{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:18:54Z","timestamp":1760242734105,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2016,5,3]],"date-time":"2016-05-03T00:00:00Z","timestamp":1462233600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>It is highly possible that future wireless communication systems will adopt ultra-dense deployment to cope with the increasing demand on spectrum efficiency and energy efficiency. The pivotal issue to achieve the potential benefits of the ultra-dense network is to deal with the complex inter-site interference. In this paper, in order to maximize the spectrum efficiency of the system, we first make a reasonable approximation on the inter-site interference to convert the problem into a convex optimization problem. Then, the Lagrangian Multiplier method is adopted to obtain the expression of the optimum power allocation, and the water filling algorithm, as one of the most classical algorithms in the information theory, can be applied to maximize the sum rate or spectrum efficiency of the system. Since the classical iteratively searching water filling algorithm needs many iterations to converge to the optimal solution, we develop a low-complexity iterative approximate water filling algorithm. Simulation results show that the developed algorithm can achieve very close performance to the classical iteratively searching water filling based power allocation with only a few iterations under different scenarios, which leads to a significant complexity reduction.<\/jats:p>","DOI":"10.3390\/e18050158","type":"journal-article","created":{"date-parts":[[2016,5,3]],"date-time":"2016-05-03T10:12:55Z","timestamp":1462270375000},"page":"158","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Low-Complexity Iterative Approximated Water-Filling Based Power Allocation in an Ultra-Dense Network"],"prefix":"10.3390","volume":"18","author":[{"given":"Xin","family":"Su","sequence":"first","affiliation":[{"name":"Tsinghua National Laboratory for Information Science and Technology, Research Institute of Information Technology, Tsinghua University, Beijing 100084, China"}]},{"given":"Bei","family":"Liu","sequence":"additional","affiliation":[{"name":"Broadband Wireless Access Laboratory, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"}]},{"given":"Xiaopeng","family":"Zhu","sequence":"additional","affiliation":[{"name":"Department of Electronic Engineering, Xiamen University, Xiamen 361005, China"}]},{"given":"Jie","family":"Zeng","sequence":"additional","affiliation":[{"name":"Tsinghua National Laboratory for Information Science and Technology, Research Institute of Information Technology, Tsinghua University, Beijing 100084, China"}]},{"given":"Chiyang","family":"Xiao","sequence":"additional","affiliation":[{"name":"Tsinghua National Laboratory for Information Science and Technology, Research Institute of Information Technology, Tsinghua University, Beijing 100084, China"}]}],"member":"1968","published-online":{"date-parts":[[2016,5,3]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/MCOM.2014.6736747","article-title":"Network Densification: The Dominant Theme for Wireless Evolution into 5G","volume":"52","author":"Bhushan","year":"2014","journal-title":"IEEE Commun. Mag."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Gelabert, X., Legg, P., and Qvarfordt, C. (2013, January 9\u201313). Small Cell Densification Requirements in High Capacity Future Cellular Netwoks. Proceedings of the 2013 IEEE International Conference on Communications Workshops (ICC), Budapest, Hungary.","DOI":"10.1109\/ICCW.2013.6649403"},{"key":"ref_3","unstructured":"Sui, Y., Guvenct, I., and Svensson, T. (2014, January 26\u201328). On the Deployment of Moving Networks In Ultra-dense Urban Scenarios. Proceedings of the IEEE International Conference on 5G for Ubiquitous Connectivity (5GU), Akaslompolo, Finland."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2078","DOI":"10.1109\/COMST.2015.2439636","article-title":"Towards 1 Gbps\/UE in Cellular Systems: Understanding Ultra-Dense Small Cell Deployments","volume":"17","author":"Ding","year":"2015","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_5","unstructured":"Chuang, M.-C., Chen, M.C., and Sun, Y.S. (2015, January 12\u201314). Resource Management Issues in 5G Ultra Dense Small cell Networks. Proceedings of the IEEE International Conference on Information Networking (ICOIN), Siem Reap, Cambodia."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Li, Y.-N.R., Li, J., Wu, H., and Zhang, W. (2014, January 8\u201312). Energy Efficient Small Cell Operation under Ultra Dense Cloud Radio Access Networks. Proceedings of the IEEE Globecom Workshops, Austin, TX, USA.","DOI":"10.1109\/GLOCOMW.2014.7063583"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Ren, Q., Fan, J., Luo, X., Xu, Z., and Chen, Y. (2015, January 8\u201312). Analysis of Spectral and Energy Efficiency in Ultra-Dense Network. Proceedings of the Communication Workshop (ICCW), London, UK.","DOI":"10.1109\/ICCW.2015.7247605"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Qian, L., Wu, G., and Hu, Q. (2013, January 8\u20139). Analytical Study on Network Spectrum Efficiency of Ultra Dense Networks. Proceedings of the IEEE International Sysmposium Personal Indoor and Mobile Radio Communications, London, UK.","DOI":"10.1109\/PIMRC.2013.6666617"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Olsson, M., Cavdar, C., Frenger, P., Tombaz, S., Sabella, D., and Jantti, R. (2013, January 7\u20139). 5GrEEn: Towards Green 5G mobile networks. Proceedings of the 2013 IEEE 9th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob), Lyon, France.","DOI":"10.1109\/WiMOB.2013.6673363"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Xiao, X., Tao, X., Jia, Y., and Lu, J. (2011, January 28\u201331). An Energy-Efficient Hybrid Structure with Resource Allocation in OFDMA Networks. Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC), Cancun, Mexico.","DOI":"10.1109\/WCNC.2011.5779346"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1849","DOI":"10.1109\/ACCESS.2015.2478863","article-title":"Energy-Efficient Context-Aware Matching for Resource Allocation in Ultra-Dense Small Cells","volume":"3","author":"Zhou","year":"2015","journal-title":"IEEE Access"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Koudourdis, G.P. (2014, January 23\u201326). On the Capacity and Energy Efficiency of Network Scheduling in Future Ultra-Dense Networks. Proceedings of the IEEE Symposium Computers and Communicatin (ISCC), Madeira, Portugal.","DOI":"10.1109\/ISCC.2014.6912645"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Dudnikova, A., Dini, P., Giupponi, L., and Panno, D. (2015, January 13\u201315). Multi-Criteria Decision for Small Cell Switch off in Ultra Dense LTE Networks. Proceedings of the 2015 13th International Conference on Telecommunications (ConTEL), Graz, Austria.","DOI":"10.1109\/ConTEL.2015.7231191"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1002\/bltj.20443","article-title":"Dynamic Idle Mode Procedures for Femtocells","volume":"15","author":"Claussen","year":"2010","journal-title":"Bell Labs Tech. J."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Cheng, L., Gao, Y., Li, Y., Yang, D., and Liu, X. (2015, January 11\u201314). A Cooperative Resource Allocation Scheme Based on Self-Organized Network in Ultra-Dense Small Cell Deployment. Proceedings of the 2015 IEEE 81st Vehicular Technology Conference (VTC Spring), Glasgow, UK.","DOI":"10.1109\/VTCSpring.2015.7145594"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Liu, L., Garcia, V., Tian, L., Pan, Z., and Shi, J. (2015, January 8\u201312). Jonit Clustering and Inter-cell Resource Allocation for CoMP in Ultra Dense Cellular Networks. Proceedings of the 2015 IEEE International Conference on Communications (ICC), London, UK.","DOI":"10.1109\/ICC.2015.7248710"},{"key":"ref_17","unstructured":"Jang, J., Lee, K.B., and Lee, Y.H. (2003, January 1\u20135). Transmit Power and Bit Allocations for OFDM Systems in a Fading Channel. Proceedings of the IEEE Global Communications Conference, San Francisco, CA, USA."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Peng, T., Wang, W., Lu, Q., and Wang, W. (2007, January 21\u201325). Subcarrier Allocation Based on Water-filling Level in OFDMA-based Cognitive Radio Networks. Proceedings of the IEEE Wireless Communications, Networking and Mobile Computing, Shanghai, China.","DOI":"10.1109\/WICOM.2007.56"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Feng, Z., Wei, Z., Shuai, T., Zhang, Q., and Li, R. (2012, January 24\u201328). An Iterative Water-fillin Based Resource Allocation Scheme in OFDMA Systems for Energy Efficiency Optimization. Proceedings of the IEEE VTC Fall, Quebec City, QC, Canada.","DOI":"10.1109\/VTCFall.2012.6399020"},{"key":"ref_20","first-page":"21","article-title":"Enhanced Power Allocation Scheme in Ultra-Dense Small Cell Network","volume":"13","author":"Gao","year":"2016","journal-title":"China Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1126","DOI":"10.1109\/JSAC.2014.2328392","article-title":"Virtual Cell Beamforming in Cooperative Networks","volume":"32","author":"Kim","year":"2014","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell System Tech. J."},{"key":"ref_23","unstructured":"Goldsmith, A. (2005). Wireless Communication, Cambridge University Press."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1109\/MCOM.2015.7010521","article-title":"Spectral and Energy Efficiency of Ultra-Dense Networks under Different Deployment Strategies","volume":"53","author":"Yunas","year":"2015","journal-title":"IEEE. Mag."},{"key":"ref_25","unstructured":"3GPP TR 36.814 Further Advancements for E-UTRA Physical Layer Aspects (Release 9). Available online: http:\/\/wenku.baidu.com\/view\/17b6456e7e21af45b307a8fd.html."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3481","DOI":"10.1109\/TWC.2015.2407355","article-title":"Resource Allocation for Cognitive Small Cell Networks: A Cooperative Bargaining Game Theoretic Approach","volume":"14","author":"Zhang","year":"2015","journal-title":"IEEE Trans. Wirel. Commun."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/5\/158\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:23:12Z","timestamp":1760210592000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/5\/158"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,3]]},"references-count":26,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2016,5]]}},"alternative-id":["e18050158"],"URL":"https:\/\/doi.org\/10.3390\/e18050158","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,5,3]]}}}