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Algorithm for Topology Search Using Dilution of Precision Criterion in Ultra-Dense Network Positioning Service Area

Mathematics 2023, 11(10), 2227; https://doi.org/10.3390/math11102227

by Grigoriy Fokin^1,* and Andrey Koucheryavy²

Reviewer 1: Anonymous

Reviewer 2: Anonymous

Mathematics 2023, 11(10), 2227; https://doi.org/10.3390/math11102227

Submission received: 31 March 2023 / Revised: 8 May 2023 / Accepted: 8 May 2023 / Published: 9 May 2023

(This article belongs to the Section Network Science)

Round 1

Reviewer 1 Report

The author presented an interesting study regarding localization of UE (User Equipment) in 5G and beyond networks, characterized with the increased spatial density of gNodeB (gNB) uspecially in urban areas. The choice of suitable topology (i.e. set) of gNB is observed, with the quantitative measure of gNB topology suitability given for horizontal (HDOP), vertical (VDOP) and position (PDOP) dilution of precision (DOP) criterion. The methods for gNB topology search using time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), combined TOA-AOA and TDOA-AOA measurements are portrayed and novel approach for topology search in positioning service area is proposed. The cliamed contribution includes algorithm and software for iterative search of all possible gNB and UE locations in space, minimizing UE geometric DOP, with the practical application in gNB topology substantiation for given positioning scenario in 5G/B5G/6G networks.

The paper is well conceived and written, with the clear and comprehensive presentation. However, some issues should be resolved:

- the contribution and distinction of here proposed solutions to the prior work in the field (authors own work and the work of other authors) should be clarified (elaborated in more detailed manner).

- the proposed approach and the forthcoming results are presented for a number of cases (for the observed case scenarios at Gazprom Arena). However, no comprison to other work or reference solution is given in order to support quality of here proposed solution.

- the referencing to prior work should be improved in chapters 2 and 3 in which background and overview on observed problem are given, as well as the state of the art in the field. The presented methods, equations, solutions and conclusions in this text should be referenced more strictly (for all instances that do not present contribution of this manuscript).

The quality of language is overall good - only final checking and error corrections are needed.

Author Response

Response to Reviewer 1

Comments and Suggestions for Authors

The author presented an interesting study regarding localization of UE (User Equipment) in 5G and beyond networks, characterized with the increased spatial density of gNodeB (gNB) especially in urban areas. The choice of suitable topology (i.e., set) of gNB is observed, with the quantitative measure of gNB topology suitability given for horizontal (HDOP), vertical (VDOP) and position (PDOP) dilution of precision (DOP) criterion. The methods for gNB topology search using time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), combined TOA-AOA and TDOA-AOA measurements are portrayed and novel approach for topology search in positioning service area is proposed. The claimed contribution includes algorithm and software for iterative search of all possible gNB and UE locations in space, minimizing UE geometric DOP, with the practical application in gNB topology substantiation for given positioning scenario in 5G/B5G/6G networks.

Thank you for your comment.

The paper is well conceived and written, with the clear and comprehensive presentation. However, some issues should be resolved:

the contribution and distinction of here proposed solutions to the prior work in the field (authors own work and the work of other authors) should be clarified (elaborated in more detailed manner).

Thank you for your comment.

Added paragraph in the end of subsection 2.1. Ultra-Dense Network Positioning Service Area Scenario

To elaborate in a more detailed manner the contribution and distinction of the proposed solution to prior own works and the works of other authors, next subsection gives background and overview on observed problem as well as the state of the art in the field.

Added subsection 2.2. Topology Search Using Dilution of Precision Criterion Background

Investigation [52] demonstrates, that the use of joint TDOA and AOA gives a significant improvement in the position accuracy, thanks to the use of antenna arrays and selective gNB exclusion method, that is able to detect and eliminate measurements, affected by Non-Line of Sight (NLOS). Simulation results revealed, that HDOP in terms of Cumulative Distribution Function (CDF) below one is attainable with probability 0,7 for TDOA, AOA and combined TDOA-AOA measurement processing. The drawback of simulation approach is an assumption on hexagonal grid for gNB deployment.

Work [53] evaluates 5G Transmission and Reception Points (TRP) standard, edge and mixed deployment strategies for 3GPP Indoor Open Office (IOO) and Indoor Factory (InF) scenarios with downlink TDOA positioning. In particular, it analyzes the effect of TRP densification on UE positioning accuracy, using simulation with two performance metrics: GDOP contour plots and Root-Mean-Square Error (RMSE) Cramer-Rao Lower Bound (CRLB). Simulation results reveals, that edge TRP deployment yields higher positioning accuracy, compared to standard deployment; also, it confirms the obvious trend, that TRP densification improves positioning accuracy. GDOP is computed for each point in the deployment area with a given step and depends on the TRP topology and positioning method. For the case of TDOA positioning it is seen, that GDOP can be lower than one for optimistic UE location in the geometric center of positioning area. Despite the conclusion about paramount influence of TRP deployment geometry on UE localization accuracy, it analyzes GDOP for only predefined TRP positions, while the problem for TRP topology search is out of scope. One more weak point of this analysis is 2D case only.

Research [54] derives GDOP factor for the proposed hybrid TDOA and TOA positioning in the 5G cellular communication system; GDOP is defined as the ratio of the accuracy limitation of a position fix to the accuracy of TOA, TDOA or AOA primary measurements. Authors analyze GDOP factor with four base stations in a simulation scenario according to the real scene of the underground parking lot in the Beijing University of Posts and Telecommunications. Simulation results for the proposed hybrid TDOA-TOA positioning reveal minimum GDOP of 0.59, and the minimum GDOP of the TDOA-only positioning is just 0.79, which confirms the possibility to attain ideal GDOP values, lower than one. From the point of view of topology search in UDN scenario, the drawback of the research in [54] is GDOP analysis of predefined gNB topology for 2D case only.

Investigation [55] argue, that localization accuracy should play prominent role in cellular infrastructure planning and consider GDOP as a metric, which could contribute to geometrically favorable 5G base stations deployment for UE positioning. It points out the importance of GDOP, which is independent of the particular positioning methodology employed and states, that even if a sophisticated positioning system is used, it can turn out to be inaccurate, if positioning architecture with physically deployed base stations, is not taken into account. At the same time investigation [55] claims, that understanding GDOP allows to substantiate the best base stations deployment. Authors contribute to a stochastic theory of location-based Fisher information in wireless networks, concerned with how to best physically deploy gNBs in order to minimize UE GDOP, and develop a closed-form probability density function (PDF) to characterize the angular difference of a pair of base stations and a UE. Then authors, using produced PDF, show some gNB deployments, that are guaranteed to yield favorable GDOP for UE positioning. The problem of practical utilization of reported results is complicated for the case of more than two gNB; also, it is only for 2D case and is limited by the assumption of hexagonal lattice model for gNB locations

Work [56] proposes to select the most appropriate subset of four BSs among the set of seven BSs. An approach is to calculate UE location for all BS subsets and to select those, which gives the smallest GDOP. Despite interesting approach of selecting the best topology, the drawback of this analysis is 2D case only and rather poor choice among the subset of seven BSs with predefined locations.

Research [57] points out proliferation of AOA primary measurements for UE positioning in cellular wireless networks due to smart antennas emergence and considers DOP expressions, that relate the primary measurement error to the position error through dilution of precision factor. An expression for AOA positioning DOP was derived and demonstrated values below one even for two base stations on the plane (2D case).

Authors in [58] investigates the DOP of a positioning system, combining AOA and TOA primary measurements, show, that the DOP value is related to the size of the deployment area (distance apart), and explain, why the DOP quantity for the AOA system is associated with the size of the deployment area, while the DOP quantity for the TOA base system is not. When the size of the deployment area scales up to infinity, the DOP is decided by the TOA part of the system, but when the deployment area size scales down to infinitely small, the DOP is decided by the AOA part of the positioning system. The drawback of the work is, that it analyses the DOP values with one, two and three different configurations of base stations only, which is weakly consistent with UDN scenario.

Dynamic Base Stations Selection (DBSS) method, proposed in [59], enables cellular system to dynamically select the positioning base station for the UE location estimation, specifically fo the case of four and five base station. The drawback of the proposed approach is the assumption about regular hexagonal base stations arrangement on the plane.

Investigation [60] performs GDOP analysis to obtain concise analytical expressions for a number of scenarios, which are generally applicable to geometries, where the UE is surrounded by gNBs. Despite the conclusion, that the results provide useful information for the design and testing of tracking systems, as well as for the determination of the geometric deployment of base stations for good GDOP in the coverage area, it covers only simple geometric 2D cases, for which analytical solutions are possible, and thus does not respond to scenarios of 5G and B5G UDN gNB distribution in 3D space.

Work [61] analyzes the impact of variable geometry and number of 5G base stations on the convergence time of precise point positioning in combined BeiDou Navigation Satellite System (BDS) and 5G mobile communication technology. Despite attainable DOP values, the weak point of investigation is, that analysis was performed on predefined geometrical configurations of gNB, however, topology search is a much more flexible tool.

Research [62] analyzes the relationship between DOP and Round-Trip Time (RTT) and angle of departure (AOD) positioning accuracy. Performed simulation experiments in two scenarios of three gNB with good and complex environments and subsequent conclusions, again, are limited to analysis on predefined geometrical configurations of gNB.

Authors in [63] examine positioning geometric dilution of precision bounds in two-dimensional (2D) scenarios and show, that lowest possible GDOP, attainable for TOA or TDOA measurements with N optimally located base stations reaches value of 2⁄√N. Intuition behind investigation is the assumption, that the best GDOP occurs, if N base stations are at the vertices of an N-sided regular polygon, and the minimum GDOP is at the center of this polygon. Simulation results for the case of 5 base stations yields lowest contour value approximately 0,9 at the center of this polygon, which agrees with the analytical estimate 2⁄√N and reaches value below one. Work [63] also points out important nuance for bearing only measurements: the matrix of partial AOA derivatives yields measurement errors in units of radians, while the position error is in units of meters. To obtain a dimensionless GDOP author in [63] defines a normalized GDOP by the additional multiplication of partial AOA derivatives on the gNB-UE distance.

Added subsection 2.3. Topology Search Using Dilution of Precision Criterion Problem Statement

Presented above analysis of the background and state of the art in the field of topology search, using dilution of precision criterion in UDN, according to open sources, revealed, that most of the observed investigations were devoted to the analysis of the predefined base stations configurations with inherent limitations of regular hexagonal layout on the plane for 2D case. However, the problem of gNB topology synthesis in the space for 3D case had not been even set up. This circumstance can be explained by the fact, that even for predefined gNB configuration obtaining HDOP, VDOP and PDOP requires a lot of calculations, the volume of which depends on the step size for UE possible location in space area for 3D case. One of the solutions for gNB topology search is its configuration enumeration with a given DOP (HDOP, VDOP or PDOP) criterion in the defined deployment area and limit on available gNB number. The task is complicated by the fact that the obtained topology, favorable for example, to HDOP criterion, is not favorable to VDOP, thus it is appropriate to perform search according to integral PDOP criterion.

Previous own author results [44, 45], concerning topology search using DOP criterion for enhanced 5G positioning service area, considered configuration of only four gNB, which is enough for UE positioning in space with either TOA, TDOA and AOA method. Work [44] proposed TOA measurement processing model for 5G UDN positioning system topology search using HDOP, VDOP and PDOP criterion, which revealed, that configuration of fixed gNB number, achieving DOP criterion in horizontal, does not provide satisfying DOP in vertical plane. In particular, gNB topology, found for HDOP, fails to meet VDOP and PDOP in analyzed enhanced positioning service area, because HDOP case requires to spread gNB from UE on the plane and VDOP case re-quires to spread gNB from UE on the heigh, being closely to UE, which is contradictive. To overcome this contradiction, further investigation [45] considered already combined TOA and DOA measurement processing for the case of fixed UE height in positioning service area, which yielded gNB topology with HDOP, VDOP and PDOP values below two for the whole 2D plane of the enhanced 5G positioning service area. However, simulation results also revealed, that for the case of variable UE height in 5G positioning service area and for the case of more than four gNB topology search algorithm should be developed.

the proposed approach and the forthcoming results are presented for a number of cases (for the observed case scenarios at Gazprom Arena). However, no comparison to other work or reference solution is given in order to support quality of here proposed solution.

Thank you for your comment.

Detailed comparison of the simulation results, presented in section 4. “Models for Topology Search Using DOP Criterion in UDN” and section 6. “Algorithm for Topology Search Using DOP Criterion in UDN Simulation” is challenging, because from the background analysis in subsection 2.2. “Topology Search Using Dilution of Precision Criterion Background” and subsection 2.3 “Topology Search Using Dilution of Precision Criterion Problem Statement” we conclude, that most of the observed investigations were devoted to the analysis of the predefined base stations configurations with inherent limitations of regular hexagonal layout on the plane for 2D case.

First paragraph in the subsection 2.3. “Topology Search Using Dilution of Precision Criterion Problem Statement” explains contribution with respect to the state-of-the-art works:

Second paragraph in the subsection 2.3. “Topology Search Using Dilution of Precision Criterion Problem Statement” explains contribution with respect to the authors own works [44, 45]:

Previous own author results [44, 45], concerning topology search using DOP criterion for enhanced 5G positioning service area, considered configuration of only four gNB, which is enough for UE positioning in space with either TOA, TDOA and AOA method. Work [44] proposed TOA measurement processing model for 5G UDN positioning system topology search using HDOP, VDOP and PDOP criterion, which revealed, that configuration of fixed gNB number, achieving DOP criterion in horizontal, does not provide satisfying DOP in vertical plane. In particular, gNB topology, found for HDOP, fails to meet VDOP and PDOP in analyzed enhanced positioning service area, because HDOP case requires to spread gNB from UE on the plane and VDOP case requires to spread gNB from UE on the heigh, being closely to UE, which is contradictive. To overcome this contradiction, further investigation [45] considered already combined TOA and DOA measurement processing for the case of fixed UE height in positioning service area, which yielded gNB topology with HDOP, VDOP and PDOP values below two for the whole 2D plane of the enhanced 5G positioning service area. However, simulation results also revealed, that for the case of variable UE height in 5G positioning service area and for the case of more than four gNB topology search algorithm should be developed.

the referencing to prior work should be improved in chapters 2 and 3 in which background and overview on observed problem are given, as well as the state of the art in the field. The presented methods, equations, solutions and conclusions in this text should be referenced more strictly (for all instances that do not present contribution of this manuscript).

Thank you for your comment.

Added paragraph in the end of subsection 2.1.

Referencing to prior works [52 – 63] is improved in section 2, in particular, added subsection 2.2. Topology Search Using Dilution of Precision Criterion Background and subsection 2.3. Topology Search Using Dilution of Precision Criterion Problem Statement. Referencing to authors own previous works [44, 45] are refined in section 3.

Comments on the Quality of English Language: The quality of language is overall good - only final checking and error corrections.

Thank you for your comment. English of this manuscript has been revised with a native English speaker.

Author Response File: Author Response.docx

Reviewer 2 Report

The primary focus of this paper is to introduce a base station topology search algorithm based on the Dilution of Precision (DOP) factor, aimed at enhancing the positioning accuracy of User Equipment (UE) in Ultra-Dense Networks (UDN). The topic is timely and relevant, addressing technical challenges related to UE positioning in 5G/B5G/6G ultra-dense networks, which aligns with the development trends and demands of the communication field. However, the paper exhibits some shortcomings in terms of theoretical analysis and simulation experiments.

1) In Section 3, the paper does not provide detailed CRLB calculation formulas and theoretical derivations for both TOA-AOA and TDOA-AOA architecture. In Section 4, the function 'f' in f(VDOP), f(HDOP), and f(PDOP) within Figures 6, 8, and 9 have not been clearly defined in the paper. Besides, the paper offers simulation results for one scenario, without discussing results under varying parameters and conditions, and does not provide specific numerical values or error ranges.

2) There is room for improvement in the paper's innovative aspects. Although a novel topology search algorithm is proposed, it does not include comparisons or outlooks with existing methods. The paper only presents a step-by-step description of the algorithm, without offering performance metrics such as complexity, convergence, and stability. In Section 6, the paper provides simulation results for another scenario but does not analyze the algorithm's advantages and limitations, nor does it discuss future work.

3) The paper employs a brute-force search method to find the optimal base station topology, which may lead to excessive computational complexity, inefficiency, or even infeasibility when dealing with a large number of base stations or an extensive spatial range, the authors have not discussed this particular aspect in the paper. Furthermore, there are several derivation and expression errors in the equations. It is recommended that the authors carefully review equations (9), (19), (10), (25), (17), (24), (30), (31), and (32) for potential issues.

4) While the paper does not contain any apparent intuitive errors in its theory, certain aspects are insufficiently rigorous or reasonable. For instance, if the Jacobi matrix for AOA is also normalized, then the unit of the covariance matrix should not be in radians but rather converted to distance units for consistency. Additionally, it is influenced by the target-to-measurement station distance, and such factors should be taken into consideration in the analysis and simulation.

5) Figure 8 and Figure 11 present DOP calculations based on distance units, while Figure 14 shows DOP derived from the normalized Jacobi matrix after measuring and normalizing angular errors. It is unclear how these two DOP values are combined into a unified error unit in Figure 17 and Figure 19. Additionally, the final DOP value being less than 1 seems unreasonable to me, and I would appreciate an explanation for this.

6) From the reviewer's perspective, one of the most important things is, if this paper is to be applied in real-time positioning scenarios, the efficiency of the optimal station layout search algorithm is crucial. However, the article does not provide an analysis in this regard, nor does it compare the approach with other optimized station layout algorithms. Furthermore, there is no comparison with the complexity of positioning calculations involving more stations and no clear conclusion on the improvement of the final DOP. The innovation of the entire work appears to be not very prominent.

7) The reviewer suggests thoroughly checking for spelling and grammatical errors.

Overall, from the reviewer's perspective, the work presented in this paper has some theoretical significance, but the design of the related methods is not fully explained. The theoretical derivation process lacks a comprehensive result of the fused formulas, and the simulation section appears to be a mere accumulation of some results without a reasonable and complete explanation of the performance gains brought by the proposed fusion method. Even if there are gains, they are not evident. Therefore, the theoretical derivation and simulation experiment sections require substantial revisions or additional work.

Comments for author File: Comments.pdf

While the paper's readability is acceptable and there are not too many direct grammatical and punctuation errors (although exist), it is recommended that the authors thoroughly check for spelling and grammatical errors.

Author Response

Response to Reviewer 2

Comments and Suggestions for Authors

In Section 3, the paper does not provide detailed CRLB calculation formulas and theoretical derivations for both TOA-AOA and TDOA-AOA architecture.

Thank you for your comment. Authors really do not give a derivation of the CRLB, because it treats only GDOP factor. From the pioneering works, for example [64], it is well known, that CRLB for RMSE can be decomposed into geometric topology factor of base stations deployment and corresponding error of primary range, range difference of angle measurements.

Added paragraph in the beginning of the Section 3:

In current section we formalize calculation of GDOP factor for TOA, TDOA, AOA, combined TOA-AOA and TDOA-AOA measurements, and do not consider derivation of the Cramer-Rao Lower Bound, because it is well known, for example from pioneering work [64], that CRLB for RMSE of UE positioning accuracy can be decomposed into geometric dilution of precision, due to topology of base stations, and corresponding error of primary range, range difference or angle measurements.

In Section 4, the function 'f' in f(VDOP), f(HDOP), and f(PDOP) within Figures 6, 8, and 9 have not been clearly defined in the paper.

The functions , and are explained in the corresponding paragraph in Subsection 4.2. “Procedures for Topology Search Using DOP Criterion in UDN”:

The processing begins with filtering out gNB placement configurations, that meet the maximum DOP factor requirements; those configurations of gNB, for which PDOP, HDOP and VDOP exceed the value of , are excluded from further analysis. After filtering the gNB configurations, they are sorted in ascending order of the DOP factor. For each gNB configuration, the distance and the angle to the UE are calculated, as well as the height of each base station (Figure 5). Based on the parameters , and , their distributions are calculated and the most probable ones are searched [44].

Additional explanation is added in that paragraph:

The resulting functions , and , plotted in Figure 6, 9, 12, 15, 18, yields obtained distributions of distance , angle and height , giving the most probable geometrical values of gNB configuration, which provides DOP in a given range.

The functions , and are explained in the corresponding paragraph in Subsection 4.2. “Procedures for Topology Search Using DOP Criterion in UDN”:

After obtaining the topology of four gNB, we calculate DOP distribution for variable UE location on the plane with a given step. In particular, , and are resulting distributions of horizontal, vertical and position GDOP factors of UE positioning in location area, plotted in Figure 8, 11, 14, 17, 20.

Besides, the paper offers simulation results for one scenario, without discussing results under varying parameters and conditions, and does not provide specific numerical values or error ranges.

Thank you for your comment. Added subsection 4.3.6. “DOP for Topology of four gNB Simulation Results Discussion” in the end of the Section 4.2.

4.3.6. DOP for Topology of four gNB Simulation Results Discussion

The choice of four gNB for topology search can be explained by the fact, that existing works in this field [52-63] usually treat layout of four base stations. To compare presented results with known ones we can use PDOP criterion with TOA and TDOA primary measurements processing. If not to take into account deployment scale, we can see, that presented PDOP contours for TOA measurements in Figure 8 and for TDOA measurements in Figure 11 are no more than one, do not contradict to corresponding contours for range and pseudo-range measurements in [63] and lowest GDOP analytical estimate in [63], thus confirming the correctness of developed mathematical and simulation models.

Comparison of a set of PDOP, HDOP, and VDOP contours for obtained gNB configurations, using AOA, combined TOA-AOA and TDOA-AOA measurements processing in Figure 14, 17, 20, is difficult due to the lack of similar results in a 3D space.

There is room for improvement in the paper's innovative aspects. Although a novel topology search algorithm is proposed, it does not include comparisons or outlooks with existing methods.

Thank you for your comment. It is rather challenging task to compare proposed algorithm with existing methods because, as far as authors know from open sources, prototypes are absent. Nevertheless, to elaborate in a more detailed manner the contribution and distinction of the proposed solution to prior own works and the works of other authors, authors added thorough background and overview on observed problem as well as the state of the art in the field in new subsection 2.2. “Topology Search Using Dilution of Precision Criterion Background” and new subsection 2.3 “Topology Search Using Dilution of Precision Criterion Problem Statement”

Added subsection 2.2. Topology Search Using Dilution of Precision Criterion Background

Added subsection 2.3. Topology Search Using Dilution of Precision Criterion Problem Statement

The paper only presents a step-by-step description of the algorithm, without offering performance metrics such as complexity, convergence, and stability.

Thank you for your comment.

Submitted paper not only presents a step-by-step description of the algorithm, but also open-source Matlab code, which can be validated; source code is available online: https://github.com/grihafokin/Topology_search_using_DOP

Performance metric is DOP factor, which decreases, when we add new gNB.

Complexity, convergence and stability is ensured by the fact, that each additional gNB should decrease DOP factor and can be evaluated empirically by the number of search cycles, plotted in Figure 30.

As for strict mathematical formalization and complexity evaluation, it is out of scope in presented submission. However, the authors are grateful to the reviewer and may perform such an assessment in the future.

Added paragraph in the section 6.1. “Algorithm Initialization”:

Proposed algorithm has open-source Matlab realization [65] and can be verified. Algorithm performance evaluation metric is DOP factor, which decreases, when we add new gNB. Complexity, convergence and stability is ensured empirically by the fact, that each additional gNB decreases DOP factor and can be evaluated at first glance by the number of search cycles, plotted in Figure 30.

In Section 6, the paper provides simulation results for another scenario...

Thank you for your comment.

In Section 6 authors provide simulation results of the same scenario for gNB topology search in space of 500×500×100 m, as it was in Section 4. May be misleading is Figure 4 with gNB grid with a step of 50 m in a space of 500×500×50 m and Figure 22 with UE placement grid on the plane when searching for gNB topology, which emphasize the fact, that we can vary step size on the plane and in the heigh during topology search.

Added paragraph in the section 6.1. “Algorithm Initialization”:

Figure 22 illustrates an example of a UE placement grid on the plane, utilized when searching for gNB topology in a space of 500×500×100 m, which is the same, as scenario, with four gNB deployment in Section 4. Figure 4 shows gNB grid with a step of 50 m in a space of 500×500×50 m and Figure 22 shows UE placement grid on the plane, when searching for gNB topology in space, which emphasize the fact, that we can vary step size on the plane and in the heigh during topology search.

…but does not analyze the algorithm's advantages and limitations, nor does it discuss future work

Thank you for your comment. Added paragraph in the section 6.1. “Algorithm Initialization”:

Comparing proposed here algorithm with an approach of four gNB topology search, presented in Section 4, we emphasize, that this empirical algorithm is intended for configuration of more than four gNB. In particular, first four gNB expectedly are located in the vertices of positioning service area, which is consistent with lowest GDOP in 2-D scenarios, derived in [63].

Added paragraph in the section 6.1. “Algorithm Initialization”:

The paper employs a brute-force search method to find the optimal base station topology, which may lead to excessive computational complexity, inefficiency, or even infeasibility when dealing with a large number of base stations or an extensive spatial range, the authors have not discussed this particular aspect in the paper.

Thank you for your comment.

Added paragraph in the section 6.1. “Algorithm Initialization”:

Furthermore, there are several derivation and expression errors in the equations. It is recommended that the authors carefully review equations (9), (19), (10), (25), (17), (24), (30), (31), and (32) for potential issues.

Thank you for your comment.

Consider Figure 2.

Figure 2. UE and gNB 3D layout geometry.

Equation (9) with respect to Figure 2 is correct:

(9)

Equation (10) with respect to Figure 2 is correct:

(10)

Equation (19) with respect to Equation (9) is correct (we just add the dependence on the vector of UE coordinates ):

(19)

Equation (25) with respect to Equation (10) is correct (we just add the dependence on the vector of UE coordinates ):

(25)

Equation (17) with respect to Equations (18) and (24) is correct:

	(17)
	(18)
	(24)

Equations (30), (31), and (32) have no potential issues.

	(30)
	(31)
	(32)

Moreover, these expressions can be tracked by the open-source matlab function, available online: https://github.com/grihafokin/Topology_search_using_DOP

% calculation PDOP, HDOP, VDOP using TOA, TDOA, AOA, TOA-AOA, TDOA-AOA

% measurement processing

function [pdop, hdop, vdop] = calculate_dop(gNB, UE, calc_case)

% gNB - gNB coordinate vector

% UE - UE coordinate vector

n = size(gNB, 1); % gNB number

dim=size(gNB, 2); % dimension 3D

r=sqrt(sum((UE-gNB).^2,2)); % ranges between gNB and UE

% TOA matrix of partial derivatives with respect to xyz coordinates

H_TOA=(UE-gNB)./r;

% TDOA matrix of partial derivatives with respect to xyz coordinates

H_TDOA=zeros(n-1, dim);

H_TOA2=H_TOA(2:n,:); % H_TOA2=(UE-gNB(2:n,:))./r(2:n);

H_TOA1=H_TOA(1,:); % H_TOA1=(UE-gNB(1,:))./r(1);

H_TDOA=H_TOA2-H_TOA1;

% DOA matrix of partial derivatives with respect to xyz coordinates

gNB2D=gNB; gNB2D(:,3)=0; UE2D=UE; UE2D(:,3)=0;

r2D=sqrt(sum((UE2D-gNB2D).^2,2));

% for this case, the rows of the matrix H are not normalized to 1

H_DOA2D = zeros(n,dim);

H_DOA2D(:,1) = -(UE(:,2)-gNB(:,2))./(r2D.^2);

H_DOA2D(:,2) = (UE(:,1)-gNB(:,1))./(r2D.^2);

H_DOA3D = zeros(n,dim);

H_DOA3D(:,1)=-(((UE(:,3)-gNB(:,3)).*(UE(:,1)-gNB(:,1)))./(r.^2))./r2D;

H_DOA3D(:,2)=-(((UE(:,3)-gNB(:,3)).*(UE(:,2)-gNB(:,2)))./(r.^2))./r2D;

H_DOA3D(:,3)=r2D./(r.^2);

H_DOA=[H_DOA2D; H_DOA3D];

% for this case, the rows of the matrix H are normalized to 1 (option 1)

azAng = atan2(UE(2)-gNB(:,2), UE(1)-gNB(:,1)); % azimuth angle

H_DOA2D_1 = [-sin(azAng), cos(azAng), zeros(size(gNB(:,1)))];

elAng = atan((UE(3)-gNB(:,3))./r2D); % elevation angle

H_DOA3D_1 = [sin(elAng).*(UE(1) - gNB(:,1))./r2D, ...

sin(elAng).*(UE(2) - gNB(:,2))./r2D, ...

cos(elAng)];

H_DOA_1 = [H_DOA2D_1; H_DOA3D_1];

% for this case, the rows of the matrix H are normalized to 1 (option 2)

H_DOA2D_2 = zeros(n,dim);

% is calculated as H_DOA2D, but there is no division by r2D

H_DOA2D_2(:,1) = -(UE(:,2)-gNB(:,2))./r2D;

H_DOA2D_2(:,2) = (UE(:,1)-gNB(:,1))./r2D;

H_DOA3D_2 = zeros(n,dim);

% is calculated as H_DOA3D, but there is no division by r

H_DOA3D_2(:,1)=-((UE(:,3)-gNB(:,3)).*(UE(:,1)-gNB(:,1)))./r./r2D;

H_DOA3D_2(:,2)=-((UE(:,3)-gNB(:,3)).*(UE(:,2)-gNB(:,2)))./r./r2D;

H_DOA3D_2(:,3)=r2D./r;

H_DOA_2=[H_DOA2D_2; H_DOA3D_2];

% choice of calculation method

if calc_case== 'TOA '

H=H_TOA;

elseif calc_case=='TDOA '

H=H_TDOA;

elseif calc_case=='DOA '

H=H_DOA_1;

elseif calc_case=='TOA-DOA '

H=[H_TOA; H_DOA_1];

elseif calc_case=='TDOA-DOA'

H=[H_TDOA; H_DOA_1];

end

% DOP matrix calculation

G = pinv(H'*H);

pdop = sqrt(G(1,1) + G(2,2) + G(3,3));

hdop = sqrt(G(1,1) + G(2,2));

vdop = sqrt(G(3,3));

end

While the paper does not contain any apparent intuitive errors in its theory, certain aspects are insufficiently rigorous or reasonable. For instance, if the Jacobi matrix for AOA is also normalized, then the unit of the covariance matrix should not be in radians but rather converted to distance units for consistency.

Thank you for your comment. It is worthwhile to note, that we do not compute error covariance matrix of the UE coordinate estimate:

(33)

Instead, we calculate Jacobi matrix of partial derivatives for TOA, TDOA, AOA, combined TOA-AOA and TDOA-AOA measurements,

(35)

which is illustrated in the Matlab function above:

function [pdop, hdop, vdop] = calculate_dop(gNB, UE, calc_case)

Jacobi matrix normalization for AOA is substantiated in the end of subsection 2.2 with reference to work [63]:

Work [63] also points out important nuance for bearing only measurements: the matrix of partial AOA derivatives yields measurement errors in units of radians, while the position error is in units of meters. To obtain a dimensionless GDOP author in [63] defines a normalized GDOP by the additional multiplication of partial AOA derivatives on the gNB-UE distance.

Additionally, it is influenced by the target-to-measurement station distance, and such factors should be taken into consideration in the analysis and simulation.

Thank you for your comment.

Taking into consideration the distance in bearing measurement case is the next step during RMSE computation and should account AOA primary measurement error. However, for DOP computation in the case of AOA measurement we operate with normalized GDOP.

Figure 8 and Figure 11 present DOP calculations based on distance units, while Figure 14 shows DOP derived from the normalized Jacobi matrix after measuring and normalizing angular errors. It is unclear how these two DOP values are combined into a unified error unit in Figure 17 and Figure 19.

Thank you for your comment. It is combined according to normalized values in the subsection 3.3.4:

3.3.4. Combined TOA-AOA and TDOA-AOA Measurement Processing

For combined range and direction TOA-AOA measurements, Jacobian matrix is:

(31)

For combined range-difference and direction TDOA-AOA measurements, Jacobi matrix is given by:

(32)

and above listed matlab function combines it into a unified error (available online: https://github.com/grihafokin/Topology_search_using_DOP

function [pdop, hdop, vdop] = calculate_dop(gNB, UE, calc_case)

Additionally, the final DOP value being less than 1 seems unreasonable to me, and I would appreciate an explanation for this.

Thank you for your comment. Explanation of about DOP value being less than 1 is given in the last paragraph of subsection 2.2:

Authors in [63] examine positioning geometric dilution of precision bounds in two-dimensional (2D) scenarios and show, that lowest possible GDOP, attainable for TOA or TDOA measurements with optimally located base stations reaches value of . Intuition behind investigation is the assumption, that the best GDOP occurs, if base stations are at the vertices of an -sided regular polygon, and the minimum GDOP is at the center of this polygon. Simulation results for the case of 5 base stations yields lowest contour value approximately 0,9 at the center of this polygon, which agrees with the analytical estimate and reaches value below one. Work [63] also points out important nuance for bearing only measurements: the matrix of partial AOA derivatives yields measurement errors in units of radians, while the position error is in units of meters. To obtain a dimensionless GDOP author in [63] defines a normalized GDOP by the additional multiplication of partial AOA derivatives on the gNB-UE distance.

Also, DOP values below 1 are reported in [52]:

From the reviewer's perspective, one of the most important things is, if this paper is to be applied in real-time positioning scenarios, the efficiency of the optimal station layout search algorithm is crucial. However, the article does not provide an analysis in this regard, nor does it compare the approach with other optimized station layout algorithms.

Thank you for your comment. Comparison with other state-of-the art algorithms is difficult due to the lack of prototypes in a 3D space. Thorough background is added in subsection 2.2. Topology Search Using Dilution of Precision Criterion Background and subsection 2.3. Topology Search Using Dilution of Precision Criterion Problem Statement.

Furthermore, there is no comparison with the complexity of positioning calculations involving more stations and no clear conclusion on the improvement of the final DOP.

Thank you for your comment. Complexity, convergence and stability of the proposed algorithm is ensured by the fact, that each additional gNB should decrease DOP factor and can be evaluated empirically by the number of search cycles, plotted in Figure 30.

The innovation of the entire work appears to be not very prominent.

Thank you for your comment. However, to authors opinion and taking into account background analysis in the added subsection 2.2. “Topology Search Using Dilution of Precision Criterion Background”, the innovation of the entire work already deserves attention and can be developed in the future investigation.

The reviewer suggests thoroughly checking for spelling and grammatical errors.

Thank you for your comment. Authors thoroughly checked manuscript for spelling and grammatical errors

Overall, from the reviewer's perspective, the work presented in this paper has some theoretical significance, but the design of the related methods is not fully explained.

Thank you for your comment.

The theoretical derivation process lacks a comprehensive result of the fused formulas, and the simulation section appears to be a mere accumulation of some results without a reasonable and complete explanation of the performance gains brought by the proposed fusion method. Even if there are gains, they are not evident.

Added conclusion 3 in subsection 4.3.6. “DOP for Topology of four gNB Simulation Results Discussion”:

HDOP, VDOP and PDOP contour plots for combined TOA-AOA and TDOA-AOA measurement processing with four gNB yields DOP values below one and is less, than lower bound with optimally located base stations for TOA or TDOA only measurement processing.

Therefore, the theoretical derivation and simulation experiment sections require substantial revisions or additional work.

Thank you for your comment. In general, the article has been thoroughly revised, and authors hope, became much more significant.

Comments on the Quality of English Language. While the paper's readability is acceptable and there are not too many direct grammatical and punctuation errors (although exist), it is recommended that the authors thoroughly check for spelling and grammatical errors.

Thank you for your comment. English of this manuscript has been revised with a native English speaker.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The authors addressed most of my previous concerns. It would be better if the authors could go through the manuscript and revise certain words to improve the readability. There are some unknown Russian words in the comment area of the uploaded revised manuscript.

Author Response

Response to Reviewer 1

Comments and Suggestions for Authors:

The authors addressed most of my previous concerns.

Thanks a lot for your valuable feedback!

It would be better if the authors could go through the manuscript and revise certain words to improve the readability. There are some unknown Russian words in the comment area of the uploaded revised manuscript.

Thank you for your comment. We again revised the manuscript to improve the readability. “Unknown Russian words in the comment area” are due to Russian version of MS Word.

Author Response File: Author Response.docx

Reviewer 2 Report

I would like to begin by acknowledging the progress made by the author in revising the paper. The updated version has addressed several questions, corrected typos, and improved the structure and readability of the manuscript. This progress is appreciated and demonstrates the author's commitment to enhancing the quality of the paper. However, I must also express my concern that some essential questions remain unanswered or inadequately addressed, with some issues potentially having a subversive impact on the research:

1) In the author's Response Letter, the explanation regarding CRLB is not satisfactory. The author argues that a detailed derivation of the CRLB is unnecessary, citing just one earlier research article as evidence. However, this issue is, in fact, the central problem of the present paper and requires a more thorough examination. DOP is essentially a result of normalizing the measurement error covariance matrix in the Cramér-Rao bound, and the error metrics for time difference, distance, and angle are different. Their contributions to positioning accuracy are inherently distinct and directly related to the error. Analyzing the positioning DOP only by normalizing the error is not reasonable, which is why I initially emphasized the importance of the CRLB theoretical derivation. It is suggested to map the measurement error to the signal-to-noise ratio (SNR) under specific antenna scenarios, normalize the SNR, and use CRLB to measure the error of hybrid positioning methods.

It must be said that the derivation and calculation of DOP in passive positioning is not difficult, as this field already has mature results. The conclusions given in the referenced literature [64] are only for TDOA systems, while the greatest challenge in mixed positioning lies in analyzing the effects of different error types on positioning accuracy. If a convincing derivation process cannot be provided for this step, it is difficult not to question the novelty of the entire paper. To achieve this, I believe there is still a considerable amount of derivation work that needs to be done, which would likely be extremely challenging to accomplish in the short term.

Of course, the current simulation results focus solely on DOP. Once the aforementioned work is completed, it will be necessary to present the corresponding simulation experiment content that demonstrates the relationship between the theoretical results and error types.

2) I recommend that the author include a more comprehensive discussion or analysis of the computational complexity of their proposed method, and if possible, address its scalability. A performance comparison with other algorithms would be an interesting addition. An approximation or estimate based on the provided Matlab implementation is acceptable, and I believe this should not be a difficult task, given that the author has already provided the Github code download link. This additional information will help readers better understand the limitations and potential applications of your method in various scenarios.

While the author might feel that it is out of scope for their current submission, I believe that considering the practical implications of their work and addressing potential concerns about its complexity or feasibility would only enhance the overall quality and impact of their paper.

3) I noticed that after addressing the reviewers' questions, the length of the paper has become quite extensive. I kindly request that the authors revise the full-text content to ensure conciseness. Please consider streamlining the text, consolidating repetitive information, and focusing on the most crucial aspects of the research while maintaining clarity and completeness. This will help improve the readability of the paper and make it more suitable for publication.

Comments for author File: Comments.pdf

The quality of the English presentation in the paper has improved, with only a few minor flaws remaining. However, these do not significantly impact readability. For now, please concentrate on addressing the key questions raised by the reviewers.

Author Response

Response to Reviewer 2

Comments and Suggestions for Authors

Thanks a lot for your valuable feedback!

Thank you for your comment. Author still argue, that

“In current section we formalize calculation of GDOP factor for TOA, TDOA, AOA, combined TOA-AOA and TDOA-AOA measurements, and do not consider derivation of the Cramer-Rao Lower Bound, because it is well known, for example from pioneering work [64], that CRLB for RMSE of UE positioning accuracy can be decomposed into geometric dilution of precision, due to topology of base stations, and corresponding error of primary range, range difference or angle measurements.”

Article is titled “Algorithm for Topology Search Using Dilution of Precision Criterion…” and considers only DOP factor, NOT CRLB. Authors could continue explanation in the manuscript, but due to the reviewer’s comment 3 about the length of the paper, we continue it here.

Additional references:

Choi, Heon Ho, et al. "Dilution of Precision Relationship between Time Difference of Arrival and Time of Arrival Techniques with No Receiver Clock Bias." Journal of Electrical Engineering and Technology 11.3 (2016): 746-750.
Li, Binghao, Andrew G. Dempster, and Jian Wang. "3D DOPs for positioning applications using range measurements." Wireless sensor network 3.10 (2011): 334.
Guvenc, Ismail, and Chia-Chin Chong. "A survey on TOA based wireless localization and NLOS mitigation techniques." IEEE Communications Surveys & Tutorials 11.3 (2009): 107-124.
Spirito, Maurizio A. "On the accuracy of cellular mobile station location estimation." IEEE Transactions on vehicular technology 50.3 (2001): 674-685.
Shin, Dong-Ho, and Tae-Kyung Sung. "Comparisons of error characteristics between TOA and TDOA positioning." IEEE Transactions on Aerospace and Electronic Systems 38.1 (2002): 307-311.
Manolakis, Dimitris E. "Efficient solution and performance analysis of 3-D position estimation by trilateration." IEEE Transactions on Aerospace and Electronic systems 32.4 (1996): 1239-1248.
Lee, Harry B. "Accuracy limitations of hyperbolic multilateration systems." IEEE Transactions on Aerospace and Electronic Systems 1 (1975): 16-29.
Lee, Harry B. "A novel procedure for assessing the accuracy of hyperbolic multilateration systems." IEEE Transactions on Aerospace and Electronic Systems 1 (1975): 2-15.
Massatt, Paul, and Karl Rudnick. "Geometric formulas for dilution of precision calculations." Navigation 37.4 (1990): 379-391.
Stansfield, R. G. "Statistical theory of DF fixing." Journal of the Institution of Electrical Engineers-Part IIIA: Radiocommunication 94.15 (1947): 762-770.

confirm, that GDOP factor is a normalized value, uniquely determined by the topology of base stations and measurement processing method (TOA, TDOA, AOA, combined). Updated paragraph with references in the beginning of the Section 3:

In current section we formalize calculation of GDOP factor for TOA, TDOA, AOA, combined TOA-AOA and TDOA-AOA measurements, and do not consider derivation of the Cramer-Rao Lower Bound, because it is well known, for example from a range of pioneering [64–69] and contemporary [70–74] works, that CRLB for RMSE of UE positioning accuracy can be decomposed into geometric dilution of precision, due to topology of base stations, and corresponding error of primary range, range difference or angle measurements.

It must be said that the derivation and calculation of DOP in passive positioning is not difficult, as this field already has mature results.

Derivation and calculation of DOP does not depend on the positioning system type: active or passive. It depends only on the topology of base stations and measurement processing method (TOA, TDOA, AOA, combined).

Despite the fact, that it is really mature topic, well-known results treat only predefined base stations positions, while the problem for base stations topology search is out of scope, as it was already substantiated in subsection 2.2.

The conclusions given in the referenced literature [64] are only for TDOA systems, while the greatest challenge in mixed positioning lies in analyzing the effects of different error types on positioning accuracy.

Work by Torrieri treated not only TDOA, but also AOA positioning, however, only for special case of three base stations and did not consider combined range and angle measurements, like other pioneering [64–69] and contemporary [70–74] works. The contribution of current work is substantiated in subsection 2.2.

If a convincing derivation process cannot be provided for this step, it is difficult not to question the novelty of the entire paper.

Convincing derivation process for DOP is presented in section 3 and with submitted open-source algorithms for validation had not yet been contributed before.

To achieve this, I believe there is still a considerable amount of derivation work that needs to be done, which would likely be extremely challenging to accomplish in the short term.

Some derivation really can be performed for CRLB, but is out of scope in this manuscript. We draw the attention of the reviewer to the fact that the article is titled “Algorithm for Topology Search Using Dilution of Precision Criterion…” and considers only DOP factor, NOT CRLB.

DOP analysis with theoretical approach is presented in previous works for predefined base stations positions, while the problem for base stations topology search is out of scope, as it was already substantiated in subsection 2.2.

Positioning system depends on several error types, including: 1) Geometric Dilution of Precision; 2) secondary measurements processing algorithm error, for example, Gauss–Newton (GN) algorithm error; 3) primary range and bearing measurements processing acquisition error, depending on SNR; 4) base stations synchronization error; 5) conditions of multipath radio waves propagation between UE and eNBs; 6) instrumental error of hardware; 7) error of eNB own coordinates.

Current works consider ONLY GDOP factor, which is a normalized value, uniquely determined by the topology of base stations and measurement processing method (TOA, TDOA, AOA, combined).

2) I recommend that the author include a more comprehensive discussion or analysis of the computational complexity of their proposed method, and if possible, address its scalability.

As authors already noted earlier: “Complexity, convergence and stability is ensured by the fact, that each additional gNB should decrease DOP factor and can be evaluated empirically by the number of search cycles, plotted in Figure. As for strict mathematical formalization and complexity evaluation, it is out of scope in presented submission. However, the authors are grateful to the reviewer and may perform such an assessment in the future.

In current manuscript we added subsection 6.2. Algorithm Procedures Simulation Analysis”

In the beginning of the algorithm, we initialize the number of algorithm search cycles; in current realization it equals to Nopt=3 and is illustrated in Figure 24 and 27 for the case of five and six gNB respectively. On the first search cycle we search topology for the case of four gNB on the plane and in the heigh. Such approach stems from the fact, that four gNB is enough for 3D positioning and we know from previous theoretical and simulation results in section 4, that the most favorable topology yields, when four gNB are located at the vertices of the square positioning area. For the case of more than four base stations, each additional gNB is added to the search process sequentially. From the Figure 24 and 27 we notice, that additional fifth and sixth gNB allows to increase PDOP approximately by 0.08 and 0.07 respectively. In this graph, the symbols H and V denote the processes of searching for the placement of gNBs on the plane and in height, respectively, and the gNB number is designated by the index. The boundaries of PDOP decrease for each search process are indicated by colors.

Main search function search_opt takes the following input parameters: nk - gNB number, whose position is being searched; gNBpos - array of coordinates of all gNBs; ueX, ueY, ueZ - arrays of UE coordinates; dSizeH - search step on the plane; dSizeV - search step in height; angStep - array of azimuths, specifying the direction of gNB position search step; Xlim, Ylim, Zlim - arrays of gNB position bounds; maxIter - the maximum number of iterations of the search process; dop_case - search criterion (HDOP, VDOP, PDOP); optDir - position search plane (on the plane, in height). Main search function search_opt produces the following output parameters: gNBpos - gNB coordinates after searching the position of the nk-th gNB; gNBposP - intermediate gNB coordinates after nk-th gNB position search; dopP - intermediate DOP values after search of the nk-th gNB position; i - number of iterations performed to search the position of the nk-th gNB.

For example, for UE placement grid, depicted in Figure 22 with UEsizeh=500 m and UEsteph=20 m, function search_opt computes 25×25 PDOP values on the fixed height and 25×25×10 PDOP values, accounting height grid with UEsizev=100 m, UEstepv=10 m.

Next, consider topology search results for the case of five and six gNB.

A performance comparison with other algorithms would be an interesting addition.

As it was already stated in subsection 4.3.6.:

“Comparison of a set of PDOP, HDOP, and VDOP contours for obtained gNB configurations, using AOA, combined TOA-AOA and TDOA-AOA measurements processing in Figure 14, 17, 20, is difficult due to the lack of similar results in a 3D space.”

For the same reason, “performance comparison with other algorithms” is complicated due to the lack of similar algorithms, especially with open-source code.

An approximation or estimate based on the provided Matlab implementation is acceptable, and I believe this should not be a difficult task, given that the author has already provided the GitHub code download link. This additional information will help readers better understand the limitations and potential applications of your method in various scenarios.

In current manuscript we added subsection 6.2. “Algorithm Procedures Simulation Analysis”

Thank you for your comment. As authors already noted earlier: “Complexity, convergence and stability is ensured by the fact, that each additional gNB should decrease DOP factor and can be evaluated empirically by the number of search cycles, plotted in Figure 30. As for strict mathematical formalization and complexity evaluation, it is out of scope in presented submission. However, the authors are grateful to the reviewer and may perform such an assessment in the future.

Thank you for your comment. Comparing original and revised versions, submission increased by four pages and resulted in 34 pages. Following the reviewer's note, we excluded the case of topology search for seven gNB, leaving only the cases of five and six gNB, because the expected trend for DOP reduction holds. After that manuscript became 33 pages. Added after the first revision cycle state-of the art analysis, to author opinion, should be retained. Some Figures in Section 4 can be excluded, if it's really critical; however, to the authors opinion, these graphical results could alleviate reader comparison of their own simulation results, obtained with open-source code, with original.

Author Response File: Author Response.docx

Round 3

Reviewer 2 Report

The authors explained the changes according to the previous review, and the effort put into addressing the questions is appreciated. However, the most critical issues which I am concerned about have not been addressed satisfactorily.

However, since the authors insist that this paper is about DOP and that CRLB does not need to be re-derived, it is only necessary to further compress the length of the paper.

No comment.

Author Response

Response to Reviewer 2

Comments and Suggestions for Authors

However, since the authors insist that this paper is about DOP and that CRLB does not need to be re-derived, it is only necessary to further compress the length of the paper.

Thanks a lot for your valuable feedback! The authors are grateful to the reviewer and may perform CRLB assessment in the future.

The length of the paper was compressed from 33 to 24 pages by omitting some illustrations.

Author Response File: Author Response.docx

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Algorithm for Topology Search Using Dilution of Precision Criterion in Ultra-Dense Network Positioning Service Area

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