diff --git "a/1NFAT4oBgHgl3EQfCxwU/content/tmp_files/load_file.txt" "b/1NFAT4oBgHgl3EQfCxwU/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/1NFAT4oBgHgl3EQfCxwU/content/tmp_files/load_file.txt" @@ -0,0 +1,759 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf,len=758 +page_content='1 Can Continuous Aperture MIMO Achieve Much Better Performance than Discrete MIMO?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhongzhichao Wan, Jieao Zhu, and Linglong Dai, Fellow, IEEE Abstract—The concept of continuous-aperture multiple- input multiple-output (CAP-MIMO) technology has been proposed recently, which aims at achieving high spectrum density by deploying extremely dense antennas or even continuous antennas in a given aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The fundamental question of CAP-MIMO is whether it can achieve much better performance than the traditional discrete MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In this paper, to model the CAP-MIMO, we use self- adjoint operators to depict the structural characteristics of the continuous random electromagnetic fields from physical laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, we propose a non-asymptotic performance comparison scheme between continuous and discrete MIMO systems based on the analysis of mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We show the consistency of the proposed scheme by proving that the mutual information between discretized transceivers converges to that between continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numeri- cal analysis verifies the theoretical results, and suggests that the mutual information obtained from the discrete MIMO with widely adopted half-wavelength spaced antennas al- most achieves the mutual information obtained from CAP- MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Index Terms—Multiple-input multiple-output (MIMO), Continuous-aperture MIMO (CAP-MIMO), mutual infor- mation, random fields, Fredholm determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' INTRODUCTION The spectrum efficiency of wireless communication systems has been greatly improved from 3G to 5G because of the use of multiple-input multiple-output (MIMO) technology [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The MIMO systems utilize multiple antennas to exploit the spatial multiplexing gain [4], where the antennas are modeled as discrete points in the continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Along with the tendency of increasing the number of antennas to achieve higher spectrum efficiency, people are considering deploying extremely dense antennas in a given aperture [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' When the number of antennas in a given aperture tends to infinity, the traditional MIMO systems with transceivers composed of discrete point antennas are equivalent to the MIMO systems with continuously controllable transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, the MIMO with extremely dense All authors are with the Department of Electronic Engineer- ing, Tsinghua University as well as Beijing National Research Center for Information Science and Technology (BNRist), Bei- jing 100084, China (E-mails: {wzzc20, zja21}@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' daill@tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This work was supported in part by the National Key Research and Development Program of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2020YFB1807201), in part by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 62031019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' antennas is called continuous-aperture MIMO (CAP- MIMO), and is also called holographic MIMO [7]–[9] or large intelligent surface [5], [10] in the recent litera- ture1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' It has attracted increasing interest in the research of MIMO technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Recent works about CAP-MIMO include pattern optimization [6], antenna design [11], channel estimation [7], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For CAP-MIMO, the fundamental question is whether the CAP-MIMO system can achieve much better performance than the traditional discrete MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Related works The structure of CAP-MIMO has been defined in the previous part but there are many structures for realizing the discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, we need to choose which structure of the discrete MIMO to compare with CAP- MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' A representative structure of discrete MIMO uses half-wavelength spaced antennas to compose the transceivers [12]–[14], because half-wavelength sampling of the electromagnetic field can reconstruct the original field according to the sampling theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' There have been several works discussing the per- formance comparison between CAP-MIMO and discrete MIMO with half-wavelength spaced antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The perfor- mance comparison is from the degrees of freedom (DoF) perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Specifically, when discarding the evanescent wave components, the Fourier transform of the received field, which is in the wavenumber domain, is concentrated in a circle or a segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This concentration phenomenon means that the field is bandlimited in the wavenumber domain, thus it can be perfectly recovered from the half- wavelength sampling points in the spatial domain [15] according to the Nyquist sampling theorem [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The above conclusion is based on the assumption that we can observe the received field in the infinitely large spatial domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' However, in practice, the destination where we can observe the field is in a finitely large aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For a rigorous analysis framework of the DoF in a finitely large aperture, the prolate spheroidal wave func- tion (PSWF) [17] is introduced to perform orthogonal expansion on the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Specifically, to 1The MIMO with extremely dense antennas can be accurately de- scribed by the name CAP-MIMO, while holographic MIMO and large intelligent surface do not focus on the continuity of the transceiver apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, in the rest part of the paper, we will prefer using the name CAP-MIMO rather than using other names like holographic MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='08411v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='IT] 20 Jan 2023 2 reconstruct the wavenumber-bandlimited electromagnetic field observed in a length-l spatial region, the PSWFs were used as the basis based on the Slepian’s concentra- tion problem [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Such an electromagnetic field can be perfectly reconstructed from infinite number of PSWFs, and approximately reconstructed from a finite number of PSWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' If the reconstruction error can be controlled within a given threshold by using N0 PSWFs, the number of DoFs of the field can be approximated by N0 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This analyzing scheme is strict for arbitrary l, but can only provide the asymptotic result of the DoF, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', the quantitative result of N0 can be obtained only when the length l or the frequency tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' However, the practical systems are with finitely large aperture and finite frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The asymptotic result can not provide quantitative number of DoFs for practical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' There- fore, a non-asymptotic performance comparison scheme between CAP-MIMO and discrete MIMO is required for the accurate performance comparison with finitely large apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Our contributions To solve this problem, in this paper, we provide a non- asymptotic performance comparison scheme between CAP-MIMO and discrete MIMO, and we further prove the rationality of the scheme2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Specifically, the contribu- tions of this paper can be summarized as follows: We build models of CAP-MIMO and discrete MIMO based on electromagnetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For CAP-MIMO with continuous transceivers, we model the structural characteristics of the continuous random electro- magnetic fields from physical laws by using self- adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Based on this model, we can utilize the spectrum theory of operators to derive the information that can be obtained from the received field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The existing models of MIMO with discrete transceivers are spatially discretized from the contin- uous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, signal-to-noise ratio (SNR) control schemes are introduced to ensure the fairness of the comparison between CAP-MIMO and discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, before comparing the performance between CAP-MIMO with continuous transceivers and tra- ditional MIMO with discrete transceivers, we first utilize the simplified model with continuous trans- mitter and discrete receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Under this simplified model, the transmitter is continuous, which is the same as that in the CAP-MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' By theo- retically analyzing the mutual information that can be obtained from the discrete receiver in this sim- plified model, we can obtain some insights about how the discretization of the receiver affects the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, the theoretical proof 2Simulation codes will be provided to reproduce the re- sults in this paper: http://oa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='cn/dailinglong/publications/ publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' of the convergence of the mutual information in the simplified model can inspire the analysis of a more practical scenario, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', the discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Finally, we extend the convergence proof from the model with discrete receiver to the model with discrete transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We prove that the mutual in- formation between the discrete transceivers con- verges to the mutual information between continuous transceivers when the number of antennas of the discretized transceivers tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, the fairness of the performance comparison is guar- anteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numerical results are provided to verify the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, it shows the near- optimality of the half-wavelength sampling of the transceivers in traditional discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Organization and notation Organization: The rest of our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' II introduces the basic model of EIT and proposes models with continuous or discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information between the transceivers is also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' III proves the convergence of the mutual information between continuous transmitter and discrete receiver when the number of discrete antennas increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, the convergence of the mutual information between discrete transceivers is illustrated in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Finally, we conclude the paper in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Notation: bold characters denote matrices and vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' j is the imaginary unit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' E [x] denotes the mean of random variable x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' x∗ denotes the conjugation of a number or a function x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' XH denotes the conjugate transpose of a vector or a matrix X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' µ0 is the permeability of a vacuum, Z0 is the free-space intrinsic impedance and c is the speed of light in a vacuum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' ∇ is the nabla operator, and ∇× is the curl operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' |φ⟩ is the quantum mechanical notation of a function φ, where the inner product is denoted by ⟨ψ| φ⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' det(·) denotes the matrix determinant or the Fredholm determinant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' tr(·) denotes the trace of a matrix or an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Im denotes the m×m identity matrix, 1 denotes the indentity operator, δ(x) denotes the delta function, and 1i=j denotes the indicator function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' |x| denotes the modulus of a complex variable, and ∥f(x)∥L∞(a,b) is the uniform norm of the function f(x) over the interval [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' C∞(K) denotes the set of smooth functions supported on a compact set K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' MODELS OF CONTINUOUS AND DISCRETE SYSTEMS In this section, we introduce the models of contin- uous and discrete systems for performance comparison between CAP-MIMO and discrete MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We control the SNR at the receiver side to ensure the fairness of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The information obtained from these models is derived from operators and matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Basic model of electromagnetic information theory To model the transceivers and the channel, we follow the approach of electromagnetic information theory (EIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The EIT is an interdisciplinary subject that integrates the classical electromagnetic theory and information theory to build an analysis framework for the ultimate performance bound of wireless communication systems [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The anal- ysis framework of EIT is based on spatially continuous electromagnetic fields, which provides us the tool to model and analyze the continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, for the consistency, the model of discrete transceivers are viewed as the discretization of the continuous model from EIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The model of EIT is built on the vector wave equa- tion [21] without boundary conditions, which is expressed by ∇×∇×E (r)−κ2 0E (r) = jωµ0J (r) = jκ0Z0J (r) , (1) where κ0 = ω√µ0ε0 is the wavenumber, and Z0 = µ0c = 120π [Ω] is the free-space intrinsic impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We assume that the transceivers are confined in two regions Vs and Vr, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The current density at the source is J(s), where s ∈ R3 is the coordinate of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The induced electric field at the destination is E(r), where r ∈ R3 is the coordinate of the field observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' To solve the linear partial differential equation (1), a general theoretical approach is to introduce the dyadic Green’s function G(r, s) ∈ C3×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to the linearity of (1), the electric field E(r) can be expressed by E(r) = � Vs G(r, s)J(s)ds, r ∈ Vr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (2) By exploiting the symmetric properties of the free space, the Green’s function in unbounded, homogeneous medi- ums at a fixed frequency point is [22] G(r, s) = jκ0Z0 4π � I + ∇r∇H r κ2 0 � ejκ0∥r−s∥ ∥r − s∥ = jκ0Z0 4π ejκ0∥r−s∥ ∥r − s∥ � � I − ˆpˆpH� + j 2π ∥r − s∥ /λ � I − 3ˆpˆpH� − 1 (2π ∥r − s∥ /λ)2 � I − 3ˆpˆpH� � [Ω/m2], (3) where ˆp = p ∥p∥ and p = r − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since there are some non-ideal factors at the receiver that corrupts the recieved field, we call them the noise field N(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The received electric field can be expressed by Y(r) = E(r) + N(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The above equations represent the deterministic model in the electromagnetic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' To satisfy the demand of wireless communication, we need to convey information through the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Specifically, the wireless communiation system encodes the information in the current J(s), and decodes the information from the noisy electric field Y(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Due to the randomness of the transmitted bit source, the electro- magnetic fields are randomly excited by the transmitter equipments before being radiated into the propagation media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, the electromagnetic fields should be modeled as random fields [23], which are random func- tions with several arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We denote the autocorre- lation function of the current and the electric field as matrix-valued functions RJ(s, s ′) = E[J(s)JH(s ′)] and RE(r, r ′) = E[E(r)EH(r ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The relationship between RJ and RE is determined by the Green’s function, which is RE(r, r′) = � Vs � Vs G(r, s)RJ(s, s′)GH(r, s)dsds′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (4) Similar definitions of the autocorrelation functions for the noise field and the noisy electric field are repre- sented as RN(r, r ′) = E[N(r)NH(r ′)] and RY(r, r ′) = E[Y(r)YH(r ′)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Continuous trasceivers In this part, we will build the model of CAP-MIMO with continuous transceivers based on the EIT model in the above subsection, and then derive the mutual information between the continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For sim- plicity, in the rest part of the paper, we assume that the transceivers are linear along the ˆz-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, since the current J can only exist on the linear source and we only observe the electric field on the linear receiver, we express all the physical quantities in a Cartesian coordinate system that satisfies s = (0, 0, s) and r = (d, 0, r), where d is the distance between the parallel source and destination line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This model corresponds to single-polarized linear antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Through this simplifica- tion scheme, we use J(s) and E(r) instead of J(s) and E(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The relationship between them can be expressed by E(r) = � l 0 G(r, s)J(s)ds, where G(r, s) is the upper left element of the matrix G(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We can derive G(r, s) as G(r, s) =jZ0ej2π √ x2+d2/λ 2λ √ x2 + d2 � j 2π √ x2 + d2/λ d2 − 2x2 x2 + d2 + d2 x2 + d2 − 1 (2π/λ)2(x2 + d2) d2 − 2x2 x2 + d2 � , (5) where x = r − s and λ = 2π/κ0 is the wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Here we consider the scenario with no channel state information, which means that the signals on the source are under equal power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The second moments (autocorrelation) of J are denoted by RJ(s, s′) = Pδ(s− s′), s, s′ ∈ [0, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since the noiseless received field is uniquely deter- mined by the source and the deterministic channel, the autocorrelation function of the electric field is expressed 4 by the source autocorrelation RJ(s, s′) and the Green’s function G(r, s), written as RE(r, r′) = � l 0 � l 0 G(r, s)RJ(s, s′)G∗(r′, s′)dsds′ = P � l 0 G(r, s)G∗(r′, s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (6) The received field on the destination is Y (r) = E(r) + N(r), where N(r) is the noise field at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In this paper, we consider thermal noise model E [N(r)N ∗(r′)] = n0 2 δ(r − r′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to [24], we can perform Mercer expansion on the electric field E(r) to obtain a set of mutually independent random variables ξk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The expansion can be written as E(r) = � k ξkφk(r), where E[ξkiξ∗ kj] = λki1i=j and ⟨φki(r), φkj(r)⟩ = δkikj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This expansion scheme has split the continuous field into independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since the white noise field can be expanded under arbitrary orthogonal bases, the continuous channel is also decomposed into independent subchannels, which makes the mutual information of the subchannels summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Next we will show that for the operator TE := φ(r) → � l 0 KE(r, r′)φ(r′)dr′, where KE(r, r′) = RE(r, r′) = P � l 0 G(r, s)G∗(r′, s)ds, all of its eigenvalues are real and nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, the sum of its eigenval- ues �∞ i=1 λi equals P � l 0 � l 0 G(r, s)G∗(r, s)drds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Notice that TE can be decomposed to T ∗T, where T := φ(r) → √ P � l 0 G(r, s)φ(r)ds and T ∗ := φ(r) → √ P � l 0 G∗(r, s)φ(r)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This decomposition means that TE = T ∗ E is a self-adjoint operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We assume that λ is an eigenvalue of TE and φ(r) is the corresponding eigenfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗ Eφ(r), φ(r)⟩ = ⟨φ(r), TEφ(r)⟩ = λ∗, (7) we know that λ is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From λ = λ⟨φ(r), φ(r)⟩ = ⟨T ∗Tφ(r), φ(r)⟩ = ⟨Tφ(r), Tφ(r)⟩ ⩾ 0, (8) we know that λ is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From [25] we know that an integral operator on [a, b] is a trace class operator if its kernel K(x, y) satisfies K(x, y) and ∂yK(x, y) are continuous on [a, b]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' There- fore TE is a trace class operator, which means that the sum of its eigenvalues is finite and can be expressed by [26] tr(TE) = � l 0 KE(r, r)dr = P � l 0 � l 0 G(r, s)G∗(r, s)drds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (9) Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The non-negative values λk n0/2 represent the SNR of the independent subchannels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information between the noisy received field and the current on the source can be expressed by I0(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Y ) = +∞ � k=1 log � 1 + λk n0/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (10) By introducing the Fredholm determinant which is the determinant of operators, we can express (10) by I0(J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Y ) = log det � 1 + TE n0/2 � , (11) where (TEφ)(r) := � L 0 RE(r, r′)φ(r′)dr′ and λk are the eigenvalues of TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Our analysis here is based on the sim- plified model with uni-polarized linear antennas as the transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This simplification reduces the dimension of the problem, where random fields degenerate to one- dimensional random processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the more general scenarios, such analyzing schemes are still effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' If the random field is defined in a region X, we can expand E(r) by E(r) = � k ξkΦk(r) and its autocorrelation function RE(r, r ′) by RE(r, r ′) = � k λkΦk(r)ΦH k (r ′) [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The expansion satisfies that λk and Φk(r) are eigenvalues and eigenfunctions of the integral equation � X RE(r, r ′)Φ(r ′)dr ′ = λkΦ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Similar expressions of the mutual information in (10) and (11) can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Continuous transmitter and discrete receiver Before building the model with discrete transceivers, in this subsection, we will first build a simplfied model with continuous transmitter and discrete receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The simplfied model analyzed here can bring some insights about the discretization of both transceivers and the SNR control schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the continuous transmitter, we still use the length-l linear transmitter along the ˆz- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the discrete receiver, we build a model with m point antennas on a segment parallel to the linear transmitter in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The ith point antenna is placed on ri ∈ [0, l].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The correlation matrices of the received signals and received noise are denoted by K ′ E and K ′ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the received signals, we assume that it is the sampling of the continuous electric field on the point ri, which means that K ′ E = KE(ri, rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' However, for the received noise on the antenna, it can not directly be assumed as the point sampling of the noise field, because of the delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' To solve this problem, we assume that K ′ N = n1 2 Im is an identity matrix, and control the signal-to-noise ratio (SNR) of this model the same as that of the continuous model to ensure the fairness of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The SNR at the receiver of the continuous model is �∞ i=1 λi n0/2, where λi is the ith eigenvalue of the operator TE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From Lemma 1 we know that �∞ i=1 λi n0/2 = P n0/2 � l 0 � l 0 G(r, s)G∗(r, s)drds is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The SNR at the receiver of the discrete model is �m i=1 λ ′ i n1/2, where λ ′ i is the ith eigenvalue of the matrix K ′ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The SNR control scheme is necessary because if we do not control the SNR, the mutual information that can be obtained from the discrete antennas in the receiver may infinitely increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Let us take a counter-example where the power of received signal and received noise on 5 each point antenna remain unchanged when the number of antennas in a given aperture increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For dense antennas we can assume that N received signals of the antennas in a small aperture are nearly the same, while the corresponding noises are independent according to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, the SNR for the N antennas will keep near-linearity increasing with N, since when we perform combing of the N received signals we have SNR = E[(�N i=1 Ei)(�N i=1 E∗ i )] E[(�N i=1 Ni)(�N i=1 N ∗ i )] ≈ N E[E1E∗ 1 ] E[N1N ∗ 1 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore the mutual information that can be obtained from the N antennas will keep near-logarithm increasing with N, which corresponds to the simulation in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to (9), the noise power in the discrete receiver model can be controlled by n1 = n0 �m i=1 KE(ri, ri) � l 0 KE(r, r)dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (12) We denote the determinant of matrix K ∈ Cm×m by det(Ki,j)m i,j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then we can express the mutual informa- tion between the transceivers by I1 = log � det(K ′ N + K ′ E) det(K ′ N) � = logdet � 1i=j + KE(ri, rj) n1/2 �m i,j=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (13) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Here the SNR on each of the point antennas in the discrete model changes with the density of point anten- nas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Notice that L m �m i=1 KE(ri, ri) is the approximation of the integral � l 0 KE(r, r)dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' When m approximates infinity, n1 will approximate mn0 2l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This phenomenon has several annotations, including the increase of the noise power on each point antenna, the reduction of antenna efficiency, and the corollary of the discretization of EIT continuous models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From the perspective of noise power, we can explain it by spatial sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the point antenna arrays, more antennas on a given aperture corresponds to a higher sampling rate in the spatial domain and a wider lowpass filter in the wavenumber domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since a wide lowpass filter can receive more noise power from the white noise field, the noise power should increase with the density of the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From the perspective of antenna efficiency, the well- known Hannan’s efficiency shows that for both transmit- ting and receiving antennas, the antenna gain is propor- tional to lxly for two-dimensional surface antennas [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, for the linear model we considered, the an- tenna gain will be inversely proportional to the sampling number when the antennas are dense enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Besides these two annotations, another perspective is viewing the model of discrete point antennas as the discretization from the EIT continuous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' If we consider m linear continuous antennas instead of point antennas in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' All the antennas are connected head to tail to occupy the [0, l] position in the space and detect the electric field by inner producting it with its eigenmode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This model fulfills the requirement of discretizing the continuous receiver to discrete receiving antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The signal received by the ith antenna is Yi = � ai+1 ai Y (r)φ(r)dr, where [ai, ai+1] is the occupied region of the ith antenna, and φ(r) is the eigenmode of the antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' If we assume φ(r) ≡ 1, the correlation matrix of the received electric field can be expressed by (KE)i,j = E �� ai+1 ai � aj+1 aj E(r)E∗(r′)drdr′ � = (ai+1 − ai)(aj+1 − aj)KE(ri, rj), (14) where ri ∈ [ai, ai+1] and rj ∈ [aj, aj+1] according to the mean value theorem for integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the noise field on the destination, we have (KN)i,j = E �� ai+1 ai � aj+1 aj N(r)N ∗(r′)drdr′ � = � (ai+1 − ai) n0 2 i = j 0 i ̸= j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (15) Therefore, the SNR after the discretization will discreases by ai+1 − ai, which is the case when the antennas are dense enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' After explaining the rationality of the SNR control scheme, we will introduce the following lemma to show the convergence of the noise power on each discrete point antenna, which will be useful for the following proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' When the number of antennas m in a given aperture increases, the noise power on each antenna n1/2 will approach mn0 2l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The difference between them is at most inverse-proportional to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Proof: From (12) and the middle point quadrature rule, we have ���� l mn1 − n0 ���� = n0 ��� � l 0 KE(r, r)dr − l/m �m i=1 KE(ri, ri) ��� ��� � l 0 KE(r, r)dr ��� ⩽ n0l3 ���K ′′ E(r, r) ��� L∞(0,l) 24m2 ��� � l 0 KE(r, r)dr ��� , (16) which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Discrete transceivers The models discussed in the above subsections keep the transmitter continuous and only perform discretization on the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' However, the commonly used model to depict wireless communication is the discrete MIMO model, in which both the transceivers are modeled as discrete point antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, in this section, we will introduce a model which discretizes the transceivers simultaneously, which is the extension of the model with continuous transmitter and discrete receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, similar 6 d Continuous Transmitter Continuous Receiver d Discrete Receiver l / 2 \uf06c l d Discrete Receiver l / 2 \uf06c Discrete Transmitter Continuous Transmitter 0I 1I 2I Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Comparison between the three models in this section with continuous transceivers and the model with discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' to the scheme in the above subsection, we will provide the corresponding SNR control scheme to ensure the fairness of the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Specifically, we build a model with m point antennas on a length-l segment in the source region and m point antennas on a length-l segment in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Similar to the above subsection, we assume that the ith point antenna is placed at si in the source region and ri in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The correlation matrix of the signals in the source region is set to be an identity matrix K ′′ J = PIm, which corresponds to the power allocation scheme with no channel state information at the transmit- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The channel gain from the ith antenna in the source region and the jth antenna in the destination region can be expressed by Hi,j = G(si, rj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The correlation matrix of the received signal is denoted by K ′′ E = HK ′′ JHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The noise matrix is denoted by K ′′ N = n2 2 Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Similar SNR control on the receiver side is used, which is expressed by n2 = n0 �m i=1 �m j=1 G(ri, sj)G∗(ri, sj) � l 0 � l 0 G(r, s)G∗(r, s)drds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (17) The mutual information between the transceivers is ex- pressed as: I2 = log � det(K ′′ N + K ′′ E) det(K ′′ N) � = logdet � 1i=j + � k G(ri, rk)G∗(rj, rk) n2/2 �m i,j=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (18) The comparison between the three models built in Sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' II-B, Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' II-C and in this subsection is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In the following two sections we will introduce the intermediate quantity I ′ 0 and I ′′ 0 to theoretically prove that I1 and I2 converge to I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The flow chart of the proof is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' PERFORMANCE COMPARISON BETWEEN DISCRETE AND CONTINUOUS RECEIVERS In the above section we have proposed the mod- els of continuous and discrete transceivers and derived the corresponding mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Before compar- ing the performance between CAP-MIMO with contin- uous transceivers and traditional MIMO with discrete transceivers, we first utilize the simplified model with continuous transmitter and discrete receiver in this sec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Under this simplified model, the transmitter is con- tinuous, which is the same as that in the CAP-MIMO system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The comparison is based on the convergence analysis of the mutual information when the number of antennas in the discrete receiver increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numer- ical analysis is provided to verify the correctness of the convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The discussion about discrete transceivers inspired by the analysis in this part will be in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Convergence analysis of the mutual information To compare the mutual information I0 and I1, we intro- duce an intermediate quantity I ′ 0 = logdet � 1 + mTE ln1/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We can bound |I0 −I1| by |I0 −I ′ 0|+|I1 −I ′ 0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to [30], I1 can be viewed as the approximation of I ′ 0 using a numerical integral scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In our discussion the point antennas in the destination region are evenly spaced, which means that ai = (i−1)l/m and ri = (i−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='5)l/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' To bound |I1 − I ′ 0|, we introduce the following lemma from [30]: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We define d(z) := det(1+zT) and dQ(z) := det (1i=j + wjzK(ri, rj))m i,j=1, where K is the kernel of the operator T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=" The difference between d(z) and dQ(z) 7 0I ' 0I 1I 2I '' 0I Discretize the receiver Discretize the transmitter Discretize the transceivers Lemma 1 Lemma 2 Lemma 3 Lemma 4 Lemma 5 Theorem 1 Theorem 2 Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Flow chart of the proof in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' is d(z) − dQ(z) = ∞ � n=1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' � Qn m(Kn) − � [a,b]n Kn(x1, · · · , xn)dx1 · · · dxn � , (19) where Kn(x1, · · · , xn) = det (K(xi, xj))n i,j=1, and Qn m(f) = �m j1=1,··· ,jn=1 �n i=1 wjif(rj1, · · · , rjn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lemma 2 provides a method to compare the difference between a Fredholm determinant of operator and a clas- sical determinant of matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In our model, the operator T corresponds to the integral operator TE, z equals 2m ln1 ,and wj = l/m according to the equally spaced antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Notice that Qn m(f) is the numerical approximation of the integral � [a,b]n Kn(x1, · · · , xn)dx1 · · · dxn, we need to use numerical integral theory to estiamte the approxi- mation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the model with equally spaced antennas, this expression corresponds to a multivariate m-point composite midpoint quadrature rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the error bound of a m-point composite midpoint quadrature [31], we have �����Qm(f) − � l 0 f(x)dx ����� ⩽ l3 24m2 ∥f ′′∥L∞(0,l) (20) According to [32], the numerical approximation error for multiple integrals in a n-dimensional unit cube can be bounded by ����� � Gn f − � n � i=1 � Qi(f) ����� ⩽ E1 + n � i=2 i−1 � j=1 WjEi, (21) where Qj(g) := � j wi,jg(xi,j), Wi = � j |wi,j| and Ei ⩾ ���Qi(f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' xi) − � 1 0 f(x1, · · · , xn)dxi ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to the models in this paper, we have wi,j = l/m and Wi = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' By simple variation of the integral band, we can bound the approximation error of the multi-dimensional numerical integral quadrature rule by �����Qn m(Kn) − � [0,l]n Kn(x1, · · · , xn)dx1 · · · dxn ����� ⩽ ni−1 n � i=1 Ei, (22) where Ei = �����Qi(Kn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' xi) − � l 0 Kn(x1, · · · , xn)dxi ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (23) Therefore, we have �����Qn m(Kn) − � [0,l]n Kn(x1, · · · , xn)dx1 · · · dxn ����� ⩽ nln+2 24m2 |Kn|2 (24) where |Kn|2 = max i ∥ ∂2Kn ∂x2 i ∥L∞((0,l)n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Similar to [30, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='4], we can bound |Kn|k by using the Hadamard’s inequality, which leads to |Kn|k ⩽ 2knn/2 � � max i+j⩽k ����� ∂i x∂j yK(x, y) ∂xi∂yj ����� L∞((0,l)2) � � n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (25) Next we will show that ��� ∂i x∂j yK(x,y) ∂xi∂yj ��� is upper-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since we have K(x, y) = � l 0 G(x, s)G∗(y, s)ds, we will first analyze the property of G(x, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We decom- pose G(x, s) as G1(x, s) + jG2(x, s), where G1, G2 ∈ C∞([0, l]2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The smoothness of G1, G2 in their domains 8 is trivial since they are compositions of polynomial func- tions, trigonometric functions and square root functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Consider the integral kernel K(x, y) expressed in terms of G1, G2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', K(x, y) = � l 0 � G1(x, s)G1(y, s) + G2(x, s)G2(y, s) � ds + j � l 0 � G1(y, s)G2(x, s) − G1(x, s)G2(y, s) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (26) Since G1(x, s) and G2(y, s) are smooth in [0, l]2, we can conclude that f1(x, y) = G1(x, s)G1(y, s) + G2(x, s)G2(y, s) and f2(x, y) = G1(y, s)G2(x, s) − G1(x, s)G2(y, s) are smooth in the same domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since compactly supported smooth functions attain their maxi- mum or minimum values, the partial derivatives of K(·, ·) are upper-bounded for any order i, j, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', ���� ∂i+jK(x, y) ∂xi∂yj ���� < ∞, ∀i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (27) Therefore, by substituting (24) and (25) into Lemma 2, we can bound the difference between the mutual information I ′ 0 and I1 by the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information I1 converges to the mutual information I ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The difference ���I1 − I ′ 0 ��� is at most inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Proof: From (6) we know that for the operator TE, the kernel function can be expressed by K(x, y) = � L 0 g(x, s)g∗(y, s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From (25) we have |Kn|2 ⩽ 4nn/2An, (28) where A = max ��� ∂i+jK(x,y) ∂xi∂yj ��� L∞((0,l)2) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore we have |d(z) − dQ(z)| ⩽ ∞ � n=1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' nln+2 24m2 max i ���� ∂2Kn ∂x2 ���� L∞((0,l)n) ⩽ ∞ � n=1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' nln+2 6m2 nn/2An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (29) According to the Stirling’s approximation, we have n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' ⩾ nne−n√ 2πn, which leads to |d(z) − dQ(z)| ⩽ l2 6m2 ∞ � n=1 � n 2π (Aezl)n nn/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (30) Since it is obvious that �∞ n=1 � n 2π (Aezl)n nn/2 is convergent, the difference between d(z) and dQ(z) is proportional to m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the difference between mutual information I1 and I ′ 0, we have |I1−I ′ 0| ⩽ |d(z) − dQ(z)| min(d(z), dQ(z)) < l2 6m2 ∞ � n=1 � n 2π (Aezl)n nn/2 , (31) where z = m ln1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From Lemma 1 we know that l mn1 ⩾ n0 − n0l3���K ′′ E(r,r) ��� L∞(0,l) 24m2| � l 0 KE(r,r)dr| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, z is upperbounded, which completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to Lemma 1 and Lemma 3, we have Theorem 1, which bounds the difference between I0 and I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information I1 that can be obtained from the discrete receiver converges to the mutual information I0 that can be obtained from the continuous receiver when the number of points increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The convergence rate is at least inverse-proportional to the square of the sampling number m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Proof: Since the Fredholm determinant f(z) = det(1 + zTE) is an analytic function, we have |det(1 + zTE) − det(1 + z1TE)| = |z − z1| ���� ∂det(1 + xTE) ∂x ���� x∈[min(z,z1),max(z,z1)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (32) In our assumption z = 2 n0 and z1 = 2m ln1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The analycity of det(1 + zTE) implies that ∂det(1+xTE) ∂x is also an anlytic function and is bounded on the interval [min(z, z1), max(z, z1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We denote M = max x ��� ∂det(1+xTE) ∂x ���, where x ∈ [min(z, z1), max(z, z1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From Lemma 1 we have ���� l mn1 − n0 ���� ⩽ n0l3 ���K ′′ E(r, r) ��� L∞(0,l) 24m2 � l 0 KE(r, r)dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (33) Since n1/m → n0/l when m approximates infinity, we denote the minimum value of n1/m by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, ���det(1 + 2 n0 TE) − det(1 + 2m ln1 TE) ��� can be bounded by ����det(1 + 2 n0 TE) − det(1 + 2m ln1 TE) ���� ⩽ Ml2 �����K ′′ E(r, r) ��� L∞(0,l) 12m2c � l 0 KE(r, r)dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (34) Similar to the direvation of (31), we know that when m increases, I ′ 0 will converge to I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The convergence rate is at least inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From Lemma 3 we know that ���I1 − I ′ 0 ��� is at most inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Since |I0 − I1| ⩽ ���I0 − I ′ 0 ��� + ���I1 − I ′ 0 ���, Theorem 1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Theorem 1 shows that the SNR control scheme between the discrete and continuous models is appropriate, since the limit of the mutual information of the discrete model is proved to be that of the continu- ous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' That is to say, our proposed model is self- consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, we can use the proposed model to compare the mutual information from the discrete and continuous receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Our analysis is based on 9 0 50 100 150 0 20 40 60 80 100 120 140 160 180 continuous receiver discrete receiver 5 10 15 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='5 3 continuous receiver discrete receiver Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information as a function of the sampling number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The transmitter is kept continuous and the receiver is discretized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' RE(r, r′) = P � l 0 G(r, s)G∗(r′, s)ds which corresponds to the scenario when no CSI can be obtained at the transmitter but not limited to this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' It can be eas- ily extended to other shapes of autocorrelation functions after power allocation at the transmitter, as long as the analyticity of RE(r, r′) is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numerical analysis about the mutual information As proven in the above subsection, the mutual infor- mation between the continuous transmitter and discrete receiver converges to the mutual information between continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, the model of the dis- crete receiver can be viewed as the discretization of the continuous receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In this subsection, we will use nu- merical analysis to show the correctness of the theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, we will show the near-optimality of the discrete receiver with half-wavelength sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We set the length l of the transceivers to 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The trans- mitter is kept continuous, while the receiver is discretized to m point antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The wavelength of the electromag- netic field is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='04 m, which correpsonds to the fre- quency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The distance between the transceivers varies from 10 m to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The simulation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From the simulation, we can observe the convergence of the mutual information between the continuous transmitter and the discrete receiver, which verifies the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For the three distances between transceivers, the half-wavelength sampling al- most achieves the supremum mutual information between continuous transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, half-wavelength sam- pling of the receiver is suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, when the distance between transceivers decreases, we can observe that the mutual information converges slower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' When the distance equals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1 m, the half wavelength sampling is at the critical state of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' If the distance is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1 m, a performance gap between the model with the continuous receiver and that with the discrete receiver may be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' This performance gap has theoretical meaning but may not be useful because the distance will be comparable to the wavelength in this scenario, where the evanescent wave components will hold a dominant position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' COMPARISON BETWEEN CONTINUOUS AND DISCRETE TRANSCEIVERS In the above section we have compared the mutual information between the models with continuous and discrete receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For both models the transmitter is kept continuous, which simplifies the analyzing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In- spired by the analysis in the above section, in this section we will compare the mutual information between contin- uous transceivers and that between discrete transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numerical analysis is then provided to show the near- optimality of the half-wavelength sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Convergence analysis of the mutual information The analysis in this section focuses on the difference between I0 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' It is an extension of the conver- gence analysis in the above section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We define I ′′ 0 = logdet � 1 + m2TE l2n2/2 � as an intermediate variable similar to I ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' First we will discuss the convergence of |I0 − I ′′ 0 | in the following lemma: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information I ′′ 0 converges to the mutual information I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The difference ���I0 − I ′′ 0 ��� is at most inverse-proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Proof: From the SNR control scheme of discrete transceivers (17) and the multivariate m-point composite midpoint quadrature rule, we have ����n0 − l2 m2 n2 ���� = n0 � l 0 K(r, r)dr ����� � l 0 � l 0 g(r, r, z)dzdr − l2 m2 m � i=1,j=1 g(ri, ri, rj) ����� ⩽ n0l4 24m2 � l 0 K(r, r)dr � ���� ∂2g(r, r, z) ∂r2 ���� L∞((0,l)2) + ���� ∂2g(r, r, z) ∂z2 ���� L∞((0,l)2) � , (35) where g(x, y, z) := G(x, z)G∗(y, z), ri = (i − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='5)l/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' It is obvious that n2/m2 converges to n0/l2 when m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We denote the minimum value of n2/m2 by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, according to (32), we know that ���det(1 + 2 n0 TE) − det(1 + 2m2 l2n2 TE) ��� converges to 0 when m → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, |I0 − I ′′ 0 | converges to 0, and the convergence rate is at least inversely proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 10 Then we will discuss the convergence of |I2 − I ′′ 0 | in the following lemma: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The difference ���I2 − I ′′ 0 ��� approaches 0 when m approaches infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, it is at most inverse- proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Proof: We denote the Fredholm determinant and its discretization by d(z) = det(1 + zT) and dV (z) = det (1i=j + wjz �m k=1 wkG(ri, rk)G∗(rj, rk))m i,j=1, where K is the kernel of the operator T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' To bound the difference between d(z) and dV (z), we define gn(x1, · · · , xn, s1, · · · , sn) as gn(x1, · · · , xn, s1, · · · , sn) = det � � g(x1, x1, s1) · · g(x1, xn, s1) · · g(xi, xj, si) · · g(xn, x1, sn) · · g(xn, xn, sn) � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (36) From the definition of g(x, y, z), we know that � l 0 g(xi, xj, si)dsi = K(xi, xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' According to the property of determinants that det(ai,j)m i,j=1 = � k1,··· ,km(−1)ka1,k1 · · · am,km, where k1 · · · km is the kth exchange of 1 · · · n, we can find that Kn(x1, · · · , xn) = � [0,l]n gn(x1, · · · , xn, s1, · · · , sn) ds1 · · · dsn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (37) If we define Cn m(gn) by (38) we have (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Here wαi correspondes to the distance between antennas in the source region and s correpsonds to the location of the antennas in the source region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' When further considering the discretization of the receiver as in (19), we should set xn 1 to the location of the antennas in the destination region, and add additional weights w which equals the distance between antennas in the destination region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Similar to the definition of Qn m in (19), we define V n m(gn) by V n m(gn) = �m j1,··· ,jn=1 � i wjiCn m(gn(rj1, · · · , rjn, s1,α1, · · · , sn,αn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' When sj,αi = rαi, we have V n m(gn) = m � j1,··· ,jn=1 m � α1,··· ,αn=1 � n � i=1 ji � � n � i=1 αi � gn(rj1, · · · , rjn, rα1, · · · , rαn) = m � j1,··· ,j2n=1 � 2n � i=1 ji � gn(rj1, · · · , rj2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (40) The difference between d(z) and dV (z) is d(z) − dV (z) = ∞ � n=1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' � V n m(gn) − � [0,l]n Kn(x1, · · · , xn)dx1 · · · dxn � = ∞ � n=1 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' � V n m(gn) − � [0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n � (41) Note that V n m(gn) is the numerical discretization of the function gn with 2n variables, we can bound V n m(gn) − � [a,b]2n gn(x1, · · · , x2n)dx1 · · · dx2n by using the multi- variate numerical integration error bound: �����V n m(gn) − � [0,l]2n gn(x1, · · · , x2n)dx1 · · · dx2n ����� ⩽ l2n−1 2n � i=1 Ei, (42) where Ei = �����Qi(gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' xi) − � l 0 gn(x1, · · · , x2n)dxi ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (43) According to the m-point composite midpoint quadrature rule, we have �����Qi(gn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' xi) − � l 0 gn(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' x2n)dxi ����� ⩽ l3 24m2 ���� ∂2gn ∂x2 i ���� L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='l)2n) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (44) From the Hadamard’s inequality [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' we can further bound it by �����V n m(gn) − � [a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='b]2n gn(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' x2n)dx1 · · · dx2n ����� ⩽ 2nl2n+2 24m2 max j ����� ∂2gn ∂x2 j ����� L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='l)2n) ⩽ 2nl2n+2 24m2 max � �nn/2 ����� ∂2g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' z) ∂z2 ���� L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='l)3) �n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 4nn/2 � � max i+j⩽2 ����� ∂i x∂j yg(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' z) ∂xi∂yj ����� L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='l)3) � � n � � ⩽ l2n+2 3m2 n(n+2)/2 � � max i+j+k⩽2 ����� ∂i x∂j y∂k z g(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' z) ∂xi∂yj∂zk ����� L∞((0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='l)3) � � n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (45) Similar to Lemma 3, we know that |I2 −I ′ 0| converges to 0, and the error is at most inverse proportional to m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, we have Theorem 2: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information I2 that can be obtained from the discrete transceivers converges to the mutual information I0 that can be obtained from the continuous transceivers when the number of antennas in the discrete transceivers increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The difference |I0−I2| is at least inverse-proportional to the square of the sampling number m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Similar to Remark 3 in Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' III, the convergence analysis in this section is not lim- ited to the scenario with equal power allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 11 Cn m(gn) = det � � � α1 wα1g(x1, x1, s1,α1) · · � α1 wα1g(xn, xn, s1,α1) · · � αi wαig(xi, xj, si,αi) · · � αn wαng(xn, x1, sn,αn) · · � αn wαng(xn, xn, sn,αn), � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (38) Cn m(gn) = � k1···kn (−1)k( m � α1=1 wα1g(x1, xk1, s1,α1)) · · · ( m � αn=1 wαng(xn, xkn, sn,αn)) = � k1···kn � (−1)k m � α1,··· ,αn=1 n � i=1 wαig(xi, xki, si,αi) � = m � α1,··· ,αn=1 �� n � i=1 wαi � � k1···kn (−1)k n � i=1 g(xi, xki, si,αi) � = m � α1,··· ,αn=1 �� n � i=1 wαi � gn(x1, · · · , xn, s1,α1, · · · , sn,αn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' (39) 0 50 100 150 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='12 continuous transceiver discrete receiver discrete transceiver 10 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1192 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information variation with different sampling num- bers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information that correpsonds to the three models with continuous and discrete transceivers is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The distance between the transceivers is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For arbitrary analytic function RJ(s, s′), the conver- gence of |I0 − I2| can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Instead of dis- cretizing � G(r, z)G∗(r′, z)dz to � i G(r, ri)G(r′, ri), we will discretize �� G(r, z)RJ(z, z′)G∗(r′, z′)dzdz′ to � i,j G(r, ri)RJ(ri, rj)G(r′, rj) in the extended sce- narios with power allocation schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Then, in- stead of g(x, y, z) we need a four-variable function h(x, y, z, ω) := G(x, z)RJ(z, ω)G(y, ω) and the deriva- tion procedure of the convergence has no essential differ- ence with Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numerical analysis about the mutual information In this subsection, we will verify the correctness of the convergence analysis in the above subsection by simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The length l of the transceivers is fixed to 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We have plotted the mutual information of the three models: continuous transceiver, continuous trans- mitter and discrete receiver, and discrete transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The transceivers are both discretized to m point antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The wavelength of the electromagnetic field is fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='04 m, which corresponds to the frequency of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' First we will show the scenarios when the distance be- tween the transceivers is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The distance between the transceivers varies from 50 m to 200 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The simulation results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' From the simulation results we find that the mutual information nearly keeps the same when the sampling number increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The reason for this phenomenon is that the DoF of the channel is nearly inverse-proportional to the distance between transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' For example, the DoF when the distance equals 50 m can be approximated by l2/(dλ) = 2, which means that when the sampling num- ber is 5, the multiplexing gain is almost fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Therefore, for large distances between transceivers, the dominant limitation is the channel DoF, which means that the suboptimal performance can be achieved by sampling sparser than half-wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Moreover, we have shown the variation of the mutual information with the sampling number when the distance between transceivers is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 5 the distance between the transceiver is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1 m and 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We can find that when the distance decreases, the dense sampling of the transceivers becomes important to fully explore the limit of the mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' However, the half-wavelength sampling of the transceivers still achieves suboptimal performance, which means that denser sampling schemes are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a comparison scheme be- tween continuous and discrete MIMO systems which is 12 0 50 100 150 0 20 40 60 80 100 120 140 160 180 continuous transceiver discrete receiver discrete transceiver Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information as a function of the sampling number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The mutual information values that correspond to the three models with continuous and discrete transceivers are plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The distance between the transceivers is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' based on a precise non-asymptotic analysis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Three information-theoretic models of the continuous and discrete transceivers were built, with the first model corresponds to the fully continuous electromagnetic infor- mation theory model, and the third model corresponds to the matrix-vector MIMO model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' We proposed physically consistent SNR control schemes to ensure the fairness of the comparison, and proved that the mutual information between discrete MIMO transceivers converges to that of continuous electromagnetic transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Numerical results verified the theoretical analysis and showed the near-optimality of the half-wavelength sampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Further works can be done by extending the lin- ear transceivers to rectangular or other two-dimensional transceivers for generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' The analysis based on the capacity after water-filling of the mutual information also remains to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' REFERENCES [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Andrews, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Buzzi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Choi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Hanly, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lozano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Soong, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang, “What will 5G be?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Areas Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1065–1082, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Boccardi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Heath, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lozano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Marzetta, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Popovski, “Five disruptive technology directions for 5G,” IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 52, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 74–80, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Andrews, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Durgin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Gupta, “Are we approaching the fundamental limits of wireless network densification?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' IEEE Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1558– 1896, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Swindlehurst, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Ashikhmin, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang, “An overview of massive MIMO: Benefits and challenges,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Topics Signal Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 742–758, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [5] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Decarli and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Dardari, “Communication modes with large intelligent surfaces in the near field,” arXiv preprint arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='10569, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [6] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Dai, “Pattern-division multiplexing for continuous-aperture mimo,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022 IEEE International Conference on Communications (ICC’22), May 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3287– 3292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [7] ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Demir, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Bj¨ornson, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sanguinetti, “Channel modeling and channel estimation for holographic massive MIMO with planar arrays,” IEEE Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Pizzo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Marzetta, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sanguinetti, “Spatially-stationary model for holographic MIMO small-scale fading,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Areas Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1964–1979, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sanguinetti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' D’Amico, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Debbah, “Wavenumber- division multiplexing in line-of-sight holographic MIMO com- munications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', early access, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='1109/TWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='3208961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Yuan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Ngo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Matthaiou, “Towards large intelligent surface (LIS)-based communications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 6568–6582, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [11] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Yurduseven, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Marks, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Fromenteze, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Smith, “Dynamically reconfigurable holographic metasurface aperture for a mills-cross monochromatic microwave camera,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Express, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 5281–5291, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Gao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Shim, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Gui, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Mao, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Adachi, “Principal component analysis-based broadband hybrid precoding for millimeter-wave massive MIMO systems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 6331–6346, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Shlezinger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Guidi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Dardari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Imani, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Eldar, “Beam focusing for near-field multi-user MIMO communications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 7476–7490, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [14] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wu and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang, “Intelligent reflecting surface enhanced wireless network via joint active and passive beamforming,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 5394–5409, Aug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Migliore, “Near field antenna measurement sampling strate- gies: From linear to nonlinear interpolation,” Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 7, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 10, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 257, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Landau, “Sampling, data transmission, and the Nyquist rate,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 55, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 10, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1701–1706, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Slepian and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty—I,” Bell System Techn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Journal, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 43–63, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Slepian, “On bandwidth,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 292–300, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Massimo, Wave Theory of Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Cambridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=': Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Press, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Dai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Debbah, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Poor, “Electromag- netic information theory: Fundamentals, modeling, applications, and open problems,” arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='02882, Dec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [21] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Dardari, “Communicating with large intelligent surfaces: Fun- damental limits and models,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Areas Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2526–2537, Nov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Poon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Brodersen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Tse, “Degrees of freedom in multiple-antenna channels: A signal space approach,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 523–536, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Yaglom, “Some classes of random fields in n-dimensional space, related to stationary random processes,” Theory of Proba- bility & Its Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 273–320, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [24] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Dai, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Chae, “Mutual infor- mation for electromagnetic information theory based on random fields,” arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content='00496, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Lax, Functional analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' John Wiley & Sons, 2002, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [26] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Gohberg and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Kre˘ın, Introduction to the theory of linear nonselfadjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' American Mathematical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', 1978, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' De Vito, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Umanit`a, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Villa, “An extension of mercer theorem to matrix-valued measurable kernels,” Applied and Com- putational Harmonic Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 339–351, May 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Pizzo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Sanguinetti, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Marzetta, “Fourier plane-wave series expansion for holographic MIMO communications,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Wireless Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 21, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 6890–6905, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [29] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Hannan, “The element-gain paradox for a phased-array an- tenna,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Antennas Propag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 423–433, Jul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [30] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Bornemann, “On the numerical evaluation of Fredholm deter- minants,” Mathematics of Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 79, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 270, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 871– 915, Sep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 13 [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Davis and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Rabinowitz, Methods of numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Courier Corporation, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Haber, “Numerical evaluation of multiple integrals,” SIAM review, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 481–526, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Meyer, Matrix analysis and applied linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' Siam, 2000, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'} +page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NFAT4oBgHgl3EQfCxwU/content/2301.08411v1.pdf'}