* Aguilar-Arevalo  _et al._ [2007] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 98, 231801 (2007), arXiv:0704.1500 [hep-ex] .
  * Aguilar-Arevalo  _et al._ [2009a] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 102, 101802 (2009a), arXiv:0812.2243 [hep-ex] .
  * Aguilar-Arevalo  _et al._ [2013] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 110, 161801 (2013), arXiv:1303.2588 [hep-ex] .
  * Aguilar  _et al._ [2001] A. Aguilar _et al._ (LSND), Phys. Rev. D 64, 112007 (2001), arXiv:hep-ex/0104049 .
  * Aguilar-Arevalo  _et al._ [2018a] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 121, 221801 (2018a), arXiv:1805.12028 [hep-ex] .
  * Aguilar-Arevalo  _et al._ [2021a] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 103, 052002 (2021a), arXiv:2006.16883 [hep-ex] .
  * Brdar and Kopp [2022] V. Brdar and J. Kopp, Phys. Rev. D 105, 115024 (2022), arXiv:2109.08157 [hep-ph] .
  * Kelly and Kopp [2023] K. J. Kelly and J. Kopp, JHEP 05, 113 (2023), arXiv:2210.08021 [hep-ph] .
  * Anselmann  _et al._ [1995] P. Anselmann _et al._ (GALLEX), Phys. Lett. B 342, 440 (1995).
  * Hampel  _et al._ [1998] W. Hampel _et al._ (GALLEX), Phys. Lett. B 420, 114 (1998).
  * Kaether  _et al._ [2010] F. Kaether, W. Hampel, G. Heusser, J. Kiko, and T. Kirsten, Phys. Lett. B 685, 47 (2010), arXiv:1001.2731 [hep-ex] .
  * Abdurashitov  _et al._ [1996] D. N. Abdurashitov _et al._ , Phys. Rev. Lett. 77, 4708 (1996).
  * Abdurashitov  _et al._ [1999] J. N. Abdurashitov _et al._ (SAGE), Phys. Rev. C 59, 2246 (1999), arXiv:hep-ph/9803418 .
  * Abdurashitov  _et al._ [2006] J. N. Abdurashitov _et al._ , Phys. Rev. C 73, 045805 (2006), arXiv:nucl-ex/0512041 .
  * Abdurashitov  _et al._ [2009] J. N. Abdurashitov _et al._ (SAGE), Phys. Rev. C 80, 015807 (2009), arXiv:0901.2200 [nucl-ex] .
  * Hamann  _et al._ [2011] J. Hamann, S. Hannestad, G. G. Raffelt, and Y. Y. Y. Wong, JCAP 09, 034 (2011), arXiv:1108.4136 [astro-ph.CO] .
  * Archidiacono  _et al._ [2013] M. Archidiacono, N. Fornengo, C. Giunti, S. Hannestad, and A. Melchiorri, Phys. Rev. D 87, 125034 (2013), arXiv:1302.6720 [astro-ph.CO] .
  * Hagstotz  _et al._ [2021] S. Hagstotz, P. F. de Salas, S. Gariazzo, M. Gerbino, M. Lattanzi, S. Vagnozzi, K. Freese, and S. Pastor, Phys. Rev. D 104, 123524 (2021), arXiv:2003.02289 [astro-ph.CO] .
  * Adamson  _et al._ [2020] P. Adamson _et al._ (MINOS+, Daya Bay), Phys. Rev. Lett. 125, 071801 (2020), arXiv:2002.00301 [hep-ex] .
  * Aartsen  _et al._ [2020a] M. G. Aartsen _et al._ (IceCube), Phys. Rev. Lett. 125, 141801 (2020a), arXiv:2005.12942 [hep-ex] .
  * Aartsen  _et al._ [2020b] M. G. Aartsen _et al._ (IceCube), Phys. Rev. D 102, 052009 (2020b), arXiv:2005.12943 [hep-ex] .
  * Dentler  _et al._ [2018] M. Dentler, A. Hernández-Cabezudo, J. Kopp, P. A. N. Machado, M. Maltoni, I. Martinez-Soler, and T. Schwetz, JHEP 08, 010 (2018), arXiv:1803.10661 [hep-ph] .
  * Diaz  _et al._ [2020] A. Diaz, C. A. Argüelles, G. H. Collin, J. M. Conrad, and M. H. Shaevitz, Phys. Rept. 884, 1 (2020), arXiv:1906.00045 [hep-ex] .
  * Acero  _et al._ [2022] M. A. Acero _et al._ , (2022), arXiv:2203.07323 [hep-ex] .
  * Acciarri  _et al._ [2015] R. Acciarri _et al._ (MicroBooNE, LAr1-ND, ICARUS-WA104), (2015), arXiv:1503.01520 [physics.ins-det] .
  * Machado  _et al._ [2019] P. A. Machado, O. Palamara, and D. W. Schmitz, Ann. Rev. Nucl. Part. Sci. 69, 363 (2019), arXiv:1903.04608 [hep-ex] .
  * Abratenko  _et al._ [2022a] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 128, 241801 (2022a), arXiv:2110.14054 [hep-ex] .
  * Abratenko  _et al._ [2022b] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 105, 112003 (2022b), arXiv:2110.14080 [hep-ex] .
  * Abratenko  _et al._ [2022c] P. Abratenko _et al._ ((The MicroBooNE Collaboration)*, MicroBooNE), Phys. Rev. D 105, 112004 (2022c), arXiv:2110.14065 [hep-ex] .
  * Abratenko  _et al._ [2022d] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 105, 112005 (2022d), arXiv:2110.13978 [hep-ex] .
  * Argüelles  _et al._ [2022a] C. A. Argüelles, I. Esteban, M. Hostert, K. J. Kelly, J. Kopp, P. A. N. Machado, I. Martinez-Soler, and Y. F. Perez-Gonzalez, Phys. Rev. Lett. 128, 241802 (2022a), arXiv:2111.10359 [hep-ph] .
  * Aguilar-Arevalo  _et al._ [2022] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 129, 201801 (2022), arXiv:2201.01724 [hep-ex] .
  * Abratenko  _et al._ [2023a] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 130, 011801 (2023a), arXiv:2210.10216 [hep-ex] .
  * Denton [2022] P. B. Denton, Phys. Rev. Lett. 129, 061801 (2022), arXiv:2111.05793 [hep-ph] .
  * Abratenko  _et al._ [2022e] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 128, 111801 (2022e), arXiv:2110.00409 [hep-ex] .
  * Bertuzzo  _et al._ [2018] E. Bertuzzo, S. Jana, P. A. N. Machado, and R. Zukanovich Funchal, Phys. Rev. Lett. 121, 241801 (2018), arXiv:1807.09877 [hep-ph] .
  * Ballett  _et al._ [2019] P. Ballett, S. Pascoli, and M. Ross-Lonergan, Phys. Rev. D 99, 071701 (2019), arXiv:1808.02915 [hep-ph] .
  * Dutta  _et al._ [2020] B. Dutta, S. Ghosh, and T. Li, Phys. Rev. D 102, 055017 (2020), arXiv:2006.01319 [hep-ph] .
  * Datta  _et al._ [2020] A. Datta, S. Kamali, and D. Marfatia, Phys. Lett. B 807, 135579 (2020), arXiv:2005.08920 [hep-ph] .
  * Abdallah  _et al._ [2020] W. Abdallah, R. Gandhi, and S. Roy, JHEP 12, 188 (2020), arXiv:2006.01948 [hep-ph] .
  * Hammad  _et al._ [2022] A. Hammad, A. Rashed, and S. Moretti, Phys. Lett. B 827, 136945 (2022), arXiv:2110.08651 [hep-ph] .
  * Abdallah  _et al._ [2022] W. Abdallah, R. Gandhi, and S. Roy, JHEP 06, 160 (2022), arXiv:2202.09373 [hep-ph] .
  * Abdullahi  _et al._ [2022] A. M. Abdullahi, J. Hoefken Zink, M. Hostert, D. Massaro, and S. Pascoli, (2022), arXiv:2207.04137 [hep-ph] .
  * Aguilar-Arevalo  _et al._ [2010a] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. Lett. 105, 181801 (2010a), arXiv:1007.1150 [hep-ex] .
  * Cheng  _et al._ [2011] G. Cheng _et al._ (SciBooNE), Phys. Rev. D 84, 012009 (2011), arXiv:1105.2871 [hep-ex] .
  * Cheng  _et al._ [2012] G. Cheng _et al._ (MiniBooNE, SciBooNE), Phys. Rev. D 86, 052009 (2012), arXiv:1208.0322 [hep-ex] .
  * Mahn  _et al._ [2012] K. B. M. Mahn _et al._ (SciBooNE, MiniBooNE), Phys. Rev. D 85, 032007 (2012), arXiv:1106.5685 [hep-ex] .
  * Aguilar-Arevalo  _et al._ [2008] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Lett. B 664, 41 (2008), arXiv:0803.3423 [hep-ex] .
  * Aguilar-Arevalo  _et al._ [2010b] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 81, 013005 (2010b), arXiv:0911.2063 [hep-ex] .
  * Wang  _et al._ [2015] E. Wang, L. Alvarez-Ruso, and J. Nieves, Phys. Lett. B 740, 16 (2015), arXiv:1407.6060 [hep-ph] .
  * Aguilar-Arevalo  _et al._ [2018b] A. A. Aguilar-Arevalo _et al._ (MiniBooNE DM), Phys. Rev. D 98, 112004 (2018b), arXiv:1807.06137 [hep-ex] .
  * Jordan  _et al._ [2019] J. R. Jordan, Y. Kahn, G. Krnjaic, M. Moschella, and J. Spitz, Phys. Rev. Lett. 122, 081801 (2019), arXiv:1810.07185 [hep-ph] .
  * Böser  _et al._ [2020] S. Böser, C. Buck, C. Giunti, J. Lesgourgues, L. Ludhova, S. Mertens, A. Schukraft, and M. Wurm, Prog. Part. Nucl. Phys. 111, 103736 (2020), arXiv:1906.01739 [hep-ex] .
  * Dasgupta and Kopp [2021] B. Dasgupta and J. Kopp, Phys. Rept. 928, 1 (2021), arXiv:2106.05913 [hep-ph] .
  * Moss  _et al._ [2018] Z. Moss, M. H. Moulai, C. A. Argüelles, and J. M. Conrad, Phys. Rev. D 97, 055017 (2018), arXiv:1711.05921 [hep-ph] .
  * Moulai  _et al._ [2020] M. H. Moulai, C. A. Argüelles, G. H. Collin, J. M. Conrad, A. Diaz, and M. H. Shaevitz, Phys. Rev. D 101, 055020 (2020), arXiv:1910.13456 [hep-ph] .
  * Akhmedov and Schwetz [2011] E. K. Akhmedov and T. Schwetz, Nucl. Phys. B Proc. Suppl. 217, 217 (2011).
  * Bramante [2013] J. Bramante, Int. J. Mod. Phys. A 28, 1350067 (2013), arXiv:1110.4871 [hep-ph] .
  * Karagiorgi  _et al._ [2012] G. Karagiorgi, M. H. Shaevitz, and J. M. Conrad, (2012), arXiv:1202.1024 [hep-ph] .
  * Asaadi  _et al._ [2018] J. Asaadi, E. Church, R. Guenette, B. J. P. Jones, and A. M. Szelc, Phys. Rev. D 97, 075021 (2018), arXiv:1712.08019 [hep-ph] .
  * Smirnov and Valera [2021] A. Y. Smirnov and V. B. Valera, JHEP 09, 177 (2021), arXiv:2106.13829 [hep-ph] .
  * Alves  _et al._ [2022] D. S. M. Alves, W. C. Louis, and P. G. deNiverville, JHEP 08, 034 (2022), arXiv:2201.00876 [hep-ph] .
  * Berryman  _et al._ [2018] J. M. Berryman, A. De Gouvêa, K. J. Kelly, and Y. Zhang, Phys. Rev. D 97, 075030 (2018), arXiv:1802.00009 [hep-ph] .
  * Palomares-Ruiz  _et al._ [2005] S. Palomares-Ruiz, S. Pascoli, and T. Schwetz, JHEP 09, 048 (2005), arXiv:hep-ph/0505216 .
  * Bai  _et al._ [2016] Y. Bai, R. Lu, S. Lu, J. Salvado, and B. A. Stefanek, Phys. Rev. D 93, 073004 (2016), arXiv:1512.05357 [hep-ph] .
  * de Gouvêa  _et al._ [2020] A. de Gouvêa, O. L. G. Peres, S. Prakash, and G. V. Stenico, JHEP 07, 141 (2020), arXiv:1911.01447 [hep-ph] .
  * Dentler  _et al._ [2020] M. Dentler, I. Esteban, J. Kopp, and P. Machado, Phys. Rev. D 101, 115013 (2020), arXiv:1911.01427 [hep-ph] .
  * Hostert and Pospelov [2021] M. Hostert and M. Pospelov, Phys. Rev. D 104, 055031 (2021), arXiv:2008.11851 [hep-ph] .
  * Fischer  _et al._ [2020] O. Fischer, A. Hernández-Cabezudo, and T. Schwetz, Phys. Rev. D 101, 075045 (2020), arXiv:1909.09561 [hep-ph] .
  * Chang  _et al._ [2021] C.-H. V. Chang, C.-R. Chen, S.-Y. Ho, and S.-Y. Tseng, Phys. Rev. D 104, 015030 (2021), arXiv:2102.05012 [hep-ph] .
  * Ballett  _et al._ [2020] P. Ballett, M. Hostert, and S. Pascoli, Phys. Rev. D 101, 115025 (2020), arXiv:1903.07589 [hep-ph] .
  * Abdullahi  _et al._ [2021] A. Abdullahi, M. Hostert, and S. Pascoli, Phys. Lett. B 820, 136531 (2021), arXiv:2007.11813 [hep-ph] .
  * Abdallah  _et al._ [2021] W. Abdallah, R. Gandhi, and S. Roy, Phys. Rev. D 104, 055028 (2021), arXiv:2010.06159 [hep-ph] .
  * Gninenko [2009] S. N. Gninenko, Phys. Rev. Lett. 103, 241802 (2009), arXiv:0902.3802 [hep-ph] .
  * Gninenko [2011] S. N. Gninenko, Phys. Rev. D 83, 015015 (2011), arXiv:1009.5536 [hep-ph] .
  * Gninenko [2012] S. N. Gninenko, Phys. Lett. B 710, 86 (2012), arXiv:1201.5194 [hep-ph] .
  * Masip  _et al._ [2013] M. Masip, P. Masjuan, and D. Meloni, JHEP 01, 106 (2013), arXiv:1210.1519 [hep-ph] .
  * Radionov [2013] A. Radionov, Phys. Rev. D 88, 015016 (2013), arXiv:1303.4587 [hep-ph] .
  * Magill  _et al._ [2018] G. Magill, R. Plestid, M. Pospelov, and Y.-D. Tsai, Phys. Rev. D 98, 115015 (2018), arXiv:1803.03262 [hep-ph] .
  * Schwetz  _et al._ [2020] T. Schwetz, A. Zhou, and J.-Y. Zhu, JHEP 21, 200 (2020), arXiv:2105.09699 [hep-ph] .
  * Vergani  _et al._ [2021] S. Vergani, N. W. Kamp, A. Diaz, C. A. Argüelles, J. M. Conrad, M. H. Shaevitz, and M. A. Uchida, Phys. Rev. D 104, 095005 (2021), arXiv:2105.06470 [hep-ph] .
  * Alvarez-Ruso and Saul-Sala [2021] L. Alvarez-Ruso and E. Saul-Sala, Eur. Phys. J. ST 230, 4373 (2021), arXiv:2111.02504 [hep-ph] .
  * Kamp  _et al._ [2023a] N. W. Kamp, M. Hostert, A. Schneider, S. Vergani, C. A. Argüelles, J. M. Conrad, M. H. Shaevitz, and M. A. Uchida, Phys. Rev. D 107, 055009 (2023a), arXiv:2206.07100 [hep-ph] .
  * Bansal  _et al._ [2023] S. Bansal, G. Paz, A. Petrov, M. Tammaro, and J. Zupan, JHEP 05, 142 (2023), arXiv:2210.05706 [hep-ph] .
  * Dutta  _et al._ [2022] B. Dutta, D. Kim, A. Thompson, R. T. Thornton, and R. G. Van de Water, Phys. Rev. Lett. 129, 111803 (2022), arXiv:2110.11944 [hep-ph] .
  * Abratenko  _et al._ [2023b] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. Lett. 130, 011801 (2023b), arXiv:2210.10216 [hep-ex] .
  * Kamp  _et al._ [2023b] N. W. Kamp, M. Hostert, C. A. Argüelles, J. M. Conrad, and M. H. Shaevitz, Phys. Rev. D 107, 092002 (2023b), arXiv:2301.12573 [hep-ph] .
  * Abratenko  _et al._ [2022f] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 106, L051102 (2022f), arXiv:2208.02348 [hep-ex] .
  * Abratenko  _et al._ [2021] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 104, 052002 (2021), arXiv:2101.04228 [hep-ex] .
  * Abratenko  _et al._ [2022g] P. Abratenko _et al._ (MicroBooNE), Phys. Rev. D 105, L051102 (2022g), arXiv:2109.06832 [hep-ex] .
  * Mention  _et al._ [2011] G. Mention, M. Fechner, T. Lasserre, T. A. Mueller, D. Lhuillier, M. Cribier, and A. Letourneau, Phys. Rev. D 83, 073006 (2011), arXiv:1101.2755 [hep-ex] .
  * Mueller  _et al._ [2011] T. A. Mueller _et al._ , Phys. Rev. C 83, 054615 (2011), arXiv:1101.2663 [hep-ex] .
  * Huber [2011] P. Huber, Phys. Rev. C 84, 024617 (2011), [Erratum: Phys.Rev.C 85, 029901 (2012)], arXiv:1106.0687 [hep-ph] .
  * Giunti  _et al._ [2022a] C. Giunti, Y. F. Li, C. A. Ternes, and Z. Xin, Phys. Lett. B 829, 137054 (2022a), arXiv:2110.06820 [hep-ph] .
  * Kopeikin  _et al._ [2021] V. Kopeikin, M. Skorokhvatov, and O. Titov, Phys. Rev. D 104, L071301 (2021), arXiv:2103.01684 [nucl-ex] .
  * Schreckenbach  _et al._ [1981] K. Schreckenbach, H. R. Faust, F. von Feilitzsch, A. A. Hahn, K. Hawerkamp, and J. L. Vuilleumier, Phys. Lett. B 99, 251 (1981).
  * Von Feilitzsch  _et al._ [1982] F. Von Feilitzsch, A. A. Hahn, and K. Schreckenbach, Phys. Lett. B 118, 162 (1982).
  * Schreckenbach  _et al._ [1985] K. Schreckenbach, G. Colvin, W. Gelletly, and F. Von Feilitzsch, Phys. Lett. B 160, 325 (1985).
  * Hahn  _et al._ [1989] A. A. Hahn, K. Schreckenbach, G. Colvin, B. Krusche, W. Gelletly, and F. Von Feilitzsch, Phys. Lett. B 218, 365 (1989).
  * Augier  _et al._ [2023] C. Augier _et al._ (Ricochet), Eur. Phys. J. C 83, 20 (2023), arXiv:2208.01760 [astro-ph.IM] .
  * Almazán  _et al._ [2023] H. Almazán _et al._ (STEREO), Nature 613, 257 (2023), arXiv:2210.07664 [hep-ex] .
  * Ashenfelter  _et al._ [2018] J. Ashenfelter _et al._ (PROSPECT), Phys. Rev. Lett. 121, 251802 (2018), arXiv:1806.02784 [hep-ex] .
  * Andriamirado  _et al._ [2021] M. Andriamirado _et al._ (PROSPECT), Phys. Rev. D 103, 032001 (2021), arXiv:2006.11210 [hep-ex] .
  * Almazán  _et al._ [2018] H. Almazán _et al._ (STEREO), Phys. Rev. Lett. 121, 161801 (2018), arXiv:1806.02096 [hep-ex] .
  * Almazán  _et al._ [2020] H. Almazán _et al._ (STEREO), Phys. Rev. D 102, 052002 (2020), arXiv:1912.06582 [hep-ex] .
  * Alekseev  _et al._ [2018] I. Alekseev _et al._ (DANSS), Phys. Lett. B 787, 56 (2018), arXiv:1804.04046 [hep-ex] .
  * Atif  _et al._ [2022] Z. Atif _et al._ (RENO, NEOS), Phys. Rev. D 105, L111101 (2022), arXiv:2011.00896 [hep-ex] .
  * Serebrov  _et al._ [2019] A. P. Serebrov _et al._ (NEUTRINO-4), Pisma Zh. Eksp. Teor. Fiz. 109, 209 (2019), arXiv:1809.10561 [hep-ex] .
  * Andriamirado  _et al._ [2020] M. Andriamirado _et al._ (PROSPECT, STEREO), (2020), arXiv:2006.13147 [hep-ex] .
  * Giunti  _et al._ [2021] C. Giunti, Y. F. Li, C. A. Ternes, and Y. Y. Zhang, Phys. Lett. B 816, 136214 (2021), arXiv:2101.06785 [hep-ph] .
  * Acero  _et al._ [2008] M. A. Acero, C. Giunti, and M. Laveder, Phys. Rev. D 78, 073009 (2008), arXiv:0711.4222 [hep-ph] .
  * Giunti and Laveder [2011a] C. Giunti and M. Laveder, Phys. Rev. C 83, 065504 (2011a), arXiv:1006.3244 [hep-ph] .
  * Barinov  _et al._ [2022] V. V. Barinov _et al._ , Phys. Rev. Lett. 128, 232501 (2022), arXiv:2109.11482 [nucl-ex] .
  * Goldhagen  _et al._ [2022] K. Goldhagen, M. Maltoni, S. E. Reichard, and T. Schwetz, Eur. Phys. J. C 82, 116 (2022), arXiv:2109.14898 [hep-ph] .
  * Giunti  _et al._ [2022b] C. Giunti, Y. F. Li, C. A. Ternes, O. Tyagi, and Z. Xin, JHEP 10, 164 (2022b), arXiv:2209.00916 [hep-ph] .
  * Giunti  _et al._ [2023] C. Giunti, Y. F. Li, C. A. Ternes, and Z. Xin, Phys. Lett. B 842, 137983 (2023), arXiv:2212.09722 [hep-ph] .
  * Brdar  _et al._ [2023] V. Brdar, J. Gehrlein, and J. Kopp, JHEP 05, 143 (2023), arXiv:2303.05528 [hep-ph] .
  * Berryman  _et al._ [2022] J. M. Berryman, P. Coloma, P. Huber, T. Schwetz, and A. Zhou, JHEP 02, 055 (2022), arXiv:2111.12530 [hep-ph] .
  * Dydak  _et al._ [1984] F. Dydak _et al._ , Phys. Lett. B 134, 281 (1984).
  * Louis [2018] W. C. Louis, (2018), arXiv:1803.11488 [hep-ex] .
  * Eskut  _et al._ [2008] E. Eskut _et al._ (CHORUS), Nucl. Phys. B 793, 326 (2008), arXiv:0710.3361 [hep-ex] .
  * Agafonova  _et al._ [2015] N. Agafonova _et al._ (OPERA), JHEP 06, 069 (2015), arXiv:1503.01876 [hep-ex] .
  * Smithers  _et al._ [2022] B. R. Smithers, B. J. P. Jones, C. A. Argüelles, J. M. Conrad, and A. Diaz, Phys. Rev. D 105, 052001 (2022), arXiv:2111.08722 [hep-ph] .
  * Aiello  _et al._ [2021] S. Aiello _et al._ (KM3NeT), JHEP 10, 180 (2021), arXiv:2107.00344 [hep-ex] .
  * Adamson  _et al._ [2017] P. Adamson _et al._ (NOvA), Phys. Rev. D 96, 072006 (2017), arXiv:1706.04592 [hep-ex] .
  * Acero  _et al._ [2021] M. A. Acero _et al._ (NOvA), Phys. Rev. Lett. 127, 201801 (2021), arXiv:2106.04673 [hep-ex] .
  * Furmanski and Hilgenberg [2021] A. P. Furmanski and C. Hilgenberg, Phys. Rev. D 103, 112011 (2021), arXiv:2012.09788 [hep-ex] .
  * Aker  _et al._ [2022] M. Aker _et al._ (KATRIN), Phys. Rev. D 105, 072004 (2022), arXiv:2201.11593 [hep-ex] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Adams  _et al._ [2020] M. Adams, F. Bezrukov, J. Elvin-Poole, J. J. Evans, P. Guzowski, B. O. Fearraigh, and S. Söldner-Rembold, Eur. Phys. J. C 80, 758 (2020), arXiv:2002.07762 [hep-ph] .
  * Dasgupta and Kopp [2014] B. Dasgupta and J. Kopp, Phys. Rev. Lett. 112, 031803 (2014), arXiv:1310.6337 [hep-ph] .
  * Hannestad  _et al._ [2014] S. Hannestad, R. S. Hansen, and T. Tram, Phys. Rev. Lett. 112, 031802 (2014), arXiv:1310.5926 [astro-ph.CO] .
  * Chu  _et al._ [2018] X. Chu, B. Dasgupta, M. Dentler, J. Kopp, and N. Saviano, JCAP 11, 049 (2018), arXiv:1806.10629 [hep-ph] .
  * Yaguna [2007] C. E. Yaguna, JHEP 06, 002 (2007), arXiv:0706.0178 [hep-ph] .
  * Saviano  _et al._ [2013] N. Saviano, A. Mirizzi, O. Pisanti, P. D. Serpico, G. Mangano, and G. Miele, Phys. Rev. D 87, 073006 (2013), arXiv:1302.1200 [astro-ph.CO] .
  * Giovannini  _et al._ [2002] M. Giovannini, H. Kurki-Suonio, and E. Sihvola, Phys. Rev. D 66, 043504 (2002), arXiv:astro-ph/0203430 .
  * Bezrukov  _et al._ [2017] F. Bezrukov, A. Chudaykin, and D. Gorbunov, JCAP 06, 051 (2017), arXiv:1705.02184 [hep-ph] .
  * Farzan [2019] Y. Farzan, Phys. Lett. B 797, 134911 (2019), arXiv:1907.04271 [hep-ph] .
  * Cline [2020] J. M. Cline, Phys. Lett. B 802, 135182 (2020), arXiv:1908.02278 [hep-ph] .
  * Archidiacono  _et al._ [2020] M. Archidiacono, S. Gariazzo, C. Giunti, S. Hannestad, and T. Tram, JCAP 12, 029 (2020), arXiv:2006.12885 [astro-ph.CO] .
  * Di Valentino  _et al._ [2022] E. Di Valentino, S. Gariazzo, C. Giunti, O. Mena, S. Pan, and W. Yang, Phys. Rev. D 105, 103511 (2022), arXiv:2110.03990 [astro-ph.CO] .
  * Gelmini  _et al._ [2004] G. Gelmini, S. Palomares-Ruiz, and S. Pascoli, Phys. Rev. Lett. 93, 081302 (2004), arXiv:astro-ph/0403323 .
  * Giunti and Zavanin [2015] C. Giunti and E. M. Zavanin, Mod. Phys. Lett. A 31, 1650003 (2015), arXiv:1508.03172 [hep-ph] .
  * Giunti and Laveder [2011b] C. Giunti and M. Laveder, Phys. Rev. D 84, 073008 (2011b), arXiv:1107.1452 [hep-ph] .
  * Conrad  _et al._ [2013] J. M. Conrad, C. M. Ignarra, G. Karagiorgi, M. H. Shaevitz, and J. Spitz, Adv. High Energy Phys. 2013, 163897 (2013), arXiv:1207.4765 [hep-ex] .
  * Kopp  _et al._ [2013] J. Kopp, P. A. N. Machado, M. Maltoni, and T. Schwetz, JHEP 05, 050 (2013), arXiv:1303.3011 [hep-ph] .
  * Giunti  _et al._ [2013] C. Giunti, M. Laveder, Y. F. Li, and H. W. Long, Phys. Rev. D 88, 073008 (2013), arXiv:1308.5288 [hep-ph] .
  * Cianci  _et al._ [2017] D. Cianci, A. Furmanski, G. Karagiorgi, and M. Ross-Lonergan, Phys. Rev. D 96, 055001 (2017), arXiv:1702.01758 [hep-ph] .
  * Pro [2019] _Neutrino Non-Standard Interactions: A Status Report_, Vol. 2 (2019) arXiv:1907.00991 [hep-ph] .
  * Akhmedov and Schwetz [2010] E. Akhmedov and T. Schwetz, JHEP 10, 115 (2010), arXiv:1007.4171 [hep-ph] .
  * Hu [2021] Z. Hu (MINOS, MINOS+, Daya Bay, Bugey-3), PoS ICHEP2020, 201 (2021).
  * Schneider  _et al._ [2021] A. Schneider, B. Skrzypek, C. A. Argüelles, and J. M. Conrad, Phys. Rev. D 104, 092015 (2021), arXiv:2106.01508 [hep-ph] .
  * Esmaili and Nunokawa [2019] A. Esmaili and H. Nunokawa, Eur. Phys. J. C 79, 70 (2019), arXiv:1810.11940 [hep-ph] .
  * Liao and Marfatia [2016] J. Liao and D. Marfatia, Phys. Rev. Lett. 117, 071802 (2016), arXiv:1602.08766 [hep-ph] .
  * Liao  _et al._ [2019] J. Liao, D. Marfatia, and K. Whisnant, Phys. Rev. D 99, 015016 (2019), arXiv:1810.01000 [hep-ph] .
  * Denton  _et al._ [2019] P. B. Denton, Y. Farzan, and I. M. Shoemaker, Phys. Rev. D 99, 035003 (2019), arXiv:1811.01310 [hep-ph] .
  * Moulai [2021] M. H. Moulai,  _Light, Unstable Sterile Neutrinos: Phenomenology, a Search in the IceCube Experiment, and a Global Picture_ , Ph.D. thesis, MIT (2021), arXiv:2110.02351 [hep-ex] .
  * Abbasi  _et al._ [2022] R. Abbasi _et al._ ((IceCube Collaboration)*, IceCube), Phys. Rev. Lett. 129, 151801 (2022), arXiv:2204.00612 [hep-ex] .
  * Hardin  _et al._ [2022] J. M. Hardin, I. Martinez-Soler, A. Diaz, M. Jin, N. W. Kamp, C. A. Argüelles, J. M. Conrad, and M. H. Shaevitz, (2022), arXiv:2211.02610 [hep-ph] .
  * Argüelles  _et al._ [2023a] C. A. Argüelles, T. Bertólez-Martínez, and J. Salvado, Phys. Rev. D 107, 036004 (2023a), arXiv:2201.05108 [hep-ph] .
  * Akhmedov and Smirnov [2022a] E. Akhmedov and A. Y. Smirnov, JHEP 11, 082 (2022a), arXiv:2208.03736 [hep-ph] .
  * Jones  _et al._ [2023] B. J. P. Jones, E. Marzec, and J. Spitz, Phys. Rev. D 107, 013008 (2023), arXiv:2211.00026 [hep-ph] .
  * Jones [2022] B. J. P. Jones, (2022), arXiv:2209.00561 [hep-ph] .
  * Akhmedov and Smirnov [2022b] E. Akhmedov and A. Y. Smirnov, (2022b), arXiv:2210.01547 [hep-ph] .
  * Schwetz [2008] T. Schwetz, JHEP 02, 011 (2008), arXiv:0710.2985 [hep-ph] .
  * Babu  _et al._ [2023] K. S. Babu, V. Brdar, A. de Gouvêa, and P. A. N. Machado, Phys. Rev. D 107, 015017 (2023), arXiv:2209.00031 [hep-ph] .
  * Pas  _et al._ [2005] H. Pas, S. Pakvasa, and T. J. Weiler, Phys. Rev. D 72, 095017 (2005), arXiv:hep-ph/0504096 .
  * Hollenberg  _et al._ [2009] S. Hollenberg, O. Micu, H. Pas, and T. J. Weiler, Phys. Rev. D 80, 093005 (2009), arXiv:0906.0150 [hep-ph] .
  * Carena  _et al._ [2017] M. Carena, Y.-Y. Li, C. S. Machado, P. A. N. Machado, and C. E. M. Wagner, Phys. Rev. D 96, 095014 (2017), arXiv:1708.09548 [hep-ph] .
  * Döring and Päs [2019] D. Döring and H. Päs, Eur. Phys. J. C 79, 604 (2019), arXiv:1808.07734 [hep-ph] .
  * Döring  _et al._ [2020] D. Döring, H. Päs, P. Sicking, and T. J. Weiler, Eur. Phys. J. C 80, 1202 (2020), arXiv:1808.07460 [hep-ph] .
  * Kostelecky and Mewes [2004] V. A. Kostelecky and M. Mewes, Phys. Rev. D 70, 076002 (2004), arXiv:hep-ph/0406255 .
  * Diaz and Kostelecky [2011] J. S. Diaz and V. A. Kostelecky, Phys. Lett. B 700, 25 (2011), arXiv:1012.5985 [hep-ph] .
  * Diaz and Kostelecky [2012] J. S. Diaz and A. Kostelecky, Phys. Rev. D 85, 016013 (2012), arXiv:1108.1799 [hep-ph] .
  * Barenboim  _et al._ [2020] G. A. Barenboim, P. Martínez-Miravé, C. A. Ternes, and M. A. Tórtola, JHEP 03, 070 (2020), arXiv:1911.02329 [hep-ph] .
  * Aguilar-Arevalo  _et al._ [2021b] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), (2021b), arXiv:2110.15055 [hep-ex] .
  * Argüelles  _et al._ [2022b] C. A. Argüelles, N. Foppiani, and M. Hostert, Phys. Rev. D 105, 095006 (2022b), arXiv:2109.03831 [hep-ph] .
  * Shoemaker and Wyenberg [2019] I. M. Shoemaker and J. Wyenberg, Phys. Rev. D 99, 075010 (2019), arXiv:1811.12435 [hep-ph] .
  * Dasgupta  _et al._ [2021] A. Dasgupta, S. K. Kang, and J. E. Kim, JHEP 11, 120 (2021), arXiv:2108.12998 [hep-ph] .
  * Bolton  _et al._ [2022] P. D. Bolton, F. F. Deppisch, K. Fridell, J. Harz, C. Hati, and S. Kulkarni, Phys. Rev. D 106, 035036 (2022), arXiv:2110.02233 [hep-ph] .
  * Coloma  _et al._ [2017] P. Coloma, P. A. N. Machado, I. Martinez-Soler, and I. M. Shoemaker, Phys. Rev. Lett. 119, 201804 (2017), arXiv:1707.08573 [hep-ph] .
  * Atkinson  _et al._ [2022] M. Atkinson, P. Coloma, I. Martinez-Soler, N. Rocco, and I. M. Shoemaker, JHEP 04, 174 (2022), arXiv:2105.09357 [hep-ph] .
  * Plestid [2021] R. Plestid, Phys. Rev. D 104, 075027 (2021), arXiv:2010.04193 [hep-ph] .
  * Huang  _et al._ [2022] G.-y. Huang, S. Jana, M. Lindner, and W. Rodejohann, JCAP 02, 038 (2022), arXiv:2112.09476 [hep-ph] .
  * Ismail  _et al._ [2022] A. Ismail, S. Jana, and R. M. Abraham, Phys. Rev. D 105, 055008 (2022), arXiv:2109.05032 [hep-ph] .
  * Brdar  _et al._ [2021] V. Brdar, A. Greljo, J. Kopp, and T. Opferkuch, JCAP 01, 039 (2021), arXiv:2007.15563 [hep-ph] .
  * Voloshin [1988] M. B. Voloshin, Sov. J. Nucl. Phys. 48, 512 (1988).
  * Barbieri and Mohapatra [1989] R. Barbieri and R. N. Mohapatra, Phys. Lett. B 218, 225 (1989).
  * Babu and Mohapatra [1990] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 64, 1705 (1990).
  * Babu and Mohapatra [1989] K. S. Babu and R. N. Mohapatra, Phys. Rev. Lett. 63, 228 (1989).
  * Leurer and Marcus [1990] M. Leurer and N. Marcus, Phys. Lett. B 237, 81 (1990).
  * Babu  _et al._ [2020] K. S. Babu, S. Jana, and M. Lindner, JHEP 10, 040 (2020), arXiv:2007.04291 [hep-ph] .
  * McKeen and Pospelov [2010] D. McKeen and M. Pospelov, Phys. Rev. D 82, 113018 (2010), arXiv:1011.3046 [hep-ph] .
  * Park  _et al._ [2016] J. Park _et al._ (MINERvA), Phys. Rev. D 93, 112007 (2016), arXiv:1512.07699 [physics.ins-det] .
  * Valencia  _et al._ [2019] E. Valencia _et al._ (MINERvA), Phys. Rev. D 100, 092001 (2019), arXiv:1906.00111 [hep-ex] .
  * Zazueta  _et al._ [2023] L. Zazueta _et al._ (MINERvA), Phys. Rev. D 107, 012001 (2023), arXiv:2209.05540 [hep-ex] .
  * Argüelles  _et al._ [2019] C. A. Argüelles, M. Hostert, and Y.-D. Tsai, Phys. Rev. Lett. 123, 261801 (2019), arXiv:1812.08768 [hep-ph] .
  * Abe  _et al._ [2019] K. Abe _et al._ (T2K), Phys. Rev. D 100, 052006 (2019), arXiv:1902.07598 [hep-ex] .
  * Argüelles  _et al._ [2023b] C. A. Argüelles, N. Foppiani, and M. Hostert, Phys. Rev. D 107, 035027 (2023b), arXiv:2205.12273 [hep-ph] .
  * Bertuzzo  _et al._ [2019] E. Bertuzzo, S. Jana, P. A. N. Machado, and R. Zukanovich Funchal, Phys. Lett. B 791, 210 (2019), arXiv:1808.02500 [hep-ph] .
  * Abi  _et al._ [2021] B. Abi _et al._ (Muon g-2), Phys. Rev. Lett. 126, 141801 (2021), arXiv:2104.03281 [hep-ex] .
  * Pospelov [2009] M. Pospelov, Phys. Rev. D 80, 095002 (2009), arXiv:0811.1030 [hep-ph] .
  * Lees  _et al._ [2017] J. P. Lees _et al._ (BaBar), Phys. Rev. Lett. 119, 131804 (2017), arXiv:1702.03327 [hep-ex] .
  * Banerjee  _et al._ [2019] D. Banerjee _et al._ , Phys. Rev. Lett. 123, 121801 (2019), arXiv:1906.00176 [hep-ex] .
  * Mohlabeng [2019] G. Mohlabeng, Phys. Rev. D 99, 115001 (2019), arXiv:1902.05075 [hep-ph] .
  * Abdullahi  _et al._ [2023] A. M. Abdullahi, M. Hostert, D. Massaro, and S. Pascoli, (2023), arXiv:2302.05410 [hep-ph] .
  * de Gouvêa  _et al._ [2019] A. de Gouvêa, P. J. Fox, R. Harnik, K. J. Kelly, and Y. Zhang, JHEP 01, 001 (2019), arXiv:1809.06388 [hep-ph] .
  * Aguilar-Arevalo  _et al._ [2012] A. A. Aguilar-Arevalo _et al._ (MiniBooNE) (2012) arXiv:1207.4809 [hep-ex] .
  * Döbrich  _et al._ [2016] B. Döbrich, J. Jaeckel, F. Kahlhoefer, A. Ringwald, and K. Schmidt-Hoberg, JHEP 02, 018 (2016), arXiv:1512.03069 [hep-ph] .
  * Fernández-Martínez  _et al._ [2023] E. Fernández-Martínez, M. González-López, J. Hernández-García, M. Hostert, and J. López-Pavón, (2023), arXiv:2304.06772 [hep-ph] .
  * Aguilar-Arevalo  _et al._ [2009b] A. A. Aguilar-Arevalo _et al._ (MiniBooNE), Phys. Rev. D 79, 072002 (2009b), arXiv:0806.1449 [hep-ex] .
  * Patterson  _et al._ [2009] R. B. Patterson, E. M. Laird, Y. Liu, P. D. Meyers, I. Stancu, and H. A. Tanaka, Nucl. Instrum. Meth. A 608, 206 (2009), arXiv:0902.2222 [hep-ex] .
  * [213] R. Wilson (SBN program), “Fermilab Short-Baseline Neutrino Program,” Snowmass Community Summer Study 2022.
  * Isaacson  _et al._ [2023] J. Isaacson, W. I. Jay, A. Lovato, P. A. N. Machado, and N. Rocco, Phys. Rev. D 107, 033007 (2023), arXiv:2205.06378 [hep-ph] .
  * Andreopoulos  _et al._ [2010] C. Andreopoulos _et al._ , Nucl. Instrum. Meth. A 614, 87 (2010), arXiv:0905.2517 [hep-ph] .
  * Patterson [2007] R. B. Patterson,  _A Search for Muon Neutrino to Electron Neutrino Oscillations at $\delta(m^{2})>0.1$ eV2_, Ph.D. thesis, Princeton U. (2007).


# Blind Diagnosis for Millimeter-wave
Large-scale Antenna Systems

Rui Sun, Weidong Wang, Li Chen, Guo Wei, and Wenyi Zhang The authors are with
the CAS Key Laboratory of Wireless-Optical Communications, University of
Science and Technology of China, Hefei 230027, China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>wenyizha@ustc.edu.cn).

###### Abstract

Millimeter-wave (mmWave) communication systems rely on large-scale antenna
arrays to combat large path-loss at mmWave band. Due to hardware
characteristics and deployment environments, mmWave large-scale antenna
systems are vulnerable to antenna element blockages and failures, which
necessitate diagnostic techniques to locate faulty antenna elements for
calibration purposes. Current diagnostic techniques require full or partial
knowledge of channel state information (CSI), which can be challenging to
acquire in the presence of antenna failures. In this letter, we propose a
blind diagnostic technique to identify faulty antenna elements in mmWave
large-scale antenna systems, which does not require any CSI knowledge. By
jointly exploiting the sparsity of mmWave channel and failure pattern, we
first formulate the diagnosis problem as a joint sparse recovery problem.
Then, the atomic norm is introduced to induce the sparsity of mmWave channel
over continuous Fourier dictionary. An efficient algorithm based on
alternating direction method of multipliers (ADMM) is proposed to solve the
formulated problem. Finally, the performance of the proposed technique is
evaluated through numerical simulations.

###### Index Terms:

Antenna array, array diagnosis, atomic norm, fault identification, millimeter-
wave communication

##  I Introduction

Millimeter-wave (mmWave) large-scale antenna system is a key technology in
current and next-generation mobile communication systems. It may suffer from
antenna element failures due to hardware characteristics and deploying
environments. MmWave active devices like amplifiers and mixers are generally
less reliable than conventional sub-6G devices due to higher operating
frequency and lower power efficiency [1, 2]. Besides, mmWave antenna elements
are susceptible to blockages with comparable sizes like dirt and precipitation
owing to the short wavelength at the mmWave band [3]. The existence of antenna
element failure will cause signal power reduction and radiation pattern
distortion, which may lead to severe degradation in system performance.
Therefore, diagnostic techniques for mmWave large-scale antenna systems are of
significant interest for system monitoring and maintenance.

Extensive related works on antenna array diagnosis have been proposed,
including compressed sensing based techniques [4, 5, 6, 7, 8, 9, 10] and deep
learning based techniques [11]. The main idea of compressed sensing based
diagnostic techniques is to use known channel state information (CSI) to
generate fault-free reference signal and then subtract it from the received
signal. Therefore, the differential signal contains the information of faulty
antenna elements, which can then be estimated from compressed measurements.
The work [4] is the first to introduce compressed sensing to array diagnosis.
Improvements over [4] include modifications in sampling methods [5, 6, 7] and
refinements in sparse recovery algorithms [8, 9, 10]. In particular, the work
in [12] extended antenna-only diagnosis to joint diagnosis of antenna, phase
shifter, and RF chain for hybrid beamforming (HBF) systems. Deep-learning
based diagnostic techniques include [11], which utilizes a convolutional
neural network (CNN) to detect the abnormality in the distribution of received
signal and then locate faulty antenna elements.

In order to distinguish antenna failure from channel fading, the above
diagnostic techniques rely on perfect CSI to generate fault-free reference
signal, which can be challenging to acquire in the presence of antenna
failures. Recently, a diagnostic technique proposed in [13] relaxed the CSI
requirement, only requiring the angle-of-arrival (AOA) of each sub-path in the
channel. However, acquiring the AOA of each sub-path can still be a
challenging task for a potentially faulty system since faulty antenna elements
will change the array geometry and distort the radiation pattern, rendering
AOA estimation highly unreliable.

Aiming at this limitation, in this letter, we propose a blind diagnostic
technique for mmWave large-scale antenna systems. The term ‘blind’ indicates
that the proposed technique does not require any knowledge of the CSI. This is
achieved by jointly exploiting the failure pattern sparsity and the mmWave
channel sparsity [14, 15, 16]. Specifically, we first formulate the diagnosis
problem as a joint sparse recovery problem. Then, the atomic norm is
introduced to induce the sparsity of mmWave channel over the continuous
Fourier dictionary. An efficient algorithm based on alternating direction
method of multipliers (ADMM) is proposed to solve the joint sparse recovery
problem. Finally, numerical simulations validate the proposed technique.

Notations: We use a lowercase and an uppercase bold letter to represent a
vector and a matrix, respectively. $\|\mathbf{a}\|_{1}$ and
$\|\mathbf{a}\|_{2}$ represent the $l_{1}$ and $l_{2}$ norm of vector
$\mathbf{a}$, respectively. In particular, $\|\mathbf{a}\|_{\mathcal{A}}$
denotes the atomic norm of $\mathbf{a}$. For a matrix $\mathbf{A}$,
$\mathbf{A}^{\mathrm{T}}$, $\mathbf{A}^{\mathrm{H}}$ denote its transpose and
conjugate transpose, respectively. $\|\mathbf{A}\|_{\mathrm{F}}$ and
$\mathrm{tr}(\mathbf{A})$ represent the Frobenius norm and the trace of
$\mathbf{A}$, respectively. $\mathbf{I}_{N}$ represents an $N\times N$
identity matrix. $\mathcal{CN}(\mu,\sigma^{2})$ denotes circularly symmetric
complex Gaussian distribution with mean $\mu$ and variance $\sigma^{2}$.
$U(a,b)$ denotes uniform distribution over the interval $[a,b]$.

##  II System Model

We consider an analog beamforming (ABF) mmWave large-scale antenna system
equipped with an $N$-element uniform linear array (ULA)111 The proposed
diagnostic technique can be extended to other system architectures (like
digital beamforming (DBF) and hybrid beamforming (HBF) systems) and other
array structures (like uniform planer array (UPA)). For HBF systems, each RF
chain can obtain independent measurements in a single time slot using
different phase shifts, which shortens the measurement time by a factor of the
number of RF chains. For DBF systems, the measurement can be accomplished in a
single time slot since each antenna has its own RF chain. , which is the
array-under-test (AUT). A single-antenna diagnostic transmitter (TX) is
adopted to transmit test symbols. The AUT receives the test symbol and
conducts the diagnosis. In the absence of antenna failure, the fault-free
received symbol can be expressed as

$y^{\prime}=\mathbf{f}^{\mathrm{T}}\mathbf{h}x+w,$ (1)

where $\mathbf{f}\in\mathbb{C}^{N}$ is the combining vector,
$\mathbf{h}\in\mathbb{C}^{N}$ is the channel vector between the TX and the
AUT, $x$ is the transmitted symbol, $w\sim\mathcal{CN}(0,1/\mathrm{SNR})$ is
the normalized noise and $\mathrm{SNR}$ is the signal-to-noise ratio.

The channel $\mathbf{h}$ is assumed as a block-fading mmWave clustered channel
[16], which can be expressed as

$\mathbf{h}=\sum_{l=1}^{L}\alpha_{l}\mathbf{a}(\theta_{l}),$ (2)

where $\mathbf{a}(\theta_{l})=[1,e^{j2\pi d\sin\theta_{l}},\cdots,e^{j2\pi
d(N-1)\sin\theta_{l}}]^{\mathrm{T}}$ is the array response vector, $L$ is the
number of sub-paths, $\alpha_{l}\sim\mathcal{CN}(0,1/L)$ and $\theta_{l}\sim
U(-\pi/2,\pi/2)$ are the complex gain and the angle-of-arrival (AOA) of the
$l$-th sub-path, respectively, and $d=1/2$ is the element spacing relative to
the wavelength.

In the presence of faulty antenna elements, the actual channel vector will
deviate from the ideal one since faults may cause additional attenuation and
phase shift to the channel response. Therefore, the received symbol under
antenna faults can be expressed as

$y=\mathbf{f}^{\mathrm{T}}(\mathbf{h}+\mathbf{h}_{\mathrm{f}})x+w,$ (3)

where $\mathbf{h}_{\mathrm{f}}\in\mathbb{C}^{N}$ is the fault-induced channel
deviation. Due to the failure pattern sparsity that usually only a small
number of antenna elements are faulty [4, 5], $\mathbf{h}_{\mathrm{f}}$ is
assumed as a sparse vector, in which non-zero entries indicate faulty antenna
elements.

To detect faults, the TX transmits test symbol $x=1$ throughout the diagnosis.
Within the channel coherence time, the AUT uses $K$ random combining vectors
to receive the signal, yielding the observation model

$\mathbf{y}=\mathbf{F}(\mathbf{h}+\mathbf{h}_{\mathrm{f}})+\mathbf{w},$ (4)

where

$\displaystyle\mathbf{y}$
$\displaystyle=[y_{1},\cdots,y_{K}]^{\mathrm{T}}\in\mathbb{C}^{K},$ (5)
$\displaystyle\mathbf{F}$
$\displaystyle=[\mathbf{f}_{1},\cdots,\mathbf{f}_{K}]^{\mathrm{T}}\in\mathbb{C}^{K\times
N},$ $\displaystyle\mathbf{w}$
$\displaystyle=[w_{1},\cdots,w_{K}]^{\mathrm{T}}\in\mathbb{C}^{K},$

in which $y_{k}$, $\mathbf{f}_{k}$, and $w_{k}$ are the received symbol, the
combining vector, and the noise corresponded to the $k$-th measurement,
respectively.

The goal of diagnosis is to recover the sparse fault-induced channel deviation
$\mathbf{h}_{\mathrm{f}}$, in which non-zero entries indicate faulty antenna
elements. From (4), it can be observed that $\mathbf{h}_{\mathrm{f}}$ is
coupled with the fault-free channel vector $\mathbf{h}$. Since the system
contains potentially faulty antenna elements, the channel vector $\mathbf{h}$
can not be estimated by conventional channel estimation techniques like pilot-
based training, and thus recovering $\mathbf{h}_{\mathrm{f}}$ under unknown
$\mathbf{h}$ can be challenging.

To cope with this issue, prior works require full or partial knowledge of the
CSI to perform the diagnosis. The diagnostic technique proposed in [5]
requires full CSI, which assumes a free-space wireless channel and proposes to
calculate the channel vector using known sub-path AOA and gain (i.e.,
$\theta_{l}$ and $\alpha_{l}$). Hence, one can generate the fault-free channel
vector $\mathbf{h}$ using (2) and subtract it from the received signal, and
the impact of channel can be eliminated. A recent work [13] relaxes the CSI
requirement in the sense that it only requires the AOA of each sub-path (i.e.,
$\theta_{l}$). There the key idea is to project the received signal onto the
null space of AOAs and thus the impact of channel can also be eliminated.

The above diagnostic techniques follow the routine of “cancel-then-recover” in
the sense that they seek to cancel the impact of channel from the received
signal first and then recover the fault-induced channel deviation. The
requirement of CSI can be challenging to satisfy, especially for the outdoor
online diagnosis in a complex multipath scattering environment. In the
following, we propose a diagnostic technique that does not require any
knowledge of the CSI (i.e., blind diagnosis). This is achieved by jointly
exploiting the mmWave channel sparsity and the failure pattern sparsity.
Besides, the fault-free channel vector $\mathbf{h}$ can also be recovered,
which provides an approach to perform channel estimation under antenna element
failures.

##  III Fault Diagnosis

###  III-A Problem Formulation

Recall that our goal is to recover $\mathbf{h}_{\mathrm{f}}$ under the
observation model (4), in which non-zero entries indicate faulty antenna
elements. At first glance, it seems impossible to distinguish channel fading
from antenna failures, and hence recovering $\mathbf{h}_{\mathrm{f}}$ under
unknown $\mathbf{h}$ appears to be a challenging task. To decouple the failure
deviation $\mathbf{h}_{\mathrm{f}}$ from the channel $\mathbf{h}$, we need to
exploit their structural characteristics. The fault-induced channel deviation
$\mathbf{h}_{\mathrm{f}}$ is sparse itself due to the failure pattern
sparsity. The ideal channel vector $\mathbf{h}$ is also sparse in the Fourier
dictionary since the mmWave channel is sparse in the angle domain (i.e., $L\ll
N$) [14, 15]. Therefore, $\mathbf{h}$ and $\mathbf{h}_{\mathrm{f}}$ are sparse
under different dictionaries, and this property allows them to be recovered
simultaneously [17].

To jointly recover $\mathbf{h}$ and $\mathbf{h}_{\mathrm{f}}$, we need to
introduce a-prior information on them. We adopt the well-known $l_{1}$ norm as
the sparsity-inducing norm for $\mathbf{h}_{\mathrm{f}}$:

$\|\mathbf{h}_{\mathrm{f}}\|_{1}=\sum_{n=1}^{N}|h_{\mathrm{f},n}|,$ (6)

where $h_{\mathrm{f},n}$ is the $n$-th entry in $\mathbf{h}_{\mathrm{f}}$.

The ideal channel vector $\mathbf{h}$ is sparse in the Fourier dictionary. To
be more specific, $\mathbf{h}$ is sparse in the continuous Fourier dictionary

$\mathcal{A}=\left\\{\mathbf{a}(\theta_{l})|\theta_{l}\in[-\pi/2,\pi/2]\right\\}.$
(7)

We adopt the atomic norm as the sparsity-inducing norm for $\mathbf{h}$. The
atomic norm of $\mathbf{h}$ over the dictionary $\mathcal{A}$ is defined as
[18]

$\|\mathbf{h}\|_{\mathcal{A}}=\inf\left\\{\sum_{l}|\alpha_{l}|\bigg{|}\mathbf{h}=\sum_{l}\alpha_{l}\mathbf{a}_{l},\mathbf{a}_{l}\in\mathcal{A}\right\\},$
(8)

which can be regarded as the $l_{1}$ norm over the continuous dictionary
$\mathcal{A}$.

Using the $l_{1}$ norm and the atomic norm as sparsity-inducing norm for
$\mathbf{h}_{\mathrm{f}}$ and $\mathbf{h}$, respectively, the optimization
problem for joint recovery can be expressed as

$\displaystyle\\{\hat{\mathbf{h}},\hat{\mathbf{h}}_{\mathrm{f}}\\}=\arg\min_{\mathbf{h},\mathbf{h}_{\mathrm{f}}}\frac{1}{2}\|\mathbf{y}-\mathbf{F}(\mathbf{h}+\mathbf{h}_{\mathrm{f}})\|_{2}^{2}+\tau\|\mathbf{h}\|_{\mathcal{A}}+\lambda\|\mathbf{h}_{\mathrm{f}}\|_{1},$
(9)

where $\tau$ and $\lambda$ are regularization parameters controlling the
sparsity penalty on $\mathbf{h}$ and $\mathbf{h}_{\mathrm{f}}$, respectively.
The optimization problem (9) tends to find a sparse vector
$\mathbf{h}_{\mathrm{f}}$ and a vector $\mathbf{h}$ that is sparse under the
dictionary $\mathcal{A}$ to fit the observation model.

Although the optimization problem (9) is clear in its form, it can not be
directly solved since it involves infinite-dimensional variable optimization
due to the continuity of the dictionary $\mathcal{A}$. To cope with this
issue, a conventional approach is to discretize the angle domain into a grid,
which forms a discrete Fourier transform (DFT) dictionary [19]. This approach,
however, induces the off-grid error since the AOAs will not lie exactly on the
grid, which may significantly degrades the recovery performance [20]. In the
following, we develop an efficient algorithm for solving the optimization
problem (9).

###  III-B An Efficient Diagnostic Algorithm

The main difficulty of solving (9) arises from the atomic norm
$\|\mathbf{h}\|_{\mathcal{A}}$ over the continuous Fourier dictionary
$\mathcal{A}$. Fortunately, the minimization of $\|\mathbf{h}\|_{\mathcal{A}}$
admits the following semidefinite program (SDP) thanks to the
Carathéodory–Fejér–Pisarenko decomposition [21, 18]

$\arg\min_{\begin{subarray}{c}\mathbf{u}\in\mathbb{C}^{N}\\\
v\in\mathbb{R}\end{subarray}}\frac{1}{2}\left(\frac{1}{N}\mathrm{tr}\big{(}T(\mathbf{u})\big{)}+v\right)\hskip
10.00002pt\mathrm{s.t.}\hskip
10.00002pt\begin{bmatrix}T(\mathbf{u})&\mathbf{h}\\\
\mathbf{h}^{\mathrm{H}}&v\end{bmatrix}\succeq 0,$ (10)

where $\mathrm{tr}(\cdot)$ denotes the trace of a matrix, and $T(\mathbf{u})$
is the Hermitian Toeplitz matrix with vector $\mathbf{u}$ as its first column.

Therefore, the optimization problem (9) has the equivalent SDP

$\displaystyle\\{\hat{\mathbf{h}},\hat{\mathbf{h}}_{\mathrm{f}}\\}=$
$\displaystyle\arg\min_{\mathbf{h},\mathbf{h}_{\mathrm{f}},\mathbf{u},v}\frac{1}{2}\|\mathbf{y}-\mathbf{F}(\mathbf{h}+\mathbf{h}_{\mathrm{f}})\|_{2}^{2}$
(11)
$\displaystyle+\frac{\tau}{2}\left(\frac{1}{N}\mathrm{tr}\big{(}T(\mathbf{u})\big{)}+v\right)+\lambda\|\mathbf{h}_{\mathrm{f}}\|_{1}$
$\displaystyle\mathrm{s.t.}\hskip
10.00002pt\begin{bmatrix}T(\mathbf{u})&\mathbf{h}\\\
\mathbf{h}^{\mathrm{H}}&v\end{bmatrix}\succeq 0.$

Related works on the atomic norm suggest using the CVX toolbox to solve the
SDP (11) [18], which can be time-consuming for large-scale problems. To solve
(11) in an efficient way, we develop an algorithm based on the alternating
direction method of multipliers (ADMM) algorithm [22]. The ADMM algorithm
integrates the augmented Lagrangian method with the dual ascent method, which
usually shows excellent efficiency in solving large-scale problems.

We first rewrite (11) into the ADMM form:

$\displaystyle\\{\hat{\mathbf{h}},\hat{\mathbf{h}}_{\mathrm{f}}\\}=$
$\displaystyle\arg\min_{\mathbf{h},\mathbf{h}_{\mathrm{f}},\mathbf{u},v}\frac{1}{2}\|\mathbf{y}-\mathbf{F}(\mathbf{h}+\mathbf{h}_{\mathrm{f}})\|_{2}^{2}$
(12)
$\displaystyle+\frac{\tau}{2}\left(\frac{1}{N}\mathrm{tr}\big{(}T(\mathbf{u})\big{)}+v\right)+\lambda\|\mathbf{h}_{\mathrm{f}}\|_{1},$
$\displaystyle\mathrm{s.t.}\hskip
10.00002pt\mathbf{Z}=\begin{bmatrix}T(\mathbf{u})&\mathbf{h}\\\
\mathbf{h}^{\mathrm{H}}&v\end{bmatrix}\succeq 0,$

where $\mathbf{Z}\in\mathbb{C}^{(N+1)\times(N+1)}$ is an auxiliary matrix.

The augmented Lagrangian function for (12) can be expressed as

$\displaystyle\mathcal{L}_{\rho}(v,\mathbf{u,h},\mathbf{h}_{\mathrm{f}},\mathbf{Z},\boldsymbol{\Lambda})$
(13)
$\displaystyle=\frac{1}{2}\|\mathbf{y}-\mathbf{F}(\mathbf{h}+\mathbf{h}_{\mathrm{f}})\|_{2}^{2}+\frac{\tau}{2}\left(\frac{1}{N}\mathrm{tr}\big{(}T(\mathbf{u})\big{)}+v\right)+\lambda\|\mathbf{h}_{\mathrm{f}}\|_{1}$
$\displaystyle+\left\langle\boldsymbol{\Lambda},\mathbf{Z}-\begin{bmatrix}T(\mathbf{u})&\mathbf{h}\\\
\mathbf{h}^{\mathrm{H}}&v\end{bmatrix}\right\rangle+\frac{\rho}{2}\left\|\mathbf{Z}-\begin{bmatrix}T(\mathbf{u})&\mathbf{h}\\\
\mathbf{h}^{\mathrm{H}}&v\end{bmatrix}\right\|_{\mathrm{F}}^{2},$

where $\boldsymbol{\Lambda}\in\mathbb{C}^{(N+1)\times(N+1)}$ is the Lagrangian
multiplier, $\rho$ is a penalty parameter, and $\langle\cdot,\cdot\rangle$
denotes the real inner product.

The ADMM algorithm minimizes the augmented Lagrangian function by iteratively
updating the following variables:

$\displaystyle v^{(l+1)}$
$\displaystyle=\arg\min_{v}\mathcal{L}_{\rho}(v,\mathbf{u}^{(l)},\mathbf{h}^{(l)},\mathbf{h}_{\mathrm{f}}^{(l)},\mathbf{Z}^{(l)},\boldsymbol{\Lambda}^{(l)}),$
(14) $\displaystyle\mathbf{u}^{(l+1)}$
$\displaystyle=\arg\min_{\mathbf{u}}\mathcal{L}_{\rho}(v^{(l+1)},\mathbf{u},\mathbf{h}^{(l)},\mathbf{h}_{\mathrm{f}}^{(l)},\mathbf{Z}^{(l)},\boldsymbol{\Lambda}^{(l)}),$
$\displaystyle\mathbf{h}^{(l+1)}$
$\displaystyle=\arg\min_{\mathbf{h}}\mathcal{L}_{\rho}(v^{(l+1)},\mathbf{u}^{(l+1)},\mathbf{h},\mathbf{h}_{\mathrm{f}}^{(l)},\mathbf{Z}^{(l)},\boldsymbol{\Lambda}^{(l)}),$
$\displaystyle\mathbf{h}_{\mathrm{f}}^{(l+1)}$
$\displaystyle=\arg\min_{\mathbf{h}_{\mathrm{f}}}\mathcal{L}_{\rho}(v^{(l+1)},\mathbf{u}^{(l+1)},\mathbf{h}^{(l+1)},\mathbf{h}_{\mathrm{f}},\mathbf{Z}^{(l)},\boldsymbol{\Lambda}^{(l)}),$
$\displaystyle\mathbf{Z}^{(l+1)}$
$\displaystyle=\arg\min_{\mathbf{Z}}\mathcal{L}_{\rho}(v^{(l+1)},\mathbf{u}^{(l+1)},\mathbf{h}^{(l+1)},\mathbf{h}_{\mathrm{f}}^{(l+1)},\mathbf{Z},\boldsymbol{\Lambda}^{(l)}),$
$\displaystyle\boldsymbol{\Lambda}^{(l+1)}$
$\displaystyle=\boldsymbol{\Lambda}^{(l)}+\rho\left(\mathbf{Z}^{(l+1)}-\begin{bmatrix}T\left(\mathbf{u}^{(l+1)}\right)&\mathbf{h}^{(l+1)}\\\
{\mathbf{h}^{(l+1)}}^{\mathrm{H}}&v^{(l+1)}\end{bmatrix}\right),$

where the superscript $(l)$ denotes the $l$-th iteration.

To perform the above updates in an explicit way, we first introduce the
partitions

$\displaystyle\mathbf{Z}^{(l)}=\begin{bmatrix}\mathbf{Z}^{(l)}_{0}&\mathbf{z}^{(l)}_{1}\\\
{\mathbf{z}^{(l)}_{1}}^{\mathrm{H}}&Z^{(l)}_{N+1,N+1}\end{bmatrix},\boldsymbol{\Lambda}^{(l)}=\begin{bmatrix}\boldsymbol{\Lambda}^{(l)}_{0}&\boldsymbol{\lambda}^{(l)}_{1}\\\
{\boldsymbol{\lambda}^{(l)}_{1}}^{\mathrm{H}}&\Lambda^{(l)}_{N+1,N+1}\end{bmatrix}.$
(15)

The augmented Lagrangian function is convex and differentiable with respect to
$v$, $\mathbf{u}$, and $\mathbf{h}$. Setting their gradients to zero, these
updates have closed-form expressions as

$\displaystyle v^{(l+1)}$
$\displaystyle=Z^{(l)}_{N+1,N+1}+\frac{1}{\rho}\left(\Lambda^{(l)}_{N+1,N+1}-\frac{\tau}{2}\right),$
(16) $\displaystyle\mathbf{u}^{(l+1)}$
$\displaystyle=\boldsymbol{\Psi}^{-1}\left(T^{*}\left(\mathbf{Z}^{(l)}_{0}+\frac{1}{\rho}\boldsymbol{\Lambda}^{(l)}_{0}\right)\right)-\frac{\tau}{2\rho}\mathbf{e}_{1},$
$\displaystyle\mathbf{h}^{(l+1)}$
$\displaystyle=(\mathbf{F}^{\mathrm{H}}\mathbf{F}+2\rho\mathbf{I}_{N})^{-1}\left(\mathbf{F}^{\mathrm{H}}(\mathbf{y}-\mathbf{F}\mathbf{h}^{(l)}_{\mathrm{f}})+2\boldsymbol{\lambda}^{(l)}_{1}+2\rho\mathbf{z}^{(l)}_{1}\right),$

where $\boldsymbol{\Psi}$ is a diagonal matrix and
$\boldsymbol{\Psi}_{i,j}=N-j+1,j=1,\cdots,N$, $\mathbf{e}_{1}$ is a zero
vector with its first entry being one, and $T^{*}(\cdot)$ generates a vector
whose $i$-th element is the trace of the $(i-1)$-th subdiagonal of the input
matrix.

The update of $\mathbf{h}_{\mathrm{f}}$ can be written as

$\displaystyle\mathbf{h}_{\mathrm{f}}^{(l+1)}$
$\displaystyle=\arg\min_{\mathbf{h}_{\mathrm{f}}}\frac{1}{2}\|(\mathbf{y}-\mathbf{F}\mathbf{h}^{(l+1)})-\mathbf{F}\mathbf{h}_{\mathrm{f}}\|_{2}^{2}+\lambda\|\mathbf{h}_{\mathrm{f}}\|_{1},$
(17)

which is the well-known LASSO problem and can be solved efficiently by the
algorithm proposed in [22].

Finally, the update of $\mathbf{Z}$ can be expressed as

$\displaystyle\mathbf{Z}^{(l+1)}$ $\displaystyle=\arg\min_{\mathbf{Z}\succeq
0}\left\|\mathbf{Z}-\left(\underbrace{\begin{bmatrix}T\left(\mathbf{u}^{(l+1)}\right)&\mathbf{h}^{(l+1)}\\\
{\mathbf{h}^{(l+1)}}^{\mathrm{H}}&v^{(l+1)}\end{bmatrix}-\frac{1}{\rho}\boldsymbol{\Lambda}^{(l)}}_{\mathbf{G}}\right)\right\|_{\mathrm{F}}^{2},$
(18)

which amounts to projecting the Hermitian matrix $\mathbf{G}$ onto the
positive semidefinite cone and can be performed by computing the eigenvalue
decomposition of $\mathbf{G}$ and setting all negative eigenvalues to zero.

The overall algorithm for solving (11) is summarized in Algorithm 1, where
$\epsilon$ and $l_{\mathrm{max}}$ are the halting threshold and the maximum
number of iterations222Empirical results suggest that setting
$\epsilon=10^{-3}$ and $l_{\mathrm{max}}=1000$ typically leads to a good
performance.. The computational complexity mainly arises from the updates of
$\mathbf{h}$, $\mathbf{h}_{\mathrm{f}}$, and $\mathbf{Z}$. The updates of
$\mathbf{h}$ and $\mathbf{h}_{\mathrm{f}}$ involve matrix inversion, which has
the complexity of $O(N^{3})$. The update of $\mathbf{Z}$ requires eigenvalue
decomposition and has the complexity of $O(N^{3})$. Therefore, the overall
computational complexity of Algorithm 1 is $O(N^{3})$.

Initialization: $\mathbf{Z}^{(0)}=\mathbf{0}$,
$\boldsymbol{\Lambda}^{(0)}=\mathbf{0}$,
$\mathbf{h}_{\mathrm{f}}^{(0)}=\mathbf{0}$;

while  _$\|\mathbf{h}_{\mathrm{f}}^{(l+1)}-\mathbf{h}_{\mathrm{f}}^{(l)}\|_{2}
>\epsilon$ and $l\leq l_{\mathrm{max}}$_ do

1$v^{(l+1)}=Z^{(l)}_{N+1,N+1}+\frac{1}{\rho}\left(\Lambda^{(l)}_{N+1,N+1}-\frac{\tau}{2}\right)$;

2$\mathbf{u}^{(l+1)}=\boldsymbol{\Psi}^{-1}\left(T^{*}\left(\mathbf{Z}^{(l)}_{0}+\frac{1}{\rho}\boldsymbol{\Lambda}^{(l)}_{0}\right)\right)-\frac{\tau}{2\rho}\mathbf{e}_{1}$;

3$\mathbf{h}^{(l+1)}=(\mathbf{F}^{\mathrm{H}}\mathbf{F}+2\rho\mathbf{I}_{N})^{-1}\left(\mathbf{F}^{\mathrm{H}}(\mathbf{y}-\mathbf{F}\mathbf{h}^{(l)}_{\mathrm{f}})+2\boldsymbol{\lambda}^{(l)}_{1}+2\rho\mathbf{z}^{(l)}_{1}\right)$;

4$\mathbf{h}_{\mathrm{f}}^{(l+1)}=\arg\min_{\mathbf{h}_{\mathrm{f}}}\frac{1}{2}\|(\mathbf{y}-\mathbf{F}\mathbf{h}^{(l+1)})-\mathbf{F}\mathbf{h}_{\mathrm{f}}\|_{2}^{2}+\lambda\|\mathbf{h}_{\mathrm{f}}\|_{1}$,
solved by the algorithm proposed in [22];

5Perform eigenvalue decomposition on
$\begin{bmatrix}T\left(\mathbf{u}^{(l+1)}\right)&\mathbf{h}^{(l+1)}\\\
{\mathbf{h}^{(l+1)}}^{\mathrm{H}}&v^{(l+1)}\end{bmatrix}-\frac{1}{\rho}\boldsymbol{\Lambda}^{(l)}$
and set all negative eigenvalues to zero, yielding $\mathbf{Z}^{(l+1)}$;

6$\boldsymbol{\Lambda}^{(l+1)}=\boldsymbol{\Lambda}^{(l)}+\rho\left(\mathbf{Z}^{(l+1)}-\begin{bmatrix}T\left(\mathbf{u}^{(l+1)}\right)&\mathbf{h}^{(l+1)}\\\
{\mathbf{h}^{(l+1)}}^{\mathrm{H}}&v^{(l+1)}\end{bmatrix}\right)$;

end while

Algorithm 1 An Efficient Algorithm for Solving (11)

##  IV Numerical Simulations

In this section, we conduct numerical simulations to evaluate the performance
of the proposed diagnostic technique. The number of antenna elements is set to
$N=64$. The number of faulty antenna elements is set to 3, whose locations are
chosen uniformly at random. The amplitude and phase of the entries in the
fault-induced channel deviation $\mathbf{h}_{\mathrm{f}}$ follow $U(0.2,1)$
and $U(0,2\pi)$, respectively. Regularization parameters are set to
$\lambda=0.4$ and $\tau=0.3$. We adopt the success probability as the
performance metric, which is defined as the probability that the states of all
antenna elements are correctly identified. For comparison, we adopt the
diagnostic techniques proposed in [5] (denoted as Eltayeb18) and [13] (denoted
as Medina20) as benchmark techniques, which require full and partial CSI,
respectively.

First, we evaluate the effect of the number of measurements. Fig. 1 shows the
success probability versus the number of measurements under different SNRs,
where the number of sub-paths is set to $L=4$. It can be observed that all
techniques perform better with a larger number of measurements and/or larger
SNR. Among all diagnostic techniques, the diagnosis with full CSI (Eltayeb18)
outperforms others since the impact of channel can be completely eliminated
using known CSI. When the SNR is sufficiently large, the proposed technique
has slightly better performance than the diagnosis with partial CSI
(Medina20).

Figure 1: Success probability versus the number of measurements under
different SNRs.

Next, we assume that the estimated sub-path gain $\hat{\alpha}_{l}$ contains
estimation error, which is defined as

$\hat{\alpha}_{l}=\alpha_{l}+\delta_{\alpha}\alpha_{\mathrm{e}},$ (19)

where $\alpha_{\mathrm{e}}\sim\mathcal{CN}(0,1)$ represents gain estimation
error and $\delta_{\alpha}$ is the error intensity. The performance of
different techniques under sub-path gain estimation error is shown in Fig. 2,
where $\mathrm{SNR}=30$dB and the number of measurements is set to $K=N$ to
ensure sufficient measurements for all techniques. We can observe that the
performance of diagnosis with full CSI (Eltayeb18) degrades significantly,
while other techniques are not affected by the estimation error since they do
not require the knowledge of sub-path gain. Besides, the more sub-paths in the
channel, the worse the performance since more errors will be introduced.

Figure 2: Success probability versus sub-path gain error intensity under
different number of sub-paths.

Finally, we assume that the estimated AOA $\hat{\theta}_{l}$ contains
estimation error, which is defined as

$\hat{\theta}_{l}=\theta_{l}+\delta_{\theta}\theta_{\mathrm{e}}\pi,$ (20)

where $\theta_{\mathrm{e}}\sim\mathcal{N}(0,1)$ represents the AOA estimation
error and $\delta_{\theta}$ is the error intensity. The success probabilities
of different techniques under AOA estimation error are shown in Fig. 3, where
$\mathrm{SNR}=30$dB and the number of measurements is set to $K=N$. It can be
observed that benchmark diagnostic techniques are highly sensitive to AOA
estimation errors. On the contrary, the performance of the proposed diagnostic
technique is not affected under all intensities of AOA estimation error since
it does not require any CSI knowledge, thereby exhibiting strong robustness
against channel estimation errors.

Figure 3: Success probability versus sub-path AOA error intensity under
different number of sub-paths.

##  V Conclusions

In this letter, we have proposed a blind diagnostic technique for mmWave
large-scale antenna systems to locate faulty antenna elements. By jointly
exploiting the sparsity of the mmWave channel and the failure pattern, the
location of faulty antenna elements can be identified without any knowledge of
the CSI. A novel atomic norm has been introduced as the sparsity-inducing norm
of the mmWave channel, and the diagnosis problem has been formulated as a
joint sparse recovery problem. An efficient ADMM-based diagnostic algorithm
has been proposed to solve the joint sparse recovery problem. Numerical
results have shown that the proposed technique has strong robustness against
channel estimation errors compared with prior works.

In future works, the diagnosis of mmWave OFDM systems can be considered. The
channels of different subcarriers share a common or similar sparsity pattern,
leading to an MMV (multiple measurement vectors) observation model. By jointly
recovering channels and fault deviations of different sub-carriers, the
diagnosis performance may be further improved.

## References

  * [1] G. G. Fischer and S. Glisic, “Temperature stability and reliability aspects of 77 GHz voltage controlled oscillators in a SiGe: C BiCMOS technology,” in _IEEE Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems_. IEEE, 2008, pp. 171–174.
  * [2] W. Chang, J. Luo, Y. Qi, and B. Wang, “Reliability and failure analysis in designing a typical operation amplifier,” in _17th IEEE International Symposium on the Physical and Failure Analysis of Integrated Circuits_. IEEE, 2010, pp. 1–4.
  * [3] E. Vinyaikin, M. Zinicheva, and A. Naumov, “Attenuation and phase variation of millimeter and centimeter radio waves in a medium consisting of dry and wet dust particles,” _Radiophys. Quantum Electron._ , vol. 37, no. 11, pp. 914–923, 1994.
  * [4] M. D. Migliore, “A compressed sensing approach for array diagnosis from a small set of near-field measurements,” _IEEE Trans. Antennas Propag._ , vol. 59, no. 6, pp. 2127–2133, 2011.
  * [5] M. E. Eltayeb, T. Y. Al-Naffouri, and R. W. Heath, “Compressive sensing for millimeter wave antenna array diagnosis,” _IEEE Trans. Commun._ , vol. 66, no. 6, pp. 2708–2721, 2018.
  * [6] R. Palmeri, T. Isernia, and A. F. Morabito, “Diagnosis of planar arrays through phaseless measurements and sparsity promotion,” _IEEE Antennas Wirel. Propag. Lett._ , vol. 18, no. 6, pp. 1273–1277, 2019.
  * [7] C. Xiong, G. Xiao, Y. Hou, and M. Hameed, “A compressed sensing-based element failure diagnosis method for phased array antenna during beam steering,” _IEEE Antennas Wirel. Propag. Lett._ , vol. 18, no. 9, pp. 1756–1760, 2019\.
  * [8] T. Ince and G. Ögücü, “Array failure diagnosis using nonconvex compressed sensing,” _IEEE Antennas Wirel. Propag. Lett._ , vol. 15, pp. 992–995, 2015.
  * [9] M. Salucci, A. Gelmini, G. Oliveri, and A. Massa, “Planar array diagnosis by means of an advanced Bayesian compressive processing,” _IEEE Trans. Antennas Propag._ , vol. 66, no. 11, pp. 5892–5906, 2018.
  * [10] S. Ma, W. Shen, J. An, and L. Hanzo, “Antenna array diagnosis for millimeter-wave MIMO systems,” _IEEE Trans. Veh. Technol._ , vol. 69, no. 4, pp. 4585–4589, 2020.
  * [11] K. Chen, W. Wang, X. Chen, and H. Yin, “Deep learning based antenna array fault detection,” in _IEEE 89th Vehicular Technology Conference (VTC2019-Spring)_. IEEE, 2019, pp. 1–5.
  * [12] R. Sun, W. Wang, L. Chen, G. Wei, and W. Zhang, “Hybrid beamforming system diagnosis: Failure modeling and identification,” _IEEE Trans. Wireless Commun._ , 2021.
  * [13] G. Medina, A. S. Jida, S. Pulipati, R. Talwar, T. Y. Al-Naffouri, A. Madanayake, M. Eltayeb _et al._ , “Millimeter-wave antenna array diagnosis with partial channel state information,” _arXiv preprint arXiv:2011.00828_ , 2020.
  * [14] X. Song, S. Haghighatshoar, and G. Caire, “A scalable and statistically robust beam alignment technique for millimeter-wave systems,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 7, pp. 4792–4805, 2018.
  * [15] ——, “Efficient beam alignment for millimeter wave single-carrier systems with hybrid MIMO transceivers,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 3, pp. 1518–1533, 2019.
  * [16] D. Zhang, A. Li, M. Shirvanimoghaddam, P. Cheng, Y. Li, and B. Vucetic, “Codebook-based training beam sequence design for millimeter-wave tracking systems,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 11, pp. 5333–5349, 2019.
  * [17] M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Sparse signal detection from incoherent projections,” in _IEEE International Conference on Acoustics Speech and Signal Processing Proceedings_ , vol. 3. IEEE, 2006, pp. III–III.
  * [18] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressed sensing off the grid,” _IEEE Trans. Inf. Theory_ , vol. 59, no. 11, pp. 7465–7490, 2013\.
  * [19] Z. Yang, L. Xie, and C. Zhang, “Off-grid direction of arrival estimation using sparse Bayesian inference,” _IEEE Trans. Signal Process._ , vol. 61, no. 1, pp. 38–43, 2012.
  * [20] Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” _IEEE Trans. Signal Process._ , vol. 59, no. 5, pp. 2182–2195, 2011.
  * [21] T. T. Georgiou, “The Carathéodory–Fejér–Pisarenko decomposition and its multivariable counterpart,” _IEEE Trans. Autom. Control_ , vol. 52, no. 2, pp. 212–228, 2007.
  * [22] S. Boyd, N. Parikh, and E. Chu, _Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers_. Now Publishers Inc, 2011.