ĐẠI HỌC QUỐC GIA THÀNH PHỐ HỒ CHÍ MINH TRƯỜNG ĐẠI HỌC BÁCH KHOA KHOA ĐIỆN - ĐIỆN TỬ BỘ MƠN VIỄN THƠNG LUẬN VĂN THẠC SĨ TỐI ƯU HĨA HIỆU NĂNG HỆ THỐNG THÔNG TIN VÔ TUYẾN ĐA NGƯỜI DÙNG MIMO VÀ MASSIVE MIMO HỌC VIÊN THỰC HIỆN: PHẠM QUỐC VIỆT - 1870092 HƯỚNG DẪN KHOA HỌC: PGS TS HÀ HỒNG KHA TP.HỒ CHÍ MINH, 01/2020 LỜI CẢM ƠN Đầu tiên, em xin gửi lời cảm ơn sâu sắc đến Thầy PGS TS Hà Hoàng Kha Thầy người hướng dẫn cho em hoàn thành luận văn Thầy ln tận tình quan tâm, hướng dẫn, truyền cảm hứng cho em suốt trình làm nghiên cứu, khơng Việt Nam mà cịn khoảng thời gian em học tập làm việc Na Uy Bên cạnh truyền đạt kiến thức chuyên môn, Thầy truyền đạt kinh nghiệm nghiên cứu, hỗ trợ tài chính, mang đến hội tầm nhìn để em phát triển thân tương lai Khơng có giúp đỡ to lớn từ Thầy, em khơng có thành cơng nho nhỏ ngày hôm Tiếp đến, em xin chân thành cảm ơn Thầy GS TS Lê Tiến Thường Thầy người hướng dẫn, truyền đạt kiến thức, rèn luyện, giúp đỡ em khoảng thời gian làm luận văn đại học học tập chương trình Việt Pháp Sự rèn luyện từ Thầy hành trang cho em đường nghiên cứu Bên cạnh đó, em xin gửi lời cảm ơn đến thầy cô trường Đại học Bách Khoa, Khoa Điện - Điện tử, Bộ môn Viễn Thông truyền đạt kiến thức cho em suốt khoảng thời gian học tập làm việc Trường Đại học Bách Khoa Hơn nữa, em xin chân thành cảm ơn gia đình, ba, mẹ, anh hai, động viên hỗ trợ vật chất lẫn tinh thần để em hoàn thành luận văn Cuối cùng, cảm ơn bạn nhóm nghiên cứu làm việc, vui chơi, giúp đỡ em khoảng thời gian làm nghiên cứu trường Đại học Bách Khoa Tp Hồ Chí Minh Tp Hồ Chí Minh, ngày 10 tháng 01 năm 2020 Phạm Quốc Việt TÓM TẮT Trong năm gần đây, phát triển bùng nổ phần cứng, số lượng, ứng dụng thiết bị viễn thơng điện thoại, máy tính, cảm biến không dây, thúc đẩy trao đổi liệu ngày lớn thiết bị Số liệu thống kê từ công ty thiết bị viễn thông Cisco hay Ericsson cho thấy lưu lượng thông tin tiếp tục gia tăng nhằm phục vụ cho nhu cầu thông minh tự động đạt gấp nhiều lần năm Bên cạnh việc gia tăng lưu lượng thông tin trao đổi thiết bị, việc truyền thông xanh, nhằm sử dụng lượng cách hiệu quả, ngành viễn thông dần nhận quan tâm từ chuyên viên nhà nghiên cứu Do đó, hệ thống thơng tin vơ tuyến hệ thứ (5G) nghiên cứu để cung cấp giải pháp nhằm đáp ứng nhu cầu dịch vụ cải thiện hiệu suất sử dụng lượng phổ Từ quan sát trên, luận văn nghiên cứu hệ thống MIMO Hetnet, MIMO song cơng, massive MIMO, nhằm tìm giải pháp thiết kế (bộ tiền mã hóa phân bổ công suất phát người dùng) để tối ưu hóa hiệu năng, hiệu suất phổ hiệu suất lượng, mạng Đầu tiên, luận văn thiết kế tiền mã hóa để tối ưu hóa hiệu suất lượng mạng MIMO khơng đồng (MIMO HetNet) với ràng buộc công suất phát antenna mức can nhiễu người dùng Giải thuật Dinkelbach Block Coordinate Ascend (BCA) sử dụng để thiết kế tiền mã hóa Tiếp theo, luận văn thiết kế tiền mã hóa để tối ưu hóa tổng tốc độ hệ MIMO hoạt động song công (FD MIMO) phục vụ nhiều người dùng Phương pháp Gradient Projection (GP) sử dụng để tìm thiết kế tối ưu cho tốn cực đại hóa hàm tổng tốc độ Cuối cùng, luận văn nghiên cứu vấn đề phân bổ công suất phát cho nhiều người dùng mạng massive MIMO nhằm tối ưu hóa cơng theo tỷ lệ hiệu suất phổ người dùng Một giải thuật lặp dựa phương pháp trị riêng tổng quát phương pháp GP sử dụng để tiếp cận tốn tối ưu Kết mơ số sử dụng để đánh giá tính hiệu thiết kế phát triển toán tối ưu hệ MIMO HetNet, FD MIMO, massive MIMO Các kết nghiên cứu luận văn đăng hội nghị quốc tế IEEE nộp tạp chí quốc tế ISI ABSTRACT There have been a growing number of communication devices, e.g mobile phones, computers, and wireless sensors, and its applications that require the powerful computational capability, which motivates exchanging data massively among the devices Statistics from telecommunications companies like Cisco or Ericsson reveal that the amount of exchanged information will continue to increase to serve intelligent and automated services and can be multiplied over the years In addition to the increase in information exchange among the devices, green communication which aims at using energy efficiently in the telecommunications industry is gradually receiving the attention from both researchers and technicians Therefore, the fifth generation communication system (5G) has been researching to improve the spectral and energy efficiency of mobile networks to meet the quality of services From the observation above, this thesis will study the MIMO Hetnet, Full-duplex MIMO, and Massive MIMO systems, to design systems, i.e precoding and power allocating, maximizing performance metrics of these networks, such as spectral and energy efficiency First, this thesis designs precoders to optimize the energy efficiency of a MIMO Heterogeneous Network (MIMO HetNet) with a power constraint on each transmit antenna and an acceptable amount of interference at the users The Dinkelbach and Block Coordinate Ascend (BCA) methods are utilized to design precoders Next, the thesis will design precoders to maximize the sum of users’ rates of a full-duplex MIMO (FD MIMO) system The Gradient Projection (GP) algorithm is deployed to resolve this optimization problem Finally, the thesis will study the problem of allocating transmit power to many users in a Massive MIMO network to optimize the proportional fairness of users’ spectral efficiency An iterative algorithm based on the generalized eigenvalue and the GP approaches is developed to solve the fairness problem Numerical results are utilized to evaluate the effectiveness of the proposed designs in the MIMO HetNet, FD MIMO, and massive MIMO networks The research results in this thesis were published in an international IEEE conference and submitted to an international ISI journal LỜI CAM ĐOAN Em tên Phạm Quốc Việt, mã số học viên 1870092, học viên cao học khóa 2018 đợt 1, chuyên ngành Kỹ thuật Viễn thông, trường Đại học Bách Khoa Tp Hồ Chí Minh Trong khn khổ luận văn này, em xin cam đoan điều sau hoàn toàn trung thực: - Quyển luận văn học viên thực viết - Các số liệu mô với giải thuật luận văn - Các tài liệu tham khảo lấy từ nguồn có độ tin cậy cao trích dẫn đầy đủ Tp Hồ Chí Minh, ngày 10 tháng 01 năm 2020 Phạm Quốc Việt Mục lục Mục lục i Danh sách hình vẽ iv Danh sách bảng vi Từ viết tắt ix Danh sách lưu đồ giải thuật x MỞ ĐẦU 1.1 Đặt vấn đề nghiên cứu 1.1.1 Xu hướng tăng trưởng lưu lượng liệu di động 1.1.2 Mức tiêu thụ lượng ngành viễn thông 1.1.3 Mạng truyền thông hệ thứ (5G) 1.2 Lý chọn đề tài 1.3 Mục tiêu nhiệm vụ luận văn 1.4 Đối tượng phạm vi nghiên cứu 1.4.1 Đối tượng nghiên cứu 1.4.2 Phạm vi nghiên cứu 1.5 Phương pháp nghiên cứu 1.6 Bố cục luận văn 1.7 Các báo hoàn thành luận văn TỔNG QUAN CÁC VẤN ĐỀ NGHIÊN CỨU VÀ LÝ THUYẾT LIÊN QUAN 2.1 Giới thiệu kỹ thuật đa ngõ vào đa ngõ MIMO 2.1.1 MIMO điểm-đến-điểm 2.1.2 MIMO đa người dùng (MU-MIMO) 2.1.3 Massive MIMO 2.2 Mạng hỗn hợp HetNet i 1 9 9 10 11 11 13 13 13 14 16 17 MỤC LỤC 2.3 2.4 2.5 2.6 2.7 Vấn đề truyền thông xanh Lý thuyết toán tối ưu 2.4.1 Bài toán tối ưu 2.4.2 Tối ưu lồi 2.4.2.1 Tập lồi 2.4.2.2 Hàm lồi 2.4.3 Bài toán tối ưu hàm mục tiêu dạng phân số 2.4.3.1 Lý thuyết hàm tựa-lõm 2.4.3.2 Giải thuật Dinkelbach cho tối ưu phân số lõm-lồi Phương pháp Coordinate Descent 2.5.1 Coordinate Descent 2.5.2 Block Coordinate Descent Phương pháp Gradient Projection 2.6.1 Ý tưởng 2.6.2 Phương pháp Gradient Descent 2.6.3 Lựa chọn kích thước bước - Quy tắc Armijo 2.6.4 Phương pháp Gradient Projection Kết luận chương 21 22 22 23 23 23 24 24 25 26 26 29 30 30 32 33 34 35 TỐI ƯU HĨA HIỆU QUẢ NĂNG LƯỢNG CỦA MẠNG KHƠNG ĐỒNG NHẤT VỚI RÀNG BUỘC CÔNG SUẤT TRÊN MỖI ANTENNA 3.1 Giới thiệu 3.2 Mơ hình hệ thống kênh đường xuống MIMO HetNet 3.3 Giải thuật lặp dưạ phương pháp BCA Dinkelbatch 3.4 Kết mô 3.5 Kết luận chương 36 37 39 41 44 46 TỐI ƯU HÓA TỔNG TỐC ĐỘ CỦA HỆ MIMO SONG CÔNG ĐA NGƯỜI DÙNG BẰNG PHƯƠNG PHÁP GRADIENT PROJECTION 4.1 Giới thiệu 4.2 Mô hình hệ thống tốn thiết kế 4.2.1 Đường lên 4.2.2 Đường xuống 4.3 Giải thuật lặp tối ưu hóa tổng tốc độ 4.3.1 Tính gradients 4.3.2 Tìm hình chiếu 4.3.2.1 Chiếu ma trận hiệp phương sai đường lên 48 49 51 51 52 54 54 56 56 ii [78] H Q Ngo, L Tran, T Q Duong, M Matthaiou, and E G Larsson, “On the total energy efficiency of cell-free massive mimo,” IEEE Transactions on Green Communications and Networking, vol 2, no 1, pp 25–39, March 2018 64, 69 [79] M Bashar, K Cumanan, A G Burr, M Debbah, and H Q Ngo, “On the uplink max-min sinr of cell-free massive mimo systems,” IEEE Trans Wireless Commun., vol 18, no 4, pp 2021–2036, April 2019 64, 66, 68, 69, 70, 75, 84, 86, 87, 88 [80] T H Nguyen, T K Nguyen, H D Han, and V D Nguyen, “Optimal power control and load balancing for uplink cell-free multi-user massive mimo,” IEEE Access, vol 6, pp 14 462–14 473, 2018 64, 65 [81] T T Vu, D T Ngo, H Q Ngo, and T Le-Ngoc, “Full-duplex cell-free massive mimo,” in ICC 2019 - 2019 IEEE International Conference on Communications (ICC), May 2019, pp 1–6 64 [82] L D Nguyen, T Q Duong, H Q Ngo, and K Tourki, “Energy efficiency in cell-free massive mimo with zero-forcing precoding design,” IEEE Commun Lett., vol 21, no 8, pp 1871–1874, Aug 2017 65 [83] G Interdonato, M Karlsson, E Bjăornson, and E G Larsson, “Downlink spectral efficiency of cell-free massive MIMO with full-pilot zero-forcing,” CoRR, vol abs/1810.00237, 2018 [Online] Available: http://arxiv.org/abs/1810.00237 65 [84] E Bjăornson, E G Larsson, and M Debbah, Massive mimo for maximal spectral efficiency: How many users and pilots should be allocated?” IEEE Transactions on Wireless Communications, vol 15, no 2, pp 1293–1308, Feb 2016 65 [85] F Kelly, “Charging and rate control for elastic traffic,” Eur Trans Telecommun., vol 8, no 1, pp 33–37, 1997 [Online] Available: https://onlinelibrary.wiley.com/ doi/abs/10.1002/ett.4460080106 65 [86] V K N Lau, “Proportional fair space-time scheduling for wireless communications,” IEEE Trans Commun., vol 53, no 8, pp 1353–1360, Aug 2005 65, 66, 75 [87] Z Hadzi-Velkov, S Pejoski, N Zlatanov, and R Schober, “Proportional fairness in aloha networks with rf energy harvesting,” IEEE Wireless Commun Lett., vol 8, no 1, pp 277–280, Feb 2019 65, 66 [88] V Pham, T Ha, H Kha, and V Son, “Sum rate maximization for full duplex multiuser mimo systems by gradient projection approach,” 10 2019 66, 77 99 [89] H Ahmadi, A Farhang, N Marchetti, and A MacKenzie, “A game theoretic approach for pilot contamination avoidance in massive mimo,” IEEE Wireless Commun Lett., vol 5, no 1, pp 12–15, Feb 2016 68 [90] E Nayebi, A Ashikhmin, T L Marzetta, H Yang, and B D Rao, “Precoding and power optimization in cell-free massive mimo systems,” IEEE Trans Wireless Commun., vol 16, no 7, pp 4445–4459, July 2017 69 [91] A Wiesel, Y C Eldar, and S Shamai, “Linear precoding via conic optimization for fixed mimo receivers,” IEEE Trans Signal Process., vol 54, no 1, pp 161–176, Jan 2006 75 [92] M Schubert and H Boche, “Solution of the multiuser downlink beamforming problem with individual sinr constraints,” IEEE Trans Veh Technol., vol 53, no 1, pp 18–28, Jan 2004 75 [93] ——, “Iterative multiuser uplink and downlink beamforming under sinr constraints,” IEEE Trans Signal Process., vol 53, no 7, pp 2324–2334, July 2005 75 [94] G H Golub and C F van Loan, Matrix Computations, 4th ed JHU Press, 2013 [Online] Available: http://www.cs.cornell.edu/cv/GVL4/golubandvanloan.htm 76 [95] B Ghojogh, F Karray, and M Crowley, “Eigenvalue and Generalized Eigenvalue Problems: Tutorial,” arXiv e-prints, p arXiv:1903.11240, Mar 2019 76 [96] Ao Tang, JiXian Sun, and Ke Gong, “Mobile propagation loss with a low base station antenna for nlos street microcells in urban area,” in IEEE VTS 53rd Vehicular Technology Conference, Spring 2001 Proceedings (Cat No.01CH37202), vol 1, May 2001, pp 333–336 vol.1 82 [97] N Zhao, Q Cao, G Gui, Y Cao, S Zhang, Y Chen, and H Sari, “Secure transmission for interference networks: User selection and transceiver design,” IEEE Syst J., vol 13, no 3, pp 2839–2850, Sep 2019 86 100 Energy Efficiency Maximization with Per-Antenna Power Constraints for Heterogeneous Networks Viet Quoc Pham1 , Tien Ngoc Ha1 , Ha Hoang Kha1 and Pham Van Quyet2 Ho Chi Minh City University of Technology, VNU-HCM, Vietnam Telecommunications University, Khanh Hoa, Vietnam Email: vietpq09@gmail.com, hntien.hcmut@gmail.com, hhkha@hcmut.edu.vn, phamquyet0607@gmail.com Abstract—This paper considers energy efficiency (EE) optimization in downlink multiple-input multiple-output (MIMO) heterogeneous networks (HetNets) in which one macro base station (MBS) and multiple small cell base stations (SBSs) communicate with multiple MIMO users The primary objective is to design the precoders to maximize the network EE under the constraints on per-antenna and per-base-station transmit power and acceptable interference levels at each user in the macro cell The design problem results in nonconvex fractional program whose optimal solutions cannot be found directly In this paper, the relationship between the achievable data rates and the minimum mean square error (MMSE) is adopted to reformulate the design problem into amenable one Then, an iterative algorithm based the Dinklebach and the block coordinate ascent (BCA) methods is employed for finding optimal precoders We conduct the numerical simulations to examine the achievable EE performance of the considered HetNets Index Terms—HetNets, energy efficiency, MIMO multiuser, per-antenna power constraint I I NTRODUCTION In wireless networks, there have been growing concerns towards improving spectral efficiency (SE) as the number of wireless devices and applications requiring high-speed data transmission has been exponentially increasing It is predicted that there will be about 12.3 billion mobile devices and global data traffic will reach 77.5 exabytes per month in 2022 [1] The authors in [2] demonstrated that deploying small cells with high frequency reuse is an effective approach to serve a large number of wireless devices and to increase SE Small cells with low transmit power can provide transmission links with high quality of service However, the great number of base stations (BSs) operating in the same frequency would result in significant interference, which may degrade the system performance if interference is not appropriately managed The research on how to efficiently handle interference and optimize SE in heterogeneous networks (HetNets) has been extensively investigated in the literature [3], [4] In [3], power control and beamforming methods were studied to minimize interference and maximize the SE in femto cells The authors of [4] introduced the optimal design for linear transceivers which maximize SE of multiple-input multipleoutput (MIMO) broadcasting channels In recent years, energy consumption issues, especially the global greenhouse gas emissions from the information and communication technology (ICT), have become an active research topic It was shown in [5] that the ICT constitutes approximately three percent of electricity consumed worldwide In addition, extensive studies have shown that designs of wireless systems aimed at maximizing SE could result in exploiting all transmit power budgets, which may cause power consumption inefficiency Therefore, the energy efficiency (EE) performance metric in wireless communication system designs has attracted considerable attentions in both academic and industrial communities [6], [7] Designing the wireless communication systems with high EE will not only lower operating costs but also reduce power consumption and, hence, prolong the operating time of energy-constrained devices [8] EE is defined as the ratio of the number of transmitted information bits to a unit of consumed energy [5], [9], [10] The wireless system designs enhancing EE under different scenarios have been investigated [11]–[13] In [11], the tradeoff between EE and SE was considered in orthogonal frequency division multiplexing (OFDM) systems The EE maximization for multiuser interference channels was studied in [13] in which the interference alignment (IA) technique and the Dinkelbach approach were used to alleviate interference and find the optimal precoders The EE of multiple-input single-output (MISO) wireless communications systems was also investigated in [14] In addition, the EE precoder designs for multicell with full-duplex communications by the pathfollowing algorithm with convex quadratic programming were introduced in [15] In [16], the max-min EE precoding designs in multicell MIMO systems were investigated through a sequential convex approximation In [17], the EE optimization for upplink HetNets was studied In [18], trade-offs between SE and harvested energy in multi-cell MIMO wireless networks were studied [18] To maximize the sum-rate and the harvested energy, reference [18] used the relationship between the MMSE and achievable data rate to develop an iterative algorithm The various constraints such as per-antenna power, sum transmit power, quality of service constraints for the design problems have been considered in the literature [19]– [23] In this paper, we consider the precoding design to maximize EE in downlink channels of multicell multiuser MIMO HetNets in which one macro base station (MBS) and several small cell base stations (SBSs) simultaneously serve multiple user equipments (UEs) at the same time and in the same frequency band [16], [21]–[24] In addition to considering 978-1-7281-5353-7/19/ $31.00 ©2019 IEEE 135 power constraints at the BSs as in the previous studies [16], [17], in this work, constraints on per-antenna and perbase-station power at the transmitters and interference at the macro-users are taken into account The relationship between achievable rates of UEs and the minimum mean square error (MMSE) is utilized to deal with the mathematical challenges of the optimization problem The resultant problem is convexconcave fractional programming with the constraints of transmit power and acceptable interference levels An iterative algorithm which is based on the block coordinate ascent (BCA) and the Dinkelbach methods is then invoked to find the optimal precoders The network EE (NEE) performance of the considered HetNets will be investigated through numerical results of simulations II S YSTEM M ODEL AND P ROBLEM F ORMULATION ×1 received signal yk ∈ CNk K yk = Hk V k sk + , Hk , V u su u=1,u=k multi−user interf erence Ki L Hk + ,i V ji s ji + n k , ∈ L, k ∈ K i=0,i= j=1 inter−cell interf erence (1) 0, σk2l where nk ∼ CN I is additive white Gaussian noise at UEk Without loss of generality, we assume that E sk sH = k Idkl As can be seen from Eq (1) that the received signal at UEk is degraded by multi-user interference in the same cell and inter-cell interference from the other BSs From Eq (1), the achievable bit rate of UEk can be given by Rk (V) = log2 INk + Hk We study downlink transmission in a MIMO wireless HetNet as depicted in Fig in which L small cells are deployed in the same coverage of one macro cell BS , which is equipped at UEk can be written as , H −1 Vk VH k Hk , Rk (V) (2) where we have defined V = {Vk |k ∈ K , ∈ L} and K Hk Rk (V) = H Vu VH u Hk , , u=1,u=k L (3) Ki Hk ,i Vji VH ji + HH k ,i +σk2l INk i=0,i= j=1 is the interference plus noise covariance matrix of UEk To consider the NEE, we first compute the power consumption of the BSs The transmit power at BS is given by K Vk VH k Pt (V) = (4) k=1 Assume that the static circuit power consumption at BS is Pc and inefficiency of power amplifier is ρ Accordingly, the total power consumption at BS is K Vk VH k P (V) = ρ + Pc (5) k=1 Fig A system model of the downlink channel in MIMO HetNets with M antennas, serves K users associated BS for ∈ L = {0, 1, , L} where BS0 is referred to as the MBS User k in cell referred to as UEk , for k ∈ K = {1, 2, , K } is equipped with Nk receive antennas The channel matrix from BS to UEk is Hk , ∈ CNk ×M Similar to works in [16], [17], global channel state information (CSI) is assumed to be perfectly known at the BSs Denote sk ∈ Cdk ×1 as dk data streams intended to UEk and Vk ∈ CM ×dk to be the transmit precoding matrix for the signal of UEk Then, the Then, the ratio of the sum achievable data rate to the total amount of power consumption, namely NEE, can be written as [25] L K =0 k=1 Rk (V) (6) ζEE (V) = L =0 P (V) In addition, it is of important practice to impose the transmit power at each antenna [19] The transmit power at antenna i of BS can be computed by K K Vk VH k pi = k=1 i,i Bk = ,i , Vk VH k (7) k=1 where Bk ,i is a diagonal matrix with the i − th diagonal element equal to one and all the others equal to zero We aim to maximize the NEE by designing the optimal precoders Specifically, in this paper, we maximize the NEE 136 with considering the constraints on the per-antenna and perbase-station transmit power and on interference power caused to macro cell users (UEk0 ) by the SBSs Denote the transmit power budget at BS by P ,max and the maximum allowable transmit power at antenna i of BS by pi ,max The design problem of interest can be mathematically stated as max L =0 ζEE (V) = {V} K s.t Vk VH k K k=1 L =0 P ≤P Rk (V) (V) ∈L ,max , Bk ,i Vk VH k ,max , i = 1, , Ml , H H s0 , V k V H k H s0 , (8d) Constraints (8b) and (8c) restrict the transmit power at each BS and each antenna less than the power budgets, respectively Constraint (8d) indicates the maximum allowable interference power caused to each macro user by SBSs III A N I TERATIVE A LGORITHM U SING BCA AND D INKELBACH M ETHODS sk = yk = Hk , V u su u=1,u=k + UH k Ki L Hk ,i V j i s j i + UH k nk i=0,i= j=1 (9) Accordingly, the mean squared error (MSE) matrix of UEk is defined by Ek = E (sk − sk ) (sk − sk ) = UH k Hk , V k − Id k UH k Hk + ,i H UH k Hk , H V k − Id k H H V ji V H ji Hk ,i Uk +σkl Uk Uk (10) where we have assumed that E sk sH ji = for ( , k) = (i, j), H = 0, E n = for ( , k) = (i, j) The n E sk n H k ji ji optimal receive matrix Uk which results in the MMSE is given by [18], [26] L Ki = i=0 j=1 −1 Hk ,i Vji VH ji HH k ,i +σk2l INk Hk , Vk (11) + dk (14) −1 (15) With Uopt and Wopt k k , the maximum of fk (V, Uk , Wk ) is mmse fk (V, Ukopt , Wopt k ) = − log Ek (16) This implies fk (V, Uk , Wk ) is a lower bound of the achievable rate function in (2) and is concave over each group of the matrices V, Uk , Wk when the other two are fixed Then, problem (8) can be equivalently reformulated as L =0 Kl k=1 (log |Wk | − Wk Ek {V,U,W} s.t L =0 + dk ) P (V) (17a) (8b), (8c), (8d) It is important to notice that it can be observed that the objective function in (17) is a concave-convex fractional function with respect to variable V while other variables {Wk , Uk } are fixed Thus, the Dinkelbach approach [27] is applied to find precoders V We introduce parameters λ and L K U (V, λ) = =0 k=1 (log |Wk | − Wk Ek + dk ) − L λ =0 P (V) , thus, problem (17) can be recast as max U (V, λ) s.t (8b), (8c), (8d) {V} (j,i)=(k, ) Uopt k log |Wk | − Wk Ek (13) is an auxiliary semidefinite matrix where Wk ∈ C variable It can be proved that fk (V, Uk , Wk ) is concave quadratic over Uk with fixed V and Wk and the maximum can be obtained from (11) By applying the BCA point Uopt k approach [18], [26], with fixed V and Uk , the MSE covariance matrix Ek is fixed, and, by the concavity of log | Wk | and the linearity of Wk Ek , fk (V, Uk , Wk ) is concave over Wk Then, by taking the gradient of fk (V, Uk , Wk ) over Wk and setting the gradient as 0, we can obtain the maximum point as max K H k , V k s k + UH k (12) Nk ×Nk Wopt = Emmse k k It can be observed that problem (8) maximizes a nonconcave and nonlinear fractional function over the constraints of the convex feasible sets To handle the difficulty in solving (8), we adopt the relationship between the achievable rate with the MMSE of UEk [16], [26] Denote Uk ∈ CNk ×dk as a receive matrix which is applied to the received signal vector yk at UEk to recover its desired signal, defined by UH k Next, to tackle with the nonconcave property of the data rate function, we consider the following function =1 k=1 UH k −1 (8b) ∈L ≤ γs0 , s ∈ K0 H −1 Vk VH k Hk , Rk Rk (V) = − log Emmse k (8c) K , In light of (2) and (12), we can derive the relationship between the achievable rate and the MMSE of UEk as fk (V, Uk , Wk ) ≤ pi k=1 L = INk + Hk Emmse k (8a) k=1 K Substitution of Uopt into the MSE matrix (10) yields k (18a) Problem (18) can be efficiently solved by CVX [28] Note that the optimal solution Vopt to (18) which satisfies U Vopt , λopt = is also that of (17) [27] The detailed procedure to combine the BCA and Dinkelbach approaches to solve problem (8) can be presented in Algorithm Similar to the discussion in [16], [17], [26], Algorithm results in a nondecreasing objective value through iterations and the objective function is upper-bounded due to the constraints of transmit power and interference Thus, the convergence of the objective value in Algorithm will be guaranteed 137 as the transmit power budget increases from 35 dBm to 40 dBm 0.45 Pt = 20 dBm Pt = 25 dBm 0.4 Pt = 30 dBm Pt = 35 dBm 0.35 EE (bits/Hz/J) Algorithm : Iterative algorithm for NEE maximization Initialization: Start with arbitrary feasible precoding matrices {V} and set iteration index n = repeat Compute {U} and {W} using (11) and (15), respectively repeat Given {V}, {U} and {W}, find λ = L K (log |W | − W E + d ) k k k k =0 k=1 L =0 P (V) Given λ, solve (18) using CVX to obtain {V} until convergence Update n = n + until convergence Output: Vopt Pt = 40 dBm 0.3 0.25 0.2 0.15 IV S IMULATION R ESULTS In this section, we demonstrate the NEE performance of the considered HetNets through numerical results The HetNet in our simulation consists of small cells in the macro cell The MBS is located at (0, 0) while SBSs are located at (R/2, R/2), (−R/2, R/2), (R/2, −R/2), (−R/2, −R/2), respectively All UEs are assumed to be randomly located in their cells Each BS has antennas (M = M = 4) and serves two UEs, Each UE equipped with antennas receives data streams from its associated BS, i.e., Nk = N = and d = The circuit power consumption is set as Pc0 = 100 W, Pc ( = 0) = Pc = 7.9 W [29], [30] The power amplifier inefficiency ρ = for ∈ L The transmit power budget of the MBS is set as 10 times higher than that of the SBS, i.e., P0,max = 10P ,max = 10Pt ( = 0) We set the constraint on the power budget at each antenna as pi ,max = P ,max /(0.9M ) Other parameters of the simulations are listed in Table I [10], [13] Unless otherwise stated, the numerical results in the following figures are averages over 100 random channel realizations TABLE I S YSTEM PARAMETERS IN SIMULATION Parameters Values Macro cell radius R 500m Small cell radius r 40m -76 dBm Noise variance Path and penetration loss at distance d (km) 148 + 37.6 log10 (d) dB Within 40m of small cells 127 + 30 log10 (d) dB First, we examine the convergence characteristic of the iterative algorithm The acceptable interference of all SBSs to macro cell UEs is set as γs0 = γ = −66 dBm Fig illustrates the convergence rate of the algorithm optimizing the NEE with different levels of the transmit power budget As can be viewed from Fig that the values of the NEE not decrease over iterations and converge quickly to optimal values NEE is also increased when increasing the transmit power budgets However, NEE will not increase significantly 0.1 20 40 60 80 100 120 140 160 180 200 Iterations Fig The convergence behavior of the iterative algorithm for NEE optimization Next, the achievable NEE obtained by Algorithm which maximizes the NEE will be compared to those achieved by using the SE maximization Fig illustrates the NEE performance obtained by these two approaches It can be observed from Fig that when the transmit power budget is low, the EE and SE maximization methods provide the same average EE performance The reason is that in this low transmit power region, the NEE objective function is dominated by the sum rate function On the other hand, for high transmit power budget, the EE maximization method provides better the achievable NEE as compared to the SE maximization The reason is that, with significant power budgets, the EE approach only uses a fraction of all the power budget which results in maximizing NEE To study the tradeoffs between EE and SE, the achievable sum-rate performance obtained by these two methods is shown in Fig As can be seen from this figure that the EE optimization scheme obtains lower sum rate for the high transmit power budgets From Figs and 4, it can conclude that the SE maximization strategy tends to use all transmit power budgets to maximize the sum-rate while the EE maximization strategy only uses a fraction of the transmit power budgets which can lead to the maximum EE V C ONCLUSION In this paper, we have studied the optimal precoding designs to maximize the NEE of the multi-user HetNets system under the constraints of per-antenna, per-BS transmit power and interference from SBSs to macro cell UEs Due to the mathematical challenges of the non-convex fractional programming, the relationship between MMSE and the achievable rate is exploited to recast the design problem into one amenable 138 0.9 0.8 EE (bits/Hz/J) 0.7 0.6 0.5 0.4 0.3 0.2 EE maximization SE maximization 0.1 10 15 20 25 30 35 40 45 50 P t (dBm) Fig The achievable NEE of the EE maximization and sum-rate maximization algorithms 140 120 Sum rate (bits/Hz) 100 80 60 40 20 EE maximization SE maximization 0 10 15 20 25 30 35 40 45 50 P t (dBm) Fig The achievable sum rate of the NEE maximization and sum-rate maximization algorithms Then, the BCA and Dinkelbach methods are applied to develop an iterative algorithm The numerical simulation results have shown the fast convergence of the iterative algorithm Also, the results have provided the interesting insights into the NEE performance for various transmit power budgets ACKNOWLEDGEMENT This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2017.308 R EFERENCES [1] Cisco, “Cisco visual networking index: Forecast and trends, 2017–2022 white paper,” Document ID, 2017 [2] M Sanjabi, M Razaviyayn, and Z Luo, “Optimal joint base station assignment and beamforming for heterogeneous networks,” IEEE Trans Signal Process., vol 62, no 8, pp 1950–1961, Apr 2014 [3] D Oh, H Lee, and Y Lee, “Power control and beamforming for femtocells in the presence of channel uncertainty,” IEEE Trans Veh Technol., vol 60, no 6, pp 2545–2554, Jul 2011 [4] S S Christensen, R Agarwal, E D Carvalho, and J M Cioffi, “Weighted sum-rate maximization using weighted MMSE for MIMOBC beamforming design,” IEEE Trans Wireless Commun., vol 7, no 12, pp 4792–4799, Dec 2008 [5] G Y Li, Z Xu, C Xiong, C Yang, S Zhang, Y Chen, and S Xu, “Energy-efficient wireless communications: tutorial, survey, and open issues,” IEEE Wireless Commun., vol 18, no 6, pp 28–35, Dec 2011 [6] C Jiang and L J Cimini, “Energy-efficient transmission for MIMO interference channels,” IEEE Trans Wireless Commun., vol 12, no 6, pp 2988–2999, Jun 2013 [7] S He, Y Huang, S Jin, F Yu, and L Yang, “Max-min energy efficient beamforming for multicell multiuser joint transmission systems,” IEEE Commun Lett., vol 17, no 10, pp 1956–1959, Oct 2013 [8] Y Chen, S Zhang, S Xu, and G Y Li, “Fundamental tradeoffs on green wireless networks,” CoRR, vol abs/1101.4343, 2011 [9] H Kwon and T Birdsall, “Channel capacity in bits per joule,” IEEE J Ocean Eng., vol 11, no 1, pp 97–99, Jan 1986 [10] Y Li, Y Tian, and C Yang, “Energy-efficient coordinated beamforming under minimal data rate constraint of each user,” IEEE Trans Veh Technol., vol 64, no 6, pp 2387–2397, Jun 2015 [11] O Amin, E Bedeer, M H Ahmed, and O A Dobre, “Energy efficiency–spectral efficiency tradeoff: A multiobjective optimization approach,” IEEE Trans Veh Technol., vol 65, no 4, pp 1975–1981, Apr 2016 [12] V Nguyen, H V Nguyen, C T Nguyen, and O Shin, “Spectral efficiency of full-duplex multi-user system: Beamforming design, user grouping, and time allocation,” IEEE Access, vol 5, pp 5785–5797, 2017 [13] T T Vu, H H Kha, O Muta, and M Rihan, “Energy-efficient interference mitigation with hierarchical partial coordination for MIMO heterogeneous networks,” IEICE Trans Commun., vol 100, no 6, pp 1023–1030, 2017 [14] K Nguyen, L Tran, O Tervo, Q Vu, and M Juntti, “Achieving energy efficiency fairness in multicell MISO downlink,” IEEE Commun Lett., vol 19, no 8, pp 1426–1429, Aug 2015 [15] Z Sheng, H D Tuan, H H M Tam, H H Nguyen, and Y Fang, “Energy-efficient precoding in multicell networks with full-duplex base stations,” EURASIP J Wireless Comm and Networking, vol 2017, no 1, p 48, Mar 2017 [16] Y Li, P Fan, and N C Beaulieu, “Cooperative downlink max-min energy-efficient precoding for multicell MIMO networks,” IEEE Trans Veh Technol., vol 65, no 11, pp 9425–9430, Nov 2016 [17] H H Kha and T N Ha, “An alternating optimization algorithm for energy efficiency in heterogeneous networks,” Journal of Science and Technology: Issue on Information and Communications Technology, vol 4, no 1, pp 1–8, 2018 [18] T N Ha and H H Kha, “Harvested energy and spectral efficiency tradeoffs in multicell MIMO wireless networks,” Radioengineering, vol 28, no 1, pp 331–339, Apr 2019 [19] M Vu, “MISO capacity with per-antenna power constraint,” IEEE Trans Commun., vol 59, no 5, pp 1268–1274, May 2011 [20] L Tran, M Juntti, M Bengtsson, and B Ottersten, “Successive zeroforcing DPC with per-antenna power constraint: Optimal and suboptimal designs,” in 2012 IEEE Int Conf Commun (ICC), Jun 2012, pp 3746– 3751 [21] L D Nguyen, H D Tuan, and T Q Duong, “Energy-efficient signalling in QoS constrained heterogeneous networks,” IEEE Access, vol 4, pp 7958–7966, 2016 [22] L D Nguyen, T Q Duong, D N Nguyen, and L Tran, “Energy efficiency maximization for heterogeneous networks: A joint linear precoder design and small-cell switching-off approach,” in 2016 IEEE Global Conference on Signal and Information Processing (GlobalSIP), Dec 2016, pp 718–722 [23] Q Vu, L Tran, R Farrell, and E Hong, “Energy-efficient zero-forcing precoding design for small-cell networks,” IEEE Trans Commun., vol 64, no 2, pp 790–804, Feb 2016 [24] S He, Y Huang, S Jin, and L Yang, “Coordinated beamforming for energy efficient transmission in multicell multiuser systems,” IEEE Trans Commun., vol 61, no 12, pp 4961–4971, Dec 2013 139 [25] S He, Y Huang, L Yang, and B Ottersten, “Coordinated multicell multiuser precoding for maximizing weighted sum energy efficiency,” IEEE Trans Signal Process., vol 62, no 3, pp 741–751, Feb 2014 [26] Q Shi, M Razaviyayn, Z Luo, and C He, “An iteratively weighted MMSE approach to distributed sum-utility maximization for a MIMO interfering broadcast channel,” in 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), May 2011, pp 3060–3063 [27] W Dinkelbach, “On nonlinear fractional programming,” Management Science, vol 13, pp 492–498, Mar 1967 [28] M Grant, S Boyd, and Y Ye, “CVX: Matlab software for disciplined convex programming (2008),” Web page and software available at http://stanford edu/ boyd/cvx, 2015 [29] D W K Ng, E S Lo, and R Schober, “Energy-efficient resource allocation in multi-cell OFDMA systems with limited backhaul capacity,” IEEE Trans Wireless Commun., vol 11, no 10, pp 3618–3631, Oct 2012 [30] M Deruyck, W Vereecken, W Joseph, B Lannoo, M Pickavet, and L Martens, “Reducing the power consumption in wireless access networks: Overview and recommendations,” Progress in electromagnetics research, vol 132, pp 255–274, 2012 140 Sum Rate Maximization for Full Duplex Multiuser MIMO Systems by Gradient Projection Approach Viet Quoc Pham, Tien Ngoc Ha, Ha Hoang Kha, and Vo Que Son Ho Chi Minh City University of Technology, VNU-HCM, Vietnam Email: vietpq09@gmail.com, hntien.hcmut@gmail.com, hhkha@hcmut.edu.vn, sonvq@hcmut.edu.vn Abstract—In this paper, we study the achievable sum-rate of a multi-user (MU) multiple-input-multiple-output (MIMO) wireless communication system in which one full-duplex (FD) base station (BS) serves a number of half-duplex (HD) mobile stations (MSs) The problem of interest is to design the precoding matrices of the uplinks and the downlinks to maximize the system sum rate under transmit power constraints at the BS and the MSs An iterative algorithm based on the gradient projection (GP) and the Armijo rule is developed to maximize the sum-rate The convergence of the algorithm and the achievable sum-rate performance of the FD MU-MIMO model as compared to those of the HD MU-MIMO one will be investigated through numerical simulation results Index Terms—Multi-user multiple-input multiple-output, fullduplex, half-duplex, precoder designs, sum-rate maximization I I NTRODUCTION In recent years, full-duplex (FD) wireless systems have become an active research area since they can potentially improve the spectral efficiency than the half-duplex (HD) ones [1], [2] In the FD systems, signals are transmitted and received at the same time and in the same frequency band and, thus, the received signals at FD devices can be suffered from strong interference from their own transmitted signals which are also known as self-interference (SI) The mitigation of SI is vitally important since SI is the dominant factor which diminishes the potential sum-rate improvement of the FD systems as compared to the HD systems In [3] the authors considered an FD multiuser multiple-input multiple-output (MU MIMO) network in which interference caused by uplink transmission to the downlinks at mobile stations (MSs) had been taken into account The authors introduced two SI suppression strategies, namely extended zero-forcing precoding and extended regularized channel inversion Alternatively, a single-cell FD massive MIMO system in which the FD BS equipped with large-scale antenna arrays serves a number of FD two-antenna MSs were also considered in [4] The work had shown that SI can be eliminated by deploying the very large number of antennas at the BS Reference [5] studied the theoretical performance of an FD bi-directional communication system and an FD relay system The authors focused on the effect of residual SI due to imperfect channel estimation and transmitter noise There have been a great number of works on FD systems FD MU MIMO systems in which base stations (BSs) and mobile stations (MSs) operate in the FD and the HD modes, respectively, were investigated in [6], [7] In [6], co-channel interference from the uplinks to the downlinks at MSs was not considered Both sequential and joint precoding schemes were deployed in [6] to obtain an approximate method of the non-convex optimization problem The authors then developed their works in [7] by considering the effects of both energy and spectral efficiency designs In the case of FD single-user (SU) MIMO systems, beamforming was utilized in [8] to minimize transmit power under the constraints of both total signal-tointerference-plus-noise ratio (SINR) and SI The authors of [9] considered a multi-cell MU-MIMO network in which FD BSs serve HD MSs The authors focused on designing filters for weighted sum rate (WSR) maximization under the effect of transmitter and receiver distortion The connection between the minimum mean squared error (MMSE) and the achievable rate was used to obtain the optimal filters with and without channel state information (CSI) errors The authors of [10] considered the proportional fairness issue in an FD MU MIMO system The paper developed a gradient projection method which offers a good balance between ensuring fairness among MSs and maximizing the sum-rate In [11], the authors focused on the maximization of the minimum weighted achievable rate of the two links in FD MIMO systems The authors used D.C (difference of two convex functions) based algorithms to address the highly non-convex optimization problem In [12], the authors proposed two different algorithms which are based on sequential convex programming (SCP) to address the sum-rate maximization problem of an FD MU-MIMO system The authors [13] proposed successive convex quadratic programming (SCQP) framework for the precoding designs to maximize the total network sum-rate or the minimum rate among cells in FD MU-MIMO multicell networks In [14], the path-following algorithms are proposed to optimize spectral and energy efficiency in FD systems with wireless information and power transfer More recently, the authors of [15] studied the robust precoding designs for MU MIMO cognitive radio networks with an FD secondary BS and multiple HD MSs In this paper, inspired from [12], we study the precoding designs for uplinks and downlinks in an FD MU-MIMO system in which an FD BS serves a number of HD MSs In addition to co-channel interference (CCI) from the uplinks to the downlinks, the effect of SI from the downlinks to the uplinks at the BS is also considered The BS and MSs are assumed to have perfect CSI An iterative algorithm which is based on the gradient projection (GP) method and the Armijo 978-1-7281-5353-7/19/$31.00 ©2019 IEEE 155 rule will be derived to maximize the sum-rate of the FD MUMIMO network The convergence of the iterative algorithm and the achievable sum-rate of the FD MU-MIMO system in comparison with the sum-rate of the HD MU-MIMO model will be examined through simulation results Notation: In this paper, upper-case bold variables refer to matrices Lower-case bold and non-bold variables stand for vectors and scalars, respectively E (.) denotes the expectation operator F represents the Frobenius norm With a complexentry matrix A, AT and AH refer to the transpose and conjugate transpose of matrix A, respectively Tr (A) represents the trace of A |A| denotes the determinant of matrix A In stands for the n-by-n identity matrix A means that A is a positive semi-definite matrix We use the notation [x]+ = max{x, 0} H H U D that E sU = I and E sD = I Accordingly, u su d sd U MT ×MT the transmit covariance matrices Qu ∈ C and QD d ∈ NT ×NT C can be expressed as U U QU u = Wu Wu QD d = H , u = 1, , K, (2a) , d = 1, , K (2b) H WdD WdD Then, the achievable sum-rate of the uplinks can be formulated as [12], [13] K RU = log2 INR + U U HU u Qu Hu H u=1 (3) −1 K B H × ΓB + GB QD d G , d=1 H II S YSTEM M ODEL We consider a single-cell FD MU-MIMO network as shown in Fig which consists of one FD base station serving K uplink MSs (UMSs) and K downlink MSs (DMSs) [12] The MSs are assumed to operate in the HD mode Each UMS has MT transmit antennas while each DMS has MR receive antennas The FD BS is assumed to have NR = KMT receive and NT = KMR transmit antennas where ΓB = E nB nB D 2) Downlink: Let Hd ∈ CMR ×NT be the channel matrix from the BS to the d-th DMS The downlink signals are also affected by the uplink ones We denote the interference matrix from the u-th UMS to the d-th DMS as Gud ∈ CMR ×MT , d = 1, , K; u = 1, , K The received signal at the d-th DMS can be written as K D D HD d W s D D yd = HD d Wd sd + =1 =d desired signal (4) downlink interference K M Gud WuU sU u + nd , + u=1 uplink interference MR ×1 with nM is the additive white Gaussian noise at d ∈ C the receiver Hence, the achievable rate of the d-th downlink can be given by D D RdD = log2 IMR + HD d Qd Hd Fig The system model of a FD MU-MIMO network NR ×MT 1) Uplink: Let HU be the channel matrix u ∈ C from the u-th UMS to the BS The u-th UMS applies the linear precoding matrix WuU ∈ CMT ×MT to its information MT ×1 signal sU to be transmitted to the BS Since u ∈ C the BS operates in the FD mode, the received signals at the BS are suffered from SI caused by the signals of the downlinks Assume that the BS uses the precoding matrix NT ×1 WdD ∈ CNT ×NT to process the downlink signal sD d ∈C B NR ×NT towards to the d-th DMS Let G ∈ C be the SI channel matrix of the BS Then, the received signal at the BS can be given by K × ΓM d + D HD HD d Q d H H u Gud QU u (Gd ) K U U HU u Wu su + y= u=1 desired signal B NR ×1 B GB WdD sD d +n , (1) where n ∈ C is the additive white Gaussian noise at the receiver of the BS Without loss of generality, we assume , (5) H M where ΓM nM The sum-rate of the downlinks d = E nd d in the FD MU-MIMO system is K RD = RdD , (6) d=1 which is written as RD = D D log2 IMR + HD d Qd Hd H d=1 d=1 self−interference H u=1 =1 =d K K −1 K + K × ΓM d D HD d Q + =1 =d H HD d −1 K Gud QU u + H (Gud ) u=1 (7) 156 The sum-rate maximization problem of the FD MU-MIMO model then can be stated as QU u U max D R=R +R , 0,QD d QU u Tr s.t P M and ∂RU = ∂QD i (8a) , u = 1, , K, ΓB + u=1 (8b) B P , GB QD d G  B H ΓB + − G GB QD d G Similarly, Eq (7) can be rewritten as K RD = This section presents the gradient projection method maximizing the sum-rate function of the FD MU-MIMO model under the power constraints The algorithm resolving problem (8) consists of two fundamental steps: computing gradients and projecting First, the gradient method is utilized to maximize the objective function through iterations Next, the projection process is deployed to address the power constraints K log2 ΓM d + =1 K u=1 K − ∂R ∂QD i ∂R ∂R + U ∂QU ∂Q j j U and ∂RD ∂QD i (9) K ∂RD = ∂QU j d=1 H K , (10b) H Gjd  u Gud QU u (Gd )   H  j  Gd − d=1 K B GB QD d G (15) K  M Γd + d=1 H HD d D HD d Q =1 =d −1 K i u Gud QU u (Gd ) + H T Gjd  , u=1 B GB QD d G − log2 ΓB + T −1 + ∗ K u=1 H u=1 D + D HD HD d Q d + =1 (10a) j U U HU u Qu H u ΓM d K , ∂R ∂R + D ∂QD ∂Q i i K K H Gjd as follows Eq (3) can be rewritten as RU = log2 ΓB + Therefore, ∗ D H u Gud QU u (Gd ) u=1 where j = 1, , K and i = 1, , K By utilizing the partial ∂RU ∂RU derivatives formula in [16] and [17], we can obtain ∂Q U , ∂QD , ∂RD , ∂QU j H HD d =1 =d + ∗ ∂f (x) ∂x U =2 D HD d Q + d=1 The gradients of the objective function in Eq (8a) are then: ∇QD R=2 i (14) K log2 ΓM d K ∂f (x) ∂f (x) +j =2 ∂x1 ∂x2 ∗ H u Gud QU u (Gd ) + The gradient of the real function f (x) with x = x1 + jx2 is defined as =2 H D HD HD d Q d d=1 A Computing the gradients ∂R ∂QU j GB  d=1 III I TERATIVE A LGORITHM FOR S UM -R ATE M AXIMIZATION ∇QUj R = (13) T −1 K B H where Eq (8b) and Eq (8c) impose the power constraints on the MSs and the BS, respectively ∗ GB  d=1 (8c) d=1 ∇x f (x) = T −1 B H + QD d H U U HU u Qu H u K K Tr K H GB and H K H ∂RD = ∂QD i d=1 HD d K H ΓM d + D HD HD d Q d H =1 −1 K d=1 H  HD d D HD d Q H HD d u Gud QU u (Gd ) + (11) T u=1 Hence, with j = 1, , K and i = 1, , K , K U ∂R = ∂QU j HU j H K ΓB + U U HU u Qu Hu −1 K B GB QD d G + d=1 H  , HU j H (16) K  M Γd + d=1 d=i u=1 T   D  Hd − H  =1 =d (12) −1 K H u Gud QU u (Gd ) + T  HD d u=1 157 The gradient of the objective function in Eq (8a) with respect to QU R, is obtained by substituting Eq (12) and j , i.e ∇QU j (15) into Eq (10a) Similarly, the gradient of the objective function with respect to QD R, is obtained by i , i.e., ∇QD i substituting Eq (13) and (16) into Eq (10b) Then, the transmit covariance matrices are updated as follows: QjU = QU R, j + sn ∇QU j (17) QiD = QD R i + sn ∇QD i (18) and The process of finding the projections will be introduced in the next subsection Algorithm :Projection algorithm Initialize: N ìN ã Given a positive semi-definite matrix Q ∈ C • Decompose Q = UDUH with D = diag(λ1 , λ2 , , λN ) and λ1 ≥ λ2 ≥ ≥ λN • Given a matrix space Ω+ (P ) • Set i = N, search = true repeat i −P Calculate: µ∗ = λ1 +λ2 + +λ i ∗ if µ ≤ λi then search = f alse else i=i−1 B Finding projections 1) Projecting the transmit covariance matrices of the uplinks: The projection is to guarantee that the obtained covariance matrices are in the feasible region Let Ω+ (P ) = {Q ∈ CN ×N |Tr (Q) = P, Q 0} denote a matrix space such that any matrix belonging to this space would satisfy the power constraint Given a positive semi-definite matrix Q ∈ CN ×N , the process of finding the projection of Q onto Ω+ (P ), denoted by proj(Q )Ω+ (P ) , can be restated as the following minimization problem: Q−Q Q s.t F (19a) Tr (Q) = P (19b) According to the Lagrangian, the solution to problem (19) can be obtained by setting ∂f (Q, µ) ∂Q = (20) where f (Q, µ) = Q − Q F + µ (Tr (Q) − P ) (21) This results in Now, applying Algorithm 1, the projection for the transmit covariance matrices of the uplinks QjU into Ω+ P M = {Q ∈ ¯ U = proj(Q U )Ω (P M ) CN ×N |Tr (Q) = P M , Q 0} is Q j j + 2) Projecting the precoder covariance matrices of the downlinks: The projection of the precoder covariance matrices of the downlinks is quite similar to the projection of the precoder covariance matrices of the uplinks However, there is a minor difference as the matrices of the downlinks has the sum power constraint, instead of individual power constraint as in the uplink case To satisfy the sum-power constraint in Eq (8c), we arrange QiD into a matrix Q ∈ CKNT ×KNT as follow:  D  Q1  Q2D    Q = (26)    Q ⇔Q = Q − µI (22a) H = U (D − µI) U , (22b) where we have used the eigenvalue decomposition of matrix Q = UDUH with D = diag(λ1 , λ2 , , λN ) and UUH = UH U = I In addition, since Q is required to be positive semi-definite and to satisfy the power constraint (19b), we use the water-filling algorithm for finding µ∗ such that: + Tr (D − µ∗ I) = P, + where (D − µ∗ I) (23) is defined as follows: [λ1 − µ∗ ]+      end if until search = f alse ¯ = proj(Q )Ω (P ) = U (D − µ∗ I)+ UH Output: µ∗ , Q + [λ2 − µ∗ ]+ [λN      − µ ∗ ]+ (24) ∗ By applying Algorithm 1, we Ω+ P B = {Q ∈ CN ×N |Tr (Q) ¯ as follows: Q  D ¯ Q   ¯ = proj(Q )Ω (P B ) =  Q +  QKD then project Q onto = P B, Q 0} to obtain ¯D Q  0     ¯D Q (27) K C Applying Armijo’s rule After projecting the covariance matrices of the uplinks and the downlinks, we use the Armijo rule to update the matrices for the next iteration [16], [18] With fixed values < σ < and < β < 1, the updated matrices of the (n + 1)-th iteration are chosen as [16], [18] The algorithm for finding µ is summarized in Algorithm The projection of Q onto Ω+ (P ) is then given by: Qj ¯ = proj(Q )Ω (P ) = U (D − µ∗ I)+ UH Q + Qi (25) U (n+1) = Qj U (n) ¯ U (n) − QU (n) , + β mn Q j j (28a) D(n+1) = Qi D(n) ¯ D(n) − QD(n) , + β mn Q i i (28b) 158 where j = 1, , K, i = 1, , K, and mn is the first nonnegative integer which satisfies: R(n+1) − R(n)  ≥ σβ mn  H  K (n) Tr ∇QUj R Tr ∇Q D R(n) i H ¯ U (n) Q j − U (n) Qj  j=1 K + σβ mn H ¯ D(n) − QD(n) Q i i i=1 (29) Let ε be the tolerance error The process of computing the gradients, projecting, and applying the Armijo rule is repeated (n+1) −R(n) until R R(n) ≤ ε The detail of the gradient projection method for maximizing the sum-rate of the FD MU-MIMO system is summarized in Algorithm Algorithm :Iterative algorithm based on gradient projection for sum-rate maximization in FD MU-MIMO systems Initialize: • Set n = 1, search = true, ε U (1) • Choose initial variables: Qj , D(1) Qi , i = 1, K repeat Calculate the gradients ∇QU R(n) , j = 1, , K j = we set the energy spent in the FD and HD modes to be the same In simulations, we adopt the simulation parameters from H [12] Particularly, we normalize E nB nB = INR and 1, K; E nM nM = IMR Elements of HU u , u = 1, , K and d d D Hd , d = 1, , K are complex Gaussian random variables with zero-mean and a variance equal to the SNR The entries of Gud are complex Gaussian random variables with zero-mean and a variance equal to the co-channel interference-to-noise ratio at MSs, denoted by INRM The SIRM = SNR/INRM is the signal-to-co-channel interference ratio at the input of the receiver of the MS Similarly, elements of GB are complex Gaussian random variables with zero-mean and a variance equal to the self-interference-to-noise ratio at the BS, INRB Then, the signal-to-self-interference ratio at the input of the receiver of the BS is SIRB = SNR/INRB In simulation, it is assumed that K = 3, MT = MR = The number of transmit and receive antennas of the BS are then NT = KMR = and NR = KMT = The maximum transmit power budgets of the MS and BS are chosen as P M = 23 dBm and P B = 30 dBm, respectively [12], [19] With regard to Algorithm 2, we choose σ = 0.1, β = 0.5 and = 10−3 j ∇QD R(n) , i = 1, , K i Compute U (n) U (n) Qj = Qj + sn ∇QU R(n) j D(n) D(n) Qi = Qi + sn ∇QD R(n) i Find the projections U (n) U (n) ¯ Q is the projection of Qj onto Ω+ P M j D(n) D(n) ¯ Qi is the projection of Qi onto Ω+ P B Applying the Armijo rule to find mn mn = repeat Computing: U (n+1) U (n) ¯ U (n) − QU (n) Qj = Qj + β mn Q j j D(n+1) D(n) ¯ D(n) − QD(n) Qi = Qi + β mn Q i i mn = mn + until Satisfying Eq (29) Considering terminating condition (n+1) −R(n) if R R(n) ≤ ε then search = false else n=n+1 end if until search = false U (n+1) D(n+1) Output: Qj , Qi IV N UMERICAL R ESULTS In this section, we compare the achievable sum-rate of the FD and the HD MU-MIMO systems by using the GP algorithm For the HD mode, two time slots are used for the uplinks and downlinks Thus, in the HD mode, the SI and CCI from the uplinks to the downlinks not exist and the total sum-rate is R = (RD + RU )/2 For fair comparison, Fig Convergence comparison of the sum-rate with SNR = 15dB, SIRB = −15dB, and SIRM = −5dB Fig illustrates the convergence characteristic of the sum-rate functions with SNR = 15 dB, SIRB = −15 dB, SIRM = −5 dB The numerical results show that the sumrate functions not decrease through iterations In addition, the GP takes under 30 iterations to converge the objective functions of both the FD MU-MIMO and the HD MU-MIMO models Fig illustrates the FD-to-HD sum-rate ratio versus SNR with SIRB = −15 dB, SIRM = −5 dB It can be observed from the figure that, at low SNR, the sum-rate of the FD system is slightly higher than the sum-rate of the HD one However, when increasing SNR, the ratio decreases dramatically Fig shows the FD-to-HD sum-rate ratio versus SIRB with SNR = dB, SIRM = −5 dB As it can be seen, at low SIRB , the sum-rate of the FD system is slightly lower 159 ACKNOWLEDGEMENT This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2017.308 R EFERENCES Fig FD-HD ratio versus SNR with SIRB = −15dB and SIRM = −5dB Fig FD-HD ratio versus SIRB with SNR = 5dB and SIRM = −5dB than the sum-rate of the HD one However, the ratio rises up drastically when at the high SIRB regime, which is caused by insignificant SI at the BS V C ONCLUSIONS In this work, the GP algorithm has been presented to maximize the sum-rate of the FD and HD MU-MIMO systems By employing the Armijo rule, the iterative algorithm only takes a few tens of iterations to converge In addition, the numerical simulation results demonstrate that the FD model does not always provide higher sum-rate than the HD one, especially when the SNR is significantly high or the SI power is high [1] M Jain, J I Choi, T Kim, D Bharadia, S Seth, K Srinivasan, P Levis, S Katti, and P Sinha, “Practical, real-time, full duplex wireless,” in Proceedings of the 17th Annual International Conference on Mobile Computing and Networking, ser MobiCom ’11 New York, NY, USA: ACM, 2011, pp 301–312 [2] M Duarte and A Sabharwal, “Full-duplex wireless communications using off-the-shelf radios: Feasibility and first results,” in 2010 Conference Record of the Forty Fourth Asilomar Conference on Signals, Systems and Computers, Nov 2010, pp 1558–1562 [3] B Yin, M Wu, C Studer, J R Cavallaro, and J Lilleberg, “Fullduplex in large-scale wireless systems,” in 2013 Asilomar Conference on Signals, Systems and Computers, Nov 2013, pp 1623–1627 [4] X Wang, D Zhang, K Xu, and C Yuan, “On the sum rate of multiuser full-duplex massive MIMO systems,” in 2016 IEEE International Conference on Communication Systems (ICCS), Dec 2016, pp 1–7 [5] A C Cirik, Y Rong, and Y Hua, “Achievable rates of full-duplex MIMO radios in fast fading channels with imperfect channel estimation,” IEEE Trans Signal Process., vol 62, no 15, pp 3874–3886, Aug 2014 [6] D Nguyen, L N Tran, P Pirinen, and M Latva-aho, “Transmission strategies for full duplex multiuser MIMO systems,” in 2012 IEEE International Conference on Communications (ICC), June 2012, pp 6825–6829 [7] ——, “Precoding for full duplex multiuser MIMO systems: Spectral and energy efficiency maximization,” IEEE Trans Signal Process., vol 61, no 16, pp 4038–4050, Aug 2013 [8] J Zhang, O Taghizadeh, and M Haardt, “Robust transmit beamforming design for full-duplex point-to-point MIMO systems,” in ISWCS 2013; The Tenth International Symposium on Wireless Communication Systems, Aug 2013, pp 1–5 [9] P Aquilina, A C Cirik, and T Ratnarajah, “Weighted sum rate maximization in full-duplex multi-user multi-cell MIMO networks,” IEEE Trans Commun., vol 65, no 4, pp 1590–1608, April 2017 [10] A C Cirik, “Fairness considerations for full duplex multi-user MIMO systems,” IEEE Wireless Communications Letters, vol 4, no 4, pp 361– 364, Aug 2015 [11] Q Cao, Y Sun, W Shi, Q Ni, and B Wang, “Max-min weighted achievable rate for full-duplex MIMO systems,” IEEE Wireless Commun Lett., vol 8, no 1, pp 37–40, Feb 2019 [12] S Huberman and T Le-Ngoc, “Full-duplex MIMO precoding for sumrate maximization with sequential convex programming,” IEEE Trans Veh Technol., vol 64, no 11, pp 5103–5112, Nov 2015 [13] H H M Tam, H D Tuan, and D T Ngo, “Successive convex quadratic programming for quality-of-service management in full-duplex MUMIMO multicell networks,” IEEE Trans Commun., vol 64, no 6, pp 2340–2353, Jun 2016 [14] V Nguyen, T Q Duong, H D Tuan, O Shin, and H V Poor, “Spectral and energy efficiencies in full-duplex wireless information and power transfer,” IEEE Trans on Commun., vol 65, no 5, pp 2220–2233, May 2017 [15] X.-X Nguyen and H H Kha, “Precoding ddesigns for full-duplex multiuser MIMO cognitive networks with imperfect CSI,” REV Journal on Electronics and Communications, vol 9, no 1-2, pp 16–22, Jan-Jun 2019 [16] Sigen Ye and R S Blum, “Optimized signaling for MIMO interference systems with feedback,” IEEE Transactions on Signal Processing, vol 51, no 11, pp 2839–2848, Nov 2003 [17] J R Magnus and H Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, 2nd ed John Wiley, 1999 [18] J Liu, Y T Hou, and H D Sherali, “Conjugate gradient projection approach for multi-antenna gaussian broadcast channels,” 2007 [Online] Available: http://arxiv.org/abs/cs/0701061 [19] G T 36.814, “Further advancements for E-UTRA physical layer aspects,” Release Std., Mar 2010 160 ĐẠI HỌC QUỐC GIA TP.HCM TRƯỜNG ĐẠI HỌC BÁCH KHOA CỘNG HÒA XÃ HỘI CHỦ NGHĨA VIỆT NAM Độc lập - Tự - Hạnh phúc TÓM TẮT LÝ LỊCH KHOA HỌC Bản thân Họ tên: Phạm Quốc Việt Phái: Nam Sinh ngày: 04/02/1994 Nơi sinh: Tây Ninh Địa liên lạc: 404 Châu Văn Liêm, Hiệp Tân, Hòa Thành, Tây Ninh Điện thoại: 0394 723 515 Email: vietpq09@gmail.com Quá trình đạo tạo: ĐẠI HỌC Tốt nghiệp chương trình PFIEV, Trường ĐH Bách Khoa – ĐHQG TP.HCM Ngành học: Viễn Thơng Loại hình đào tạo: Chính quy Thời gian đào tạo: 2012 đến năm 2018 Xếp loại tốt nghiệp: Khá Quá trình học tập làm việc thân (từ đại học đến nay) Từ Đến Thành tích Học làm việc Ở đâu học tập 07/2019 Nay Học viên ngành Kỹ Đại Học Bách Khoa Thuật Viễn Thơng Thành Phố Hồ Chí Minh 08/2018 06/2019 Học viên chương trình Đại học South-Eastern trao đổi bậc thạc sĩ Norway 04/2018 07/2018 Học viên ngành Kỹ Đại Học Bách Khoa Thuật Viễn Thơng Thành Phố Hồ Chí Minh 09/2012 03/2018 Sinh viên Khoa Điện – Đại Học Bách Khoa Khá Điện Tử Thành Phố Hồ Chí Minh Lời cam đoan Tôi xin cam đoan nội dung khai thật xin chịu trách nhiệm trước pháp luật nội dung lý lich khoa học thân Tp HCM, ngày….tháng….năm 2019 Người khai ký tên Phạm Quốc Việt ... bách nói trên, nhiệm vụ luận văn hướng tới việc tối ưu hóa hiệu (hiệu suất phổ hiệu suất lượng) hệ thống thông tin vô tuyến đa người dùng MIMO Massive MIMO Các định hướng luận văn trình bày tóm tắt... gồm có: • Hệ thống MIMO vấn đề tối ưu hóa hiệu suất phổ thiết kế ma trận tiền mã hóa phát • Hệ thống MIMO HetNet vấn đề tối ưu hóa hiệu suất lượng thiết kế ma trận tiền mã hóa phát • Hệ thống Cell-free... dùng MIMO Massive MIMO • Mơ hình tốn, phân tích phát triển giải thuật tối ưu hóa hiệu (hiệu suất phổ, hiệu suất lượng) hệ thống MIMO đa người Mô đánh giá thiết kế đưa so sánh hiệu suất hệ thống với