{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,6]],"date-time":"2026-01-06T13:07:03Z","timestamp":1767704823542,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"17","license":[{"start":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T00:00:00Z","timestamp":1630540800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004837","name":"Ministerio de Ciencia e Innovaci\u00f3n","doi-asserted-by":"publisher","award":["TEC2017-82807-P"],"award-info":[{"award-number":["TEC2017-82807-P"]}],"id":[{"id":"10.13039\/501100004837","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The operation of the power amplifier (PA) in wireless transmitters presents a trade-off between linearity and power efficiency, being more efficient when the device exhibits the highest nonlinearity. Its modeling and linearization performance depend on the quality of the underlying Volterra models that are characterized by the presence of relevant terms amongst the enormous amount of regressors that these models generate. The presence of PA mechanisms that generate an internal state variable motivates the adoption of a bivariate Volterra series perspective with the aim of enhancing modeling capabilities through the inclussion of beneficial terms. In this paper, the conventional Volterra-based models are enhanced by the addition of terms, including cross products of the input signal and the new internal variable. The bivariate versions of the general full Volterra (FV) model and one of its pruned versions, referred to as the circuit-knowledge based Volterra (CKV) model, are derived by considering the signal envelope as the internal variable and applying the proposed methodology to the univariate models. A comparative assessment of the bivariate models versus their conventional counterparts is experimentally performed for the modeling of two PAs driven by a 30 MHz 5G New Radio signal: a class AB PA and a class J PA. The results for the digital predistortion of the class AB PA under a direct learning architecture reveal the benefits in linearization performance produced by the bivariate CKV model structure compared to that of the univariate CKV model.<\/jats:p>","DOI":"10.3390\/s21175897","type":"journal-article","created":{"date-parts":[[2021,9,2]],"date-time":"2021-09-02T23:05:12Z","timestamp":1630623912000},"page":"5897","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Bivariate Volterra Series Model for the Design of Power Amplifier Digital Predistorters"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0879-5891","authenticated-orcid":false,"given":"Carlos","family":"Crespo-Cadenas","sequence":"first","affiliation":[{"name":"Departamento de Teor\u00eda de la Se\u00f1al y Comunicaciones, Escuela T\u00e9cnica Superior de Ingenier\u00eda, Universidad de Sevilla, Camino de los Descubrimientos, s\/n, 41092 Seville, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6614-2771","authenticated-orcid":false,"given":"Mar\u00eda J.","family":"Madero-Ayora","sequence":"additional","affiliation":[{"name":"Departamento de Teor\u00eda de la Se\u00f1al y Comunicaciones, Escuela T\u00e9cnica Superior de Ingenier\u00eda, Universidad de Sevilla, Camino de los Descubrimientos, s\/n, 41092 Seville, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4351-7830","authenticated-orcid":false,"given":"Juan A.","family":"Becerra","sequence":"additional","affiliation":[{"name":"Departamento de Teor\u00eda de la Se\u00f1al y Comunicaciones, Escuela T\u00e9cnica Superior de Ingenier\u00eda, Universidad de Sevilla, Camino de los Descubrimientos, s\/n, 41092 Seville, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2021,9,2]]},"reference":[{"key":"ref_1","first-page":"2381","article-title":"Machine-Learning Assisted Optimisation of Free-Parameters of a Dual-Input Power Amplifier for Wideband Applications","volume":"21","author":"Wang","year":"2021","journal-title":"Sensors"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Borel, B., Barzd\u00e8nas, V., and Vasjanov, A. (2021). Linearization as a Solution for Power Amplifier Imperfections: A Review of Methods. Electronics, 10.","DOI":"10.3390\/electronics10091073"},{"key":"ref_3","unstructured":"Apartsyn, A.S. (2011). Nonclassical Linear Volterra Equations of the First Kind, De Gruyter."},{"key":"ref_4","unstructured":"Chua, L.O. (2015). Integral Dynamical Models: Singularities, Signals & Control, World Scientific Publ. Pte Ltd."},{"key":"ref_5","unstructured":"Maas, S. (2003). Nonlinear Microwave and RF Circuits, Artech House. [2nd ed.]. Artech House Microwave Library."},{"key":"ref_6","unstructured":"Wood, J. (2014). Behavioral Modeling and Linearization of RF Power Amplifiers, Artech House. [1st ed.]. Artech House Microwave Library."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"494","DOI":"10.1109\/TAES.1979.308734","article-title":"Modeling and performance evaluation of nonlinear satellite links\u2014A Volterra series approach","volume":"15","author":"Benedetto","year":"1979","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1049\/el:20010940","article-title":"Digital predistortion of wideband signals based on power amplifier model with memory","volume":"37","author":"Kim","year":"2001","journal-title":"Electron. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"3852","DOI":"10.1109\/TSP.2006.879264","article-title":"A generalized memory polynomial model for digital predistortion of RF power amplifiers","volume":"54","author":"Morgan","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"156","DOI":"10.1109\/TVT.2003.822001","article-title":"Effects of even-order nonlinear terms on power amplifier modeling and predistortion linearization","volume":"53","author":"Ding","year":"2004","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lima, E.G., Cunha, T.R., Teixeira, H.M., Pirola, M., and Pedro, J.C. (2009, January 7\u201312). Base-Band Derived Volterra Series for Power Amplifier Modeling. Proceedings of the 2009 IEEE\/MTT-S International Microwave Symposium Digest, Boston, MA, USA.","DOI":"10.1109\/MWSYM.2009.5165958"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1109\/TMTT.2007.895155","article-title":"Pruning the Volterra Series for Behavioral Modeling of Power Amplifiers Using Physical Knowledge","volume":"55","author":"Zhu","year":"2007","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2747","DOI":"10.1109\/TMTT.2007.909873","article-title":"Design of a Power-Amplifier Feed-Forward RF Model With Physical Knowledge Considerations","volume":"55","author":"Cunha","year":"2007","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3022","DOI":"10.1109\/TCSI.2013.2252691","article-title":"A Volterra-based procedure for multi-port and multi-zone GaN FET amplifier CAD simulation","volume":"60","year":"2013","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Crespo-Cadenas, C., Madero-Ayora, M.J., and Becerra, J.A. (2021, January 17\u201320). A bivariate Volterra series approach to modeling and linearization of power amplifiers. Proceedings of the 2021 IEEE Topical Conference on RF\/Microwave Power Amplifiers for Radio and Wireless Applications, San Diego, CA, USA.","DOI":"10.1109\/PAWR51852.2021.9375591"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1255","DOI":"10.1002\/j.1538-7305.1973.tb02019.x","article-title":"Volterra systems with more than one input port\u2014Distortion in a frequency converter","volume":"52","author":"Rice","year":"1973","journal-title":"Bell Syst. Tech. J."},{"key":"ref_17","first-page":"2660","article-title":"An electrothermal model for AlGaN\/GaN power HEMTs including trapping effects to improve large-signal simulation results on high VSWR","volume":"55","author":"Jardel","year":"2007","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1109\/22.780397","article-title":"High-Efficiency Power Amplifier Using Dynamic Power-Supply Voltage for CDMA Applications","volume":"47","author":"Hanington","year":"1999","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1109\/TMTT.2014.2387852","article-title":"Behavioral modeling and predistortion of power amplifiers under sparsity hypothesis","volume":"63","author":"Chao","year":"2015","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"726","DOI":"10.1109\/LMWC.2018.2845947","article-title":"A Doubly Orthogonal Matching Pursuit Algorithm for Sparse Predistortion of Power Amplifiers","volume":"28","author":"Becerra","year":"2018","journal-title":"IEEE Microw. Wirel. Compon. Lett."},{"key":"ref_21","unstructured":"Moon, J., Lee, J., Son, J., Kim, J., Jee, S., Kim, S., and Kim, B. (2011, January 9\u201314). Effects of even-order terms on behavior model of envelope tracking transmitters. Proceedings of the 2011 6th European Microwave Integrated Circuit Conference, Manchester, UK."},{"key":"ref_22","unstructured":"Dahlman, E., Parkvall, S., and Skold, J. (2020). 5G NR: The Next Generation Wireless Access Technology, Academic Press. [2nd ed.]."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1109\/TSP.2006.882058","article-title":"Novel adaptive nonlinear predistorters based on the direct learning algorithm","volume":"55","author":"Zhou","year":"2007","journal-title":"IEEE Trans. Signal Process."}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/17\/5897\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:54:40Z","timestamp":1760165680000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/21\/17\/5897"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,2]]},"references-count":23,"journal-issue":{"issue":"17","published-online":{"date-parts":[[2021,9]]}},"alternative-id":["s21175897"],"URL":"https:\/\/doi.org\/10.3390\/s21175897","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2021,9,2]]}}}