{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:32:12Z","timestamp":1760239932676,"version":"build-2065373602"},"reference-count":16,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,1,24]],"date-time":"2019-01-24T00:00:00Z","timestamp":1548288000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002347","name":"Bundesministerium f\u00fcr Bildung und Forschung","doi-asserted-by":"publisher","award":["Alexander von Humboldt Professorship"],"award-info":[{"award-number":["Alexander von Humboldt Professorship"]}],"id":[{"id":"10.13039\/501100002347","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Standard upper and lower bounds on the capacity of relay channels are cut-set (CS), decode-forward (DF), and quantize-forward (QF) rates. For real additive white Gaussian noise (AWGN) multicast relay channels with one source node and one relay node, these bounds are shown to be quasi-concave in the receiver signal-to-noise ratios and the squared source-relay correlation coefficient. Furthermore, the CS rates are shown to be quasi-concave in the relay position for a fixed correlation coefficient, and the DF rates are shown to be quasi-concave in the relay position. The latter property characterizes the optimal relay position when using DF. The results extend to complex AWGN channels with random phase variations.<\/jats:p>","DOI":"10.3390\/e21020109","type":"journal-article","created":{"date-parts":[[2019,1,24]],"date-time":"2019-01-24T11:12:48Z","timestamp":1548328368000},"page":"109","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Quasi-Concavity for Gaussian Multicast Relay Channels"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8314-500X","authenticated-orcid":false,"given":"Mohit","family":"Thakur","sequence":"first","affiliation":[{"name":"Independent Researcher, Amalienstr. 49A, 80799 Munich, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3904-9181","authenticated-orcid":false,"given":"Gerhard","family":"Kramer","sequence":"additional","affiliation":[{"name":"Institute for Communications Engineering, Technical University of Munich, 80333 Munich, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,1,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3037","DOI":"10.1109\/TIT.2005.853304","article-title":"Cooperative strategies and capacity theorems for relay networks","volume":"51","author":"Kramer","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1109\/TMC.2009.114","article-title":"Optimal relay station placement in broadband wireless access networks","volume":"9","author":"Lin","year":"2010","journal-title":"IEEE Trans. Mob. Comput."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"2518","DOI":"10.1109\/TIT.2009.2018337","article-title":"Calderbank. On maximizing coverage in Gaussian relay channels","volume":"55","author":"Aggarwal","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Joshi, G., and Karandikar, A. (2011, January 28\u201330). Optimal relay placement for cellular coverage extension. 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Nonlinear Programming, Wiley.","DOI":"10.1002\/0471787779"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/2\/109\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:28:25Z","timestamp":1760185705000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/2\/109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1,24]]},"references-count":16,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,2]]}},"alternative-id":["e21020109"],"URL":"https:\/\/doi.org\/10.3390\/e21020109","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,1,24]]}}}