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We first propose a frequency variational Bayesian algorithm, which leverages common sparsity of different sub\u2010carriers in the frequency domain. The algorithm shares all the information of the support sets from the measurement matrices, significantly improving channel estimation accuracy. To enhance robustness of the frequency variational Bayesian algorithm, we develop a hierarchical Gaussian prior channel model, which employs an identify\u2010and\u2010reject strategy to deal with random outliers imposed by hardware impairments. A support selection frequency variational Bayesian channel estimation algorithm is also proposed, which adaptively selects support sets from the measurement matrices. As a result, the overall computational complexity can be reduced. Validated by the Bayesian Cram\u00e9r\u2010Rao bound, simulation results show that, both frequency variational Bayesian and support selection\u2010frequency variational Bayesian algorithms can achieve higher channel estimation accuracy than existing methods. Furthermore, compared with frequency variational Bayesian, support selection\u2010frequency variational Bayesian requires significantly lower computational complexity, and hence, it is more practical for channel estimation\u00a0applications.<\/jats:p>","DOI":"10.1049\/cmu2.12088","type":"journal-article","created":{"date-parts":[[2020,12,26]],"date-time":"2020-12-26T11:15:59Z","timestamp":1608981359000},"page":"566-579","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Channel estimation using variational Bayesian learning for multi\u2010user mmWave MIMO systems"],"prefix":"10.1049","volume":"15","author":[{"given":"Bo","family":"Xiao","sequence":"first","affiliation":[{"name":"School of Information and Communication Engineering Beijing University of Posts and Telecommunications (BUPT)  Beijing 100876 China"}]},{"given":"Pingmu","family":"Huang","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering Beijing University of Posts and Telecommunications (BUPT)  Beijing 100876 China"}]},{"given":"Zhipeng","family":"Lin","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering Beijing University of Posts and Telecommunications (BUPT)  Beijing 100876 China"}]},{"given":"Jie","family":"Zeng","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering Tsinghua University  Beijing 100084 China"}]},{"given":"Tiejun","family":"Lv","sequence":"additional","affiliation":[{"name":"School of Information and Communication Engineering Beijing University of Posts and Telecommunications (BUPT)  Beijing 100876 China"}]}],"member":"265","published-online":{"date-parts":[[2020,12,26]]},"reference":[{"key":"e_1_2_7_2_1","doi-asserted-by":"publisher","DOI":"10.1109\/MCOM.2014.6736761"},{"key":"e_1_2_7_3_1","doi-asserted-by":"publisher","DOI":"10.1109\/COMST.2018.2843719"},{"key":"e_1_2_7_4_1","doi-asserted-by":"publisher","DOI":"10.1109\/JSAC.2016.2549418"},{"key":"e_1_2_7_5_1","doi-asserted-by":"publisher","DOI":"10.1109\/LWC.2016.2598777"},{"key":"e_1_2_7_6_1","doi-asserted-by":"publisher","DOI":"10.1109\/TCOMM.2018.2797993"},{"key":"e_1_2_7_7_1","doi-asserted-by":"publisher","DOI":"10.1109\/MWC.2017.1700147"},{"key":"e_1_2_7_8_1","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2012.2225405"},{"key":"e_1_2_7_9_1","doi-asserted-by":"publisher","DOI":"10.1109\/JSTSP.2014.2334278"},{"key":"e_1_2_7_10_1","first-page":"1","article-title":"Compressive channel estimation exploiting block sparsity in multi\u2010user massive MIMO systems","author":"Xu W.","year":"2017","journal-title":"In: Proceedings IEEE Wireless Communications and Networking Conference (WCNC), San Francisco"},{"key":"e_1_2_7_11_1","doi-asserted-by":"publisher","DOI":"10.1109\/TWC.2017.2751046"},{"key":"e_1_2_7_12_1","doi-asserted-by":"publisher","DOI":"10.1109\/JSTSP.2018.2814008"},{"key":"e_1_2_7_13_1","doi-asserted-by":"publisher","DOI":"10.1109\/LCOMM.2016.2555299"},{"key":"e_1_2_7_14_1","doi-asserted-by":"publisher","DOI":"10.1109\/JSAC.2017.2720856"},{"key":"e_1_2_7_15_1","doi-asserted-by":"publisher","DOI":"10.1109\/TWC.2017.2718502"},{"key":"e_1_2_7_16_1","doi-asserted-by":"publisher","DOI":"10.1109\/TWC.2018.2804943"},{"key":"e_1_2_7_17_1","first-page":"1","article-title":"3D wideband mmWave localization for 5G massive MIMO systems","author":"Lin Z.","year":"2019","journal-title":"In: Proc. 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