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Access Point Deployment for Localizing Accuracy and User Rate in Cell-Free Systems
无蜂窝系统中接入点部署以提高定位精度和用户速率

Fanfei Xu 徐凡飞Southeast University School of
东南大学Information Science and Engineering
信息科学与工程Nanjing 210096, China 南京 210096，中国

Shangqing Shi 上清诗Southeast University School of
东南大学Information Science and Engineering
信息科学与工程Nanjing 210096, China 南京 210096，中国

Shengheng Liu* 刘胜恒*Southeast University School of
东南大学Information Science and Engineering
信息科学与工程Nanjing 210096, China 南京 210096，中国s.liu@seu.edu.cn

Zihuan Mao 自欢猫Southeast University School of
东南大学Information Science and Engineering
信息科学与工程Nanjing 210096, China 南京 210096，中国

Dazhuan Xu 大转序
Purple Mountain Laboratories
紫山实验室
Nanjing 211111, China 南京 211111，中国

Dongming Wang 王东明Southeast University School of
东南大学Information Science and Engineering
信息科学与工程Nanjing 210096, China 南京 210096，中国

Yongming Huang* 永明黄*Southeast University School of
东南大学Information Science and Engineering
信息科学与工程Nanjing 210096, China 南京 210096，中国

Abstract 摘要

Evolving next-generation mobile networks is designed to provide ubiquitous coverage and networked sensing. With utility of multiview sensing and multi-node joint transmission, cell-free is a promising technique to realize this prospect. This paper aims to tackle the problem of access point (AP) deployment in cell-free systems to balance the sensing accuracy and user rate. By merging the D optimality with Euclidean criterion, a novel integrated metric is proposed to be the objective function for both max-sum and max min min min\min problems, which respectively guarantee the overall and lowest performance in multi-user communication and target tracking scenario. To solve the corresponding high dimensional non-convex multi-objective problem, the Soft actor-critic (SAC) is utilized to avoid risk of local optimal result. Numerical results demonstrate that proposed SAC-based APs deployment method achieves 20 % 20 % 20%20 \% of overall performance and 120 % 120 % 120%120 \% of lowest performance.
演进的下一代移动网络旨在提供无处不在的覆盖和网络感知。利用多视角感知和多节点联合传输，无基站技术是一种实现这一前景的有前途的技术。本文旨在解决无基站系统中接入点（AP）部署的问题，以平衡感知精度和用户速率。通过将 D 最优性与欧几里得标准相结合，提出了一种新颖的综合指标作为最大和最小 min min min\min 问题的目标函数，分别保证多用户通信和目标跟踪场景中的整体性能和最低性能。为了解决相应的高维非凸多目标问题，采用软演员-评论家（SAC）方法以避免局部最优结果的风险。数值结果表明，所提出的基于 SAC 的 AP 部署方法在整体性能上达到了 20 % 20 % 20%20 \% ，在最低性能上达到了 120 % 120 % 120%120 \% 。

CCS CONCEPTS CCS 概念

  • Networks → → rarr\rightarrow Network performance modeling; Wireless access points, base stations and infrastructure; ⋅ ⋅ *\cdot Computing methodologies → → rarr\rightarrow Policy iteration; ⋅ ⋅ *\cdot Hardware → → rarr\rightarrow Wireless integrated network sensors.
    网络 → → rarr\rightarrow 网络性能建模；无线接入点、基站和基础设施； ⋅ ⋅ *\cdot 计算方法 → → rarr\rightarrow 策略迭代； ⋅ ⋅ *\cdot 硬件 → → rarr\rightarrow 无线集成网络传感器。
    *Both authors are also affiliated to the Purple Mountain Laboratories, Nanjing 211111, China.
    两位作者也隶属于中国南京 211111 的紫金山实验室。
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ACM MobiCom '24, November 18-22, 2024, Washington D.C., DC, USA
ACM MobiCom '24，2024 年 11 月 18 日至 22 日，美国华盛顿特区
© 2024 Copyright held by the owner/author(s). Publication rights licensed to ACM. ACM ISBN 979-8-4007-0489-5/24/11
© 2024 版权所有，归所有者/作者持有。出版权已授权给 ACM。ACM ISBN 979-8-4007-0489-5/24/11
https://doi.org/10.1145/3636534.3698221

KEYWORDS 关键词

Cell-free, integrated sensing and communication, access point deployment, soft actor-critic, deep reinforcement learning
无细胞的集成传感与通信、接入点部署、软演员-评论家、深度强化学习

ACM Reference Format: ACM 参考格式：

Fanfei Xu, Shengheng Liu, Zihuan Mao, Shangqing Shi, Dazhuan Xu, Dongming Wang, and Yongming Huang. 2024. Access Point Deployment for Localizing Accuracy and User Rate in Cell-Free Systems. In The 30th Annual International Conference on Mobile Computing and Networking (ACM MobiCom '24), November 18-22, 2024, Washington D.C., DC, USA. ACM, New York, NY, USA, 7 pages. https://doi.org/10.1145/3636534.3698221
Fanfei Xu, Shengheng Liu, Zihuan Mao, Shangqing Shi, Dazhuan Xu, Dongming Wang, 和 Yongming Huang. 2024. 无蜂窝系统中定位精度和用户率的接入点部署. 在第 30 届国际移动计算与网络会议 (ACM MobiCom '24), 2024 年 11 月 18-22 日, 华盛顿特区, 美国. ACM, 纽约, NY, 美国, 7 页. https://doi.org/10.1145/3636534.3698221

1 INTRODUCTION 1 引言

The integrated sensing and communication (ISAC) is expected to become a key usages scenario of the future six-th generation (6G) networks, which is mentioned in newest recommendation concerned with framework and objective of future network of the International Telecommunication Union [10]. Telecommunication base stations, as a widely deployed infrastructure, can significantly enhance situational awareness when equipped with radar sensing capabilities, enabling a variety of novel applications, such as low-altitude drone intrusion detection and accurate virtual environment construction for digital twins [12]. Moreover, as wireless communication networks continue to evolve towards higher frequency bands, more spectrum resources can be obtained to satisfy the high bandwidth requirements of sensing functions. However, high-frequency signals with poor diffraction and high attenuation, makes traditional single station sensing performance significantly limited to shadow effects of obstructions.
集成感知与通信（ISAC）预计将成为未来第六代（6G）网络的关键应用场景，这在国际电信联盟最新的关于未来网络框架和目标的建议中提到。作为广泛部署的基础设施，电信基站在配备雷达感知能力时，可以显著增强态势感知，支持多种新颖应用，如低空无人机入侵检测和数字双胞胎的精确虚拟环境构建。此外，随着无线通信网络不断向更高频段演进，可以获得更多频谱资源，以满足感知功能的高带宽需求。然而，高频信号具有较差的衍射性和高衰减性，使得传统单站感知性能受到遮挡阴影效应的显著限制。
Cell-free networks [9, 14], an innovative architecture for 6G networks, may tackle this issue. Unlike conventional cellular networks, where each user equipment (UE) is served by a single base station within a designated cell, cell-free networks deploy a large number of distributed access points (APs) that collaboratively serve all users within the area. Widely-spread APs connected to a central processing unit (CPU) enable seamless sensing and ubiquitous coverage by use of key techniques such as dynamic user association and
无基站网络[9, 14]，作为 6G 网络的一种创新架构，可能会解决这个问题。与传统的蜂窝网络不同，在传统网络中，每个用户设备（UE）由指定小区内的单个基站服务，而无基站网络则部署大量分布式接入点（AP），这些接入点协同为区域内的所有用户提供服务。广泛分布的接入点连接到中央处理单元（CPU），通过动态用户关联等关键技术实现无缝感知和普遍覆盖。
APs deployment [1]. Different APs location seriously affects the channel condition, intuitively, to fully leverage the advantages such as multi-view sensing information and joint transmission of the cell-free ISAC systems, investigating optimal APs deployment is vitally necessary.
APs 部署 [1]。不同 AP 的位置严重影响信道条件，直观上，为了充分利用如多视角感知信息和无小区 ISAC 系统的联合传输等优势，研究最佳 AP 部署是至关重要的。
Mathematical solutions such as vector quantization and gradient descent [ 2 , 4 , 13 ] [ 2 , 4 , 13 ] [2,4,13][2,4,13] were proposed to optimize the two dimension (2D) location of APs to improve of spectral efficiency in cell-free networks. On-demand service capability concerned with sum throughput was achieved by solving APs deployment [7] based on multiple linear regression model. Moreover, APs deployment not only affects communication performance, but also impacts sensing performance. Cramér-Rao lower bound (CRLB) was developed [6] for target velocity estimation in multiple-input multiple output (MIMO) radar, shown that the antenna placement affects the estimation accuracy significantly. Also in MIMO radar systems [3], CRLB for target localization in both coherent and non-coherent processing was developed. Additionally, based on the best unbiased linear unbiased estimator it derived, a closed-form localization estimation that revealed the relationship between antennas location, target location, and localization accuracy was provided. Furthermore, geometry gain of antennas deployment for target localizing is [11] analyzed in MIMO radar systems. In summary, substantial researches have separately analyzed the impacts of APs deployment on communication and sensing performance. However, the area of jointly considering both of them still remains blanket.
数学解决方案，如向量量化和梯度下降 [ 2 , 4 , 13 ] [ 2 , 4 , 13 ] [2,4,13][2,4,13] ，被提出用于优化接入点（AP）的二维（2D）位置，以提高无蜂窝网络中的频谱效率。通过基于多元线性回归模型解决接入点部署问题，实现了与总吞吐量相关的按需服务能力[7]。此外，接入点的部署不仅影响通信性能，还影响感知性能。为多输入多输出（MIMO）雷达中的目标速度估计开发了克拉美-罗下界（CRLB）[6]，显示天线位置显著影响估计精度。在 MIMO 雷达系统中[3]，也开发了针对目标定位的 CRLB，适用于相干和非相干处理。此外，基于最佳无偏线性无偏估计器，提供了一种闭式形式的定位估计，揭示了天线位置、目标位置和定位精度之间的关系。此外，针对目标定位的天线部署几何增益在 MIMO 雷达系统中进行了分析[11]。 总之，大量研究分别分析了接入点部署对通信和感知性能的影响。然而，联合考虑这两者的领域仍然空白。
Thus, to simultaneously measure sensing and communication performance, in this work we propose a unified evaluation metric in cell-free ISAC systems, merging user rate with localizing accuracy [8] derived from Euclidean distance and D-optimal criterion, and utilize it as the objective function for APs deployment optimization. In addition, considering fairness for all UEs and localizing accuracy throughout the target moving trajectory, we provide the deployment results of both max-sum and max-min problem. Due to the non-convex and high-dimensional property of the original multi-objective problem, mathematical algorithms are challenging to solve it. Soft actor-critic (SAC) [5], a deep reinforcement learning (DRL) based APs deployment method is proposed, with the utility of additive maximum AP deployment entropy term to avoid local optimum. Numerical results show the superior performance of our proposed SAC-based deployment method compared with other DRL algorithms such as deep deterministic policy gradient (DDPG) and twin-delayed DDPG (TD3).
因此，为了同时测量感知和通信性能，在本研究中，我们提出了一种统一的评估指标，用于无小区 ISAC 系统，将用户速率与基于欧几里得距离和 D-最优准则的定位精度[8]相结合，并将其作为 AP 部署优化的目标函数。此外，考虑到所有用户设备（UEs）的公平性以及目标移动轨迹中的定位精度，我们提供了最大和最小问题的部署结果。由于原始多目标问题的非凸性和高维特性，数学算法难以解决。我们提出了一种基于深度强化学习（DRL）的 AP 部署方法——软演员评论家（SAC）[5]，利用附加的最大 AP 部署熵项以避免局部最优。数值结果表明，与其他 DRL 算法（如深度确定性策略梯度（DDPG）和双延迟 DDPG（TD3））相比，我们提出的基于 SAC 的部署方法具有优越的性能。

2 ISAC SIGNAL MODEL AND PROBLEM FORMULATION
2 ISAC 信号模型与问题表述

As illustrated in Fig. 1, we consider a cell-free ISAC system consists of M M MM single-antenna transmitter APs and N N NN single-antenna receiver APs which collaboratively serve K K KK single-antenna UEs and estimate position of one target moving with specific trajectory. The 2D position of them is denoted by t m = [ x m t , y m t ] , r n = [ x n r , y n r ] , u k = t m = x m t , y m t , r n = x n r , y n r , u k = t_(m)=[x_(m)^(t),y_(m)^(t)],r_(n)=[x_(n)^(r),y_(n)^(r)],u_(k)=\mathbf{t}_{m}=\left[x_{m}^{\mathrm{t}}, y_{m}^{\mathrm{t}}\right], \mathbf{r}_{n}=\left[x_{n}^{\mathrm{r}}, y_{n}^{\mathrm{r}}\right], \mathbf{u}_{k}= [ x k , y k ] , p = [ x p , y p ] x k , y k , p = x p , y p [x_(k),y_(k)],p=[x^(p),y^(p)]\left[x_{k}, y_{k}\right], \mathbf{p}=\left[x^{\mathrm{p}}, y^{\mathrm{p}}\right] respectively, where m ∈ M = { 1 , … , M } m ∈ M = { 1 , … , M } m inM={1,dots,M}m \in \mathbb{M}=\{1, \ldots, M\}, n ∈ N = { 1 , … , N } , k ∈ K = { 1 , … , K } n ∈ N = { 1 , … , N } , k ∈ K = { 1 , … , K } n inN={1,dots,N},k inK={1,dots,K}n \in \mathbb{N}=\{1, \ldots, N\}, k \in \mathbb{K}=\{1, \ldots, K\}.
如图 1 所示，我们考虑一个无细胞的 ISAC 系统，由 M M MM 个单天线发射器 AP 和 N N NN 个单天线接收器 AP 组成，它们协同服务 K K KK 个单天线用户设备，并估计一个沿特定轨迹移动的目标的位置。它们的二维位置分别用 t m = [ x m t , y m t ] , r n = [ x n r , y n r ] , u k = t m = x m t , y m t , r n = x n r , y n r , u k = t_(m)=[x_(m)^(t),y_(m)^(t)],r_(n)=[x_(n)^(r),y_(n)^(r)],u_(k)=\mathbf{t}_{m}=\left[x_{m}^{\mathrm{t}}, y_{m}^{\mathrm{t}}\right], \mathbf{r}_{n}=\left[x_{n}^{\mathrm{r}}, y_{n}^{\mathrm{r}}\right], \mathbf{u}_{k}= 和 [ x k , y k ] , p = [ x p , y p ] x k , y k , p = x p , y p [x_(k),y_(k)],p=[x^(p),y^(p)]\left[x_{k}, y_{k}\right], \mathbf{p}=\left[x^{\mathrm{p}}, y^{\mathrm{p}}\right] 表示，其中 m ∈ M = { 1 , … , M } m ∈ M = { 1 , … , M } m inM={1,dots,M}m \in \mathbb{M}=\{1, \ldots, M\} ， n ∈ N = { 1 , … , N } , k ∈ K = { 1 , … , K } n ∈ N = { 1 , … , N } , k ∈ K = { 1 , … , K } n inN={1,dots,N},k inK={1,dots,K}n \in \mathbb{N}=\{1, \ldots, N\}, k \in \mathbb{K}=\{1, \ldots, K\} 。

2.1 Fisher Information 2.1 费舍尔信息

When system is executing sensing function module, M M MM transmitting APs send probe signal and N N NN receiving APs capture the echo from
当系统执行感知功能模块时， M M MM 发送探测信号的接入点和 N N NN 接收回波的接入点
Figure 1: Cell-free ISAC systems.
图 1：无细胞 ISAC 系统。
target. In this distributed detection system, the low-pass equivalent of the narrow-band signal transmitted from i i ii-th APs at time t t tt is represented as E / M s i ( t ) E / M s i ( t ) sqrt(E//M)s_(i)(t)\sqrt{E / M} s_{i}(t), where E E EE denotes the total transmission energy. We assume that the transmission signal is uncorrelated in any time delay,
目标。在这个分布式检测系统中，从 i i ii -th APs 在时间 t t tt 传输的窄带信号的低通等效表示为 E / M s i ( t ) E / M s i ( t ) sqrt(E//M)s_(i)(t)\sqrt{E / M} s_{i}(t) ，其中 E E EE 表示总传输能量。我们假设传输信号在任何时间延迟下都是不相关的，
∫ T s i ( t ) s j ∗ ( t − τ ) d t ≈ { 1 , if i = j 0 , if i ≠ j ∫ T   s i ( t ) s j ∗ ( t − τ ) d t ≈ 1 ,       if       i = j 0 ,       if       i ≠ j int_(T)s_(i)(t)s_(j)^(**)(t-tau)dt~~{[1","," if ",i=j],[0","," if ",i!=j]:}\int_{T} s_{i}(t) s_{j}^{*}(t-\tau) d t \approx\left\{\begin{array}{lll} 1, & \text { if } & i=j \\ 0, & \text { if } & i \neq j \end{array}\right.
where ( ⋅ ) ∗ ( ⋅ ) ∗ (*)^(**)(\cdot)^{*} represent the conjugate operator. In addition, the signal is normalized in the whole signal processing interval T T TT, i.e., ∫ T | s i ( t ) | 2 d t = 1 ∫ T   s i ( t ) 2 d t = 1 int_(T)|s_(i)(t)|^(2)dt=1\int_{T}\left|s_{i}(t)\right|^{2} d t=1.
其中 ( ⋅ ) ∗ ( ⋅ ) ∗ (*)^(**)(\cdot)^{*} 代表共轭算子。此外，信号在整个信号处理区间 T T TT 内被归一化，即 ∫ T | s i ( t ) | 2 d t = 1 ∫ T   s i ( t ) 2 d t = 1 int_(T)|s_(i)(t)|^(2)dt=1\int_{T}\left|s_{i}(t)\right|^{2} d t=1 。
Non-coherent processing, a more practical operation is selected due to its low requirement of time synchronization compared with coherent processing. The echo signal accepted at n n nn-th receiving AP is denoted by
非相干处理由于对时间同步的低要求，相较于相干处理，选择了一种更实用的操作。在 n n nn -th 接收 AP 接收到的回波信号表示为
y n ( t ) = ∑ m = 1 M η m n s m ( t − τ m n ) + w n ( t ) y n ( t ) = ∑ m = 1 M   η m n s m t − τ m n + w n ( t ) y_(n)(t)=sum_(m=1)^(M)eta_(mn)s_(m)(t-tau_(mn))+w_(n)(t)y_{n}(t)=\sum_{m=1}^{M} \eta_{m n} s_{m}\left(t-\tau_{m n}\right)+w_{n}(t)
where w n ( t ) ∼ i.i.d. C N ( 0 , 1 ) w n ( t ) ∼  i.i.d.  C N ( 0 , 1 ) w_(n)(t)∼^(" i.i.d. ")CN(0,1)w_{n}(t) \stackrel{\text { i.i.d. }}{\sim} \mathcal{C N}(0,1) represents additive Gaussian noise, η = η = eta=\eta= [ η 11 , η 12 , … , η m n , … , η M N ] T η 11 , η 12 , … , η m n , … , η M N T [eta_(11),eta_(12),dots,eta_(mn),dots,eta_(MN)]^(T)\left[\eta_{11}, \eta_{12}, \ldots, \eta_{m n}, \ldots, \eta_{M N}\right]^{\mathrm{T}} is the coefficient referring to target reflection and channel propagation fading. τ m n τ m n tau_(mn)\tau_{m n} denotes time delay of signal transmission and reflection between m m mm-th transmitting AP and n n nn-th receiving AP
其中 w n ( t ) ∼ i.i.d. C N ( 0 , 1 ) w n ( t ) ∼  i.i.d.  C N ( 0 , 1 ) w_(n)(t)∼^(" i.i.d. ")CN(0,1)w_{n}(t) \stackrel{\text { i.i.d. }}{\sim} \mathcal{C N}(0,1) 代表加性高斯噪声， η = η = eta=\eta= [ η 11 , η 12 , … , η m n , … , η M N ] T η 11 , η 12 , … , η m n , … , η M N T [eta_(11),eta_(12),dots,eta_(mn),dots,eta_(MN)]^(T)\left[\eta_{11}, \eta_{12}, \ldots, \eta_{m n}, \ldots, \eta_{M N}\right]^{\mathrm{T}} 是指目标反射和信道传播衰落的系数。 τ m n τ m n tau_(mn)\tau_{m n} 表示第 m m mm 个发射接入点与第 n n nn 个接收接入点之间信号传输和反射的时间延迟。
τ m n = d m + d n c τ m n = d m + d n c tau_(mn)=(d_(m)+d_(n))/(c)\tau_{m n}=\frac{d_{m}+d_{n}}{c}
where c c cc is speed of light and d m = ‖ t m − p ‖ 2 , d n = ‖ r n − p ‖ 2 d m = t m − p 2 , d n = r n − p 2 d_(m)=||t_(m)-p||_(2),d_(n)=||r_(n)-p||_(2)d_{m}=\left\|\mathbf{t}_{m}-\mathbf{p}\right\|_{2}, d_{n}=\left\|\mathbf{r}_{n}-\mathbf{p}\right\|_{2} denote the distance between target p p p\mathbf{p} and transmitter or receiver AP.
其中 c c cc 是光速， d m = ‖ t m − p ‖ 2 , d n = ‖ r n − p ‖ 2 d m = t m − p 2 , d n = r n − p 2 d_(m)=||t_(m)-p||_(2),d_(n)=||r_(n)-p||_(2)d_{m}=\left\|\mathbf{t}_{m}-\mathbf{p}\right\|_{2}, d_{n}=\left\|\mathbf{r}_{n}-\mathbf{p}\right\|_{2} 表示目标 p p p\mathbf{p} 与发射器或接收器 AP 之间的距离。
As for the localizing accuracy, we utilize D-optimal criterion to evaluate the performance of parameter estimation. The product of the CRLB matrix eigenvalues (or determinant) is minimize, which is equivalent to minimize the area of the Elliptical error probable. Fisher information matrix (FIM) is the inverse of CRLB matrix, therefore, minimizing the determinant of CRLB matrix converts to maximizing the FIM. The FIM of the estimation for the parameter
关于定位精度，我们利用 D-最优准则来评估参数估计的性能。最小化 CRLB 矩阵特征值（或行列式）的乘积，这等同于最小化椭圆误差概率的面积。费舍尔信息矩阵（FIM）是 CRLB 矩阵的逆，因此，最小化 CRLB 矩阵的行列式转化为最大化 FIM。参数估计的 FIM
vector ϑ ϑ vartheta\vartheta by use of the measurement vector x x x\mathbf{x} can be expressed as:
向量 ϑ ϑ vartheta\vartheta 可以通过测量向量 x x x\mathbf{x} 表示为：
Φ = E { [ ∂ ∂ ϑ ln ⁡ f ( x ∣ ϑ ) ] [ ∂ ∂ ϑ ln ⁡ f ( x ∣ ϑ ) ] T } . Φ = E ∂ ∂ ϑ ln ⁡ f ( x ∣ ϑ ) ∂ ∂ ϑ ln ⁡ f ( x ∣ ϑ ) T . Phi=E{[(del)/(del vartheta)ln f(x∣vartheta)][(del)/(del vartheta)ln f(x∣vartheta)]^(T)}.\Phi=\mathrm{E}\left\{\left[\frac{\partial}{\partial \vartheta} \ln \mathrm{f}(\mathrm{x} \mid \vartheta)\right]\left[\frac{\partial}{\partial \vartheta} \ln \mathrm{f}(\mathrm{x} \mid \vartheta)\right]^{\mathrm{T}}\right\} .
To be specific, when the error of the estimation is Gaussian noise, the FIM for in cell-free ISAC system, non-coherent sensing FIM for target detection is equal to
具体来说，当估计误差为高斯噪声时，细胞自由 ISAC 系统中的 FIM，非相干传感的目标检测 FIM 等于
Φ = [ ϕ 11 ϕ 12 ϕ 21 ϕ 22 ] = J 0 T Σ − 1 J 0 Φ = ϕ 11      ϕ 12 ϕ 21      ϕ 22 = J 0 T Σ − 1 J 0 Phi=[[phi_(11),phi_(12)],[phi_(21),phi_(22)]]=J_(0)^(T)Sigma^(-1)J_(0)\Phi=\left[\begin{array}{ll} \phi_{11} & \phi_{12} \\ \phi_{21} & \phi_{22} \end{array}\right]=\mathbf{J}_{0}^{\mathrm{T}} \Sigma^{-1} \mathrm{~J}_{0}
where J 0 J 0 J_(0)\mathbf{J}_{0} is the Jacobian matrix of the target localizing at p = p = p=\mathbf{p}= [ x p , y p x p , y p [x^(p),y^(p):}\left[x^{\mathrm{p}}, y^{\mathrm{p}}\right. ]
其中 J 0 J 0 J_(0)\mathbf{J}_{0} 是定位于 p = p = p=\mathbf{p}= [ x p , y p x p , y p [x^(p),y^(p):}\left[x^{\mathrm{p}}, y^{\mathrm{p}}\right. 的雅可比矩阵
J 0 = [ ( α 1 t + α 1 r ) ( α 1 t + α 2 r ) ⋯ ( α 1 t + α N r ) ⋯ ( α M t + α N r ) ] T J 0 = α 1 t + α 1 r α 1 t + α 2 r ⋯ α 1 t + α N r ⋯ α M t + α N r T J_(0)=[(alpha_(1)^(t)+alpha_(1)^(r))(alpha_(1)^(t)+alpha_(2)^(r))cdots(alpha_(1)^(t)+alpha_(N)^(r))cdots(alpha_(M)^(t)+alpha_(N)^(r))]^(T)\mathrm{J}_{0}=\left[\left(\alpha_{1}^{t}+\alpha_{1}^{r}\right)\left(\alpha_{1}^{t}+\alpha_{2}^{r}\right) \cdots\left(\alpha_{1}^{t}+\alpha_{N}^{r}\right) \cdots\left(\alpha_{M}^{t}+\alpha_{N}^{r}\right)\right]^{\mathrm{T}}
where α m t = [ cos ⁡ θ m t sin ⁡ θ m t ] T , α n r = [ cos ⁡ θ n r sin ⁡ θ n r ] T , θ α m t = cos ⁡ θ m t sin ⁡ θ m t T , α n r = cos ⁡ θ n r sin ⁡ θ n r T , θ alpha_(m)^(t)=[cos theta_(m)^(t)sin theta_(m)^(t)]^(T),alpha_(n)^(r)=[cos theta_(n)^(r)sin theta_(n)^(r)]^(T),theta\alpha_{m}^{t}=\left[\cos \theta_{m}^{t} \sin \theta_{m}^{t}\right]^{\mathrm{T}}, \alpha_{n}^{r}=\left[\cos \theta_{n}^{r} \sin \theta_{n}^{r}\right]^{\mathrm{T}}, \theta represents the bearing angle of the AP relative to the target, measured from the horizontal axis. According to [11], the expression of determinant of the angular FIM is
其中 α m t = [ cos ⁡ θ m t sin ⁡ θ m t ] T , α n r = [ cos ⁡ θ n r sin ⁡ θ n r ] T , θ α m t = cos ⁡ θ m t sin ⁡ θ m t T , α n r = cos ⁡ θ n r sin ⁡ θ n r T , θ alpha_(m)^(t)=[cos theta_(m)^(t)sin theta_(m)^(t)]^(T),alpha_(n)^(r)=[cos theta_(n)^(r)sin theta_(n)^(r)]^(T),theta\alpha_{m}^{t}=\left[\cos \theta_{m}^{t} \sin \theta_{m}^{t}\right]^{\mathrm{T}}, \alpha_{n}^{r}=\left[\cos \theta_{n}^{r} \sin \theta_{n}^{r}\right]^{\mathrm{T}}, \theta 代表 AP 相对于目标的方位角，从水平轴测量。根据 [11]，角度 FIM 的行列式表达式为
| Φ | = { ∑ m = 1 M ∑ n = 1 N ( cos ⁡ θ m t + cos ⁡ θ n r ) 2 ∑ m = 1 M ∑ n = 1 N ( sin ⁡ θ m t + sin ⁡ θ n r ) 2 − [ ∑ m = 1 M ∑ n = 1 N ( cos ⁡ θ m t + cos ⁡ θ n r ) ( sin ⁡ θ m t + sin ⁡ θ n r ) ] 2 } | Φ | = ∑ m = 1 M   ∑ n = 1 N   cos ⁡ θ m t + cos ⁡ θ n r 2 ∑ m = 1 M   ∑ n = 1 N   sin ⁡ θ m t + sin ⁡ θ n r 2 − ∑ m = 1 M   ∑ n = 1 N   cos ⁡ θ m t + cos ⁡ θ n r sin ⁡ θ m t + sin ⁡ θ n r 2 {:[|Phi|={sum_(m=1)^(M)sum_(n=1)^(N)(cos theta_(m)^(t)+cos theta_(n)^(r))^(2)sum_(m=1)^(M)sum_(n=1)^(N)(sin theta_(m)^(t)+sin theta_(n)^(r))^(2):}],[{:-[sum_(m=1)^(M)sum_(n=1)^(N)(cos theta_(m)^(t)+cos theta_(n)^(r))(sin theta_(m)^(t)+sin theta_(n)^(r))]^(2)}]:}\begin{aligned} |\Phi| & =\left\{\sum_{m=1}^{M} \sum_{n=1}^{N}\left(\cos \theta_{m}^{t}+\cos \theta_{n}^{r}\right)^{2} \sum_{m=1}^{M} \sum_{n=1}^{N}\left(\sin \theta_{m}^{t}+\sin \theta_{n}^{r}\right)^{2}\right. \\ & \left.-\left[\sum_{m=1}^{M} \sum_{n=1}^{N}\left(\cos \theta_{m}^{t}+\cos \theta_{n}^{r}\right)\left(\sin \theta_{m}^{t}+\sin \theta_{n}^{r}\right)\right]^{2}\right\} \end{aligned}
Next, we transform the determinant of the angular FIM into twodimensional Cartesian coordinates, and determine optimal APs deployment positions based on the maximum value of the FIM determinant.
接下来，我们将角度 FIM 的行列式转换为二维笛卡尔坐标，并根据 FIM 行列式的最大值确定最佳 AP 部署位置。
| Φ | = { ∑ m = 1 M ∑ n = 1 N ( x p − x m t ‖ p − t m ‖ 2 + x p − x n r ‖ p − r n ‖ 2 ) 2 ∑ m = 1 M ∑ n = 1 N ( y p − y m t ‖ p − t m ‖ 2 + y p − y n r ‖ p − r n ‖ 2 ) 2 − [ ∑ m = 1 M ∑ n = 1 N ( x p − x m t ‖ p − t m ‖ 2 + x p − x n r ‖ p − r n ‖ 2 ) ( y p − y m t ‖ p − t m ‖ 2 + y p − y n r ‖ p − r n ‖ 2 ) ] 2 } | Φ | = ∑ m = 1 M   ∑ n = 1 N   x p − x m t p − t m 2 + x p − x n r p − r n 2 2 ∑ m = 1 M   ∑ n = 1 N   y p − y m t p − t m 2 + y p − y n r p − r n 2 2 − ∑ m = 1 M   ∑ n = 1 N   x p − x m t p − t m 2 + x p − x n r p − r n 2 y p − y m t p − t m 2 + y p − y n r p − r n 2 2 {:[|Phi|={sum_(m=1)^(M)sum_(n=1)^(N)((x^(p)-x_(m)^(t))/(||p-t_(m)||_(2))+(x^(p)-x_(n)^(r))/(||p-r_(n)||_(2)))^(2)sum_(m=1)^(M)sum_(n=1)^(N)((y^(p)-y_(m)^(t))/(||p-t_(m)||_(2))+(y^(p)-y_(n)^(r))/(||p-r_(n)||_(2)))^(2):}],[{:-[sum_(m=1)^(M)sum_(n=1)^(N)((x^(p)-x_(m)^(t))/(||p-t_(m)||_(2))+(x^(p)-x_(n)^(r))/(||p-r_(n)||_(2)))((y^(p)-y_(m)^(t))/(||p-t_(m)||_(2))+(y^(p)-y_(n)^(r))/(||p-r_(n)||_(2)))]^(2)}]:}\begin{aligned} |\boldsymbol{\Phi}|= & \left\{\sum_{m=1}^{M} \sum_{n=1}^{N}\left(\frac{x^{p}-x_{m}^{t}}{\left\|\mathbf{p}-\mathbf{t}_{m}\right\|_{2}}+\frac{x^{p}-x_{n}^{r}}{\left\|\mathbf{p}-\mathbf{r}_{n}\right\|_{2}}\right)^{2} \sum_{m=1}^{M} \sum_{n=1}^{N}\left(\frac{y^{p}-y_{m}^{t}}{\left\|\mathbf{p}-\mathbf{t}_{m}\right\|_{2}}+\frac{y^{p}-y_{n}^{r}}{\left\|\mathbf{p}-\mathbf{r}_{n}\right\|_{2}}\right)^{2}\right. \\ & \left.-\left[\sum_{m=1}^{M} \sum_{n=1}^{N}\left(\frac{x^{p}-x_{m}^{t}}{\left\|\mathbf{p}-\mathbf{t}_{m}\right\|_{2}}+\frac{x^{p}-x_{n}^{r}}{\left\|\mathbf{p}-\mathbf{r}_{n}\right\|_{2}}\right)\left(\frac{y^{p}-y_{m}^{t}}{\left\|\mathbf{p}-\mathbf{t}_{m}\right\|_{2}}+\frac{y^{p}-y_{n}^{r}}{\left\|\mathbf{p}-\mathbf{r}_{n}\right\|_{2}}\right)\right]^{2}\right\} \end{aligned}

2.2 User Sum Rate 2.2 用户总速率

An cell-free uplink communication model is studied in this section, where all M + N M + N M+NM+N APs simultaneously accept signals from all K K KK UEs.The uplink receiving signal at l l ll-th AP is
在本节中研究了一种无细胞上行通信模型，其中所有 M + N M + N M+NM+N 个接入点同时接收来自所有 K K KK 个用户设备的信号。第 l l ll 个接入点的上行接收信号为
y l = ∑ k = 1 K ρ h l k x k + w l y l = ∑ k = 1 K   ρ h l k x k + w l y_(l)=sum_(k=1)^(K)sqrtrhoh_(lk)x_(k)+w_(l)y_{l}=\sum_{k=1}^{K} \sqrt{\rho} h_{l k} x_{k}+w_{l}
where ρ , x k ρ , x k sqrtrho,x_(k)\sqrt{\rho}, x_{k} is transmitter power and data symbol of k k kk-th UE, l ∈ { 1 , 2 , … , M + N } l ∈ { 1 , 2 , … , M + N } l in{1,2,dots,M+N}l \in\{1,2, \ldots, M+N\}. The received signal of all M + N M + N M+NM+N APs can be represented as
其中 ρ , x k ρ , x k sqrtrho,x_(k)\sqrt{\rho}, x_{k} 是 k k kk -th UE 的发射功率和数据符号， l ∈ { 1 , 2 , … , M + N } l ∈ { 1 , 2 , … , M + N } l in{1,2,dots,M+N}l \in\{1,2, \ldots, M+N\} 。所有 M + N M + N M+NM+N 个 AP 的接收信号可以表示为
y = ρ H x + w y = ρ H x + w y=sqrtrhoHx+w\mathbf{y}=\sqrt{\rho} \mathbf{H x}+\mathbf{w}
where x = [ x 1 , x 2 , … , x K ] T , w = [ w 1 , w 2 , … , w M + N ] T x = x 1 , x 2 , … , x K T , w = w 1 , w 2 , … , w M + N T x=[x_(1),x_(2),dots,x_(K)]^(T),w=[w_(1),w_(2),dots,w_(M+N)]^(T)\mathbf{x}=\left[x_{1}, x_{2}, \ldots, x_{K}\right]^{\mathrm{T}}, \mathbf{w}=\left[w_{1}, w_{2}, \ldots, w_{M+N}\right]^{\mathrm{T}} and H [ l , k ] = H [ l , k ] = H[l,k]=\mathbf{H}[l, k]= h l k , H ∈ C ( M + N ) × K h l k , H ∈ C ( M + N ) × K h_(lk),HinC^((M+N)xx K)h_{l k}, \mathbf{H} \in \mathbb{C}^{(M+N) \times K} is channel coefficients matrix. The narrowband fading channel coefficient is written as h l k = β l k g l k h l k = β l k g l k h_(lk)=sqrt(beta_(lk))g_(lk)h_{l k}=\sqrt{\beta_{l k}} g_{l k}, where g l k ∼ i.i.d. C N ( 0 , 1 ) g l k ∼  i.i.d.  C N ( 0 , 1 ) g_(lk)∼^(" i.i.d. ")CN(0,1)g_{l k} \stackrel{\text { i.i.d. }}{\sim} \mathcal{C N}(0,1)