is obtained by an important increase in computational complexity. Nevertheless, the proposed algorithms could represent useful benchmarks in the field.<\/jats:p>","DOI":"10.3390\/a12080147","type":"journal-article","created":{"date-parts":[[2019,7,24]],"date-time":"2019-07-24T10:48:19Z","timestamp":1563965299000},"page":"147","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Optimized Differential Step-Size LMS Algorithm"],"prefix":"10.3390","volume":"12","author":[{"given":"Alexandru-George","family":"Rusu","sequence":"first","affiliation":[{"name":"Department of Telecommunications, University Politehnica of Bucharest, 1-3, Iuliu Maniu Blvd., 061071 Bucharest, Romania"},{"name":"Department of Research and Development, Rohde &amp; Schwarz Topex, 014142 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Silviu","family":"Ciochin\u0103","sequence":"additional","affiliation":[{"name":"Department of Telecommunications, University Politehnica of Bucharest, 1-3, Iuliu Maniu Blvd., 061071 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0379-2360","authenticated-orcid":false,"given":"Constantin","family":"Paleologu","sequence":"additional","affiliation":[{"name":"Department of Telecommunications, University Politehnica of Bucharest, 1-3, Iuliu Maniu Blvd., 061071 Bucharest, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0036-5865","authenticated-orcid":false,"given":"Jacob","family":"Benesty","sequence":"additional","affiliation":[{"name":"INRS-EMT, University of Quebec, Montreal, QC H5A 1K6, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Haykin, S.S., and Widrow, B. (2003). Least-Mean-Square Adaptive Filters, Wiley.","DOI":"10.1002\/0471461288"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"508","DOI":"10.1109\/89.861368","article-title":"Proportionate normalized least-mean-squares adaptation in echo cancelers","volume":"8","author":"Duttweiler","year":"2000","journal-title":"IEEE Trans. Speech Audio Process."},{"key":"ref_3","unstructured":"Gay, S.L. (1998, January 1\u20134). An efficient, fast converging adaptive filter for network echo cancellation. Proceedings of the Conference Record of Thirty-Second Asilomar Conference on Signals, Systems and Computers (Cat. No.98CH36284), Pacific Grove, CA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Benesty, J., and Gay, S.L. (2002, January 13\u201317). An improved PNLMS algorithm. Proceedings of the 2002 IEEE International Conference on Acoustics, Speech, and Signal Processing, Orlando, FL, USA.","DOI":"10.1109\/ICASSP.2002.1006134"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1794","DOI":"10.1109\/TSP.2006.872533","article-title":"Proportionate adaptive algorithms for network echo cancellation","volume":"54","author":"Deng","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2036","DOI":"10.1109\/TSP.2009.2038420","article-title":"A PNLMS algorithm with individual activation factors","volume":"58","author":"Tobias","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"623","DOI":"10.1109\/TASLP.2015.2499602","article-title":"Proportionate adaptive filtering for block-sparse system identification","volume":"24","author":"Liu","year":"2016","journal-title":"IEEE\/ACM Trans. Audio Speech Lang. Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1109\/LSP.2009.2024736","article-title":"\u21130 norm constraint LMS algorithm for sparse system identification","volume":"16","author":"Gu","year":"2009","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1591","DOI":"10.1109\/TASL.2009.2025903","article-title":"A class of sparseness-controlled algorithms for echo cancellation","volume":"17","author":"Loganathan","year":"2009","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Li, Y., Wang, Y., and Sun, L. (2018). A proportionate normalized maximum correntropy criterion algorithm with correntropy induced metric constraint for identifying sparse systems. Symmetry, 10.","DOI":"10.3390\/sym10120683"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Rusu, A.G., Ciochin\u0103, S., and Paleologu, C. (2019, January 1\u20133). On the step-size optimization of the LMS algorithm. Proceedings of the 42nd International Conference on Telecommunications and Signal Processing, Budapest, Hungary.","DOI":"10.1109\/TSP.2019.8768842"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ciochin\u0103, S., Paleologu, C., Benesty, J., Grant, S.L., and Anghel, A. (September, January 29). A family of optimized LMS-based algorithms for system identification. Proceedings of the 2016 24th European Signal Processing Conference (EUSIPCO), Budapest, Hungary.","DOI":"10.1109\/EUSIPCO.2016.7760559"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"807","DOI":"10.1016\/B978-0-12-396501-1.00030-3","article-title":"Acoustic echo control","volume":"Volume 4","author":"Enzner","year":"2014","journal-title":"Academic Press Library in Signal Processing"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1109\/LSP.2003.821722","article-title":"Variable step-size NLMS and affine projection algorithms","volume":"11","author":"Shin","year":"2004","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"581","DOI":"10.1109\/LSP.2006.876323","article-title":"A nonparametric VSS-NLMS algorithm","volume":"13","author":"Benesty","year":"2006","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1109\/LSP.2009.2026197","article-title":"Scheduled-stepsize NLMS algorithm","volume":"16","author":"Park","year":"2009","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2055","DOI":"10.1109\/TSP.2011.2181505","article-title":"A new variable step-size NLMS algorithm and its performance analysis","volume":"60","author":"Huang","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"817","DOI":"10.1109\/LSP.2012.2221699","article-title":"A normalized least-mean-square algorithm based on variable-step-size recursion with innovative input data","volume":"19","author":"Song","year":"2012","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1093\/biomet\/12.1-2.134","article-title":"On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables","volume":"12","author":"Isserlis","year":"1918","journal-title":"Biometrika"},{"key":"ref_20","unstructured":"Iqbal, M.A., and Grant, S.L. (April, January 31). Novel variable step size NLMS algorithms for echo cancellation. Proceedings of the 2008 IEEE International Conference on Acoustics, Speech and Signal Processing, Las Vegas, NV, USA."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1109\/LSP.2007.910276","article-title":"Variable step-size NLMS algorithm for under-modeling acoustic echo cancellation","volume":"15","author":"Paleologu","year":"2008","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1007\/BF02165411","article-title":"Gaussian Elimination is not Optimal","volume":"13","author":"Strassen","year":"1969","journal-title":"Numer. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"251","DOI":"10.1016\/S0747-7171(08)80013-2","article-title":"Matrix multiplication via arithmetic progressions","volume":"9","author":"Coppersmith","year":"1990","journal-title":"J. Symb. Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.sigpro.2015.06.016","article-title":"An optimized NLMS algorithm for system identification","volume":"118","author":"Paleologu","year":"2016","journal-title":"Signal Process."},{"key":"ref_25","unstructured":"ITU (2002). Digital Network Echo Cancellers, ITU-T Recommendation G 168, ITU."},{"key":"ref_26","first-page":"1208","article-title":"Non-negative matrix factorization with sparseness constraints","volume":"49","author":"Hoyer","year":"2001","journal-title":"J. Mach. Learn. Res."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/8\/147\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T13:09:11Z","timestamp":1760188151000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/8\/147"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,24]]},"references-count":26,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2019,8]]}},"alternative-id":["a12080147"],"URL":"https:\/\/doi.org\/10.3390\/a12080147","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2019,7,24]]}}}