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23 pages, 672 KiB  
Article
Metaheuristic Optimization for Robust RSSD-Based UAV Localization with Position Uncertainty
by Yuanyuan Zhang, Jiping Li, T. Aaron Gulliver, Huafeng Wu, Guangqian Xie, Xiaojun Mei, Jiangfeng Xian, Weijun Wang and Linian Liang
Drones 2025, 9(2), 147; https://doi.org/10.3390/drones9020147 - 18 Feb 2025
Viewed by 388
Abstract
Unmanned aerial vehicles (UAVs) have garnered significant research interest across various fields due to their excellent maneuverability, scalability, and flexibility. However, potential collisions and other issues can disrupt communication and hinder functionality in real-world applications. Therefore, accurate localization of UAVs is crucial. Nonetheless, [...] Read more.
Unmanned aerial vehicles (UAVs) have garnered significant research interest across various fields due to their excellent maneuverability, scalability, and flexibility. However, potential collisions and other issues can disrupt communication and hinder functionality in real-world applications. Therefore, accurate localization of UAVs is crucial. Nonetheless, environmental factors and inherent stability issues can lead to node positional errors in UAV networks, compounded by inaccuracies in transmit power estimation, complicating the effectiveness of signal strength-based localization methods in achieving high accuracy. To mitigate the adverse effects of these issues, a novel received signal strength difference (RSSD)-based localization scheme based on a robust enhanced salp swarm algorithm (RESSA) is presented. In this algorithm, an elitism strategy based on tent opposition-based learning (TOL) is proposed to promote the leader to move around the food source. Differential evolution (DE) is then used to enhance the exploration ability of each agent and improve global search. In addition, a dynamic movement mechanism for followers is designed, enabling the swarm to swiftly converge towards the food source, thereby accelerating the overall convergence process. The RSSD-based Cramér–Rao lower bound (CRLB) with position uncertainty is derived to evaluate the performance. Experimental results are presented, which show that the proposed RESSA provides better localization performance than related methods in the literature. Full article
(This article belongs to the Special Issue Space–Air–Ground Integrated Networks for 6G)
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<p>Control parameter <math display="inline"><semantics> <msub> <mi>c</mi> <mn>1</mn> </msub> </semantics></math> versus the iteration number <math display="inline"><semantics> <mrow> <mi>i</mi> <mi>t</mi> </mrow> </semantics></math>.</p>
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<p>Tent map <math display="inline"><semantics> <mi mathvariant="bold-italic">t</mi> </semantics></math> versus the iteration number <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p>
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<p>Convergence parameter <math display="inline"><semantics> <msup> <mi>c</mi> <mo>′</mo> </msup> </semantics></math> versus the iteration number <math display="inline"><semantics> <mi>τ</mi> </semantics></math>.</p>
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<p>Flowchart of the RESSA.</p>
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<p>RMSE for 7 methods and the CRLB versus <math display="inline"><semantics> <msup> <mi>σ</mi> <mn>2</mn> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>, and (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math> randomly deployed nodes.</p>
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<p>RMSE for 7 methods and the CRLB versus the path loss exponent <math display="inline"><semantics> <mi>ξ</mi> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>, and (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math> randomly deployed nodes.</p>
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<p>RMSE for 7 methods and CRLB versus position error <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>a</mi> </msub> </semantics></math> with <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math>, and (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math> randomly deployed nodes.</p>
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<p>CDF of <math display="inline"><semantics> <mfenced separators="" open="&#x2225;" close="&#x2225;"> <mover accent="true"> <mi mathvariant="bold-italic">x</mi> <mo stretchy="false">^</mo> </mover> <mo>−</mo> <mi mathvariant="bold-italic">x</mi> </mfenced> </semantics></math> for 7 methods with <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>a</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>; <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics></math>; and (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> m, (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math> m, (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math> m, and (<b>d</b>) <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>e</mi> <mi>n</mi> <mi>g</mi> <mi>t</mi> <mi>h</mi> <mo>=</mo> <mn>80</mn> </mrow> </semantics></math> m.</p>
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26 pages, 1166 KiB  
Article
Preamble-Based Signal-to-Noise Ratio Estimation for Adaptive Modulation in Space–Time Block Coding-Assisted Multiple-Input Multiple-Output Orthogonal Frequency Division Multiplexing System
by Shahid Manzoor, Noor Shamsiah Othman and Mohammed W. Muhieldeen
Algorithms 2025, 18(2), 97; https://doi.org/10.3390/a18020097 - 9 Feb 2025
Viewed by 379
Abstract
This paper presents algorithms to estimate the signal-to-noise ratio (SNR) in the time domain and frequency domain that employ a modified Constant Amplitude Zero Autocorrelation (CAZAC) synchronization preamble, denoted as CAZAC-TD and CAZAC-FD SNR estimators, respectively. These SNR estimators are invoked in a [...] Read more.
This paper presents algorithms to estimate the signal-to-noise ratio (SNR) in the time domain and frequency domain that employ a modified Constant Amplitude Zero Autocorrelation (CAZAC) synchronization preamble, denoted as CAZAC-TD and CAZAC-FD SNR estimators, respectively. These SNR estimators are invoked in a space–time block coding (STBC)-assisted multiple-input multiple-output (MIMO) orthogonal frequency division multiplexing (OFDM) system. These SNR estimators are compared to the benchmark frequency domain preamble-based SNR estimator referred to as the Milan-FD SNR estimator when used in a non-adaptive 2×2 STBC-assisted MIMO-OFDM system. The performance of the CAZAC-TD and CAZAC-FD SNR estimators is further investigated in the non-adaptive 4×4 STBC-assisted MIMO-OFDM system, which shows improved bit error rate (BER) and normalized mean square error (NMSE) performance. It is evident that the non-adaptive 2×2 and 4×4 STBC-assisted MIMO-OFDM systems that invoke the CAZAC-TD SNR estimator exhibit superior performance and approach closer to the normalized Cramer–Rao bound (NCRB). Subsequently, the CAZAC-TD SNR estimator is invoked in an adaptive modulation scheme for a 2×2 STBC-assisted MIMO-OFDM system employing M-PSK, denoted as the AM-CAZAC-TD-MIMO system. The AM-CAZAC-TD-MIMO system outperformed the non-adaptive STBC-assisted MIMO-OFDM system using 8-PSK by about 2 dB at BER = 104. Moreover, the AM-CAZAC-TD-MIMO system demonstrated an SNR gain of about 4 dB when compared with an adaptive single-input single-output (SISO)-OFDM system with M-PSK. Therefore, it was shown that the spatial diversity of the MIMO-OFDM system is key for the AM-CAZAC-TD-MIMO system’s improved performance. Full article
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<p>STBC-assisted MIMO-OFDM system with adaptive modulation block diagram.</p>
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<p>Suparna preamble structure proposed for time synchronization [<a href="#B32-algorithms-18-00097" class="html-bibr">32</a>].</p>
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<p>Proposed modified preamble structure for CAZAC-TD and CAZAC-FD SNR estimators.</p>
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<p>Proposed modified preamble structure with cyclic prefix.</p>
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<p>Preamble structure used in Milan SNR estimator in [<a href="#B18-algorithms-18-00097" class="html-bibr">18</a>].</p>
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<p>Flowchart of CAZAC-TD SNR estimation algorithm.</p>
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<p>At <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>N</mi> <mi>R</mi> <mo>=</mo> <mn>12</mn> </mrow> </semantics></math> dB, the autocorrelation plots of (<b>a</b>) the transmitted OFDM signal and (<b>b</b>) the received STBC-decoded signal.</p>
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<p>The non-adaptive STBC-assisted MIMO-OFDM system’s BER performance when employing <span class="html-italic">M</span>-PSK modulation for transmission over the SUI-5 channel.</p>
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<p>Autocorrelation plots of OFDM received signal, transmitted over AWGN channel: (<b>a</b>) the Suparna preamble structure; (<b>b</b>) the modified CAZAC preamble structure.</p>
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<p>The NMSE performance invoking Suparna preamble structure and the modified CAZAC preamble structure for the AWGN channel.</p>
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<p>The estimated SNR performance for the AWGN channel with a zoomed-in view in the inset.</p>
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<p>The estimated SNR performance for the SUI-5 channel with a zoomed-in view in the inset.</p>
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<p>The NMSE performance of the non-adaptive STBC-assisted MIMO-OFDM system for the AWGN channel.</p>
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<p>The NMSE performance of the non-adaptive STBC-assisted MIMO-OFDM system for the SUI-5 channel.</p>
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<p>The BER performance of the non-adaptive STBC-assisted MIMO-OFDM system for the AWGN channel.</p>
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<p>The BER performance of the non-adaptive STBC-MIMO-OFDM system for the SUI-5 channel.</p>
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<p>The proposed AM-CAZAC-TD-MIMO system’s BER performance for the SUI-5 channel.</p>
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<p>The proposed AM-CAZAC-TD-MIMO system’s channel capacity performance for the SUI-5 channel.</p>
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<p>A comparison of the BER performance of the AM-CAZAC-TD-MIMO system and AM-CAZAC-TD-SISO system employing <span class="html-italic">M</span>-PSK for the SUI-5 channel.</p>
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<p>A comparison of the channel capacity performance of the AM-CAZAC-TD-MIMO system and AM-CAZAC-TD-SISO system employing <span class="html-italic">M</span>-PSK for SUI-5 channel.</p>
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29 pages, 40870 KiB  
Article
Ground-Based RFI Source Localization via Single-Channel SAR Using Pulse Range Difference of Arrival
by Jiaxin Wan, Bing Han, Jianbing Xiang, Di Yin, Shangyu Zhang, Jiazhi He, Jiayuan Shen and Yugang Feng
Remote Sens. 2025, 17(4), 588; https://doi.org/10.3390/rs17040588 - 8 Feb 2025
Viewed by 572
Abstract
Radio Frequency Interference (RFI) significantly degrades the quality of spaceborne Synthetic Aperture Radar (SAR) images, and RFI source localization is a crucial component of SAR interference mitigation. Single-station, single-channel SAR, referred to as single-channel SAR, is the most common operational mode of spaceborne [...] Read more.
Radio Frequency Interference (RFI) significantly degrades the quality of spaceborne Synthetic Aperture Radar (SAR) images, and RFI source localization is a crucial component of SAR interference mitigation. Single-station, single-channel SAR, referred to as single-channel SAR, is the most common operational mode of spaceborne SAR. However, studies on RFI source localization for this system are limited, and the localization accuracy remains low. This paper presents a method for locating the ground-based RFI source using spaceborne single-channel SAR echo data. First, matched filtering is employed to estimate the range and azimuth times of the RFI pulse-by-pulse in the SAR echo domain. A non-convex localization model using Pulse Range Difference of Arrival (PRDOA) is established based on the SAR observation geometry. Then, by applying Weighted Least Squares and Semidefinite Relaxation, the localization model is transformed into a convex optimization problem, allowing for the solution of its global optimal solution to achieve RFI source localization. Furthermore, the error analysis on the PRDOA localization model is conducted and the Cramér–Rao Lower Bound is derived. Based on the simulation platform and the SAR level-0 raw data of Gaofen-3, we conduct several verification experiments, with the Pulse Time of Arrival localization selected for comparison. The results demonstrate that the proposed method achieves localization accuracy with a hundred-meter error in azimuth and a kilometer-level total error, with the total localization errors reduced to approximately 1/4 to 1/3 of those of the Pulse Time of Arrival method. Full article
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<p>Examples of SAR images affected by RFI.</p>
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<p>Schematic diagram of the ECEF coordinate system and the geometric relationship between SAR and RFI sources: (<b>a</b>) ECEF coordinate system; (<b>b</b>) Geometric relationship between SAR and the RFI source.</p>
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<p>Representations of signals with different modulation (linear and sinusoidal frequency modulation) at an SNR of −10 dB, along with the corresponding results after matched filtering: (<b>a</b>) Time-frequency domain image of the original signals; (<b>b</b>) Time-frequency domain image of the signals after matched filtering; (<b>c</b>) Time-domain amplitude plot of the signals after matched filtering.</p>
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<p>Processing steps and results of matched filtering and range time <math display="inline"><semantics> <mi>τ</mi> </semantics></math> estimation for fully and partially received LFM signals.</p>
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<p>Reception of RFI signals during SAR operation.</p>
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<p>Estimation of the total interval of RFI pulses <math display="inline"><semantics> <msub> <mi>T</mi> <mi>y</mi> </msub> </semantics></math> during the simulated SAR non-reception time: (<b>a</b>) Example of the time-frequency diagram of the simulated signal; (<b>b</b>) Estimation error of the <math display="inline"><semantics> <msub> <mi>T</mi> <mi>y</mi> </msub> </semantics></math>.</p>
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<p>Schematic diagram of RFI source grid point setting in the simulation experiment.</p>
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<p>2-D contour plots of the CRLB and RMSE for PRDOA localization, and CRLB for PTOA localization under different satellite orbit parameters: (<b>a</b>–<b>c</b>) Distribution of the CRLB and RMSE for the proposed algorithm and the CRLB for PTOA localization under the Satellite A orbit parameters; (<b>d</b>–<b>f</b>) Distribution of the CRLB and RMSE for the proposed algorithm and the CRLB for PTOA localization under the Satellite B orbit parameters.</p>
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<p>Variation in RMSE and CRLB for PRDOA localization and CRLB for PTOA localization with azimuth and range under different satellite orbit parameters: (<b>a</b>,<b>b</b>) Variation in CRLB and RMSE for the proposed algorithm and CRLB for PTOA localization with azimuth and range under Satellite A orbit parameters; (<b>c</b>,<b>d</b>) Variation in CRLB and RMSE for the proposed algorithm and CRLB for PTOA localization with azimuth and range under Satellite B orbit parameters.</p>
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<p>Projections of PRDOA and PTOA localization metrics in azimuth and range directions as functions of azimuth and range: (<b>a</b>,<b>b</b>) The azimuth CRLB and RMSE of the proposed algorithm, and the azimuth CRLB of PTOA localization as functions of azimuth and range; (<b>c</b>,<b>d</b>) The range CRLB and RMSE of the proposed algorithm, and the range CRLB of PTOA localization as functions of azimuth and range.</p>
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<p>Variation in RMSE and CRLB for PRDOA localization and CRLB for PTOA localization with parameter variations: (<b>a</b>) Localization error variations with different RFI signal reception durations; (<b>b</b>) Localization error variations with different noise variances <math display="inline"><semantics> <msub> <mi>σ</mi> <mrow> <mi>P</mi> <mi>R</mi> <mi>D</mi> <mi>O</mi> <mi>A</mi> </mrow> </msub> </semantics></math>; (<b>c</b>) Localization error variations with different PRIs of RFI; (<b>d</b>) Localization error variations with different PRFs of SAR.</p>
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<p>Differences between the PRDOA localization errors based on the real <math display="inline"><semantics> <msub> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">R</mi> </msub> </semantics></math> and those using the approximated <math display="inline"><semantics> <msub> <mi mathvariant="bold">C</mi> <mi mathvariant="bold">R</mi> </msub> </semantics></math> for a 21 × 20 grid under the observation of satellite A.</p>
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<p>Simulated radar RFI source and its deployment environment used in the experiment.</p>
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<p>Time-frequency images of SAR raw echo with RFI signal and after matched filtering: (<b>a</b>) Time-frequency image of single-frame raw echo with interference signal; (<b>b</b>) Time-frequency image after matched filtering.</p>
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<p>PRDOA estimation for each frame in the SAR echo domain with RFI signal.</p>
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<p>Schematic of the RFI source at three grid points in the azimuth or range direction.</p>
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<p>PRDOA of three points in the same azimuth or range direction: (<b>a</b>) PRDOA of three points in the same range direction; (<b>b</b>) PRDOA of three points in the same azimuth direction.</p>
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16 pages, 951 KiB  
Technical Note
Angle and Range Unambiguous Estimation with Nested Frequency Diverse Array MIMO Radars
by Zhengxi Wang, Ximin Li, Shengqi Zhu, Fa Wei and Congfeng Liu
Remote Sens. 2025, 17(3), 446; https://doi.org/10.3390/rs17030446 - 28 Jan 2025
Viewed by 388
Abstract
This paper proposes an unambiguous method for joint angle and range estimation in colocated multiple-input multiple-output (MIMO) radar using the nested frequency diverse array (NFDA). Unlike a conventional phased array (PA), the transmission beampattern of FDA-MIMO radar depends not only on angle but [...] Read more.
This paper proposes an unambiguous method for joint angle and range estimation in colocated multiple-input multiple-output (MIMO) radar using the nested frequency diverse array (NFDA). Unlike a conventional phased array (PA), the transmission beampattern of FDA-MIMO radar depends not only on angle but also on range, which enables the precise identification of ambiguous regions in the two-dimensional frequency space. As a result, we can simultaneously estimate the angle and range of targets using FDA-MIMO radar, even when range ambiguity exists. By employing a nested array configuration, the degrees of freedom (DOFs) of the FDA are expanded. This expansion leads to improved accuracy in parameter estimation and enables a greater number of identifiable targets. In addition, the Cramér–Rao lower bound (CRLB) and the algorithm complexity are obtained to facilitate performance analysis. The simulation outcomes are presented to showcase the superior performance of the suggested approach. Full article
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<p>The configuration of transmitter.</p>
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<p>The receiver processing program.</p>
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<p>Spectral distributions for different numbers of targets. (<b>a</b>–<b>c</b>) Target number = 3. (<b>d</b>–<b>f</b>) Target number = 4. (<b>g</b>–<b>i</b>) Target number = 5.</p>
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<p>Resolvability of two point targets in the same range bin.</p>
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<p>Parameter estimation results. (<b>a</b>) Number of range ambiguity. (<b>b</b>) Principal range difference.</p>
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<p>RMSE versus SNR. (<b>a</b>) Angle estimation. (<b>b</b>) Range estimation.</p>
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<p>RMSE versus the number of snapshots. (<b>a</b>) Angle estimation. (<b>b</b>) Range estimation.</p>
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26 pages, 1270 KiB  
Article
Node Selection and Path Optimization for Passive Target Localization via UAVs
by Xiaoyou Xing, Zhiwen Zhong, Xueting Li and Yiyang Yue
Sensors 2025, 25(3), 780; https://doi.org/10.3390/s25030780 - 28 Jan 2025
Viewed by 435
Abstract
The performance of passive target localization is affected by the positions of unmanned aerial vehicles (UAVs) at a large scale. In this paper, to improve resource utilization efficiency and localization accuracy, the node selection problem and the path optimization problem are jointly investigated. [...] Read more.
The performance of passive target localization is affected by the positions of unmanned aerial vehicles (UAVs) at a large scale. In this paper, to improve resource utilization efficiency and localization accuracy, the node selection problem and the path optimization problem are jointly investigated. Firstly, the target passive localization model is established and the Chan-based time difference of arrival (TDOA) localization method is introduced. Then, the Cramer–Rao lower bound (CRLB) for Chan-TDOA localization is derived, and the problems of node selection and path optimization are formulated. Secondly, a CRLB-based node selection method is proposed to properly divide the UAVs into several groups, localizing different targets, and a CRLB-based path optimization method is proposed to search for the optimal UAV position configuration at each time step. The proposed path optimization method also effectively handles no-fly-zone (NFZ) constraints, ensuring operational safety while maintaining optimal target tracking performance. Also, to improve the efficiency of path optimization, particle swarm algorithm (PSO) is applied to accelerate the searching process. Finally, numerical simulations are performed to verify the validity and effectiveness of the proposed methods in this paper. Full article
(This article belongs to the Special Issue Radar Target Detection, Imaging and Recognition)
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Figure 1
<p>Geometric schematic of the TDOA localization model. The distance between target <span class="html-italic">j</span> and each UAV is denoted as <math display="inline"><semantics> <msub> <mi>r</mi> <mrow> <mi>j</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </semantics></math>, where UAV 0 serves as the reference node (or master UAV), and UAV <span class="html-italic">i</span> (<math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>) represents other nodes in the swarm. The concentric circles around the target indicate the signal propagation.</p>
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<p>Flow chart of the Chan-TDOA algorithm.</p>
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<p>Diagram of UAV motion constraints. The UAV moves from position <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mi>t</mi> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mi>i</mi> <mi>t</mi> </msubsup> <mo stretchy="false">)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi>x</mi> <mi>i</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo>,</mo> <msubsup> <mi>y</mi> <mi>i</mi> <mrow> <mi>t</mi> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <mo stretchy="false">)</mo> </mrow> </semantics></math> along a circular arc, where <math display="inline"><semantics> <msubsup> <mi>L</mi> <mi>i</mi> <mi>t</mi> </msubsup> </semantics></math> represents the path length, <math display="inline"><semantics> <msubsup> <mi>θ</mi> <mi>i</mi> <mi>t</mi> </msubsup> </semantics></math> is the turning angle, and <span class="html-italic">v</span> denotes the velocity vector of the UAV.</p>
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<p>Initial positions of the UAVs and the targets. The blue triangles represent the initial positions of the 9 UAVs (UAV0–UAV8), and the orange stars represent the initial positions of the 3 targets.</p>
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<p>UAV grouping results and their movement directions after node selection. Different colors represent different UAV groups assigned to their respective targets, and arrows indicate the planned movement directions.</p>
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<p>Evolution of best fitness value (CRLB) during PSO iterations for the first UAV group at the initial time step.</p>
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<p>Three selected UAVs localizing target 1 and their moving direction. (<b>a</b>) Movement direction of the three selected UAV groups for target 1. (<b>b</b>) Trajectories of the three selected UAV groups for target 1.</p>
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<p>Three selected UAVs localizing target 2 and their moving directions. (<b>a</b>) Movement direction of the three selected UAV groups for target 2. (<b>b</b>) Trajectories of the three selected UAV groups for target 2.</p>
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<p>Three selected UAVs localizing target 3 and their moving direction. (<b>a</b>) Movement direction of the three selected UAV groups for target 3. (<b>b</b>) Trajectories of the three selected UAV groups for target 3.</p>
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<p>Comparison of CRLB and RMSE under path optimization versus fixed configuration for target 1.</p>
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<p>Comparison of RMSE and CRLB between the proposed path optimization method and the traditional method.</p>
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<p>Comparison of RMSE and CRLB between the proposed path optimization method and the traditional method.</p>
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<p>CRLB comparison and single-iteration computation time comparisons with different optimization algorithms localizing different targets. (<b>a</b>) CRLB comparison with different optimization algorithms localizing target 1. (<b>b</b>) CRLB comparison with different optimization algorithms localizing target 2. (<b>c</b>) CRLB comparison with different optimization algorithms localizing target 3. (<b>d</b>) Single-iteration computation time comparisons between different optimization methods.</p>
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<p>Optimized UAV paths under different minimum turning radii and their RMSE comparisons. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> = 5000 m. (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> = 7500 m. (<b>c</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> = 10,000 m. (<b>d</b>) Comparison of RMSE and CRLB.</p>
Full article ">Figure 14 Cont.
<p>Optimized UAV paths under different minimum turning radii and their RMSE comparisons. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> = 5000 m. (<b>b</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> = 7500 m. (<b>c</b>) <math display="inline"><semantics> <msub> <mi>L</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> = 10,000 m. (<b>d</b>) Comparison of RMSE and CRLB.</p>
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<p>Optimized UAV paths and comparison of RMSE and CRLB under different no-fly-zone radii. (<b>a</b>) <math display="inline"><semantics> <msub> <mi>R</mi> <mrow> <mi>N</mi> <mi>F</mi> <mi>Z</mi> </mrow> </msub> </semantics></math> = 1000 m. (<b>b</b>) <math display="inline"><semantics> <msub> <mi>R</mi> <mrow> <mi>N</mi> <mi>F</mi> <mi>Z</mi> </mrow> </msub> </semantics></math> = 2000 m. (<b>c</b>) <math display="inline"><semantics> <msub> <mi>R</mi> <mrow> <mi>N</mi> <mi>F</mi> <mi>Z</mi> </mrow> </msub> </semantics></math> = 3000 m. (<b>d</b>) Comparisons of RMSE and CRLB.</p>
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39 pages, 2003 KiB  
Article
Estimation of the Measurement Accuracy of Wireless Passive Resonance Sensors
by Leonhard M. Reindl, Taimur Aftab, Thomas Schaechtle, Thomas Ostertag, Wei Luo and Stefan Johann Rupitsch
Sensors 2025, 25(3), 747; https://doi.org/10.3390/s25030747 - 26 Jan 2025
Viewed by 460
Abstract
Resonators are passive devices that respond to an excitation signal by oscillating at their natural frequency with exponentially decreasing amplitudes. Physical, chemical and electrical variables can modify the natural frequencies of resonators. If resonators are connected to antennas or other transducers that couple [...] Read more.
Resonators are passive devices that respond to an excitation signal by oscillating at their natural frequency with exponentially decreasing amplitudes. Physical, chemical and electrical variables can modify the natural frequencies of resonators. If resonators are connected to antennas or other transducers that couple into a communication channel, they enable purely passive sensors that can be read wirelessly. In this manuscript, we use maximum likelihood estimation to analyze the measurement accuracy that can be achieved by the wireless readout of passive resonant sensors as a function of the read signal, the stimulation power and noise figure of the reader, the distance and transducer gain of the transmission channel, and the natural frequency and quality factor of the resonant passive sensor. The Crámer–Rao lower bound characterizes the minimum variance of the natural frequency and decay constant of the resonator. We show the derivation of the Crámer–Rao lower bounds from the Fisher information matrix based on a maximum likelihood estimation of discrete-time samples of an exponentially decaying phasor. This theoretical lower limit of accuracy is almost achieved by an iterative algorithm that approximates the maximum of the measured resonator spectrum with a Lorentz curve. Full article
(This article belongs to the Special Issue Feature Papers in Physical Sensors 2024)
Show Figures

Figure 1

Figure 1
<p>Schematic of the operation of a transponder system using passive resonant sensors: A reader sends a read signal (blue arrow to the right) to the sensor node over a wireless channel. This signal is received there and then stored in an oscillation in the resonator. When the read signal is switched off, the oscillation resonates with an exponentially decreasing amplitude at a frequency, which is modified by the quantity to be measured. A part of the stored energy is sent back to the reader as a backscatter signal (blue arrow to the left). There, it is received, sampled and evaluated.</p>
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<p>Schematic representation of the stimulation signal <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> delivered from the antenna to the resonator and the stored oscillation <math display="inline"><semantics> <msub> <mi>s</mi> <mrow> <mi>d</mi> <mi>e</mi> <mi>c</mi> <mi>a</mi> <mi>y</mi> </mrow> </msub> </semantics></math> in the resonator. The amplitude and frequency of the excitation signal were normalized to one, which is the time to <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>·</mo> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> </semantics></math>. The resonator in this example has a quality factor of 10. Shown is the real part of <math display="inline"><semantics> <mrow> <msub> <mi>s</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <msub> <mi>s</mi> <mrow> <mi>d</mi> <mi>e</mi> <mi>c</mi> <mi>a</mi> <mi>y</mi> </mrow> </msub> </semantics></math>. The stimulation signal stops at <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>d</mi> </mrow> </msub> </semantics></math>, and the oscillation decays. Half of the signal from the decaying resonator is applied to the source impedance of the antenna, the other half to the internal resistance of the resonator. The figure is taken from [<a href="#B49-sensors-25-00747" class="html-bibr">49</a>].</p>
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<p>Typical reception power <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </semantics></math> in the reader when reading passive resonant transponders. Electromagnetic propagation in free space was assumed (<math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mi>m</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>). The transmission power <math display="inline"><semantics> <msub> <mi>P</mi> <mi>t</mi> </msub> </semantics></math> is 10 mW, the antennas radiate into the half-space (<math display="inline"><semantics> <mrow> <msub> <mi>g</mi> <mi>t</mi> </msub> <mo>=</mo> <msub> <mi>g</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>) and the stimulation was carried out at resonance frequency (<math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mi>R</mi> </msub> <mfenced open="(" close=")"> <mi>ω</mi> </mfenced> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>) over the duration of <span class="html-italic">Q</span> oscillations (<math display="inline"><semantics> <mrow> <mi>α</mi> <mo>·</mo> <msub> <mi>t</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <mi>π</mi> </mrow> </semantics></math>). The curves are drawn from top to bottom for the frequencies 433 MHz (red line), 868 MHz (green line), 915 MHz (blue line), 2.45 GHz (magenta line) and 5.8 GHz (black line) starting from a distance of 1 <math display="inline"><semantics> <mi>λ</mi> </semantics></math>.</p>
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<p>Schematic representation of the complex input signals of the reader, real part on the left and real and imaginary parts on the right, both as a function of time. Frequency and time are scaled as in <a href="#sensors-25-00747-f002" class="html-fig">Figure 2</a>. Initially, the ambient echoes are larger than the resonator’s decaying signal, but they have a smaller quality factor and soon disappear. In the next phase, the resonator signals dominate, which later also disappear into the noise. The resonator in this schematic has a quality factor of 10.</p>
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<p>Remaining initial power level <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mi>g</mi> <mi>a</mi> <mi>t</mi> <mi>e</mi> <mi>d</mi> </mrow> </msub> </semantics></math> in the reader when reading passive resonant transponders with a quality factor of 10, 100 and 1000 after a time gate to eliminate ambient echoes. The quality factor of the transmission channel was set to 15. There are three curves for each of the frequencies 433 MHz (red line), 868 MHz (green line), 915 MHz (blue line), 2.45 GHz (magenta line) and 5.8 GHz (black line) for the resonator quality factors of 1000 (solid line), 100 (dashed line) and 10 (dotted line). All other parameters correspond to those in <a href="#sensors-25-00747-f003" class="html-fig">Figure 3</a>.</p>
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<p>Remaining signal-to-noise ratio <math display="inline"><semantics> <mi>η</mi> </semantics></math> in the reader after the time gate at the start of the sampling. The same frequency ranges and resonator qualities were used as in <a href="#sensors-25-00747-f005" class="html-fig">Figure 5</a>. The noise bandwidths in the left graph were selected according to the regulations for the ISM [<a href="#B41-sensors-25-00747" class="html-bibr">41</a>] and SRD [<a href="#B42-sensors-25-00747" class="html-bibr">42</a>] bands, to 1.75 MHz for the 433 MHz band (red line), 2 MHz for the 868 MHz band (green line), 26 MHz for the 915 MHz band (blue line), 100 MHz for the 2.45 GHz band (magenta line) and 150 MHz for the 5.8 GHz band (black line). The resonator quality factors used were 1000 (solid line), 100 (dashed line) and 10 (dotted line). In the right graph, constant relative bandwidths of <math display="inline"><semantics> <mrow> <mn>0.5</mn> <mo>%</mo> </mrow> </semantics></math> of the carrier frequency are assumed. All other parameters correspond to those in <a href="#sensors-25-00747-f003" class="html-fig">Figure 3</a>.</p>
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<p><b>Left chart</b>: reduction in the residual <math display="inline"><semantics> <mrow> <mi>r</mi> <mfenced separators="" open="(" close=")"> <mi>α</mi> <mi>N</mi> <mi>T</mi> </mfenced> </mrow> </semantics></math> according to Equation (<a href="#FD39-sensors-25-00747" class="html-disp-formula">39</a>), <b>right chart</b>: increase in the resulting measurement resolution, both graphs as a function of <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>N</mi> <mi>T</mi> </mrow> </semantics></math>. A resonator signal with a quality factor of 3 and a sampling of 2 samples per oscillation was used.</p>
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<p>Crámer–Rao lower bound for the relative frequency error <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>σ</mi> <msub> <mi>f</mi> <mn>0</mn> </msub> </msub> <msub> <mi>f</mi> <mn>0</mn> </msub> </mfrac> </mstyle> </semantics></math> as a function of the readout distance. The same frequency ranges, resonator quality factors and noise bandwidth as in <a href="#sensors-25-00747-f006" class="html-fig">Figure 6</a> were used in the left diagram, in the right graph, constant relative bandwidths of <math display="inline"><semantics> <mrow> <mn>0.5</mn> <mo>%</mo> </mrow> </semantics></math> of the carrier frequency are assumed. All other parameters correspond to those in <a href="#sensors-25-00747-f003" class="html-fig">Figure 3</a>.</p>
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<p>Relative frequency error <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <msub> <mi>f</mi> <mn>0</mn> </msub> </msub> <mo>/</mo> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> </semantics></math> for one readout cycle as a function of the quality factor of a resonator in selected ISM or SRD bands. The read distance <span class="html-italic">R</span> is set to <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>=</mo> <mn>1</mn> <mi>λ</mi> </mrow> </semantics></math>. The diagrams start with the quality factor at which the resonance bandwidth fits into the assigned band. The resonator was stimulated and recorded for the time of <span class="html-italic">Q</span> oscillations. For quality factors where the resonator bandwidth became smaller than one fifth of the assigned bandwidth, the sampled bandwidth was limited to five times the resonator bandwidth. All other parameters correspond to those in <a href="#sensors-25-00747-f003" class="html-fig">Figure 3</a>. The approximation in Equation (<a href="#FD46-sensors-25-00747" class="html-disp-formula">46</a>) is also shown in the dotted lines.</p>
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<p><b>Left</b>: Detail enlargement of the peak in the frequency domain that was calculated via Fourier transformation of the signal of a decaying resonator over <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac bevelled="true"> <mi>Q</mi> <mi>f</mi> </mfrac> </mstyle> </semantics></math> oscillations. The initial signal-to-noise ratio is 40 dB. The blue circles represent the sample values for a calculation without zero padding. The red line was calculated with a zero padding of 10. The center frequency is marked with a small cross. The slight ripple results from the truncation of the exponential function. <b>Right</b>: Measurement signal in magnitude and phase that was demodulated with the frequency of the coarse resolved maximum in the left spectrum without zero padding. Since the actual frequency is about a quarter step size apart from the maximum, a phase error of 90° remains.</p>
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<p>Detailed enlargement of the peak in the frequency domain, which was calculated by Fourier transforming the signal of a decaying resonator for <math display="inline"><semantics> <mstyle scriptlevel="0" displaystyle="true"> <mfrac bevelled="true"> <mi>Q</mi> <mi>f</mi> </mfrac> </mstyle> </semantics></math> oscillations. The initial signal-to-noise ratio is 40 dB. The blue line connects the samples which were calculated without zero padding. The circles mark the maximum and its left and right sampling points. The dashed black line is a parabola approximation through these 3 points. The red line was calculated with a zero padding of 20 times the number of points. The actual resonance frequency and the maximum of the parabola are marked with small crosses.</p>
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<p>Convergence of above algorithm for 20 dB (<b>left</b>), 10 dB (<b>middle</b>) and 0 dB (<b>right</b>) signal-to-noise ratio according to Equation (<a href="#FD19-sensors-25-00747" class="html-disp-formula">19</a>). The resonator has a center frequency of 1 and a quality of 100. <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> of the Crámer–Rao lower bound is shown in blue dashed lines. The estimated center frequencies start on the left at the nearest sampling point of the spectrum calculated via FFT and then converge very quickly to the best estimate for this data set. Each time, 10 examples with varied noise are shown. Note the different scaling of the ordinates of the graphs.</p>
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<p>Histogram of 5000 runs of the simulations shown in <a href="#sensors-25-00747-f012" class="html-fig">Figure 12</a>, plotted against <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>6</mn> <mi>σ</mi> </mrow> </semantics></math> of the Crámer–Rao lower bound. The signal-to-noise ratios again are 20 dB (<b>left</b>), 10 dB (<b>middle</b>) and 0 dB (<b>right</b>). The normalized Gaussian functions of the corresponding Crámer–Rao lower bounds are also shown in red.</p>
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<p>Cumulative density functions in blue of the 5000 simulation runs shown in <a href="#sensors-25-00747-f013" class="html-fig">Figure 13</a> together with the cumulative density functions of the corresponding Gaussians in red. The signal-to-noise ratios again are 20 dB (<b>left</b>), 10 dB (<b>middle</b>) and 0 dB (<b>right</b>).</p>
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<p>Probability that the algorithms in Equation (<a href="#FD50-sensors-25-00747" class="html-disp-formula">50</a>) to Equation (<a href="#FD53-sensors-25-00747" class="html-disp-formula">53</a>) result within an interval of <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>6</mn> <mi>σ</mi> </mrow> </semantics></math> at the actual resonant frequency depicted as a function of the signal-to-noise ratio for a resonator with quality factors of 10 (red line), 100 (blue line) and 1000 (black line).</p>
Full article ">Figure 16
<p>Convergence trajectories for each of the 10 runs at signal-to-noise powers where the detection probabilities are only 80% for a resonator with quality factors of 10 (<b>left</b>), 100 (<b>middle</b>) and 300 (<b>right</b>). <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>1</mn> <mi>σ</mi> </mrow> </semantics></math> of the Crámer–Rao lower bound is again shown in blue dashed lines.</p>
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<p><b>Left</b>: Reduction in the convergence behavior of the algorithm as a function of the sample length for sample lengths between <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>·</mo> <mi>Q</mi> <mo>/</mo> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>20</mn> <mo>·</mo> <mi>Q</mi> <mo>/</mo> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </semantics></math>. The quality factor of the resonator signal is 100 and the initial signal-to-noise ratio is 0 dB. The probability of convergence of the algorithm decreases as the sample length increases from 100% to 20%. The <b>right</b> graph shows for the sample length of <math display="inline"><semantics> <mrow> <mn>20</mn> <mo>·</mo> <mi>Q</mi> <mo>/</mo> <msub> <mi>f</mi> <mi>d</mi> </msub> </mrow> </semantics></math> the few remaining correct detections of the natural frequency for 5000 runs.</p>
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<p>The left diagram shows the weighting coefficient <math display="inline"><semantics> <mi>γ</mi> </semantics></math> according to Equation (<a href="#FD56-sensors-25-00747" class="html-disp-formula">56</a>) as a function of <math display="inline"><semantics> <mrow> <mi>α</mi> <mi>N</mi> <mi>T</mi> </mrow> </semantics></math>. The improvement in the convergence probability of the algorithm by the weighting function is shown in the right diagram for a resonator signal with a quality factor of 100 and an initial signal-to-noise ratio of 0 dB. The red line was calculated without weighting and the blue line with weighting. The detection probability shifts by approximately 3 dB to lower signal-to-noise ratios.</p>
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<p>Cumulative density functions from 5000 simulation runs using the weighting function Equation (<a href="#FD56-sensors-25-00747" class="html-disp-formula">56</a>) together with the cumulative density functions of the corresponding Gaussian functions. All other parameters correspond to those in <a href="#sensors-25-00747-f014" class="html-fig">Figure 14</a>. The curves are so close together that they can hardly be distinguished. The signal-to-noise ratios are again 20 dB (<b>left</b>), 10 dB (<b>middle</b>) and 0 dB (<b>right</b>).</p>
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<p>The left graph shows the remaining phase angles of the individual sampling points <math display="inline"><semantics> <msub> <mi>z</mi> <mi>n</mi> </msub> </semantics></math>, where the natural frequency was eliminated by multiplying with <math display="inline"><semantics> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>j</mi> <mover accent="true"> <mi>ω</mi> <mo stretchy="false">˜</mo> </mover> <mi>n</mi> <mi>T</mi> </mrow> </msup> </semantics></math> and where the impact of noise was reduced by weighting with <math display="inline"><semantics> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>α</mi> <mi>n</mi> <mi>T</mi> </mrow> </msup> </semantics></math>. In addition, the phase angle <math display="inline"><semantics> <mover accent="true"> <mi>φ</mi> <mo stretchy="false">˜</mo> </mover> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> is plotted according to Equation (<a href="#FD31-sensors-25-00747" class="html-disp-formula">31</a>). For the center graph, the amplitudes of the sample points weighted with <math display="inline"><semantics> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mi>α</mi> <mi>n</mi> <mi>T</mi> </mrow> </msup> </semantics></math> were plotted together with the amplitude <math display="inline"><semantics> <mover accent="true"> <mi>b</mi> <mo stretchy="false">˜</mo> </mover> </semantics></math> at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> calculated according to Equation (<a href="#FD32-sensors-25-00747" class="html-disp-formula">32</a>). The noisy signal from the decaying resonator with a signal-to-noise ratio of 20 dB is shown in red on the right along with the echoes in the transmission channel. The blue curve shows the remaining signal when the estimated values <math display="inline"><semantics> <msub> <mi>μ</mi> <mi>n</mi> </msub> </semantics></math> have been subtracted from the measured values <math display="inline"><semantics> <msub> <mi>z</mi> <mi>n</mi> </msub> </semantics></math>.</p>
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<p>Sum of the complex values of 50 measurement signals where the estimated values of the decaying resonator have been subtracted; on the left in linear scale and on the right in logarithmic scale. The systematic interference signals disappear into the noise at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mn>15</mn> </mrow> </semantics></math>, and the signal evaluation could be started earlier by this interval, resulting in a higher resolution.</p>
Full article ">Figure A1
<p>Left chart: Normalized <math display="inline"><semantics> <msub> <mi>σ</mi> <mrow> <msub> <mi>ω</mi> <mi>d</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>A</mi> </msub> </mrow> </msub> </semantics></math> as a function of the sampling duration <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>d</mi> <mi>a</mi> <mi>q</mi> </mrow> </msub> </semantics></math> according to Equation (A50). The value of <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>/</mo> <msub> <mi>t</mi> <mrow> <mi>t</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math> was set to 1, the natural frequency of the resonator is 1 Hz and its quality factor is 100. The quality factor of the communication channel was set to 10, i.e., <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>a</mi> <mi>t</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>t</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>a</mi> <mi>d</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow> <mi>d</mi> <mi>a</mi> <mi>q</mi> </mrow> </msub> </mrow> </semantics></math>. The right graph shows the shift in the optimal timing for the sampling period and the increase in the corresponding <math display="inline"><semantics> <msub> <mi>σ</mi> <mrow> <msub> <mi>ω</mi> <mi>d</mi> </msub> <mo>,</mo> <msub> <mi>N</mi> <mi>A</mi> </msub> </mrow> </msub> </semantics></math> as a function of the length of the gate <math display="inline"><semantics> <msub> <mi>t</mi> <mrow> <mi>g</mi> <mi>a</mi> <mi>t</mi> <mi>e</mi> </mrow> </msub> </semantics></math>.</p>
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13 pages, 421 KiB  
Article
Hyperbolic Diffusion Functionals on a Ring with Finite Velocity
by Marco Nizama
Entropy 2025, 27(2), 105; https://doi.org/10.3390/e27020105 - 22 Jan 2025
Viewed by 491
Abstract
I study a lattice with periodic boundary conditions using a non-local master equation that evolves over time. I investigate different system regimes using classical theories like Fisher information, Shannon entropy, complexity, and the Cramér–Rao bound. To simulate spatial continuity, I employ a large [...] Read more.
I study a lattice with periodic boundary conditions using a non-local master equation that evolves over time. I investigate different system regimes using classical theories like Fisher information, Shannon entropy, complexity, and the Cramér–Rao bound. To simulate spatial continuity, I employ a large number of sites in the ring and compare the results with continuous spatial systems like the Telegrapher’s equations. The Fisher information revealed a power-law decay of tν, with ν=2 for short times and ν=1 for long times, across all jump models. Similar power-law trends were also observed for complexity and the Fisher information related to Shannon entropy over time. Furthermore, I analyze toy models with only two ring sites to understand the behavior of the Fisher information and Shannon entropy. As expected, a ring with a small number of sites quickly converges to a uniform distribution for long times. I also examine the Shannon entropy for short and long times. Full article
(This article belongs to the Special Issue Theory and Applications of Hyperbolic Diffusion and Shannon Entropy)
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Figure 1
<p>The Fisher information over time in arbitrary units is depicted for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>401</mn> </mrow> </semantics></math> sites in (<a href="#FD9-entropy-27-00105" class="html-disp-formula">9</a>) of the ring. The solid lines represent the NN, GJ, and PJ jump models (black, red, and blue, respectively). In all cases, the dashed lines show a power-law fit of <math display="inline"><semantics> <msup> <mi>t</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≪</mo> <mi>τ</mi> </mrow> </semantics></math> (ballistic regime), while for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≫</mo> <mi>τ</mi> </mrow> </semantics></math>, the Fisher measure fits as <math display="inline"><semantics> <mrow> <msup> <mi>D</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>t</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </semantics></math> in dotted and dashed lines (diffusive regime). The parameters used are <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math> (GJ model), and <math display="inline"><semantics> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0.4</mn> </mrow> </semantics></math> (PJ model). The plots display the fitted functions, such as <math display="inline"><semantics> <mrow> <mi>β</mi> <msup> <mi>t</mi> <mrow> <mo>−</mo> <mi>ν</mi> </mrow> </msup> </mrow> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> in the ballistic regime and <math display="inline"><semantics> <mrow> <mi>ν</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> in the diffusive regime, along with the jump models that were studied.</p>
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<p>Complexity <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> as a function of time for three different jump models: NN, GJ, and PJ (represented by solid black, red, and blue lines, respectively). The dashed lines depict the power-law fit <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>fitted</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>a</mi> <msup> <mi>t</mi> <mrow> <mo>−</mo> <mi>b</mi> </mrow> </msup> </mrow> </semantics></math> for all models, the manuscript providing the values of <span class="html-italic">a</span> and <span class="html-italic">b</span>. The parameters used are the same as in <a href="#entropy-27-00105-f001" class="html-fig">Figure 1</a>. For small <span class="html-italic">t</span> compared to <math display="inline"><semantics> <mi>τ</mi> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> scales as <math display="inline"><semantics> <msup> <mi>t</mi> <mrow> <mo>−</mo> <mi>b</mi> </mrow> </msup> </semantics></math> with <span class="html-italic">b</span> close to 2, indicating a ballistic regime. As <span class="html-italic">t</span> becomes larger than <math display="inline"><semantics> <mi>τ</mi> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>C</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> approaches a value of 1, signifying the diffusive regime.</p>
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<p>Fisher information is shown plotted against Shannon entropy for the NN model (black line), GJ model (red line), and PJ model (blue line). The dashed lines represent a power-law fit <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <mi>a</mi> <msup> <mi>S</mi> <mrow> <mo>−</mo> <mn>1.12</mn> </mrow> </msup> </mrow> </semantics></math> for small values of <span class="html-italic">S</span>, where <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>9.25</mn> </mrow> </semantics></math> (NN model), <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>18.67</mn> </mrow> </semantics></math> (GJ model), and <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>12.73</mn> </mrow> </semantics></math> (PJ model). For large values of <span class="html-italic">S</span>, the Fisher information for all models follows a diffusive regime given by <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>=</mo> <mrow> <mo>(</mo> <mn>2</mn> <mi>π</mi> <mi>e</mi> <mo>)</mo> </mrow> <msup> <mi>e</mi> <mrow> <mo>−</mo> <mn>2</mn> <mi>S</mi> </mrow> </msup> </mrow> </semantics></math>, shown in green lines. The parameters used are consistent with those in <a href="#entropy-27-00105-f001" class="html-fig">Figure 1</a>.</p>
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<p>The Cramér–Rao bound (CR) is plotted as a function of <span class="html-italic">t</span> for the NN model (black line), GJ model (red line), and PJ model (blue line). The green dashed line indicates the CR value in the diffusive regime. The parameters are the same as those in <a href="#entropy-27-00105-f001" class="html-fig">Figure 1</a>. CR exceeds one for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≪</mo> <mi>τ</mi> </mrow> </semantics></math> (ballistic regime) and approaches one for <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>≫</mo> <mi>τ</mi> </mrow> </semantics></math> (diffusive regime).</p>
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<p>The Shannon entropy <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is plotted as a function of time in arbitrary units. The solid lines represent the NN (black), GJ (red), and PJ (blue) jump models. For small time values (ballistic regime), all cases show a power-law fit of <math display="inline"><semantics> <msup> <mi>t</mi> <mrow> <mo>−</mo> <mn>1.67</mn> </mrow> </msup> </semantics></math> represented by dashed lines. In contrast, for long time values (diffusive regime), the entropy fits as <math display="inline"><semantics> <mrow> <mo form="prefix">ln</mo> <msqrt> <mrow> <mo>[</mo> <mn>2</mn> <mi>π</mi> <mi>e</mi> <mi>D</mi> <mi>t</mi> <mo>]</mo> </mrow> </msqrt> </mrow> </semantics></math> shown by dotted and dashed lines. The parameters used are consistent with those in <a href="#entropy-27-00105-f001" class="html-fig">Figure 1</a>.</p>
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17 pages, 3584 KiB  
Article
Accurate Joint Estimation of Position and Orientation Based on Angle of Arrival and Two-Way Ranging of Ultra-Wideband Technology
by Di Zhang, Hongbiao Xu, Li Zhan, Ye Li, Guangqiang Yin and Xinzhong Wang
Electronics 2025, 14(3), 429; https://doi.org/10.3390/electronics14030429 - 22 Jan 2025
Viewed by 456
Abstract
In wireless sensor networks (WSNs), ultra-wideband (UWB) technology is essential for robot localization systems, especially for methods of the simultaneous estimation of position and orientation. However, current approaches frequently depend on rigid body models, which require multiple base stations and lead to substantial [...] Read more.
In wireless sensor networks (WSNs), ultra-wideband (UWB) technology is essential for robot localization systems, especially for methods of the simultaneous estimation of position and orientation. However, current approaches frequently depend on rigid body models, which require multiple base stations and lead to substantial equipment costs. This paper presents a cost-effective UWB SL model utilizing the angle of arrival (AOA) and double-sided two-way ranging (DS-TWR). To improve localization accuracy, we propose a self-localization algorithm based on constrained weighted least squares (SL-CWLS), integrating a weighted matrix derived from a measured noise model. Additionally, we derive the constrained Cramér–Rao lower bound (CCRLB) to analyze the performance of the proposed algorithm. Simulation results indicate that the proposed method achieves high estimation accuracy, while real-world experiments validate the simulation results. Full article
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<p>Main scenarios of robot localization. (<b>a</b>) SL. (<b>b</b>) RBL.</p>
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<p>Message-passing and timing measurement of UWB for AOA-TWR localization model.</p>
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<p>RMSE v.s. noise standard deviation <math display="inline"><semantics> <mi>σ</mi> </semantics></math> for algorithms. (<b>a</b>) Location <math display="inline"><semantics> <mi mathvariant="bold">s</mi> </semantics></math>. (<b>b</b>) Rotation <math display="inline"><semantics> <mi mathvariant="bold">R</mi> </semantics></math>.</p>
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<p>RMSE v.s. number of tags, <span class="html-italic">M</span>, for algorithms. (<b>a</b>) Location <math display="inline"><semantics> <mi mathvariant="bold">s</mi> </semantics></math>. (<b>b</b>) Rotation <math display="inline"><semantics> <mi mathvariant="bold">R</mi> </semantics></math>.</p>
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<p>Dynamic trajectories of algorithms. (<b>a</b>) Trajectories. (<b>b</b>) Enlarged view of (<b>a</b>).</p>
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<p>Dynamic orientations of algorithms. (<b>a</b>) Orientations. (<b>b</b>) Enlarged view of (<b>a</b>).</p>
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<p>CDFs of algorithms. (<b>a</b>) Trajectories. (<b>b</b>) Orientations.</p>
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<p>UWB devices for AOA-TWR-based SL. (<b>a</b>) Three-antenna UWB base station, (<b>b</b>) UWB label.</p>
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<p>AOA-TWR-based SL scenario based on UWB. (<b>a</b>) Real scenario, (<b>b</b>) Test positions of base station. (<b>c</b>) Measurement dial for orientation angles of base station in (<b>a</b>).</p>
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<p>RMSEs for different AOAs in simulation for <a href="#electronics-14-00429-f009" class="html-fig">Figure 9</a>c. (<b>a</b>) Trajectories. (<b>b</b>) Orientations.</p>
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15 pages, 3552 KiB  
Article
Fast Hadamard-Encoded 7T Spectroscopic Imaging of Human Brain
by Chan Hong Moon, Frank S. Lieberman, Hoby P. Hetherington and Jullie W. Pan
Tomography 2025, 11(1), 7; https://doi.org/10.3390/tomography11010007 - 13 Jan 2025
Viewed by 787
Abstract
Background/Objectives: The increased SNR available at 7T combined with fast readout trajectories enables accelerated spectroscopic imaging acquisitions for clinical applications. In this report, we evaluate the performance of a Hadamard slice encoding strategy with a 2D rosette trajectory for multi-slice fast spectroscopic [...] Read more.
Background/Objectives: The increased SNR available at 7T combined with fast readout trajectories enables accelerated spectroscopic imaging acquisitions for clinical applications. In this report, we evaluate the performance of a Hadamard slice encoding strategy with a 2D rosette trajectory for multi-slice fast spectroscopic imaging at 7T. Methods: Moderate-TE (~40 ms) spin echo and J-refocused polarization transfer sequences were acquired with simultaneous Hadamard multi-slice excitations and rosette in-plane encoding. The moderate spin echo sequence, which targets singlet compounds (i.e., N-acetyl aspartate, creatine, and choline), uses cascaded multi-slice RF excitation pulses to minimize the chemical shift dispersion error. The J-refocused sequence targets coupled spin systems (i.e., glutamate and myo-inositol) using simultaneous multi-slice excitation to maintain the same TE across all slices. A modified Hadamard slice encoding strategy was used to decrease the peak RF pulse amplitude of the simultaneous multi-slice excitation pulse for the J-refocused acquisition. Results: The accuracy of multi-slice and single-slice rosette spectroscopic imaging (RSI) is comparable to conventional Cartesian-encoded spectroscopic imaging (CSI). Spectral analyses for the J-refocused studies of glutamate and myo-inositol show that the Cramer Rao lower bounds are not significantly different between the fast RSI and conventional CSI studies. Linear regressions of creatine/N-acetyl aspartate and glutamate/N-acetyl aspartate with tissue gray matter content are consistent with literature values. Conclusions: With minimal gradient demands and fast acquisition times, the 2.2 min to 9 min for single- to four-slice RSI acquisitions are well tolerated by healthy subjects and tumor patients, and show results that are consistent with clinical outcomes. Full article
(This article belongs to the Section Neuroimaging)
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<p>Pulse sequences for the (<b>A</b>) Hadamard slice encoding acquisition using a moderate-TE single-spin-echo with cascaded excitation pulses and the (<b>B</b>) J-refocused acquisition with simultaneous multi-slice excitation pulses. Four slices are shown in the cascaded sequence. For both sequences, two RF distributions are used, with the homogeneous distribution applied in the water suppression and spin echo components (RF #1) and the ring distribution (RF #2) applied for outer volume suppression of extracerebral signal. (<b>C</b>) The phases for the Hadamard-encoded excitation slice profiles of the sequence are shown in black for each scan (S<sub>1</sub> through S<sub>4</sub>). The reconstruction scheme shows the summation of scans S<sub>1</sub> to S<sub>4</sub> needed to generate each of the slices (S<sub>A</sub> through S<sub>D</sub>, red–blue–green–black). The pulses in the sequences are not shown to scale in magnitude or time.</p>
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<p>Comparison of data from matching CSI and four-slice Hadamard moderate-echo single-spin-echo RSI studies showing scout T1 anatomy (<b>A</b>), magnitude NAA images (<b>B</b>), spectra (<b>C</b>), and a voxel-to-voxel plot of NAA and Cr amplitudes (<b>D</b>). Four representative ROIs numbered 1 to 4 (yellow numbered circles in (<b>A</b>) match the spectra in (<b>C</b>). The regression analysis in (<b>D</b>) has R<sup>2</sup> = 0.92 (<span class="html-italic">p</span> &lt; 0.001). (<b>E</b>) Regression data for Cr/NAA vs. fraction GM (fGM) acquired from a control subject.</p>
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<p>J-refocused single-slice spectroscopic imaging. (<b>A</b>) Scout T1, (<b>B</b>) magnitude NAA image, (<b>C</b>) comparison of matched spectra acquired with the RSI vs. the CSI from sampling points (yellow numbering, 1, 2, 3, 4, 5 in (<b>A</b>)), the corresponding numbered spectra in (<b>C</b>), and (<b>D</b>) regression of Glu/NAA with tissue fraction of GM (fGM) (every other pixel sampled), which has an R = 0.64, with a dependence of 0.57 + 0.61 × fGM (<span class="html-italic">p</span> &lt; 0.001).</p>
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<p>Multi-slice Hadamard J-refocused RSI. (<b>A</b>) Scout T1, (<b>B</b>) magnitude NAA images, and (<b>C</b>) sample spectra as indicated from four-slice J-refocused RSI using simultaneous Hadamard slice encoding; 7 mm slice thickness, 2 mm slice gap, ~9 min total duration, and TE 38 ms. Spectra at yellow numbered points (1, 2, 3, 4) in (<b>A</b>) for slice 1–4 are shown in (<b>C</b>); the red-boxed spectra shown in (<b>C</b>) are from the matched single-slice RSI for slice 3 for a 2.2 min duration.</p>
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<p>Moderate-TE spin echo cascaded Hadamard RSI acquisitions from two oligodendroglioma tumor patients, patient #1 (for slice 1 (<b>A1</b>), 2 (<b>A2</b>), and 3 (<b>A3</b>) in the top and middle panels); patient #2 ((<b>B</b>) in the bottom panel). In the (<b>A1</b>–<b>A3</b>) top panel, scout T1 anatomy is shown, while in the (<b>B</b>) two left panels, T2 FLAIR at 3T and T1 anatomy at 7T are shown. Spectra were from ROIs (yellow and green rectangle in (<b>A1</b>–<b>A3</b>) and (<b>B</b>), respectively). Both patients were clinically identified to have experienced tumor progression, consistent with spectroscopic imaging, although the 69 yo patient #2 shows a more aggressive worsening, with a Ch/NAA ratio of greater than 2 seen at the edge of the brain and lesion.</p>
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<p>Spectral analyses of Inositol from 2.2 min and 6.6 min J-refocused RSI acquisitions at 9 mm isotropic resolution. (<b>A</b>,<b>B</b>) Comparison of 2.2 min vs. 6.6 min results for the SNR and CRLB. The red data point with error bars shows the mean and standard deviation for the plotted data; the black line shows the identity diagonal. Combining both the 2.2 min (filled circles) and 6.5 min (open circles) data, the relationship between 1/SNR and CRLB (<b>C</b>), linewidth (Hz), and the CRLB are shown (<b>D</b>). The plot in (<b>D</b>) appears to have fewer data points than in (<b>C</b>) due to the forced binned outputs of the LCM for both the CRLB and linewidth values (thus, each datapoint is representative of multiple points). The linear regressions calculated from all data points are significant for both 1/SNR and LW, with R<sup>2</sup> = 0.67 and R<sup>2</sup> = 0.38, <span class="html-italic">p</span> &lt; 0.001 for both.</p>
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16 pages, 2617 KiB  
Article
Integrated Spectral Sensitivity as Physics-Based Figure of Merit for Spectral Transducers in Optical Sensing
by Felix L. McCluskey, Anne van Klinken and Andrea Fiore
Sensors 2025, 25(2), 440; https://doi.org/10.3390/s25020440 - 13 Jan 2025
Viewed by 620
Abstract
The design of optical sensors aims at providing, among other things, the highest precision in the determination of the target measurand. Many sensor systems rely on a spectral transducer to map changes in the measurand into spectral shifts of a resonance peak in [...] Read more.
The design of optical sensors aims at providing, among other things, the highest precision in the determination of the target measurand. Many sensor systems rely on a spectral transducer to map changes in the measurand into spectral shifts of a resonance peak in the reflection or transmission spectrum, which is measured by a readout device (e.g., a spectrometer). For these spectral transducers, figures of merit have been defined which are based on specific assumptions on the readout and the data analysis. In reality, however, different transducers achieve optimal performance with different types of readout. Additionally, some transducers present a more complex spectral response for which existing figures of merit do not apply. In this paper, we investigate an approach to quantifying the potential performance of a given transducer for a more general class of readout methods. Starting from the Cramér–Rao lower bound, we define a new figure of merit, the integrated spectral sensitivity, which is directly related to the physical limit of precision and applicable to a wide variety of sensing systems. We apply this analysis to two different examples of transducers. The results bring useful insights into the design of optical sensors. Full article
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<p>(<b>a</b>) Schematic representation of the sensing system. Reflectance spectrum (blue) of (<b>b</b>) a photonic crystal (insets with biolayer in red) and (<b>c</b>) a Si/SiO<sub>2</sub> multilayer (inset with biolayer in red) and its derivative with respect to the measurand (red).</p>
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<p>(<b>a</b>,<b>b</b>) Reflectance spectrum of PhC transducer (blue line) with the transmission functions for the readout channels of a spectrometric readout (green lines) for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>. (<b>c</b>) Calculated integrated spectral sensitivity for PhC sensor with spectrometric readout of various linewidths.</p>
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<p>(<b>a</b>,<b>b</b>) Reflectance spectrum of multilayer stack (blue line) with the transmission functions for the readout channels of a spectrometric readout (green lines) for (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> and (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>50</mn> </mrow> </semantics></math>. (<b>c</b>) Calculated integrated spectral sensitivity for multilayer stack with spectrometric readout of various linewidths.</p>
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<p>Reflectance spectrum of (<b>a</b>) the PhC transducer and (<b>b</b>) a multilayer transducer, together with the two respective transmission functions for the ideal readout.</p>
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<p>Reflectance spectrum of (<b>a</b>) PhC transducer with different normalized radii <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>/</mo> <mi>a</mi> </mrow> </semantics></math> and (<b>b</b>) multilayer structures with different thicknesses. In (<b>c</b>,<b>d</b>), the corresponding integrated spectral sensitivity for an ideal readout (red crosses, left axis) and a high-resolution spectrometric readout (blue circles, left axis) are shown in comparison with the traditional figure of merit <math display="inline"><semantics> <mrow> <mi>F</mi> <mi>o</mi> <msub> <mrow> <mi>M</mi> </mrow> <mrow> <mo>ν</mo> </mrow> </msub> </mrow> </semantics></math> (black squares, right axis).</p>
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22 pages, 865 KiB  
Article
Secrecy-Constrained UAV-Mounted RIS-Assisted ISAC Networks: Position Optimization and Power Beamforming
by Weichao Yang, Yajing Wang, Dawei Wang, Yixin He and Li Li
Drones 2025, 9(1), 51; https://doi.org/10.3390/drones9010051 - 13 Jan 2025
Viewed by 782
Abstract
This paper investigates secrecy solutions for integrated sensing and communication (ISAC) systems, leveraging the combination of a reflecting intelligent surface (RIS) and an unmanned aerial vehicle (UAV) to introduce new degrees of freedom for enhanced system performance. Specifically, we propose a secure ISAC [...] Read more.
This paper investigates secrecy solutions for integrated sensing and communication (ISAC) systems, leveraging the combination of a reflecting intelligent surface (RIS) and an unmanned aerial vehicle (UAV) to introduce new degrees of freedom for enhanced system performance. Specifically, we propose a secure ISAC system supported by a UAV-mounted RIS, where an ISAC base station (BS) facilitates secure multi-user communication while simultaneously detecting potentially malicious radar targets. Our goal is to improve parameter estimation performance, measured by the Cramér–Rao bound (CRB), by jointly optimizing the UAV position, transmit beamforming, and RIS beamforming, subject to constraints including the UAV flight area, communication users’ quality of service (QoS) requirements, secure transmission demands, power budget, and RIS reflecting coefficient limits. To address this non-convex, multivariate, and coupled problem, we decompose it into three subproblems, which are solved iteratively using particle swarm optimization (PSO), semi-definite relaxation (SDR), majorization–minimization (MM), and alternating direction method of multipliers (ADMM) algorithms. Our numerical results validate the effectiveness of the proposed scheme and demonstrate the potential of employing UAV-mounted RIS in ISAC systems to enhance radar sensing capabilities. Full article
(This article belongs to the Special Issue Physical-Layer Security in Drone Communications)
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<p>A secure ISAC system supported by a UAV-mounted RIS.</p>
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<p>CRB versus the number of RIS reflecting the <span class="html-italic">M</span> element.</p>
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<p>CRB versus the SINR requirement <math display="inline"><semantics> <mi>γ</mi> </semantics></math>.</p>
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<p>CRB versus the transmit power <math display="inline"><semantics> <msub> <mi>P</mi> <mi>BS</mi> </msub> </semantics></math>.</p>
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<p>CRB versus the number of antennas <span class="html-italic">N</span>.</p>
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22 pages, 5134 KiB  
Article
Reinforcement Learning-Based Resource Allocation Scheme of NR-V2X Sidelink for Joint Communication and Sensing
by Zihan Li, Ping Wang, Yamin Shen and Song Li
Sensors 2025, 25(2), 302; https://doi.org/10.3390/s25020302 - 7 Jan 2025
Viewed by 539
Abstract
Joint communication and sensing (JCS) is becoming an important trend in 6G, owing to its efficient utilization of spectrums and hardware resources. Utilizing echoes of the same signal can achieve the object location sensing function, in addition to the V2X communication function. There [...] Read more.
Joint communication and sensing (JCS) is becoming an important trend in 6G, owing to its efficient utilization of spectrums and hardware resources. Utilizing echoes of the same signal can achieve the object location sensing function, in addition to the V2X communication function. There is application potential for JCS systems in the fields of ADAS and unmanned autos. Currently, the NR-V2X sidelink has been standardized by 3GPP to support low-latency high-reliability direct communication. In order to combine the benefits of both direct communication and JCS, it is promising to extend existing NR-V2X sidelink communication toward sidelink JCS. However, conflicting performance requirements arise between radar sensing accuracy and communication reliability with the limited sidelink spectrum. In order to overcome the challenges in the distributed resource allocation of sidelink JCS with a full-duplex, this paper has proposed a novel consecutive-collision mitigation semi-persistent scheduling (CCM-SPS) scheme, including the collision detection and Q-learning training stages to suppress collision probabilities. Theoretical performance analyses on Cramér–Rao Lower Bounds (CRLBs) have been made for the sensing of sidelink JCS. Key performance metrics such as CRLB, PRR and UD have been evaluated. Simulation results show the superior performance of CCM-SPS compared to similar solutions, with promising application prospects. Full article
(This article belongs to the Special Issue Communication, Sensing and Localization in 6G Systems)
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<p>NG-RAN architecture supporting the PC5 interface.</p>
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<p>Process flow of sensing-based semi-persistent scheduling (SB-SPS).</p>
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<p>Markov chain for state transition of SPS.</p>
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<p>Reinforcement learning framework.</p>
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<p>CCM-SPS accelerate reselection.</p>
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<p>Empirical CDF of the root CRLB for a range using SB-SPS, FD-enhanced and CCM-SPS.</p>
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<p>Bar graph of root CRLB (at CCDF = 95-percentile) for a range using different schemes with varying vehicle densities.</p>
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<p>PRR over distance using SB-SPS, FD-enhanced and CCM-SPS.</p>
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<p>The maximum distance allowing PRR larger than 0.95 is evaluated using conventional SB-SPS, FD-enhanced methods and CCM-SPS.</p>
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<p>Empirical CDF of root CRLB for range estimation.</p>
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<p>PRR vs. distance performance of SB-SPS with different pack sizes in case of density = 50, 150, 250 veh/km.</p>
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<p>CCM-SPS’s range sensing performance evaluation on empirical CDF of root CRLB with different pack sizes in the case of density = 50, 150, 250 veh/km.</p>
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<p>PRR vs. distance performance of CCM-SPS with different pack sizes in case of density = 50, 150, 250 veh/km.</p>
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<p>CCM-SPS’s communication performance evaluation on empirical CDF of update delay with different pack sizes in the case of density = 50, 150, 250 veh/km.</p>
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20 pages, 5738 KiB  
Article
Time-of-Arrival and Angle-of-Arrival Measurement-Assisted 3D Inter-Unmanned Aerial Vehicle Relative Localization Under Distance-Dependent Noise Model
by Jiawei Tang, Tian Chang, Qinglong Jiang, Xuhui Ding and Dekang Liu
Electronics 2025, 14(1), 90; https://doi.org/10.3390/electronics14010090 - 28 Dec 2024
Viewed by 522
Abstract
This paper addresses the 3D relative localization problem for two unmanned aerial vehicles (UAVs) using a combination of time-of-arrival (TOA) and angle-of-arrival (AOA) measurements across varied flight trajectories. We commenced by examining the problem of relative attitude estimation using only time-of-arrival (TOA) measurements, [...] Read more.
This paper addresses the 3D relative localization problem for two unmanned aerial vehicles (UAVs) using a combination of time-of-arrival (TOA) and angle-of-arrival (AOA) measurements across varied flight trajectories. We commenced by examining the problem of relative attitude estimation using only time-of-arrival (TOA) measurements, taking into account a distance-dependent noise model. To address this issue, we constructed a constrained weighted least squares (CWLS) problem and applied semidefinite relaxation (SDR) techniques for its resolution. Furthermore, we extended our analysis to incorporate AOA measurements and scrutinize the Cramer–Rao Lower Bound (CRLB) to illustrate enhanced localization accuracy through TOA-AOA integration compared to TOA alone under stable trajectory conditions. Ultimately, numerical simulations substantiate the efficacy of the proposed methodologies. Full article
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<p>Scenario of UAV relative localization estimation.</p>
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<p>Four trajectories of two UAVs.</p>
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<p>Acomparison of the influence of noise level on the SDP under different flight trajectories.</p>
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<p>Acomparison of the influence of noise level on the SDP under different flight trajectories.</p>
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<p>A comparison of the influence of noise levels on the SDR TOA and TOA-AOA methods under four different flight trajectories.</p>
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<p>A comparison of the influence of number of measurements on the SDR TOA and TOA-AOA methods under four different flight trajectories.</p>
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14 pages, 337 KiB  
Article
Limiting Performance of Radar-Based Positioning Solutions for the Automotive Scenario
by Francesco Bandiera and Giuseppe Ricci
Sensors 2024, 24(24), 7940; https://doi.org/10.3390/s24247940 - 12 Dec 2024
Viewed by 472
Abstract
Road safety applications for automotive scenarios rely on the ability to estimate vehicle positions with high precision. Global navigation satellite systems (GNSS) and, in particular, the global positioning system (GPS), are commonly used for self localization. But, especially in urban vehicular scenarios, due [...] Read more.
Road safety applications for automotive scenarios rely on the ability to estimate vehicle positions with high precision. Global navigation satellite systems (GNSS) and, in particular, the global positioning system (GPS), are commonly used for self localization. But, especially in urban vehicular scenarios, due to obstructions, they may not provide the requirements for crucial position-based applications. In this paper, we investigate the potential of GPS-free positioning schemes and, in particular, we compute the ultimate performance, i.e., Cramér–Rao lower bounds (CRLB), of localization schemes in which each vehicle estimates its position exploiting range and/or angle measurements of an assigned set of landmarks with a known position. Full article
(This article belongs to the Section Radar Sensors)
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<p>Geometric model of the system with only one landmark. Vehicle is in point <span class="html-italic">P</span> and the angle <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>i</mi> </msub> </semantics></math> is measured on the <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>−</mo> <mi>y</mi> </mrow> </semantics></math> plane.</p>
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<p>RMS estimation error curves for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error curves for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>9</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error curves for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error curves for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>+</mo> <mn>4.5</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error curves for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error curves for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>9</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error vs. <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>r</mi> </msub> </semantics></math> (in m), for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>45</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>θ</mi> </msub> <mo>=</mo> <mn>2</mn> <mo>°</mo> </mrow> </semantics></math>. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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<p>RMS estimation error vs. <math display="inline"><semantics> <msub> <mi>σ</mi> <mi>θ</mi> </msub> </semantics></math> (in degrees), for <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math> m, <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mo>−</mo> <mn>45</mn> <mo>)</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>r</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> m. (1,1)th: red, (2,2)th: black; solid: range and azimuth measurements, dashed: range measurements only, dash-dotted: azimuth measurements only.</p>
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28 pages, 22621 KiB  
Article
A Ray-Tracing-Based Single-Site Localization Method for Non-Line-of-Sight Environments
by Shuo Hu, Lixin Guo and Zhongyu Liu
Sensors 2024, 24(24), 7925; https://doi.org/10.3390/s24247925 - 11 Dec 2024
Viewed by 685
Abstract
Localization accuracy in non-line-of-sight (NLOS) scenarios is often hindered by the complex nature of multipath propagation. Traditional approaches typically focus on NLOS node identification and error mitigation techniques. However, the intricacies of NLOS localization are intrinsically tied to propagation challenges. In this paper, [...] Read more.
Localization accuracy in non-line-of-sight (NLOS) scenarios is often hindered by the complex nature of multipath propagation. Traditional approaches typically focus on NLOS node identification and error mitigation techniques. However, the intricacies of NLOS localization are intrinsically tied to propagation challenges. In this paper, we propose a novel single-site localization method tailored for complex multipath NLOS environments, leveraging only angle-of-arrival (AOA) estimates in conjunction with a ray-tracing (RT) algorithm. The method transforms NLOS paths into equivalent line-of-sight (LOS) paths through the generation of generalized sources (GSs) via ray tracing. A novel weighting mechanism for GSs is introduced, which, when combined with an iteratively reweighted least squares (IRLS) estimator, significantly improves the localization accuracy of non-cooperative target sources. Furthermore, a multipath similarity displacement matrix (MSDM) is incorporated to enhance accuracy in regions with pronounced multipath fluctuations. Simulation results validate the efficacy of the proposed algorithm, achieving localization performance that approaches the Cramér–Rao lower bound (CRLB), even in challenging NLOS scenarios. Full article
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<p>A flowchart of the proposed RT algorithm.</p>
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<p>Binary tree structure of ray nodes.</p>
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<p>Schematic diagram of ray-splitting structure. Red nodes indicate split nodes that will be deleted, while blue nodes represent newly generated split nodes.</p>
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<p>Schematic diagram of ray tube determination and reception. Red lines represent virtual ray tubes, while blue lines indicate the edge rays of the ray tube.</p>
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<p>An overview of the overall technical roadmap of the RT-LBS algorithm.</p>
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<p>Power measurement system architecture and key equipment. The <b>upper half</b> of the figure is the block diagram of the channel sounder used in this paper. The <b>lower half</b> is the key equipment of the sounder, including the signal generator, power amplifier, spectrum analyzer, power supplier, RTK, and antennas.</p>
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<p>Localization test system architecture and key equipment. The <b>upper half</b> of the figure is the block diagram of the localization test system used in this paper. The <b>lower half</b> is the key equipment in the signal transmitter system, UCA direction-finding equipment, the Rx antenna array, and the RF processing circuit.</p>
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<p>Measurement scenario. (<b>a</b>) The raw point cloud image of the measurement scenario. (<b>b</b>) The geometric building model extracted from the point cloud.</p>
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<p>Measurement path and power distribution at (<b>a</b>) 3 GHz frequency, (<b>b</b>) 3.6 GHz frequency, (<b>c</b>) 4 GHz frequency, (<b>d</b>) 5 GHz frequency, and (<b>e</b>) 5.9 GHz frequency.</p>
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<p>Raw power measurement data and power measurement data after applying the sliding filter at (<b>a</b>) 3 GHz frequency, (<b>b</b>) 3.6 GHz frequency, (<b>c</b>) 4 GHz frequency, (<b>d</b>) 5 GHz frequency, and (<b>e</b>) 5.9 GHz frequency.</p>
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<p>RSS predictions and measurements in the scenario at (<b>a</b>) 3 GHz frequency, (<b>b</b>) 3.6 GHz frequency, (<b>c</b>) 4 GHz frequency, (<b>d</b>) 5 GHz frequency, and (<b>e</b>) 5.9 GHz frequency. The basic RT method refers to the approach presented in [<a href="#B39-sensors-24-07925" class="html-bibr">39</a>].</p>
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<p>The angle measurement scenario and the positions of the NCTS (denoted by T1, T2, and T3) and sensor (denoted by R).</p>
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<p>The AOA spectrum measured for the source located at T1.</p>
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<p>The AOA spectrum measured for the source located at T2.</p>
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<p>The AOA spectrum measured for the source located at T3.</p>
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<p>Comparison between measured AS and simulated multipath at (<b>a</b>) T1 position, (<b>b</b>) T2 position, and (<b>c</b>) T3 position.</p>
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<p>NCTS and sensor positions and a geometrical map of the scenario. The line segments represent the multipath between the source and the sensor, distinguished using different colors.</p>
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<p>A comparison of the proposed localization algorithm’s accuracy with the CRLB. (<b>a</b>) The source at location A; (<b>b</b>) the source at location B; (<b>c</b>) the source at location C.</p>
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<p>Localization error at point A with different AOA and RSSD errors.</p>
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<p>Localization error at point B with different AOA and RSSD errors.</p>
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<p>Localization error at point C with different AOA and RSSD errors.</p>
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<p>MSD distribution at (<b>a</b>) 0.1° AOA error, (<b>b</b>) 0.5°AOA error, (<b>c</b>) 1°AOA error, (<b>d</b>) 2°AOA error, (<b>e</b>) 4°AOA error, and (<b>f</b>) 6°AOA error.</p>
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<p>MSD distribution at (<b>a</b>) 0.1° AOA error, (<b>b</b>) 0.5°AOA error, (<b>c</b>) 1°AOA error, (<b>d</b>) 2°AOA error, (<b>e</b>) 4°AOA error, and (<b>f</b>) 6°AOA error.</p>
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<p>Schematic diagram of displacement compensation expansion method.</p>
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<p>Planar Localization Error Distribution with 0.1° AOA error. (<b>a</b>) Original localization algorithm; (<b>b</b>) localization algorithm with MSDM.</p>
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<p>Planar Localization Error Distribution with 0.5° AOA error. (<b>a</b>) Original localization algorithm; (<b>b</b>) localization algorithm with MSDM.</p>
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<p>Planar Localization Error Distribution with 1° AOA error. (<b>a</b>) Original localization algorithm; (<b>b</b>) localization algorithm with MSDM.</p>
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<p>Planar Localization Error Distribution with 2° AOA error. (<b>a</b>) Original localization algorithm; (<b>b</b>) localization algorithm with MSDM.</p>
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<p>Planar Localization Error Distribution with 4° AOA error. (<b>a</b>) Original localization algorithm; (<b>b</b>) localization algorithm with MSDM.</p>
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<p>Planar Localization Error Distribution with 6° AOA error. (<b>a</b>) Original localization algorithm; (<b>b</b>) localization algorithm with MSDM.</p>
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<p>Schematic diagram of GPU acceleration algorithm.</p>
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<p>Power coverage map.</p>
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<p>Efficiency comparison of different acceleration methods.</p>
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