diff --git "a/2dE1T4oBgHgl3EQflgSY/content/tmp_files/load_file.txt" "b/2dE1T4oBgHgl3EQflgSY/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2dE1T4oBgHgl3EQflgSY/content/tmp_files/load_file.txt" @@ -0,0 +1,1103 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf,len=1102 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='03286v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='SP] 9 Jan 2023 1 A Dual-Function Radar-Communication System Empowered by Beyond Diagonal Reconfigurable Intelligent Surface Bowen Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Hongyu Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Student Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Ziyang Cheng,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Shanpu Shen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Member,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and Bruno Clerckx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fellow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IEEE Abstract—This work focuses on the use of reconfigurable intelligent surface (RIS) in dual-function radar-communication (DFRC) systems to improve communication capacity and sensing precision,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and enhance coverage for both functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In contrast to most of the existing RIS aided DFRC works where the RIS is modeled as a diagonal phase shift matrix and can only reflect signals to half space, we propose a novel beyond diagonal RIS (BD-RIS) aided DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, the proposed BD-RIS supports the hybrid reflecting and transmitting mode, and is com- patible with flexible single/group/fully-connected architectures, enabling the system to realize full-space coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To achieve the expected benefits, we jointly optimize the transmit waveform, the BD-RIS coefficients, and sensing receive filters, by maximizing the minimum signal-to-clutter-plus-noise ratio for fair target detection, subject to the constraints of the communication quality of service, different BD-RIS architectures and power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To solve the non-convex and non-smooth max-min problem, a general solution based on the alternating direction method of multipliers is provided for all considered BD-RIS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Numerical simulations validate the efficacy of the proposed algorithm and show the superiority of the BD-RIS aided DFRC system in terms of both communication and sensing compared to conventional RIS aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Index Terms—Beyond diagonal reconfigurable intelligent sur- faces, dual-function radar-communication, full-space coverage, max-min optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' INTRODUCTION In recent years, spectrum resources are becoming increas- ingly limited and valuable due to the exponential growth of services in wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Meanwhile, radar sys- tems are competing for the same scarce sources, which moti- vates the emergence of the dual-function radar-communication (DFRC) technology to achieve spectrum sharing between communication and radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In DFRC systems, communication and radar functionalities are integrated on a common platform, which brings the benefit of enhanced spectrum efficiency while (Corresponding author: Ziyang Cheng, Shanpu Shen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Wang and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Cheng are with the School of Information & Communica- tion Engineering, University of Electronic Science and Technology of China, Chengdu, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (email: B W Wang@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='com, zycheng@uestc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Li is with the Department of Electrical & Electronic Engineering, Impe- rial College London, London SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (email: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='li21@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Shen is with the Department of Electronic and Computer Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (email: sshenaa@connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='ust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Clerckx is with the Department of Electrical & Electronic Engineering, Imperial College London, London, SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and with Silicon Austria Labs (SAL), Graz A-8010, Austria (email: b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='clerckx@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' reducing power consumption and hardware costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, DFRC is envisioned to play an important role in emerging environment-aware applications [1], such as vehicular net- works, environmental monitoring, and smart houses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Due to the benefits of DFRC, plenty of technical efforts have been devoted to designing DFRC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The design method- ology can be roughly divided into three categories: radar- centric design [2]–[4], communication-centric design [5]–[7], and joint waveform design [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Radar-centric approaches utilize the radar waveform as the information carrier, where the communication symbols are embedded in conventional radar signals, such as linear frequency modulation [2] and frequency hopping [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' On the other hand, communication- centric approaches realize the radar sensing tasks by modifying existing communication protocols [5] and waveforms [6], [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In contrast to the first two categories [2]–[7], the DFRC waveforms can be jointly designed to provide more design freedoms so as to enhance both functionalities [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Despite the above works [2]–[9] achieve satisfactory sensing and communication performance, one limitation is that they rely on the line-of-sight (LoS) links between the base station (BS) and communication users/sensing targets, which however yields the following two issues in practice: 1) The LoS link toward sensing targets or communication users can be easily blocked by obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) The LoS channels may suffer from severe path loss especially for high frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To overcome these issues, a promising technology named reconfigurable intelligent surface (RIS) [10]–[13] can be lever- aged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, RIS consists of numerous passive reconfig- urable scattering elements with low hardware cost and power consumption [10]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By properly placing and adjusting the RIS, it can establish virtual non-LOS (NLoS) links to “bypass” obstacles, and therefore compensate for the path loss and enhance system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Due to its advantages, RIS has been investigated for communications [14]–[16] and sensing [17]–[20] fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Furthermore, RIS has been explored in various DFRC systems [21]–[27] to enhance both the communication and sensing performance, which are classified into the following two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The first category assumes LoS links exist from BS to users and targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this category, the RIS is used to compensate for the propagation loss and to improve the performance [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The second category focuses on the scenario where either communication users or sensing targets are blocked by barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this category, RIS 2 is utilized to establish a NLoS link to bypass the barriers and thus enable DFRC [24]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The limitation of the aforementioned works [21]–[27] is that they assume the RIS can only reflect signals towards the same side as the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this case, both communication users and sensing targets should be located at the same side of RIS, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', within the same half-space, which limits the coverage and beam control flexibility of the RIS enabled DFRC sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To address this limitation, a novel hybrid transmissive and reflective RIS, namely simultaneously transmitting and reflecting RIS (STAR-RIS) [28] or intelligent omni-surface [29], is proposed to support signal reflection and transmission and thus extend the coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The integration of STAR- RIS and DFRC is first studied in [30], where the system is designed by minimizing the Cram´er-Rao bound (CRB) for radar target estimation subject to communication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, a STAR-RIS is deployed at the vehicle to improve both sensing and communication performance [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Nevertheless, the achievable performance of STAR-RIS aided DFRC in [30], [31] is limited by the simple architecture of STAR-RIS without fully exploiting the architecture of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To enhance the performance of RIS, a novel branch, namely beyond diagonal RIS (BD-RIS) [32]–[35], is proposed by exploring different architectures/modes of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BD-RIS with group/fully-connected architectures under the reflective mode is first proposed in [32], which provides more controllable scattering matrices than conventional RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, the hybrid reflective and transmissive BD-RIS is proposed in [33] to achieve full-space coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is proved that STAR-RIS is essentially a particular instance of two-port group-connected reconfigurable impedance network when each two antenna ports are connected to each other, namely cell-wise single- connected (CW-SC) architecture in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' More general cell- wise group/fully-connected (CW-GC/FC) architectures are also proposed based on the flexible connections among more antenna ports, which achieves better performance than STAR- RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Furthermore, a multi-sector BD-RIS is proposed in [35], which not only achieves full-space coverage but also provides higher performance gain than hybrid BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Due to the benefits of BD-RIS, in this paper, we propose to adopt BD-RIS in DFRC systems to achieve full-space cov- erage and better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To the best of our knowledge, adopting BD-RIS in DFRC has not been investigated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, in contrast to [30], [31] which ignore the signal-dependent clutters, we consider a more general and practical multi-target detection scenario with the presence of multiple clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The main contributions of this work are summarized as follows: Proposing BD-RIS aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We propose a BD-RIS aided DFRC system, which consists of a BD-RIS enabling the full-space coverage, multiple users, and multiple sensing targets corrupted by multiple clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The BD-RIS divides the space into two sides and establishes virtual NLoS links for communication and sensing, where the dual-function BS (DFBS) performs communication tasks in one half space and sensing tasks in another side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To avoid multi-step path loss, we implement the radar sensing receiver on the BD-RIS for multi-target detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Formulating Max-min fairness problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We formulate the optimization problem to jointly design the transmit wave- form at the DFBS, the reflective and transmissive beamforming at the BD-RIS, and matched filters at the radar sensing receiver, to maximize the minimum radar output signal-to- clutter-plus-noise ratio (SCNR), subject to the communication quality of service (QoS) requirement for downlink communi- cations, the transmit power constraint at the DFBS, and the BD-RIS constraints with different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Developing joint design framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The joint design of BD-RIS aided DFRC is challenging due to the complicated and non-smooth objective, and newly introduced non-convex constraints of BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To overcome these difficulties, we propose to decouple the BD-RIS constraints by the alternat- ing direction method of multipliers (ADMM) framework so that the resulting sub-problems are reformulated into easily handled forms and iteratively solved until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Providing insights and numerical validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We provide simulation results to illustrate the performance improvement achieved by BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is shown that benefiting from the high flexibility of BD-RIS, and the joint design of transmit waveform, BD-RIS, and the matched filters, the CW-GC/FC BD-RISs can achieve higher radar SCNR than CW-SC (STAR- RIS) ones under the same communication requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is also shown the BD-RIS can substantially improve the performance and coverage compared to the conventional RIS, which shows the high flexibility of BD-RIS in manipulating the incident signal for enhancing the DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Organization: Section II presents the system model of the proposed BD-RIS aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Section III formulates the max-min fairness problem and provides a joint design algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Section IV evaluates the performance of the pro- posed algorithm and compares different BD-RIS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Section V concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Notation: Scalars, vectors and matrices are denoted by stan- dard lowercase letter a, lower case boldface letter a and upper case boldface letter A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Cn and Cm×n denote the n-dimensional complex-valued vector space and m × n complex-valued matrix space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (·)T , (·)H, and (·)−1 denote the transpose, conjugate-transpose operations, and inversion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ℜ{·} and ℑ{·} denote the real and imaginary part of a complex number, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∥ · ∥F and | · | denote the Frobenius norm and magnitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Diag(·) denotes a diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BlkDiag(·) denotes a block matrix such that the main-diagonal blocks are matrices and all off-diagonal blocks are zero matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IL indicates an L × L identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' \uf6be denotes imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∠(·) represent the phase values of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Tr(·) denotes the summation of diagonal elements of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ⌊·⌋ is the round-down operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' SYSTEM MODEL As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1, we consider a DFRC system, where an NT-antenna DFBS simultaneously sends communication symbols to U single-antenna users and detects K targets in the presence of Q strong clutters with the assistance of an NS-cell BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The BD-RIS adopts the hybrid transmissive and reflective mode, which divides the whole space into two 3 Target Target Clutter Clutter Transmissive Area for Radar Cell 1 BD-RIS Target 1 Target K Clutter 1 Clutter Q Reflective Area for Communication Transmissive Area for Radar User User DFBS Reflective Area for Communication RIS elements Sensor elements User 1 User NU DFBS NT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Illustration of a BD-RIS aided DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' half areas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', the transmissive and reflective areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The DFBS provides communication services at the reflective area while performing radar sensing at the transmissive area aided by BD- RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The radar sensing receiver with NR antennas is installed adjacent to the BD-RIS to collect target echos and conduct target detection tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In the following subsections, we will review the modeling of BD-RIS with different architectures, and establish the communication and radar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BD-RIS Architecture Model According to [33], the hybrid reflective and transmissive mode is essentially based on the group-connected reconfig- urable impedance network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, each two antenna ports are connected to each other, constructing one cell as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Within each cell, two antennas with uni- directional radiation pattern are back to back placed such that each antenna covers half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Mathematically, the BD-RIS with hybrid reflective and transmissive mode is characterized by two matrices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', ΦR ∈ CNS×NS and ΦT ∈ CNS×NS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Depending on the inter-cell connection strategies, the BD-RIS can be categorized into the following three architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1) CW-SC BD-RIS Architecture: As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2(a), we provide a simple example of CW-SC BD-RIS with 2 cells, from which we can observe that different RIS cells are not connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, matrices ΦT, ΦR are all restricted to be diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', ΦT = Diag(φT,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , φT,NS) and ΦR = Diag(φR,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , φR,NS), and satisfy |φT,i|2 + |φR,i|2 = 1, ∀i = 1, · · · , NS, (1) which conforms to the STAR-RIS constraints, indicating that the STAR-RIS is a special case of BD-RIS with CW-SC architecture [28], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) CW-FC BD-RIS Architecture: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2(b) depicts an exam- ple of CW-FC BD-RIS with 2 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In contrast to CW-SC case, all cells of the CW-FC BD-RIS are connected to each other through reconfigurable impedance components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Accordingly, ΦT, ΦR are all full matrices satisfying ΦH T ΦT + ΦH R ΦR = INS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (2) 3) CW-GC BD-RIS Architecture: As a balance between the above two extreme cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' CW-GC divides all cells into several Cell� User� User� BD�RIS DFBS Target� Target� Clutter� Clutter� Reflective�Area�for� Communication Transmissive�Area� for�Radar Antenna�3 Antenna�1 Antenna�4 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z3 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='3 Z1 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='3 Z2 Z4 Antenna�2 2�Cell�CW�FC�BD�RIS (b) Cell�1 Cell�2 Antenna�4 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z2 Z4 Antenna�2 2�Cell�CW�SC�BD�RIS� (a) Cell�2 Antenna�5 Antenna�1 Antenna�6 Z5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 Z5 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='5 Z1 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 Z2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='5 Z2 Z6 Antenna�2 4�Cell�CW�GC�BD�RIS Antenna�7 Antenna�3 Antenna�8 Z7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 Z7 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='7 Z3 Z4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 Z3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 Z4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='7 Z4 Z8 Antenna�4 Group�1 Group�2 (c) Cell�1 Cell�2 Cell�3 Cell�4 Antenna�3 Antenna�1 Z3 Z1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='3 Z1 Cell�1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Examples of (a) CW-SC BD-RIS, (b) CW-FC BD-RIS, and (c) CW- GC BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' groups and cells in each group adopt the the fully-connected architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Depending on the group division strategies, there are plenty of CW-GS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For simplicity, here we consider the case where NS cells of the BD-RIS are uniformly divided into G groups and each group has the same size M = NS/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For ease of understanding, an example of a 4-cell BD- RIS with CW-GC architecture having 2 groups is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Hence, the model for CW-GC BD-RIS can be expressed as ΦT = BlkDiag(ΦT,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , ΦT,G), ΦR = BlkDiag(ΦR,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' , ΦR,G), ΦH T,gΦT,g + ΦH R,gΦR,g = IM, ∀g = 1, · · · , G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (3) where ΦT,g ∈ CM×M and ΦR,g ∈ CM×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The CW-GC architecture of BD-RIS is a general case, which becomes the CW-SC architecture (STAR-RIS) with a simple circuit when G = NS, and the CW-FC architecture achieving the best performance as G = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This means that CW-SC and CW-FC architectures are special cases of CW-GC architecture and the beam control flexibility/ability of CW-GC BD-RIS can be improved by decreasing G, but at the expense of increasing circuit complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Communication Model In this paper, we consider a standard multiuser multiple input single output (MISO) downlink scenario, where the DFBS provides communication services to the reflective area aided by the BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We assume the direct links between the DFBS and downlink users are blocked and the channel state information (CSI) is available at the DFBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The data symbol vector sl = [sl [1] , · · · , sl [U]]T ∈ CU contains the overall U data symbols in the l-th time slot, which are assumed to 4 be drawn from a standard M order phase-shift keying (M- PSK) modulation constellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Furthermore, the data symbol vector sl is mapped to the transmit waveform w [l] ∈ CNT at the DFBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Accordingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the received signal of the u-th user at symbol time t is yu (t) = e\uf6be2πfct L � l=1 hH u ΦRGw [l] rect (t − l∆t) + nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='u (t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (4) where fc is the carrier frequency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' L is the number of time slots during one transmission duration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' G ∈ CNS×NT and hu ∈ CNS stand for the channel coefficients of the communication links DFBS→BD-RIS and BD-RIS→u-th user,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∆t stands for symbol duration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' rect (t) is the rectangle window function that takes the value 1 for t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∆t] and 0 otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and nc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='u (t) is the additive white Gaussian noise (AWGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By down converting the signal into baseband and sampling received signal yu (t) at the rate fs = 1/∆t within the symbol duration, the discrete baseband signal at the l-th time slot is yu [l] = hH u ΦRGw [l] + nc,u [l] , (5) where nc,u [l] is the AWGN with zero mean and variance σ2 C,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this work, we adopt the recently emerged symbol level beamforming (SLB) technology for communication in DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, SLB technology utilizes the constructive inter- ference (CI), which is defined as the multi-user interference (MUI) that pushes the received symbols away from the detec- tion thresholds of the modulation constellation, to enhance the communication QoS while reducing BER [36], [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Here we briefly review the concept of SLB as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3 takes quadrature-PSK (QPSK) as an example, where point A stands for the desired symbol sl [u] with the required signal-to-noise-ratio (SNR) threshold Γu,l of the u-th user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', −→ OA = � σ2 C,uΓu,lsl [u], and point D is the received noise- free signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', −→ OD = ˜yu [l] = hH u ΦRGw [u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The CI region refers to a polyhedron bounded by hyperplanes parallel to decision boundaries of the constellation, which is depicted as blue-shaded area in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The key of SLB is to enforce the received signal located in the CI region, which means the received signal is pushed away from decision boundaries and the SNR is guaranteed to be no less than the SNR threshold Γu,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To mathematically depict the SLB constraint, we project point D into the direction of −→ OA at point C, and extend −→ CD to intersect with the nearest boundary of CI region at point B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Consequently, one of the criteria that specifies the location of −→ OD in the CI region is |−→ CD| |−→ AC| = ��ℑ � hH u ΦRGw [l] e\uf6be∠(su[l])��� ℜ � hH u ΦRGw [l] e\uf6be∠(su[l])� − � σ2 C,uΓu,l ≤ tan Ω, (6) where Ω = π/M is half of the angular range of the CI resign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In this work, we adopt SLB instead of con- ventional block-level beamforing (BLB) due to the following two reasons: 1) By adopting SLB technology in our con- sidered DFRC system, we directly design transmit waveform W ∈ CNT×L for L time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, the BLB in the D B � � � � u l y � � C Received Symbol � � � � 2 , u c u l l y � � � � � � � u l y� CI�Region 2 , c u l � � O � Imag Real A Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Description of the CI region for a QPSK symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' same scenario requires the design of the transmit beamformer Wl ∈ CNT×U, ∀l for all data symbols and time slots due to the linear mapping, which results in an increasing com- putational complexity [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) BLB regards the MUI as a harmful component and suppresses the MUI to guarantee communication QoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, the SLB utilizes the MUI to enhance the communication QoS, which provides additional design flexibility in DFRC [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Radar Model To improve the sensing performance of the BD-RIS aided DFRC system, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1, we adopt a novel sensor- at-RIS architecture [20], where the radar receiving sensors are installed adjacent to the BD-RIS to collect the echo signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This architecture greatly reduces the multi-step path- loss compared with the sensor-at-DFBS architecture [21]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, we consider a scenario where the radar receiver attempts to detect K targets in the presence of Q strong clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, the k-th target of interest is characterized by angle ϕk and time delay τ k T , respectively, while the q-th clutter is characterized by angle ϑq and delay τq C, respectively1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The backscattered signal at the radar receiver after down conversion is thus [39]–[41] r (t) = K � k=1 L � l=1 αkA (ϕk) ΦTGw [l] rect � t − l∆t − τk T � + Q � q=1 L � l=1 βqA (ϑq) ΦTGw [l] rect (t − l∆t − τq C) + nr (t) , (7) where αk and βq, respectively, denote the propagation co- efficient for the k-th target and q-th clutter consisting of radar cross section (RCS) and channel propagation effects with E(|αk|2) = ζ2 k and E(|βq|2) = ξ2 q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A (ϕ) = aR (ϕ) aH T (ϕ) ∈ CNR×NS is the effective radar channel, where aT (ϕ) = 1 √NS [1, · · · , ej 2π λ d(NS−1) sin ϕ]T and aR (ϕ) = 1 √NR [1, · · · , ej 2π λ d(NR−1) sin ϕ]T denote the the transmit and 1In this paper, we assume the targets and clutters are slowly moving or stay still, whose Doppler frequencies equal to zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5 receive steering vector, respectively, with d and λ being element spacing and wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' nr (t) denotes AWGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' we select the first target echo as the reference and sample the received signal r (t) at fs = 1/∆t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' yielding the following received baseband signal R = K � k=1 αkA (ϕk) ΦTGWJrk T � �� � Target Echos + Q � q=1 βqA (ϑq) ΦTGWJrq C � �� � Clutter Returns + Nr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (8) where Jr = [0L×r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 0L×(Lobs−L−r)] ∈ CL×Lobs is the shift matrix with Lobs = L + {maxk rk T} − {mink rk T} being the receiver observation length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' rk T = ⌊(τ k T − {min˜k τ ˜k T })fs⌋ the rang ring of the k-th target,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and rq C = ⌊(τ q C − {mink τ k T })fs⌋ the rang ring of the q-th clutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Nr = [nr [1] , · · · , nr [L]] ∈ CNR×L is the Gaussian noise matrix with nr [l] ∼ CN � 0, σ2 RINR � , ∀l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' by performing the matched filter Uk ∈ CNR×Lobs to the k-th target at radar receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the k-th target detection problem can formulated as a binary hypothesis test [39]–[41]: \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \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 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 Hk 1 : αkUH k A (ϕk) ΦTGWJrk T (9a) + K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k αpUH k A (ϕp) ΦTGWJrp T + Q � q=1 βqUH k A (ϑq) ΦTGWJrq C + Nr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (9b) Hk 0 : K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k αpUH k A (ϕp) ΦTGWJrp T (9c) + Q � q=1 βqUH k A (ϑq) ΦTGWJrq C + Nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (9d) According to the above binary hypothesis test (9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the detection probability P k D of the k-th target can be evaluated as [41] P k D = Q �� 2SCNRk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' � −2 ln (Pfa) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (10) where Q (·,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ·) is the Marcum Q-function of order 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Pfa is the false alarm probability,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and the radar output SCNR of the k-th target after the matched filtering is given by SCNRk(W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Uk) = ς−1 k |Tr(αkUH k A(ϕk)ΦTGWJrk T )| 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (11) where ςk = �K p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k |Tr(αpUH p A (ϕp) ΦTGWJrp T )| 2 + �Q q=1 |Tr(βqUHA (ϑq) ΦTGWJrq C)| 2 + σ2 R ∥Uk∥2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' MAX-MIN FAIRNESS FOR BD-RIS AIDED DFRC In this section, we first formulate the joint design problem for BD-RIS aided DFRC, followed by a general algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Finally, we propose an initialization scheme and analyze the computational complexity of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem Formulation Given that (10) is strictly increasing in SCNRk, for a specified value of false alarm probability Pfa, improving the detection probability P k D of the k-th target is equivalent to maximize the radar output SCNR of the k-th target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, for multiple target detection cases, beamforming design usually aims to improve the detection probability for all targets, especially for the weakest targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, to improve the overall target detection probability and guarantee target detection fairness, we propose to maximize the minimal radar output SCNR among the K targets by jointly designing the transmit beamformer W, the BD-RIS matrices {ΦT, ΦR}, and radar receiver filters {Uk}∀k, subject to communication QoS constraints, transmit power constraint, and BD-RIS con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The joint design problem is thus formulated as2 P1 \uf8f1 \uf8f4 \uf8f4 \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 \uf8f4 \uf8f4 \uf8f3 max W,ΦT,ΦR,{Uk} � min ∀k SCNRk (W, ΦT, Uk) � (12a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ���ℑ � ˜hH u w [l] ���� ℜ � ˜hH u w [l] � − � σ2 C,uΓu,l ≤ tan Ω, (12b) ∥W∥2 F = E, (12c) ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , (12d) ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , (12e) ΦH T,gΦT,g + ΦH R,gΦR,g = ING, ∀g, (12f) where ˜hH u = hH u ΦRG is the equivalent channel for DFBS→DB-RIS→ u-th user and E is the transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P1 is a challenging non-convex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Partic- ularly, the non-convexity stems from the complicated frac- tional SCNR expression in the objective and highly coupled optimization variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To simplify the joint design, in the following subsection, we propose a series of transformations and an ADMM based framework to decouple problem (12) into multiple more tractable sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Overview of Proposed Joint Design Framework To facilitate the joint design, we propose to re-arrange the SCNR (11) into explicit and compact forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By defining uk = Vec(Uk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' w = Vec(W) and φT = Vec (ΦT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and applying basic vectorization properties [42],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' the SCNR in (11) shares the following three equivalent expressions SCNRk (W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Uk) = uH k ΨT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kuk uH k (ΨC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k + σ2 RINRL) uk ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (13a) = wHΥT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kw wHΥC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kw + σ2 R ∥Uk∥2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (13b) = φH T ΞT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kφT φH T ΞC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='kφT + σ2 R ∥Uk∥2 F ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (13c) where ΨT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k =ζ2 k � ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � wwH� ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2Based on the discussion in Remark 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' herein we focus on the design when the BD-RIS has CW-GC architecture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' which is a general case including both CW-SC and CW-FC cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6 ΨC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k = K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k ζ2 p � ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � wwH� ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �H + Q � q=1 ξ2 q � ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � wwH� ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΥT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k =ζ2 k � ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �HukuH k � ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΥC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k = K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k ζ2 p � ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �HukuH k � ¯ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � + Q � q=1 ξ2 q � ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) �HukuH k � ¯ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΞT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k =ζ2 k � ˜ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) �H ukuH k � ˜ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΞC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='k = K � p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='p̸=k ζ2 p � ˜ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) �H ukuH k � ˜ MT (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) � + Q � q=1 ξ2 q � ˜ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) �H ukuH k � ˜ MC (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' with ¯ MT (k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) = JT rk T ⊗ (A (ϕk) ΦTG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ¯ MC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT) = JT rq C ⊗ (A (ϑq) ΦTG),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜ MT(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) = (JT rk T WT GT ) ⊗ A(ϕk),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜ MC(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' W) = (JT rq CWT GT ) ⊗ A(ϑk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Based on the above derivations, the objective in problem P1 is more tractable with respect to uk, w, or φT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, it is still difficult to find the solution to P1 due to non-convex and coupled constraints (12b), (12c), and (12f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To tackle constraint (12f), we first define Φg = [ΦH T,g, ΦH R,g]H and rewrite (12f) as ΦH g Φg = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, we introduce auxiliary variables Θg = [ΘH T,g, ΘH R,g]H = Φg and decouple constraint (12f) into two separate constraints by adding the equality, which yields the following problem: P2 \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max {Uk},W,{Φg},{Θg} � min ∀k SCNRk (W, ΦT, Uk) � (15a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b), (12c), (12d), (12e), (15b) ΘH g Θg = IM, ∀g, (15c) Φg = Θg, ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (15d) Problem P2 is a typical multi-variable optimization, which could be solved based on the ADMM framework using block coordinate descent (BCD) methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To facilitate ADMM, we place the equality constraints Φg = Θg, ∀g into the objective function, and obtain the augmented Lagrangian (AL) as L ({Uk} , W, {Φg} , {Θg}) = −{min ∀k SCNRk (W, ΦT, Uk)} + G � g=1 ℜ � Tr � ΛH g (Φg − Θg) �� + ̺ 2 G � g=1 ∥Φg − Θg∥2 F , (16) where Λg ∈ C2M×M, ∀g are dual variables associated with Φg = Θg, and ̺ ≥ 0 is the corresponding penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Replacing the original objective function with AL function (16), we obtain the AL minimization problem as P2 AL � min {Uk},W,{Φg},{Θg} L ({Uk} , W, {Φg} , {Θg}) (17a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b)-(12e), (15c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (17b) Now, the ADMM framework is constructed as follows, where the superscript of notations refers to the iteration index: Un+1 k = arg min Uk L � {Uk} , Wn, � Φn g � , � Θn g �� (18a) Wn+1 = arg min W L �� Un+1 k � , W, � Φn g � , � Θn g �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b), (12c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (18b) � Φn+1 g � = arg min Φg L �� Un+1 k � , Wn+1, {Φg} , � Θn g �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (12b), (12d), (12e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (18c) � Θn+1 g � = arg min Θg L �� Un+1 k � , Wn+1, � Φn+1 g � , {Θg} � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (15c), (18d) Λn+1 g = Λn g + ̺ � Φn+1 g − Θn+1 g � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (18e) Variables (18a) to (18e) are successively updated by solving corresponding sub-problems until some stopping conditions are reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In the following subsection3, we will elaborate on the solutions to sub-problems (18a) to (18d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solution to Sub-problems 1) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t Uk: Given other variables, the opti- mization problem for updating Uk can be expressed as P2 AL,{Uk} � max {Uk} � min ∀k uH k ΨT,kuk uH k (ΨC,k + σ2 RINRL) uk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (19) It can be observed that P2 AL,{Uk} is an unconstrained optimiza- tion problem and has K separable objective functions, each of which has the following form max Uk uH k ΨT,kuk uH k (ΨC,k + σ2 RINRL) uk , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (20) Problem (20) is a classical generalized fractional quadratic optimization problem, whose optimal solution can be obtained by taking the generalized eigenvalue decomposition as [39] uk = EIG �� ΨCN,k + σ2 RINRL �−1 × ΨT,k � , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (21) where EIG (·) represents the eigenvector operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t W: Given other variables, the opti- mization problem for updating W can be expressed as P2 AL,W \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max W � min ∀k wHΥT,kw wHΥC,kw + σ2 R ∥Uk∥2 F � (22a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ���ℑ � ˜hH u w [l] ���� ℜ � ˜hH u w [l] � − � σ2 C,uΓu,l ≤ tan Ω, (22b) ∥W∥2 F = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (22c) Problem P2 AL,W is hard to settle due to the non-smooth objective function and complicated non-convex constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To simplify the design, we first equivalently transform the 3When introducing solutions to sub-problems, we omit the superscript of notations for conciseness unless otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7 objective into a smooth form by introducing an auxiliary variable γ, which yields the following problem P2−1 AL,W \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 max W,γ γ (23a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' min ∀k wHΥT,kw wHΥC,kw + σ2 R ∥Uk∥2 F ≥ γ, (23b) γ ≥ 0, (23c) (22b), (22c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (23d) Then, we deal with constraints (23b), (22b), and (22c) step- by-step detailed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step 1: Majorization minimization (MM) to (23b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We first rewrite constraint (23b) as wHΥC,kw − wHΥT,kw γ + σ2 R ∥Uk∥2 F ≤ 0, ∀k, (24) where the second term is a composite function with both w and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To simplify the joint design of problem (27), we perform MM and propose the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Assume Υ is positive definite and γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A majorizer of f (w, γ) = wHΥw γ is f (w, γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' wn, γn) = 2ℜ � (wn)HΥw � γn − γ (wn)HΥwn (γn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Proof: Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Using Lemma 1, we conduct the majorization on constraint (24) at point (wn, γn), yielding wHΥC,kw − 2ℜ � (wn)HΥT,kw � γn + γ (wn)HΥT,kwn (γn)2 + σ2 R ∥Uk∥2 F ≤ 0, ∀k, (25) where γn is computed by γn = min ∀k (wn)HΥT,kwn (wn)HΥC,kwn + σ2 R ∥Uk∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (26) Step 2: Reformulation to (22b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' After some algebraic ma- nipulations, we rewrite (22b) as (22b) ⇔ \uf8f1 \uf8f2 \uf8f3 ℜ �¯hH u,1 (ΦR) w [l] � ≥ � σ2 C,uΓu,l sin Ω, (27a) ℜ �¯hH u,2 (ΦR) w [l] � ≥ � σ2 C,uΓu,l sin Ω, (27b) where ¯hu,1 (ΦR) = GHΦH R hu(sin Ω + e\uf6be π 2 cos Ω)e−\uf6be∠(su[l]) and ¯hu,2 (ΦR) = GHΦH R hu(sin Ω − e\uf6be π 2 cos Ω)e−\uf6be∠(su[l]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step 3: Simplification to (22c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We first scale the equality constraint (22c) as E−ǫ ≤ ∥W∥2 F ≤ E+ǫ, where ǫ ≥ 0 is an auxiliary variable whose value approaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It is easy to notice that the right-hand side ∥W∥2 F ≤ E + ǫ is convex, while the left-hand side E − ǫ ≤ ∥W∥2 F is non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To convexify the non-convex part, we perform MM and transform (22c) as two convex constraints � ∥W∥2 F − E − ǫ ≤ 0, (28a) 2ℜ {Tr (WnW)} − ∥Wn∥2 F − E + ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (28b) Replacing non-convex contraints in (23) with (25), (27) and (28) based on Steps 1-3, and penalizing the slack variable ǫ into the objective function, we minimize the following problem P2−2 AL,W \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 min W,γ,ǫ −γ + κǫ (29a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' γ ≥ 0, ǫ ≥ 0, (29b) (25), (27a), (27b), (28a), (28b) , (29c) where κ ≥ 0 represents the penalty parameter to scale the impact of the penalty term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P2−2 AL,W is a convex second-order cone programming (SOCP) problem and can be globally solved by the interior point method (IPM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t {Φg}: Given other variables, the sub- problem for updating {Φg} is P2 AL,{Φg} \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \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 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min Φg − � min ∀k φH T ΞT,kφT φH T ΞC,kφT + σ2 R ∥Uk∥2 F � + G � g=1 ℜ � Tr � ΛH g (Φg − Θg) �� +̺ 2 G � g=1 ∥Φg − Θg∥2 F (30a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ���ℑ � ˜hH u w [l] ���� ℜ � ˜hH u w [l] � − � σ2 C,uΓu,l ≤ tan Ω, (30b) ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , (30c) ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , (30d) where ΦR and ΦT are separable in both objective and con- straints, and thus can be designed in parallel as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solution to ΦR: The problem regarding ΦR is P2 AL,ΦR \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 min ΦR G � g=1 ℜ � Tr � ΛH R,g (ΦR,g − ΘR,g) �� +̺ 2 G � g=1 ∥ΦR,g − ΘR,g∥2 F (31a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ��ℑ � Tr � ¯Hu,lΦR ���� ℜ � Tr � ¯Hu,lΦR �� − � σ2 C,uΓu,l ≤ tan Ω, (31b) ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) , (31c) where ΛR,g is extracted from the last M rows of Λg, ¯Hu,l = e\uf6be∠(su[l])Gw [l] hH u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The difficulty of solving problem (31) comes from constraints (31b) and (31c), which can be tackled based on the following matrix arrangements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, we partition ¯Hu,l as ¯Hu,l = \uf8ee \uf8ef\uf8f0 ¯H11 u,l · · ¯H1G u,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ¯HG1 u,l · · ¯HGG u,l \uf8f9 \uf8fa\uf8fb , ∀u, l, (32) where ¯Hij u,l ∈ CM×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' By defining ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l = [ ¯H11 u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ¯HGG u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' and re-arranging (31c) as ˜ΦR = [ΦR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='G],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' constraints (31b) and (31c) are merged into the following constraint ���ℑ � Tr � ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ˜ΦR ����� ℜ � Tr � ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ˜ΦR �� − � σ2 C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uΓu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l ≤ tan Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∀u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (33a) 8 ⇔ \uf8f1 \uf8f2 \uf8f3 ℜ � Tr � ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='1 ˜ΦR �� ≥ � σ2 C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uΓu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l sin Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ℜ � Tr � ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 ˜ΦR �� ≥ � σ2 C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='uΓu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l sin Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ∀u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (33b) where ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='1 = ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l � sin Ω + e−\uf6be π 2 cos Ω � and ˆHu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 = ˜Hu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='l � sin Ω − e−\uf6be π 2 cos Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This brings the following opti- mization problem P2−1 AL,˜ΦR \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ˜ΦR ℜ � Tr � ˜ΛH R � ˜ΦR − ˜ΘR ��� +̺ 2∥˜ΦR − ˜ΘR∥2 F (34a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (33b), (34b) where ˜ΘR = [ΘR,1, · · · , ΘR,G] and ˜ΛR = [ΛH R,1, · · · , ΛH R,G]H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P2−1 AL,˜ΦR is a quadratic program (QP) with linear constraints and can be efficiently tackled via many existing optimization tools, such as the active set method and the primal-dual subgradient method [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solution to ΦT: The problem regarding ΦT is P2 AL,ΦT \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 min ΦT − � min ∀k φH T ΞT,kφT φH T ΞC,kφT + σ2 R ∥Uk∥2 F � + G � g=1 ℜ � Tr � ΛH T,g (ΦT,g − ΘT,g) �� +̺ 2 G � g=1 ∥ΦT,g − ΘT,g∥2 F (35a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) , (35b) Similarly, we re-organize P2 AL,ΦT into a concise form as P2−1 AL,˜ΦT \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ˜ΦT,η −η + ℜ � Tr � ˜ΛH T � ˜ΦT − ˜ΘT ��� +̺ 2 ���˜ΦT − ˜ΘT ��� 2 F (36a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' min ∀k ˜φH T ˜ΞT,k ˜φT ˜φH T ˜ΞC,k ˜φT + σ2 R ∥Uk∥2 F ≥ η, (36b) η ≥ 0, (36c) where ˜ΦT = [ΦT,1, · · · , ΦT,G], ˜ΘT = [ΘT,1, · · · , ΘT,G], ˜ΛT = [ΛH T,1, · · · , ΛH T,G]H with ΛT,g extracted from the first M rows of Λg, and ˜φT = Vec(˜ΦT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜ΞT,k = KGΞT,kKH G and ˜ΞC,k = KGΞC,kKH G, where KG = BlkDiag([IM ⊗ [0M,(g−1)M, IM, 0M,(G−g)M]]G g=1) ∈ {0, 1}MNS×N 2 S denotes the linear mapping matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Using Lemma 1 to simplify constraint (36b), we have P2−2 AL,˜ΦT \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \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 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min ˜ΦT,η −η + ℜ � Tr � ˜ΛH T � ˜ΦT − ˜ΘT ��� +̺ 2 ���˜ΦT − ˜ΘT ��� 2 F (37a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ˜φH T ˜ΞC,k ˜φT − 2ℜ � ( ˜φn T)H ˜ΞT,k ˜φT � ηn +η 2ℜ � ( ˜φn T)H ˜ΞT,k ˜φn T � (ηn)2 +σ2 R ∥Uk∥2 F ≤ 0, ∀k, (37b) η ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (37c) Algorithm 1 Max-Min Fairness for BD-RIS Aided DFRC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Input: hu, ∀u, G, ̺ and system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1: Initialize � U0 k � , W0, Φ0 T, and Φ0 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2: Set n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3: repeat 4: Calculate radar receive filters {Un k} by (21) in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5: Update transmit waveform Wn by solving (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6: Compute BD-RIS matrix Φn R by solving (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7: Update BD-RIS matrix Φn T by solving (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 8: Obtain auxiliary variables � Θn g � by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9: Update dual variables � Λn g � by (18e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 10: n = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 11: until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 12: Return {Un k}, Wn, Φn T and Φn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Output: {U⋆ k} = {Un k}, W⋆ = Wn, Φ⋆ T = Φn T, Φ⋆ R = Φn R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Problem P2−2 AL,˜ΦT is a convex SOCP and can be solved by IPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 4) Sub-problem w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t {Θg}: Given the other variables, the sub-problem for updating {Θg} is P2 AL,{Θg} \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min Φg G � g=1 ℜ � Tr � ΛH g (Φg − Θg) �� +̺ 2 G � g=1 ∥Φg − Θg∥2 F (38a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΘH g Θg = IM, ∀g, (38b) Problem P2 AL,{Θg} can be split into G sub-problems, each of which has the following form P2−1 AL,Θg \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 min Φg ℜ � Tr � ΛH g (Φg − Θg) �� +̺ 2 ∥Φg − Θg∥2 F (39a) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΘH g Θg = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (39b) Now, the remaining challenge of solving problem P2 AL,Θg lies in the unitary constraint (39b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The unitary constraint (39b) forms a 2M dimensional complex Stiefel manifold [44], which can be approximately solved via manifold based algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', Riemannian conjugate gradient (RCG) and Riemannian trust regions (RTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' However, the iterative procedure of man- ifold methods might cause a lot of computational burdens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To speed-up the design, we provide a closed-form solution of problem P2 AL,Θg in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' With the unitary constraint (39b), the optimal solution for Θg is given by Θg = Bg [IM×M, 0M×M] DH g (40) where BgΣgDH g = Λg + ̺Φg is the singular value decom- position (SVD) of Λg + ̺Φg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Proof: Please refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Based on the above derivations, the procedure of the above ADMM based algorithm is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Initialization Scheme Given that the ADMM procedure is usually sensitive to initial values, we present a 2-step initialization strategy to accelerate the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step1: Since it is not that straightforward to quickly find proper ΦT and ΦR, we randomly generate ΦT and ΦR, which satisfy the BD-RIS constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Step2: With initialized ΦR, we obtain the cascaded channel ˜hH u (ΦR) = hH u ΦRG for the communication link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' To provide a feasible and “good” initial point satisfying the constraint (12b), we initialize the transmit waveform W by solving the following QoS-constrained problem max W,Γ Γ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ℜ �¯hH u,1 (ΦR) w [l] � ≥ � σ2 C,uΓ sin Ω, ∀u, l, ℜ �¯hH u,2 (ΦR) w [l] � ≥ � σ2 C,uΓ sin Ω, ∀u, l, ∥W∥2 F ≤ E, (41) which is a convex problem and can be efficiently solved by many numerical approaches [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Complexity Analysis We provide a broad complexity analysis for Algorithms 1, which is summarized as follows 1) Initialization: The main computational complexity of this stage comes from step 2 by solving the SOCP problem (41) with IPM, which requires approximately O � N 3 TL3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) ADMM: This stage includes the iterative design of the radar receive filters Uk, transmit beamformer W, BD-RIS coefficients (ΦT, ΦR) and auxiliary variable {Θg}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Updating radar receive filters Uk requires O � KN 3 R � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Solving problem (29) for updating W with IPM method needs complexity O � N 3 TL3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The complexity of updating BD-RIS coefficients (ΦT, ΦR) can be upper bounded by O � GN 3 S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Using Theo- rem 1 to update auxiliary variable {Θg} requires complexity of O � GM 3� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, the overall complexity of the ADMM framework is O(N0(KN 3 R + N 3 TL3 + GN 3 S + GM 3)), where N0 denotes the maximum number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' PERFORMANCE EVALUATION In this section, we provided extensive simulation results to validate the effectiveness of the proposed algorithm and the performance of the proposed BD-RIS aided DFRC system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' System Setup We assume that the DFBS equipped with NT = 8 antennas transmits QPSK symbols (M = 4) to U = 4 downlink users and detects K = 3 targets with the assistance of a BD-RIS having NS = 16 cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The radar sensing receiver colocated with the BD-RIS has NR = 8 receive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The code length is L = 16 and the power budget at the DFBS is set as E = 10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The noise power at the users and radar sensing receiver are set as σ2 C,u = σ2 R = −100 dBm, ∀u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The communication QoS threshold is set the same for all users, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=', Γu,l = Γ, ∀u, l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, the distance-dependent TABLE I INFORMATION OF K TARGETS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target Index Range (m) Azimuth (◦) RCS (dB) Target 1 10 30 5 Target 2 14 0 8 Target 3 19 20 10 TABLE II INFORMATION OF Q CLUTTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' of clutters Range (m) Azimuth (◦) RCS (dB) 5 15 [20:2:28] 25 4 20 [-3:2:3] 25 9 [6:1:14] 10 25 5 [16:1:20] 30 25 path loss is modeled as η (d) = ℵ (d/d0)−ℓ, where ℵ = −30 dB denotes the signal attenuation at the reference distance d0 = 1 m, and ℓ represents the path loss exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We set the path loss exponents for the DFBS→BD-RIS, BD- RIS→user, BD-RIS→target, and BD-RIS→clutter as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2, 2, and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The DFBS and BD-RIS are located as (−20 m, 0 m) and (0 m, 0 m), respectively, which results in the distance between DFBS and BD-RIS as dBR = 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The U users are randomly located at reflective side with the same distance dRU = 16 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The DFBS→BD-RIS and BD-RIS→user channels are assumed to follow the Rician fading model with the Rician factor being 3 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For the radar function, we assume K = 3 targets and 4 groups (Q = 23) of strong clutters are located in the transmissive side, whose detailed information is presented in Tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, we assume the range resolution as ∆d = 1 m, which indicates the radar sampling rate fs = 150 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Combining Table I and the path loss model, the ratio of the propagation coefficients of the three radar targets is ζ2 1 : ζ2 2 : ζ2 3 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='7 [17]–[19], [21], indicating that target 3 is the weakest target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Benchmark Schemes For comparison, we consider the following two benchmark schemes in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 1) Benchmark 1: The radar-only case is selected as the up- per bound of the radar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We obtain this benchmark by changing the BD-RIS into transmissive mode and removing the downlink users, where the resultant problem can be tackled by modifying the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) Benchmark 2: We consider a doulbe-RIS case where one diagonal RIS working on the reflective mode while another working on the transmissive mode are adja- cently placed to achieve full-space coverage [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This baseline is a special case of BD-RIS with CW-SC where ΦT = Diag([φT,1, · · · , φT, NS 2 ], 01× NS 2 ) and ΦR = Diag(01× NS 2 , [φR,1, · · · , φR, NS 2 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Therefore, we can obtain this benchmark by modifying the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Simulation Results 1) Convergence Performance: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' we investigate the convergence of the proposed Algorithm 1 for different 10 0 20 40 60 80 100 120 140 160 180 200 Number of Iteration 0 5 10 15 20 Radar Output SCNR (dB) CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 (a) 0 50 100 150 200 250 300 Number of Iteration 5 0 5 10 15 20 Radar Output SCNR (dB) CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-FC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-GC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 1 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 2 CW-SC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Target 3 (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Radar output SCNR versus the number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (a) communica- tion threshold Γ = 0 dB, (b) communication threshold Γ = 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' BD-RIS architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It can be observed that the proposed algorithm quickly converges to a stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Specifically, after several iterations, all targets have nearly the same SCNR value, demonstrating that our algorithm can achieve fairness for multiple targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, the CW-FC architecture enjoys faster convergence than other architectures under the same communication threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' At the same time, the CW-SC re- quires nearly twice as many iterations of CW-FC to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' For the same architecture, the proposed algorithm with a large communication threshold Γ needs more iterations to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is due to the fact that if the intended communication threshold Γ is higher, fewer degrees of freedom (DoFs) in the optimization problem can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 2) System Performance with Varying Parameters: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5, we study the minimum radar output SCNR versus the communication threshold Γ for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' As ex- pected, the radar output SCNR monotonically decreases with Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is because when the intended Γ is higher, less resource can be used to maximize the radar SCNR, which indicates that there is a trade-off between communication QoS and radar output SCNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Meanwhile, the proposed algorithm with different architectures outperform the conventional RIS, which validates the advantage of deploying BD-RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, the output SCNR gap between CW-FC/GC and CW-SC becomes large with increasing communication QoS requirement, which indicates that the advantage of CW-FC/GC BD-RIS is more prominent in high communication QoS requirement scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6 displays the minimum radar output SCNR as a 0 5 10 15 20 Communication QoS Threshold (dB) 2 4 6 8 10 12 14 16 18 20 Minimum Radar Output SCNR (dB) Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC Double-RIS, CW-SC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Minimum radar output SCNR versus the communication threshold Γ for different architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 10 20 30 40 50 Transmit Power (W) 5 10 15 20 25 Minimum Radar Output SCNR (dB) Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC Double-RIS, CW-SC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Minimum radar output SCNR versus the transmit power E wit communication threshold Γ = 15 dB for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' function of transmit power E under different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' It can be observed that the output SCNR for all schemes grows with the increase of transmit power E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Meanwhile, the growth of SCNR becomes slow when the transmit power is substantially large for all considered architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is because we can improve transmit power to boost system performance to some degree, but excessive power will not improve performance further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, the slope variation of the BD-RIS scheme with CW-FC/GC/SC architectures is more significant than its competitors, indicating that CW-FC/GC/SC architectures are more sensitive to power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7, we present the minimum radar SCNR versus the number of groups G with different numbers of BD-RIS cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We observe that with the same number of groups, the radar output SCNR increases with the increasing number of BD-RIS cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The performance enhancement comes from the additional DoF of passive beamforming induced by the increasing number of cells, and the joint design of transmit waveform, the BD-RIS with more general constraints, and the 11 1 2 4 8 12 16 20 24 32 40 Number of groups, G 0 2 4 6 8 10 12 14 16 18 20 Minimum Radar Output SCNR (dB) CW-SC CW-FC Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Minimum radar output SCNR versus the number of groups G with different BD-RIS cells NS and communication threshold Γ = 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 80 60 40 20 0 20 40 60 80 Angle (Degree) 40 35 30 25 20 15 10 5 0 Normalized Transmit Beampattern (dB) Target 1 Target 2 Target 3 Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC (a) 80 60 40 20 0 20 40 60 80 Angle (Degree) 40 35 30 25 20 15 10 5 0 Normalized Transmit Beampattern (dB) Target 1 Target 2 Target 3 Radar Only, CW-FC BD-RIS, CW-FC BD-RIS, CW-GC BD-RIS, CW-SC (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Transmit beampattern of BD-RIS obtained via proposed algorithm for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (a) communication threshold Γ = 0 dB, (b) communication threshold Γ = 15 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' matched filters, which also confirms the results in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' More importantly, the slope of each carve becomes steeper with the increasing number of groups, which indicates that the number of non-zero elements of BD-RIS matrices plays a significant role in increasing system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 3) Radar Performance: In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 8, we present the transmit beampattern obtained by the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Results show that regardless of BD-RIS architectures, the transmit power (a) Radar-only, CW-FC (b) BD-RIS, CW-FD (c) BD-RIS, CW-GD (d) BD-RIS, CW-SD Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The space-range beampattern behavors of the receive weights for the target 3 detection with communication threshold Γ = 10 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' mainly concentrates around the three targets, which guarantees a high SCNR output at target directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Moreover, the BD- RIS with CW-FC/GC architectures can focus more energy toward targets and has a lower sidelobe than that with CW-SC architecture, thanks to the more flexible passive beamfomring control provided by the CW-FC/GC architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We also observe that the transmit power towards target 3 is much high than other targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This is because, as mentioned early, target 3 is the weakest one, which needs more energy to improve the output radar SCNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' In addition, the transmit beampattern performance for BD-RIS with all architectures gets worse with larger communication QoS thresholds, which confirms the conclusion in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 9 shows the space-range beampattern of the designed waveform when BD-RIS has different architectures, where the beampattern of the k-th target is computed as P k R (θ, l) = |Tr{(U⋆ k)H A (θ) ΦTGW⋆Jrl}|2 [39]–[41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Without loss of generality, we take target 3 (k = 3) as an example to illustrate the space-range behavior of the designed waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Results show that the space-range beampattern can form a mainlobe at the location of the target k = 3 (green circle), but achieve null points at the locations of the other non-of-interest targets (red circles) and strong clutter sources (black rectangles) for all proposed architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' This phenomenon can be explained as follows: i) To detect target k, the other targets are regarded as interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ii) BD-RIS with more general architectures can provide more DoFs to resist strong clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' CONCLUSION This paper considers the use of BD-RIS in the DFRC system in the presence of multiple targets and strong clutters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' We start by reviewing the BD-RIS architectures, and deriving the communication and radar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Our objective is to maximize the minimum radar output SCNR subject to the constraints of communication QoS, BD-RIS coefficients, and power budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, a general algorithm utilizing the ADMM 30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy (sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)-30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy (sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)-30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy(sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)-30 40 50 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 ncy (sino) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4-70 dB 80 90 100 110 120 5 30Normalized Spatial freque 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='8 1 5 10 15 20 Range (m)12 approach is developed to solve the resulting complicated non- convex max-min optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Finally, simulation results demonstrate the effectiveness of the proposed design algorithm, and the superiority of employing the BD-RIS in DFRC systems in terms of enhancing both communication and radar performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Based on this initial work, there are many issues worth studying for future research on BD-RIS aided DFRC, such as wideband waveform design, the scenarios for target estimation, as well as exploring the application of multi- sector BD-RIS in DFRC systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' APPENDIX A PROOF OF LEMMA 1 Given that f (w, γ) = wHΥw γ is jointly concave in w and γ when Υ ⪰ 0 and γ ≥ 0 [43], the first order approximation of f (w, γ), denoted by f (w, γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' wn, γn), is a majorizer of f (w, γ) at the point (wn, γn), which is f (w, γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' wn, γn) = f (wn, γn) + ( ∂f ∂w|w=wn)T (w − wn) (42a) + ( ∂f ∂w∗ |w=(wn)∗)T (w − (wn)∗) + (∂f ∂γ |γ=γn)T (γ − γn) + ( ∂f ∂γ∗ |γ=(γn)∗)T (γ − (γn)∗) = (wn)HΥwn γn + 2ℜ \uf8f1 \uf8f2 \uf8f3 � 2Υwn γn (wn)HΥwn (γn)2 �H � w − wn γ − γn �\uf8fc \uf8fd \uf8fe = 2ℜ � (wn)HAw � γn − γ (wn)HΥwn (γn)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The proof is thereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 1 We start by rewriting objective (39a) as [43] ℜ � Tr � ΛH g (Φg − Θg) �� + ̺ 2 ∥Φg − Θg∥2 F = −ℜ � Tr � ΘH g (Λg + ̺Φg) �� + ̺ 2 ∥Φg∥2 F + ̺M � �� � constant .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' Then, problem P2−1 AL,Θg can be symplified as max Φg ℜ � Tr � ΘH g (Λg + ̺Φg) �� s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' ΘH g Θg = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (43) Performing SVD to Λg + ̺Φg as BgΣgDH g = Λg +̺Φg, we can re-arrange the objective of (43) as ℜ � Tr � ΘH g (Λg + ̺Φg) �� = ℜ {Tr (ΣgZg)} = M � i=1 Σg [i, i] Zg [i, i] , (44) where Zg = DH g ΘH g Bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' (44) achieves its maximum when Zg = IM×2M, yielding the optimal solution Θg = Bg [IM×M, 0M×M] DH g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' The proof is thus completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' REFERENCES [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dE1T4oBgHgl3EQflgSY/content/2301.03286v1.pdf'} +page_content=' 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