diff --git "a/DNE0T4oBgHgl3EQfggEA/content/tmp_files/load_file.txt" "b/DNE0T4oBgHgl3EQfggEA/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/DNE0T4oBgHgl3EQfggEA/content/tmp_files/load_file.txt" @@ -0,0 +1,1368 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf,len=1367 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='02417v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='IT] 6 Jan 2023 1 Uplink Precoding Design for Cell-Free Massive MIMO with Iteratively Weighted MMSE Zhe Wang, Jiayi Zhang, Senior Member, IEEE, Hien Quoc Ngo, Senior Member, IEEE, Bo Ai, Fellow, IEEE and M´erouane Debbah, Fellow, IEEE Abstract In this paper, we investigate a cell-free massive multiple-input multiple-output system with both access points and user equipments equipped with multiple antennas over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We study the uplink spectral efficiency (SE) for the fully centralized processing scheme and large-scale fading decoding (LSFD) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further improve the SE performance, we design the uplink precoding schemes based on the weighted sum SE maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since the weighted sum SE maximization problem is not jointly over all optimization variables, two efficient uplink precoding schemes based on Iteratively Weighted sum-Minimum Mean Square Error (I-WMMSE) algorithms, which rely on the iterative minimization of weighted MSE, are proposed for two processing schemes investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, with maximum ratio combining applied in the LSFD scheme, we derive novel closed-form achievable SE expressions and optimal precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Numerical results validate the proposed results and show that the I-WMMSE precoding schemes can achieve excellent sum SE performance with a large number of UE antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Index Terms Cell-free massive MIMO, uplink precoding, weighted sum-rate maximization, spectral efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' INTRODUCTION Cell-free massive multiple-input multiple-output (CF mMIMO) has attracted a lot of research interest and is regarded as a promising technology for future wireless communications, for its ability to achieve This article was presented in part at IEEE International Conference on Communications 2022 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Zhang are with the School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China, and also with the Frontiers Science Center for Smart High-speed Railway System, Beijing Jiaotong University, Beijing 100044, China (e-mail: {zhewang 77, jiayizhang}@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ngo is with the Institute of Electronics, Communications, and Information Technology, Queen’s University Belfast, BT3 9DT Belfast, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (email: hien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ngo@qub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ai is with the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing 100044, China, also with the Frontiers Science Center for Smart High-Speed Railway System and the Henan Joint International Research Laboratory of Intelligent Networking and Data Analysis, Zhengzhou University, Zhengzhou 450001, China, and also with the Research Center of Networks and Communications, Peng Cheng Laboratory, Shenzhen 518066, China (e-mail: boai@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Debbah is with the Technology Innovation Institute, Abu Dhabi, United Arab Emirates, and also with CentraleSup´elec, University Paris-Saclay, 91192 Gif-sur-Yvette, France (e-mail: merouane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='debbah@tii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 uniformly high spectral efficiency (SE) [2]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Basically, a large number of access points (APs), arbitrarily distributed in a wide coverage area and connected to one or several central processing units (CPUs), jointly serve all user equipments (UEs) on the same time-frequency resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Compared with the traditional cellular mMIMO system, the CF mMIMO system operates with no cell boundaries and many more APs than UEs [8]–[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Relying upon the prominent network topology of CF mMIMO, four uplink (UL) signal processing schemes, distinguished from levels of the mutual cooperation between all APs and the assistance from the CPU, can be implemented as [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Among these schemes, the “Level 4” and “Level 3” are viewed as efficient processing techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The so-called Level 4 is a fully-centralized processing scheme where all the pilot and data signals received at APs are transmitted to the CPU via the fronthaul links and the CPU performs channel estimation and data detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The similar scheme was also investigated in [11]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The so-called Level 3 stands for a two layer decoding scheme: in the first layer, each AP estimates channels and decodes the UE data locally by applying an arbitrary combining scheme based on the local channel state information (CSI);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' in the second layer, all the local estimates of the UE data are gathered at the CPU in which they are linearly weighted by the optimal large-scale fading decoding (LSFD) coefficient to obtain the final decoding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The LSFD scheme has been widely investigated in [14]–[17] since it can make full use of the prominent network topology for CF mMIMO and achieve excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To promote the practical implementation of the CF mMIMO network, a new framework of scalable CF mMIMO system and its respective processing algorithms were proposed in [9] by exploiting the dynamic cooperation cluster (DCC) concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the scalability aspects in a realistic scenario with multiple CPUs were considered in [18], where the data processing, network topology and power control strategies with multiple CPUs were discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the authors of [19] considered the uplink of a radio-strip- based CF mMIMO network architecture with sequential fronthaul links between APs and proposed MMSE- based sequential processing schemes, which significantly reduced the fronthaul requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, when the CF mMIMO network is operated in practice, a more practical capacity-constrained fronthaul network would have a great effect on the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [20] and [21] discussed the uplink performance of a CF mMIMO system with limited capacity fronthaul links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, it is worth noting that the CF mMIMO architecture has been co-designed with another promising future wireless technology: Reconfigurable Intelligent Surface (RIS) [22], [23], which would undoubtedly provide vital tutorials for the future wireless network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3 The vast majority of scientific papers on CF mMIMO focus on the scenario with single-antenna UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, in practice, contemporary UEs with moderate physical sizes have already been equipped with multiple antennas to achieve higher multiplexing gain and boost the system reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [24] investigated the UL performance of a CF mMIMO system with multi-antenna UEs over maximum ratio (MR) combining and zero-forcing (ZF) combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [25] considered a user-centric (UC) approach for CF mMIMO with multi-antenna UEs and proposed power allocation strategies for either sum- rate maximization or minimum-rate maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the authors of [26] analyzed the downlink SE performance for a CF mMIMO system with multi-antenna UEs and computed SE expressions in closed- form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, the SE performance for a CF mMIMO system with multi-antenna UEs and low-resolution DACs was investigated in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Nevertheless, these works only investigated a simple distributed processing scheme and are based on the overly idealistic assumption of independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=') Rayleigh fading channels, neglecting the spatial correlation that has a significant impact on practical CF mMIMO systems [15], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [28] considered a CF mMIMO system with multi-antenna UEs over the jointly-correlated Weichselberger model [29] and analyzed four UL processing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As observed in [26], [28], increasing the number of antennas per UE may not always benefit the SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The SE would reach the maximum value with particular number of antennas per UE, then decrease with the increase of number of antennas per UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One main reason for this phenomenon is that the UEs cannot make full use of the benefit of equipping with multiple antennas to achieve higher SE performance without UL precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So it is undoubtedly vital to design the UL precoding scheme to further improve the performance of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, it is worth noting that the design of UL precoding for CF mMIMO has not been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the traditional mMIMO or MIMO systems, one popular optimization objective for the uplink/downlink precoding design is to maximize the weighted sum rate (WSR) [30]–[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The authors of [30] and [32] discussed the equivalence between the WSR maximization problem and the Weighted sum-Minimum Mean Square Error (WMMSE) problem in MIMO systems and proposed an iteratively downlink transceiver design algorithm for the WSR maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the algorithm relies on the iterative minimization of weighted MSE since the WMMSE problem are not jointly convex over all optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the authors of [31] investigated the UL precoding scheme optimization based on [30] under sum-power-constraint or individual-power-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Motivated by the above observations, we investigate a CF mMIMO system with both multi-antenna APs and UEs over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Two pragmatic processing schemes: 1) 4 the fully centralized processing scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) the large-scale fading decoding scheme are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The main contributions are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We design an efficient UL precoding scheme to maximize the WSR for the fully centralized processing scheme based on an iteratively WMMSE (I-WMMSE) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the design of I-WMMSE precoding scheme for the fully centralized processing scheme is implemented at the CPU and based on the instantaneous CSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the LSFD processing scheme, we derive a UL precoding scheme for the WSR maximization based on an iteratively WMMSE algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The design of I-WMMSE precoding scheme for the LSFD scheme is implemented at the CPU but based only on channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, we compute achievable SE expressions and optimal precoding schemes in novel closed-form for the LSFD scheme with MR combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We analyze the practical implementation and computation complexity for the proposed I-WMMSE precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is found that the proposed I-WMMSE precoding schemes can be guaranteed to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, the proposed UL precoding schemes are efficient to achieve excellent sum SE/rate performance and the average rate benefits from the multiple antennas at the UE-side, which undoubtedly provides vital insights for the practical implementation of multi-antenna UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that this paper differs from the conference version [1] in the following aspects: i) we investigate the fully centralized processing/LSFD schemes and design their respective UL precoding schemes, while only the LSFD scheme was considered in [1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ii) we provide details for the derivation of the I-WMMSE precoding schemes, which are omitted in [1] due to the lack of space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' iii) we analyze the practical im- plementation and convergence behavior of the proposed precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, numerical results show vital insights for the CF mMIMO system with the proposed UL precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In Section II, we consider a CF mMIMO system with the Weichselberger Rayleigh fading channel, and describe the channel estimation and data detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, Section III introduces the fully centralized processing and LSFD processing schemes, and provides their respective achievable SE expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Novel closed-form SE expressions for the LSFD scheme with MR combining are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, based on the achievable SE expressions, we propose UL I- WMMSE precoding schemes for two processing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, Section IV provides some insights for the practical implementation and computation complexity of proposed I-WMMSE precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In Section V, numerical results and performance analysis for the I-WMMSE precoding schemes are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 CPU Fronthaul Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A cell-free massive MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, the major conclusions and future directions are drawn in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Notation: Lowercase letters x and boldface uppercase letters X denote the column vectors and matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' E {·}, tr {·} and ≜ are the expectation operator, the trace operator, and the definitions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' |·|, ∥·∥ and ∥·∥F are the determinant of a matrix or the absolute value of a number, the Euclidean norm and the Frobenious norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' vec (A) denotes a column vector formed by the stack of the columns of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The n×n identity matrix is represented by In×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The Kronecker products and the element-wise products are denoted by ⊗ and ⊙, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, x ∼ NC (0, R) is a circularly symmetric complex Gaussian distribution vector with correlation matrix R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' SYSTEM MODEL In this paper, we investigate a CF mMIMO system consisting of M APs and K UEs, where all APs are connected to one or several CPUs via fronthaul links as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For simplicity, there is only one CPU and all APs serve all UEs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The numbers of antennas per AP and UE are L and N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A standard block fading model is investigated, in which the channel response is constant and frequency flat in a coherence block of τc-length (channel uses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Let τp and τc − τp denote channel uses dedicated for the channel estimation and data transmission, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We denote by Hmk ∈ CL×N the channel response between AP m and UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We assume that Hmk for different AP-UE pairs are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Channel Model Based on the jointly-correlated (also known as the Weichselberger model [29]) Rayleigh fading channel2, Hmk is modeled as Hmk = Umk,r � ˜Ωmk ⊙ Hmk,iid � UH mk,t (1) 1As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1, a more practical network topology is with multiple CPUs and dynamic cooperation clusters, where each UE is only served by a cluster of APs and the APs are grouped into cell-centric clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Each cell-centric cluster is connected to a particular CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2Note that the Rayleigh fading channel is a special case of the Rician fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' And the performance gap between the Rician channel and the Rayleigh channel is small [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, the focus of this paper is not on the channel model but on the UL precoding scheme design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So for the simplicity of analysis, we investigate an essential Rayleigh fading channel by assuming there is no line-of-sight (LoS) link between each UE and AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 0000000000006 where Umk,r = [umk,r,1, · · · , umk,r,L] ∈ CL×L and Umk,t = [umk,t,1, · · · , umk,t,N] ∈ CN×N are the eigen- vector matrices of the one-sided correlation matrices Rmk,r ≜ E � HmkHH mk � and Rmk,t ≜ E � HT mkH∗ mk � , and Hmk,iid ∈ CL×N is composed of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' NC (0, 1) random entries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, we denote by Ωmk ≜ ˜Ωmk ⊙ ˜Ωmk ∈ RL×N the “eigenmode coupling matrix” with the (l, n)-th element [Ωmk]ln specifying the average amount of power coupling from umk,r,l to umk,t,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Hmk can also be formed as Hmk = [hmk,1, · · · , hmk,N] with hmk,n ∈ CL being the channel between AP m and n-th antenna of UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By stacking the columns of Hmk on each other, we define hmk ≜ vec (Hmk) = [hT mk,1, · · · , hT mk,N]T ∼ NC (0, Rmk), where Rmk ≜ E{hmkhH mk} is the full correlation matrix Rmk = (U∗ mk,t ⊗ Umk,r)diag (vec (Ωmk)) (U∗ mk,t ⊗ Umk,r)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (2) Moreover, note that Rmk can be structured into the block form as [28] with the (n, i)-th submatrix being Rni mk = E{hmk,nhH mk,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the large-scale fading coefficient βmk can be extracted from Rmk as βmk = 1 LN tr (Rmk) = 1 LN ∥Ωmk∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is worth mentioning that the motivations for adopting the Weichselberger channel model are: 1) The Weichselberger model investigated in (1) not only captures the correlation features at both the AP-side and UE-side but models the joint correlation dependence between each AP-UE pair through the coupling matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) The coupling matrix Ωmk reflects the practical spatial arrangement of scattering objects between AP m and UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More significantly, the Weichselberger model can reduce to most channel models of great interest by adjusting the coupling Ωmk to particular formulation, such as the Kronecker model and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Rayleigh fading model [28], [29];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3) Compared with other stochastic channel models, the Weichselberger model displays significantly less modeling error, which is validated based on the practical measurement in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Channel Estimation For the channel estimation, mutually orthogonal pilot matrices are constructed and each pilot matrix is composed of N mutually orthogonal pilot sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We denote by Φk the pilot matrix assigned to UE k with ΦH k Φl = τpIN, if l = k and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' And Pk is the index subset of UEs using the same pilot matrix as UE k including itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When all UEs transmit their pilot matrices, the received signal at AP m Yp mk ∈ CL×τp is Yp m = �K k=1 HmkFk,pΦT k + Np m, where Fk,p ∈ CN×N is the precoding matrix for UE k under the phase of pilot transmission, Np m ∈ CL×τp is the additive noise at AP m with independent NC(0, σ2) entries and σ2 being the noise power, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The pilot transmission should be implemented under the power constraint as tr(Fk,pFH k,p) ⩽ pk, where pk is the maximum transmit power for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To derive sufficient statistics for hmk, AP m projects Yp mk onto Φ∗ k as Yp mk = Yp mΦ∗ k = 7 �K l=1 HmlFl,p � ΦT l Φ∗ k � +Np mΦ∗ k = � l∈Pk τpHmlFl,p + Qp mk, where Qp mk ≜ Np mΦ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, following the standard MMSE estimation steps in [35] and [36], AP m can compute the MMSE estimation of hmk as ˆhmk = vec( ˆHmk) = Rmk˜FH k,pΨ−1 mkyp mk, (3) where ˆHmk is the MMSE estimation of Hmk, ˜Fk,p = FT k,p⊗IL, yp mk ≜ vec (Yp mk) = � l∈Pk τp˜Fl,phml + qp m, qp m = vec (Qp mk) and Ψmk = � l∈Pk τp˜Fl,pRml˜FH l,p + σ2ILN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the estimate ˆhmk and estimation error ˜hmk = hmk − ˆhmk are independent random vectors distributed as ˆhmk ∼ NC(0, ˆRmk) and ˜hmk ∼ NC(0, Cmk), where ˆRmk ≜ τpRmk˜FH k,pΨ−1 mk˜Fk,pRmk and Cmk ≜ Rmk − ˆRmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We can also form ˆRmk and Cmk in the block structure with the (n, i)-th submatrix being ˆRni mk = E{ˆhmk,nˆhH mk,i} and Cni mk = E{˜hmk,n˜hH mk,i}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Data Transmission For the data transmission, all antennas of all UEs simultaneously transmit their data symbols to all APs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The received signal ym ∈ CL at AP m is ym = K � k=1 Hmksk + nm, (4) where nm ∼ NC(0, σ2IL) is the independent receiver noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The transmitted signal from UE k sk ∈ CN can be constructed as sk = Fk,uxk, where xk ∼ NC(0, IN) is the data symbol for UE k and Fk,u ∈ CN×N is the precoding matrix for the data transmission which should satisfy the power constraint of UE k as tr(Fk,uFH k,u) ⩽ pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' SPECTRAL EFFICIENCY ANALYSIS AND I-WMMSE PRECODING DESIGN In this section, we investigate two promising signal processing schemes, called “fully centralized pro- cessing” and “LSFD processing”, and analyze their corresponding SE performance and design respective iteratively WMMSE precoding schemes3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Fully Centralized Processing 1) Spectral Efficiency Analysis: For the fully centralized processing scheme, all M APs send all the received pilot signals and data signals to the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Indeed, both the channel estimation and data detection are implemented at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The collective channel hk ∈ CMLN for UE k can be constructed as hk = [vec(H1k)T, · · · , vec(HMk)T]T ∼ NC(0, Rk) with Rk = diag (R1k, · · · , RMk) ∈ CMLN×MLN being the 3We only optimize the precoding matrices for the phase of data transmission Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The optimization of Fk,p is left for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Although we do not design Fk,p in this paper, we try to keep the derived equations more generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So a scenario with arbitrary Fk,p instead of limiting Fk,p to a particular form is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is worth noting that all equations in this paper hold for any Fk,p so undoubtedly provide some important guidelines for the investigation of optimization design for Fk,p in the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8 whole block-diagonal correlation matrix for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Similar to (3), the CPU can derive the channel estimate for UE k as4 ˆhk ≜ � ˆhT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ˆhT Mk �T ∼ NC � 0, τpRk¯FH k,pΨ−1 k ¯Fk,pRk � where ¯Fk,p = diag(˜Fk,p, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ˜Fk,p � �� � M ) and Ψ−1 k = diag(Ψ−1 1k , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Ψ−1 Mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The channel estimation error is ˜hk ∼ NC (0, Ck) where Ck ≜ Rk − τpRk¯FH k,pΨ−1 k ¯Fk,pRk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the received data signal at the CPU can be denoted as [yT 1 , · · · , yT M]T � �� � = y = K � k=1 [HT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , HT Mk]T � �� � = Hk Fk,uxk + [nT 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , nT M]T � �� � = n , (5) or a compact form as y = �K k=1 HkFk,uxk + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Under the setting of “fully centralized processing”, we assume that UL precoding matrices (Fk,u and Fk,p) are available at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on the collective channel estimates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' the CPU designs an arbitrary receive combining matrix Vk ∈ CLM×N for UE k to detect xk as ˇxk = VH k y = VH k ˆHkFk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uxk + VH k ˜HkFk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uxk + K � l̸=k VH k HlFl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uxl + VH k n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (6) and the conditional MSE matrix for UE k is Ek,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = E{(xk − ˇxk)(xk − ˇxk)H|{ ˆHk},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' {Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u}} = IN − VH k ˆHkFk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u − FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ˆHH k Vk + VH k � K � l=1 � ˆHl¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ˆHH l + C′ l � + σ2IML � Vk (7) where ¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ≜ Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uFH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C′ l ≜ diag (C′ 1l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' C′ Ml) ∈ CML×ML and C′ ml = E{ ˜Hml¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u ˜HH ml} ∈ CL×L with the (j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' q)-th element of C′ ml being [C′ ml]jq = �N p1=1 �N p2=1 �¯Fl � p2p1 [Cp2p1 ml ]jq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By implementing the per-user-basis minimum mean-squared error-based successive interference cancel- lation (MMSE-SIC) detector while treating co-user interference as uncorrelated Gaussian noise, we derive the achievable SE for UE k as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' An achievable for UE k under the setting of “fully centralized processing” with the MMSE estimator is SEk,(1) = � 1 − τp τc � E � log2 ���IN + DH k,(1)Σ−1 k,(1)Dk,(1) ��� � , (8) where Dk,(1) ≜ VH k ˆHkFk,u and Σk,(1) ≜ VH k ��K l=1 ˆHl¯Fl,u ˆHH l − ˆHk¯Fk,u ˆHH k + �K l=1 C′ l + σ2IML � Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4Note that the pilot signals received at the APs are first transmitted to the CPU and then the CPU estimates the channels, where τpML complex scalars are sent from the APs to the CPU at each coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Alternatively, all APs can first estimate the channels as (3), and then send their channel estimates to the CPU, where MKLN complex scalars are sent from the APs to the CPU at each coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since the pilot contamination is investigated (τp < KN) in this paper, we consider the first transmission protocol due to its lower fronthaul overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 The expectations are with respect to all sources of randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof follows from the similar approach as [28, Corollary 1] and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that Corollary 1 holds for any combining schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One promising combining scheme is the MMSE combining as VMMSE k = � K � l=1 � ˆHl¯Fl,u ˆHH l + C′ l � + σ2IML �−1 ˆHkFk,u, (9) which can minimize the mean-squared error MSEk,(1) = tr(Ek,(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' With the MMSE combining scheme, the conditional MSE matrix in (7) is Eopt k,(1) = IN − FH k,u ˆHH k � K � l=1 � ˆHl¯Fl,u ˆHH l + C′ l � + σ2IML �−1 ˆHkFk,u (10) More importantly, the MMSE combining in (9) can also maximize the achievable SE in (8) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The achievable SE for UE k in (8) can be maximized by the MMSE combining scheme in (9) with the maximum value SEopt k,(1) = � 1 − τp τc � E \uf8f1 \uf8f2 \uf8f3log2 ������ IN + FH k,u ˆHH k � K � l=1 � ˆHl¯Fl,u ˆHH l + C′ l � − ˆHk¯Fk,u ˆHH k + σ2IML �−1 ˆHkFk,u ������ \uf8fc \uf8fd \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof can be found in [28, Appendix B] and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) Iteratively WMMSE Precoding Design: In this part, we design the uplink precoding scheme for the “fully centralized processing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One popular weighted sum-rate maximization problem is investigated as5 max {F} K � k=1 µk,(1)SEk,(1) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (12) where µk,(1) represents the priority weight of UE k and SEk,(1) is given by (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5The notation F is short for {Fk,u}k=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=',K, denoting all variables Fk,u with k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Similar definitions are applied for V, A, W, S in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In this section, we denote by Fk,u,(1) and Fk,u,(2) the UL precoding matrix of UE k for the fully centralized processing and LSFD scheme, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10 As [30] and [32], the matrix-weighted sum-MSE minimization problem as min {F,V,W} K � k=1 µk,(1) � tr � Wk,(1)Ek,(1) � − log2 ��Wk,(1) ��� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (13) is equivalent to the weighted sum-rate maximization problem (12), where Wk,(1) ∈ CN×N is the weight matrix for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that (13) is convex over each optimization variable F, V, W but is not jointly convex over all optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Following the method in [30], we can solve (13) by sequentially fixing two of the three optimization variables F, V, W and updating the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Fixing the other variables, the update of Vk is given by the MMSE solution as (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Under the MMSE combining, the MSE matrix is given by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, note that optimal Wk,(1) for (13) is Wopt k,(1) = E−1 k,(1), (14) which can be easily derived through the first order optimality condition for Wk,(1) by fixing F and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When the MMSE combining VMMSE k and Wopt k,(1) for all UEs are implemented in (13), we have tr(Wk,(1)Ek,(1))−log2 ��Wk,(1) �� = tr (IN)−log2 |(Eopt k,(1))−1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the matrix-weighted sum-MSE minimization problem in (13) would reduce to the equivalent optimization problem of (12) as6: max {F} K � k=1 µk,(1) log2 ���� � Eopt k,(1) �−1���� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (15) which is a well-known relationship between Eopt k,(1) and SEopt k,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, fixing V and W, the update of Fk,u,(1) for (13) results in the optimization problem as7 6Note that “SE” is equivalent to “rate” except from having one scaling factor (τc − τp)/τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since τc and τp are constants in this paper, so we ignore the difference between SE and rate in the optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7It is worth mentioning that the updates of optimization variables are based on the preliminary of fixing the other optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, when updating Fk,u,(1), we should fix the other optimization variables (Vk and Wk,(1)) but not only limited to their respective optimal solutions VMMSE k and Wopt k,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we update Fk,u,(1) based on (16) with generalized Vk and Wk,(1) instead of (15) with optimal VMMSE k and Wopt k,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 11 min {F} K � k=1 µk,(1)tr � Wk,(1) � IN − VH k ˆHkFk,u,(1) � � IN − VH k ˆHkFk,u,(1) �H� + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l̸=k ˆHlFl,u,(1)FH l,u,(1) ˆHH l � Vk � − K � k=1 µk,(1) log2 ��Wk,(1) �� + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l=1 E � ˜HlFl,u,(1)FH l,u,(1) ˜HH l ��� F � + σ2IML � Vk � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(1) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (16) which is a convex quadratic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the classic Lagrange multipliers methods and Karush-Kuhn-Tucker (KKT) conditions can be applied to derive an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The Lagrange function of (16) is f � F1,u,(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , FK,u,(1) � = K � k=1 µk,(1)tr � Wk,(1) � IN − VH k ˆHkFk,u,(1) � � IN − VH k ˆHkFk,u,(1) �H� + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l̸=k ˆHlFl,u,(1)FH l,u,(1) ˆHH l � Vk � + K � k=1 µk,(1)tr � Wk,(1)VH k � K � l=1 E � ˜HlFl,u,(1)FH l,u,(1) ˜HH l ��� F � + σ2IML � Vk � + K � k=1 λk,(1) � tr � Fk,u,(1)FH k,u,(1) � − pk � (17) Finally, we derive the optimal precoding scheme as the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By fixing other optimization variables and applying the first-order optimality condition of (17) with respect to each Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' the optimal precoding scheme is given by Fopt k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � K � l=1 µl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � ˆHH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˆHk + E � ˜HH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˜Hk ��� V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W �� + λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)IN �−1 ˆHH k VkWk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = µk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � K � l=1 µl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � ˆHH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˆHk + ¯Ckl � + λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)IN �−1 ˆHH k VkWk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (18) where λk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) ⩾ 0 is the Lagrangian multiplier and the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' n)-th element of ¯Ckl ≜ E{ ˜HH k VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l ˜Hk|V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' W} ∈ CN×N is �¯Ckl � in = tr( ¯VlE{˜hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='n˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i}) = tr �¯VlCk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='in � with ¯Vl ≜ VlWl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1)VH l and Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ni ≜ E{˜hk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='n˜hH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i} = 12 diag (Cni 1k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Cni Mk) ∈ CML×ML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' According to the KKT condition, λk,(1) and Fk,u,(1) should also satisfy ��Fk,u,(1) ��2 ⩽ pk, λk,(1) ���Fk,u,(1) ��2 − pk � = 0, λk,(1) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (19) Proof: The proof is given in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We denote by Fk,u,(1)(λk,(1)) the right-hand side of (18), when �K l=1 µl,(1)( ˆHH k VlWl,(1)VH l ˆHk + ¯Ckl) is invertible and tr[Fk,u,(1)(0)Fk,u,(1)(0)H] ⩽ pk, then Fopt k,u,(1) = Fk,u,(1) (0), otherwise we have tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] = pk to satisfy (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] is a monotonically decreasing function of λk,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof: Let DΛDH denote the eigendecomposition of �K l=1 µl,(1)( ˆHH k VlWl,(1)VH l ˆHk + ¯Ckl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Fol- lowing the method in [30], we define Φ = µ2 k,(1)DH ˆHH k VkW2 k,(1) ˆHkVH k D and we have tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] = tr �� DΛDH + λk,(1)IN �−1 DΦDH � DΛDH + λk,(1)IN �−1� = tr �� DΛDH + λk,(1)IN �−2 DΦDH� = tr �� Λ + λk,(1)IN �−2� = N � n=1 [Φ]nn � [Λ]nn + λk,(1) �2, (20) so tr[Fk,u,(1)(λk,(1))Fk,u,(1)(λk,(1))H] is a monotonically decreasing function of λk,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on Corollary 3, optimum λk,(1) (denoted by λopt k,(1)) can be easily obtained by a one-dimensional (1-D) bisection algorithm so we derive the solution for Fk,u,(1)(λopt k,(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, an iterative opti- mization algorithm for Fk,u,(1), called “iteratively WMMSE (I-WMMSE) algorithm”, is summarized in Algorithm 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The convergence of Algorithm 1 is proven in [30, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the design of Fk,p is a valuable future direction to further improve the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' One valuable optimization problem is to minimize the total MSE of the channel estimators of all UEs as min {Fk,p} K � k=1 tr (Ck) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ∥Fk,p∥2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (21) where the optimization goal is only based on the statistical knowledge so Fk,p is also based on the statistical knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8To balance the efficiency and the computational complexity of the proposed algorithm, we also include the stopping criterion “R(i) (1) < R(i−1) (1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the I-WMMSE precoding scheme is derived at iteration (i − 1), which may achieve higher sum SE than the one at iteration i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 13 Algorithm 1: I-WMMSE Algorithm for the Design of Fk,u,(1) Input: Collective channel estimates ˆHk for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Estimation error covariance matrices Cml for all possible pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' UE weights µk,(1) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Output: Optimal precoding matrices Fk,u,(1) for all UEs (F(i) k,u,(1) for the first or third stopping criterion and F(i−1) k,u,(1) for the second stopping criterion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 Initiation: i = 0, F(0) k,u,(1) and R(0) (1) = �K k=1 µk,(1)SE(0) k,(1) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' maximum iteration number I(1),max and threshold ε(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 repeat 3 i = i + 1 4 Update the MMSE combining scheme V(i) k with F(i−1) k,u,(1) based on (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 Update optimal MSE matrix E(i) k,(1) with F(i−1) l,u,(1) based on (10), and update W(i) k,(1) based on (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6 Update optimal precoding matrix F(i) l,u,(1) with V(i) k and W(i) k,(1) based on (18), where λ(i) k,(1) is found by a bisection algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7 Update sum weighted rate R(i) (1) = �K k=1 µk,(1)SE(i) k,(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8 until ���R(i) (1) − R(i−1) (1) ��� /R(i−1) (1) ⩽ ε(1) or R(i) (1) < R(i−1) (1) or i ⩾ I(1),max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Large-Scale Fading Decoding 1) Spectral Efficiency Analysis: Another promising processing scheme is “large-scale fading decoding”, which is a two-layer decoding scheme to decode the data symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that UL precoding matrices (Fk,u and Fk,p) are assumed to be available at all APs and the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In the first layer, AP m applies an arbitrary combining matrix Vmk ∈ CL×N to derive local detection of xk as ˜xmk = VH mkym = VH mkHmkFk,uxk + K � l=1,l̸=k VH mkHmlFl,uxl + VH mknm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (22) We notice that Vmk is designed based on local channel estimates at AP m and one handy choice is MR combining Vmk = ˆHmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, local MMSE (L-MMSE) combining Vmk = � K � l=1 � ˆHml¯Fl,u ˆHH ml + C′ ml � + σ2IL �−1 ˆHmkFk,u, (23) is also regarded as a promising scheme, since (23) can minimize E{∥ xk − VH mkym ∥2 |{ ˆHmk}, {Fk,u}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In the second layer, the “LSFD” method is implemented at the CPU [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The CPU weights all the local estimates ˜xmk from all APs by the LSFD coefficient matrix as ˆxk = M � m=1 AH mk˜xmk = M � m=1 AH mkVH mkHmkFk,uxk + M � m=1 K � l=1,l̸=k AH mkVH mkHmlFl,uxl+n′ k, (24) where Amk ∈ CN×N is the complex LSFD coefficient matrix for AP m-UE k and n′ k = �M m=1 AH mkVH mknm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 14 Moreover, we can rewrite ˆxk in a more compact form as ˆxk = AH k GkkFk,uxk + K � l=1,l̸=k AH k GklFl,uxl + n′ k = AH k � GkkFk,uxk + K � l=1,l̸=k GklFl,uxl + ˜n′ k � � �� � ˜xk (25) where Ak ≜ [AT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , AT Mk]T ∈ CMN×N, Gkl ≜ [VH 1kH1l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' VH MkHMl] ∈ CMN×N and ˜n′ k = � VH 1kn1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' VH MknM � ∈ CMN×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the CPU does not have the knowledge of channel estimates and is only aware of channel statistics [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The conditional MSE matrix for UE k Ek,(2) ≜ E � (xk − ˆxk) (xk − ˆxk)H |Θ � is Ek,(2) = IN − FH k,uE{GH kk}Ak − AH k E{Gkk}Fk,u + AH k � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk � Ak, (26) where Θ denotes all the channel statistics and Sk = diag(E{VH 1kV1k}, · · · , E{VH MkVMk}) ∈ CMN×MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, we apply classical use-and-then-forget bound to obtain the following ergodic achievable SE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the “LSFD” scheme, an achievable SE for UE k can be written as SEk,(2) = � 1 − τp τc � log2 ���IN + DH k,(2)Σ−1 k,(2)Dk,(2) ��� , (27) where Σk,(2) = �K l=1 AH k E{Gkl¯Fl,uGH kl}Ak − Dk,(2)DH k,(2) + σ2AH k SkAk and Dk,(2) = AH k E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof follows similar steps as the proof of [28, Corollary 2] and is therefore omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that Ak can be optimized by the CPU based on channel statistics to maximize the achievable SE in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on the theory of optimal receivers as in [37], we derive the optimal LSFD coefficient matrix, which not only maximizes the achievable SE but minimizes the conditional MSE, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The achievable SE in (27) is maximized by Aopt k = � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk �−1 E{Gkk}Fk,u, (28) leading to the maximum value as SEopt k,(2) = � 1 − τp τc � log2 ������ IN + FH k,uE {Gkk} � K � l=1 E � Gkl¯Fl,uGH kl � − E {Gkk} ¯Fk,uE � GH kk � + σ2Sk �−1 E {Gkk} Fk,u ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (29) 15 Note that the optimal LSFD coefficient matrix in (28) can also minimize the conditional MSE for UE k MSEk,(2) = tr(Ek,(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The proof is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' If the optimal LSFD coefficient matrix is applied, the MSE matrix for UE k can be written as Eopt k,(2) = IN − FH k,uE � GH kk � � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk �−1 E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (31) Furthermore, if MR combining Vmk = ˆHmk is applied, we derive closed-form SE expressions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For MR combining Vmk = ˆHmk, (27) can be computed in closed-form as SEk,(2),c = � 1 − τp τc � log2 ���IN + DH k,(2),cΣ−1 k,(2),cDk,(2),c ��� , (32) where Σk,(2),c = AH k (�K l=1 Tkl,(1) + � l∈Pk Tkl,(2))Ak − Dk,(2),cDH k,(2),c + σ2AH k Sk,cAk and Dk,(2),c = AH k ZkFk,u, with E{Gkk} = Zk = [ZT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ZT Mk]T and Sk,c = diag(Z1k, · · · , ZMk) with the (n, n′)- th element of Zmk ∈ CN×N being [Zmk]nn′ = tr(ˆRn′n mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) ≜ diag(Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M) ∈ CMN×MN and Tmm′ kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) = Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m − Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m if m = m′ and Λmkl¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='uΛm′lk otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' where Tmm′ kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) de- notes (m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m′)-submatrix of Tkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ∈ CMN×MN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' the (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' n′)-th element of N × N-dimension complex matrices Λmkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Λm′lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Γ(1) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m and Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m are [Λmkl]nn′ = tr(Ξn′n mkl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [Λm′lk]nn′ = tr(Ξn′n m′lk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' [Γ(1) mkl]nn′ = �N i=1 �N i′=1 [¯Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='u]i′itr(Ri′i ml ˆRn′n mk) and [Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m]nn′ given by � Γ(2) kl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='m � nn′ = N � i=1 N � i′=1 �¯Fl � i′i � tr � Ri′i mlPn′n mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � +τ 2 p N � q1=1 N � q2=1 � tr � ˜Pq1n mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Ri′q2 ml ˜Rq2i ml ˜Pn′q1 mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) � + tr � ˜Pq1n mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Ri′q2 ml � tr � ˜Pn′q2 mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rq2i ml ��� (33) with Ξmkl = τpRml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 mk˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRmk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ξm′lk = τpRm′k˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 m′k˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRm′l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Pmkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = τpSmk(Ψmk − τp˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='p)SH mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Smk = Rmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Pmkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) = Smk˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pSH mk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ˜Rni ml and ˜Pni mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) being (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' i)- submatrix of R 1 2 ml and P 1 2 mkl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, the optimal LSFD coefficient matrix in (28) and MSE matrix in (31) can also be computed in closed-form as \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Aopt k,c = ��K l=1 Tkl,(1) + � l∈Pk Tkl,(2) + σ2Sk,c �−1 ZkFk,u, Eopt k,(2),c = IN − FH k,uZH k ��K l=1 Tkl,(1) + � l∈Pk Tkl,(2) + σ2Sk,c �−1 ZkFk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (34) Proof: The proof is given in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 16 2) Iteratively WMMSE Precoding Design: For the LSFD scheme, we also investigate a weighted sum- rate maximization problem as max {F} K � k=1 µk,(2)SEk,(2) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (35) where µk,(2) represents the priority weight of UE k for the “LSFD” scheme and SEk,(2) is given in (27) with arbitrary combining structure in the first decoding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Similarly, the matrix-weighted sum-MSE minimization problem as9 min {F,A,W,G,S} K � k=1 µk,(2) � tr � Wk,(2)Ek,(2) � − log2 ��Wk,(2) ��� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (36) is equivalent to the weighted sum-rate maximization problem (35), where Wk,(2) is the weight matrix for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that (36) is convex over each optimization variable F, A, W, G, S but is not jointly convex over all optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we can solve (36) by sequentially fixing four of the five optimization variables F, A, W, G, S and updating the fifth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='10 The update of Ak and Ek,(2) are given by the optimal LSFD scheme (28) and MSE matrix with optimal LSFD scheme (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that optimal Wk,(2) for (36) is Wopt k,(2) = E−1 k,(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When Aopt k and Wopt k,(2) for all UEs are applied in (36), we notice that (36) becomes to the equivalent optimization problem of (35) as max {F,G,S} K � k=1 µk,(2) log2 ���� � Eopt k,(2) �−1���� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (37) which is a well-known relationship between Eopt k,(2) and SEopt k,(2) and proven in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Last but not least, fixing other variables, the update of Fk,u,(2) for (36) results in the optimization 9The notation G denotes all G-relevant variables, like E{Gkl¯Fl,u,(2)GH kl} and E{Gkk}, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10As for G and S, if L-MMSE combining scheme applied, E {Gkk} and Sk are relevant to Fk,u,(2) so we should also update them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' On the contrary, E{Gkk} and Sk with MR combining structure are irrelevant to F so we only need to update E{Gkl¯Fl,u,(2)GH kl}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 17 problem as min {F} K � k=1 µk,(2) � tr � Wk � IN − FH k,u,(2)E � GH kk � Ak � � IN − FH k,u,(2)E � GH kk � Ak �H�� + K � k=1 µk,(2) � tr � Wk,(2)AH k � K � l̸=k E � Gkl¯Fl,u,(2)GH kl � + σ2Sk � Ak �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ��Fk,u,(2) ��2 ⩽ pk ∀k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , K (38) which is a convex quadratic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Thus, we can also derive the optimal precoding scheme by applying classic Lagrange multipliers methods and KKT conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The Lagrange function of (38) is f � F1,u,(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , FK,u,(2) � = K � k=1 µk,(2) � tr � Wk,(2) � IN − FH k,u,(2)E � GH kk � Ak � � IN − FH k,u,(2)E � GH kk � Ak �H�� + K � k=1 µk,(2) � tr � Wk,(2)AH k � K � l̸=k E � Gkl¯Fl,u,(2)GH kl � + σ2Sk � Ak �� + K � k=1 λk,(2) � tr � Fk,u,(2)FH k,u,(2) � − pk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (39) Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' By applying the first-order optimality condition of (39) with respect to each Fk,u,(2) and fixing other optimization variables, we obtain the optimal precoding scheme as Fopt k,u,(2) = µk,(2) � K � l=1 µl,(2)E � GH lkAlE−1 l,(2)AH l Glk � + λk,(2)IN �−1 E � GH kk � AkE−1 k,(2), (40) where λk,(2) ⩾ 0 is the Lagrangian multiplier during the phase of “LSFD” scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' According to the KKT condition, λk,(2) and Fk,u,(2) should also satisfy ��Fk,u,(2) ��2 ⩽ pk, λk,(2) ���Fk,u,(2) ��2 − pk � = 0, λk,(2) ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (41) Note that when �K l=1 µl,(2)E{GH lkAlE−1 l,(2)AH l Glk} is invertible and tr � Fk,u,(2)(0)Fk,u,(2)(0 �H] ⩽ pk, then Fopt k,u,(2) = Fk,u,(2) (0), otherwise we must have tr[Fk,u,(2)(λk,(2))Fk,u,(2)(λk,(2))H] = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Following the similar method in Corollary 3, we notice that λk,(2) can be easily found by a 1-D bisection algorithm since tr[Fk,u,(2)(λk,(2))Fk,u,(2)(λk,(2))H] is a monotonically decreasing function of λk,(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, if MR combining Vmk = ˆHmk is applied in the first layer, we can compute expectations in (40) in closed-form as following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' With MR combining Vmk = ˆHmk and the optimal LSFD scheme applied, we can compute 18 E{GH kk}, Aopt k , and Eopt k,(2) in closed-form as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' we have ¯Tlk = E{GH lkAlE−1 l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2)AH l Glk} ∈ CN×N where the (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' n)-th element of ¯Tlk is tr(¯Al ¯Glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ni) with ¯Al ≜ AlE−1 l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2)AH l and the [(m − 1) N + p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (m′ − 1) N + p′]-th (or [o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' j]-th briefly) entry of ¯Glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ni ≜ E{glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ngH lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i} ∈ CMN×MN being E{glk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='ngH lk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i}oj = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l /∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m ̸= m′ tr(Rni mk ˆRp′p ml),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l /∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m = m′ tr(Ξnp mlk)tr(Ξp′i m′kl),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l ∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m ̸= m′ tr � Rni mkPp′p mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) � + τ 2 p �N q1=1 �N q2=1 tr � ˜Pq1p mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rnq2 mk ˜Rq2i mk ˜Pp′q1 mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) � +τ 2 p �N q1=1 �N q2=1 tr � ˜Pq1n mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rnq1 mk � tr � ˜Pp′q2 mlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) ˜Rq2i mk � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' l ∈ Pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' m = m′ (42) where Ξmlk = τpRmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 mk˜Fl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRml,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Ξm′kl = τpRm′l˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 m′l˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRm′k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sml = Rml˜FH l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pΨ−1 ml,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Pmlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(1) = τpSml(Ψml − τp˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='p)SH ml and Pmlk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(2) = Sml˜Fk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pRmk˜FH k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='pSH ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Plugging the derived results into (40), we can compute Fopt k,u,(2) in closed-form as Fopt k,u,(2),c = µk,(2) � K � l=1 µl,(2) ¯Tlk + λk,(2)IN �−1 ZH k Aopt k,c Eopt,−1 k,(2),c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (43) Proof: The proof is given in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, an iterative optimization algorithm for Fk,u,(2) is summarized in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The con- vergence of Algorithm 2 is proven in [30, Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Relying on the iterative minimization of weighted MSE, two efficient uplink I-WMMSE precoding schemes to maximize the weighted sum SE are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding schemes for the “FCP” and “LSFD” schemes are investigated in Algorithm 1 and Algorithm 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the design of I-WMMSE precoding scheme for the FCP/LSFD is based on instantaneous CSI/channel statistics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, we can compute I-WMMSE precoding schemes in novel closed- form only for the LSFD scheme with MR combining based on Theorm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' PRECODING IMPLEMENTATION AND COMPLEXITY ANALYSIS In this section, we discuss the practical implementation and analyze computational complexity for the UL precoding schemes investigated in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Precoding Implementation 1) Precoding Characteristics: As described above, we investigate a standard block fading model, where the channel response is constant and frequency flat in a coherence block, which contains τc channel 19 Algorithm 2: I-WMMSE Algorithm for the Design of Fk,u,(2) Input: Channel statistics Θ for all possible pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' UE weights µk,(2) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Output: Optimal precoding matrices Fk,u,(2) for all UEs (F(i) k,u,(2) for the first or third stopping criterion and F(i−1) k,u,(2) for the second stopping criterion);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 Initiation: i = 0, F(0) k,u,(2) and R(0) (2) = �K k=1 µk,(2)SE(0) k,(2) for all UEs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' maximum iteration number I(2),max and threshold ε(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 repeat 3 i = i + 1 4 Update channel statistics Θ(i), such as E{G(i) kk}, E{G(i) kl ¯F(i−1) l,u (G(i) kl )H} and S(i) k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 Update optimal LSFD matrix A(i) k with F(i−1) l,u,(2) and Θ(i) based on (28);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6 Update optimal MSE matrix E(i) k,(2) with F(i−1) l,u,(2), A(i) k and E{G(i) kk} based on (31) and update W(i) k,(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7 Update optimal precoding matrix F(i) k,u,(2) with A(i) k , W(i) k,(2) and Θ(i) based on (40), where λ(i), k,(2) is found by a bisection algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8 Update sum weighted rate R(i) (2) = �K k=1 µk,(2)SE(i) k,(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 until |R(i) (2) − R(i−1) (2) |/R(i−1) (2) ⩽ ε(2) or R(i) (2) < R(i−1) (2) or i ⩾ I(2),max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the “fully centralized processing” scheme, we notice that the I-WMMSE precoding design is implemented at the CPU based on the instantaneous CSI as (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, to guarantee the convergence of Algorithm 1, only MMSE combining as (9) is advocated to detect the UL data since the equivalent relationship between Eopt k,(1) and SEopt k,(1), which only satisfies with MMSE combining, should be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the LSFD scheme, the optimal design of Fopt k,u,(2) as (40) can only be implemented at the CPU, but relies only on channel statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, L-MMSE or MR combining can be applied at each AP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When MR combining is applied, all terms in Algorithm 2 can be computed in closed-form as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2) Fronthaul Requirements: For the FCP scheme with the I-WMMSE precoding, in each coherence block, all APs should relay their received signals to the CPU and the CPU requires precoding matrices Fk,u,(1) feedback to all UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' All APs need to send τcML complex scalars (τpML complex scalars for the pilot signals and (τc − τp)ML complex scalars for the received data signals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the full correlation matrices {Rmk} are available at the CPU, which contains MKL2N2/2 complex scalars for each realization of the AP/UE locations/statistics11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the CPU transmits optimal precoding matrices to all UEs, which are described by KN2 complex scalars per coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In summary, for the FCP scheme with the I-WMMSE precoding implemented, total τcMLNr + MKL2N2/2 + KN2Nr complex scalars are transmitted via fronthaul links for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For comparison, when 11Note that the channel statistics remain constant for each realization of the AP/UE locations and each realization of the AP/UE locations contains Nr channel realizations (coherence blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 20 the FCP scheme without the I-WMMSE precoding is implemented, all APs should also transmit τcML complex scalars for the received signals to the CPU in each coherence block and MKL2N2/2 complex scalars for {Rmk} to the CPU for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So for the CFP scheme without the I-WMMSE precoding, total τcMLNr + MKL2N2/2 complex scalars are transmitted via fronthaul links for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the LSFD scheme with the I-WMMSE precoding, all APs transmit their local data estimates ˜xmk, described by (τc − τp)MKN complex scalars, to the CPU per coherence block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, E{Gkk} ∈ CMN×N, described by MKN2 complex scalars for each realization of the AP/UE locations, are also required at the CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for E{Gkl¯Fl,u,(2)GH kl} ∈ CMN×MN, following the formulation method investigated in Appendix F, the optimization of ¯Fl,u,(2) requires the knowledge of � E � vH ml,phmk,nhH m′k,ivm′l,p′�� , described by M2K2N4/2 complex scalars for each realization of the AP/UE locations, where vml,p denotes the p-th column of Vml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the CPU requires optimal precoding matrices Fk,u,(2) feedback to all APs and UEs only for each realization of the AP/UE locations, which are KN2 complex scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the LSFD scheme without the I-WMMSE precoding, local data estimates ˜xmk, described by (τc − τp)MKN complex scalars per coherence block, E{Gkk}, described by MKN2 complex scalars for each realization of the AP/UE locations, and E{GklGH kl} ∈ CMN×MN, described by M2K2N2/2 complex scalars for each realization of the AP/UE locations, are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' That is total (τc −τp)MKNNr + MKN2 + M2K2N2/2 complex scalars transmitted via fronthaul links for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3) Practical Implementation: Note that the basic motivation of the investigated I-WMMSE precoding schemes is to achieve as good the sum uplink SE performance as possible so we ignore some practical issues, which are vital for the realistic implementation of the investigated precoding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' When the precoding schemes are implemented in practice, these realistic issues should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Capacity-constrained fronthaul network As discussed above, the I-WMMSE precoding require more fronthaul requirements than the case without the I-WMMSE precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' It is quite vital to consider a more practical capacity-constrained fronthaul network [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the wireless fronthaul [39], which is more flexible than the conventional wire fronthaul, would also be regarded as a promising solution to boost the practical implementation of the I-WMMSE precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Scalability aspects with dynamic cooperation clusters When the precoding schemes are implemented in practice, a more realistic network architecture with 21 multiple CPUs and dynamic cooperation clusters should be advocated, where each UE is only served by a cluster of APs (that a is user-centric cluster) and the APs are grouped into cell-centric clusters as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note a user-centric cluster might consist of APs connecting with different CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on the signal processing schemes in [9], [18], the analytical framework in this paper can be implemented in a scalable paradigm where the fronthaul requirements and computational complexity can be relieved with an anticipated modest performance loss compared with canonical architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding design with these two practical aspects is left in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To bring valuable technical insights for the study of I-WMMSE precoding schemes with the DCC strategy and the capacity-constrained fronthaul link, we provide two tutorials for the FCP and LSFD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 based on [9], [10], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tutorials to investigate the I-WMMSE precoding scheme with the DCC strategy and the capacity-constrained fronthaul 1: Joint initial access, pilot assignment, and cluster formation for the DCC topology based on a classical algorithm in [9, Sec V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A] or a more efficient algorithm as [10, Algorithm 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2: Each AP transmits the quantized versions of the local detection signals in (22) to the CPU based on Case 2 in [38] called "Quantized Weighted Signal Available at the CPU" as [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (20)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (1), generate the DCC based processing scheme for the LSFD (the local combining design, P-LSFD, and achievable SE computation) motivated by [10, Sec Ⅱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (2), formulate the I-WMMSE precoding design optimization problem with a capacity-constrained fronthaul motivated by [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (24)] and [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (26)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5: Obtain the optimal precoding scheme based on potential methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tutorial 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding for the LSFD with the DCC strategy and the capacity-constrained fronthaul 1: Joint initial access, pilot assignment, and cluster formation for the DCC topology based on a classical algorithm in [9, Sec V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A] or a more efficient algorithm as [10, Algorithm 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2: Each AP transmits the quantized versions of its received pilot signals and data signals to the CPU based on Case 1 in [38] called "Quantized Estimate of the Channel and Quantized Signal Available at the CPU" as [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (11) ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (1), generate the DCC based processing scheme for the FCP (the receive combining and achievable SE computation) motivated by [9, Sec V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4: Based on Section Ⅲ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (2), formulate the I-WMMSE precoding design optimization problem with a capacity-constrained fronthaul motivated by [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (24)] and [38, eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (26)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5: Obtain the optimal precoding scheme based on potential methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Tutorial 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The I-WMMSE precoding for the FCP with the DCC strategy and the capacity-constrained fronthaul Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Two tutorials to investigate the I-WMMSE precoding schemes with the DCC strategy and the capacity-constrained fronthaul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Complexity Analysis In this subsection, we analyze the computational complexity of two precoding schemes investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Since the bisection step for λk,{(1),(2)} generally takes few iterations compared with other steps, we ignore bisection steps for λk,{(1),(2)} in the complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the fully centralized processing scheme and each realization of the AP/UE locations, the per-iteration complexity of iterative optimiza- tion is O (M3K2N5Nr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the LSFD scheme and each realization of the AP/UE locations, the per- iteration complexity of iterative optimization based on L-MMSE combining with the Monte-Carlo method, MR combining with the Monte-Carlo method and MR combining with the closed-form expressions are O (M2K2N3Nr), O (M2K2N3Nr + M3KN3) and O (M3K2N5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further reduce the computation complexity, it’s quite necessary to apply the asymptotic analysis method [40], [41] to compute the terms, which cannot be computed in closed-form, in approximation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 22 TABLE I COMPARISON OF TWO PRECODING SCHEMES IN THIS PAPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' THE NUMBER OF COMPLEX SCALARS IS COMPUTED FOR EACH REALIZATION OF THE AP/UE LOCATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' THE SUM SE IMPROVEMENT IS COMPUTED WITH M = 20, K = 10, L = 1 AND N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='FCP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='LSFD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Instantaneous CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Statistical CSI ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Detection scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='MMSE combining ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='L-MMSE/MR combining + Optimal LSFD scheme ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Number of complex scalars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='sent from APs to the CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='with I-WMMSE precoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='τcMLNr + MKL2N 2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(τc − τp)MKNNr + MKN 2 + M 2K2N 4/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Number of complex scalars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='sent from APs to the CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='without I-WMMSE precoding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='τcMLNr + MKL2N 2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='(τc − τp)MKNNr + MKN 2 + M 2K2N 2/2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Number of complex scalars ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='feedback sent from the CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='KN 2Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='KN 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Per-iteration computational ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='complexity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 3K2N 5Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='L-MMSE: O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 2K2N 3Nr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='MR (Monte-Carlo): O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 2K2N 3Nr + M 3KN 3� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='MR (Analytical): O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='M 3K2N 5� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='Sum SE improvement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='93% L-MMSE: 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='74% MR: 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='13% V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' NUMERICAL RESULTS In this paper, a CF mMIMO system is investigated, where all APs and UEs are uniformly distributed in a 1×1 km2 area with a wrap-around scheme [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The pathloss and shadow fading are modeled similarly as [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In practice, Umk,r, Umk,t and Ωmk are estimated through measurements [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, we generate them randomly in this paper, where the coupling matrix Ωmk consists of one strong transmit eigendirection capturing dominant power [43]12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, we have Fk,p = F(0) k,u,{(1),(2)} = � pk N IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for Algorithm 1 and Algorithm 2, balancing the convergence and accuracy, we assume that I(1),max = I(2),max = 20, ε(1) = ε(2) = 5 × 10−4, and weights for all UEs are equal (µk,(1) = µk,(2) = 1) without losing generality, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, we consider communication with 20 MHz bandwidth and σ2 = −94 dBm noise power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' All UEs transmit with 200 mW power constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Each coherence block contains τc = 200 channel uses and τp = KN/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, a pilot assignment approach similar as that in [28] is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 3 shows the cumulative distribution function (CDF) of the achievable sum SE over different realizations of the AP/UE locations for two processing schemes investigated (we shortly call “fully centralized processing” as “FCP” in the following) over “I-WMMSE precoding” or “w/o precoding”13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that the FCP scheme undoubtedly achieves higher SE than that of the LSFD scheme since the 12In this paper, we choose one eigendirection capturing dominant channel power (randomly accounting for 80% ∼ 95% of the total channel power) and other eigendirections contain the remaining power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 13The “w/o precoding” scenario denotes that identity precoding matrices Fk,u,{(1),(2)} = � pk N IN are implemented without optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 23 0 10 20 30 40 50 60 70 80 90 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='8 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' CDF of the sum SE over different processing schemes and precoding schemes with M = 20, K = 10, L = 2, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 2 3 4 5 6 0 20 40 60 80 100 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Sum SE against the number antennas per AP L over different processing schemes and precoding schemes with M = 20, K = 10, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 2 3 4 5 6 1 2 3 4 5 6 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average rate against the number of antennas per UE N over different processing schemes and precoding schemes with M = 20, K = 10, and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 1 2 3 4 5 6 1 2 3 4 5 6 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average SE with I-WMMSE precoding schemes against the number of antennas per UE N over different τc with M = 20, K = 10, and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' FCP with MMSE combining is a competitive scheme in CF mMIMO [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, the proposed I-WMMSE schemes are efficient to improve the respective achievable sum SE performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='78%, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='54% and 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='13% sum SE improvement for the FCP, the LSFD with MR combining and the LSFD with L-MMSE combining, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, for the LSFD scheme with MR combining, markers “◦” generated by analytical results overlap with the curves generated by simulations, respectively, validating our derived closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 4 shows the achievable sum SE as a function of the number of antennas per AP with two processing schemes investigated and different precoding schemes14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We notice that, for the FCP or LSFD with (L-)MMSE combining, the performance gap between the “I-WMMSE” and “w/o precoding” becomes smaller with the increase of L, which implies that (L-)MMSE combining can use all antennas on each 14Note that the achievable sum SE investigated is the average sum SE value taken over many AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 24 10 20 30 40 50 60 1 2 3 4 5 6 7 8 9 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average SE against the number of APs M for the LSFD scheme with K = 10, L = 4, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 3 4 5 6 8 10 12 2 3 4 5 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='5 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Average SE against the number of antennas per UE N for different channel models with M = 40, K = 8, and L = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 4 6 8 10 12 14 16 18 25 26 27 28 29 30 31 32 33 34 (a) FCP 2 4 6 8 10 12 14 16 16 18 20 22 24 26 28 (b) LSFD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Convergence examples of the I-WMMSE algorithm for the FCP and LSFD with M = 20, K = 10, L = 2, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' AP to suppress interference and achieve excellent SE performance even without any precoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, the performance gap between the “I-WMMSE” and “w/o precoding” for the LSFD with L-MMSE combining is 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='74% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='17% over L = 1 and L = 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Meanwhile, for the LSFD with MR combining, the performance gap between the “I-WMMSE” and “w/o precoding” becomes large with the increase of L, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='13% and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='48% for L = 1 and L = 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, for the LSFD scheme with MR combining, markers “✷” generated by analytical results overlap with the curves generated by simulations, respectively, validating our derived closed-form expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further show the advantage of the proposed I-WMMSE precoding schemes, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 shows the average rate15 as a function of the number of antennas per UE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We find that the average rates for all schemes with I-WMMSE precoding schemes grow with N and the average rates for the case without UL precoding may also suffer the degradation with the increase of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The implementation of the I-WMMSE precoding 15Note that one main reason for the phenomenon that additional UE antennas may give rise to the SE degradation is that increasing N will increase the channel estimation overhead and reduce the pre-log factor “(τc − τp) /τc” in all SE expressions [26], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we investigate “the average rate” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5, ignoring the effect of “(τc − τp) /τc”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 25 50 100 150 200 250 300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='8 4 104 50 100 150 200 250 300 0 2 4 6 105 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Total number of complex scalars sent via the fronthaul per channel use for each realization of the AP/UE locations with M = 20, K = 10, L = 2, and N = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' schemes undoubtedly makes UEs benefit from multiple antennas and achieve excellent rate performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, we observe that the I-WMMSE precoding schemes perform more efficiently with a larger number of UE antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, the average rate improvements achieved by the I-WMMSE precoding for the LSFD with L-MMSE combining are 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='91% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='43% for N = 6 and N = 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' However, the average SE (with scaling factor (τc − τp)/τc) with I-WMMSE precoding implemented may also degrade with the increase of N as the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 2 in [1] since, with the increase of N, the prerequisite of “mutually orthogonal pilot matrices” still requires huge channel uses for the pilot transmission and the inter-user interference also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the design of non-orthogonal pilot matrices and per-antenna power control scheme are quite necessary, which are regarded as promising ways to reduce the cost of pilot transmission and further improve the SE performance [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 6 discusses the average SE with I-WMMSE precoding schemes against N over different τc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5 can be viewed as a special case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 6 with the coherence block with infinite length τc = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We observe that the average SE with I-WMMSE precoding schemes increases with N over τc = 500 or ∞, which means the SE performance can benefit from having additional UE antennas when the coherence block resource is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 7 investigates the average SE as a function of M for the LSFD scheme over different precoding schemes16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For MR combining, markers “✷” generated by analytical results overlap with the curves generated by simulations, respectively, validating our derived closed-form expressions again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the I-WMMSE algorithm is more efficient to improve the SE performance for MR combining than that of L-MMSE combining for the scenario over large L and M, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='03% and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='21% SE improvement for L- MMSE combining and MR combining with M = 60, respectively, implying that the L-MMSE combining 16The “WMMSE precoding” denotes the precoding schemes generated by the I-WMMSE algorithm with only single iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 26 based on large L and M can achieve excellent SE performance even without any precoding scheme and the proposed I-WMMSE precoding scheme is handy to mitigate the weakness of MR combining17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 8 considers the average SE as a function of N over the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' and the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As observed, the proposed I-WMMSE precoding schemes are more efficient over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For instance, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='89% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='77% average SE improvement can be achieved when N = 6 over the “Weichselberger” scenario for the LSFD scheme with MR combining and the FCP scheme, respectively, but only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='29% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='63% average SE improvement can be achieved for “I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Rayleigh channel”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, compared with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5, we notice that the I-WMMSE precoding scheme for the FCP scheme is more efficient in the highly loaded system (the scenario in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 5) where the number of total AP-antennas is comparable with the number of total UE-antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 9 illustrates the convergence behavior of the I-WMMSE algorithms for the FCP scheme and the LSFD scheme with L-MMSE/MR combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note the convergence example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 (a) for the FCP is given by a particular channel realization and the convergence example for the LSFD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 (b) is given by a particular realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the algorithms investigated can be guaranteed to converge and are efficient to achieve excellent sum SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 9 (b) for the LSFD scheme with MR combining validates our derived closed-form expressions in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Figure 10 investigates the total number of complex scalars sent via the fronthaul per channel use against τc for each realization of the AP/UE locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As observed, total number of complex scalars per channel use for the FCP/LSFD scheme becomes smaller/larger, which can also be easily found from Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, the LSFD scheme requires more fronthaul signaling than the FCP scheme since APs under the LSFD scheme need to transmit all received data signals to the CPU, which requires a huge fronthaul load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, with the increase of τc, the gap between “I-WMMSE precoding” and “W/O precoding” becomes smaller for either the FCP scheme or the LSFD scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Considering the SE performance improvement of the I-WMMSE precoding, additional fronthaul loads can be acceptable, especially when the coherence resource is abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Although the computational complexity of Algorithm 1 for the FCP scheme is much higher than that of Algorithm 2 for the LSFD scheme, the FCP scheme needs much less fronthaul signaling than that of the LSFD scheme and can achieve better SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So two processing schemes and their respective precoding schemes can be chosen based on different requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 17MR combining is a simple combining scheme but cannot efficiently suppress the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' 27 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' CONCLUSION We consider a CF mMIMO system with both APs and UEs equipped with multiple antennas over the Weichselberger Rayleigh fading channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The FCP scheme and LSFD scheme are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' To further improve the sum SE performance, efficient UL precoding schemes based on iteratively WMMSE algorithms are investigated to maximize weighted sum SE for the two processing schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that we compute achievable SE expressions and optimal precoding schemes in novel closed-form for the LSFD scheme with MR combining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Numerical results show that the investigated I-WMMSE precoding schemes are efficient to achieve excellent sum SE performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' More importantly, it can be seen that the proposed I-WMMSE precoding schemes are more efficient with a larger number of UE antennas, which means the I-WMMSE precoding schemes can achieve excellent performance even with a large number of UE antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The derived results undoubtedly provides vital insights for the practical implementation of multi- antenna UEs in CF mMIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In future work, we will investigate the design of UL precoding scheme for the phase of pilot transmission and consider the practical implementation of the investigated I-WMMSE precoding schemes with capacity-constrained fronthaul network and dynamic cooperation clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Moreover, the non-orthogonal pilot matrix design will also be considered to further improve the performance for the CF mMIMO system with multi-antenna UEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Last but not least, the UL precoding performance over a more practical Rician fading channel with phase-shifts will also be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX A SOME USEFUL LEMMAS Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Let X ∈ CM×N be a random matrix and Y is a deterministic M × M matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' The (n, i)-th element of E � XHYX � is tr � Y · E � xixH n �� where xi and xn are the i-th and n-th column of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For matrices A ∈ CN1×N1, B ∈ CN1×N2, C ∈ CN2×N2, and D ∈ CN2×N1, we have (A + BCD)−1 = A−1−A−1B � DA−1B + C−1�−1 DA−1, which is a well-known matrix inversion lemma [36, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX B PROOF COROLLARY 5 Since the CPU is only aware of channel statistics, we need to treat E{Gkk}Fk,u as the true deterministic channel and rewrite ˜xk in (25) as ˜xk = E {Gkk} Fk,uxk+(GkkFk,u − E {Gkk} Fk,u) xk + K � l=1,l̸=k GklFl,uxl + n′ k � �� � v where v is a complex circular symmetric noise with an invertible covariance matrix Ξk = E{vvH|Θ} = 28 �K l=1 E{GklFk,uFH k,uGH kl} − E{Gkk}Fk,uFH k,uE{GH kk} + σ2Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Firstly, we whiten the noise as Ξ − 1 2 k ˆxk = Ξ − 1 2 k E {Gkk} Fk,uxk + ˜v, where ˜v ≜ Ξ − 1 2 k v becomes white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Next, we project Ξ − 1 2 k ˆxk in the direction of Ξ − 1 2 k E {Gkk} Fk,u to obtain an effective scalar channel as � Ξ − 1 2 k E {Gkk} Fk,u �H Ξ − 1 2 k ˆxk = (E {Gkk} Fk,u)H Ξ−1 k E {Gkk} Fk,uxk + (E {Gkk} Fk,u)H Ξ−1 k v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (44) Based on theories of optimal receivers [37], we derive optimal LSFD matrix Ak=Ξ−1 k E {Gkk}Fk,u as Ak = � K � l=1 E � GklFk,uFH k,uGH kl � − E {Gkk} Fk,uFH k,uE � GH kk � + σ2Sk �−1 E {Gkk} Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (45) Moreover, based on the the standard results of matrix derivation in [45], we can easily obtain the LSFD matrix minimizing the conditional MSE for UE k MSE(2) k = tr(E(2) k ) as Ak = � K � l=1 E{Gkl¯Fl,uGH kl} + σ2Sk �−1 E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (46) We notice that the LSFD matrix in (45) is equivalent to the LSFD matrix in (46), except from having another scaling matrix IN − � CHB−1C + IN �−1 CHB−1C on the right side, which would not affect the value of (27), where B = �K l=1 E{GklFk,uFH k,uGH kl} + σ2Sk and C = E{Gkk}Fk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So the LSFD matrix in (46) cannot maximize the achievable SE but minimize the MSE for UE k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX C PROOF OT THEOREM 2 In this part, we compute terms of (27) in closed-form for the LSFD scheme with MR combining Vmk = ˆHmk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For the first term Dk,(2) = AH k E{Gkk}Fk,u, we have E{Gkk} = [E{VH 1kH1k};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' E{VH MkHMk}] = [ZT 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ZT Mk]T ≜ Zk, where Zmk = E{VH mkHmk} = E{ ˆHH mk ˆHmk} ∈ CN×N and the (n, n′)-th el- ement of Zmk can be denoted as [Zmk]nn′ = E{ˆhH mk,nˆhmk,n′} = tr(ˆRn′n mk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we derive the closed- form for Dk,(2) as Dk,(2),c = AH k ZkFk,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for the second term Sk ∈ CMN×MN, we have Sk = diag(E{VH 1kV1k}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , E{VH MkVMk}) = diag(Z1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ZMk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For E{Gkl¯Fl,uGH kl}, we notice that the (m, m′)-submatrix of E{Gkl¯Fl,uGH kl} is E{VH mkHml¯Fl,uHH m′lVm′k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on [28], we compute E{VH mkHml¯Fl,uHH m′lVm′k} for four possible AP-UE combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “m ̸= m′, l /∈ Pk”, we have E{VH mkHml¯Fl,uHH m′lVm′k} = 0 for the independence between Vmk and Hml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “m ̸= m′, l ∈ Pk”, we have E{VH mkHml¯Fl,uHH m′lVm′k} = E{VH mkHml}¯Fl,uE{HH m′lVm′k} = Λmkl¯Fl,uΛm′lk, where the (n, n′)-th element of N ×N-dimension complex matrices Λmkl ≜ E{VH mkHml}, 29 Λm′lk ≜ E{HH m′lVm′k} are [Λmkl]nn′ = E{ˆhH mk,nˆhml,n′} = tr(Ξn′n mkl) and [Λm′lk]nn′ = E{ˆhH m′l,nˆhmk,n′} = tr(Ξn′n m′lk) with Ξmkl ≜ E{ˆhmlˆhH mk} = τpRml˜FH l,pΨ−1 mk˜Fk,pRmk, Ξm′lk ≜ E{ˆhm′kˆhH m′l} = τpRm′k˜FH k,pΨ−1 m′k˜Fl,pRm′l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “m = m′, l /∈ Pk”, we define Γ(1) mkl ≜ E{VH mkHml¯Fl,uHH mlVmk} ∈ CN×N with the (n, n′)-th element [Γ(1) mkl]nn′ = �N i=1 �N i′=1 [¯Fl,u]i′iE{ˆhH mk,nhml,i′hH ml,iˆhmk,n′} being [Γ(1) mkl]nn′ = N � i=1 N � i′=1 [¯Fl,u]i′itr(E � hml,i′hH ml,i � E{ˆhmk,n′ˆhH mk,n}) = N � i=1 N � i′=1 [¯Fl,u]i′itr(Ri′i ml ˆRn′n mk) (47) since ˆHmk and Hml are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, for “m = m′, l ∈ Pk”, ˆHmk and Hml are no longer inde- pendent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We define Γ(2) mkl ≜ E{VH mkHml¯Fl,uHH mlVmk} ∈ CN×N whose (n, n′)-th element is [Γ(2) mkl]nn′ = �N i=1 �N i′=1 [¯Fl,u]i′iE{ˆhH mk,nhml,i′hH ml,iˆhmk,n′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We follow the similar method in [28] and derive [Γ(2) kl,m]nn′ = �N i=1 �N i′=1 [¯Fl,u]i′itr(Ri′i mlPn′n mkl,(1))+τ 2 p �N q1=1 �N q2=1 [¯Fl,u]i′i[tr(˜Pq1n mkl,(2) ˜Ri′q2 ml ˜Rq2i ml ˜Pn′q1 mkl,(2))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='+ τ 2 p �N q1=1 �N q2=1 [¯Fl,u]i′itr(˜Pq1n mkl,(2) ˜Ri′q2 ml )tr(˜Pn′q2 mkl,(2) ˜Rq2i ml), where Pmkl,(1) ≜ τpSmk(Ψmk−τp˜Fl,pRml˜FH l,p)SH mk, Smk ≜ Rmk˜FH k,pΨ−1 mk and Pmkl,(2) ≜ Smk˜Fl,pRml˜FH l,pSH mk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, ˜Rni ml and ˜Pni mkl,(2) denote (n, i)-submatrix of R 1 2 ml and P 1 2 mkl,(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' In summary, combining all the cases, we have E{Gkl¯Fl,uGH kl} = Tkl,(1) + Tkl,(2) if l ∈ Pk and Tkl,(1) otherwise, where Tkl,(1) ≜ diag(Γ(1) kl,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Γ(1) kl,M) ∈ CMN×MN and Tmm′ kl,(2) = Γ(2) kl,m − Γ(1) kl,m if m = m′ and Λmkl¯Fl,uΛm′lk otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Plugging the derived results into (28) and (31), we can easily compute the optimal LSFD coefficient matrix and MSE matrix in closed-form as (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So we have finished the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For more details on the derived expression, please refer to [28, Appendix D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX D PROOF OF THEOREM 1 When other optimization variables are fixed, we derive the partial derivative of (17) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='t F(1) k,u as ∂f � F1,u,(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , FK,u,(1) � ∂Fk,u,(1) = K � l=1 µl,(1) � ˆHH k VlWl,(1)VH l ˆHk + E � ˜HH k VlWl,(1)VH l ˜Hk ��� V, W �� + λk,(1)IN − µk,(1) ˆHH k VH k Wk,(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' (48) By applying the first-order optimality condition and setting ∂f(F1,u,(1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=',FK,u,(1)) ∂Fk,u,(1) = 0, we can easily obtain the optimal precoding scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Besides, λk,(1) and Fk,u,(1) should also satisfy KKT condition as (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' As for ¯Ckl ≜ E{ ˜HH k VlWl,(1)VH l ˜Hk|V, W} ∈ CN×N, by applying Lemma 1, the (i, n)-th element of ¯Ckl is tr( ¯VlE{˜hk,n˜hH k,i}) where ¯Vl ≜ VlWl,(1)VH l and ˜hk,n = [˜hT 1k,n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , ˜hT Mk,n]T ∈ CML is the n-th column of ˜Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Finally, we derive Ck,ni ≜ E{˜hk,n˜hH k,i} = diag (Cni 1k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Cni Mk) ∈ CML×ML since ˜hmk,n 30 and ˜hm′k,n for m ̸= m′ are independent and both have zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' So Ck,ni is a block-diagonal matrix with the square matrices Cni 1k = E{˜h1k,n˜hH 1k,i}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' , Cni Mk = E{˜hMk,n˜hH Mk,i} on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX E PROOF OF (15) For the LSFD scheme, the conditional MSE matrix for UE k can be written as (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Based on [28, Appendix C], we prove that (28) can also minimize MSEk,(2) = tr � Ek,(2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' With (28) implemented, Ek,(2) is given by (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Then, by applying Lemma 2, we have � Eopt k,(2) �−1 = IN + FH k,u,(2)E � GH kk � � K � l=1 E � Gkl¯Fl,u,(2)GH kl � − E {Gkk} ¯Fk,u,(2)E � GH kk � + σ2Sk �−1 × E {Gkk} Fk,u,(2), where A ≜ IN, B ≜ −FH k,u,(2)E{GH kk}, C ≜ (�K l=1 E{Gkl¯Fl,u,(2)GH kl}+σ2Sk)−1 and D ≜ E{Gkk}Fk,u,(2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' We show the equivalence between SEopt k,(2) and log2 |(Eopt k,(2))−1| without a factor (1 − τp/τc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' APPENDIX F PROOF OF THEOREM 4 When MR combining Vmk = ˆHmk and the optimal LSFD scheme applied, we can easily compute E{GH kk}, Aopt k , and Eopt k,(2) in closed-form as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Furthermore, by applying Lemma 1, the (i, n)-th entry of ¯Tlk = E{GH lkAlE−1 l,(2)AH l Glk} ∈ CN×N can be denoted as tr(¯AlE{glk,ngH lk,i}), where ¯Al ≜ AlE−1 l,(2)AH l and glk,n ∈ CMN is the n-th column of Glk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' Note that the (m − 1) N + p-th element of glk,n is ˆhH ml,phmk,n so the [(m − 1) N + p, (m′ − 1) N + p′]-th (or [o, j]-th briefly) entry of ¯Glk,ni ≜ E{glk,ngH lk,i} ∈ CMN×MN can be denoted as E{ˆhH ml,phmk,nhH m′k,iˆhm′l,p′}, which can be computed for four AP-UE combinations as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l /∈ Pk, m ̸= m′”, we have E{ˆhH ml,phmk,nhH m′k,iˆhm′l,p′} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l ∈ Pk, m ̸= m′”, we have E{ˆhH ml,phmk,nhH mk,iˆhml,p′} = tr(Rni mk ˆRp′p ml).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l /∈ Pk, m = m′”, we have E{ˆhH ml,phmk,nhH m′k,iˆhm′l,p′} = E{ˆhH ml,phmk,n}E{hH m′k,iˆhm′l,p′} = tr(Ξnp mlk)tr(Ξp′i m′kl), where Ξmlk = τpRmk˜FH k,pΨ−1 mk˜Fl,pRml and Ξm′kl = τpRm′l˜FH l,pΨ−1 m′l˜Fk,pRm′k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE0T4oBgHgl3EQfggEA/content/2301.02417v1.pdf'} +page_content=' For “l ∈ Pk, m = m′”, we obtain E{ˆhH ml,phmk,nhH mk,iˆhml,p′} = tr(Rni mkPp′p mkl,(1))+ τ 2 p �N q1=1 �N q2=1 tr(˜Pq1p mlk,(2) ˜Rnq2 mk ˜Rq2i mk ˜Pp′q1 mlk,(2)) + τ 2 p �N q1=1 �N q2=1 tr(˜Pq1n mlk,(2) ˜Rnq1 mk)tr(˜Pp′q2 mlk,(2) ˜Rq2i mk), where Sml = Rml˜FH l,pΨ−1 ml, Pmlk,(1) = τpSml(Ψml−τp˜Fk,pRmk˜FH k,p)SH ml and Pmlk,(2) = Sml˜Fk,pRmk˜FH k,pSH ml with ˜Rni mk and ˜Pni mkl,(2) being (n, i)-submatrix of R 1 2 mk and P 1 2 mkl,(2), respectively.' metadata={'source': 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