{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:01:54Z","timestamp":1760241714832,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2018,7,24]],"date-time":"2018-07-24T00:00:00Z","timestamp":1532390400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["No. 61301178 and No. 61271174"],"award-info":[{"award-number":["No. 61301178 and No. 61271174"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Consider a network consisting of two independent single-antenna sources, a single-antenna destination and a helping multiple-antenna relay. This network is called a dual-hop multiple access relay network (MARN). In this network, sources transmit to the relay simultaneously in the first time slot. The relay retransmits the received sum-signal to the destination using a linear beamforming scheme in the second time slot. In this paper, we characterize the achievable rate region of MARN under linear beamforming. The achievable rate region characterization problem is first transformed to an equivalent \u201ccorner point\u201d optimization problem with respect to linear beamforming matrix at the relay. Then, we present an efficient algorithm to solve it via only semi-definite programming (SDP). We further derive the mathematical close-forms of the maximum individual rates and the sum-rate. Finally, numerical results demonstrate the performance of the proposed schemes.<\/jats:p>","DOI":"10.3390\/e20080547","type":"journal-article","created":{"date-parts":[[2018,7,24]],"date-time":"2018-07-24T11:51:38Z","timestamp":1532433098000},"page":"547","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Achievable Rate Region under Linear Beamforming for Dual-Hop Multiple-Access Relay Network"],"prefix":"10.3390","volume":"20","author":[{"given":"Guiguo","family":"Feng","sequence":"first","affiliation":[{"name":"The State Key Laboratory of Integrated Services Network, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Wangmei","family":"Guo","sequence":"additional","affiliation":[{"name":"The State Key Laboratory of Integrated Services Network, Xidian University, Xi\u2019an 710071, China"}]},{"given":"Binyue","family":"Liu","sequence":"additional","affiliation":[{"name":"Bell Labs., Alcatel-Lucent, Shanghai 201206, China"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"3062","DOI":"10.1109\/TIT.2004.838089","article-title":"Cooperative diversity in wireless networks: Efficient protocols and outage behavior","volume":"50","author":"Laneman","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","first-page":"3114","article-title":"Improving amplify-and-forward relay networks: Optimal power allocation versus selection","volume":"6","author":"Zhao","year":"2007","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"765","DOI":"10.1109\/TIT.2004.842566","article-title":"On the capacity of large Gaussian relay networks","volume":"51","author":"Gastpar","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Steinwandt, J., Radhakrishnan, V., and Haardt, M. (2015, January 19\u201324). Distributed beamforming for cooperative networks with widely-linear processing at the relays and the receiver. Proceedings of the IEEE International Conference on Acoustics, Brisbane, QLD, Australia.","DOI":"10.1109\/ICASSP.2015.7178464"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Xu, Y., Shen, C., Ding, Z., Sun, X., Yan, S., and Zhu, G. (2017, January 21\u201325). Joint beamforming design and power splitting control in cooperative SWIPT NOMA systems. Proceedings of the IEEE International Conference on Communications (ICC), Paris, France.","DOI":"10.1109\/ICC.2017.7996560"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1398","DOI":"10.1109\/TWC.2007.348336","article-title":"Optimal design of non-regenerative mimo wireless relays","volume":"6","author":"Tang","year":"2007","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"2499","DOI":"10.1109\/TIT.2009.2018175","article-title":"Network beamforming using relays with perfect channel information","volume":"55","author":"Jing","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1880","DOI":"10.1109\/TIT.2010.2040875","article-title":"Bandwidth and power allocation for cooperative strategies in Gaussian relay networks","volume":"56","author":"Maric","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1109\/TSP.2014.2373332","article-title":"Matrix-Monotonic Optimization for MIMO Systems","volume":"63","author":"Xing","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"3930","DOI":"10.1109\/TSP.2015.2425800","article-title":"A Framework for Transceiver Designs for Multi-Hop Communications With Covariance Shaping Constraints","volume":"63","author":"Xing","year":"2015","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1519","DOI":"10.1109\/TSP.2015.2504344","article-title":"Transceiver Optimization for Multi-Hop Communications With Per-Antenna Power Constraints","volume":"64","author":"Xing","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1145\/1282427.1282425","article-title":"Embracing wireless interference: Analog network coding","volume":"37","author":"Katti","year":"2007","journal-title":"ACM SIGCOMM Comput. Commun. Rev."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1109\/TIT.2011.2173721","article-title":"Multihop analog network coding via amplify-and-forward: The high SNR regime","volume":"58","author":"Maric","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"3015","DOI":"10.1109\/TWC.2015.2399671","article-title":"Analog Network Coding in the Multiple Access Relay Channel: Error Rate Analysis and Optimal Power Allocation","volume":"14","author":"Ntontin","year":"2015","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3350","DOI":"10.1109\/TIT.2007.904984","article-title":"Duality and Rate Optimization for Multiple Access and Broadcast Channels with Amplify-and-Forward Relays","volume":"60","author":"Jafar","year":"2007","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/978-3-642-36899-8_3","article-title":"Optimal rate region of two-hop multiple access channel via amplify-and-forward scheme","volume":"7777","author":"Liu","year":"2013","journal-title":"Lect. Notes Compt. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"625","DOI":"10.4218\/etrij.14.0113.1135","article-title":"Design of Distributed Beamforming for Dual-Hop Multiple-Access Relay Networks","volume":"36","author":"Liu","year":"2014","journal-title":"ETRI J."},{"key":"ref_18","unstructured":"Cover, T., and Thomas, J. (2012). Elements of Information Theory, John Wiley & Sons."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"409","DOI":"10.1109\/TIT.1981.1056382","article-title":"Asynchronous multiple access channel capacity","volume":"27","author":"Cover","year":"1981","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1109\/MSP.2010.936019","article-title":"Semidefinite relaxation of quadratic optimization problems","volume":"27","author":"Luo","year":"2010","journal-title":"IEEE Signal Process. Mag."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Boyd, S.P., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1287\/moor.1070.0268","article-title":"Complex matrix decomposition and quadratic programming","volume":"32","author":"Huang","year":"2007","journal-title":"Math. Oper. Res."},{"key":"ref_23","unstructured":"Grant, M., and Boyd, S. (2018, July 24). CVX Users\u2019 Guide for CVX. Available online: http:\/\/cvxr.com\/cvx\/doc\/."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.A. (1985). Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511810817"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/547\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:14:05Z","timestamp":1760195645000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/8\/547"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,24]]},"references-count":24,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2018,8]]}},"alternative-id":["e20080547"],"URL":"https:\/\/doi.org\/10.3390\/e20080547","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,7,24]]}}}