{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:07:57Z","timestamp":1760242077180,"version":"build-2065373602"},"reference-count":45,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,12,20]],"date-time":"2018-12-20T00:00:00Z","timestamp":1545264000000},"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>The combination of adaptive filters is an effective approach to improve filtering performance. In this paper, we investigate the performance of an adaptive combined scheme between two adaptive multiple-input single-output (MISO) filters, which can be easily extended to the case of multiple outputs. In order to generalize the analysis, we consider the multichannel affine projection algorithm (APA) to update the coefficients of the MISO filters, which increases the possibility of exploiting the capabilities of the filtering scheme. Using energy conservation relations, we derive a theoretical behavior of the proposed adaptive combination scheme at steady state. Such analysis entails some further theoretical insights with respect to the single channel combination scheme. Simulation results prove both the validity of the theoretical steady-state analysis and the effectiveness of the proposed combined scheme.<\/jats:p>","DOI":"10.3390\/a12010002","type":"journal-article","created":{"date-parts":[[2018,12,20]],"date-time":"2018-12-20T12:54:36Z","timestamp":1545310476000},"page":"2","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Steady-State Performance of an Adaptive Combined MISO Filter Using the Multichannel Affine Projection Algorithm"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4067-4504","authenticated-orcid":false,"given":"Danilo","family":"Comminiello","sequence":"first","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, 00185 Roma RM, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3164-6256","authenticated-orcid":false,"given":"Michele","family":"Scarpiniti","sequence":"additional","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, 00185 Roma RM, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2458-0914","authenticated-orcid":false,"given":"Luis A.","family":"Azpicueta-Ruiz","sequence":"additional","affiliation":[{"name":"Department of Signal Theory and Communications, Universidad Carlos III de Madrid, 28903 Getafe, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5793-0917","authenticated-orcid":false,"given":"Aurelio","family":"Uncini","sequence":"additional","affiliation":[{"name":"Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, 00185 Roma RM, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2018,12,20]]},"reference":[{"key":"ref_1","first-page":"120","article-title":"Combinations of Adaptive Filters: Performance and Convergence Properties","volume":"1","author":"Silva","year":"2016","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2685","DOI":"10.1109\/78.790651","article-title":"Universal Linear Prediction by Model Order Weighting","volume":"47","author":"Singer","year":"1999","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_3","unstructured":"Kozat, S.S., and Singer, A.C. (2000, January 17). Multi-Stage Adaptive Signal Processing Algorithms. Proceedings of the IEEE Sensor Array Multichannel Signal Workshop (SAM), Cambridge, MA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1109\/78.902113","article-title":"A Unified Approach to the Steady-State and Tracking Analyzes of Adaptive Filters","volume":"49","author":"Yousef","year":"2001","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2430","DOI":"10.1016\/j.sigpro.2005.11.008","article-title":"Plant Identification via Adaptive Combination of Transversal Filters","volume":"86","year":"2006","journal-title":"Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1078","DOI":"10.1109\/TSP.2005.863126","article-title":"Mean-Square Performance of a Convex Combination of Two Adaptive Filters","volume":"54","author":"Sayed","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1109\/TASL.2009.2019925","article-title":"Adaptive Combination of Proportionate Filters for Sparse Echo Cancellation","volume":"17","year":"2009","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3306","DOI":"10.1016\/j.sigpro.2013.05.014","article-title":"Combined Adaptive Beamforming Schemes for Nonstationary Interfering Noise Reduction","volume":"93","author":"Comminiello","year":"2013","journal-title":"Signal Process."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1109\/TASL.2010.2045185","article-title":"Adaptive Combination of Volterra Kernels and its Application to Nonlinear Acoustic Echo Cancellation","volume":"19","author":"Zeller","year":"2011","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1502","DOI":"10.1109\/TASL.2013.2255276","article-title":"Functional Link Adaptive Filters for Nonlinear Acoustic Echo Cancellation","volume":"21","author":"Comminiello","year":"2013","journal-title":"IEEE Trans. Audio Speech Lang. Process."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Comminiello, D., Scardapane, S., Scarpiniti, M., Parisi, R., and Uncini, A. (2013, January 4\u20136). Convex Combination of MIMO Filters for Multichannel Acoustic Echo Cancellation. Proceedings of the IEEE International Symposium on Image and Signal Processing (ISPA), Trieste, Italy.","DOI":"10.1109\/ISPA.2013.6703842"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1109\/TSP.2012.2236835","article-title":"A Soft-Switching Blind Equalization Scheme via Convex Combination of Adaptive Filters","volume":"61","author":"Silva","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Comminiello, D., Scarpiniti, M., Azpicueta-Ruiz, L.A., Arenas-Garc\u00eda, J., and Uncini, A. (2016, January 13\u201316). A Block-Based Combined Scheme Exploiting Sparsity in Nonlinear Acoustic Echo Cancellation. Proceedings of the IEEE 26th International Workshop on Machine Learning for Signal Processing (MLSP), Salerno, Italy.","DOI":"10.1109\/MLSP.2016.7738822"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"168","DOI":"10.1016\/j.sigpro.2017.01.009","article-title":"Combined Nonlinear Filtering Architectures Involving Sparse Functional Link Adaptive Filters","volume":"135","author":"Comminiello","year":"2017","journal-title":"Signal Process."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Comminiello, D., Scarpiniti, M., Scardapane, S., Azpicueta-Ruiz, L.A., and Uncini, A. (2018, January 3\u20137). Combined Sparse Regularization for Nonlinear Adaptive Filters. Proceedings of the 26th European Signal Processing Conference (EUSIPCO), Rome, Italy.","DOI":"10.23919\/EUSIPCO.2018.8552955"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3137","DOI":"10.1109\/TSP.2008.919105","article-title":"Improving the Tracking Capability of Adaptive Filters Via Convex Combination","volume":"56","author":"Silva","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1853","DOI":"10.1109\/TSP.2007.911486","article-title":"An Affine Combination of two LMS Adaptive Filters\u2014 Transient Mean-Square Analysis","volume":"56","author":"Bershad","year":"2008","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"4064","DOI":"10.1109\/TSP.2010.2048210","article-title":"Transient and Steady-State Analysis of the Affine Combination of Two Adaptive Filters","volume":"58","author":"Candido","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4050","DOI":"10.1109\/TSP.2010.2049650","article-title":"Steady-State MSE Perfromance Analysis of Mixture Approaches to Adaptive Filtering","volume":"58","author":"Kozat","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3314","DOI":"10.1109\/TSP.2012.2189110","article-title":"Steady State and Transient MSE Analysis of Convexly Constrained Mixture Methods","volume":"60","author":"Donmez","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Li, J., and Stoica, P. (2005). Robust Adaptive Beamforming, Wiley.","DOI":"10.1002\/0471733482"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Uncini, A. (2015). Fundamentals of Adaptive Signal Processing, Springer.","DOI":"10.1007\/978-3-319-02807-1"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"3163","DOI":"10.1109\/TWC.2011.081011.102169","article-title":"Peak-to-Average Power Ratio Reduction for MISO OFDM Systems with Adaptive All-Pass Filters","volume":"10","author":"Hong","year":"2011","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_24","unstructured":"Slock, D.T.M. (1996, January 7\u201310). Spatio-Temporal Training-Sequence Based Channel Equalization and Adaptive Interference Cancellation. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Atlanta, GA, USA."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Gay, S.L., and Benesty, J. (2000). Acoustic Signal Processing for Telecommunications, Kluwer Academic Publishers.","DOI":"10.1007\/978-1-4419-8644-3"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Th\u00fcne, P., and Enzner, G. (2013, January 4\u20136). Trends in Adaptive MISO System Identification for Multichannel Audio Reproduction and Speech Communication. Proceedings of the 8th International Symposium on Image and Signal Processing and Analysis (ISPA), Trieste, Italy.","DOI":"10.1109\/ISPA.2013.6703840"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1002\/ecja.4400670503","article-title":"An Adaptive Filtering Algorithm Using an Orthogonal Projection to an Affine Subspace and its Properties","volume":"67","author":"Ozeki","year":"1984","journal-title":"Electron. Commun. Jpn. (Part I Commun.)"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1109\/97.484209","article-title":"A Multichannel Affine Projection Algorithm with Applications to Multichannel Acoustic Echo Cancellation","volume":"3","author":"Benesty","year":"1996","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Albu, F., Paleologu, C., and Benesty, J. (2011, January 22\u201327). A Variable Step Size Evolutionary Affine Projection Algorithm. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic.","DOI":"10.1109\/ICASSP.2011.5946432"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Paleologu, C., Benesty, J., Albu, F., and Ciochin\u0103, S. (2011, January 22\u201327). An Efficient Variable Step-Size Proportionate Affine Projection Algorithm. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Prague, Czech Republic.","DOI":"10.1109\/ICASSP.2011.5946332"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1049\/iet-spr.2012.0213","article-title":"Evolutionary and Variable Step Size Strategies for Multichannel Filtered-x Affine Projection Algorithms","volume":"7","author":"Gonzalez","year":"2012","journal-title":"IET Signal Process."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Albu, F., Coltuc, D., Comminiello, D., and Scarpiniti, M. (2012, January 15\u201316). The Variable Step Size Regularized Block Exact Affine Projection Algorithm. Proceedings of the 10th International Symposium on Electronics and Telecommunications (ISETC), Timisoara, Romania.","DOI":"10.1109\/ISETC.2012.6408092"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"3349","DOI":"10.1109\/TSP.2014.2324997","article-title":"Robust Shrinkage Affine-Projection Sign Adaptive-Filtering Algorithms for Impulsive Noise Environments","volume":"62","author":"Bhotto","year":"2014","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_34","unstructured":"Ferrer, M., de Diego, M., Gonzalez, A., and Pi\u00f1ero, G. (2009, January 24\u201328). Convex Combination of Affine Projection Algorithms. Proceedings of the European Signal Processing Conference (EUSIPCO), Glasgow, UK."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2771","DOI":"10.1109\/TSP.2012.2189390","article-title":"Steady-State Mean Square Performance of the Multichannel Filtered-X Affine Projection Algorithm","volume":"60","author":"Ferrer","year":"2012","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_36","unstructured":"Ar\u00e9valo, L., Apolin\u00e1rio, J.A., de Campos, M.L.R., and Sampaio-Neto, R. (2013, January 27\u201330). Convex Combination of Three Affine Projections Adaptive Filters. Proceedings of the IEEE International Symposium on Wireless Communication Systems (ISWCS), Ilmenau, Germany."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"466","DOI":"10.1049\/el.2013.3997","article-title":"Combination of Affine Projection Sign Algorithms for Robust Adaptive Filtering in Non-Gaussian Impulsive Interference","volume":"50","author":"Shi","year":"2014","journal-title":"Electron. Lett."},{"key":"ref_38","first-page":"493","article-title":"Combined-Step-Size Affine Projection Sign Algorithm for Robust Adaptive Filtering in Impulsive Interference Environments","volume":"63","author":"Huang","year":"2016","journal-title":"IEEE Trans. Circuits Syst. II Express Briefs"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"64","DOI":"10.1016\/j.sigpro.2014.01.015","article-title":"Adaptive Combination of Affine Projection and NLMS Algorithms","volume":"100","author":"Choi","year":"2014","journal-title":"Signal Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1016\/j.neunet.2015.05.002","article-title":"Improving Nonlinear Modeling Capabilities of Functional Link Adaptive filters","volume":"69","author":"Comminiello","year":"2015","journal-title":"Neural Netw."},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Sayed, A.H. (2008). Adaptive Filters, Wiley.","DOI":"10.1002\/9780470374122"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1109\/TSP.2003.820077","article-title":"Mean-Square Performance of a Family of Affine Projection Algorithms","volume":"52","author":"Shin","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1086","DOI":"10.1109\/78.827542","article-title":"Convergence Behavior of Affine Projection Algorithms","volume":"48","author":"Sankaran","year":"2000","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"3890","DOI":"10.1109\/TSP.2010.2047501","article-title":"Adaptively Biasing the Weights of Adaptive Filters","volume":"58","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_45","unstructured":"Azpicueta-Ruiz, L.A., Figueiras-Vidal, A.R., and Arenas-Garc\u00eda, J. (April, January 31). A Normalized Adaptation Scheme for the Convex Combination of Two Adaptive Filters. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, NV, USA."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/1\/2\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:35:16Z","timestamp":1760196916000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/12\/1\/2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,12,20]]},"references-count":45,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1]]}},"alternative-id":["a12010002"],"URL":"https:\/\/doi.org\/10.3390\/a12010002","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2018,12,20]]}}}