{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,17]],"date-time":"2026-06-17T16:27:49Z","timestamp":1781713669605,"version":"3.54.5"},"reference-count":22,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T00:00:00Z","timestamp":1768694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2023YFA1008600"],"award-info":[{"award-number":["2023YFA1008600"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012166","name":"National Key R&amp;D Program of China","doi-asserted-by":"publisher","award":["2021YFA1000500"],"award-info":[{"award-number":["2021YFA1000500"]}],"id":[{"id":"10.13039\/501100012166","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Integrated Sensing and Communication (ISAC) has emerged as a cornerstone technology for next-generation wireless networks, where accurate performance evaluation is essential. In such systems, the capacity\u2013distortion function provides a fundamental measure of the trade-off between communication and sensing performance, making its computation a problem of significant interest. However, the associated optimization problem is often constrained by non-convexity, which poses considerable challenges for deriving effective solutions. In this paper, we propose extended Arimoto\u2013Blahut (AB) algorithms to solve the non-convex optimization problem associated with the capacity\u2013distortion trade-off in bistatic ISAC systems. Specifically, we introduce auxiliary variables to transform non-convex distortion constraints in the optimization problem into linear constraints, prove that the reformulated linearly constrained optimization problem maintains the same optimal solution as the original problem, and develop extended AB algorithms for both squared error distortion and logarithmic loss distortion. The numerical results validate the effectiveness of the proposed algorithms.<\/jats:p>","DOI":"10.3390\/e28010115","type":"journal-article","created":{"date-parts":[[2026,1,19]],"date-time":"2026-01-19T08:23:56Z","timestamp":1768811036000},"page":"115","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Extended Arimoto\u2013Blahut Algorithms for Bistatic Integrated Sensing and Communications Systems"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-0393-1042","authenticated-orcid":false,"given":"Tian","family":"Jiao","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4451-7242","authenticated-orcid":false,"given":"Yanlin","family":"Geng","sequence":"additional","affiliation":[{"name":"State Key Laboratory of ISN, Xidian University, Xi\u2019an 710071, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3400-5590","authenticated-orcid":false,"given":"Zhiqiang","family":"Wei","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zai","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Mathematics and Statistics, Xi\u2019an Jiaotong University, Xi\u2019an 710049, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1823","DOI":"10.1109\/JSAC.2022.3155509","article-title":"Waveform design and performance analysis for full-duplex integrated sensing and communication","volume":"40","author":"Xiao","year":"2022","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"5951","DOI":"10.1109\/TWC.2020.2998583","article-title":"On the effectiveness of OTFS for joint radar parameter estimation and communication","volume":"19","author":"Gaudio","year":"2020","journal-title":"IEEE Trans. Wireless Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1745","DOI":"10.1109\/TWC.2022.3206614","article-title":"Integrated Sensing and Communication With mmWave Massive MIMO: A Compressed Sampling Perspective","volume":"22","author":"Gao","year":"2023","journal-title":"IEEE Trans. Wireless Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1109\/MNET.128.2200446","article-title":"The rise of intelligent reflecting surfaces in integrated sensing and communications paradigms","volume":"37","author":"Elbir","year":"2022","journal-title":"IEEE Netw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4017","DOI":"10.1109\/TWC.2023.3313938","article-title":"Beamforming in integrated sensing and communication systems with reconfigurable intelligent surfaces","volume":"23","author":"Sankar","year":"2024","journal-title":"IEEE Trans. Wireless Commun."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1109\/TIT.2022.3176139","article-title":"An information-theoretic approach to joint sensing and communication","volume":"70","author":"Ahmadipour","year":"2022","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5723","DOI":"10.1109\/TIT.2023.3284449","article-title":"On the fundamental tradeoff of integrated sensing and communications under Gaussian channels","volume":"69","author":"Xiong","year":"2023","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Liu, F., Xiong, Y., Wan, K., Han, T.X., and Caire, G. (2023, January 25\u201330). Deterministic-random tradeoff of integrated sensing and communications in Gaussian channels: A rate-distortion perspective. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Taipei, Taiwan.","DOI":"10.1109\/ISIT54713.2023.10206462"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"10161","DOI":"10.1109\/TVT.2022.3179869","article-title":"Information-theoretic limits of integrated sensing and communication with correlated sensing and channel states for vehicular networks","volume":"71","author":"Liu","year":"2022","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Ahmadipour, M., Wigger, M., and Shamai, S. (2023, January 25\u201330). Strong converses for memoryless bi-static ISAC. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Taipei, Taiwan.","DOI":"10.1109\/ISIT54713.2023.10207009"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"9302","DOI":"10.1109\/TIT.2025.3621465","article-title":"Information-theoretic limits of bistatic integrated sensing and communication","volume":"71","author":"Jiao","year":"2025","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Chen, J., Yu, L., Li, Y., Shi, W., Ge, Y., and Tong, W. (2025). On the Fundamental Limits of Integrated Sensing and Communications Under Logarithmic Loss. arXiv.","DOI":"10.1109\/TCOMM.2025.3624157"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"14","DOI":"10.1109\/TIT.1972.1054753","article-title":"An algorithm for computing the capacity of arbitrary discrete memoryless channels","volume":"18","author":"Arimoto","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1109\/TIT.1972.1054855","article-title":"Computation of channel capacity and rate-distortion functions","volume":"18","author":"Blahut","year":"1972","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Yasui, K., and Matsushima, T. (2010, January 13\u201318). Toward computing the capacity region of degraded broadcast channel. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Austin, TX, USA.","DOI":"10.1109\/ISIT.2010.5513525"},{"key":"ref_16","unstructured":"Liu, Y., and Geng, Y. (July, January 26). Blahut-Arimoto Algorithms for Computing Capacity Bounds of Broadcast Channels. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Espoo, Finland."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Dou, Y., Liu, Y., Niu, X., Bai, B., Han, W., and Geng, Y. (2024). Blahut\u2013Arimoto Algorithms for Inner and Outer Bounds on Capacity Regions of Broadcast Channels. Entropy, 26.","DOI":"10.3390\/e26030178"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Joudeh, H., and Caire, G. (2024, January 19\u201321). Joint communication and state sensing under logarithmic loss. Proceedings of the IEEE International Symposium on Joint Communications & Sensing (JC&S), Leuven, Belgium.","DOI":"10.1109\/JCS61227.2024.10646257"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIT.1976.1055508","article-title":"The rate-distortion function for source coding with side information at the decoder","volume":"22","author":"Wyner","year":"2003","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1109\/JSAIT.2020.2991561","article-title":"The information bottleneck problem and its applications in machine learning","volume":"1","author":"Goldfeld","year":"2020","journal-title":"IEEE J. Sel. Areas Inf. Theory"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Huang, T.H., and El Gamal, A. (2021, January 12\u201320). A provably convergent information bottleneck solution via admm. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Melbourne, Australia.","DOI":"10.1109\/ISIT45174.2021.9518141"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Chen, L., Shitong, W., Jiachuan, Y., Huihui, W., Zhang, W., and Hao, W. (2024, January 9\u201313). Efficient and Provably Convergent Computation of Information Bottleneck: A Semi-Relaxed Approach. Proceedings of the IEEE International Conference on Communications (ICC), Denver, CO, USA.","DOI":"10.1109\/ICC51166.2024.10623121"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/115\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T05:10:41Z","timestamp":1769058641000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/1\/115"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,18]]},"references-count":22,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["e28010115"],"URL":"https:\/\/doi.org\/10.3390\/e28010115","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,18]]}}}