{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:56:36Z","timestamp":1760241396901,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2018,2,21]],"date-time":"2018-02-21T00:00:00Z","timestamp":1519171200000},"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":["61601071,61301124"],"award-info":[{"award-number":["61601071,61301124"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Scientific 300 and Technological Research Program of Chongqing Municipal Education Commission","award":["KJ1600412"],"award-info":[{"award-number":["KJ1600412"]}]},{"DOI":"10.13039\/501100005230","name":"Municipal Natural Science Foundation of Chongqing","doi-asserted-by":"publisher","award":["CSTC2016JCYJA2197"],"award-info":[{"award-number":["CSTC2016JCYJA2197"]}],"id":[{"id":"10.13039\/501100005230","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Dr. Startup Founds 302 of Chongqing University of Posts and Telecommunications","award":["A2016-12"],"award-info":[{"award-number":["A2016-12"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>Heterogeneous wireless networks are a promising technology in next generation wireless communication networks, which has been shown to efficiently reduce the blind area of mobile communication and improve network coverage compared with the traditional wireless communication networks. In this paper, a robust power allocation problem for a two-tier heterogeneous wireless networks is formulated based on orthogonal frequency-division multiplexing technology. Under the consideration of imperfect channel state information (CSI), the robust sum-rate maximization problem is built while avoiding sever cross-tier interference to macrocell user and maintaining the minimum rate requirement of each femtocell user. To be practical, both of channel estimation errors from the femtocells to the macrocell and link uncertainties of each femtocell user are simultaneously considered in terms of outage probabilities of users. The optimization problem is analyzed under no CSI feedback with some cumulative distribution function and partial CSI with Gaussian distribution of channel estimation error. The robust optimization problem is converted into the convex optimization problem which is solved by using Lagrange dual theory and subgradient algorithm. Simulation results demonstrate the effectiveness of the proposed algorithm by the impact of channel uncertainties on the system performance.<\/jats:p>","DOI":"10.3390\/s18020639","type":"journal-article","created":{"date-parts":[[2018,2,21]],"date-time":"2018-02-21T12:40:47Z","timestamp":1519216847000},"page":"639","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Robust Rate Maximization for Heterogeneous Wireless Networks under Channel Uncertainties"],"prefix":"10.3390","volume":"18","author":[{"given":"Yongjun","family":"Xu","sequence":"first","affiliation":[{"name":"School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"},{"name":"State Key Laboratory of Integrated Services Networks, Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yuan","family":"Hu","sequence":"additional","affiliation":[{"name":"School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guoquan","family":"Li","sequence":"additional","affiliation":[{"name":"School of Communication and Information Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,2,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"14616","DOI":"10.1109\/ACCESS.2017.2731423","article-title":"Robust MISO beamforming under the deterministic model in two-tier heterogeneous networks","volume":"5","author":"Wang","year":"2017","journal-title":"IEEE Access"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1109\/JSAC.2012.120401","article-title":"Femtocells: Past, Present, and Future","volume":"30","author":"Andrews","year":"2012","journal-title":"IEEE J. Sel. Area. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1109\/MWC.2011.5876497","article-title":"Enhanced intercell interference coordination challenges in heterogeneous networks","volume":"18","author":"Guvenc","year":"2011","journal-title":"IEEE Wirel. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"6509","DOI":"10.1109\/TWC.2015.2456055","article-title":"Distributed power control in a two-tier heterogeneous network","volume":"14","author":"Wang","year":"2015","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"9272","DOI":"10.1109\/TVT.2016.2517700","article-title":"Power-efficient resource allocation in a heterogeneous network with cellular and D2D capabilities","volume":"65","author":"Romero","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2366","DOI":"10.1109\/TCOMM.2014.2328574","article-title":"Resource allocation in spectrum-sharing OFDMA femtocells with heterogeneous services","volume":"62","author":"Zhang","year":"2014","journal-title":"IEEE Trans. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"4906","DOI":"10.1109\/TWC.2009.080457","article-title":"Interference mitigation using uplink power control for two-tier femtocell networks","volume":"8","author":"Jo","year":"2009","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"718","DOI":"10.1109\/TVT.2015.2402618","article-title":"Distributed energy-efficient power control for macrofemto networks","volume":"65","author":"Mao","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Gershman, A.B., and Sidiropoulos, N.D. (2005). Space-Time Processing for MIMO Communications, John Wiley and Sons.","DOI":"10.1002\/0470010045"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1007\/s10107-006-0092-2","article-title":"Selected Topics in Robust Convex Optimization","volume":"112","author":"Nemirovski","year":"2007","journal-title":"Math. Program."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1834","DOI":"10.1109\/COMST.2015.2425040","article-title":"Robust power control and beamforming in cognitive radio networks: A survey","volume":"17","author":"Xu","year":"2015","journal-title":"IEEE Commun. Surv. Tutor."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.comcom.2017.05.004","article-title":"Robust distributed resource allocation in OFDMA femtocell networks","volume":"109","author":"Vaezpour","year":"2017","journal-title":"Comput. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1109\/TCOMM.2014.2382095","article-title":"Downlink power control in two-tier cellular OFDMA networks under uncertainties: A robust stackelberg game","volume":"63","author":"Zhu","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1109\/TWC.2017.2723399","article-title":"Robust resource allocation in full-duplex-enabled OFDMA femtocell networks","volume":"16","author":"Xiao","year":"2017","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3167","DOI":"10.1109\/TWC.2014.042314.130981","article-title":"Robust power control for heterogeneous users in shared unlicensed bands","volume":"13","author":"Parsaeefard","year":"2014","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Xu, Y.J., Hu, Y., Chen, Q., Song, T., and Lai, R. (2017, January 21\u201325). Robust resource allocation for multi-tier cognitive heterogeneous networks. Proceedings of the IEEE International Conference on Communications (ICC), Paris, France.","DOI":"10.1109\/ICC.2017.7996534"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1109\/TMC.2012.28","article-title":"Robust distributed power control in cognitive radio networks","volume":"12","author":"Parsaeefard","year":"2013","journal-title":"IEEE Trans. Mob. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/j.comnet.2017.05.001","article-title":"Robust power allocation based on hierarchical game with consideration of different user requirements in two-tier femtocell networks","volume":"122","author":"Liu","year":"2017","journal-title":"Comput. Netw."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"145","DOI":"10.1016\/j.comnet.2016.02.013","article-title":"Robust power control for femtocell networks under outage-based QoS constraints","volume":"102","author":"Liu","year":"2016","journal-title":"Comput. Netw."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2090","DOI":"10.1109\/JSAC.2014.141112","article-title":"Quantized ergodic radio resource allocation in cognitive femto networks with controlled collision and power outage probabilities","volume":"32","author":"Mokari","year":"2014","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1109\/TWC.2013.022013.121111","article-title":"Robust MISO transmit optimization under outage-based QoS constraints in two-tier heterogeneous networks","volume":"12","author":"Wang","year":"2013","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_22","first-page":"1","article-title":"Downlink energy efficiency of power allocation and wireless backhaul bandwidth allocation in heterogeneous small cell networks","volume":"95","author":"Zhang","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"9287","DOI":"10.1109\/TVT.2017.2696974","article-title":"Distributed learning for energy efficient resource management in selforganizing heterogeneous networks","volume":"66","author":"Arani","year":"2017","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"7719","DOI":"10.1109\/TVT.2015.2500902","article-title":"Resource allocation for cognitive radio-enabled femtocell networks with imperfect spectrum sensing and channel uncertainty","volume":"65","author":"Zhang","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_25","unstructured":"Shenouda, M.B., and Davidson, T.N. (2008, January 2\u20135). Probabilistically-constrained approaches to the design of the multiple antenna downlink. Proceedings of the IEEE 42nd Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, USA."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1761","DOI":"10.1109\/TVT.2015.2405538","article-title":"Interference-limited resource optimization in cognitive femtocells with fairness and imperfect spectrum sensing","volume":"65","author":"Zhang","year":"2016","journal-title":"IEEE Trans. Veh. Technol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Boyd, S., and Vandenberghe, L. (2004). Convex Optimization, Cambridge University Press.","DOI":"10.1017\/CBO9780511804441"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Horst, R., Pardalos, P.M., and Thoai, N.V. (2000). Introduction to Global Optimization, Kluwer Academic Publishers. [2nd ed.].","DOI":"10.1007\/978-1-4615-0015-5"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"2726","DOI":"10.1109\/TWC.2005.858010","article-title":"Adaptive resource allocation in multiuser OFDM systems with proportional rate constraints","volume":"4","author":"Shen","year":"2005","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1310","DOI":"10.1109\/TCOMM.2006.877962","article-title":"Dual methods for nonconvex spectrum optimization of multicarrier systems","volume":"54","author":"Yu","year":"2006","journal-title":"IEEE Trans. Commun."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"915","DOI":"10.1109\/JPROC.2009.2015718","article-title":"Robust transmit power control for cognitive radio","volume":"97","author":"Setoodeh","year":"2009","journal-title":"Proc. IEEE"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Cacuci, D.G. (2003). Sensitivity and Uncertainty Analysis, Volume II: Applications to Large-scale Systems, Chapman and Hall\/CRC.","DOI":"10.1201\/9780203498798"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/2\/639\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:55:47Z","timestamp":1760194547000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/18\/2\/639"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,21]]},"references-count":32,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,2]]}},"alternative-id":["s18020639"],"URL":"https:\/\/doi.org\/10.3390\/s18020639","relation":{},"ISSN":["1424-8220"],"issn-type":[{"type":"electronic","value":"1424-8220"}],"subject":[],"published":{"date-parts":[[2018,2,21]]}}}