Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Koppisetti, Naga Raghavendra Surya Vara Prasad
Doctor of Philosophy - PhD
University of British Columbia
Radio frequency (RF) wireless systems generate large amounts of data every day on the signal content and the received signal strength (RSS) information. However, limited efforts have been put into analyzing the RF data for security applications, such as the detection of unauthorized transmissions from an intruder. To bridge this gap, the Ph.D. thesis presents machine learning algorithms which detect and localize the intruder devices from their RF transmissions. First, we study the problem of detecting the intruder RF signals in wideband RF traces. An example setup is considered, with the intruder transmitting Wi-Fi signals and there exists interference from Bluetooth and microwave oven signals. We show that the conventional energy-thresholding algorithms are sensitive to noise variations and they require handcrafted parameters for each RF trace. To address this concern, we develop a deep learning solution, which employs convolutional neural networks to perform the signal de- tection. Experiments on both synthetic and real RF traces confirm the superior performance of the proposed solution in terms of the achieved mean average precision. Second, we study the problem of locating the intruder device from the RSS measured passively in the system. We work with the difference of RSS (DRSS), calculated with respect to a reference sensor, in order to handle the unknown transmission power and device heterogeneity of the intruder. The localization problem is formulated as a Gaussian process regression (GP) task to obtain the location estimates and the associated confidence intervals in closed-form. We propose two GP methods which take the stochastic nature of the test DRSS into account and provide more accurate confidence intervals on the test locations than the conventional GP method. Third, to improve the localization accuracy of the proposed GP methods, we present data reconstruction techniques which exploit the low-dimensionality exhibited by the DRSS vectors. Fourth, we study intruder localization for the case when the rate of signal strength decay with distance, also called the path loss exponent, is Gaussian distributed in the area. We propose two low-cost linear least squares estimators for the device location, which employ multilateration on the maximum-likelihood estimates of the distances to the sensors.
Machine Learning Algorithms for IntruderSignal Detection and Device Localization inWireless Radio Frequency SystemsbyNaga Raghavendra Surya Vara Prasad KoppisettiB.Tech., Indian Institute of Technology Bhubaneswar, India, June 2012M.A.Sc., The University of British Columbia, Canada, May 2016A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2021c© Naga Raghavendra Surya Vara Prasad Koppisetti 2021The following individuals certify that they have read, and recommend to the Faculty of Graduate and Post-doctoral Studies for acceptance, the dissertation entitled:Machine Learning Algorithms for Intruder Signal Detection and Device Localization in Wireless RadioFrequency Systems,submitted by Naga Raghavendra Surya Vara Prasad Koppisetti, in partial fulfillment of the requirements forthe degree of Doctor of Philosophy in Electrical and Computer EngineeringExamining Committee:Vijay K. Bhargava, Professor, Electrical and Computer Engineering, UBC VancouverSupervisorVincent Wong, Professor, Electrical and Computer Engineering, UBC VancouverSupervisory Committee MemberRyozo Nagamune, Associate Professor, Mechanical Engineering, UBC VancouverUniversity ExaminerVictor Leung, Professor Emeritus, Electrical and Computer Engineering, UBC VancouverUniversity ExaminerOctavia Dobre, Professor, Electrical and Computer Engineering, Memorial UniversityExternal ExaminerAdditional Supervisory Committee Members:Sudip Sekhar, Associate Professor, Electrical and Computer Engineering, UBC VancouverSupervisory Committee MemberLutz Lampe, Professor, Electrical and Computer Engineering, UBC VancouverSupervisory Committee MemberiiAbstractRadio frequency (RF) wireless systems generate large amounts of data every day on the signal content andthe received signal strength (RSS) information. However, limited efforts have been put into analyzing theRF data for security applications, such as the detection of unauthorized transmissions from an intruder. Tobridge this gap, the Ph.D. thesis presents machine learning algorithms which detect and localize the intruderdevices from their RF transmissions.First, we study the problem of detecting the intruder RF signals in wideband RF traces. An examplesetup is considered, with the intruder transmitting Wi-Fi signals and there exists interference from Bluetoothand microwave oven signals. We show that the conventional energy-thresholding algorithms are sensitiveto noise variations and they require handcrafted parameters for each RF trace. To address this concern, wedevelop a deep learning solution, which employs convolutional neural networks to perform the signal de-tection. Experiments on both synthetic and real RF traces confirm the superior performance of the proposedsolution in terms of the achieved mean average precision.Second, we study the problem of locating the intruder device from the RSS measured passively in thesystem. We work with the difference of RSS (DRSS), calculated with respect to a reference sensor, inorder to handle the unknown transmission power and device heterogeneity of the intruder. The localizationproblem is formulated as a Gaussian process regression (GP) task to obtain the location estimates and theassociated confidence intervals in closed-form. We propose two GP methods which take the stochasticnature of the test DRSS into account and provide more accurate confidence intervals on the test locationsthan the conventional GP method.Third, to improve the localization accuracy of the proposed GP methods, we present data reconstructiontechniques which exploit the low-dimensionality exhibited by the DRSS vectors. Fourth, we study intruderlocalization for the case when the rate of signal strength decay with distance, also called the path lossexponent, is Gaussian distributed in the area. We propose two low-cost linear least squares estimators forthe device location, which employ multilateration on the maximum-likelihood estimates of the distances toiiithe sensors.ivLay SummaryMachine learning (ML) algorithms can help us build radio frequency (RF) surveillance systems in which thewireless signals received passively at the sensors are analyzed for security-sensitive patterns. In this thesis,we propose ML algorithms to detect and position an intruder device transmitting RF signals in unautho-rized areas such as airports. First, we propose an algorithm which detects the intruder presence from its RFtransmissions. Second, we propose algorithms which estimate the intruder location from the received signalstrength information but also provide the relevant error margins. Third, we propose data transformation tech-niques to enhance the positioning accuracy of the proposed algorithms. Fourth, we propose low-complexitypositioning algorithms for the special case when we know how frequently the path loss exponent, whichdenotes the rate of signal strength decay with distance, takes a predefined range of numerical values in theservice area.vPrefacePublications that resulted from the research presented in this thesis are as follows:• K. N. R. S. V. Prasad, K. D’Souza, V. K. Bhargava, “A downscaled faster RCNN framework forsignal detection and time-frequency localization in wideband RF systems," IEEE Transactions onWireless Communications, vol. 19, no. 7, pp. 4847-4862, July 2020. (Linked to Chapter 2)• K. N. R. S. V. Prasad, K. D’Souza, H. Boostanimehr, S. Mallick, Wireless threat detection device,system, and methods to detect signals in wideband RF systems and localize related time and frequencyinformation based on deep learning, U.S. Patent 16/567630, published June 2020. (Linked to Chapter2)• K. N. R. S. V. Prasad, K. D’Souza, V. K. Bhargava, S. Mallick, H. Boostanimehr, “A deep learningframework for blind time-frequency localization in wideband RF systems," in Proc. of IEEE VehicularTechnology Conference (VTC Spring 2020), Virtual Conference, May 2020, pp. 1-6. (Linked toChapter 2)• K. N. R. S. V. Prasad, E. Hossain, V. K. Bhargava, “Machine learning methods for RSS-based userpositioning in distributed massive MIMO,"IEEE Transactions on Wireless Communications, vol. 17,no. 12, pp. 8402-8417, Oct. 2018. (Linked to Chapter 3)• K. N. R. S. V. Prasad, E. Hossain, V. K. Bhargava, “A numerical approximation method for RSS-based user positioning in distributed massive MIMO," in Proc. of IEEE International Conference onAdvanced Networks and Telecommunications Systems (ANTS), Bhubaneswar, India, Dec. 2017, pp.1-6. (Linked to Chapter 3)• K. N. R. S. V. Prasad, E. Hossain, V. K. Bhargava, “Analytical approximation-based machine learningmethods for user positioning in distributed massive MIMO," IEEE Access Special Issue on GNSS,viLocalization and Navigation Technologies, vol. 6, pp. 18431-18452, Feb. 2018. (Linked to Chapter3 and Chapter 4)• K. N. R. S. V. Prasad, E. Hossain, V. K. Bhargava, “Low-dimensionality of noise-free RSS and itsapplication in distributed massive MIMO," IEEE Wireless Communications Letters, vol. 7, no. 4, pp.486-489, Dec. 2017. (Linked to Chapter 4)• K. N. R. S. V. Prasad, V. K. Bhargava, “Rss localization under Gaussian distributed path loss expo-nent model," IEEE Wireless Communications Letters, vol. 10, no. 1, pp. 111-115, Jan. 2021. (Linkedto Chapter 5)I am the primary researcher and author of all the contributions made in this thesis. I held the main responsi-bility in developing the original ideas, the mathematical analysis, and the software codes (in Matlab, Python,MySQL) for the computer simulations. I also wrote the manuscripts for each publication. The contributionsof my co-authors are as follows. Prof. Vijay K. Bhargava is my thesis supervisor. He provided valuableguidance and directions in identifying the research problems and gave constructive feedback during thepreparation of the manuscripts. I have also actively collaborated with Prof. Ekram Hossain from Universityof Manitoba, who has contributed to four of the above publications. He has guided me in formulating theresearch problems and has reviewed the mathematical analysis presented in the papers. He also guided meon the results to present in the papers.Kevin D’Souza was a former Masters student in our research group. In three of the above publica-tions, he has provided important technical feedback during the problem formulation stage and has helped inrunning a few simulations. He also reviewed the manuscripts.Dr. Mallick and Dr. Boostanimehr are our industry collaborators from Skycope Technologies Inc.Burnaby, Canada, and are the co-authors in two of the above publications. They helped with identifyingthe time-frequency localization problem presented in the paper and its equivalent box detection problemin the spectrograms. They also provided inputs on the spectrogram and Wi-Fi signal simulation, and gavetechnical feedback during the problem formulation and paper submission stages.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixList of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview of Intruder Signal Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Overview of Intruder Device Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Related Work, Motivation, and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 A Deep Learning Framework for Signal Detection in Wideband RF Systems . . . . . . . . . 212.1 A Downscaled Faster RCNN Framework for Signal Detection and Time-Frequency Local-ization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2 Data Generation Framework for Wi-Fi Signals in the 2.4GHz ISM Band . . . . . . . . . . . 29viii2.3 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 Gaussian Process Regression Algorithms for Intruder Localization from Uplink DRSS . . . 503.1 System Model and Localization with Conventional GP Method . . . . . . . . . . . . . . . 523.2 Numerical Approximation GP Method for Location Prediction . . . . . . . . . . . . . . . . 603.3 Gaussian Approximation GP (GaGP) Method for Location Prediction . . . . . . . . . . . . 633.4 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724 Gaussian Process Regression Algorithms with Data Reconstruction for Intruder Localizationfrom Uplink DRSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.1 Localization Setup and the Low-dimensionality of Noise-free DRSS . . . . . . . . . . . . . 754.2 Reconstruction GP (RecGP) Method for Location Prediction . . . . . . . . . . . . . . . . 784.3 Reconstruction Gaussian Approximation GP (RecGaGP) Method for Location Prediction . 834.4 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985 Linear Least Squares Algorithms for Intruder Localization under Gaussian Distributed PathLoss Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.1 Localization Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.2 Proposed Estimators for Source Location . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3 Performance Bound on the Localization Performance . . . . . . . . . . . . . . . . . . . . . 1085.4 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1146 Summary, Conclusions, and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120ixAppendicesA Appendix for Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138A.1 Analysing STFT vectors on Each Time Bin for Different Frequencies . . . . . . . . . . . . 138B Appendices for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147B.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147B.2 Proof of Remark 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150B.3 Mathematical Formulas for Simplifications in Chapter 3 . . . . . . . . . . . . . . . . . . . 152B.4 Complexity Analysis for the GP Methods and Baselines in Chapter 3 . . . . . . . . . . . . 154C Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157C.1 Complexity Analysis for the GP Methods and baselines in Chapter 4 . . . . . . . . . . . . . 157D Appendix for Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159D.1 On the Inability to Pursue Weighted Least Squares for Location Estimation in Chapter 5 . . 159D.2 Mathematical Simplifications to Obtain β in the CLLS Estimator . . . . . . . . . . . . . . 160D.3 Complexity Analysis for the Proposed Location Estimation Algorithms and Baselines inChapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162E List of Other Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164xList of Tables3.1 Simulation parameters to generate DRSS vectors and study the performance of the NaGPand GaGP methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.1 Simulation setup and parameters to study DRSS-based localization under Gaussian dis-tributed PLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2 Average computation time of the considered localization algorithms when σm = 6 dB. . . . . 1135.3 Average estimation error in the PLE, as reported by the MLE-TLLS and MAP-TLLS algo-rithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114A.1 Summary of the detection performance of the RuLSIF method for the four example timebins in Fig. A.5. None of the (n, k) perform consistently well across all the time-bins. . . . . 146xiList of Figures1.1 An overview of the wireless RF surveillance system of interest. When intruder devices, suchas drones and their remote controllers (RCs), send communication signals to each other,the wideband sensors deployed across the service area passively receive the signals on theuplink. A computing unit connected to the sensor nodes hosts learning-based models whichanalyze the received signals to passively detect the intruder presence and estimate their time-frequency span, so as to employ narrowband jamming. It also hosts learning-based modelswhich estimate the intruder locations from the uplink received signal strength informationand also report the associated confidence intervals, so as to facilitate device recovery. . . . . 21.2 Proposed framework for signal detection and time-frequency localization . . . . . . . . . . 41.3 The time content, frequency content, and spectrogram of an example wideband RF captureof duration 90 ms, center frequency 2.457 GHz, and bandwidth 56 MHz respectively. Thesignals of interest are the Wi-Fi signals and the RFI constitutes Bluetooth low energy andmicrowave oven signals. Signal detection in the absence of RFI is a simplified case underthis example setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 System model for passive intruder device localization. When an intruder device, such asa drone, is transmitting RF signals to an intended receiver such as a remote controller, thesensor nodes in the service area passively receive the signals on the uplink and record theRSS information. The proposed localization algorithms in the thesis utilize the differentialRSS (DRSS) values, i.e., the difference of RSS with respect to a reference sensor, to estimatethe intruder device location and report the associated 2σ error bars in closed-form. We usethe DRSS instead of the RSS in order to handle the unknown transmit power and the deviceheterogeneity of the intruder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8xii2.1 The downscaled FRCNN model. For each spectrogram chunk, the base network yields afeature map which is fed into the region proposal network (RPN) and the Detector network.The RPN and the Detector host a bunch of feature extraction (FE) layers, the outputs ofwhich are used for classification and regression to yield bounding boxes. . . . . . . . . . . 232.2 (a) Training loss convergence for the classification and regression tasks in the RPN and theDetector networks. (b)-(c) Example positive targets for the RPN and the Detector networks.The rectangular boxes are the targets, while the signals underneath the boxes are the groundtruths. To illustrate noise and RFI, we consider colored noise and a signal each from Blue-tooth (right of the targets) and microwave oven (left of the targets). . . . . . . . . . . . . . . 312.3 Performance of the morphological processing method [7] with the best 5 hyperparametercombinations obtained through grid search . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.4 Top 5 mAP performances of the SVM-HOG method. Apart from the hyperparameters listedin the legend, hyperparameters common to all five curves are: (pix = 8, b = 16, bstride = 8,and wstep = 8). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 Precision vs. recall when the RPN and Detector in the FRCNN are downscaled by a factor of2,4, and 8 respectively. Downscaling by 2 gives the best mAP performance. The downscaledFRCNN model is also seen to significantly outperform the two baseline methods. . . . . . . 392.6 Example predictions on the test spectrograms with the downscaled FRCNN model when theBN is the VGG-13 (trainable, initialized with pretrained weights). The blue and red rectan-gles indicate the signal ground truth and the predictions respectively. The model successfullyseparates out the RFI and provides tight bounding boxes for most signals of interest, evenif there is an overlap with the RFI signals. The loss in mAP occurs mainly due to falsepositives arising from microwave oven signals. . . . . . . . . . . . . . . . . . . . . . . . . 41xiii2.7 Example predictions from the morphological processing method [7] and the proposed deeplearning method. The optimized hyperparameters in [7] do not generalize well across cap-tures with different noise and interference types. As a result, we observe that the optimizedhyperparameter combination (γbin = 0.8,nerode = 2, s = 9) results in missed detections, par-tially as in captures 3 and 4 or completely as in captures 5 and 6. Handcrafted hyperparam-eters, found through trial and error, can provide improved performance but handcrafting isnot a practical solution. The proposed deep learning method, on the other hand, providesuniformly good performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.8 Example predictions from the SVM-HOG method [115] with the optimized hyperparame-ters (w f = 32,wt = 80,ori = 7, pix = 8,b = 16,bstride = 8,wstep = 8). We observe much fewernumber of missed detections than with [7] due to the supervised nature of the method. How-ever, the SVM-HOG method does not perform very well in separating out the RFI, resultingin a significant number of false positives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.9 Impact of depth of the base network. The legend indicates the BN name, whether the ini-tialization is pretrained or Xavier normal [171], and whether the BN is trainable or not. Wenote that the VGG-13 model, initialized with pretrained weights and configured as trainable,performs better than the others. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.10 The mAP with different RPN and Detector thresholds. An mAP of up to 0.8 is achievedthrough a simple alternate-once search strategy. . . . . . . . . . . . . . . . . . . . . . . . . 452.11 Example predictions on real test captures when the synthetically-trained FRCNN model isfinetuned with real RF traces. When compared to the case of synthetic test traces, the realtest captures contain new types of background noise and RFI (compare the spectrogrambackgrounds with Fig. 2.6). The model is still able to successfully detect the signals ofinterest, but experiences a marginal loss in the mAP. . . . . . . . . . . . . . . . . . . . . . 462.12 Precision vs. recall when the test captures are obtained through real measurements. Wenotice a drop in the mAP if the model is trained purely on synthetic data. Data mixing andmodel finetuning are seen to alleviate the drop in the mAP. . . . . . . . . . . . . . . . . . . 462.13 Precision vs. recall curves for datasets with different SNR ranges. We notice a drop in mAPas we widen the SNR range and enter into negative SNR values. . . . . . . . . . . . . . . . 48xiv3.1 Example setup with three sensors, labelled B1, B2, and B3, and five devices to be localized,labelled U1, U2, U3, U4, and U5. The radius dth is the distance beyond which the PLE ηchanges in value (for example, dth = 45 m in 3GPP UMi [97] , as the η changes from 2 to6.7). For any pair of devices, φ(., .) should be high if the x−coordinates are closely-spacedand it should decrease with the separation. . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Localization setup with M = 31 anchors (left-bottom as the reference for DRSS), L˜ = 100training locations, and L̂ = 25 test locations. . . . . . . . . . . . . . . . . . . . . . . . . . . 653.3 Average localization RMSE achieved by the CGP, NaGP, and GaGP methods at differentshadowing noise levels σm when (a) trained with noisy DRSS and (b) trained with noise-free DRSS respectively. From (a), we note that for most σ, the three GP methods performbetter than the existing algorithms when trained with noisy DRSS. From (b), we note thatit is also worthwhile training with noise-free DRSS, particularly for σ < −5 dB, where theGP methods perform better than the MLE and RSDPE algorithms, and they achieve compa-rable performance as the kNN algorithm. Note that, unlike the baseline algorithms, the GPmethods additionally provide confidence intervals around the estimated test locations. . . . 683.4 [Training with noise-free DRSS] Plots of the average 2σ error-bars on the estimated loca-tions, as provided by the CGP, GaGP, and NaGP methods. The CGP method provides 2σerror-bars that are quite optimistic, with less than 40% of the true test device locations withinthe 2σ confidence range of the estimated locations. The GaGP and NaGP methods providemore realistic 2σ error-bars on the location estimates, with more than 90% of the true testdevice locations within the 2σ confidence range of the estimated locations. . . . . . . . . . 703.5 [Training with noisy DRSS] Plots of the average 2σ error-bars on the estimated locations,as provided by the CGP, GaGP, and NaGP methods. The CGP method provides 2σ error-bars that are relatively optimistic, resulting in less than 85% of the true test device locationswithin the 2σ error-bars around the location estimates. The GaGP and NaGP methodsprovide more realistic 2σ error-bars on the location estimates, with more than 95% of thetrue test device locations within the 2σ confidence range of the estimated locations. . . . . 713.6 Plot of the average localization RMSE of the three GP methods when the number of DRSSsensor nodes is increased from 10 to 50. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72xv4.1 Plots of the singular values and reconstruction errors (with the first M0 PCs) for the numberof DRSS sensor nodes (M − 1) = 20 and (M − 1) = 30, averaged over 200 different noise-free DRSS matrices. In Figs. 4.1a-4.1b, we observe that the first few singular values explainmost of the energy contained in P∗. In Figs. 4.1c-4.1d, we observe that the first few PCsreconstruct more than 95% of the data contained in P∗. . . . . . . . . . . . . . . . . . . . . 774.2 Overview of the training and prediction procedures in the proposed RecGP method. (a)When the training DRSS P˜∗ is noise-free, it is directly used to train the GP model. Whenotherwise, we reconstruct the noisy training DRSS P˜ from a low-dimensional principal sub-space spanned by the noise-free DRSS P¯∗ in the service area using (4.13) and train the GPmodel with the reconstructed DRSS P˜(rec). (b) The test DRSS P̂ is noisy due to shadowingeffects of the wireless channel and is reconstructed either from P˜∗ or P¯∗ prior to deriving theposterior distribution of the test location coordinates x̂. The reconstructed test DRSS P̂(rec)is treated as a deterministic input when deriving the posterior distribution of x̂ (c.f. (4.19)). . 794.3 Overview of the training and prediction procedures in the proposed RecGaGP method. (a)The training procedure is the same as in the RecGP method. (b) The reconstructed test DRSSP̂(rec), obtained in the same fashion as in RecGP, is treated as a stochastic input when deriv-ing the posterior distribution of x̂. The true posterior density is analytically intractable, soit is approximated as a Gaussian distribution with the same first and second order moments(c.f. Theorem 4.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.4 Average data loss and the average residual noise variance in the test DRSS, when the numberof PCs M0 for the reconstruction step is varied from 0 to M −1, for M ranging from 21 to 61. 904.5 Average localization RMSE achieved by the proposed GP methods. In (a), the number ofPCs M0 in the reconstruction step is chosen as the minimum number of singular valueswhich explain 99% of the energy contained in the matrix P¯∗ used for the reconstruction step(c.f. (4.12)). In (b), M0 is chosen through empirical studies as the one that most frequentlygave the lowest RMSE over a hold-out validation dataset containing DRSS vectors at 25uniformly distributed device locations in the service area. . . . . . . . . . . . . . . . . . . . 93xvi4.6 Plots of the average RMSE vs. M0 performance of the RecGP method and the histogramof the RMSE-minimizing M0 values, when the GP model is trained with noise-free DRSS.From Fig. 4.6a, we observe that the RMSE decreases with M0 initially, attains a minimum,followed by a gradual increase. From Fig. 4.6b, we observe that the histograms have multi-ple peak values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.7 [Training with noise-free DRSS] Plots of the average 2σ error-bars on the estimated lo-cations, as provided by the RecGP and RecGaGP methods. The RecGP method performssimilar to the CGP method and provides unrealistically small 2σ error-bars, with less than40% of the true device locations within the 2σ confidence range of the predicted locations.The RecGaGP method performs similar to the NaGP and GaGP method and provides morerealistic 2σ error-bars on the location estimates, with more than 95% of the true devicelocations within the 2σ confidence range of the estimated locations. . . . . . . . . . . . . . 954.8 [Training with noisy DRSS] Plots of the average 2σ error-bars on the estimated locations, asprovided by the RecGP and RecGaGP methods. Although the 2σ error-bars from the RecGPmethod are not as optimistic as with noise-free training DRSS, we note that the RecGaGPmethod continues to perform better than the RecGP. . . . . . . . . . . . . . . . . . . . . . 964.9 Plot of the average localization RMSE of the RecGP and RecGaGP methods when the num-ber of DRSS sensor nodes (M −1) is increased from 10 to 50. . . . . . . . . . . . . . . . . 975.1 Error performance of the proposed ratio-of-distance estimator vs. shadowing standard devi-ation σm for different PLE γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.2 Localization RMSE performance of the proposed least squares and MAP estimators vs.shadowing noise standard deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112A.1 Example RF capture 1 to expose the difficulty in hyperparameter tuning for the RuLSIF[107] method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141A.2 Changepoint detection for the three WiFi signals in Fig. A.1, when (n, k) = (50,10). Reduc-ing α from 0.1 to 0.01 decreases the number of false positives. . . . . . . . . . . . . . . . . 142A.3 Changepoint detection for the three WiFi signals in Fig. A.1, when α = 0.01. Decreasing(n, k) from (50,10), while keeping the 5 : 1 ratio, does not affect the performance until (n, k)=(15,3), where false positives begin to appear (see (a)). . . . . . . . . . . . . . . . . . . . . . 143xviiA.4 Example RF capture 2 to expose the difficulty in hyperparameter tuning for the RuLSIF[107] method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144A.5 Changepoint detection for four example time-bins (see Fig. A.4 for the frequency edges).The edges are successfully detected if the (n, k) is appropriately chosen but there is no single(n, k) which correctly detects the edges across all four time-bins. . . . . . . . . . . . . . . . 145xviiiList of AbbreviationsBLE Bluetooth Low EnergyBN Base NetworkBS Base StationCG Conjugate GradientCGP Conventional Gaussian Process regressionCLLS Constrained Linear Least SquaresCNN Convolutional Neural NetworkCQT Constant-Q TransformCWT Continuous Wavelet TransformDRSS Differential RSS/Difference of RSSDSP Digital Signal ProcessingFE Feature ExtractionFFT Fast Fourier TransformFN False NegativesFP False PositivesFRCNN Faster Region-based Convolutional Neural NetworkGaGP Gaussian Approximation GPxixGP Gaussian Process regressionGPS Global Positioning SystemGPU Graphical Processing UnitGT Ground TruthHCRLB Hybrid Cramer-Rao Lower BoundHLF Hyperbolic Location FingerprintHOG Histogram of Ordered GradientsIoT Internet of ThingsIoU Intersection over UnionIP Inner ProductkNN k Nearest NeighboursLM Levenberg-MarquardtLS Least SquaresLTE Long Term EvolutionMAC Medium Access ControlmAP mean Average PrecisionMAP Maximum A Posteriori estimatorMIMO Multiple-Input Multiple-OutputML Machine LearningMLE Maximum Likelihood EstimationMO Microwave OvenxxNaGP Numerical approximation GPNLOS Non-Line-of-SightPCA Principal Component AnalysisPLE Path Loss ExponentRC Remote ControllerRecGP Reconstruction GPRecGaGP Reconstruction Gaussian approximation GPRF Radio FrequencyRFI Radio Frequency InterferenceRMSE Root Mean Squared ErrorRPN Region Proposal NetworkRSDPE Robust Semi Definite Programming EstimatorRSS Received Signal StrengthRuLSIF Relative unconstrained Least Squares Importance FittingSDP Semi-definite ProgrammingSDR Semidefinite RelaxationSE Squared ExponentialSTFT Short-Time Fourier TransformSVM Support Vector MachineTLLS Two-step Linear Least SquaresTN True NegativesxxiTP True PositivesUSRP Universal Software Radio PeripheralWi-Fi Wireless FidelityWSN Wireless Sensor NetworkWVT Wigner-Ville TransformYOLO You Only Look OncexxiiList of Algorithms2.1 RF data generation with Wi-Fi as the signals of interest, under interference from Bluetoothlow energy (BLE) and microwave oven (MO) signals. . . . . . . . . . . . . . . . . . . . . . 302.2 The mean average precision (mAP) metric for signal detection performance evaluation . . . 33xxiiiAcknowledgementsI was fortunate to meet and learn from many incredible people at the University of British Columbia (UBC)during my Ph.D. study. Foremost, I would like to thank my Ph.D. supervisor Professor Vijay K. Bhargavaand my research collaborator Professor Ekram Hossain from the University of Manitoba, for their guidance,support, and inspiration over the years. I am very grateful for the tremendous opportunity they have givento me. Professor Bhargava’s motivation, vision, generosity and integrity inspire me every day.I also thank Professor Lutz Lampe, Professor Vincent Wong, Professor Sudip Sekhar, Professor VictorLeung, and Professor Ryozo Nagamune for serving on my candidacy, departmental defence and universitydefence exams held at various stages of my Ph.D. program . I thank them for providing insightful commentson improving the quality of the thesis.I am also grateful to my colleagues and alumni from the CITR research group at UBC, for their in-valuable support, collaboration, and friendship. Special thanks to Kevin D’Souza, with whom I had thegood fortune to collaborate over multiple interesting projects. I am also very grateful to Dr. Mallick and Dr.Boostanimehr at Skycope Technologies Ltd., for bringing in valuable industry insights and guiding me in ourcollaborative work. I have also greatly benefited from the many hours of stimulating discussions I had withJiaming Cheng and Dr. Duong Nguyen. I am also deeply indebted to my former and current colleagues, Dr.Rindranirina Ramamonjison, Dr. Gautham Prasad, Elena Dobre, Jiaming Cheng, and Tarannum Nisha, whotook the time to review the thesis and have given detailed comments. I also thank the faculty and staff at theUBC ECE department for their great support, including Ross Shepherd, Kristie Henriksen, Shelby Murray,Sabrina Ho and many others.Words would not be enough to thank Annet Khashchyna for bringing in a ton of positivity and helpingme manage exam stress during the final stages of the Ph.D. I also thank my best friend Gurpreet Kaur foralways encouraging me and providing a great deal of support throughout the Ph.D. years. I am gratefulto Anahita Shojaei, Amit Kumar Mishra, Arefin Rashid Ayon, and Rahul Krishna, for their friendship andgood vibes during my stay in Canada.xxivI also thank my teaching mentors Dr. Saloome Motavas, Dr. Jonatan Schroeder, Dr. Siamik Najarian, Dr.Paul Lusina, and Fatimah Mahmood, with whom I worked as a Teaching Assitant (TA) for several coursesat UBC. I thank them for providing wonderful teaching opportunities to me and for facilitating many greatinteractions with the students.Finally, I thank the Natural Sciences and Engineering Research Council (NSERC) Canada and UBCfor supporting my PhD study through the Alexander Graham Bell Doctoral scholarship. I also thank thegovernment of British Columbia (BC) for the BC Graduate Scholarship. I am also grateful to the Com-puter Science department at UBC for recognizing my work as a teaching assistant (TA) with the 2019 BestGraduate TA award and for also nominating me for the 2021 UBC Killam Graduate TA award.xxvDedicationI dedicate this thesis to my parents, Surya Kumari Koppisetti and Gopala Krishna Koppisetti, and my dearsister Pavani Sindhura Koppisetti, with deep gratitude for their love, continuous support, and many sacrifices.xxviChapter 1IntroductionWith the emergence of the Internet of things (IoT), we are currently witnessing a steep surge in the numberof wireless devices around us. With time, more and more IoT devices will coexist with mobile phones. Ittherefore becomes imperative to distinguish between these devices from a security and spectrum manage-ment point of view. Technology that can detect and categorize heterogeneous wireless signals can be partic-ularly useful in building wireless radio frequency (RF) surveillance systems. For example, if we detect thepresence of an unexpected narrowband RF transmitter in a given wideband spectrum of interest and we arealso able to localize its transmissions in time and frequency, we can build wireless security products to sendemergency alerts about the presence of the unexpected wireless device. We refer to the unexpected wirelessdevice from here on as the intruder device. We can also potentially use jammers to block the signals from theintruder device. The time-frequency localization feature also informs us on which time-frequency resourcesare under-utilized in the wideband spectrum. We may then build smart spectrum allocation products to servean increased number of devices per unit area. For example, if we decode the frequency-hopping pattern of agiven device, we may use a time delay and re-allocate the same set of time-frequency resources to a differentdevice operating with the same dwell-time, bandwidth, and hopping period. Such smart spectrum allocationproducts would be of great value in the IoT era, where we will continue to witness increasing density of thewireless devices. In order to enable the applications mentioned above, there is a need to develop a signaldetection algorithm which can perform two major tasks simultaneously: detect the presence of a wirelesssignal in a wideband RF spectrum and localize its time-frequency span.Upon successfully detecting the wireless signals from an intruder device, a second important aspect ofa surveillance system is to estimate the intruder device location from the measured signal properties. Overthe past few years, the received signal strength (RSS) has been a popular choice for the device localizationtask in wireless sensor networks (WSN) because it is simple and easy to implement when compared to angleand time-based methods. This advantage becomes particularly handy in security and wireless surveillance1Figure 1.1: An overview of the wireless RF surveillance system of interest. When intruder devices, such as drones and their re-mote controllers (RCs), send communication signals to each other, the wideband sensors deployed across the service area passivelyreceive the signals on the uplink. A computing unit connected to the sensor nodes hosts learning-based models which analyze thereceived signals to passively detect the intruder presence and estimate their time-frequency span, so as to employ narrowband jam-ming. It also hosts learning-based models which estimate the intruder locations from the uplink received signal strength informationand also report the associated confidence intervals, so as to facilitate device recovery.applications, where the RSS information is easily measurable at the sensor nodes. The goal in such sys-tems is to passively estimate the intruder location from the RSS information. The localization algorithmsneed to work around the unknown transmit power of the intruder device. For example, even if the intruderis successfully detected and the signal patterns are analyzed to reverse engineer the device specifications,the intruder might be intentionally deviating from the power specifications and transmitting with a differ-ent antenna gain. Besides, the intruder device can exhibit heterogeneity, i.e., the devices procured fromthe same vendor can exhibit different transmit powers due to hardware imperfections and manufacturing er-rors. Surveillance systems also require that the localization algorithms provide accurate confidence intervalsaround the location estimates, in order to identify a finite amount of area within which the intruder devicecan be found and recovered. An overview of the wireless RF surveillance system of interest is presented inFig. 1.1.In this thesis, we address the above research problems, in part, by proposing (i) a deep learning solutionfor the signal detection task when there exists radio frequency interference (RFI) from friendly devicesoperating in the same spectrum, (ii) probabilistic machine learning algorithms which estimate the intruderdevice location from the RSS information and also report the associated confidence intervals, and (iii) linearleast squares algorithms which estimate the intruder device location from the RSS when the rate of signal2strength decay with distance, also called the path loss exponent, is known to follow a Gaussian distributionin the service area. Before presenting further details on our contributions, we present an overview of theintruder signal detection task in Section 1.1, an overview of the intruder device localization task in Section1.2, a detailed literature review on these topics, along with our motivation and research objectives in Section1.3, and the outline of the thesis in Section 1.4.1.1 Overview of Intruder Signal DetectionWe are interested in detecting the intruder signals and simultaneously extracting their time and frequencyspan information in a wideband RF spectrum. To achieve this task, we need to start with choosing anappropriate time-frequency representation of the spectrum where both the time and frequency spans aresimultaneously exposed. The short-time Fourier transform (STFT) [1] is a popular choice in this regard,for it is a single time-frequency resolution method, i.e., the same time-frequency resolution is maintainedacross the wideband spectrum irrespective of the center-frequency of the narrowband transmissions. Wecannot however directly work with the raw STFT matrices because they are complex-valued. Instead, wecan work with the spectrograms, which are the log-transformed STFT magnitude matrices. While the STFTmagnitudes are known to be sufficient for separating and reconstructing the underlying signals [1], a fur-ther log-transformation offers better control over the numerical range of the STFT magnitudes for differentfrequencies. It is therefore commonplace to work with the spectrogram matrices. Alternatives to the STFTinclude the continuous wavelet transform (CWT), the constant-Q transform (CQT), the Wigner-Ville trans-form (WVT), and the S-method [2, 3]. The CWT and CQT have a frequency-dependant time-frequencyresolution, wherein the lowest frequency components are least smeared in frequency and the highest fre-quency components are the most smeared in frequency. Such variable resolution methods are not preferablefor spectrum monitoring and smart spectrum allocation applications, where all the frequencies within thewideband spectrum are equally important. The WVT and the S-methods, on the other hand, offer a singletime-frequency resolution but have other shortcomings. The WVT provides a finer time-frequency resolu-tion than the STFT but introduces undesirable artifacts (cross-terms) and an increased amount of aliasing.The S-method minimizes the cross-terms introduced by the WVT but it is non-invertible, making it difficultto recover the underlying narrowband signals. We therefore work with the STFT spectrogram matrices inthis thesis.3Spectrogram creation, pre-processingBox detection in spectrogramsTime-frequency information extractionRF time-series captureFigure 1.2: Proposed framework for signal detection and time-frequency localizationLimited literature exists, however, on the joint signal detection and time-frequency localization in thespectrograms. Existing works [4–7] have considered RF captures that are limited to interference-free scenar-ios. Also, their performance is subject to handcrafted hyperparameters which do not generalize well acrossthe diverse types of noise and interference. The resulting arbitrariness in performance may be overcomethrough supervised learning, in which a machine learning model is trained with spectrograms containing thesignals of interest, alongside various types of RFI and noise. The model can learn features which facilitatethe signal detection task despite the imperfections caused by the noise and interference. Below, we proposea generic supervised machine learning framework to detect and estimate the time-frequency span of thesignals of interest in a wideband RF spectrum.Framework for signal detection and time-frequency localizationThe proposed framework takes the wideband RF time-series data as the input and provides the time andfrequency information of the signals of interest as the output, as outlined in Fig.1.2. The role of each stagein the framework is explained below.RF time-series captureIn the first stage, we employ a wideband RF sensor with center frequency fc and bandwidth W to recordtime-series RF data in fragments of T milliseconds each. The time and frequency content of an examplewideband capture with fc = 2.457 GHz, W = 56 MHz, and T = 90 ms is given in Fig. 1.3. In this examplecapture, let us suppose that we need to localize the Wi-Fi signals in time and frequency when there existsRF interference from friendly devices transmitting Bluetooth and microwave oven signals. Signal detectionin the absence of RFI is a special case under this example setup and is an easier problem to solve. The signalamplitude is plotted as a function of time in Fig. 1.3a and the Fast Fourier Transform (FFT) magnitudes areplotted as a function of frequency in Fig. 1.3b.40 10 20 30 40 50 60 70 80 90Time (ms)020040060080010001200Signal magnitude(a) Signal amplitude vs. time-20 -10 0 10 20Frequency (MHz)00.020.040.060.080.10.120.14FFT Magnitude(b) FFT magnitude vs. frequency10 20 30 40 50 60 70 80Time (ms)-20-1001020Frequency (MHz)-140-120-100-80-60-40-20Power/frequency (dB/Hz)Bluetooth interferenceMicrowave oven interferenceSignals of interest (Wi-Fi)(c) Spectrogram (d) Spectrogram (3D-view)Figure 1.3: The time content, frequency content, and spectrogram of an example wideband RF capture of duration 90 ms, centerfrequency 2.457 GHz, and bandwidth 56 MHz respectively. The signals of interest are the Wi-Fi signals and the RFI constitutesBluetooth low energy and microwave oven signals. Signal detection in the absence of RFI is a simplified case under this examplesetup.5Spectrogram creation and pre-processingIn order to detect the signals of interest and extract their joint time-frequency information, we use thespectrogram matrices as the input format. An illustration corresponding to the capture in Figs. 1.3a-1.3bis presented in Figs. 1.3c-1.3d, with the STFT obtained using a Hann window of size 1400, 50% windowoverlap, and FFT size of 1024. As seen in Fig. 1.3c, the problem of signal detection and time-frequencylocalization boils down to that of obtaining rectangular bounding boxes in the time-frequency domain.Bounding box detection in spectrogramsTo detect bounding boxes for the signals of interest, we propose a supervised learning approach, in whichwe train an object detection model which takes the spectrograms as input and provides the bounding boxcoordinates as the output. For example, in Chapter 2, we build a synthetic dataset containing RF captureswith Wi-Fi as the signal of interest and there exists RFI from Bluetooth and microwave oven signals. Wethen train a modified form of the Faster RCNN model [8] to obtain bounding boxes for the Wi-Fi signals inthe spectrogram of each RF capture. The trained model, when input with a test spectrogram matrix, detectsthe bounding boxes for the signals of interest and reports their time-frequency span.An inherent limitation of this approach is that the bounding box detection algorithm reports all thedetected Wi-Fi signals (more generally, all the detected signals from the same wireless protocol), regardlessof whether they are transmitted by the intruder device or by other friendly Wi-Fi access points operatingon the same channel. Additional device-level classification would be required as a post-processing step, forexample based on inter-arrival time between the probe request frames [191,192] in the case of Wi-Fi signals,in order to separate out the intruder signals from those transmitted by the friendly devices.Time-frequency information extractionThe machine learning model is trained to report the dimensions of the bounding box of the signals andnot the absolute time and frequency span information, in order to keep the learned attributes at a similarscale. The absolute time and frequency span attributes otherwise have vastly different scales, in the orderof milliseconds and megahertz respectively. Therefore, as the final step, we use the time and frequencyresolution information from the STFT parameters to linearly scale each predicted bounding box into thestart time, end time, start frequency, and end frequency of the detected signals of interest.6Upon successfully detecting the intruder signals, a second important aspect of the wireless RF surveil-lance systems is to estimate the geographic location of the intruder device. We now present an overview ofthis topic.1.2 Overview of Intruder Device LocalizationTraditionally, devices in a WSN are positioned using the satellite-based global positioning system (GPS)technology, wherein GPS sensors are installed on the device of interest to frequently transmit its locationcoordinates to a central server connected to the WSN. However, in the case of an intruder device, we do notnecesarily have access to the GPS information, as this information may be encrypted in a format unknownto the WSN, or the device may not host GPS sensors at all because they are power-hungry [13] and drain thebattery on board. It therefore becomes important to develop alternative positioning techniques which relyon the signal properties of the device, such as the angle-of-arrival, time-of-arrival, and the RSS information.Time-based methods, such as [17–26] and references therein, require accurate timing synchronization withthe sensor nodes but this is not possible in the case of a non-cooperating intruder device. Besides, time-based methods require a direct line of sight between the transmitter and the receiver. Angle-based methods( [23, 25–33] and references therein) require multiple antennas to be deployed at each sensor and theyonly provide coarse location estimates under non-line-of-sight conditions. In contrast, the RSS informationcan be readily measured at the sensor nodes and it does not require additional hardware or tight timingsynchronization efforts. RSS is therefore the simpler and cost-effective choice for device positioning.The passive device localization setup under study in the thesis is illustrated in Fig. 1.4. When an intruderdevice, such as a drone, is transmitting RF signals to an intended receiver such as a remote controller, thesensor nodes in the service area passively receive the signals on the uplink and record the RSS information.The RSS measured at a sensor node depends on the source location through a non-invertible function gov-erned by the distance-dependant path loss. A standard way to model the RSS pdBm at a sensor node m in dBunits is to negate the log-distance path loss from the source transmit power ρ and add a shadowing noiseterm, as done below [34–41]pdBm = 10log10(ρg0)−10γ log10 dm+ zm, where,dm =√(x− xm)2+(y− ym)2.(1.1)7Figure 1.4: System model for passive intruder device localization. When an intruder device, such as a drone, is transmitting RFsignals to an intended receiver such as a remote controller, the sensor nodes in the service area passively receive the signals onthe uplink and record the RSS information. The proposed localization algorithms in the thesis utilize the differential RSS (DRSS)values, i.e., the difference of RSS with respect to a reference sensor, to estimate the intruder device location and report the associated2σ error bars in closed-form. We use the DRSS instead of the RSS in order to handle the unknown transmit power and the deviceheterogeneity of the intruder.In (1.1), (x, y) and (xm, ym) denote the 2D coordinates of the source location and the sensor m respectively,dm is the distance to the sensor m, g0 is the path loss at a reference distance, γ is the path loss exponent(PLE), and zm is the shadowing noise coefficient. We may note that it is difficult to invert the relation in(1.1) and directly obtain the source location (x, y) from the RSS values. We need to therefore use indirectmethods to estimate the source location. For example, in ranging methods, the distances to each sensor arefirstly estimated and the source location is then obtained by formulating a linear system of equations in thedistances. Alternatively, in machine learning methods, the RSS data is collected at various locations in theservice area and a model is trained to make the location predictions when input with the RSS information.When using the RSS for the intruder device localization, the transmit power ρ needs to be consideredas an unknown. This is because the transmission power can vary across the device units due to differentantenna gains and imperfections in the circuitry and packaging [42–46]. The goal of wireless RF surveillancesystems is therefore to passively estimate the intruder location from the RSS, despite the lack of knowledgeon the transmit power. The difficulty of this task is further compounded by the noisy nature of the RSSarising from the small-scale fading and shadowing effects of the wireless channel. Typically, in RSS-based8positioning, the small-scale fading effects are averaged out over multiple coherence intervals [34]. Theshadowing noise, on the other hand, cannot be mitigated because spatial averaging is necessary and thisrequires prior knowledge of the source location. The presence of shadowing noise is known to adverselyaffect the accuracy of RSS-based localization methods [47–50].To handle the unknown transmit power and the device heterogeneity, a recently popular approach[54–58] is to use the differential RSS (DRSS), i.e., the differences of RSS values with respect to a ref-erence sensor, in place of the actual RSS for the localization. Note from (1.1) that the transmit power term10log10(ρg0) is eliminated when we assign a reference node, say m = 1, and work with the difference ofRSS values with respect to the reference, i.e., pdBm − pdB1 , for the localization. We take this approach in thethesis and resort to DRSS-based localization for the intruder device.Existing works on DRSS-based localization can be broadly categorized into four types: maximum-likelihood estimation (MLE) [55,59,60] , semi-definite relaxation (SDR) methods [56,61–63], least squares(LS)-based ranging [55, 57, 58, 64], and fingerprinting [65–71] methods. The localization accuracy in theMLE, SDR, and LS-based ranging methods is typically limited: due to convergence to a local optimum inthe MLE methods, and due to the approximations introduced as part of the solution procedure in the SDRand LS-based ranging methods. Also, these methods do not generalize well for multi-slope path loss regions.Fingerprinting methods, on the other hand, deliver superior localization performance at the cost of buildinga DRSS radiomap in the service area and using a machine learning model to make the location predictions.Keeping the localization accuracy in mind, we pursue the development of fingerprinting solutions in thisthesis.Existing fingerprinting solutions on DRSS localization ( [65–71] and references therein) have mainlyfocused on cooperative scenarios, where the device is part of the system. Moreover, the available solutionsdo not provide accurate confidence intervals around the location estimates. The localization is performed onthe downlink, in which the device measures the DRSS from multiple transmitters and estimates its own loca-tion. In addition, popular solutions such as the k-nearest neighbours [71], do not provide accurate confidenceintervals around the location estimates and their performance is sensitive to the density and distribution ofthe training locations. These algorithms are therefore not suitable for use in RF surveillance systems. Weneed to develop fingerprinting algorithms that operate on the uplink and can passively estimate the intruderdevice location without any cooperation from the device. The developed algorithms also need to report ac-curate confidence intervals around each location estimate, in order to facilitate device recovery. Moreover,9to reduce labor costs, the developed algorithms need to make location predictions in the continuous space,without having to rely on a dense grid of training points. Keeping this requirements in mind, in Chapter 3and 4 of the thesis, we propose fingerprinting solutions based on the Gaussian Process regression (GP) [98]framework. The proposed algorithms not only estimate the intruder device location, but also provide closed-form expressions for the associated confidence intervals. In addition, they pursue non-linear regression andtherefore do not require a dense grid of training locations to accurately localize the test devices.The improved localization accuracy in fingerprinting methods comes mainly at the expense of conduct-ing measurement campaigns to collect datapoints across the service area. Fortunately, the cost of building atraining database can be alleviated, in part, by using synthetic data generated via computer simulations. Thisis made possible mainly due to the availability of mathematical models for the RSS. For example, the previ-ous works [34–41] have conducted empirical studies and found that the RSS can be modelled as in (1.1) in avariety of wireless systems. We can generate RSS data using (1.1) and train a machine learning model whichtakes the noise-free DRSS vectors as input and provides the device location as the output. When doing so,there is naturally a mismatch between the training and test dataset because the test DRSS is inevitably noisydue to fading effects of the wireless channel. It is unclear from the existing literature whether localization ispossible at all when the training is done with noise-free DRSS, and if yes, how does the mismatch betweenthe training and test DRSS impact the localization error. In addition, when probabilistic machine learningalgorithms such as GPs are used, it is unclear how the mismatch between the training and test DRSS af-fects the confidence intervals provided by the algorithms. Besides, when we pursue the more traditionalcase of training with noisy DRSS, it is unclear whether we can utilize the synthetically generated noise-freeDRSS vectors for data reconstruction purposes, so as to achieve potential enhancements in the localizationaccuracy. We explore these research directions in Chapters 3 and 4 of the thesis.Lastly, in some service areas, it is possible to obtain statistical knowledge of the PLE. For example,the previous works [99–101] conducted empirical studies in specific urban and sub-urban environments andfound that the PLE follows a Gaussian distribution. However, existing works have not investigated DRSS-based localization when the PLE is known to be Gaussian distributed in the area. We pursue this researchdirection in Chapter 5 of the thesis. We develop low-complexity localization algorithms which make use ofthe statistical knowledge of the PLE to firstly derive maximum-likelihood estimates of the distances to thesensors and subsequently obtain the source location from them through linear least squares.We now review the prior works on the topics of intruder signal detection and device localization in more10detail. The shortcomings in the existing literature are highlighted to provide the motivation and objectivesof our research.1.3 Related Work, Motivation, and Objectives1.3.1 Intruder signal detectionTo report the presence of unexpected signals and localize them in time and/or frequency domains, fewauthors have developed changepoint detection algorithms which can localize only the frequency span or thetime span information, while others have directly studied the joint time-frequency localization problem.Changepoint detection for frequency-span or time-span localizationRecent works [103–106] have studied the problem of signal detection and frequency span localizationthrough changepoint detection. To report the occupied frequencies over a predefined time period, an energy-based thresholding was pursued in [103,104], an energy-based statistical significance testing procedure waspursued in [105], and a more involved multi-scale wavelet edge detection method was pursued in [106].These methods require additional work to simultaneously report the time span of each signal. For example,the same changepoint detection algorithms need to be run on each detected frequency to obtain the timespan. The results over multiple frequencies need to be grouped into disjoint signals. In addition, due to thenature of the wireless channel, the energy-based methods [103,105,106] do not achieve robust performanceunder noisy and fading conditions, whereas the wavelet method [106] requires handcrafted hyperparameterswhich do not generalize well beyond a handful of captures with varied noise and interference types. Also,none of these works study RFI separation, so additional classification work is required to filter out the inter-ference. The work [104] classifies the detected signals into different wireless technologies, but it requiresprior knowledge of the features exhibited by the expected signals.More generally, the works [107,112–114] have studied statistical feature-based changepoint detection ingeneral-purpose sequential data. These methods are more tolerant to noise and interference when comparedto energy-based or wavelet-based edge detection methods. Although, similar to energy-based thresholdingmethods, a certain amount of post-processing is required in order to extend them for the time-frequencylocalization task. For example, we can study one STFT time-bin (frequency-bin) at a time and detect allthe frequency (time) changepoints. To achieve the time-frequency localization, we need to then develop a11multi-time-bin (multi-frequency-bin) changepoint combining algorithm to combine the changepoints overmultiple time-bins (frequency-bins) and a signal classification algorithm to separate out the RFI from thedetected signals. Such a per-time-bin (per-frequency-bin) STFT analysis is not efficient because each timebin is analyzed independently of each other and so, the analysis does not make use of the knowledge ofwhether the adjacent time bins contain a signal or not. Also, separate hyperparameter tuning is required foreach data sequence (see Appendix A.1 for an example illustration).Joint time-frequency localization in spectrogramsOnly a limited number of works have studied the problem of joint signal detection and time-frequency lo-calization in spectrograms. The works [5–7] have employed simple border detection methods based ondigital signal processing to obtain the bounding boxes. Different sequences of morphological image pro-cessing operations are proposed in these works to localize the signals in an interference-free environment.When compared to [5] and [6], the work [7] offers improved signal detection with fewer number of hyper-parameters. Numerical experiments however reveal that the detection performance of [7] is poor on datasetscontaining more than a handful of RF captures (c.f. Section 2.3.4 and Fig. 2.7). The same is the case withother morphological processing methods [5,6]. This is because the morphological processing, by its nature,suffers from limited hyperparameter generalizability, i.e., the hyperparameters which work well for a givenRF capture do not work well for other captures experiencing different noise/interference levels.Since the major concern is the limited hyperparameter generalizability across captures with differentnoise and interference levels, a natural solution is to devise a supervised machine learning approach, inwhich a model is trained to detect the signals of interest in the presence of varied levels of noise and inter-ference. Unfortunately, previous works have not explored this research direction. Deep learning methodsare the state-of-the-art for supervised machine learning but we need to firstly verify if the signal detectiontask can be achieved using simpler and more traditional machine learning methods. For the purpose of ex-position, in Chapter 2, we considered a support vector machine (SVM) classification method [115] whichuses the histogram of gradient (HOG) features for object detection. The SVM-HOG method, which is quitepopular in the computer vision community for object detection in daily-life images, is adopted for the signaldetection task in the spectrogram matrices. We notice through experiments on example RF traces (Section2.3.4 and 2.3.5) that the SVM-HOG method achieves a detection performance which is much better thanthe morphological processing method [7] but is still worse when compared to the deep learning method12proposed in Chapter 2. In addition, there are no clear guidelines available for the hyperparameter tuning.An exhaustive grid search is not only sub-optimal but also computationally prohibitive because the train-ing HOG feature set needs to be regenerated for each hyperparameter combination. Recent advances indeep learning for time-series and matrix analysis [8] allow us to extract rich features out of RF data andutilize convolutional neural networks (CNNs) for improved and generalizeable signal detection. However,this research direction has not been explored in detail in the existing literature on joint signal detection andtime-frequency localization.From among the several CNN models available in the computer vision literature, for the signal detectiontask in this thesis, we have chosen to work with the Faster region-based CNN (RCNN) [8] frameworkbecause it was the state-of-the-art at the time of initiating our project on deep learning for signal detection.More advanced frameworks, such as you-only-look-once (YOLO) [9] and Efficient-Det [10], have emergedrecently and they are known to provide better performance on general-purpose object detection datasetssuch as ImageNet [11] and COCO [12]. A similar improvement in performance may be expected whenthese frameworks are adopted for the signal detection task.To alleviate the shortcomings in the existing literature on intruder signal detection and time-frequencylocalization, we pursue the following research objectives in the thesis:(i) Explore the use of deep learning on spectrogram matrices to achieve the joint signal detection andtime-frequency localization, and compare the performance with standard morphological processingand support vector machine algorithms.(ii) Provide insights on how the Faster RCNN [8], which is a standard deep learning archiecture for objectdetection in computer vision, can be adopted for the signal detection task. Expose the various com-ponents within the Faster RCNN architecture that can be engineered for performance improvement.(iii) Conduct experiments on real RF traces, captured using a software-defined radio device [117]. Developmethods that allow us to use a model trained on synthetically generated data for signal detection onthe real RF traces.The contributions made in Chapter 2 address the above objectives. Details are presented at the beginning ofthe chapter.131.3.2 Intruder device localizationWe now review the existing literature on intruder device localization and motivate the objectives of ourresearch in Chapters 3-5 of the thesis.The works [35, 48–50, 78] and references therein study source localization under perfect knowledgeof the transmission power. In the case of an intruder source, the transmit power is unknown. Even ifthe intruder device specifications are reverse-engineered, for example, based on inference from the signalpatterns exhibited by the intruder, the actual transmit power can vary across the devices due to differentantenna gain and/or hardware component losses and variations in the battery power [42–45]. The works[51, 52, 121] have investigated the joint estimation of the transmit power alongside the source location. Insuch methods, inaccurate estimation of the source transmit power leads to further inaccuracies in the positionestimation. With this perspective, few works such as [53] have circumvented estimating the transmit powerand have instead attempted localization by operating on the ratio of RSS values with respect to a referenceanchor. In [65], the authors propose a fingerprinting algorithm using the hyperbolic location fingerprint(HLF), defined as the normalized logarithm of RSS ratios between all access point pairs, for the localizationtask. The HLF fingerprint is experimentally motivated to handle hardware imperfections and it does notperform well when the devices have different transmit powers. The works [82–86, 125–134, 138, 140–148,153] propose machine learning algorithms which directly use the RSS as the fingerprint for the localizationand therefore perform poorly when they encounter differences in the transmit power. The more recentworks [54–58] have resorted to DRSS-based localization, wherein the differences of RSS measurementswith respect to a reference sensor node are utilized for the localization task. Note from (1.1) that the transmitpower term 10log10(ρg0) is eliminated when we work with the difference of RSS with respect to a referencenode. The ratio and DRSS approaches are essentially equivalent, with the latter being the more popular onedue to the convenience of working in dB units. In this thesis, we utilize DRSS values for the localizationtask, in order to handle the unknown transmit power and the heterogeneity of the intruder devices.Existing works on DRSS-based localization can be broadly categorized into four types: maximum-likelihood estimation (MLE) [55,59,60] , semi-definite relaxation (SDR) methods [56,61–63], least squares(LS)-based ranging [55, 57, 58, 64], and fingerprinting [65–71] methods. The MLE methods formulate anoptimization problem to maximize the joint likelihood of the RSS values, with the device location as theoptimization variable. Since the problem is highly non-convex, an appropriate initialization is required to14ensure convergence to the global optimum. The SDR methods, on the other hand, focus on convexifying themaximum-likelihood objective through constraint relaxation in order to obtain sub-optimal solutions. Whenthe relaxation is sufficiently tight, the solution from semi-definite relaxation is generally an approximateestimate of the maximum-likelihood solution, but the resulting computational complexity is still very high.As a low-cost alternative, the LS-based ranging methods take a two-step approach of firstly estimating thedistances to each sensor from the measured RSS, followed by multilaterating on the distance estimates toderive a linear system of equations in terms of the source location. Linear least squares is then employedto obtain the location estimate. LS-based ranging methods are computationally simple to implement, buthave the disadvantage that the errors from the distance estimation propagate into the localization phase,thus adversely affecting their performance when the shadowing noise is large. As may be noted from theabove discussion, the localization accuracy in the MLE, SDR, and LS-based ranging methods is typicallylimited. This is due to the choice of the starting point in the MLE methods, and due to the approximationsintroduced as part of the solution procedure in the SDR and LS-based ranging methods. Moreover, thesemethods typically involve multiple matrix inversions, and can therefore result in numerical noise wheneverthe matrices to be inverted have low condition number. To overcome these limitations and achieve improvedlocalization accuracy, machine learning solutions based on fingerprinting have emerged. In these methods,a radiomap of DRSS fingerprints is firstly built through measurement campaigns in the service area. Thetest device location is then determined through machine learning techniques which match the measuredfingerprint to the existing DRSS radiomap. Such data-driven approaches achieve improved localizationaccuracy because they have access to additional information in the form of a radiomap of fingerprints in theservice area. The MLE, SDR, and LS-based ranging methods, on the other hand, operate on the test DRSSalone.A review of the most recent fingerprinting solutions for DRSS-based localization is presented in [71].The most popular fingerprinting approach is to build a radiomap containing a dense grid of training pointsand determine the test device location as the point which is closest in the DRSS space to the test input[66–71]. A rank-based fingerprinting solution is proposed in [66], where the authors assign scores forthe test fingerprints when matching with the training fingerprint database. The location with the highestmatching score in the training radio map is returned. In the works [67, 68], a discrete grid of radiomappoints is considered in the service area and the test location is determined as the point on the grid thathas the maximum likelihood of representing the test DRSS. In other words, the test device is classified to15be at the location lj , if P(DRSStest |lj) > P(DRSStest |li), for all locations li in the training radiomap. Thelikelihood function P(DRSStest |lj) is estimated from the training dataset using kernel or histogram methods.In [69–71], the k nearest neighbours (kNN) algorithm is used to achieve improved localization performanceover the maximum-likelihood methods in [67, 68]. In these works, the test device location is determinedas the average of the nearest neighbours in the DRSS space as per the Euclidean distance metric. In [67],the DRSS vectors comprise of the RSS differences between all possible sensor node pairs. In [70, 71], theDRSS vectors contain the difference between the RSS at each sensor node and the mean RSS across thesensor nodes. The localization performance of the above algorithms is sensitive to the existence of a densegrid of DRSS vectors in the service area [135, 136], thus making them most suited for indoor positioningwhere extensive measurement surveys are possible.We notice that the existing fingerprinting solutions for DRSS-based localization restrict the test devicepredictions to a discrete space defined by the training locations. Also, the focus is on obtaining the locationestimates alone, and the proposed algorithms do not provide the confidence intervals around the locationestimates. The existing algorithms also assume cooperative localization on the downlink, where the devicesparticipate in the localization procedure.In the thesis, we develop probabilistic machine learning algorithms which pursue non-linear regressionand directly learn the function that maps the DRSS vectors to the device locations. This allows us to makepredictions in the continuous space, without restricting ourselves to the discrete space defined by the traininglocations. We work with GPs [98] for the non-linear regression because GP methods provide us with closed-form expressions for the estimated device locations and also the associated 2σ error-bars. Moreover, GPmethods are non-parametric in nature and the hyperparameters can be obtained in a straightforward mannerby optimizing the marginal likelihood of the training observations. GP methods have been used in the pastfor time-series analysis [155–159] and wireless channel prediction [160–163]. However, to the best of ourknowledge, there are no existing works on the use of GPs for uplink DRSS-based localization in a non-cooperative setting such as the intruder device positioning. We cannot directly use the conventional GPmethod [98] for the intruder device localization task because it treats the DRSS vectors as deterministicinputs. In the thesis, we show that it provides inaccurate 2σ error bars around the location estimates as aresult. To address this issue, we need to take the stochastic nature of the test DRSS vectors into accountwhen deriving the posterior density of the test locations. We pursue this research direction in Chapters 3 and4 of the thesis.16A challenge with the use of machine learning for the localization task is the cost of conducting mea-surement campaigns to build the training database. Depending on the number of training points requiredfor the localization, the task can become labor-intensive and is therefore a bottleneck for the practicalityand scalability of machine learning-based solutions. To reduce the calibration effort and costs of buildingthe training database, the works [138, 139, 146, 147, 150–154] have constructed prediction models for theRSS information, in order to be able to synthetically augment the training database from a small amount ofmeasured data. The existing works have however not explored the limiting case of using a noise-free RSSor DRSS database for training the machine learning models for the localization task. Training on the noise-free DRSS data can be beneficial when the shadowing noise variance in the test DRSS is small. Besides,a database containing synthetic noise-free DRSS vectors can be used for data reconstruction purposes. Forexample, we can reconstruct both the training and test DRSS vectors from a principal subspace spanned bythe noise-free DRSS vectors and investigate whether the reconstruction step facilitates a reduction in thelocalization error. We explore this research direction in Chapter 4 of the thesis.Lastly, in service areas where the statistical properties of the PLE are available, it is unclear from theexisting literature if this information can be utilized to develop low-complexity localization algorithms. Inthe recent work [102], the authors assumed that the PLE values are equally likely and have modelled the PLEas a uniform random variable in a predefined range. In areas where measurement campaigns are possible,more accurate statistical models can be developed for the PLE. For example, the authors in [99–101] foundthat the PLE is Gaussian distributed in specific urban and sub-urban environments. For such areas wherethe PLE is known to be Gaussian distributed, it is possible to develop linear least squares algorithms whichmake use of the statistical knowledge of the PLE when deriving the location estimates. We explore thisdirection in Chapter 5 of the thesis.In summary, to alleviate the shortcomings in the existing literature on intruder device localization fromthe DRSS information, we pursue the following research objectives in the thesis:(i) Employ Gaussian process regression to estimate the intruder device location from the uplink DRSSvectors. When predicting the test device location, take the statistical distribution of the DRSS vectorsinto account and derive more accurate 2σ error bars on the location estimates than those given by theconventional GP method.(ii) Apply principal component analysis and investigate the dimensionality of the noise-free DRSS vec-17tors.(iii) Explore the use of noise-free DRSS for the intruder localization task in the following two ways:– Train the GP models with noise-free DRSS and derive both the location estimates and the as-sociated confidence intervals in closed-form. Reconstruct the test DRSS vectors from a low-dimensional prinicipal subspace of the noise-free training DRSS and report the localization rootmean squared error (RMSE) performance.– Train the GP models with noisy DRSS and utilize the noise-free DRSS dataset to reconstructboth the training and test DRSS vectors. Investigate the performance benefit of the data recon-struction step in terms of the achieved localization RMSE performance.(iv) Develop linear least squares algorithms for DRSS-based localization for service areas where the PLEis known to follow a Gaussian distribution in the area of interest.The contributions made in Chapter 3 address the objectives (i) and (iii). The contributions in Chapter 4address the objectives (ii) and (iii). The contributions in Chapter 5 address the objective (iv). Details arepresented at the beginning of the respective chapters. The remainder of the thesis is organized as follows.1.4 Thesis OutlineIn Chapter 2, we propose a deep learning algorithm to detect the presence of a wireless intruder devicefrom the time-series data captured by wideband RF sensors. An example setup is considered, where theintruder is a drone transmitting Wi-Fi signals and there exists RF interference from Bluetooth and microwaveoven signals. We demonstrate that conventional energy thresholding algorithms such as [7] do not providea scalable performance beyond a few RF captures. Similar is the case with the support vector machine(SVM) classification algorithm in [115], which uses the histogram of ordered gradient (HOG) features todetermine whether a sliding window contains Wi-Fi signals or not. To address this concern, we developa deep learning solution based on the Faster RCNN architecture [8], which employs convolutional neuralnetworks to extract and analyze more advanced features than the HOG for the signal detection task. Wepropose a data generation framework for the task at hand and create a synthetic dataset to evaluate the signaldetection performance. Experiments on both synthetic and real RF traces reveal that the deep learning18approach achieves superior performance when compared to the energy thresholding and support vectormachine algorithms in terms of the mean average precision metric.In Chapter 3, we propose machine learning algorithms to estimate the intruder device location from theDRSS information measured at the wireless sensor nodes in the service area. We pursue probabilistic ma-chine learning using the GP framework in order to derive both the location estimates and the associated 2σerror-bars in closed-form. The conventional GP method treats the test DRSS vectors as deterministic inputsand provides optimistic 2σ error-bars on the location estimates. To address this concern, we propose twoGP methods, namely the numerical approximation GP (NaGP) and the Gaussian approximation GP (GaGP)methods, which take the stochastic nature of the test DRSS into account when deriving the true posteriordistribution of the test locations. In the two methods, the posterior distribution is approximated as a Gaussiandistribution using numerical and analytical moment-matching respectively. Numerical studies reveal that theproposed NaGP and GaGP methods provide more realistic 2σ error-bars than the conventional GP method,both when training with noise-free and noisy DRSS vectors. They achieve improved localization RMSEperformance when compared to the existing maximum-likelihood and semidefinite relaxation methods. Theproposed GP methods also serve as a cost-efficient alternative to the standard kNN fingerprint-matchingalgorithm, in that they do not require a dense grid of training locations to achieve similar performance.In Chapter 4, we apply principal component analysis on the noise-free DRSS vectors and demonstratethe low-dimensionality of the subspace spanned by these vectors. We then propose two GP methods, namelythe reconstruction GP (RecGP) and reconstruction Gaussian approximation GP (RecGaGP), which recon-struct the noisy training and test DRSS data from a principal subspace spanned by the noise-free DRSSvectors. When training with noise-free DRSS, the reconstruction step is applied to the test DRSS alone. Onthe other hand, when training with noisy DRSS vectors, both the training and test DRSS are reconstructedprior to deriving the posterior distribution of the test locations. The RecGP method treats the reconstructedtest DRSS as deterministic inputs and provides optimistic 2σ error-bars on the estimated locations. TheRecGaGP method addresses this concern by treating the reconstructed test DRSS as stochastic inputs. Thetrue posterior predictive distribution is derived and analytically approximated as a Gaussian distributionwith the same first and second order moments. Numerical studies reveal that the data reconstruction step inthe RecGP and RecGaGP methods allows them to achieve improved localization RMSE performance whencompared to the conventional GP, NaGP, and GaGP methods. The RecGaGP method also provides morerealistic 2σ error-bars on the estimated locations when compared to the RecGP method.19In Chapter 5, we propose linear least squares algorithms to estimate the intruder device location from theDRSS information when the PLE in the service area is known to follow a Gaussian distribution. We derivemaximum-likelihood estimates of the ratio-of-distances with respect to the reference sensor. Using theratio-of-distance estimates, we then formulate a linear system of equations in the device location and solveit using constrained linear least squares (CLLS) and an approximate two-step linear least squares (TLLS)algorithms. Numerical examples reveal that the TLLS and CLLS estimators provide reasonably accuratelocation estimates under small noise variance. They also serve as a warm start for achieving improvedlocalization accuracy when recursive estimation is pursued in the maximum a posteriori (MAP) estimator ofthe intruder location. The achieved localization performance is also close to the hybrid Cramer-Rao lowerbound.In Chapter 6, we summarize the contributions of the thesis and provide concluding remarks, along withpossible future research directions on the topic of the thesis.20Chapter 2A Deep Learning Framework for SignalDetection and Time-Frequency Localizationin Wideband RF SystemsIn the current chapter, we propose a wideband spectrum sensing technique to detect and localize the timeand frequency span of wireless radio frequency (RF) signals from an intruder device when friendly devicesoperating in the same spectrum cause RF interference (RFI). Spectrograms are used for the joint time-frequency representation of the wideband RF data. We note that traditional energy-based signal detectionmethods such as morphological processing [7] require handcrafted parameters for each spectrogram becausethese methods are very sensitive to variations in the noise and interference. To address this concern andachieve uniform performance that scales across multiple spectrograms, we show for the first time that deeplearning-based object detection frameworks can be used. Specifically, we adopt and modify the Fasterregion-based convolutional neural network (FRCNN) [8] framework for the signal detection task and showthat deep learning solutions can achieve significantly better performance when compared to the existingalgorithms. For performance evaluation, we consider an example setup in the 2.4GHz license-free ISMband with IEEE 802.11g (Wi-Fi) signals to be detected under interference from Bluetooth and microwaveoven signals. Since a dataset is required, we propose a data generation framework which provides detailson the wideband spectrum composition and the simulation parameters that can be used for the signal andintereference generation. When the model trained purely on synthetic RF traces is used for the signaldetection on real RF traces, we encounter a drop in performance due to the inherent data mismatch. We showthat the performance drop can be alleviated using the following two methods: (i) training with a mixture ofsynthetic and real RF traces, and (ii) finetuning the synthetically-trained model with an additional round oftraining on a small number of real RF traces.21Our main contributions in the chapter are summarized below:• We identify that existing energy-based edge detection algorithms such as the morphological process-ing method [7] achieve poor performance on datasets containing more than a handful of captures be-cause they suffer from limited hyperparameter generalizability, i.e., the hyperparameters which workwell for a given RF capture do not work well for other captures. To overcome this issue and achieveuniformly good performance on a wide range of RF captures, we propose a deep learning solutionbased on the FRCNN framework.• To perform supervised learning on the signal detection task, we need a labelled dataset. To facilitatethe same, we present a data generation framework for an example setup where the signals of interestare from the IEEE 802.11g (Wi-Fi) protocol and the RFI is from the Bluetooth low energy [168] andmicrowave oven [169] signals. Experiments on a synthetic dataset reveal that the proposed deep learn-ing framework achieves a mean average precision (mAP) of up to 0.8, significantly outperforming thestandard morphological processing [7] and support vector machine classification [115] methods.• When adopting the FRCNN framework for the time-frequency localization task, multiple designchoices need to be made, for example, on the size of the framework, the type and the depth of thefeature extraction network, and the anchor sizes. We provide insights on choosing these parame-ters for the signal detection task and conduct numerical experiments to illustrate the impact of eachparameter.• For real RF traces, directly deploying the model trained on synthetic data results in performance lossdue to the inherent mismatch in the data. To alleviate the performance loss, we propose two solutions,namely data mixing and model finetuning. In data mixing, we train the model with a mixture ofsynthetic and real measurement captures. In model finetuning, the synthetically-trained model isfinetuned through an additional round of training on a small number of real RF traces. Experimentson traces captured using the NI USRP device [117] reveal that model finetuning is the more effectivesolution.The rest of the chapter is organized as follows. We proposed a general-purpose framework for the signaldetection and time-frequency localization task in Section 1.1 of Chapter 1. We use the same framework here,with the downscaled FRCNN framework for the bounding box detection in the spectrograms. In Section 2.1,22Figure 2.1: The downscaled FRCNN model. For each spectrogram chunk, the base network yields a feature map which is fed intothe region proposal network (RPN) and the Detector network. The RPN and the Detector host a bunch of feature extraction (FE)layers, the outputs of which are used for classification and regression to yield bounding boxes.we present an overview of the downscaled FRCNN architecture, with insights on the design choices that wemake for the signal detection task. A data generation framework is then presented in Section 2.2, followedby numerical evaluations in Section 2.3 and concluding remarks in Section 2.4.2.1 A Downscaled Faster RCNN Framework for Signal Detection andTime-Frequency LocalizationThe downscaled FRCNN framework is composed of three major modules, as illustrated in Fig 2.1. The firstmodule, which is the base network (BN), takes the 2D matrix as the input (spectrogram in our case), extractsfeatures that are relevant to the signal detection task and outputs a downscaled feature matrix. The secondmodule, which is the region proposal network (RPN), takes as input the down-scaled feature matrix, a set ofanchors and the ground truths (GTs). The RPN firstly performs feature extraction (FE) on the input featuremap using a 2D convolution layer containing 128 filters of size 3× 3. On each pixel of the output featuremap, the RPN positions the anchors and performs a classification and a regression task to output regionproposals, which are the likely candidate regions for the signals to be present, along with the probabilitiesof signal presence. The region proposals from the RPN, along with the feature map from the BN, are fedinto the Detector network, which is the third module. The Detector network performs feature extraction on23the region proposals using two densely-connected neural network layers, each containing 2048 filters of size1×1. The Detector then assigns object class labels to each region proposal and also performs regression onthe positive region proposals to tighten them into bounding boxes. Also provided are the probabilities withwhich the assigned labels are true. When spectrograms are provided as input, the FRCNN can be thoughtof as a single unified framework for detecting and localizing the bounding boxes for the signals of interest.Next, we present design insights on tailoring each module within the downscaled FRCNN model for thesignal detection and localization task.2.1.1 Spectrogram InputsAs inputs, we need to generate spectrograms such that the time and frequency resolution is sufficient tocapture the time-frequency span of the signals of interest. Also, care should be taken to ensure that thesize of the input matrix is not too large. When we increase the size of the input matrix to the model,each pass of training takes longer time to complete because of the larger input feature maps, resulting inlonger convergence times. In addition, the base network that extracts the features has a particular receptivefield depending on the choice of the base network, for example, the VGG-13 and RESNET-50. When thereceptive fields are large when compared to the size of the signals to be detected, it is difficult for the basenetwork to make sense of the features to be extracted because the input would mostly contain backgroundnoise and only a small portion is occupied by the signals of interest. In light of these two observations,we restrict the length of our input matrix to 600 pixels in time. This is achieved by chopping the inputspectrogram into chunks of 600 time-pixels each and working with the relative positions of the ground truths.Doing so ensures that the size of the signals of interest is comparable to the size of the input spectrogramchunk.2.1.2 The Base NetworkThe base network (BN) in the FRCNN model handles the crucial task of feature extraction and needs to bechosen based on the input format and the type of signals to be detected. Typically, the BN is a convolutionalneural network (CNN) whose depth depends on the complexity of features to be extracted. Standard featureextraction models in computer vision include VGG-13 [165] and RESNET-50 [170], among others. It isunclear whether these standard models can extract necessary features for the signal detection in RF datasets.24We therefore conduct numerical experiments in Section 2.3 to choose an appropriate BN.2.1.3 Region Proposal NetworkIn the RPN, the feature map obtained from the BN is passed through a 2D convolution layer, which contains128 filters of size 3× 3, to obtain a low-dimensional feature map. The RPN positions the anchors on eachpixel in the input feature map to create raw region proposals. The low-dimensional feature map, along withthe raw region proposals created from the anchors, are fed into two 2D convolution layers to perform aclassification and a regression task respectively. The classification task assigns probabilistic labels to eachraw proposal as positive or negative, to denote whether the proposal contains the signal of interest or not.For proposals that are deemed positive, the regression task fine-tunes the size to suit the dimensions of thesignal.To train the RPN, if any given anchor has an intersection over union (IoU) greater than a certain thresholdwith the ground truth, namely the RPN max overlap, we treat that anchor as a positive proposal. Similarly,if the IoU is smaller than a certain threshold, namely the RPN min overlap, we treat the anchor as a negativeregion proposal. For pixels that are on or close to the feature map boundary, we clip the anchors to limit theirspan to the matrix size. The clipped sizes are used for the IoU calculations. On the other hand, ground truthswhich are on the feature map boundary are ignored from the training procedure because we experiencedconvergence issues when they were included. This is not a major concern though, because we can alwaysdetect signals that have been abruptly cut at the boundary of a spectrogram chunk by feeding an overlappingspectrogram chunk to the FRCNN model and combining the results through simple post-processing. Theclassification layer is then trained to separate positive and negative proposals. The regression layer tightensthe positive proposals to bound the ground truths. Since many of the positive proposals would have a highoverlap with each other, a non-max suppression (NMS) operation is employed.2.1.4 AnchorsThe anchors serve as raw region proposals for the RPN. We may therefore choose the anchor sizes andaspect ratios to match the dimensions of the signals of interest. For example, if we are interested in the IEEE802.11g (Wi-Fi) protocol, we know that the signals have a bandwidth of 20 MHz and duration from about0.05 to 15 milliseconds [164]. We may therefore choose multiple anchors, each with a frequency span of 2025MHz and a time-span chosen uniformly randomly in the range [0.05,15] ms.2.1.5 Detector NetworkThe RoIs from the RPN may have varied sizes, depending on our choice of the anchors and the output ofthe regression. An RoI convolution pooling operation [8] is performed to convert the RoIs into a fixed size,typically 7×7, so as to be able to use convolution layers for further classification and regression. After theRoI convolutional pooling, the fixed size RoIs are fed into two densely-connected layers, each containing2048 filters of size 3×3, to convert them into low-dimensional feature vectors. These feature vectors are theninput to two densely-connected layers to perform a classification and a regression task. The classificationassigns probabilistic positive and negative labels to each RoI, depending on the presence of the signal. Theregression fine-tunes each positive-labelled RoI to match the signal dimensions and outputs the four cornercoordinates of the regressed RoI. When training, any RoI whose IoU with the ground truth is greater thanDetector max overlap is a positive target. If the IoU is less than the Detector max overlap but greater thanDetector min overlap, the RoI is a negative target.2.1.6 Downscaling the default FRCNNThe default FRCNN model hosts (i) one 2D convolution layer of 256 3×3 filters for feature extraction withinthe RPN, and (ii) two densely-connected neural network layers of 4096 1×1 filters each for feature extractionwithin the Detector. While these sizes were configured for general-purpose object detection in the ImageNetdataset, it is unclear whether such large feature extraction layers are required for the signal detection andlocalization task at hand. The signal detection task requires detecting signals of a much smaller variety thanin general-purpose object detection. Also, unlike in ImageNet, the input matrices for signal detection arepredominantly occupied by different types of background noise. In addition, the numerical range of valuesis much smaller in spectrograms and the numerical fluctuations within the signal are much more rapid andpredominant (c.f. Fig. 1.3(d) for example). This follows from the nature of how the wireless signals aregenerated and transmitted. We therefore conduct numerical experiments on the default FRCNN architecturein Section 2.3 with downscaling factors of 2,4, and 8 for the RPN and the Detector. Downscaling by a factorof 2 is seen to give the best performance, thus leading us to choose 128−filter feature extraction layer in theRPN and 2048−filter feature extraction layers in the Detector.262.1.7 RPN and Detector ThresholdsThe RPN max overlap needs to be chosen such that at least a few anchors per spectrogram have a highenough IoU with the ground-truth to be considered as positive targets for the RPN training. Typically, theRPN max overlap is chosen to be greater than or equal to 0.5 because an IoU of 0.5 or more gives usconfidence that the anchor indeed contains the signal of interest. Similarly, the RPN min overlap needs tobe chosen such that at least a few anchors per spectrogram have a low enough IoU with ground-truth to beconsidered as negative targets for the RPN training. Typically, the RPN min overlap needs to be less than0.5 because an IoU less than 0.5 denotes that the anchor may not contain the signal of interest. Higher RPNmax overlap and lower RPN min overlap values would deem fewer anchors as positive and negative targetsrespectively.The Detector max overlap needs to be chosen so that at least a few RoIs per spectrogram have a highenough IoU with the ground-truth to be considered as positive targets for training the Detector. Typically,the Detector max overlap is chosen to be at least 0.5 because an IoU of 0.5 or more gives us confidencethat the RoI indeed contains the signal of interest. On a similar note, the Detector min overlap needs to bechosen such that at least a few RoIs per spectrogram have an IoU value between the Detector max and minoverlap to be considered as negative targets for the training. Typically, the Detector min overlap needs tobe less than 0.5 because an IoU less than 0.5 denotes that the RoI may not contain the signal of interest.The higher the Detector max overlap, the fewer the number of positive target RoIs. Similarly, the smallerthe difference between the Detector max and min overlap, the fewer the number of negative target RoIs. InSection 2.3, we attempt a simple one-pass search algorithm to choose the RPN and Detector min and maxoverlap values.2.1.8 Training procedure and complexityFor the classification and regression tasks in both the RPN and the Detector networks, we use the standardweighted-sum multi-task loss function proposed in [8]. The loss function is a weighted sum of a binarycross-entropy loss term for classification and a robust loss term for regression. Similar to [8], we employthe approximate joint training method to train our system. To prevent overfitting, we implement the earlystopping criterion where the mult-task loss threshold is set to 0.1, and also set the maximum number oftraining epochs to 10. Due to the number of parameters involved, it is difficult to quantify the complexity27of CNN models in terms of the Big-O notation. An abstract yet commonplace alternative is to report theinference times on test data. The Faster RCNN architecture with VGG-13 as the base network processesabout 5 frames per second (fps) on an NVIDIA Tesla K40 GPU. It can therefore perform real-time signaldetection.2.1.9 DatasetTo perform supervised learning for the Wi-Fi signal detection, we need a dataset comprising a variety oftime-series RF captures measured in the ISM 2.4 GHz wideband spectrum. Each RF capture needs to containthe RFI experienced in the ISM band and needs to be accurately labeled for the Wi-Fi signals. Training withreal RF traces is obviously the most desired option because it allows us to build a tailor-made model forfield deployments, with the added convenience of not having to explicitly build mathematical simulationmodels for the noise and interference. However, a major challenge with training on real RF traces is the dataavailability. Extensive measurement campaigns need to be conducted in order to capture a sufficiently widevariety of RF traces. If we do not capture enough data and instead choose to train with a small amount ofreal data, the model would simply overfit and provide poor test performance. In the event that one is able tocollect a large amount (and variety) of real RF traces, an additional challenge is to accurately establish theground truths and label them. Standard packet demodulation procedures are non-ideal and their performanceis sensitive to the received SNR [37]. Missed or incorrect labels would confuse the model during the trainingand therefore result in poor test performance. Manual labeling is an alternative solution but is not scalable.Consequently, in order to pursue supervised learning, we train with synthetic traces.Training with synthetic data, although more of a requirement than a choice we make, offers a variety ofbenefits. Firstly, synthetic traces allow us to confidently establish the ground truths. The ability to establishground truths empowers us to create large amounts of accurately labelled data, which is a fundamentalrequirement for supervised learning. Secondly, by using synthetic traces for training, one can expose themodel to a wider variety of noise, interference, and fading types that are difficult in general to capturethrough a finite amount of field measurements. Lastly, one may note that the synthetic traces are generatedusing simulation models which were originally developed from real measurement studies. Consequently,training with synthetic data tailors the model to the task at hand, albeit with a certain data mismatch thatneeds to be overcome prior to field deployment. For on-field deployments, we consider two solutions toovercome the data mismatch between synthetic and real RF traces: (i) data mixing, wherein, we train the28model with a mixture of synthetic and real traces, and (ii) model finetuning, wherein, we further train thesynthetically-trained model on a small amount of real measurements.In the next section, we present a detailed data generation framework to create synthetic RF capturescontaining Wi-Fi signals in the 2.4GHz ISM band. For the RFI, we consider the ubiquitous Bluetooth lowenergy [168] and microwave oven [169] signals. Multiple fading and coloured noise types are considered forthe signal degradation. While the basic simulation models for Wi-Fi, Bluetooth, and microwave oven signalsare available online, there is no available framework which connects these models together in a meaningfulmanner, with appropriate numerical values for the variables involved, for example, the transmission pow-ers, narrowband frequencies, inter-packet delay times, frequency hopping indices, and the microwave ovenproperties. The proposed framework bridges this gap.2.2 Data Generation Framework for Wi-Fi Signals in the 2.4GHz ISMBandWe consider RF transmissions in the 2.4 GHz ISM band as per the IEEE 802.11g (Wi-Fi) protocol [164] andgenerate the time-series data synthetically using MATLAB WLAN toolbox. Each generated RF capture iscentered in the 2.4 GHz range, has a wideband bandwidth of 56 MHz and spans a duration of 90 ms. Sincethe channels 1,6, and 11 are the most popular and each channel spans 20 MHz, depending on the centerfrequency, the capture may contain one or two channels of Wi-Fi transmissions. On an average, each RFcapture contains about 4.5 Wi-Fi signal packets, with the average packet spacing obtained upon observingreal Wi-Fi captures from a NI USRP-2901 device [117]. We also randomly add radio frequency interference(RFI) comprising Bluetooth low energy (BLE) and microwave oven (MO) signals. All the signals, whetherWi-Fi or the RFI, are subject to distance-dependent signal attenuation, small-scale fading, and additivewhite/colored noise. The small-scale fading effects are simulated using the MATLAB LTE toolbox to beone among the tap-delay line implementation of the SISO Extended ITU outdoor to indoor and pedestrianmodel or the SISO Extended typical urban model [172]. The additive noise is randomly chosen to be oneamong the white, pink, red, and purple types and is generated using the MATLAB DSP System toolbox.Details on the data generation are presented in Algorithm 2.1. As inputs to the Algorithm 1, we providefs = 56 MHz, l = 90 ms, Nc = 200 captures, PWi-Fi = −10 dB, PBLE ∈ {−30,−26,−10} dB, PMO = −10 dB,dWi-Fi ∈ U[5,50] m1, dBLE ∈ U[2,30] m, and dMO ∈ U[2,30] m. The 200 captures amount to a total of 88529Algorithm 2.1 RF data generation with Wi-Fi as the signals of interest, under interference from Bluetoothlow energy (BLE) and microwave oven (MO) signals.Require: Sampling rate fs, wideband capture length l, number of wideband captures Nc.Require: Transmit powers and distances of the Wi-Fi source (PWiFi, dWiFi), BLE source (PBLE, dBLE), andmicrowave oven (PMO, dMO).1: Choose a center frequency for the wideband RF capturefc (GHz)← 2.412+5e6∗UI[0, 10]2: Choose one or more of the channels {1,6,11} for Wi-Fi transmissions3: for each Wi-Fi channel, until the capture length is l, do4: generate Wi-Fi packets as per IEEE 802.11g [164] with power PWiFi and inter-packet delay times∈ U[0.001,30]ms5: record the start and end points in time and frequency for each packet6: end for7: Generate and add BLE packets as per Bluetooth 5 [168] with power PBLE by randomly:• alternating between advertising channel and data channel types,• alternating between the LE1M and LE2M physical layer modes• introducing delays ∈ U[0.1,1] ms between successive packets, and• frequency-hopping among the available channels (79 channels are available for LE1M and 40 forLE2M respectively).8: Generate and add MO signal bursts with power PMO, following [169], with• voltage frequency fAC = 60Hz,• signal inter-arrival time parameters α1 = −72 and α0 = 180• magnetron on-state times TM1 ∈ U[1.4,1.9] ms and TM2 ∈ U[0.3,0.9] ms, and magnetronfrequency-drift times TFD ∈ U[4,4.25] ms.9: Apply distance-based signal attenuation, small-scale fading, and additive white/colored noise to all ofthe Wi-Fi, BLE, and MO signals.10: return Wideband RF captures with recorded time-frequency labels for the Wi-Fi signalsWi-Fi signals. For this (PWi-Fi,dWi-Fi) combination, the received signal-to-noise ratio spans the range [15,50]dB. From the total dataset, 150 captures are used for training and the remaining 50 for test purposes. Forthe purpose of generalizability, the following composition is used when creating both the training and testdatasets: 16% contain Wi-Fi only, 16% Wi-Fi with BLE RFI, 16% Wi-Fi with MO RFI, and the remaining52% Wi-Fi with BLE and MO RFI.1The notationsU[x, y] andUI[x, y] refer to uniformly random values and uniformly random integers in [x, y] respectively.300 200 400 600 800 1000Time steps0123456Multi-task loss RPNDetector(a) Training loss convergence(b) RPN Targets (c) Detector TargetsFigure 2.2: (a) Training loss convergence for the classification and regression tasks in the RPN and the Detector networks. (b)-(c)Example positive targets for the RPN and the Detector networks. The rectangular boxes are the targets, while the signals underneaththe boxes are the ground truths. To illustrate noise and RFI, we consider colored noise and a signal each from Bluetooth (right ofthe targets) and microwave oven (left of the targets).2.3 Numerical StudiesWe now present numerical studies on the performance of the downscaled FRCNN model for the signaldetection and time-frequency localization task. Firstly, we provide details on the training and test datasets,the spectrogram generation, the numerical thresholds chosen for the FRCNN model, and the metric forperformance evaluation.312.3.1 Spectrogram generation and numerical choices for the downscaled FRCNNSTFT is applied on each RF capture with a Hann-type window of size 1400 with 50% overlap and anFFT size of 1024. The resulting spectrograms have 7199 time bins and 1024 frequency bins each, i.e,with a resolution of 0.0125 ms and 54.7 kHz respectively. We chop the spectrograms into fixed chunks of600 time-bins in order to speed up the training. Each RF capture is therefore split into 12 input matricesfor the FRCNN model. When presenting example spectrograms, we focus on one input at a time, i.e,one spectrogram of time-span 7.5 ms at a time. Regarding the numerical choices for the FRCNN, weconsider Na = 3 anchors defined such that the time-axis sizes are chosen from the set {2,10,80} time-bins,to represent signal time-spans of {0.025,0.125,1} ms respectively, and the frequency-axis size is chosen tobe 366 frequency-bins, to represent a frequency-span of 20 MHz. The RPN max overlap and the Detectormax overlap are chosen from the set {0.5,0.7,0.9}. The RPN min overlap and the Detector min overlap arechosen from the set {0.1,0.3}. Also, following [8], we set the number of RoIs Nr from the NMS operationto 300, the output RoI size from the RoI pooling network to 7× 7, and the Detector positiveness thresholdto 0.5. The λ value in the multi-task loss function (c.f. [8]) is set to 1 for the classification and regression.The mini-batch size is set to one spectrogram chunk.2.3.2 Training convergenceTo verify the training convergence, we consider an example experiment with the base network set to VGG-13 initialized with pre-trained weights and configured as trainable. Each training mini-batch contains onespectrogram chunk and we set RPN min overlap to 0.3, RPN max overlap to 0.5, Detector min overlap to0.1, Detector max overlap to 0.5, and the number of training epochs to 10. Stochastic gradient descent isused to optimize both the RPN and the Detector networks, with a learning rate of 10−5, zero momentum andzero learning rate decay. Optimal tuning of these hyperparameters can lead to improved performance of thedownscaled FRCNN, which is an interesting avenue for future work. In Fig. 2.2a, we plot the multi-task lossfor the RPN and Detector networks as a function of time. Moving average was applied over a window of 5time steps to focus on the trends. We notice that the sum-loss converges with time to zero. The convergencerates and the loss fluctuations depend on the learning rate of the stochastic gradient descent algorithm and thenumber of positive and negative targets available per mini-batch. Example positive targets on a spectrogramchunk for the RPN and the Detector networks are shown in Figs. 2.2b and 2.2c respectively. As may be32Algorithm 2.2 The mean average precision (mAP) metric for signal detection performance evaluationRequire: SNR levels, ground truths, bounding box predictions, detection probabilities1: Sort the predictions in decreasing order of detection probabilities2: for Each bounding box prediction i do3: Assign a True label if it has IoU ≥ 0.5 with any GT (or False if otherwise)4: Calculate number of true positives (TP), false positives (FP), and false negatives (FN) so far5: Calculate precision preci until the current prediction aspreci = TP/(TP + FP)6: Calculate recall ri until the current prediction asri = TP/(TP + FN)7: end for8: for recall levels rj= {0,0.1,0.2, . . .,1} do9: Calculate the maximum achieved precision p˜recj for recall rj asp˜recj = maxrk ≥rjprec (rk), where prec(rk) is the precision at recall rk10: end for11: return The average of the maximum precisions for the 11 recall levels, calculated as mAP =11∑j=1p˜recjnoted from Sections 2.1.3 and 2.1.5, the RPN targets are the anchors whose IoU with the ground truth is≥ 0.5 and the detector targets are the RoI proposals chosen after NMS operation.2.3.3 Prediction performance metrics and baseline methodsTo evaluate the performance of the FRCNN, we consider the mean average precision (mAP) metric [8],which is a standard and widely-used metric for evaluating object detection algorithms in computer vision.An overview of the mAP calculation is presented in Algorithm 2.2. We begin by sorting all the boundingbox predictions from the FRCNN model in the decreasing order of the Detector classification probabilities.Next, we assign a True or False label for each prediction, depending on whether it has an IoU ≥ 0.5 with anyground truth or not, and calculate the precision and recall values until the current prediction. The definitionof the precision and recall metrics are provided in Steps 5 and 6 of the algorithm. We then consider 11recall levels ranging from 0 to 1 in steps of 0.1 and record the maximum achieved precision for each recalllevel. The mAP is then obtained as the average of the precision values recorded for the 11 recall levels.Higher mAP values denote better prediction performance. The precision vs. recall curves may also beused to evaluate the performance of the model, with larger area under the curves denoting better predictionperformance.33Baseline 1: Morphological processing [7]As the first baseline, we consider the energy-based morphological processing method [7]. In [7], the spec-trograms are processed for time-frequency localization through (i) binarization with an energy thresholdγbin, (ii) a modified morphological opening operation on the binary image with nerode successive erode andndilate successive dilate operations using a structure element of size s, (iii) a connected-component labellingoperation on the opened image, and (iv) an energy filtering step where the isolated regions with the top Ksum-energies are reported as the signals of interest. The first step serves as an energy-based denoising step.The second step compensates, partially, for the loss of important information from the binarization step andalso serves as a second level of denoising. The amount of denoising and energy compensation is governedby nerode, ndilate, and s. The third step isolates high-energy regions corresponding to the signals present inthe capture. The final energy filtering step serves as an interference removal step.The authors in [7] have only considered the detection of bird sound signals. As a result, we cannotdirectly use their recommended hyperparameter values for our task of detecting Wi-Fi-type signals. Weneed to optimize the hyperparameters but unfortunately, the authors in [7] do not provide much insight onhow the hyperparameters can be chosen for a generic dataset containing captures with varied SNRs. Forexample, only two sample RF captures are considered in [7], with SNR 0 dB and 40 dB respectively, and thebinarization thresholds are set via trial-and-error as γ = 1 and γ = 0.2 respectively for the two captures. Nofurther insight is provided on how theγ can be automatically chosen for an arbitrary capture. Similarly, thenumber of erode and dilate operations and the size of the structure element are set as nerode = 7, ndilate = 8,and s = 4, without much insight on how these hyperparameters need to be set for an arbitrary capture. Thetopics of hyperparameter optimization and the scalability to varied types of noise and interference are notstudied and are simply acknowledged as important works for the future. Although we are investigating thespecific case of [7], the following concerns are also true for other morphological processing algorithms,such as the ones proposed in [5, 6]: (i) the hyperparameters are chosen on a trial-and-error basis for a smallnumber of spectrograms, and (ii) the topics of hyperparameter optimization and the scalability to variedtypes of noise and interference are not studied and are instead referred to as avenues for future work.Since an optimal hyperparameter tuning algorithm is unavailable, we have resorted to a simple butcomputationally intensive grid search method. The following range of values are considered: (i) the bina-rization threshold γ ∈ [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9], (ii) the number of erode operations nerode ∈34[1, 2, 3.4, 5, 6, 7, 8, 9, 10, 11, 12], (iii) the number of dilate operations ndilate = nerode + 1 ( ndilate needs to belarger than nerode by atleast 1 unit, in order for the modified morphological opening operation in [7] towork effectively; we have therefore set ndilate = nerode + 1, as done in [7]), (iv) the structure element lengths ∈ [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], and (v) the connectivity length m = 4; the connectivity length m is typicallychosen to be 4 for isolating simple shapes such as rectangles and disks, and 8 for more complex and non-uniform shapes such as humans and daily-life objects.To separate out the noise and interference in the final energy filtering step, i.e., step (iv), we cannotset a maximum K on the number of signals. This is because Wi-Fi signals have randomized time-delaysbetween consecutive signals. The number of signals per capture will differ from spectrogram to spectrogram.Consequently, instead of setting a maximum number of signals K at a per-spectrogram level, we filter thedetected signals based on the time-frequency span of the isolated regions to separate out the noise andinterference. That is, from all the isolated regions in step (iii), we filtered out regions that are similar in sizeas the interference signals. We also filter out regions whose frequency spans are less than 10 MHz becauseall the considered Wi-Fi signals have a bandwidth of 20 MHz.In Fig. 2.3, we plot the precision vs. recall curves for the top 5 hyperparameter combinations obtainedfrom the grid-search. Details on the hyperparameters are included in the legend. For the mAP calculation,the predictions were considered as True if the intersection over union (IoU) with the ground truth is at least0.5. We notice that the best achievable mAP performance is 0.3676, for the above mentioned range of searchfor the hyperparameters. When we further investigate the poor mAP performance of the morphologicalprocessing method [7], we realize that it is a result of the limited hyperparameter generalizability across thediverse RF captures (please see example RF captures in Fig. 2.7 ). That is, the hyperparameters {γ,nerode, s}which work very well for a few captures fail to provide a similar performance for the remaining captures.This is in turn because the hyperparameters need to be carefully handcrafted for each capture, based on thenoise, fading, and interference conditions, in order to achieve the signal detection.Baseline 2: Support vector machine classification algorithm [115]As the second baseline, we consider the SVM-HOG classification method proposed in [115] for objectdetection in images. The SVM-HOG method runs a sliding detection window of size w f ×wt across eachspectrogram, in steps of length wstep along each axis. A histogram of ordered gradients (HOG) featurevector is assigned to each detection window as follows. Each window is divided into smaller regions, called350 0.1 0.2 0.3 0.4 0.5Recall0.50.60.70.80.91PrecisionmAP = 0.3676 ( = 0.8, nerode = 2, s = 9)mAP = 0.3613 ( = 0.8, nerode = 1, s = 10)mAP = 0.3602 ( = 0.8, nerode = 2, s = 8)mAP = 0.3538 ( = 0.8, nerode = 1, s = 9)mAP = 0.3530 ( = 0.8, nerode = 2, s = 7)Figure 2.3: Performance of the morphological processing method [7] with the best 5 hyperparameter combinations obtained throughgrid searchcells, of size pix × pix. For each cell, an HOG vector is formed by calculating the gradients at each pixeland quantizing them into ori orientation bins (see [115] for details). To reduce the impact of illuminationand shadowing, the HOG of each cell is locally normalized over somewhat larger regions, called blocks,of size blk × blk. The HOG descriptor for the window is then obtained by accumulating the HOGs of allthe blocks within itself. Typically, the blocks are overlapped with a stride of length bstride, so that each cellcontributes multiple HOGs to the final feature vector, each normalized with respect to a different block.Positive labels are then assigned to the windows which have IoU≥ ioumin with the signal of interest, whilenegative labels are assigned to the ones having zero IoU with all the signals of interest. A linear SVMclassifier is then trained with the positive and negative labelled feature vectors, in order to predict whethera window contains the signal of interest or not. Non-max suppression is then employed on the positivewindows because multiple overlapping prediction windows will be encountered for each signal.To adopt the SVM-HOG method for the signal detection and time-frequency localization task, we dothe following. For assigning the positive labels, while it is commonplace to use ioumin = 0.5, we haveset ioumin to a significantly smaller value of 0.05 because unlike in [115], our ground truths have variabletime-span. Next, the authors in [115] have reported marginal improvements in performance when an SVMwith a Gaussian kernel is used in place of a linear SVM for the classification, although at much highertraining times. To speed up the training, a commonplace solution is to use explicit kernel approximationwith the Nystroem method [166], followed by classification on the transformed HOG vectors using linearSVMs. This training approach was used in our work, with the Gaussian kernel width and the number of data36points used for constructing the basis of the kernel set to the standard values of 0.2 and 100 respectively.For the training, the authors in [115] have considered a ratio of 1 : 10 for the positive to negative trainingsamples. This ratio is suitable for the images of size 320× 240 considered in [115], but the same ratioresults in memory problems when training the SVM for our spectrograms of size 600×1024 because of thesignificant increase in the size of the training dataset. We have reduced the ratio to 1 : 2 in order to be ableto fit the training dataset into the 8 GB of RAM on our computer. We have later experimented with the ratio1 : 10 by partitioning the training dataset into 5 subsets and sequentially training the SVM model with onesubset at a time, but this did not yield an improvement in performance. We have also included hard negativemining [167] so as to achieve improvements in performance beyond regular hyperparameter tuning. In thisprocedure, an SVM model is initially trained with the original training feature set and all the false positivesfrom the training are gathered. We then re-train the same model on an augmented dataset containing boththe original feature set and the false positives. This is iterated a few times until the number of false positivesceases to decrease. Such a hard negative mining technique is seen to provide significant improvements inthe performance. We train an ensemble of 20 SVM models, each over 10000 positive samples drawn fromthe full training dataset independently across the models and randomly without repetition for each model.The average ensemble performance is reported.The authors in [115] do not provide much insight on how the hyperparameters can be tuned and haveinstead relied on experimental evaluations over a range of numerical values. This is a significant draw-back of the SVM-HOG method, because we need to generate the training features for each hyperparam-eter combination and this can get quite cumbersome if an exhaustive grid search is done for the hyper-arameter tuning. In our simulations, we have conducted a search over the following range of values:(i) w f ∈ [320,400,480,640,800], corresponding to signals of bandwidth [17.5,21.875,26.25,35, 43.75]MHz respectively, (ii) wt ∈ [24,32,40,48,56,64,72,80,160], corresponding to signals of time-span [0.3,0.4,0.5,0.6, 0.7,0.8,0.9,1,2] ms respectively, (iii) pix ∈ [2,4,6,8,16], and (iv) the number of orientation binsori ∈ [3,5,7,9,11,13]. Besides, following [115], we have set the block size to b = 2× pix (each block istherefore a grid of 4 cells), the block stride to bstride = 0.5× b, the window step size to wstep = 8, and theblock normalization scheme to L2-Hys [116].The results from the hyperparameter tuning are included in Fig. 2.4, where we have presented the mAPperformance of the top 5 hyperparameter combinations. Details on the hyperparameters are included in thelegend and the caption of the figure. For the mAP calculation, the predictions were considered as True if the370 0.1 0.2 0.3 0.4 0.5 0.6 0.7Recall00.20.40.60.81PrecisionmAP = 0.4835 ((wf , wt, ori) = (320, 80, 7))mAP = 0.4380 ((wf, wt, ori) = (400, 64, 7))mAP = 0.4367 ((wf, wt, ori) = (400, 48, 5))mAP = 0.3498 ((wf, wt, ori) = (400, 48, 9))mAP = 0.3202 ((wf,wt,ori) = (320, 48, 5))Figure 2.4: Top 5 mAP performances of the SVM-HOG method. Apart from the hyperparameters listed in the legend, hyperparam-eters common to all five curves are: (pix = 8, b = 16, bstride = 8, and wstep = 8).IoU with ground truth is at least 0.05. We have chosen a very small IoU threshold because the Wi-Fi signalshave variable time span but the SVM-HOG method considers a fixed time span wt for the detection windows.The small IoU threshold allows us to automatically incorporate the effect of a regression operation on thepositive prediction windows, which is ideally required for the successful detection of variable time-spansignals.We notice that an mAP of up to 0.4835 is achievable with the SVM-HOG method, which is much betterthan that of the morphological processing method [7]. The superior performance of the proposed SVM clas-sification algorithm comes from its supervised nature. In other words, since we train on a dataset containingdiverse types of noise and interference, the model learns to achieve scalability and a more uniform perfor-mance across diverse captures. This is unlike the morphological processing method, where the optimizedhyperparameters work very well only for a select few captures and result in partially or completely misseddetections in other captures.2.3.4 Design ChoicesBefore evaluating the performance of the deep learning model, we need to make three important designchoices: (i) downscale the default RPN and Detector networks by an appropriate factor, (ii) choose anappropriate BN for feature extraction, and (iii) choose an appropriate combination of the (RPN min, RPNmax, Detector min, Detector max) overlap values.380 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Recall0.40.50.60.70.80.91PrecisionDefault FRCNN (RPN:256, Det:4096)RPN: 128, Det: 2048RPN: 64, Det: 1024RPN: 32, Det: 512Morphological processing methodSVM-HOG methodmAP = 0.694mAP = 0.67mAP = 0.7mAP = 0.4835mAP = 0.713mAP = 0.3676Figure 2.5: Precision vs. recall when the RPN and Detector in the FRCNN are downscaled by a factor of 2,4, and 8 respectively.Downscaling by 2 gives the best mAP performance. The downscaled FRCNN model is also seen to significantly outperform thetwo baseline methods.Size of the RPN and Detector NetworksWe downscale the size of the feature extraction layers in the RPN and the Detector from the default(256,4096) filters to (128,2048), (64,1024), and (32,512) respectively, i.e., by downscaling factors of 2,4, and 8 respectively. The precision vs. recall performance is plotted in Fig. 2.5. We note that the best mAPis achieved when the default FRCNN is downscaled by a factor of 2. Downscaling is expected to provide per-formance improvement because the signal detection task at hand is simpler than the general-purpose objectdetection task for which the FRCNN was originally designed. We stick to the (128,2048) combination for theremainder of the experiments. For comparison with the state-of-the-art, in Fig. 2.5, we also plot the preci-sion vs. recall performance of the two baseline methods. We note that the downscaled FRCNN significantlyoutperforms the morphological processing and the SVM-HOG methods, which achieve an mAP of 0.3676and 0.4835 respectively when their hyperparameters are optimized through grid search. The hyperparametercombinations which delivered the best mAP for the two baselines were (γbin = 0.8,nerode = 2, s = 9,m = 4)and (w f = 320,wt = 80,ori = 7, pix = 8,b = 16,bstride = 8,wstep = 8) respectively. Example predictions fromthe FRCNN and the two baseline methods are illustrated in Figs. 2.6, 2.7, and 2.8 respectively. The superiorperformance of the downscaled FRCNN may be attributed to the sophisticated feature extraction procedureand the superior learning ability of the deep neural networks when compared to the two baselines. We notethat the SVM-HOG method yields an excessive number of false positives that result from the model’s limited39ability in learning and separating the signals of interest from the RFI. Also, the loss in mAP in the morpho-logical processing method comes mainly from a large number of missed detections and false positives, bothof which result from the limited generalizability of the hyperparameters involved.Depth of the base networkIn Fig. 2.9(a)-(b), we plot the precision vs. recall curves and the mAP vs. depth values respectively, whenthe BN is chosen to be VGG-10, VGG-13, and RESNET-50, where the numbers 10,13, and 50 denotethe number of weight layers in the BN. We try three different combinations for feature extraction: (i) usepretrained weights as the initialization for the BN and configure it as trainable (Pretrained, Trainable), (ii)use pretrained weights as the initialization for the BN and configure it as non-trainable (Pretrained, Non-trainable), and (iii) use Xavier normal initializer [171] for the BN and configure it as trainable (Xavier,Trainable). The VGG-13, set as trainable and initialized with the pretrained weights, gives the best precisionvs. recall as well as mAP performance. When the depth is appropriate, as with VGG-13, there is only amarginal drop in the mAP if the BN weights are initialized using the Xavier normal method, instead ofthe pretrained weight initialization. In other words, we note that the VGG-13 provides an appropriatearchitecture for feature extraction, but the FRCNN model only benefits marginally when the starting weightsare the pretrained weights. The VGG-10, which is a slightly shallower network than the VGG-13, achievedlower mAP because it has insufficient depth for feature extraction. Henceforth, we fix the BN to VGG-13,initialize it with pretrained weights and set it as trainable.RPN and Detector Min and Max OverlapsWe need to choose an appropriate combination of the RPN and Detector min and max overlap values.Since a grid search can be computationally exhaustive, we pursue a simple alternate-once search strategyfor choosing these numerical thresholds. We first fix the Detector Min and Max overlaps to be 0.1 and0.5 respectively and search over different combinations of RPN min and max overlap values as shown inFig 2.10a. Among the different thresholds, we observe that the (RPN Min, RPN Max) overlap combination(0.1,0.7) achieves better mAP values than the rest. We therefore fix this combination and search over thedifferent Detector min and max overlap values, as shown in Fig 2.10b. We notice that the best (Detectormin, Detector max) overlap combination is the (0.1,0.5), with an achievable mAP of 0.8. This simple searchmethod exposes an important avenue to improve the mAP of the model. Note that the mAP achieved with40Figure 2.6: Example predictions on the test spectrograms with the downscaled FRCNN model when the BN is the VGG-13 (train-able, initialized with pretrained weights). The blue and red rectangles indicate the signal ground truth and the predictions respec-tively. The model successfully separates out the RFI and provides tight bounding boxes for most signals of interest, even if there isan overlap with the RFI signals. The loss in mAP occurs mainly due to false positives arising from microwave oven signals.41(a) Example capture 1 (b) Example capture 2(c) Example capture 3 (d) Example capture 442(e) Example capture 5 (f) Example capture 6Figure 2.7: Example predictions from the morphological processing method [7] and the proposed deep learning method. Theoptimized hyperparameters in [7] do not generalize well across captures with different noise and interference types. As a result,we observe that the optimized hyperparameter combination (γbin = 0.8,nerode = 2, s = 9) results in missed detections, partially asin captures 3 and 4 or completely as in captures 5 and 6. Handcrafted hyperparameters, found through trial and error, can provideimproved performance but handcrafting is not a practical solution. The proposed deep learning method, on the other hand, providesuniformly good performance.Figure 2.8: Example predictions from the SVM-HOG method [115] with the optimized hyperparameters (w f = 32,wt = 80,ori =7, pix = 8,b = 16,bstride = 8,wstep = 8). We observe much fewer number of missed detections than with [7] due to the supervisednature of the method. However, the SVM-HOG method does not perform very well in separating out the RFI, resulting in asignificant number of false positives.430 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8Recall00.20.40.60.81PrecisionRESNET-50, Pretrained, TrainableRESNET-50, Pretrained, Non-trainableRESNET-50, Xavier, TrainableVGG-10, Pretrained, TrainableVGG-10, Pretrained, Non-trainableVGG-10, Xavier, TrainableVGG-13, Pretrained, TrainableVGG-13, Pretrained, Non-trainableVGG-13, Xavier, Trainable(a) Precision vs. Recall for different BNs10 15 20 25 30 35 40 45 50 55Depth of the base network00.10.20.30.40.50.60.70.8mAPPretrained, TrainablePretrained, Non-trainableXavier, TrainableRESNET-50VGG-13VGG-10(b) mAP for different BNsFigure 2.9: Impact of depth of the base network. The legend indicates the BN name, whether the initialization is pretrained orXavier normal [171], and whether the BN is trainable or not. We note that the VGG-13 model, initialized with pretrained weightsand configured as trainable, performs better than the others.440.1 0.3RPN min overlap00.10.20.30.40.50.60.70.80.9mAPRPN max overlap = 0.5RPN max overlap = 0.7RPN max overlap = 0.9(a) Detector (min, mix) overlap = (0.1, 0.5)0.1 0.3Detector min overlap00.20.40.60.8mAPDetector max overlap = 0.5Detector max overlap = 0.7Detector max overlap = 0.9(b) RPN (min, max) overlap = (0.1, 0.7)Figure 2.10: The mAP with different RPN and Detector thresholds. An mAP of up to 0.8 is achieved through a simple alternate-oncesearch strategy.the different combinations of the RPN and Detector min and max overlaps are consistently better than thoseachieved by the two baselines. Naturally, a more exhaustive grid search method may help us choose betterthreshold values but would require many more experimental runs.2.3.5 Performance evaluation on real traces and varied SNRsWe now proceed to evaluate the performance of the downscaled FRCNN on real RF traces and also validatethe robustness with regard to variations in the SNR. For all the results presented henceforth, we set theRPN feature extraction layer to 128 3× 3 filters, the Detector feature extraction layer to 2048 1× 1 filters,the BN to VGG-13 configured as trainable and initialized with pretrained weights, and the combination(0.3,0.5,0.1,0.5) for the (RPN min, RPN max, Detector min, Detector max) overlap values.Real RF capturesThe downscaled FRCNN model, when trained on synthetic datasets, exhibits a drop in mAP performancewhen the test captures are from real measurements because of the inherent mismatch in the data contained.To illustrate the same, we create a test dataset using the NI USRP-2901 device [117] with one or more Wi-Fisources located up to 20 m away and a microwave oven source located up to 2 m away from the receiver.Multiple Bluetooth devices are present within a 5 m radius around the receiver. The receiver gain is setto 20 dB and a total of 50 test captures are recorded, each containing RFI from both the microwave oven45Figure 2.11: Example predictions on real test captures when the synthetically-trained FRCNN model is finetuned with real RFtraces. When compared to the case of synthetic test traces, the real test captures contain new types of background noise and RFI(compare the spectrogram backgrounds with Fig. 2.6). The model is still able to successfully detect the signals of interest, butexperiences a marginal loss in the mAP.0 0.2 0.4 0.6 0.8Recall00.20.40.60.81PrecisionTrain synthetic, test syntheticTrain synthetic, test realTrain synthetic (80%)+ real (20%), test realTrain synthetic, finetune real, test realmAP = 0.484mAP = 0.713mAP = 0.125mAP = 0.6204Figure 2.12: Precision vs. recall when the test captures are obtained through real measurements. We notice a drop in the mAP ifthe model is trained purely on synthetic data. Data mixing and model finetuning are seen to alleviate the drop in the mAP.46and the Bluetooth signals. In Fig. 2.12, we plot the precision vs. recall performance when the model istrained on the synthetic dataset generated as per Algorithm 2.1. We notice that the mAP drops significantlyfrom 0.713 for synthetic test data to 0.125 for real test data. The drop is mainly because the model is notexposed to the varied types of background noise and interference experienced in real measurements (c.f. thebackground in Fig. 2.11).To alleviate the drop in mAP, we propose two solutions, namely data mixing and model finetuning. Indata mixing, we train the downscaled FRCNN with a mixture of synthetic and real measurement data. Weexpect improvements in the mAP because the model benefits from the accurately labelled ground truthsin the synthetic data and also from the exposure to the varied types of background noise and unknowninterference which cannot be modelled accurately or have not been accounted for in the synthetic data.Synthetic traces would still constitute a majority of the training dataset because it is not scalable to manuallylabel a large amount of real RF traces. In Fig. 2.12, we plot the precision vs. recall when the model istrained with 120 synthetic and 30 real captures, i.e., 80% synthetic and 20% real captures. The test capturesare fully from real measurements. We notice that the mAP improves from 0.125 to 0.484, corroborating ourhypothesis that the model would perform better with data mixing because it is now exposed to the variety ofbackground noise and unknown interference experienced in real RF traces.In model finetuning, we initially train the downscaled FRCNN with synthetic RF traces and then finetunethe trained weights through an additional round of training on a small amount of real RF traces. Since thesynthetic RF traces are similar to the real RF traces to a certain extent, the model from the initial syntheticdata training serves as a warm start for the finetuning and therefore facilitates an improvement in the mAPperformance. The additional computational workload from model finetuning is only marginal because thesize of the real training dataset is much smaller than the synthetic training dataset. Also, a smaller numberof training epochs is sufficient to achieve convergence. In Fig. 2.12, we plot the precision vs. recallperformance when model finetuning is performed with the same 30 real RF traces as used in the data mixing.For the finetuning, convergence was achieved with just 2 epochs. We notice that the mAP improves from0.125 to 0.6204. This indicates that the model finetuning is a viable solution for field deployments, whereby,we train a downscaled FRCNN model offline with large amounts of synthetic data and then finetune theweights over a small amount of real measurements to achieve a good mAP performance. We also note thatthe two baselines perform much worse than the proposed deep learning method on the real RF traces. Themorphological processing method [7] achieves an mAP of 0.1 when the combination (γbin = 0.4,nerode =470 0.2 0.4 0.6 0.8Recall0.20.30.40.50.60.70.80.91PrecisionSNR [15, 50] dBSNR [5, 50] dBSNR [-5, 5] dBmAP = 0.58mAP = 0.63mAP = 0.713Figure 2.13: Precision vs. recall curves for datasets with different SNR ranges. We notice a drop in mAP as we widen the SNRrange and enter into negative SNR values.9, s = 6), obtained via grid search over the 30 real RF traces within the data mixing dataset, is employed. Ifwe instead employ the combination (γbin = 0.8,nerode = 2, s = 9) optimized for synthetic traces, none of thesignals in the real RF traces are detected. The SVM-HOG method, on the other hand, provides a non-zeromAP of 0.0114 when tested directly on real RF traces. For better mAP, we attempted model finetuning andonly experienced a marginal increase from 0.0114 to 0.134. Since the two baselines perform significantlyworse than the proposed deep learning method, we avoid clutter and do not include their precision vs. recallcurves in Fig. 2.12.Variations in the signal-to-noise ratioWe now proceed to evaluate the mAP performance for different SNR levels in the dataset. In the syntheticdataset so far, the signals span the SNR range of [15,50] dB. We now create two additional datasets of thesame size, but with the received SNR in the range [5,50] dB and [−5,5] dB respectively. The same trainingand test composition as given in Section 2.2 is maintained. In Fig. 2.13, we plot the precision vs. recallachieved with the three datasets. We notice that the mAP drops marginally to 0.63 when the SNR range isexpanded from [15,50] dB to [5,50] dB. The drop becomes more significant as we move to the [−5,5] dBSNR range, with the mAP being 0.58. These trends expose the limited generalizability of the model acrosswider SNR ranges and negative SNR regions. A customized and more advanced denoising mechanism istherefore required as a preprocessing step to improve the SNR before using the proposed deep learningmodel.482.4 SummaryIn this chapter, we proposed the use of deep learning with a downscaled form of the Faster RCNN (FRCNN)[8] framework to perform signal detection and time-frequency localization in a wideband radio frequency(RF) spectrum when friendly but uninteresting signals cause interference. Multiple design insights areprovided on adopting the downscaled FRCNN for the task at hand, all based on the signals of interest.For synthetic traces, a mean average precision (mAP) of up to 0.8 is observed when the signals of interestare from Wi-Fi and the RF interference is from Bluetooth and microwave oven signals, thus significantlyoutperforming the standard morphological processing and support vector machine (SVM) algorithms. Whencompared to the default FRCNN architecture, improvements in performance are achieved when the featureextraction is done using a pretrained VGG-13 [165] model configured as trainable, the anchor sizes aresimilar to those of the signals of interest, the region proposal network (RPN) and Detector networks aredownscaled from the default architecture by a factor of 2, and the numerical thresholds within the RPNand Detector networks are optimized through a simple alternate-once search method. For real RF traces,we propose to (i) train with a mixture of synthetic and real RF traces, or (ii) finetune the synthetic-trainedmodel through an additional round of training on a small amount of real RF traces. Both strategies offerimprovements in the mAP when compared to directly using the model trained on synthetic data alone.49Chapter 3Fingerprinting Algorithms based onGaussian process regression for IntruderDevice Localization from UplinkDifferential RSSIn this chapter, we propose supervised ML algorithms to position the intruder devices in a WSN from theiruplink RSS information. To circumvent the problem of unknown transmit power of the intruder device,we work with the DRSS with respect to a reference sensor. We firstly consider the conventional Gaussianprocess regression (CGP) algorithm for the localization task and demonstrate that it provides optimistic 2σerror-bars on the location estimates, resulting in confidence intervals that do not fully encompass the truelocations of the test devices. To address this concern and provide more realistic 2σ error-bars on the locationestimates, we propose two GP methods, namely the numerical approximation GP (NaGP) and Gaussianapproximation GP (GaGP) methods. The NaGP method does so by numerically approximating the posteriorpredictive distribution as a Gaussian distribution with the same first and second order moments. This methoddoes not however lend itself to any analytical insights on the derived location estimates and their 2σ error-bars. The GaGP method, on the other hand, analytically approximates the posterior predictive distributionas a Gaussian distribution, in order to derive closed-form expressions for both the location estimates andtheir 2σ error-bars. The derived expressions lend themselves to analytical insights, for example, the GaGPmethod is seen to introduce multiple correction factors into the expressions provided by the CGP methodso as to account for the stochastic nature of the test DRSS vectors. Numerical studies with both noise-free and noisy training DRSS vectors reveal that the proposed GP methods achieve improved localizationperformance over the existing maximum likelihood and semidefinite programming solutions. They also50serve as a cost-effective alternative to the standard k nearest neighbours fingerprint-matching algorithm, inthat they do not require a dense grid of training locations to achieve a similar RMSE performance. Theproposed GP methods do not require prior knowledge of the transmit power of the device, nor do they entailcooperation from the device during the localization procedure.Our main contributions in this chapter are as follows.• We study intruder device localization from the uplink DRSS information and propose machine learn-ing algorithms which provide both the location estimates and the associated 2σ error bars in closed-form expressions.• When formulating the localization problem as a Gaussian process regression task, the covariancebetween the x−coordinates (or y−coordinates) of the devices needs to be modelled as a functionof their DRSS vectors. To make this design choice, we study the mathematical relation betweenthe DRSS and the device locations and recommend the use of a weighted-sum covariance functioncomprising the squared exponential and inner-product terms.• We conduct numerical studies and demonstrate that the conventional GP method provides optimistic2σ error-bars when estimating the device locations from their uplink DRSS.• To derive more realistic 2σ error-bars on the location estimates, we propose two GP methods, namelythe NaGP and GaGP methods. These methods take the statistical distribution of the test DRSS vectorsinto account to derive the posterior predictive density and approximate it as a Gaussian distributionwith the same first and second-order moments. While the NaGP method takes a numerical approachfor the moment-matching, the GaGP method resorts to analytical moment-matching and providesclosed-form expressions for both the location estimates and the associated 2σ error-bars.• We simplify the mean and variance expressions provided by the GaGP method, and bring them to asimilar form as those provided by the conventional GP method. The following analytical insights areobtained as a result: (i) the mean and variance expressions in the GaGP method are similar in structureto those from the conventional GP method, with several additive and multiplicative correction factorsintroduced by the GaGP method in order to account for the stochastic nature of the test DRSS, and (ii)in the case of the test DRSS being noise-free, the mean and variance expressions for the GaGP turnout to be exactly the same as those provided by the conventional GP method.51• We consider both noisy and noise-free DRSS vectors for training the GP models and illustrate throughnumerical studies that the proposed algorithms achieve smaller localization RMSE than the existingmaximum-likelihood and semidefinite programming solutions. They also serve as a cost-effectivealternative to the k nearest neighbours fingerprinting algorithm, in that they do not require a densegrid of training locations. Besides, unlike the existing solutions, the proposed algorithms provide the2σ error bars on the location estimates in closed-form.Mathematical notationWe use regular font small letters for scalars, boldface small letters for vectors, and boldface capital lettersfor matrices, for example, a, a, and A, respectively. The notations [a]i, [A]i, and [A]i j refer to the element iin vector a, column i in matrix A, and the element (i, j) in matrix A, respectively. The overhead symbol (˜.)refers to training data and the overhead symbol (̂.) refers to test data, respectively. An additional superscript(.)∗ is used to refer to noise-free data. The symbol ≈ denotes approximation. The notations ∇a(.) and ∇a,a(.)refer to the gradient and the Hessian with respect to the vector a. Also, ∇[a]i (.) refers to the partial derivativewith respect to the element i of vector a. A random vector a that is Gaussian distributed with mean u andcovariance U is referred to as a ∼ N(u,U), and its probability density function is denoted as N(a;u,U).3.1 System Model and Localization with Conventional GP Method3.1.1 Localization SetupWe consider a localization setup where M distributed sensors, also referred to as anchors interchangeably,record the RSS of the signals from an unknown source k. When the transmit power is ρ, the RSS qmk atanchor m under frequency-flat fading and log-normal shadowing is given byqmk = ρg0d−ηmk10zmk10 |hmk |2, where,dmk = | |sk− âm | |,∀m = 1, . . .M, and zmk ∼ N(0,σ2m).(3.1)In (3.1), η is the path loss exponent (PLE), dmk is the distance from source k to anchor m, g0 is the pathloss at reference distance of 1 m, σm is the shadowing noise standard deviation, and zmk is the shadowingcoefficient. The hmk is the small-scale fading coefficient, the in-phase and quadrature-phase components of52which are assumed to be independent and identically distributed Gaussian random variables with mean zeroand variance of 0.5. The power term |hmk |2 is then exponentially distributed with unit mean. The vectorssk = [xk ; yk]T and âm = [x̂m; ŷm]T denote the locations of the source and the anchor m respectively. Fromthe RSS in (3.1), we assume that the small-scale fading effects are averaged out over time [34, 48, 61] andthe resulting mean value is converted to dB scale to obtainqdBmk = qdB0 −10η log10(dmk)+ zmk, (3.2)where qdB0 = 10log10(ρg0) is the uplink RSS at the reference distance d0.To estimate the source location sk from the RSS values loge qmk , we would require knowledge of thetransmit power ρ. However, for the case of an intruder device, the transmit power ρ is not necessarily knownbeforehand. As a workaround, we calculate the DRSS w.r.t. a reference sensor node so as to eliminate thetransmit power term from the RSS. When the anchor m = 1 is the reference, the DRSS pmk at the remaininganchors is given byp(m−1)k = 10log10(qmk)−10log10(q1k), ∀m = 2, . . .M, (3.3)Substituting (3.2) into (3.3), we have the per-anchor DRSS values given byp(m−1)k = −10η log10(dmkd1k)+ zmk − z1k, where zmk ∼ N(0,σ2m), z1k ∼ N(0,σ21 ), (3.4)Once the pmk , ∀m = 2, . . .,M, are extracted as above at a central computing unit connected to the M sensors,the M − 1 DRSS values of each device k are accumulated into an (M − 1) × 1 uplink DRSS vector pk suchthatpk = [p1k p2k p3k . . . p(M−1)k]T . (3.5)533.1.2 Machine Learning SetupWe seek to learn the functions fx(.) and fy(.) which take the uplink DRSS vector pk of a device k as inputand provide the device’s location coordinates xk and yk as output respectively, i.e.,xk = fx(pk) and yk = fy(pk) ∀xk, yk . (3.6)We follow a supervised machine learning approach, with a training phase and a test phase, to learn fx(.)and fy(.). During the training phase, a machine learning model is trained with a database comprising uplinkDRSS vectors at several known device locations. During the test phase, we feed as input to the trained ma-chine learning model, the DRSS vector of a test device whose location coordinates are unknown. The trainedmodel provides as output, an estimate of the test device’s location coordinates, along with the associated 2σerror-bars.Remark 3.1. [On the choice of Gaussian process regression to estimate device locations] The problem ofestimating the x−coordinate xk (and y−coordinate yk) of a device k from its DRSS vector pk is a non-linear regression problem in the machine learning context. This is because xk is a non-linear function ofpk , with the additional presence of shadowing noise zmk (c.f. (3.2)). From among several available non-linear regression methods, we choose Gaussian process regression for our study because (i) GP methods arenon-parametric and can be used to model any arbitrary function [98], (ii) GP methods are known to per-form as good as most other machine learning methods in terms of multiple performance metrics, includingthe squared prediction error [174, 175] and the correlation between prediction error and the 2σ error-barsprovided by the ML model [174], (iii) GP methods provide probabilistic estimates in the form of a fullpredictive distribution, wherein we obtain closed-form expressions for the estimated parameters (the devicelocation coordinates in our case), as well as their 2σ error-bars.We begin with the conventional GP regression method for the device localization task and observethrough numerical studies that it provides optimistic 2σ error-bars on the location estimates. To address thisissue, we propose two GP methods, namely the NaGP and GaGP methods, which provide more realistic 2σerror-bars on the location estimates. The localization performance is investigated both when the GP modelsare trained with noise-free DRSS vectors and when they are trained with noisy DRSS vectors. While itis intuitive to consider noisy DRSS for the localization task due to the presence of shadowing, it is also54worthwhile to consider the use of synthetically generated noise-free DRSS for the training, as summarizedin Remark 3.2 below. Numerical studies in Section 3.4 illustrate that the three GP methods achieve superiorperformance over existing maximum-likelihood estimation (MLE) and semi-definite programming (SDP)solutions when trained with noisy DRSS vectors. The localization performance is also competitive for smallshadowing noise variance levels when we train with noise-free DRSS vectors.We will now present details on modelling the GP methods for the localization task, with focus on es-timating the x−coordinates. Extension for the y−coordinates can be made in a straightforward mannerfollowing the same approach.Remark 3.2. [On the use of noise-free DRSS for training the ML model] A major concern with the use ofmachine learning for device positioning is the availability of appropriate training data. Particularly, to trainwith noisy DRSS, we need to conduct measurement campaigns and maintain the training database. This isnot only labor intensive, but can sometimes also be infeasible due to harsh environmental conditions. Tocircumvent this challenge, we consider the use of noise-free DRSS for the training purposes. Instead ofrelying on extensive measurement campaigns, we can generate noise-free DRSS datasets synthetically viasoftware simulations. To do so, we would only require knowledge of the training locations, which can bechosen as random points within the service area, and the path loss exponent. For simplicity of exposition, werestrict ourselves to the areas such as open air environments where the path loss exponent is constant and itsnumerical value is obtained through a pre-calibration procedure among the sensor nodes [93–96]. In a moregeneral setting, where the statistical properties of the PLE are available, we can generate the noise-free DRSSvectors for different PLE realizations and use them for training the ML model. Our work serves as proofthat we can indeed obtain location estimates, as well as their 2σ error-bars, through machine learning withnoise-free training DRSS inputs. Numerical studies in Section 3.4 confirm that the proposed GP methods,when trained with noise-free DRSS vectors, achieve competitive performance for small shadowing noisevariance levels.3.1.3 Training PhaseAt the core of Gaussian process regression is the assumption that the function to be learned, i.e., fx(.), isdrawn from a zero-mean Gaussian process prior specified by a user-defined covariance function φ(., .) [98].That is, any finite set of realizations of the function to be learned (e.g. fx(.) in the case of x−coordinates)55B1B2B3U1U2U3U4dddU5Figure 3.1: Example setup with three sensors, labelled B1, B2, and B3, and five devices to be localized, labelled U1, U2, U3, U4,and U5. The radius dth is the distance beyond which the PLE η changes in value (for example, dth = 45 m in 3GPP UMi [97] , asthe η changes from 2 to 6.7). For any pair of devices, φ(., .) should be high if the x−coordinates are closely-spaced and it shoulddecrease with the separation.is Gaussian distributed with mean zero and covariance matrix Φ, the entries of which are given by a user-defined function φ(., .). We refer to this assumption asfx(.) ∼ GP(0, φ(., .)). (3.7)The function φ(., .) models the covariance of x−coordinates of any two device locations in the system asa function of their DRSS vectors. We model φ(., .) as a weighted-sum of squared-exponential (SE), innerproduct (IP) and delta terms, defined for any two devices k and k ′ with DRSS vectors pk and pk′, respec-tively, asφ(pk,pk′) =αe− 12 (pk−pk′ )T B−1(pk−pk′ )+γpTk pk′ +σ2erδkk′,where B = diag{βm},m = 1, . . .,M −1,and δkk′ = {1 if k = k ′, 0 if otherwise}.(3.8)The SE term αe− 12 (pk−pk′ )T B−1(pk−pk′ ) serves as the stationary component because it captures the correlationbetween the x−coordinates of the devices k and k ′ as a function of the distance between their DRSS vec-tors. The IP term γpTkpk′ serves as the non-stationary component because it captures the correlation amongx−coordinates as a function of the actual DRSS vectors. The delta term σ2erδkk′ serves as the measurementerror component. The rationale behind our choice of φ(., .) is as follows.In Fig. 3.1, we present an example setup to motivate our choice of the covariance model. Any pair ofdevices in our system can be categorized into three feasible types: (i) Type I: the devices are closely-spaced56in both x−coordinate space X and DRSS space P. The devices U1 and U2 in Fig. 2 fall under this category.(ii) Type II: the devices are well-separated in both X and P. The devices U1 and U5 fall under this category.(iii) Type III: the devices are closely-spaced in X but are well-separated in P. The devices U2 and U3 fallunder this category because their y−coordinates are well-separated.Our choice of φ(., .) needs to satisfy certain criteria. Firstly, φ(., .) needs to generate entries of a positivedefinite matrix because the resulting matrix is the covariance of a Gaussian distribution. Secondly, the φ(., .)should exhibit local stationarity, i.e., for a given pair of devices, the covariance between their x−coordinatesshould decrease with the distance between their DRSS vectors. This will allow φ(., .) to accommodatedevice pairs of Type I and II. Example functions that can model the local stationarity include the Matern-class, exponential, squared exponential, and rational quadratic covariance functions [98]. Lastly, the φ(., .)should also exhibit some non-stationarity, i.e., the φ(., .)model should allow devices to have high covariancebetween their x−coordinates even if they are well-separated in the DRSS space. This will allow φ(., .)to accommodate device pairs of Type III. Example functions that can model the non-stationarity includethe inner dot product, polynomials of the inner dot product, and neural network covariance functions [98].Since our choice of φ(., .) needs to exhibit local stationarity as well as some non-stationarity, we follow thestandard approach of using a weighted sum of stationary and non-stationary covariance functions. Fromamong the various stationary and non-stationary functions that are available in the GP literature, we choosethe squared exponential function to model the local stationarity and the inner dot-product function to modelthe non-stationarity. This choice of φ(., .) allows us to analytically solve the multiple integral expressionsin the proposed GaGP method (c.f. (B.1)-(B.6)). Further investigation is needed to deem whether otherexample combinations, such as a weighted sum of Matern-class and neural network covariance functions,facilitate a closed-form solution in the GaGP method or not.For notational ease, we transform the covariance model in (3.8) as follows:φ(pk,pk′) = α′N(pk ;pk′,B)+γpTk pk′ +σ2erδkk′,∀pk,pk′, (3.9)where we introduce a new parameter α′ = α(2pi)(M−1)/2 |B|1/2 to convert the SE term in (3.8) into a Gaussianterm. The free parameters α, β1, . . . , βM , and γ introduced by the covariance model in (3.8) are to belearned during the training phase. We accumulate them into an (M +1)×1 vector θ defined such that57θ = [α β1 . . . βM−1 γ]T . (3.10)In order to learn the vector θ, we specify the training database as follows. Let there be a total of L˜ trainingdevice locations, whose x−coordinates are accumulated into an L˜ ×1 vector x˜ and their noise-free trainingDRSS vectors into an L˜×(M −1) matrix P˜, defined such thatx˜ = [x˜1 x˜2 . . . x˜L˜]T ,P˜ = [˜p1 p˜2 . . . p˜L˜]T .(3.11)Note from (4.8) that the DRSS vector p˜l in P˜ corresponds to the training x−coordinate x˜l in x˜, ∀ l = 1, . . ., L˜.The vector θ can then be obtained via maximum log-marginal likelihood of the training x−coordinate vectorx˜ asθopt = argmaxθlog(p(˜x|P˜,θ)),(a)= argmaxθlog(N (˜x;0, Φ˜)),(3.12)where θopt is the learned parameter vector, (a) follows from the standard GP assumption in (3.7) that x˜ isGaussian distributed asx˜|P˜,θ ∼ N(0, Φ˜), where[Φ˜]ll′ = φ(˜pl, p˜l′),∀ l, l ′ = 1, . . ., L˜.(3.13)The optimization problem in (4.9), to be solved during the training phase, is well-known to be non-convexin the GP literature [98, 140–143]. Nevertheless, we can obtain the first-order gradients with respect to θ ina closed-form and can therefore employ gradient ascent methods, such as the conjugate gradient [180], toobtain a local optimum. The training phase is complete upon obtaining the optimum vector θ¯.When presenting numerical studies in Section 3.4, we follow the same training procedure as givenabove in (3.7)-(4.10) for both the cases of noise-free training DRSS and noisy training DRSS. A morerigorous training procedure for the case of noisy DRSS vectors would be to take their stochastic natureinto account, instead of treating them as deterministic inputs, and maximize the posterior log-marginallikelihood log(∫ p(˜x|P˜,θ)p(P˜)). This is a direction to investigate in the future. We now proceed to explain58the test location prediction procedure in the conventional GP method.3.1.4 Prediction with the Conventional GP MethodWe now consider CGP - a GP method which employs conventional GP principles [98] to predict the devicelocations. Let us say that we need to predict L̂ test device locations from their noisy DRSS vectors. The testdevices’ x−coordinates are accumulated into an L̂×1 vector x̂ and their DRSS vectors are accumulated intoan L̂×(M −1) matrix P̂ such thatP̂ = [̂p1 p̂2 . . . p̂L̂]T ,x̂ = [x̂1 x̂2 . . . x̂L̂]T .(3.14)Note from (3.14) that the test DRSS vector p̂l belongs to the test device l with x−coordinate [̂x]l = x̂l,∀ = 1, . . ., L̂. The CGP method treats the noisy test DRSS vectors as deterministic inputs and uses thestandard GP assumption (3.7) to obtain the joint distribution of the training and test x−coordinate vectors x˜and x̂ asx˜x̂ P˜, P̂ ∼ N©­­«00ª®®¬,©­­«Φ˜ (Φ†)TΦ† Φ̂ª®®¬ , (3.15)where Φ˜ ∈ RL˜×L˜ , Φ† ∈ RL˜×L̂ , and Φ̂ ∈ RL̂×L̂ are the covariance matrices between the training and test DRSSvectors, defined such that [Φ˜]ll′ = φ(˜pl, p˜l′),∀l, l ′ = 1, . . ., L˜, [Φ†]ll′ = φ(̂pl, p˜l′),∀l = 1, . . ., L̂, l ′ = 1, . . ., L˜, and[Φ̂]ll′ = φ(̂pl, p̂l′),∀l, l ′ = 1, . . ., L̂. Conditioning the joint distribution in (3.15) over x˜ gives usx̂ |˜x, P˜, P̂ ∼ N(µ̂(CGP)x , Ĉ(CGP)x ), whereµ̂(CGP)x = Φ†Φ˜−1x˜, Ĉ(CGP)x = Φ̂−Φ†Φ˜−1(Φ†)T .(3.16)Eq. (3.16) gives us the predicted mean µ̂(CGP)x and the associated covariance Ĉ(CGP)x of the test x−coordinatevector x̂ when CGP is employed. The distribution of the x−coordinate [̂x]l of any particular test device lcan be obtained, through marginalization of x̂ |˜x, P˜, P̂, as[̂x]l |˜x, P˜, p̂l ∼ N([µ̂(CGP)x ]l, [Ĉ(CGP)x ]ll), where59[µ̂(CGP)x ]l =L˜∑i=1φ(̂pl, p˜i)[Φ˜−1x˜]i,[Ĉ(CGP)x ]ll = φ(̂pl, p̂l)−L˜∑i=1L˜∑j=1φ(̂pl, p˜i)[(Φ˜)−1]i jφ(˜pj, p̂l). (3.17)In (3.17), the terms [µ̂(CGP)x ]l and [Ĉ(CGP)x ]ll refer to the predicted mean and variance of the x−coordinate[̂x]l of any test device l. Since the predictive distribution in (3.17) is Gaussian, the term [µ̂(CGP)x ]l denotesthe maximum-a-posteriori (MAP) estimate of the test x−coordinate [̂x]l. Also, the predictive variance bearsa strong correlation with the empirical mean-squared prediction error on the test points [98, 174]. There-fore, [Ĉ(CGP)x ]ll serves as a direct indicator of the prediction error associated with the estimate [µ̂(CGP)x ]l.The 2σ error-bars, calculated as ±2√[Ĉ(CGP)x ]ll, provide us with the 95% confidence interval[[µ̂(CGP)x ]l ±2√[Ĉ(CGP)x ]ll]on choosing [µ̂(CGP)x ]l as the estimate of [̂x]l. The CGP method treats the noisy test DRSSvectors as deterministic inputs and consequently provides optimistic 2σ error-bars on the estimated loca-tions. To address this concern, we now propose two GP methods, namely, the NaGP and the GaGP methods,both of which derive more realistic 2σ error-bars on the estimated locations by accounting for the stochasticnature of the test DRSS data.3.2 Numerical Approximation GP Method for Location PredictionWe now propose the numerical approximation GP method (NaGP) method for the intruder device locationestimation from the uplink DRSS vectors. The proposed method takes the stochastic nature of the noisy testDRSS vectors into account and numerically approximates the posterior predictive distribution as a Gaussiandistribution, in order to furnish more realistic 2σ error-bars on the location estimates than the CGP method.Details are as follows.We observe from (3.2) that any noisy DRSS value pdBmkrecorded at the sensor m is the sum of a noise-free component, i.e., −10η log10( dmkd1k ), and a shadowing noise component, i.e., zmk − z1k . This allows us toexpress the test DRSS vector p̂l asp̂l = p̂∗l + ẑl, such that ẑl ∼ N(0, Σ̂l), (3.18)where p̂∗lis the noise-free component in p̂l and ẑl is the shadowing noise with covariance Σ̂l, given by60Σl = Γ{diag([σ21 ; . . . ;σ2M ])}ΓT ,Γ = [−1;I],where 1 denotes a column vector of (M − 1) ones and I denotes the (M − 1) × (M − 1) identity matrix. Weassume that the shadowing variances [σ21 ; . . . ;σ2M ] of the M uplink channels of the test device are known tothe CU. We then have the conditional distribution of p̂∗lfrom (3.18) asp̂∗l |p̂l, Σ̂l ∼ N(̂pl, Σ̂l). (3.19)We can now treat p̂∗las a deterministic latent variable that is input to the trained GP model, and use (3.17)to obtain an estimate of [̂x]l in terms of p̂∗l . After this, we can use (3.19) to integrate out the latent variablep̂∗land obtain the posterior predictive distribution of [̂x]l in terms of p̂l as followsp([̂x]l |˜x, P˜, p̂l) =∫p([̂x]l |˜x, P˜, p̂∗l )p(̂p∗l |p̂l, Σ̂l)dp̂∗l , (3.20)where p([̂x]l |˜x, P˜, p̂∗l ) is obtained from (3.17) and p(̂p∗l |p̂l, Σ̂l) from (3.19), respectively. The predictivedistribution p([̂x]l |˜x, P˜, p̂l) in (3.20) is non-Gaussian and cannot be obtained in closed-form because theintegral on the right hand side is intractable. As explained below, this is in turn because the first termp([̂x]l |˜x, P˜, p̂∗l ) inside the integral in (3.20) is a complicated non-linear function of the Gaussian randomvector p̂∗l, over which we integrate. Observe from (3.17) that p([̂x]l |˜x, P˜, p̂∗l ) is given byp([̂x]l |˜x, P˜, p̂∗l ) =N([̂x]l;L˜∑i=1φ(̂p∗l , p˜i)[Φ˜−1x˜]i, φ(̂p∗l , p̂∗l )−L˜∑i=1L˜∑j=1φ(̂p∗l , p˜i)[(Φ˜)−1]i jφ(˜pj, p̂∗l )), (3.21)where the training covariance matrix Φ˜ is defined as in (3.13). From (3.21), we note that p([̂x]l |˜x, P˜, p̂∗l )is a complicated non-linear function of p̂∗lbecause (i) any Gaussian distribution is non-linear in its meanand covariance, and (ii) the mean and covariance of the Gaussian distribution in the r.h.s. of (3.21) areboth non-linear functions of p̂∗l(c.f. (3.8)). This makes the integral in (3.20) analytically intractable . As aconsequence, we can only obtain an approximation to the true predictive distribution p([̂x]l |˜x, P˜, p̂l).The NaGP method takes a numerical approach and approximates p([̂x]l |˜x, P˜, p̂l) using Markov-ChainMonte-Carlo sampling [176] as follows. We draw S i.i.d samples p̂∗l(s), 1 ≤ s ≤ S, from p̂∗l|p̂l, Σ̂l ∼N(̂pl, Σ̂l)1For notational ease, all integrals are written as indefinite integrals, but in reality, they are definite over appropriate sets.61and approximate the integral in (3.20) asp([̂x]l |˜x, P˜, p̂l) (a)≈S∑s=11Sp([̂x]l |˜x, P˜, p̂∗l (s))(b)=S∑s=11SN([̂x]l; [µ̂CGPx (s)]l, [ĈCGPx (s)]ll).(3.22)In (3.22), (a) follows from the Monte-Carlo approximation procedure [176], and (b) from (3.17), with the[µ̂CGPx (s)]l and [ĈCGPx (s)]ll being the same as [µ̂CGPx ]l and [ĈCGPx ]ll but with the test DRSS vector p̂l replacedby the Monte-Carlo sample p̂∗l(s). Since the r.h.s of (3.22) is a mixture of S Gaussians with identical weights,we know from [177] (eq. (14.10)-(14.11)) that the l.h.s can be approximated as a Gaussian distribution withthe same first and second order moments, i.e.,p([̂x]l |˜x, P˜, p̂l) ≈ N([̂x]l; [µ̂(NaGP)x ]l, [Ĉ(NaGP)x ]ll),where,[µ̂(NaGP)x ]l =1SS∑s=1[µ̂CGPx (s)]l,[Ĉ(NaGP)x ]ll = 1SS∑s=1([µ̂CGPx (s)]l −[µ̂(NaGP)x ]l)2 +1SS∑s=1[ĈCGPx (s)]ll,∀l = 1, . . ., L̂.(3.23)In (3.23), [µ̂(NaGP)x ]l refers to the estimate of the test x−coordinate [̂x]l from NaGP and [Ĉ(NaGP)x ]ll refersto the associated predictive variance. As is the case with with CGP, we know that the 2σ error-bars onthe estimate [µ̂(NaGP)x ]l, calculated from the predictive variance as ±2√[Ĉ(NaGP)x ]ll, provide us with the 95%confidence interval[[µ̂(NaGP)x ]l±2√[Ĉ(NaGP)x ]ll]on choosing [µ̂(NaGP)x ]l as the estimate of [̂x]l. By increasingS, we can increase the accuracy of the [µ̂(NaGP)x ]l and [Ĉ(NaGP)x ]ll values because the numerical approximationin (3.22) becomes tighter with increasing S [176].We may note from (3.20) that, unlike the CGP method which naively treats the noisy test DRSS vectorsas deterministic inputs, the NaGP method treats the noise-free components in the test DRSS vectors aslatent variables and integrates them out using the statistical knowledge of the noise present. By doing so,the NaGP method learns from and incorporates the noise covariance matrices {Σ̂l} into the predicted meanand variance expressions (c.f. (3.23)). This learning enables NaGP to provide more realistic 2σ error-barson the predicted locations than the CGP method. The NaGP is a numerical moment-matching GP methodthat does not lend itself to much analytical insight on the obtained mean and variance expressions in (3.23).62To address this concern, we now propose an analytical moment-matching GP method for the uplink DRSS-based localization task.3.3 Gaussian Approximation GP (GaGP) Method for Location PredictionWe now propose the Gaussian approximation GP method, which employs analytical moment-matching andapproximates the posterior predictive density as a Gaussian distribution, in order to derive both the intruderdevice location and the associated 2σ error bars in closed-form, in terms of the uplink DRSS vectors. Foreach test device l, the GaGP method firstly derives the true predictive distribution p([̂x]l |˜x, P˜, p̂l) by takingthe noisy nature of the test DRSS p̂l into account. The derived predictive distribution cannot be expressedin closed-form and is therefore approximated analytically as a Gaussian distribution with the same first andsecond order moments. Specifics are given below.From the previous section (c.f. (3.18)-(3.21)), we know that we can only obtain an approximation to thetrue predictive distribution p([̂x]l |˜x, P˜, p̂l), using either numerical or analytical approximation procedures.While the numerical approximation is pursued in the NaGP method, here we take an analytical approxi-mation approach because it incurs lower computational complexity and also allows us to obtain analyticalinsights on how the resulting mean and variance compare against the conventional GP method.The GaGP method analytically approximates the true predictive distribution p([̂x]l |˜x, P˜, p̂l) in (3.20) asa Gaussian distribution with the same first and second order moments, as follows:p([̂x]l |˜x, P˜, p̂l) ≈ N([̂x]l; [µ̂(GaGP)x ]l, [Ĉ(GaGP)x ]ll), where[µ̂(GaGP)x ]l = E[̂x]l ([̂x]l |˜x, P˜, p̂l) and [Ĉ(GaGP)x ]ll = E[̂x]l (([̂x]l |˜x, P˜, p̂l)2)− ([µ̂(GaGP)x ]l)2.(3.24)In (3.24), [µ̂(GaGP)x ]l and [Ĉ(GaGP)x ]ll are the estimated mean and variance of the test x−coordinate [̂x]l.Lemma 3.1 provides closed-form expressions for the mean and variance, [µ̂(GaGP)x ]l and [Ĉ(GaGP)x ]ll, reportedby the GaGP method for the uplink DRSS-based device localization task.Lemma 3.1. When the weighted-sum covariance model given in (3.8) is employed, the GaGP method yieldsthe following closed-form expressions for the predicted mean [µ̂(GaGP)x ]l and variance [Ĉ(GaGP)x ]ll of thex−coordinate [̂x]l of the test device l63[µ̂(GaGP)x ]l =L˜∑i=1α′[ψ]iN (̂pl; p˜i,B+ Σ̂l)+γ[ψ]ip̂Tl p˜i, and[Ĉ(GaGP)x ]ll = σ2er +α+γp̂Tl p̂l +γTr(Σ̂l)−L˜∑i=1L˜∑j=1[ξ]i j{(α′)2N (˜pi; p˜j,2B)N (̂pl;p˜i + p˜j2,B2+ Σ̂l)+ [Υ]Tj p˜i+(˜pj)T [Υ]i +γ2p̂Tl p˜i (˜pj)T p̂l +γ2 Tr(˜pi (˜pj)T Σ̂l)}−([µ̂(GaGP)x ]l)2,(3.25)respectively, where the vector ψ ∈ RL˜ and the matrices Υ ∈ RM−1×L˜ and ξ ∈ RL˜×L˜ are defined such thatψ = Φ˜−1x˜,[Υ]i = α′γN (̂pl; p˜i,B+ Σ̂l)(Σ̂lB−1 + I)−1Σ̂l(B−1p˜i + Σ̂−1l p̂l), and[ξ]i j = [Φ˜−1]i j −[ψ]i[ψ]j, ∀i, j = 1, . . ., L˜.(3.26)Proof. See Appendix B.1. Besides obtaining closed-form expressions for the predicted mean and variance of the test devices’ x−coordinates,we also derive few analytical insights from these expressions, as summarized in Remark 3.3 below.Remark 3.3. Although not obvious from an initial observation, the expressions for [µ̂(GaGP)x ]l and [Ĉ(GaGP)x ]llreported in (3.25) in 3.1 are similar in structure as the [µ̂(CGP)x ]l and [Ĉ(CGP)]ll terms obtained from (3.17),with several multiplicative and additive correction factors. The GaGP method introduces these correctionfactors so as to account for the stochastic nature of the test DRSS p̂l. In addition, if the test DRSS vectorsare noise-free, i.e., if Σ̂l = 0, we verify that [µ̂(GaGP)x ]l and [Ĉ(GaGP)x ]ll are exactly the same as [µ̂(CGP)x ]l and[Ĉ(CGP)x ]ll, respectively.Proof. See Appendix B.2. Observe from (3.20)-(3.25) that, unlike the CGP method, the GaGP method accounts for the stochasticnature of the test DRSS. It handles the noise-free terms in the test DRSS vectors as latent variables andintegrates them out using the statistical properties of the shadowing noise present in the test DRSS. Thisallows the GaGP method to provide more realistic 2σ error-bars on the estimated locations than the CGPmethod.64-10 -5 0 5 10-10-8-6-4-20246810Sensor nodeTraining pointTest pointFigure 3.2: Localization setup with M = 31 anchors (left-bottom as the reference for DRSS), L˜ = 100 training locations, and L̂ = 25test locations.Table 3.1: Simulation parameters to generate DRSS vectors and study the performance of the NaGP and GaGP methodsSystem parameter ValueTransmit power (ρ) 26 dBm (400 mW),path loss parameters g0 = −47.6 dB, d0 = 1 m,(WINNER-II FSPL [179]) η = 0 if dmk < 1 m, 2 otherwise.3.4 Numerical StudiesWe now present numerical examples to evaluate the performance of the proposed GP methods for the uplinkDRSS localization task. As shown in Fig. 3.2, we consider a setup of size 20 m ×20 m, with L˜ = 100training locations, M = 31 anchor nodes, and Lˆ = 25 test device locations distributed uniformly across theservice area. The RSS vectors are generated using (3.2) with the simulation parameters as per Table 3.1.The transmit power ρ is set to 26 dBm as per the remote controller specifications for the DJI Phantom 4 ProV2.0 drone [178]. The reference path loss g0 and the PLE η are chosen as per the WINNER-II free spacepath loss model to be −47.6 dB and 2 respectively, with the reference distance d0 set to 1 m [179]. Theanchor at the left-bottom corner is then used as the reference to obtain the DRSS vectors for each devicelocation. We assume that all available device locations have a measurement error standard deviation (σer )of −20 dB. When training with noise-free DRSS, the training dataset comprises of a 100×30 DRSS matrixand the corresponding x-coordinate and y-coordinate vectors of the training device locations. When trainingwith noisy DRSS, we generate 200 Monte-Carlo realizations of the DRSS vector for each training location.65The resulting dataset constitutes a 20000× 30 training DRSS matrix and the corresponding x-coordinateand y-coordinate vectors. Since the dataset is quite large in the case of noisy training DRSS, we train anensemble of 20 GP models, each with 400 DRSS vectors sampled randomly without repetition from the fulltraining dataset, and report the average performance.Once the training DRSS data is ready, the GP models are trained by solving the log marginal-likelihoodmaximization problem in (4.9) using conjugate gradient method [180]. We run 20 trials with random ini-tializations and choose the local optimum which gives the best training objective, i.e., log(p(˜x|P˜,θ)) value.The same learned parameter vector θ¯ is reused for evaluating the prediction performance of all three GPmethods under study because they use the same training procedure. For the test dataset, we generate 200Monte-Carlo realizations of the DRSS vectors for each test location, amounting to a total of 5000 test DRSSvectors. The average performance over the RSS shadowing standard deviation range σm ∈ [−8,8] dB isreported. For the NaGP method, we set the number of samples S in (3.23) to 50.3.4.1 Performance MetricsWe measure prediction performance in terms of the root-mean-squared prediction error (RMSE) defined asRMSE =1L̂L̂∑l=1√([̂x]l −[µ̂(.)x ]l)2 + ([̂y]l −[µ̂(.)y ]l)2 (3.27)where [̂x]l and [̂y]l are the actual x and y coordinates of the test device l, [µ̂(.)x ]l and [µ̂(.)y ]l are the estimatesof [̂x]l and [̂y]l as given by the chosen GP method, and [Ĉ(.)x ]ll and [Ĉ(.)y ]ll are the variances associated withthe estimates [µ̂(.)x ]l and [µ̂(.)y ]l, respectively. For example, if we choose CGP, [µ̂(.)x ]l = [µ̂(CGP)x ]l, [µ̂(.)y ]l =[µ̂(CGP)y ]l, [Ĉ(.)x ]ll = [Ĉ(CGP)x ]ll and [Ĉ(.)y ]ll = [Ĉ(CGP)y ]ll. Smaller RMSE values indicate better predictionperformance.3.4.2 Baselines for performance comparisonAs the first baseline, we consider the maximum-likelihood estimator (MLE) in [55], which is known to beasymptotically optimal when working with the test DRSS alone. The test device locations are estimated bymaximizing the joint likelihood of the test DRSS values received at the sensors. The resulting optimizationproblem is highly non-linear and non-convex with multiple local minima, so an appropriate intial point isrequired. As suggested in [55], we utilize the linear least squares estimate in eq. (10) of [55] as the initial66point. As a second baseline, we consider the robust semidefinite programming-based estimator (RSDPE)proposed in [61]. This work also pursues the maximum-likelihood objective but resorts to constraint re-laxations so as to obtain a convexified problem, which can be solved using the Matlab CVX toolbox withMosek solver. The RSDPE is robust to sensor node placements and model uncertainities, and is the state-of-the-art among solutions which do not use machine learning for the DRSS-based localization task. A fewleast-squares solutions, such as [57] [58], have emerged in the recent times, with focus on bias reduction inthe location estimates while achieving a similar localization error as in [61]. The performance of the solu-tions in [57] [58] is also sensitive to sensor node placements, while this is not the case for RSDPE. As thethird baseline, we consider the k-nearest-neighbour (kNN) algorithm considered in [71] for fingerprinting-based localization using the DRSS values. The test device location is obtained as the average of the four,i.e., k = 4, nearest neighbours in the training database, in terms of the Euclidean distance between the DRSSvectors.3.4.3 Localization RMSE PerformanceIn Fig. 3.3, we illustrate the average localization RMSE performance achieved by the GP methods understudy, when the shadowing standard deviation (σ) in the RSS vectors is increased from −8 dB to 8 dB.Several interesting observations can be made from Fig. 3.3. Firstly, we note that the CGP, NaGP, and GaGPmethods provide similar RMSE values for different shadowing noise levels. Secondly, when comparedto the MLE and RSDPE solutions, the GP methods facilitate a smaller RMSE over the entire range ofthe shadowing level σ in the case of noisy training DRSS. When trained with noise-free DRSS, a similarobservation is made for small values of σ. The improvements in the RMSE come primarily from employingmachine learning for the localization task. The MLE and RSDPE solutions work on the test DRSS alone,whereas the GP models additionally learn the mapping functions fx(.) and fy(.) from the training DRSSinformation. Thirdly, the GP methods perform better than the kNN algorithm for most σ in the case oftraining with noisy DRSS and for small σ in the case of training with noise-free DRSS. This is mainlybecause the GP methods pursue non-linear regression to learns the functions fx(.) and fy(.). The kNNalgorithm, on the other hand, relies on finding a neigbour set in the training data that accurately surroundsthe test location, and is therefore sensitive to the distribution and the density of the training locations. Theuniformly distributed training locations, as considered in Fig. 3.2, presents an optimistic case for the kNNalgorithm. Lastly, when training with the noise-free DRSS, the obtained RMSE values are higher than when67Shadowing noise standard deviation (dB)Localization RMSE (m)CGPGaGPNaGPRSDPEMLEkNN(a) Training with noisy DRSSShadowing noise standard deviation (dB)Localization RMSE (m)CGPGaGPNaGPRSDPEMLEkNN(b) Training with noise-free DRSSFigure 3.3: Average localization RMSE achieved by the CGP, NaGP, and GaGP methods at different shadowing noise levels σmwhen (a) trained with noisy DRSS and (b) trained with noise-free DRSS respectively. From (a), we note that for most σ, thethree GP methods perform better than the existing algorithms when trained with noisy DRSS. From (b), we note that it is alsoworthwhile training with noise-free DRSS, particularly for σ < −5 dB, where the GP methods perform better than the MLE andRSDPE algorithms, and they achieve comparable performance as the kNN algorithm. Note that, unlike the baseline algorithms, theGP methods additionally provide confidence intervals around the estimated test locations.trained with noisy DRSS because there is an inherent mismatch between the training and test data. The GPmethods end up projecting the noise present in the test DRSS onto the output location coordinate space.For the same reason, we also notice that the rate of increase in the RMSE with σ is higher in the case ofnoise-free training DRSS.3.4.4 2σ Error-Bar PerformanceWe now plot the average 2σ error-bars on the test devices’ x− coordinates and y−coordinates, as givenby the proposed NaGP and GaGP methods, for the case of noise-free training DRSS in Fig. 3.4, andfor the case of noisy training DRSS in Fig. 3.5 respectively. The MLE, RSDPE, and kNN algorithmsdo not provide 2σ error-bars on the location estimates and are therefore not included in the plot. In thesame figures, we also plot the number (%) of true test device locations which are within the 2σ confidencerange ([µ̂(.)x ]l ±2√[Ĉ(.)x ]ll, [µ̂(.)y ]l ±2√[Ĉ(.)y ]ll) of the estimated locations. We observe that the CGP methodprovides optimistic 2σ error-bars both when trained with noise-free and noisy DRSS vectors. Particularly,in the case of noise-free training DRSS, the 2σ error-bars are unrealistically small and are not comparableto the corresponding RMSE values in Fig. 3.3. As a result, we note that less than 40% of the true test devicelocations fall inside the 2σ confidence range of the CGP method when trained with noise-free DRSS (c.f.68Fig. 3.4c). The performance improves when we train with noisy DRSS, but the number of true locationswithin the confidence range is still less than 85% (c.f. Fig. 3.5c). The NaGP and GaGP methods, on theother hand, consistently provide more realistic 2σ error-bars on the estimated locations for varied shadowingnoise levels. In the case of noise-free training DRSS, more than 90% of the true test device locations arewithin the 2σ confidence range provided by these methods. The performance is improved further in thecase of noisy training DRSS, with more than 95% of the true test device locations consistently within theconfidence range of the location estimates (c.f. Fig. 3.4c and 3.5c). Overall, we note that both the NaGP andGaGP methods achieve improved performance over the CGP method, in terms of the estimated 2σ error-bars. The GaGP method provides similar performance as the NaGP method, and it serves a low-complexityalternative because it employs analytical moment-matching and provides closed-form expressions for the2σ error-bars.3.4.5 Impact of the number of sensor nodesIn Fig. 3.6, we plot the average RMSE performance of the CGP, NaGP, and GaGP methods when thenumber of DRSS sensor nodes (M − 1) is increased from 10 to 50. As M increases, we notice that theRMSE performance of the three GP methods decreases with M . We attribute the decrease in RMSE withM to two reasons: (i) the increased average signal strength at the sensor nodes, which is a consequence ofthe reduced average sensor-to-device distance, and (ii) the extra DRSS information that becomes availableto the GP models due to the increase in the number of sensor nodes.We now make the Remark 4.3 below about estimating multiple test device locations simultaneously.Remark 3.4. A common attribute of the proposed NaGP and GaGP methods is that they allow us to es-timate multiple test device locations simultaneously, without affecting the localization accuracy for anygiven device. Also, for cost savings, we can pre-compute certain mathematical terms and reuse them whencalculating the location estimates in parallel.Proof. The GaGP method allows us to parallelize the computation of test device x−cordinate estimates[µ̂(GaGP)x ]l and their 2σ error-bars ±2√[Ĉ(GaGP)x ]ll for the L̂ test devices. As may be observed from (3.25)-(3.26), this is because the computation of [µ̂(GaGP)x ]l and [Ĉ(GaGP)x ]ll for any device l does not rely on thecomputation of [µ̂(GaGP)x ]l′ and [Ĉ(GaGP)x ]l′l′ of any other device l ′. Similar is the case with the NaGP method.When parallelizing the computation of the location estimates, we can also reduce the cost of each GP69Shadowing noise standard deviation (dB)CGPGaGPNaGP(a) Average 2σ error-bar on the estimated x-coordinatesShadowing noise standard deviation (dB)CGPGaGPNaGP(b) Average 2σ error-bar on the estimated y-coordinatesShadowing noise standard deviation (dB)True locations within confidence interval (%)CGPGaGPNaGP(c) True locations within the confidence range of the location estimatesFigure 3.4: [Training with noise-free DRSS] Plots of the average 2σ error-bars on the estimated locations, as provided by the CGP,GaGP, and NaGP methods. The CGP method provides 2σ error-bars that are quite optimistic, with less than 40% of the true testdevice locations within the 2σ confidence range of the estimated locations. The GaGP and NaGP methods provide more realistic2σ error-bars on the location estimates, with more than 90% of the true test device locations within the 2σ confidence range of theestimated locations.70Shadowing noise standard deviation (dB)CGPGaGPNaGP(a) Average 2σ error-bar on the estimated x-coordinatesShadowing noise standard deviation (dB)CGPGaGPNaGP(b) Average 2σ error-bar on the estimated y-coordinatesShadowing noise standard deviation (dB)True locations within confidence interval (%)CGPGaGPNaGP(c) True locations within the confidence range of the location estimatesFigure 3.5: [Training with noisy DRSS] Plots of the average 2σ error-bars on the estimated locations, as provided by the CGP,GaGP, and NaGP methods. The CGP method provides 2σ error-bars that are relatively optimistic, resulting in less than 85% ofthe true test device locations within the 2σ error-bars around the location estimates. The GaGP and NaGP methods provide morerealistic 2σ error-bars on the location estimates, with more than 95% of the true test device locations within the 2σ confidencerange of the estimated locations.71Number of the DRSS sensorsLocalization RMSE (m)CGPGaGPNaGP = -4dB = 8dB(a) Training with noisy DRSSNumber of the DRSS sensorsLocalization RMSE (m)CGPGaGPNaGP = 0dB = 2dB = -2dB(b) Training with noise-free DRSSFigure 3.6: Plot of the average localization RMSE of the three GP methods when the number of DRSS sensor nodes is increasedfrom 10 to 50.method by pre-computing few mathematical terms which are re-usable across the L̂ devices. For example,we can pre-compute the matrix Φ˜−1 and the vector Φ˜−1x˜ when computing the [µ̂(CGP)x ]l and [Ĉ(CGP)x ]ll in theconventional GP method (c.f. (3.17)). Similarly, we can pre-compute the α′, ψ, ξ , and N (˜pi; p˜j,2B) whenusing (3.25)-(3.26) to calculate the [µ̂(GaGP)x ]l and [Ĉ(GaGP)x ]ll in the GaGP method. A detailed analysis onthe computational complexity of the studied GP methods is presented in Appendix B.4 3.5 SummaryWe proposed a supervised machine learning approach based on Gaussian process regression to localizeintruder devices from their DRSS information. Both noise-free and noisy DRSS data were considered fortraining the GP model, while the test DRSS vectors are always considered noisy due to shadowing effectsof the wireless channel. We proposed two GP methods for the localization problem, namely the NaGP andGaGP methods. These methods approximate the posterior predictive density as a Gaussian distribution,using numerical and analytical moment-matching respectively, in order to provide more realistic 2σ error-bars on the location estimates than the conventional GP method. When compared to NaGP, the GaGPmethod serves as a low-complexity alternative and lends itself to a few analytical insights, for it providesclosed-form expressions for both the location estimates and the associated 2σ error bars. Numerical studies72confirm that (i) the proposed GP methods achieve smaller localization RMSE when compared to the existingmaximum-likelihood and semi-definite programming solutions; they also serve as a cost-effective alternativeto the standard k nearest neighbours fingerprint-matching algorithm, in that they do not require a dense gridof training locations to achieve a similar RMSE performance. (ii) the NaGP and GaGP methods consistentlyprovide more realistic 2σ error-bars than the conventional GP method, both when trained with noise-freeand noisy DRSS vectors, and (iii) the localization RMSE of both the proposed GP methods decreases withan increase in the number of sensor nodes in the area. Despite the improvement in providing more accurate2σ error-bars, the RMSE performance of the NaGP and GaGP methods is found to be similar to that ofthe conventional GP method. Improvements in the RMSE are possible if, in addition to accounting for thestochastic nature of the test DRSS, the GP methods attempt noise reduction in the DRSS. We explore thisdirection in the next chapter and propose two new GP methods which incorporate a data reconstruction stepinto the localization procedure in order to achieve a smaller localization RMSE.73Chapter 4Gaussian Process Regression with DataReconstruction for Intruder Localizationfrom Uplink Differential RSSIn this chapter, we continue to study the problem of positioning intruder devices from their uplink DRSS in-formation. We note that the NaGP and GaGP methods proposed in Chapter 3 report more realistic 2σ error-bars on the location estimates than the CGP method, but achieve similar localization root mean squarederror (RMSE) performance. To address this shortcoming, in this chapter, we propose two new GP algo-rithms, namely the reconsruction GP (RecGP) and reconstruction Gaussian approximation GP (RecGaGP),both of which incorporate a data reconstruction step into the localization procedure. The proposed GPmethods are seen to achieve improved localization RMSE performance when compared to the GP methodsproposed in Chapter 3.Our main contributions in this chapter are as follows.• We apply principal component analysis (PCA) to the noise-free DRSS vectors in a wireless sensornetwork and demonstrate through numerical examples that they span a low-dimensional principalsubspace.• We make use of the low-dimensionality of the DRSS vectors to propose two new GP methods for theintruder device localization. These methods reconstruct the noisy training and test DRSS vectors froma low-dimensional subspace spanned by the noise-free DRSS vectors in the service area, and utilizethe reconstructed DRSS vectors for the localization task.• We demonstrate through numerical studies that the data reconstruction step allows the RecGP andRecGaGP methods to achieve improved localization RMSE performance when compared to the CGP,74NaGP, and GaGP methods presented in Chapter 3.• When estimating the test device locations, the RecGP method is seen to provide optimistic 2σ errorbars, for it treats the reconstructed test DRSS vectors as deterministic inputs. This concern is ad-dressed in the RecGaGP method, as it treats the reconstructed DRSS vectors as stochastic inputs andtakes their statistical distribution into account.Mathematical notationThe notation in the current chapter is the same as in Chapter 3. The overhead symbol (˜.) refers to trainingdata and the overhead symbol (̂.) refers to test data, respectively. An additional superscript (.)∗ is used torefer to noise-free data. Also, we use the overhead symbol ¯(.) to refer to the noise-free DRSS data generatedto facilitate the data reconstruction step when we train the RecGP and RecGaGP models with noisy DRSSvectors.4.1 Localization Setup and the Low-dimensionality of Noise-free DRSS4.1.1 Localization SetupSimilar to the previous chapter, we consider a setup where M distributed sensor nodes record the RSS ofthe signals from an unknown source k. Following the steps (3.1)-(3.4), one of the sensors is chosen as areference and the M − 1 DRSS values of each device k are accumulated into an (M − 1) × 1 uplink DRSSvector pk such thatpk = [p1k p2k . . . p(M−1)k]T . (4.1)The machine learning setup is also similar to the previous chapter, in that we seek to learn the functionsfx(.) and fy(.) which take the uplink DRSS vector pk of a device k as input and provide the device’s locationcoordinates xk and yk as output respectively, i.e.,xk = fx(pk) and yk = fy(pk) ∀xk, yk . (4.2)75Similar to Chapter 3, we follow a supervised machine learning approach to learn fx(.) and fy(.). Duringthe training phase, a machine learning model is trained with a database comprising uplink DRSS vectorsat several known device locations. During the test phase, we feed as input to the trained machine learningmodel, the DRSS vector of a test device whose location coordinates are unknown. The trained modelprovides as output, an estimate of the test device’s location coordinates, along with the associated 2σ error-bars.Before proceeding further, we firstly examine and demonstrate the low-dimensionality of the noise-freeuplink DRSS vectors in a wireless sensor network. Specifically, we apply principal component analysis(PCA) to the noise-free uplink DRSS and observe that it spans over a low-dimensional principal subspace.This observation allows us to propose two new Gaussian process regression (GP) methods, namely, theRecGP and the RecGaGP method, both of which exploit the low-dimensionality aspect of the noise-freeDRSS vectors and incorporate a data reconstruction step into their localization procedure.4.1.2 Low-dimensionality of the noise-free DRSSLet us consider N randomly chosen device locations in the service area and accumulate their noise-freeDRSS vectors, simulated using (3.4), into an N ×(M −1) noise-free DRSS matrix P∗, such thatP∗ = [p∗1; p∗2; . . . ;p∗N ]. (4.3)We take a simulation approach to demonstrate that the noise-free DRSS matrix P∗ has a low-dimensionalprincipal subspace. We consider two example scenarios with the number of DRSS sensor nodes M −1 = 20and M − 1 = 30 sensors distributed randomly over a 10× 10 m service area. We then choose N = 100randomly distributed locations in the service area and build a sample noise-free DRSS matrix P∗ using (3.2)with the simulation parameters as per Table 3.1 in the previous chapter and the first denoted sensor as thereference. This is repeated to build 200 P∗ matrices each for M−1= 20 and M−1= 30. We then decomposeeach sample matrix P∗ into three parts via singular value decomposition [183] to obtainP∗ = UDVT , (4.4)where the matrices U and V contain the left and right singular vectors of P∗ and the diagonal matrix Dcontains the singular values of P∗ in decreasing order. In Fig. 4.1a and 4.1b, we plot the singular values,760 5 10 15 20Singular Value Index00.20.40.60.81Normalized Singular ValueAverage singular valuesMaximum observed deviation(a) Number of DRSS sensors=200 5 10 15 20 25 30Singular Value Index00.10.20.30.40.50.60.70.8Normalized Singular ValueAverage singular valuesMaximum observed deviation(b) Number of DRSS sensors=300 5 10 15 20Number of PCs chosen (L)010203040506070Reconstruction error (%)Average reconstruction errorMaximum observed deviation(c) Number of DRSS sensors=200 5 10 15 20 25 30Number of PCs chosen (L)0102030405060Reconstruction error (%)Average reconstruction errorMaximum observed deviation(d) Number of DRSS sensors=30Figure 4.1: Plots of the singular values and reconstruction errors (with the first M0 PCs) for the number of DRSS sensor nodes(M −1) = 20 and (M −1) = 30, averaged over 200 different noise-free DRSS matrices. In Figs. 4.1a-4.1b, we observe that the firstfew singular values explain most of the energy contained in P∗. In Figs. 4.1c-4.1d, we observe that the first few PCs reconstructmore than 95% of the data contained in P∗.77averaged over the 200 different P∗ matrices, for M − 1 = 20 and M − 1 = 30, respectively. The error barsrepresent the maximum observed deviation from average values. For both M − 1 = 20 and M − 1 = 30, wenotice that the first few singular values represent most of the energy contained in P∗.For further insight, we study the error incurred upon reconstructing the noise-free DRSS matrices fromthe subspace spanned by the first M0 principal components (PCs). Using truncated SVD [183], we canreconstruct each sample matrix P∗ from its first M0 PCs as U[M0]D[M0]V[M0]T , where U[M0] and V[M0] arematrices formed by the first M0 columns of U and V, respectively, and D[M0] is the diagonal matrix formedby the first M0 singular values of P∗. The reconstruction error | |P∗ −U[M0]D[M0]V[M0]T | |2, averaged overthe 200 different P∗ matrices, is plotted in Figs. 4.1c and 4.1d against the number of chosen PCs M0. Weobserve that for both M −1 = 20 and M −1 = 30, the first few PCs are consistently able to reconstruct morethan 95% of the data contained in P∗. These plots confirm that we can form a low-dimensional principalsubspace of the noise-free DRSS by combining its first few PCs.In the next section, we propose two GP methods which utilize the low-dimensionality of the noise-freeDRSS vectors to reconstruct the training/test DRSS vectors prior to the localization task. The reconstructionstep allows them to achieve lower localization error when compared to the CGP, NaGP, and GaGP methodspresented in Chapter 3. The presented GP methods are probabilstic in nature, in that we obtain both the testdevice locations as well as the associated 2σ error-bars. Details on the proposed GP methods are presentedbelow, with focus on estimating the x−coordinates. Extension for the y−coordinates can be made in astraightforward manner following the same approach. When training the GP models, we investigate the useof both noise-free DRSS as well as noisy DRSS vectors.4.2 Reconstruction GP (RecGP) Method for Location PredictionWe have illustrated in Section 4.1.2 that the noise-free uplink DRSS in a wireless sensor network spans alow-dimensional principal subspace. We now propose RecGP, a reconstruction-based GP method, whichmakes use of this property for the device localization task as follows. If we train with a noise-free DRSSmatrix P˜∗, we reconstruct the test DRSS vectors from a principal subspace spanned by P˜∗. If we instead trainwith a noisy DRSS matrix P˜, we firstly use (3.2) to generate a noise-free DRSS matrix P¯∗ for a few randomlychosen device locations in the service area. We then reconstruct both the training and test DRSS matrices P˜and P̂ from the subspace spanned by the newly generated matrix P¯∗ . The reconstructed DRSS vectors are78(a) Training procedure in the RecGP method(b) Prediction procedure in the RecGP methodFigure 4.2: Overview of the training and prediction procedures in the proposed RecGP method. (a) When the training DRSS P˜∗is noise-free, it is directly used to train the GP model. When otherwise, we reconstruct the noisy training DRSS P˜ from a low-dimensional principal subspace spanned by the noise-free DRSS P¯∗ in the service area using (4.13) and train the GP model withthe reconstructed DRSS P˜(rec). (b) The test DRSS P̂ is noisy due to shadowing effects of the wireless channel and is reconstructedeither from P˜∗ or P¯∗ prior to deriving the posterior distribution of the test location coordinates x̂. The reconstructed test DRSSP̂(rec) is treated as a deterministic input when deriving the posterior distribution of x̂ (c.f. (4.19)).used to derive the posterior density of the test locations, which is unlike the case with the CGP, NaGP, andGaGP methods, where we only use the original test DRSS for the location estimation.. An overview of theRecGP method is presented in Fig. 4.2. Details on the training and test procedures in the RecGP method arepresented below.4.2.1 Training with noise-free DRSS vectorsWe consider L˜ training locations in the service area, and generate a noise-free training DRSS matrix of sizeL˜×(M −1), denoted asP˜∗ = [p˜∗1 p˜∗2 . . . p˜∗ L˜].T (4.5)The training procedure with the noise-free DRSS is similar to the one followed in the conventional GPmethod. Any finite set of the device x-coordinates is assumed to be Gaussian distributed with mean zeroand covariance matrix Φ, the entries of which are given by a user-defined function φ(., .), i.e., fx(.) ∼GP(0, φ(., .)). As in Chapter 3, we model the covariance function φ(., .) as a weighted-sum of squared-exponential (SE), inner product (IP) and delta terms, defined for any two devices k and k ′ with DRSS79vectors pk and pk′, respectively, asφ(pk,pk′) =αe− 12 (pk−pk′ )T B−1(pk−pk′ )+γpTk pk′ +σ2erδkk′,where B = diag{βm},m = 1, . . .,M −1,and δkk′ = {1 if k = k ′, 0 if otherwise}.(4.6)The free parameters α, β1, . . . , βM , and γ introduced by the covariance model in (4.6) are to be learnedduring the training phase. We accumulate them into an (M +1)×1 vector θ defined such thatθ = [α β1 . . . βM−1 γ]T . (4.7)In order to learn the vector θ, we specify the training database as follows. Let there be a total of L˜ trainingdevice locations, whose x−coordinates are accumulated into an L˜ ×1 vector x˜ and their noise-free trainingDRSS vectors into an L˜×(M −1) matrix P˜∗, defined such thatx˜ = [x˜1 x˜2 . . . x˜L˜]T ,P˜∗ = [p˜∗1 p˜∗2 . . . p˜∗ L˜]T ,(4.8)where the noise-free DRSS vector p˜∗l in P˜∗ corresponds to the training x−coordinate x˜l in x˜, ∀ l = 1, . . ., L˜.The vector θ can then be obtained via maximum marginal likelihood of the training x−coordinate vector x˜asθ¯ = argmaxθlog(p(˜x|P˜∗,θ)),(a)= argmaxθlog(N (˜x;0, Φ˜∗)),(4.9)where θ¯ is the learned parameter vector, (a) follows from the standard GP assumption that x˜ is Gaussiandistributed asx˜|P˜∗,θ ∼ N(0, Φ˜), where [Φ˜]ll′ = φ(p˜∗l, p˜∗l′),∀ l, l ′ = 1, . . ., L˜. (4.10)The optimization problem in (4.9) is well-known to be non-convex in the GP literature [98, 140–143], butfirst-order gradients with respect to θ can be obtained in a closed-form. We can therefore employ gradientascent methods, such as the conjugate gradient [180], to obtain a local optimum. The training is complete80upon obtaining the optimum vector θ¯.4.2.2 Training with noisy DRSS vectorsWhen training with noisy DRSS vectors, we firstly reconstruct them from a principal subspace spanned bythe noise-free DRSS in the service area. For the L˜ training locations, let us suppose that the noisy trainingDRSS matrix of size L˜×(M −1) is given byP˜ = [˜p1 p˜2 . . . p˜L˜].T (4.11)We now consider L¯ randomly distributed locations in the service area, drawn independently from the traininglocations, and generate the corresponding noise-free DRSS matrix using (3.2). The same reference sensornode as in P˜ is used for the DRSS calculations, to obtainP¯∗ = [ ¯p∗1 ¯p∗2 . . . p¯∗ L¯].T (4.12)The training DRSS matrix P˜ is then reconstructed from the first M0 principal components of P¯ as follows,to obtain [183]P˜(rec) = P˜V[M0]V[M0]T , where, P¯∗ = UDVT (4.13)In (4.13) V[M0] is the matrix of the first M0 right singular vectors of the noise-free DRSS matrix P¯∗ . Thetraining database is then formed with the L˜ × 1 training x-coordinate vector x˜ and the reconstructed noisytraining DRSS matrix P˜(rec) of size L˜×(M −1) , i.e.,x˜ = [x˜1 x˜2 . . . x˜L˜]T , and, P˜(rec) = [˜p(rec)1 p˜(rec)2 . . . p˜(rec)L˜ ]T , (4.14)where the reconstructed noisy DRSS vector p˜(rec)lin P˜(rec) corresponds to the training x−coordinate x˜l in x˜,∀ l = 1, . . ., L˜. The vector θ of hyperaparameters in the covariance model can then be obtained via maximummarginal likelihood of the training x−coordinate vector x˜ as81θopt = argmaxθlog(p(˜x|P˜(rec),θ)),(a)= argmaxθlog(N (˜x;0, Φ˜(rec))),(4.15)where θopt is the learned parameter vector, and (a) follows from the GP assumption that x˜ is Gaussiandistributed asx˜|P˜(rec),θ ∼ N(0, Φ˜(rec)), where [Φ˜(rec)]ll′ = φ(˜p(rec)l , p˜(rec)l′ ),∀ l, l ′ = 1, . . ., L˜. (4.16)The optimization problem in (4.15) is again non-convex and can be solved for a local optimum using theconjugate gradient [180] method.4.2.3 Prediction with reconstructed test DRSS vectorsIn the prediction phase of the RecGP method, we reconstruct the noisy test DRSS matrix P̂ from a principalsubspace formed by the first M0 PCs of the noise-free DRSS. When training with noise-free DRSS, thereconstruction is done from the training DRSS matrix P˜∗ in as follows [183]P̂(rec) = P̂V[M0]V[M0]T , where, P˜∗ = UDVT . (4.17)In (4.17), V[M0] is the matrix of the first M0 right singular vectors of the noise-free training DRSS matrixP˜∗. When we instead train with the noisy DRSS, we utilize the simulated noise-free DRSS matrix P¯∗ from(4.12) for the reconstruction, as followsP̂(rec) = P̂V[M0]V[M0]T , where, P¯∗ = UDVT . (4.18)In (4.18), V[M0] is the matrix of the first M0 right singular vectors of the simulated noise-free DRSS matrixP¯∗. The reconstruction steps in (4.17) and (4.18) facilitate noise reduction in the test DRSS, as illustratedthrough numerical examples in Section 4.4.To predict the test x−coordinate vector x̂, the same procedure as in (3.15) - (3.17) is followed, but withthe reconstructed DRSS P̂(rec) in place of the original test DRSS P̂. In short, the joint distribution of thetraining and test x−coordinate vectors x˜ and x̂ is firstly obtained from the standard GP assumption, and theposterior predictive density of x̂ is then obtained by conditioning the joint distribution over x˜. When the GP82model is trained with the noise-free DRSS P˜∗, the posterior predictive density of the test x-coordinate [̂x]lis given by[̂x]l |˜x, P˜∗, p̂(rec)l ∼ N([µ̂(RecGP)x ]l, [Ĉ(RecGP)x ]ll), where[µ̂(RecGP)x ]l =L˜∑i=1φ(̂p(rec)l, p˜∗i )[(Φ˜)−1x˜]i, and[Ĉ(RecGP)x ]ll = φ(̂p(rec)l , p̂(rec)l )−L˜∑i=1L˜∑j=1{φ(̂p(rec)l, p˜∗i )[(Φ˜)−1]i jφ(p˜∗j , p̂(rec)l )}. (4.19)Remark 4.1. [Location estimates from the RecGP method when trained with noisy DRSS vectors] Whenthe RecGP model is instead trained with noisy DRSS and we reconstruct both the training and test DRSSmatrices P˜ and P̂ from the noise-free DRSS matrix P¯∗ in (4.12), the posterior density of the test x-coordinate[̂x]l is similar to (4.19), but with p˜∗i replaced by p˜i(rec) ∀i = 1, . . . L˜ and Φ˜ replaced by Φ˜(rec).In (4.19), [µ̂(RecGP)x ]l and [Ĉ(RecGP)x ]ll are the predicted mean and variance of the x−coordinate [̂x]l ofthe test device l. Since [̂x]l is Gaussian distributed, [µ̂(RecGP)x ]l gives us the maximum-a-posteriori estimateof [̂x]l. Also, the ±2√[Ĉ(RecGP)x ]ll values represent the 2σ error-bars on choosing [µ̂(RecGP)x ]l as the estimateof [̂x]l. The RecGP method treats the reconstructed test DRSS as deterministic vectors, even though theyare stochastic in nature (due to the presence of shadowing noise). As a consequence, the RecGP methodprovides optimistic 2σ error bars on the estimated locations. This issue is addressed in the next section,where we propose the RecGaGP method. The RecGaGP method takes the statistical distribution of thereconstructed test DRSS vectors into account when estimating the test device location, and is consequentlyable to provide more realistic 2σ error-bars on the estimated locations when compared to the RecGP method.4.3 Reconstruction Gaussian Approximation GP (RecGaGP) Method forLocation PredictionWe now propose the RecGaGP method, which (i) reconstructs the training and test DRSS vectors from alow-dimensional principal subspace spanned by the noise-free training DRSS in the service area (as donein the RecGP method), and (ii) applies the Gaussian approximation moment-matching procedure followedin the GaGP method (c.f. Section (3.3)) to the reconstructed test DRSS vectors when estimating the test83(a) Training procedure in the RecGaGP method(b) Prediction procedure in the RecGaGP methodFigure 4.3: Overview of the training and prediction procedures in the proposed RecGaGP method. (a) The training procedure isthe same as in the RecGP method. (b) The reconstructed test DRSS P̂(rec), obtained in the same fashion as in RecGP, is treatedas a stochastic input when deriving the posterior distribution of x̂. The true posterior density is analytically intractable, so it isapproximated as a Gaussian distribution with the same first and second order moments (c.f. Theorem 4.1).device locations. While the reconstruction step seeks noise reduction in the DRSS vectors, the Gaussianapproximation moment-matching procedure allows the GP method to take the statistical properties of thereconstructed test DRSS vectors into account, instead of treating them as deterministic vectors as done inthe RecGP method. An overview of the RecGaGP method is presented in Fig. (4.3).The data reconstruction steps in the RecGaGP method are the same as in the RecGP method. In short,when the GP model is trained with a noise-free DRSS matrix P˜∗ (c.f. (4.5)), the RecGaGP method recon-structs the noisy test DRSS P̂ from a low-dimensional principal subspace spanned by P˜∗. When the GPmodel is instead trained with a noisy DRSS matrix P˜, the reconstruction procedure is applied to both thetraining and test DRSS matrices, P˜ and P̂, from a subspace spanned by an independently generated noise-free DRSS matrix P¯∗ (c.f. (4.12)). The training procedure in the RecGaGP method is the same as in RecGPmethod, for both the cases of training with noise-free and noisy DRSS vectors (c.f. Sections (4.2.1) and(4.2.2)).The test location prediction procedure is however different from RecGP in that we no longer treat thereconstructed test DRSS vectors as deterministic inputs to the trained GP model. Instead, we derive thestatistical properties of the reconstructed test DRSS vectors and incorporate this information when derivingthe posterior predictive density. Details are presented below.844.3.1 Location prediction with reconstructed test DRSS vectorsWe firstly reconstruct the test DRSS vectors, following (4.17) for the case of training with noise-free DRSSvectors and (4.18) for the case of training with noisy DRSS vectors, respectively. From (4.17) and (4.18),we know that the reconstructed test DRSS matrix P̂(rec) is given byP̂(rec) = P̂V[M0](V[M0])T , (4.20)where, depending on whether we are training with noise-free or noisy DRSS, the columns of the matrixV[M0] are the first M0 right singular vectors of P˜∗ or P¯∗ (c.f. (4.5) and (4.12)). Since the reconstruction stepin (4.20) is a linear algebraic operation, we can derive the statistical properties of the noise present in P̂(rec)from that of the original test DRSS matrix P̂ as follows. Observe from (4.20) that any reconstructed testDRSS vector p̂reclin P̂(rec) can be expressed in terms of its original counterpart p̂l in P̂ asp̂(rec)l= (̂pTl V[M0](V[M0])T )T ∀l = 1, . . . L̂,= (̂p∗Tl V[M0](V[M0])T )T + (̂zTl V[M0](V[M0])T )T , (from (3.18))= p̂(rec)∗l+ ẑ(rec)l,(4.21)where we have defined p̂(rec)∗l= (̂p∗TlV[M0](V[M0])T )T as the noise-free component and ẑ(rec)l= (̂zTlV[M0](V[M0])T )Tas the residual noise in p̂(rec)lrespectively. Statistical properties of ẑ(rec)lare given by Lemma 4.1 below.Lemma 4.1. The residual noise ẑ(rec)lin the reconstructed DRSS vector p̂(rec)lis Gaussian distributed withmean 0 and covariance Σ̂(rec)l , whose elements are given by[Σ̂(rec)l ]i j =M∑m=1M∑m′=1[Σ̂l]mm′(M0∑m′′=1[V[M0]]mm′′[(V[M0])T ]m′′i)(M0∑m′′=1[V[M0]]m′m′′[(V[M0])T ]m′′ j) ∀i, j = 1, . . ., L̂.(4.22)Consequently, the noise-free component p̂(rec)∗lin p̂(rec)lis conditionally Gaussian distributed as (̂p(rec)∗l|p̂(rec)l, Σ̂l) ∼N(̂p(rec)l, Σ̂(rec)l ).Proof. By definition, since we know ẑ(rec)l= (̂zTlV[M0](V[M0])T )T , we can obtain elements of ẑ(rec)las[̂z(rec)l]m =M∑m′=1M0∑m′′=1[̂zl]m′[V[M0]]m′m′′[(V[M0])T ]m′′m,∀m = 1, . . .,M −1. (4.23)85We conclude from (4.23) and (3.18) that ẑ(rec)lis also Gaussian distributed because each element [̂z(rec)l]min ẑ(rec)lis a weighted sum of MM0 Gaussian random variables. Also, we can obtain the elements of themean of ẑ(rec)lasE([̂z(rec)l]m) = E(M∑m′=1M0∑m′′=1[̂zl]m′[V[M0]]m′m′′[(V[M0])T ]m′′m),=M∑m′=1M0∑m′′=1E([̂zl]m′)[V[M0]]m′m′′[(V[M0])T ]m′′m= 0,∀m = 1, . . .,M −1, (since E([̂zl]) = 0) (4.24)Similarly, if Σ̂(rec)l is the covariance matrix of ẑ(rec)l, elements of Σ̂(rec)l can be obtained as, ∀i, j = 1, . . .,M−1,[Σ̂(rec)l ]i j = [E(̂z(rec)l ẑ(rec)Tl )−E(̂z(rec)l )E(̂z(rec)Tl )]i j,= [E(̂z(rec)lẑ(rec)Tl)]i j (since E(̂z(rec)l ) = 0)(a)= E{M∑m=1M0∑m′′=1[̂zl]m[V[M0]]mm′′[(V[M0])T ]m′′iM∑m′=1M0∑m′′=1[̂zl]m′[V[M0]]m′m′′[(V[M0])T ]m′′ j}(b)= E{M∑m=1M∑m′=1[̂zl]m[̂zl]m′M0∑m′′=1[V[M0]]mm′′[(V[M0])T ]m′′iM0∑m′′=1[V[M0]]m′m′′[(V[M0])T ]m′′ j}=M∑m=1M∑m′=1[Σ̂l]mm′(M0∑m′′=1[V[M0]]mm′′[(V[M0])T ]m′′i)(M0∑m′′=1[V[M0]]m′m′′[(V[M0])T ]m′′ j), (4.25)where (a) is obtained by substituting (4.23) and (b) follows from rearranging the terms in order to arrive atan expression in terms of the elements [Σ̂l]mm′ of the original test covariance matrix. Combining (4.24) and(4.25) gives us the probability distribution ẑ(rec)l∼ N(0, Σ̂(rec)l ), with elements of Σ̂(rec)l given by Lemma(4.1). In addition, since we also know from (4.21) that p̂(rec)l= p̂(rec)∗l+ ẑ(rec)l, we know that p̂(rec)∗lisconditionally distributed as(̂p(rec)∗l|p̂(rec)l, Σ̂(rec)l ) ∼ N(̂p(rec)l , Σ̂(rec)l ). (4.26)This completes the proof. 86Lemma 4.1 gives us the probability distribution of the residual noise ẑ(rec)lpresent in the reconstructedtest DRSS vectors, along with the conditional distribution of the noise-free components (̂p(rec)∗l|p̂(rec)l, Σ̂l)in the reconstructed test DRSS. In order to estimate the test device locations, we can use this statisticalknowledge to apply the Gaussian moment-matching approximation procedure followed in the GaGP method(c.f. (3.24)) to the reconstructed test DRSS vectors. Doing so gives us closed-form expressions for theestimated mean and variance of the test devices’ x−coordinates, as summarized in Theorem 4.1 below.Theorem 4.1. Using the weighted-sum covariance model in (4.6), the RecGaGP method provides the fol-lowing closed-form expressions for the estimated mean [µ̂(RGP)x ]l and variance [Ĉ(RGP)x ]ll of the x−coordinate[̂x]l. When the RecGaGP is trained with noise-free DRSS P˜∗, we have[µ̂(RGP)x ]l =L˜∑i=1α′[ψ]iN (̂p(rec)l ; p˜∗i,B+ Σ̂(rec)l )+γ[ψ]i (̂p(rec)l )T p˜∗i, and[Ĉ(RGP)x ]ll = σ2er + (α+γ(̂p(rec)l )T p̂(rec)l +γTr(Σ̂(rec)l ))−L˜∑i=1L˜∑j=1[ξ]i j{(α′)2N(p˜∗i; p˜∗ j,2B)N (̂p(rec)l ;p˜∗i + p˜∗ j2,B2+(4.27)Σ̂(rec)l )+ [Υ(rec)]Tj p˜∗i + (p˜∗ j)T [Υ(rec)]i +γ2(̂p(rec)l )T p˜∗i(p˜∗ j)T p̂(rec)l +γ2 Tr(p˜∗i(p˜∗ j)T Σ̂(rec)l )}−([µ̂(RGP)x ]l)2,respectively, where the matrix Υ(rec) ∈ RM−1×L˜ is defined such that, ∀i = 1, . . ., L˜,[Υ(rec)]i = α′γN (̂p(rec)l ; p˜∗i,B+ Σ̂(rec)l )(B−1 + (Σ̂(rec)l )−1)−1(B−1p˜∗i + (Σ̂(rec)l )−1p̂(rec)l ), (4.28)and the vector ψ and matrix ξ are given byψ = Φ˜−1x˜, and, [ξ]i j = [Φ˜−1]i j −[ψ]i[ψ]j, ∀i, j = 1, . . ., L˜. (4.29)Proof. (Sketch) We apply the analytical moment-matching approximation procedure followed in the GaGPmethod ( c.f. 3.24) to predict the test device l’s x−coordinate [̂x]l from its reconstructed DRSS vector p̂(rec)l .We treat the noise-free component p̂(rec)∗lin p̂(rec)las the deterministic input to the trained RecGaGP modeland use (4.19) to obtain the expressions for the mean and variance of [̂x]l in terms of p̂(rec)∗l . The p̂(rec)∗l87is then integrated out, using the conditional distribution (̂p(rec)∗l|p̂(rec)l, Σ̂l) ∼ N(̂p(rec)l , Σ̂(rec)l ) from Lemma4.1, by following the steps (3.20) and (3.24), so as to obtain the closed-form expressions for [µ̂(RGP)x ]l and[Ĉ(RGP)x ]ll in terms of p̂(rec)l , as given by (4.27). Notice that the [µ̂(RGP)x ]l and [Ĉ(RGP)x ]ll in (4.27) are the sameas those given by the GaGP method in (3.25), but with the original DRSS vectors replaced by the appropriatereconstructed DRSS vectors. This is because the RecGaGP method employs the data reconstruction steps(4.13) and (4.17)-(4.18) prior to applying the analytical moment-matching approximation on the posteriorpredictive density, whereas the GaGP method directly works with the original DRSS vectors. Remark 4.2. [Location estimates from the RecGaGP method when trained with noisy DRSS vectors] Whenthe RecGaGP model is trained with noisy DRSS and we reconstruct both the training and test DRSS matricesP˜ and P̂ from the noise-free DRSS matrix P¯∗ in (4.12), the posterior density of the test x-coordinate [̂x]l issimilar to (4.27), but with p˜∗i replaced by p˜i(rec) ∀i = 1, . . . L˜ and Φ˜ replaced by Φ˜(rec).Location estimates obtained from the RecGP and RecGaGP method, as given by (4.19) and (4.27)respectively, incur smaller RMSE values when compared to the CGP, NaGP, and GaGP methods presentedin Chapter 3 because the data reconstruction step facilitates noise reduction in the DRSS vectors. Also, theRecGaGP method provides more realistic 2σ error-bars on the estimated locations than the RecGP methodbecause, similar to the GaGP method, it takes the stochastic nature of the input test DRSS vectors intoaccount. Numerical studies in Section 4.4 confirm the improved localization performance achieved by theRecGP and RecGaGP methods.4.4 Numerical StudiesWe now present numerical examples to evaluate the performance of the proposed GP methods. We considerthe same localization setup as in Chapter 3, spanning a service area of 20 m ×20 m, with L˜ = 100 traininglocations, M = 31 anchor nodes, and Lˆ = 25 test device locations distributed uniformly across the servicearea. The RSS vectors are generated using (3.2) with the simulation parameters as per Table 3.1. The anchorat the left-bottom corner is used as the reference to obtain the 30×1 DRSS vectors for each device location.The shadowing standard deviation in the RSS vectors σm is varied from −8 dB to 8 dB. All available devicelocations are assumed to have a measurement error standard deviation (σer ) of −20 dB. We generate 200Monte-Carlo test DRSS matrices each for shadowing standard deviation σm = −8,−6, . . .,8 dB in the test88RSS. The average performance over the Monte-Carlo realizations is reported.When training with noise-free DRSS, the training dataset comprises of a 100×30 DRSS matrix and thecorresponding x-coordinate and y-coordinate vectors of the training device locations. The same matrix isutilized for the data reconstruction step (c.f. (4.17)) during the test prediction phase. On the other hand,when training with noisy DRSS, we generate 200 Monte-Carlo realizations of the DRSS vector for eachtraining location. The resulting dataset constitutes a 20000×30 training DRSS matrix and the correspondingx-coordinate and y-coordinate vectors. For the data reconstruction steps (4.13) and (4.18), we considerL¯ = 100 randomly chosen locations in the service area and generate the noise-free DRSS matrix P¯∗ using(3.2). Since the dataset is quite large in the case of noisy training DRSS, we train an ensemble of 20 GPmodels, each with 400 DRSS vectors sampled randomly without repetition from the full training dataset,and report the average performance.Once the training DRSS data is ready, each GP model is trained by solving the log-likelihood max-imization problem in (4.9) using conjugate gradient (CG) method [180]. We run 20 trials with randominitializations and choose the local optimum which gives the best objective value, i.e., log(p(˜x|P˜,θ)).We measure the localization performance in terms of the RMSE, defined asRMSE =1L̂L̂∑l=1√([̂x]l −[µ̂(.)x ]l)2 + ([̂y]l −[µ̂(.)y ]l)2 (4.30)where [̂x]l and [̂y]l are the actual x and y coordinates of the test device l, [µ̂(.)x ]l and [µ̂(.)y ]l are the estimatesof [̂x]l and [̂y]l as given by the GP method under study, and [Ĉ(.)x ]ll and [Ĉ(.)y ]ll are the variances associatedwith the estimates [µ̂(.)x ]l and [µ̂(.)y ]l, respectively. For example, if we choose RecGP, [µ̂(.)x ]l = [µ̂(RecGP)x ]l,[µ̂(.)y ]l = [µ̂(RecGP)y ]l, [Ĉ(.)x ]ll = [Ĉ(RecGP)x ]ll and [Ĉ(.)y ]ll = [Ĉ(RecGP)y ]ll. Smaller RMSE values indicate betterprediction performance.For performance comparison, we firstly consider the three baseline algorithms used in the previouschapter, namely the maximum-likelihood estimator (MLE) in [55], the robust semidefinite programming-based estimator (RSDPE) in [61], and the k-nearest-neighbour (kNN) algorithm in [71] with the numberof nearest neighbours set to 4. As additional baselines, we consider the CGP, NaGP, and GaGP methodspresented in Chapter 3.89Number of principal components (M0)Increasing MFigure 4.4: Average data loss and the average residual noise variance in the test DRSS, when the number of PCs M0 for thereconstruction step is varied from 0 to M −1, for M ranging from 21 to 61.4.4.1 Impact of the reconstruction stepLet us first study the impact of the reconstruction step (4.17) on the test DRSS. We know from (3.18) thatwe can express any noisy test DRSS p̂l as the sum of a noise-free term p̂∗l and a shadowing noise term ẑl asp̂l = p̂∗l + ẑl, such that ẑl ∼ N(0, Σ̂l), (4.31)where Σ̂l is the noise covariance matrix. The corresponding reconstructed DRSS p̂(rec)l contains a noise-freeterm p̂(rec)∗land a noise term ẑ(rec)l, as shown in (4.21). From Lemma 4.1, we know that ẑ(rec)lis Gaussiandistributed with zero mean and covariance Σ̂(rec)l . In Fig. 4.4, we plot the average data loss in the test DRSS,calculated as | |p̂∗l− p̂(rec)∗l| |2/| |p̂∗l| |2, and the average variance of the residual noise in the reconstructed testDRSS, calculated as (Trace(Σ̂(rec)l )/Trace(Σ̂l)), for σm =−8, . . .,8 dB. We notice that the reconstruction step(4.17) facilitates noise reduction: when the first few PCs are used, the noise variance in the reconstructedtest DRSS is lower than the original shadowing variance whereas the data loss is negligible.4.4.2 Localization RMSE PerformanceIn Fig. 4.5, we illustrate the average localization RMSE performance achieved by the proposed GP methods,when the shadowing standard deviation (σm) in the RSS is increased from −8 dB to 8 dB. Several interesting90observations can be made from Fig. 4.5. Firstly, when compared to the CGP, NaGP, and GaGP methods,the RecGP and the RecGaGP methods achieve a superior RMSE performance both when trained with noise-free and noisy DRSS vectors. We attribute this improvement in performance to the noise reduction inthe DRSS due to the data reconstruction steps (c.f. (4.22)). Secondly, when compared to the MLE andRSDPE solutions, the RecGP and RecGaGP methods facilitate a smaller RMSE over the entire range of theshadowing standard deviation σ in the case of noisy training DRSS. When trained with noise-free DRSS,a similar observation is made for small values of σm. The improvements in the RMSE come primarilyfrom employing machine learning for the localization task. The MLE and RSDPE solutions work on thetest DRSS alone, whereas the GP models additionally learn the mapping functions fx(.) and fy(.) fromthe training DRSS information. Thirdly, the RecGP and RecGaGP methods perform better than the kNNalgorithm for most σ in the case of noisy training DRSS and for small σ in the case of noise-free trainingDRSS. This is mainly because the proposed GP methods pursue non-linear regression to learns the functionsfx(.) and fy(.). The kNN algorithm, on the other hand, relies on finding a neighbour set in the training datathat accurately surrounds the test location, and is therefore sensitive to the distribution and the density of thetraining locations. The uniformly distributed training locations considered in our localization setup presentsan optimistic case for the kNN algorithm. Fourthly, we note that the RecGP and RecGaGP methods providesimilar RMSE values for different shadowing noise levels. In Chapter 3, we found that the CGP, and itsnumerical and analytical moment-matching counterparts, namely the NaGP and GaGP methods respectively,achieve similar RMSE performance. We make a similar observation here, in the case of RecGP and itsanalytical moment-matching counterpart, the RecGaGP method. Lastly, when the GP models are trainedwith noise-free DRSS, the obtained RMSE values from the RecGP and RecGaGP methods are higher thanwhen trained with noisy DRSS vectors. This is because there is an inherent mismatch between the trainingand test data in the former case, and as a result, both the RecGP and RecGaGP methods end up projectingthe noise present in the test DRSS onto the output location coordinate space. For the same reason, we alsonote that the rate of increase in the RMSE with σ is higher in the case of training with noise-free DRSS.Choosing the number of principal components (M0)We may note from Section 4.4.1 that the reconstruction step not only facilitates noise reduction in the testDRSS, but also incurs data loss. Increasing the number of principal components (M0) increases the amountof information retained in the test DRSS upon reconstruction and also the amount of residual noise present in91the reconstructed test DRSS. Consequently, we expect the RMSE performance of the RecGP and RecGaGPmethods to vary with M0.When training with noisy DRSS vectors, we choose M0 in the reconstruction step as the minimum num-ber of singular values which explain 99% of the energy contained in the matrix P¯∗ used for the reconstructionstep (c.f. (4.12)). When training with noise-free DRSS, we take a more elaborate approach, as describedbelow.In Fig. 4.6a, we consider noise-free training DRSS and plot the average RMSE values obtained fromRecGP when M = 30, for M0 ranging from 2 to 30. For low M0, we observe large RMSE values becausemost of the information contained in the test DRSS P̂ is lost in the reconstruction step (4.17). When M0 isincreased, the RMSE values decrease initially because of the increase in the information retained in the testDRSS upon reconstruction. This trend continues until a certain M0 and the RMSE values attain a minimumlevel, followed by a gradual upward trend. This is because the noise-free training DRSS spans a low-dimensional principal subspace and any further increase beyond a certain M0 would not increase the amountof information retained in the test DRSS but the amount of residual noise would continue to increase. Theincrease in RMSE with M0 is more prominent for higher noise levels because of the increase in the amountof residual noise in the reconstructed test DRSS (c.f. (4.22)). Lastly, we observe that the RMSE-minimizingM0 is different for different noise levels. Choosing an appropriate M0 is, therefore, an important decisionwhich we make using the following heuristic approach.We generate a hold-out validation set containing 200 Monte-Carlo realizations of the DRSS vectors at25 device locations which are distributed uniformly across the service area. As an example, we chooselocations that are at an offset of 1 m along the positive x and y directions from the test device locations. Wethen plot, in Fig. 4.6b, the histograms of the RMSE-minimizing M0 for this dataset. We observe that thehistograms generally have multiple peaks for each shadowing variance level. We therefore use the followingheuristic to choose an M0: for a given noise level, we run the GP algorithms on the hold-out validationdataset and select the most-frequent RMSE-minimizing M0 value from the histograms. When reporting theaverage RMSE performance of the RecGP method in Fig. 4.5, we followed this heuristic approach to chooseM0. For a fair comparison, the same set of M0 values are chosen for the RecGaGP method as well.92Shadowing noise standard deviation (dB)Localization RMSE (m)CGPGaGPNaGPRecGPRecGaGPRSDPEMLEkNN(a) Training with noisy DRSSShadowing noise standard deviation (dB)Localization RMSE (m)CGPGaGPNaGPRecGPRecGaGPRSDPEMLEkNN(b) Training with noise-free DRSSFigure 4.5: Average localization RMSE achieved by the proposed GP methods. In (a), the number of PCs M0 in the reconstructionstep is chosen as the minimum number of singular values which explain 99% of the energy contained in the matrix P¯∗ used forthe reconstruction step (c.f. (4.12)). In (b), M0 is chosen through empirical studies as the one that most frequently gave the lowestRMSE over a hold-out validation dataset containing DRSS vectors at 25 uniformly distributed device locations in the service area.Number of principal components (M 0)Localization RMSE of RecGP (m) = -8 dB = 8 dB(a) Average RMSE vs. M0 in the RecGP method whentrained with noise-free DRSS(b) Histogram of the RMSE-minimizing M0 over avalidation dataset containing DRSS vectors at 25uniformly distributed device locations in the service area.Figure 4.6: Plots of the average RMSE vs. M0 performance of the RecGP method and the histogram of the RMSE-minimizingM0 values, when the GP model is trained with noise-free DRSS. From Fig. 4.6a, we observe that the RMSE decreases with M0initially, attains a minimum, followed by a gradual increase. From Fig. 4.6b, we observe that the histograms have multiple peakvalues.934.4.3 2σ Error-Bar PerformanceWe now plot the average 2σ error-bars given by the proposed RecGP and RecGaGP methods when trainedwith noise-free DRSS in Fig. 4.7 and when trained with noisy DRSS in Fig. 4.8. In the figures, we also plotthe number of true test device locations that are within the 2σ confidence range ([µ̂(.)x ]l ±2√[Ĉ(.)x ]ll, [µ̂(.)y ]l ±2√[Ĉ(.)y ]ll) of the estimated locations. From among the baselines, we only include the CGP, NaGP, andGaGP methods because the others do not provide the 2σ error-bars around the location estimates.We observe that, similar to the CGP method, the RecGP method provides optimistic 2σ error-bars bothwhen trained with noise-free and noisy DRSS vectors. Particularly, in the case of noise-free training DRSS,the 2σ error-bars are unrealistically small and are not comparable to the corresponding RMSE values inFig.4.5. As a result, we note that less than 40% of the true test device locations fall inside the 2σ confidencerange of the RecGP method when trained with noise-free DRSS (c.f. Fig. 4.7c). The RecGaGP methodconsistently provides more realistic 2σ error-bars on the estimated locations, with more than 95% of thetrue test device locations within the confidence range of the location estimates (c.f. Fig. 4.7c). When trainedwith noisy DRSS, the performance of the RecGP method improves, with up to 90% of the true locationswithin the confidence range of the estimated locations (c.f. Fig. 4.8c). The performance improvementoffered by the RecGaGP method is now diminished, but it still performs at least as well as, if not betterthan the RecGP method (c.f. Fig. 4.8c). Overall, when compared to the RecGP method, we note that theRecGaGP method achieves improved performance, in terms of the estimated 2σ error-bars. In addition, theRecGaGP method performs similar to the NaGP and GaGP methods in providing realistic 2σ error-bars,while incurring smaller RMSE values both in the case of noise-free and noisy training DRSS (c.f. Fig. 4.5).4.4.4 Impact of the number of sensor nodesIn Fig. 4.9 we plot the average RMSE performance of the RecGP and RecGaGP methods when the numberof DRSS sensor nodes M−1 is increased from 10 to 50. As M increases, we notice a decrease in the reportedRMSE values for both the GP methods. This may be attributed to two reasons: (i) the increased averagesignal strength at the sensor nodes, which is a consequence of the reduced average sensor-to-device distance,and (ii) the extra DRSS information that becomes available to the GP models due to the increase in the num-ber of sensor nodes. Also, we note that the RecGP and RecGaGP methods consistently achieve improvedRMSE performance over the CGP, NaGP, and GaGP methods. The data reconstruction step facilitates the94Shadowing noise standard deviation (dB)CGPGaGPNaGPRecGPRecGaGP(a) Average 2σ error-bar on the estimated x-coordinatesShadowing noise standard deviation (dB)CGPGaGPNaGPRecGPRecGaGP(b) Average 2σ error-bar on the estimated y-coordinatesShadowing noise standard deviation (dB)True locations within confidence interval (%)CGPGaGPNaGPRecGPRecGaGP(c) True locations within the confidence range of the location estimatesFigure 4.7: [Training with noise-free DRSS] Plots of the average 2σ error-bars on the estimated locations, as provided by theRecGP and RecGaGP methods. The RecGP method performs similar to the CGP method and provides unrealistically small 2σerror-bars, with less than 40% of the true device locations within the 2σ confidence range of the predicted locations. The RecGaGPmethod performs similar to the NaGP and GaGP method and provides more realistic 2σ error-bars on the location estimates, withmore than 95% of the true device locations within the 2σ confidence range of the estimated locations.95Shadowing noise standard deviation (dB)CGPGaGPNaGPRecGPRecGaGP(a) Average 2σ error-bar on the estimated x-coordinatesShadowing noise standard deviation (dB)CGPGaGPNaGPRecGPRecGaGP(b) Average 2σ error-bar on the estimated y-coordinatesShadowing noise standard deviation (dB)True locations within confidence interval (%)CGPGaGPNaGPRecGPRecGaGP(c) True locations within the confidence range of the location estimatesFigure 4.8: [Training with noisy DRSS] Plots of the average 2σ error-bars on the estimated locations, as provided by the RecGPand RecGaGP methods. Although the 2σ error-bars from the RecGP method are not as optimistic as with noise-free training DRSS,we note that the RecGaGP method continues to perform better than the RecGP.96Number of the DRSS sensorsLocalization RMSE (m)CGPGaGPNaGPRecGPRecGaGP = 2dB = 0dB = -2dB(a) Training with noise-free DRSSNumber of the DRSS sensorsLocalization RMSE (m)CGPGaGPNaGPRecGPRecGaGP = -4dB = 8dB(b) Training with noisy DRSSFigure 4.9: Plot of the average localization RMSE of the RecGP and RecGaGP methods when the number of DRSS sensor nodes(M −1) is increased from 10 to 50.reduction in RMSE, and the impact is more significant with increasing shadowing noise variance.Taking both the RMSE and 2σ error bar plots into consideration, we note the following about theproposed RecGP and RecGaGP methods. Firstly, the RecGP method achieves lower localization RMSEwhen compared to the CGP, NaGP, and GaGP methods, owing to the data reconstruction step introducedinto the localization procedure. Secondly, despite the improvement in RMSE performance, the RecGPmethod provides optimistic 2σ error-bars because, similar to the CGP method, it does not take the stochasticnature of the reconstructed test DRSS into account when deriving the posterior predictive density of the testdevice locations. Thirdly, the above limitation is overcome in the RecGaGP method, in that it takes thestatistical distribution of the reconstructed test DRSS into account and provides more realistic 2σ error-bars, as is the case with the NaGP and GaGP methods. Also, similar to the RecGP method, the RecGaGPmethod achieves improved RMSE performance over the CGP, NaGP, and GaGP methods, owing to thenoise reduction from reconstructng the test DRSS. Overall, the RecGaGP method is seen to deliver the bestlocalization performance among the considered GP methods.We now make the Remark 4.3 below about estimating multiple test device locations simultaneouslyusing the RecGP and RecGaGP methods.Remark 4.3. A common attribute of the proposed RecGP and RecGaGP methods is that they allow us topredict multiple test device locations simultaneously, without affecting the localization accuracy for anygiven test device. Also, for cost savings, we can pre-compute certain terms and reuse them when calculatingthe location estimates in parallel.97Proof. We note from (4.20) and (4.22) that we can compute the reconstructed test DRSS vectors p̂(rec)land the residual noise covariance matrices Σ̂(rec)l for the L̂ test devices in parallel. Once the p̂(rec)landΣ̂(rec)l values are available, we can also parallelize the computation of the [µ̂(RecGP)x ]l, [Ĉ(RecGP)x ]ll, [µ̂(RGP)x ]land [Ĉ(RGP)x ]ll values for the L̂ test devices. As may be observed from (4.19) and (4.27), this is becausethe computation of [µ̂(RecGP)x ]l, [Ĉ(RecGP)x ]ll, [µ̂(RGP)x ]l and [Ĉ(RGP)x ]ll does not rely on the correspondingcomputations of any other device l ′. When parallelizing the computation of the location estimates andtheir 2σ error-bars, we can also reduce the cost of each GP method by pre-computing few terms whichare re-usable across the L̂ devices. For example, we can pre-compute the matrix Φ˜−1 and the vector Φ˜−1x˜when computing the [µ̂(RecGP)x ]l and [Ĉ(RecGP)x ]ll in the RecGP method (c.f. (4.19)). Similarly, we can pre-compute the terms α′, ψ, ξ , and N (˜pi; p˜j,2B) when using (4.27) to calculate the [µ̂(RGP)x ]l and [Ĉ(RGP)x ]llin the RecGaGP method. In Appendix B.4, we have presented a detailed analysis on the computationalcomplexity of the proposed RecGP and RecGaGP methods. 4.5 SummaryIn this chapter, we proposed two GP methods, namely the RecGP and the RecGaGP, for the intruder devicelocalization from the uplink DRSS. The proposed GP methods exploit the low-dimensionality of the noise-free DRSS vectors to incorporate a data recontruction step into the localization procedure. Specifically, theRecGP method reconstructs the noisy test/training DRSS vectors from a principal subspace spanned by thenoise-free DRSS vectors. The reconstructed test DRSS vectors are treated as deterministic inputs, resultingin optimistic 2σ error bars on the estimated locations. The RecGaGP method addresses this concern by tak-ing the stochastic nature of the reconstructed test DRSS into account when deriving the posterior predictivedistribution of the test locations. Analytical moment-matching is employed in order to approximate the pos-terior distribution as a Gaussian distribution and subsequently derive closed-form expressions for both themean and variance of the test locations. Numerical studies reveal that (i) the data reconstruction step facili-tates improvements in the localization RMSE performance when compared to the GP methods presented inChapter 3, (ii) the RecGaGP method provides more realistic 2σ error-bars on the estimated locations whencompared to the RecGP method, both when training with noise-free and noisy DRSS vectors, and (iii) theRMSE performance of the proposed GP methods improves with the increase in the number of sensor nodes.In the localizations algorithms presented so far in Chapters 3 and 4, for simplicity of exposition, we98have restricted ourselves to open air environments and considered a constant PLE in the numerical studies.In practice, when more complex environments are considered, the PLE varies across the service area dueto changes in the structure and geometry of the environment. For example, the previous works [99–101]conducted empirical studies in specific urban and suburban environments and found that the measured PLEsfollow a Gaussian distribution. In the next chapter, we investigate DRSS-based localization when the PLEis known to follow a Gaussian distribution in the service area.99Chapter 5Linear Least Squares Algorithms forIntruder Localization under GaussianDistributed Path Loss ExponentIn this chapter, we investigate intruder device localization when the PLE, which denotes the rate of signalstrength decay with distance, is known to follow a Gaussian distribution in the service area. Similar tothe Chapter 2 and 3, we assume that the transmit power of the intruder is unknown and we handle thisproblem by working with the DRSS with respect to a reference sensor. Keeping the computational costin mind, we pursue the following ranging approach. First, we take the statistical distribution of the PLEinto account and derive maximum-likelihood estimates of the ratio-of-distances to the sensors in terms ofthe DRSS values. Next, we use the ratio-of-distance estimates to formulate a linear system of equations inthe source location. We then propose a two-step linear least squares (TLLS) estimator and a constrainedlinear least squares (CLLS) estimator which exploit the known relation between the source coordinates andthe range variable to derive the location estimates. Finally, we derive the maximum a posteriori (MAP)estimator for the source location which maximizes the posterior likelihood of the DRSS values in orderto obtain the location estimates. Numerical studies confirm that the TLLS and CLLS estimators achievesimilar localization performance as the existing semidefinite programming solution, while incurring smallercomputational cost. Moreover, the TLLS estimates serve as a warm start for the MAP estimator and facilitatea smaller localization RMSE when compared to other standard choices for the initial point.Our main contributions in this chapter are as follows.• We investigate the intruder device localization from the uplink DRSS when the PLE in the servicearea is known to be Gaussian distributed. We propose a ranging-based solution, in which we derivemaximum-likelihood estimates of the ratio-of-distances to the sensors from the DRSS values and100formulate a linear system of equations in the source location. Two low-complexity linear least squaresalgorithms are then proposed to estimate the device location from the ratio-of-distances.• We derive the maximum a posteriori estimator for the source location which maximizes the jointposterior likelihood of the DRSS values, given the statistical distribution of the PLE. We also derivethe hybrid Cramer-rao lower bound (HCRLB) on the achievable RMSE performance.• We demonstrate through numerical studies that the location estimates from the proposed linear leastsquares algorithms achieve similar localization RMSE as the existing semidefinite programming so-lution, but at reduced computational cost. Besides, they serve as a warm start for more accuratelocalization when recursive updates are pursued in the MAP estimator.The rest of the chapter is organized as follows. In Section 5.1, we present details on the localizationsetup. The maximum-likelihood ratio-of-distance estimator is proposed in Section 5.2. The linear leastsquares and maximum a posteriori estimators for the device location are presented in Section 5.2. Numericalstudies are then presented in Section 5.4 to validate the performance of the proposed estimators, followedby concluding remarks in Section 5.5.The mathematical notation in the current chapter is the same as in Chapter 3.5.1 Localization SetupWe consider a setup where M distributed anchors record the RSS of the signals from an unknown source.When the transmit power is ρ, the RSS pm at anchor m under log-normal shadowing and frequency-flatfading ispm = ρg0d−γm 10σmzm10 |hm |2, where,dm = | |s− âm | |,∀m = 1, . . .M, and zm ∼ N(0,1).(5.1)In (5.1), γ is the PLE, dm is the distance from source to anchor m, g0 is the path loss at a reference distanceof 1 m, σm is the shadowing standard deviation, and zm is the shadowing noise coefficient. The hm is thesmall-scale fading coefficient, the in-phase and quadrature-phase components of which are assumed to beindependent and identically distributed Gaussian random variables with mean zero and variance of 0.5. Thepower term |hmk |2 then follows an exponential distribution with unit mean. The vectors s = [x; y]T andâm = [x̂m; ŷm]T denote locations of the source and the anchor m respectively. From the RSS in (5.1), we101assume that the small-scale fading coefficient is averaged out over time [34, 48, 61] and convert to log-scaleto obtainloge pm = loge(ρg0)−γ loge dm+0.1loge 10σmzm. (5.2)To estimate the source location s from the RSS values loge pm, a standard approach in least squares-basedranging algorithms is to firstly estimate the source-to-anchor distances dm and then employ multilateration.However, when ρ and γ are unknown, we cannot estimate dm from pm (c.f. (5.2)). As a workaroundfor the unknown ρ, we calculate the DRSS w.r.t. a reference anchor and use the ratio-of-distances for thelocalization. When the anchor m = 1 is the reference, the DRSS qm at the remaining anchors areqm−1 = loge(pm)− loge(p1), ∀m = 2, . . .M,= −γ loge δm+λmζm, where, δm =dmd1, ζm = zm− z1. (5.3)In (5.3), δm is the ratio-of-distances w.r.t. the reference, ζm ∼ N(0,1) is a standard normal r.v., and λ2m =(0.1loge(10))2(σ2m+σ21 ) is the noise variance. As may be noted from (5.2), the DRSS qm in (5.3) does notdepend on the transmit power ρ. In addition, from (5.3), we haveqm;δm |γ ∼ N(−γ loge δm, λ2m). (5.4)We study the case when the PLE is Gaussian distributed as γ ∼ N(µγ,σ2γ ), where µγ and σ2γ are the meanand variance of the PLE, and propose a maximum likelihood estimation procedure to obtain the ratio-of-distances δm.5.2 Proposed Estimators for Source LocationWe derive maximum-likelihood estimates of the ratio-of-distances w.r.t. the reference sensor and use themto formulate a linear system of equations in the source location. Details on the ratio-of-distance estimationare presented first.Using (5.4) and the statistical model γ ∼N(µγ,σ2γ ) for the PLE, we obtain the probability density of the102DRSS qm asp(qm;δm) =∫ ∞−∞N(qm;−γ loge δm, λ2m)N(γ; µγ,σ2γ )dγ (5.5)=e−σ2γq2m−λ2mµ2γ2λ2mσ2γ2piλmσγ∫e−σ2γγ2(loge δm )2−λ2mγ2−2σ2γqmγ loge δm+2λ2mγµγ )2λ2mσ2γ dγ.The maximum-likelihood ratio-of-distance estimates δ∗m can then be obtained by setting (∂(p(qm ;δm))∂δm )|δ∗m = 0,i.e., ∫ ∞−∞2σ2γγ2 loge δ∗m+2σ2γqmγδ∗mexp{−σ2γγ2(loge δm)22λ2mσ2γ+−(λ2mγ2 +2γσ2γqm loge δm−2λ2mγµγ)2λ2mσ2γ}dγ = 0.(5.6)We now eliminate constant terms, introduce the auxiliary variables a=√σ2γ (loge δ∗m)2+λ2m√2λmσγand b=σ2γqm loge δ∗m−λ2mµγλmσγ√2(σ2γ loge δ∗m+λ2m)to simplify the exponential term, and multiply by e−b2/2 to complete the quadratic, to obtain∫ ∞−∞(γ2 loge δ∗m+γqm)e−(aγ+b)2dγ = 0. (5.7)Next, using the change of variable u = aγ+ b gives us∫ ∞−∞{(u− ba)2 loge δ∗me−u2+ (u− ba)qme−u2}1a du = 0. (5.8)Re-arranging and simplifying with∫ ∞−∞ u2e−u2du =√pi2 ,∫ ∞−∞ ue−u2du = 0, and∫ ∞−∞ e−u2du =√pi, we obtainloge δ∗ma√pi2+ (b2aloge δ∗m− qmb)√pi = 0 (5.9)Substituting the auxiliary variables a and b back and rearranging gives usλ2mσ4γ (loge δ∗m)3− µγσ2γqmλ2m(loge δ∗m)2+{(σ2γ + µ2γ)λ4m−σ2γq2mλ2m} loge δ∗m+ µγqmλ4m = 0.(5.10)Eq. (5.10) is a cubic polynomial in loge δ∗m, the roots of which can be obtained using the MATLAB rootsfunction [184]. Out of the three roots, at least one is real-valued because the coefficients in (5.10) arereal-valued. In order to select the root which best represents the log-ratio-of-distance, the following root103selection is pursued. When we only have one real root, it is selected because the log-ratio-of-distancecannot be complex-valued. When we have three real roots, we select the one which incurs the smallest errorin the least squares objective of our localization, i.e., | |Aϑ−b| | (c.f. (5.13)). Since the vector ϑ is unknown,we use the estimate ϑ∗WLS obtained from (5.14) with W = I. Note that | |Aϑ −b| | requires the δm estimatesfor all the M −1 anchors, so we need to run through each possible combination of real roots and choose theone that results in the smallest error. We now utilize the obtained ratio-of-distance estimates to formulate alinear system of equations in the source location. This allows us to employ linear least squares to obtain thesource location from the ratio-of-distance estimates δ∗m.5.2.1 Two-step linear least squares (TLLS) estimatorWe seek to obtain the source location s = [x; y]T from the ratio-of-distance estimates δ∗m. By definition, weknowδm =√(x̂m+1− x)2 + (ŷm+1− y)2√(x̂1− x)2 + ((ŷ1− y)2,∀m = 1, . . .,M −1. (5.11)Squaring the terms on both sides and introducing the ratio-of-distance estimate δ∗m, we haveδ2m+ (δ∗m)2−(δ∗m)2 =(x̂m+1− x)2 + (ŷm+1− y)2(x̂1− x)2 + (ŷ1− y)2 ,∀m. (5.12)Defining the vector of unknowns ϑ = [x; y; x2+ y2]T and re-arranging (5.12), we obtain the linear system ofequationsAϑ = b+ e, where,[A]m,: =[2(x̂m+1− x̂1δ∗m2) 2(ŷm+1− ŷ1δ∗m2) δ∗m2−1],[b]m = [x̂2m+1 + ŷ2m+1− δ∗m2(x̂21 + ŷ21)], and,[e]m = [(δ∗m2− δ2m)((x̂1− x)2 + (ŷ1− y)2)]. (5.13)In (5.13), [A]m,: denotes the row m of coefficient matrix A, [b]m the element m of the observation vector b,and [e]m the element m of the residual error vector e. The weighted least squares solution for (5.13) is givenbyϑ∗WLS = (ATWA)−1ATWb, (5.14)104where W is the weight matrix. The optimum W, which is the inverse of the error covariance in linearleast squares, cannot be derived in closed-form (c.f. Appendix D.1). As a result, we make the sub-optimalchoice of W = I, where I is the identity matrix. The resulting linear least squares (LLS) estimate is ϑ∗LLS =(ATA)−1ATb. Next, we introduce a second least squares step for the localization.We know that the unknowns in the vector ϑ satisfy[ϑ]3 = [ϑ]21 + [ϑ]22. (5.15)Following the framework in [78], a second least squares step is derived as follows. When the LLS estimate[[ϑ∗LLS]1; [ϑ∗LLS]2]T is close to the source location [x; y]T , we can make the following linear approximations[ϑ∗LLS]21− x2 ≈ 2x([ϑ∗LLS]1− x), and [ϑ∗LLS]22− y2 ≈ 2y([ϑ∗LLS]2− y). (5.16)Putting (5.15) and (5.16) together, we obtain the following linear system in the unknown vector ϕ = [x2; y2]Tf = Dϕ+n, where, (5.17)f = [[ϑ∗LLS]21; [ϑ∗LLS]22; [ϑ∗LLS]3],D = [1 0;0 1;1 1], andn = [2x([ϑ∗LLS]1− x); 2y([ϑ∗LLS]2− y); [ϑ∗LLS]3− x2− y2].The TLLS estimator solves (5.17) via least squares and then uses the sign of ϑ∗ to obtain the source locationass∗TLLS = sgn([ϑ∗LLS]1:2) ◦√(DTD)−1DT f, (5.18)where s∗TLLS is the estimate of the source location, sgn(.) is the sign function, and ◦ denotes the element-wiseproduct.In the proposed TLLS estimator, we have incorporated the range constraint by introducing a linearapproximation step (c.f. (5.16)). The main advantage of such linear approximation is the low computationalcomplexity, achieved by simply working with a second step of linear least squares. Instead of resorting to alinear approximation of the range variable constraint, we can instead formulate a constrained optimizationproblem, as done in Section 5.2.2 below. To solve the constrained optimization problem, we incur increasedcomputational complexity because an extra optimization variable is introduced, namely, the Lagrangian105multiplier. Below, we present details on how the constrained optimization problem can be formulated andwe can subsequently derive a closed-form solution for the source location. The derived constrained linearleast squares (CLLS) estimator is found to only achieve a marginal improvement in performance over theTLLS estimator, despite the added workload of optimizing the Lagrangian multiplier.5.2.2 Constrained Linear Least Squares EstimatorWe begin by expressing the ratio of distances δm in terms of the source coordinates (x, y), as done in (5.11),and then formulating a linear system of equations in terms of the unknown vector ϑ = [x; y; x2 + y2], asdone in (5.12)-(5.13). Next, we note that the range variable x2 + y2 in ϑ depends on the source locationcoordinates x and y as[ϑ]3 = [ϑ]21 + [ϑ]22. (5.19)Following the linear system in (5.13), we follow the framework proposed in [185] and formulate a con-strained optimization problem to estimate the unknown vector ϑ as followsϑ∗CLLS =argminϑ(Aϑ −b)T (Aϑ −b),s.t. ϑTHϑ +gTϑ = 0, where, H=1 0 00 1 00 0 0and g=00−1(5.20)The problem in (5.20) can now be transformed into an unconstrained one through the use of a Lagrangianparameter β as followsϑ∗CLLS = argminϑ{ (Aϑ −b)T (Aϑ −b)+ β(ϑTHϑ +gTϑ)}. (5.21)The optimum estimate ϑ∗CLLS, in terms of β, is then obtained in closed-form, by setting the gradient w.r.t. ϑto zero, asϑ∗CLLS=(ATA+ βH)−1(ATb−0.5βg), (5.22)106The optimum Lagrangian parameter β∗ can then be obtained by substituting ϑ∗CLLS from (5.22) into theconstraint in (5.20) and solving the resulting fifth degree polynomial (see Appendix D.2 for details). Lastly,we substitute β∗ into (5.22) in order to obtain the optimal estimate ϑ∗CLLS. The source location is then givenby [[ϑ∗CLLS]1; [ϑ∗CLLS]2]T . This completes the localization procedure in the CLLS estimator.Despite working with the constrained optimization problem, as opposed to linearization of the con-straints as done in (5.16), numerical studies reveal that the performance improvement offered by the CLLSestimator is only marginal when compared to the TLLS estimator. Next, we propose a recursive updateestimator for improved localization accuracy over the TLLS and CLLS estimators.5.2.3 Maximum a Posteriori (MAP) EstimatorWe now seek to estimate the source location by maximizing the joint posterior likelihood of the uplinkDRSS values. Denoting the unknown vector as ψ = [s; γ], the MAP estimator of ψ, given the statisticaldistribution of γ, is obtained asψMAP = argmaxψloge p(ψ |q), (5.23)(a)= argminψ(q+γ loge δ)TΣq−1(q+γ loge δ)+(γ− µγ)2σ2γ,where (a) follows from applying Bayes rule, ignoring the constant∫p(q|ψ)p(ψ) in the denominator, notingthat p(ψ) = p(γ) because γ is the only r.v. in ψ, substituting p(γ), and noting from (5.3) that the distributionof q isq|ψ ∼ N(−γ loge δ,Σq), where,Σq = Γ{diag([σ21 ; . . . ;σ2M ])}ΓT ,Γ = [−1;I]. (5.24)In (5.24), 1 denotes a column vector of (M − 1) ones and I denotes the (M − 1) × (M − 1) identity matrix.The problem in (5.23) is non-linear and non-convex in ψ and can be solved for a local optimum usinggradient descent algorithms. The Levenberg-Marquardt (LM) algorithm [186], which is the standard choicefor non-linear least squares, updates the estimate ψ∗k at the end of iteration k as in (5.25) below.107ψ∗k+1 = ψ∗k + {(JTΣ−1q J+hTh+ωkI)−1JTΣ−1q (q+γ loge δ)+µγ −γσγhT }|ψ=ψk , (5.25)where h = [0; 0; −1σγ ]T , and[J]m,: = [ γ(x− x̂m+1)(x− x̂m+1)2 + (y− ŷm+1)2 −γ(x− x̂1)(x− x̂1)2 + (y− ŷ1)2 ;γ(y− ŷm+1)(x− x̂m+1)2 + (y− ŷm+1)2 −γ(y− ŷ1)(x− x̂1)2 + (y− ŷ1)2 ;logδm],∀m (5.26)In (5.25), ωk is the damping factor for iteration k, h is the Jacobian vector ∂∂ψ(γ−µγ )σγ, J is the Jacobian matrix∂∂ψ (q+ γ loge δ) with rows as given in (5.25), and the subscript {.}|ψ=ψk denotes the evaluation at ψ = ψk .The LM algorithm progresses as follows. If the objective in (5.23) decreases with the update ψ∗k+1, thedamping factor ωk is scaled down by a fixed factor τ, and if otherwise, the update is reset to ψk and the ωkis scaled up by τ. Convergence is achieved when the change in ψ or the change in the objective in (5.23)becomes smaller than a negligible  . The convergence rate of the LM algorithm is quadratic when ψk isclose to the optimal value and linear when otherwise. This is because the LM algorithm is similar to theGauss-Newton algorithm when ωk is small and to the standard gradient descent algorithm if otherwise. Theinitial point ψ0 therefore governs the speed of convergence. We use the location estimates from the proposedTLLS algorithm as the starting point for s in the MAP estimator.The complexity of the TLLS, CLLS, and MAP estimators are analyzed and reported in Appendix D.3.5.3 Performance Bound on the Localization PerformanceTo evaluate the performance of the proposed TLLS, CLLS, and MAP estimators, we need a theoretical lowerbound on the achievable localization root-mean-square error (RMSE) performance. Within the unknownvector ψ = [x; y; γ]T , we know that the path loss exponent γ ∼ N(µγ,σ2γ ) is a random variable. For a lowerbound on the joint estimation performance, we derive the hybrid Cramer-Rao lower bound (HCRLB), for ittakes the statistical distribution of γ into account. In terms of the hybrid information matrix H, the RMSEof the source location s = [x ; y] is lower bounded as108√E((s∗− s)(s∗− s)T ≥√[H−1]1,1 + [H−1]2,2, where,H = Eγ{Eq;s |γ{−∇ψ(∇ψ loge p(q;s|γ))} −∇ψ(∇ψ loge p(γ))}.(5.27)In (5.27), s∗ and s denote the estimate and the true value of the source location, Eγ(.) and Eq;s |γ(.) denotethe expectation w.r.t. γ and q;s|γ respectively, and ∇ψ(.) denotes the Jacobian matrix w.r.t ψ. Before weproceed further, we note that the gradient vectors of loge δ w.r.t. x and y, denoted by ∇x loge δ and ∇y loge δ,are given by[∇x loge δ]m =x− x̂m+1(x− x̂m+1)2 + (y− ŷm+1)2 −x− x̂1(x− x̂1)2 + (y− ŷ1)2 ,[∇y loge δ]m =y− ŷm+1(x− x̂m+1)2 + (y− ŷm+1)2 −y− ŷ1(x− x̂1)2 + (y− ŷ1)2 ,∀m = 1, . . ., (M −1).(5.28)The top-left element [H]1,1 of the matrix H can then be obtained as[Eγ{Eq;s |γ{−∇ψ(∇ψ(loge p(q;s|γ)))} −∇ψ(∇ψ loge p(γ))]1,1(a)= Eγ{Eq:s |γ{−∇x(∇x(loge p(q;s|γ)))} −∇x(∇x loge p(γ))}(b)= Eγ{Eq;s |γ{∇x(12 (q+γ loge δ)T (Σq)−1γ∇x(loge δ))} −∇x(0)(c)= Eγ{Eq;s |γ{(γ(∇x loge δ)T (Σq)−1γ∇x(loge δ)+γ(q+γ loge δ)T (Σq)−1∇x(∇x(loge δ))},(d)= Eγ{γ2(∇x(loge δ))T (Σq)−1∇x(loge δ)},(e)= (µ2γ +σ2γ )(∇x(loge δ))T (Σq)−1∇x(loge δ).(5.29)In (5.29), (a) follows from noting that [ψ]1 = x, (b) from substituting the densities p(q;s|γ) and p(γ) andevaluating the inner gradient ∇x(.), (c) from evaluating the outer gradient ∇x(.), (d) from evaluating theexpectation w.r.t q and noting that Eq;s |γ(q)=−γ loge δ, and (e) from noting that Eγ(γ2)= µ2γ+σ2γ . Followinga similar procedure as in (5.29), the remaining elements of the HIM matrix H are given by109Table 5.1: Simulation setup and parameters to study DRSS-based localization under Gaussian distributed PLESystem Parameters ValueDevice transmit power (ρ) 26 dBm (400 mW) [178],Reference path loss and distance g0 = −45 dB, d0 = 1 m, [179]Anchor locations âm (−5,5), (10,0), (0,10), (10,10),Source locations (s) (2,8), (8,4), (6,9),(2,12), (−3,10), (12,8)PLE mean and variance (µγ,σγ) (3.5,0.97) [101][H]1,2=[H]2,1 = (µ2γ +σ2γ )(∇y(loge δ))T (Σq)−1∇x(loge δ),[H]1,3=[H]3,1 = µγ(∇x(loge δ))T (Σq)−1 loge δ,[H]2,3=[H]3,2 = µγ(∇y(loge δ))T (Σq)−1 loge δ,[H]2,2 = (µ2γ +σ2γ )(∇y(loge δ))T (Σq)−1(∇y(loge δ)), and[H]3,3 = (loge δ)T (Σq)−1 loge δ+1σ2γ(5.30)Substituting (5.29) and (5.30) into (5.27) gives us the HCRLB for the localization RMSE in closed-form.5.4 Numerical StudiesWe now present numerical studies to evaluate the performance of the proposed estimators. The simulationsetting is per Table 5.1. The transmit power ρ is set to 26 dBm as per the remote controller specificationsfor the DJI Phantom 4 Pro V2.0 drone [178]. The path loss parameters are chosen as per the WINNERII A1-NLOS model with the reference path loss and distance set to −45 dB and 1 m respectively. The 4anchor locations are the same as in the previous work [102] which considers a statistical path loss exponentmodel. The 6 source locations are chosen so that three are inside and three outside the convex hull ofthe anchors, and the PLE mean and variance are as per empirical studies for non line-of-sight (NLOS)conditions in [101]. Lastly, the MATLAB levenbergMarquardt function is used as the off-the-shelf LMalgorithm implementation, with an initial damping rate ω0 of 0.01 and a damping scaling factor τ of 10.All simulations are run on an 8−core Intel i7-9700K 3.60GHz CPU. The average performance over 5000independent runs is reported.For the localization performance, we consider five baselines: (i) WLS-known γ, which is the two-stepWLS proposed in [55] for known γ; we set γ = µγ, (ii) RSDPE, which is the semidefinite programmingsolution proposed in [54] to estimate s under imperfect γ; we set µγ as the imperfect γ and use the MATLAB110-10 -5 0 5 10Shadowing standard deviation m (dB)00.050.10.150.20.25Absolute error |m - m*|m = 0.1, = 2.5m = 0.1, = 3.5m = 0.1, = 4.5m = 0.9, = 2.5m = 0.9, = 3.5m = 0.9, = 4.5Figure 5.1: Error performance of the proposed ratio-of-distance estimator vs. shadowing standard deviation σm for different PLEγ.CVX with Mosek solver for the implementation, (iii) ReducedBiasDrss, which is the reduced bias leastsquares algorithm proposed in [57] for known γ; we set γ = µγ, (iv) the ratio and numerical search methodproposed in [53]; we search for γ within [µγ −2σγ, µγ +2σγ] in steps of 0.01, and (v) MLE, the maximumlikelihood method proposed in [73] for joint estimation of [s;γ; ρ] from RSS; we extend it for estimationof ψ = [s;γ] from the DRSS. Similar to MAP, the resulting problem is non-convex and can be solved for alocal optimum using the LM algorithm. For both MAP and MLE, we use the initial point ψ0 = [s∗TLLS; µγ]and refer to them as MLE-TLLS and MAP-TLLS in Fig. 5.2. To demonstrate the impact of TLLS as awarm start, we also report the performance of MAP under two conventional choices of s for the initial pointψ0: (i) MAP-random, wherein s is a randomly chosen point within the convex hull of the anchors, and (ii)MAP-maxdrss, wherein s is the anchor with the highest DRSS. Lastly, we include the hybrid Cramer-Raolower bound (HCRLB) for the localization RMSE.In Fig. 5.1, we plot the ratio-of-distance estimation error for different σm and γ, with δm = 0.1 andδm = 0.9 as the example values. We vary the PLE from its mean µγ = 3.5 and see that (i) the error increasesas we move away from µγ, with larger error noted for γ < µγ, and (ii) the impact of variations in thePLE is diminished when we increase the δm from 0.1 to 0.9. Also, the estimation error increases withσm, with a higher rate of increase for larger δm. Overall, the estimation performance is more sensitiveto variations in the PLE for small δm and to variations in the shadowing standard deviation for large δm.Further insight can be obtained with a theoretical analysis on the impact of γ and σm on the estimation111-10 -5 0 5 10Shadowing standard deviation m (dB)024681012Localization RMSE (m)WLS-Known RedBiasDrssPLSearchTLLSCLLSRSDPEMLE-TLLSMAP-randomMAP-maxdrssMAP-TLLSHCRLBFigure 5.2: Localization RMSE performance of the proposed least squares and MAP estimators vs. shadowing noise standarddeviation .error, but we do not pursue this direction. We note here that the errors in the ratio-of-distance estimationprocedure would propagate into the localization procedure, as is the case with most least squares-basedranging algorithms [47, 55, 57, 58], and proceed to study the localization performance.In Fig. 5.2, we plot the average root mean squared error (RMSE) for different σm when the γ for eachindependent run is drawn fromN(µγ,σ2γ ). Table 5.2 reports the average computation time when σm = 6 dB.Firstly, we note from Fig. 5.2 that the CLLS and the TLLS estimators perform similarly for most σm. Whenσm is increased beyond 5 dB, the CLLS estimator performs marginally better than the TLLS estimator. Thisis expected because the latter resorts to an approximation of the range variable constraint (c.f. (5.16)). Themarginal improvement in performance from the CLLS estimator comes at the cost of an extra step of solvinga fifth degree polynomial in the Lagrangian multiplier (c.f. (D.5) ). For example, when σm = 6 dB, which isthe shadowing standard deviation considered in Table 5.2 for reporting the computational times, the CLLSestimator requires 0.24 ms, whereas the TLLS estimator only requires 0.19 ms to completion. The MLE andMAP estimators, both of which are iterative in nature, require a close initial point for quick convergence.Fortunately, the TLLS and CLLS estimators resolve the issue of a choosing a close initial point by servingas the warm start. Since the TLLS estimator incurs lower complexity and performs similarly to the CLLSestimator, we use the TLLS estimates as the warm start for the MLE and MAP algorithms.Revisiting Fig. 5.2, we see that the proposed MAP-TLLS achieves the smallest localization RMSE,112Table 5.2: Average computation time of the considered localization algorithms when σm = 6 dB.WLS-Known γ 0.14 ms PLSearch 18 ms RedBiasDRSS 0.16 msRSDPE 480 ms TLLS 0.19 ms CLLS 0.24 msMAP-maxdrss 16.78 ms MAP-TLLS 10 ms MLE-TLLS 9.1 mswhich is also close to the HCRLB. The MAP-TLLS performs similar to the MLE-TLLS for small σm andbecomes marginally better as σm increases beyond 5 dB. Also, when compared to the MLE-TLLS, the MAP-TLLS consistently achieves smaller estimation error in the PLE (c.f. Table 5.3). These improvements comefrom the ability to make use of the statistical knowledge of γ (c.f. (5.23)), at the cost of a marginal increasein the computation time. The proposed TLLS estimator performs similar to the MAP-TLLS and MLE-TLLSwhen σm is low and experiences degradation as σm increases beyond 5 dB. While the MAP-TLLS and MLE-TLLS are recursive in nature, the TLLS is available in closed-form and is about 5 times less expensive atσm = 6 dB (c.f. Table 5.2). We also notice from Fig. 5.2 that the TLLS serves as a warm start for the MAP,by delivering (i) significantly better performance than the MAP-random (which is often seen to converge toa bad local optimum and so its computation time is not reported) and (ii) marginally better performance thanthe MAP-maxdrss. The TLLS start also facilitates faster convergence than the MAP-maxdrss: the overallruntime is smaller, despite the added workload of computing the TLLS estimates. The RSDPE performsmarginally worse than the MAP-TLLS and is also computationally more expensive (about 48 times moreexpensive at σm = 6 dB) because it requires solving a semidefinite programming problem. The PLsearchperforms similar to the remaining algorithms when σm is low but diverges away as σm increases. We canimprove the performance of PLSearch with smaller search steps for γ, but the current runtime is alreadylarger than that of the proposed estimators. Lastly, the WLS-known γ and RedBiasDRSS perform worsethan the proposed estimators because they always assume γ = µγ. In contrast, the TLLS estimator takesinto account the probability distribution of the PLE. Doing so increases the runtime when compared toWLS-known γ and RedBiasDRSS (c.f. Table II). Nevertheless, the approximate estimates from the TLLSachieve similar RMSE as the RSDPE at a lower computational cost. Also, they serve as a warm start forimproved localization accuracy when recursive updates are made in the MAP-TLLS estimator. Lastly, theRMSE trends in Fig. 5.2 suggest that the WLS-known γ and the RedBiasDRSS algorithms would eventuallyoutperform the TLLS estimator when the σm is well beyond 10 dB. This case is however not very interestingbecause σm ≤ 10 dB for most indoor and outdoor environments [100, 101].113Table 5.3: Average estimation error in the PLE, as reported by the MLE-TLLS and MAP-TLLS algorithms.σm (dB)→ -10 -6 -2 2 6 10MAP-TLLS 0.05 0.06 0.09 0.15 0.18 0.20MLE-TLLS 0.05 0.07 0.13 0.31 0.64 1.295.5 SummaryIn this chapter, we studied intruder device localization using the DRSS with respect to a reference sensornode, when the PLE is known to follow a Gaussian distribution in the area of interest. We firstly proposeda maximum-likelihood estimation procedure to obtain the ratio-of-distance estimates from the DRSS val-ues. Next, for estimating the source location given the ratio-of-distance estimates, we took a least squaresapproach and proposed the TLLS and CLLS estimators. Lastly, we also derived the MAP estimator of thesource location, given the statistical distribution of the PLE. Numerical examples revealed that the TLLSand CLLS estimators achieve similar localization performance as the existing semidefinite programming so-lution, while incurring smaller computational costs. Moreover, the TLLS estimates serve as a warm start forthe MAP estimator and facilitate a smaller localization RMSE when compared to other standard choices forthe initial point. The MAP estimator, although recursive in nature, is also seen to achieve smaller localizationRMSE when compared to the existing semidefinite programming solution.114Chapter 6Summary, Conclusions, and Future WorkIn this thesis, we proposed machine learning algorithms for the detection of intruder devices and the local-ization of their geographical position from the wireless signals received in a wideband RF sensor network.In Chapter 2, we proposed a deep learning solution to detect and localize the time-frequency span of theintruder signals present in wideband RF time-series captures, when friendly but uninteresting signals causeRF interference. The problem at hand is equivalent to detecting bounding boxes of the signals of interestin spectrogram matrices. Existing energy-based thresholding algorithms are sensitive to noise and requirehandcrafted hyperparameter tuning for each RF capture. To address this shortcoming, we adopted a modi-fied form of the FRCNN architecture [8] for the signal detection task and exposed the various componentsthat can be engineered within it for improvements in performance. To generate a dataset for performanceevaluation, we proposed a data generation framework in which the signals of interest are from the Wi-Fiprotocol and the RFI is from Bluetooth and microwave oven signals. Experiments on synthetic RF tracesrevealed that the proposed deep learning solution achieved significantly better mean average precision per-formance when compared to the conventional energy-thresholding and support vector machine classificationalgorithms. When working with real RF traces, we experience a drop in performance when we directly usea model trained on synthetic data. To alleviate the drop, we proposed a data mixing and a model finetuningstrategy. Our conclusion was that the proposed deep learning approach can be implemented in RF surveil-lance systems to detect the presence of an intruder. The reported time-frequency span information can beused to employ narrowband jamming on the intruder device. The proposed solution is semi-automated inthat the learning material needs to be gathered and the neural network model needs to be trained offline.Once the training is done, the system becomes fully automated and can facilitate real-time detection of theintruder signals. Apart from detecting the presence of intruder signals, if we are also interested in the infor-mation transmitted by the intruder, we will need to include additional information decoding blocks into theRF surveillance system.115In Chapter 3, we proposed machine learning algorithms based on GPs to localize the intruder devicesfrom their uplink RSS information. Instead of directly working with the RSS, we used the DRSS values asthe fingerprints for the localization because the actual transmit power of the intruder can be different fromits device specifications. We demonstrated that the conventional GP method provides optimistic 2σ errorbars on its location estimates because the test DRSS vectors are treated as deterministic inputs. To addressthis concern, we proposed two GP methods, namely the NaGP and GaGP, which treat the test DRSS vectorsas stochastic inputs. The proposed methods approximate the posterior distribution of the test locations as aGaussian distribution using numerical and analytical moment-matching techniques respectively. Numericalstudies revealed that the proposed NaGP and GaGP methods provided more realistic 2σ error-bars on theirlocation estimates while achieving a similar localization RMSE as the conventional GP method. Our con-clusion was that the proposed GP methods can be implemented in RF surveillance systems for positioningthe intruder devices from their RSS. The proposed GP methods achieve smaller localization RMSE thanthe standard maximum-likelihood and semidefinite relaxation methods. They also serve as a cost-effectivealternative to the standard k nearest neighbours fingerprint-matching algorithms, in that they do not requirea dense grid of training locations to achieve a similar RMSE performance. Also, unlike the kNN algorithm,the proposed GP methods provide 2σ error bars around their location estimates and therefore allow us todefine an area within which we can find the intruder device. The 2σ error bar information can also be usedto track the intruder device. From among the NaGP and GaGP, we recommend using the GaGP method forthe localization because it incurs lower computational cost.In Chapter 4, we conducted principal component analysis on the noise-free DRSS vectors and demon-strated the low-dimensionality exhibited by them. We then proposed two reconstruction-based GP methods,namely the RecGP and RecGaGP, which utilize this property to incorporate a data reconstruction step intothe localization procedure. Specifically, the noisy training and test DRSS vectors are reconstructed froma principal subspace spanned by the noise-free DRSS vectors in the service area, and the reconstructedDRSS vectors are used for the localization task. We demonstrated that the RecGP method provides opti-mistic 2σ error bars on the estimated locations because, similar to the conventional GP method, it treatsthe reconstructed test DRSS vectors as deterministic inputs. We addressed this concern in the RecGaGPmethod, wherein the stochastic nature of the reconstructed test DRSS is taken into account when derivingthe posterior distribution of the test locations. Analytical moment-matching was employed in order to ap-proximate the posterior distribution as a Gaussian distribution and obtain closed-form expressions for both116the location estimates and the 2σ error bars. Numerical examples revealed that the RecGP and RecGaGPmethods achieve a smaller localization RMSE when compared to the GP methods in Chapter 3. Also, theRecGaGP method provides more realistic 2σ error bars on the estimated locations than the RecGP method.Our conclusion was that the proposed RecGaGP method can be implemented for intruder localization in RFsurveillance systems. The RecGaGP method requires that the data reconstruction step facilitates noise re-duction in the DRSS vectors, in order to deliver a smaller localization RMSE than the GP methods in Chapter3. We therefore recommend using it in RF surveillance systems in which the number of deployed sensornodes is sufficiently large for the noise-free DRSS vectors to span a low-dimensional principal subspace.In Chapter 5, we proposed ranging-based algorithms for DRSS-based localization in service areas wherethe PLE is known to follow a Gaussian distribution. We derived maximum-likelihood estimates of theratio-of-distances to the sensors and used them to formulate a linear system of equations in the sourcelocation. We then proposed two linear least squares algorithms, namely the TLLS and CLLS, to obtainapproximate estimates of the source location. We also derived the MAP estimator of the source location,which requires solving a non-convex optimization problem. The LM algorithm was employed, with theTLLS estimate as the initial point for the source location, in order to obtain a good local optimum. Numericalexamples confirmed that the TLLS and CLLS estimators achieve similar localization performance as theexisting semidefinite programming solution, while incurring smaller computational costs. Moreover, theTLLS estimates serve as a warm start for the MAP estimator and facilitate a smaller localization RMSEwhen compared to other standard choices for the initial point. The MAP estimator, although recursivein nature, is also seen to achieve smaller localization RMSE when compared to the existing semidefiniteprogramming solution. Our conclusion was that the MAP estimator, with the TLLS estimate as the initialpoint, can be implemented as a low-cost localization algorithm in areas where the PLE is known to follow aGaussian distribution. Furthermore, if the shadowing noise variance is small, TLLS estimator can be usedas a cheaper alternative to obtain the source location.Overall, the contributions made in the thesis advocate the use of machine learning techniques for theintruder signal detection and device localization tasks in wireless RF surveillance systems. The signaldetection and time-frequency localization task not only facilitates the detection of the presence of an intruderdevice, but also aids in spectrum management. The time-frequency span estimation allows us to buildanti-drone surveillance products, in which narrowband jamming is employed to neutralize security threatsfrom rogue drones. Besides, the developed localization algorithms will aid in successful recovery of the117intruder devices in security-sensitive areas. The localization algorithms proposed in the thesis can alsobe extended for use in general-purpose wireless communication systems. Using the proposed algorithms,network operators can provide location-aware services such as emergency alerts, advertisements, and low-latency internet access to the users.The following research directions can be pursued as follow-up work to the thesis.Future research directions• In Chapter 2, we proposed a deep learning framework based on the FRCNN architecture for thesignal detection and time-frequency localization task. Our experiments on varied SNRs revealedthat the presented architecture performs better at higher SNRs. A custom denoising algorithm canbe developed, for example using autoencoder architectures [188], in order to reduce the amount ofnoise present in the spectrograms before feeding them to the signal detection and time-frequencylocalization framework. A joint training of the autoencoder and the Faster RCNN framework couldpotentially result in a more uniform signal detection performance across varied SNRs.• The proposed deep learning framework in Chapter 2 focuses on detecting the signals of a singlemodulation type, while classifying the other known but uninteresting signals as interference. Toextend this work, we can consider signals of multiple modulation types and train the model to performjoint time-frequency localization and modulation classification. This would enable the RF surveillancesystem to distinguish between intruder devices transmitting RF signals with different modulationschemes. Also, a device-level classification algorithm can be developed in order to separate out theintruder devices from the friendly devices that are transmitting RF signals using the same wirelessprotocol. An example scenario is when the intruder devices are transmitting Wi-Fi signals and thereare other Wi-Fi accesspoints and hotspots operating in the same spectrum. To classify between Wi-Fi devices, we can use distinguishing features such as the medium access control (MAC) addresses[189], contents in the probe request frames [190], and the inter-arrival time between the probe requestframes [191, 192]. Experimental studies need to be conducted on a system that uses the FRCNNframework for the Wi-Fi signal detection and one or more of the abovementioned techniques for thedevice-level classification. A performance analysis can then be conducted to report the overall successrate in detecting the intruder devices.118• In Chapters 3 and 4, we have proposed Gaussian process regression algorithms to estimate the intruderdevice location from the uplink DRSS information. For the performance evaluation, numerical studieswere conducted using synthetic data generated using computer simulations. An extension to this workwould be to conduct experimental studies, for example using the NI USRP devices [117]. This wouldhelp analyze the impact of practical aspects, such as changes in the geometry of the environment andfluctuations in the battery power of the intruder, which have not been explicitly accounted for in ournumerical studies.• The localization algorithms proposed in Chapters 3 and 4 can be extended for user positioning inmodern telecommunication systems, such as those enabled by the massive MIMO technology [193,194]. In massive MIMO systems, multiple users are scheduled to transmit signals simultaneously onthe same time-frequency resource. In [148], the authors have investigated the case where the userstransmit orthogonal pilots for the channel estimation and have used the RSS measured during thechannel estimation for the user positioning. As a follow-up work, we can study localization underdevice heterogeneity, using the differential RSS in place of the RSS. We can also investigate theuse of non-orthogonal pilots and study the resulting impact of pilot interference on the localizationalgorithms.• The localization algorithms presented in the thesis have focused on the use of DRSS information as thefingerprint to position the intruder devices. As a follow-up work, we can develop fusion algorithms,where more than one type of fingerprints are collectively used for the localization. For example, inthe case of antenna arrays being deployed at the wideband sensors, we could use the angle of arrivalinformation alongside the DRSS in order to estimate the intruder location. The statistical propertiesof each fingerprint type needs to be taken into account when deriving the location estimates and theconfidence intervals.• The localization algorithms in Chapter 5 study intruder localization in areas where the PLE is knownto be Gaussian distributed. The developed algorithms only report the estimates of the intruder devicelocation. 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Wireless Commun., vol.16, no. 3, pp. 1834-1850, Mar. 2017.137Appendix AAppendix for Chapter 1A.1 Analysing STFT vectors on Each Time Bin for Different FrequenciesLogically, the time-frequency localization task in Chapter 2 can also be achieved by analyzing each STFTtime bin to detect the occupied frequencies. This is essentially a simplification of our proposed methodology,wherein, instead of analyzing one two-dimensional STFT matrix containing STFT values for multiple timebins at a time, we analyze one STFT vector containing the STFT values for one time bin at a time.Is this simplified approach worth pursuing? Below, we present an overview of the three algorithmswhich need to be implemented in sequence, in order to achieve the time-frequency localization task. Wethen investigate the difficulty in hyperparameter tuning. Our observation is that the per-time-bin STFTanalysis is sensitive to the choice of the underlying hyperparameters, the tuning of which is non-trivial. Wetherefore believe that the simplified approach of analyzing each STFT time bin, although logically valid, isthe less preferred option when compared to the proposed deep learning solution.Overview of the per-time-bin STFT analysisTo achieve the time-frequency localization by analyzing one STFT time bin at a time, we need to implementthree main algorithms in sequential order. Firstly, we need to implement a changepoint detection algorithmat a per-time-bin level in order to detect the frequency edges. Following this, the frequency edges need tobe categorized into start and stop frequencies, in order to deduce the frequency span of each signal (theedge categorization becomes involved when we have an odd number of detections because we need to eitherignore one of the detections assuming it to be a false positive or verify that the last edge corresponds to asignal whose other edge is beyond the limits of the spectrum). Secondly, since most signals span acrossmultiple time-bins, we would need a multi-time-bin changepoint combining algorithm, wherein, the fre-quency spans over multiple consecutive time-bins need to be combined in order to report the time-frequency138span of the detected signal. The output of the algorithm would be a set of time-frequency localized signals,with the set containing both the signals of interest (e.g: Wi-Fi signals) and the interference signals (e.g.:the Bluetooth and microwave oven signals). Thirdly, we need a signal classification algorithm which canseparate the signals of interest from the interference signals. If we employ supervised machine learning forthe classification task, we would in turn require a feature extraction algorithm which extracts fixed-lengthfeatures from the detected signals. This is essential because the time-frequency spans can be different fordifferent signals but the inputs to the machine learning algorithm need to be of a fixed length. The signalclassification may also be attempted at a per-time-bin level, i.e, before we combine the changepoints overmultiple consecutive time-bins, but doing so would be sub-optimal and inefficient. This is because the clas-sification algorithm would be forced to rely solely on the STFT changes within a single time bin to performthe interference separation for that time bin. In other words, the classification algorithm would be unable tomake use of the temporal changes in the signals to separate out the noise and interference. Finally, after thesignal classification, we may calculate the mAP and report the overall performance.In order to verify whether the above approach is worth pursuing, below, we analyze and expose the needto tailor the hyperparameters for each time bin. There are no clear guidelines however, on automaticallyoptimizing the hyperparameters for each time bin. Hyperparameter tuning is therefore a difficult task whenthe per-time-bin STFT analysis is pursued.Difficulty in hyperparameter tuning of the per-time-bin STFT analysisLet us consider the changepoint detection algorithm, wherein we detect the frequency edges of the signals,to be run as the first step in the per-time-bin STFT analysis. As an example changepoint detection algo-rithm, we consider the relative unconstrained least squares importance fitting (RuLSIF) method proposedin [107]. The RuLSIF method is popular for its linear complexity in the number of samples and for its easeof implementation. It is known to deliver state-of-the-art performance for the CENSREC speech recognitiondataset [108], WSU smart home activity monitoring dataset [109], and the HACS human activity sensingdataset [110] (see Table 4 in [111] for a comparison study with other existing algorithms). Alternative al-gorithms which achieve a similar performance are the BinSeg method [112] and the KCP [113] method,both of which incur the same or worse complexity in the number of samples (c.f. Table 4 in [114] for acomparison study). We therefore consider the RuLSIF method for the changepoint detection and present anoverview next.139Overview of the RuLSIF method:In terms of methodology, the RuLSIF method runs a sliding window of length n along the signal vector andcomputes the discrepancy between two adjacent windows to detect the changepoints. More specifically, foran index i in the signal vector, a sequence of length n starting at i is treated as a window and an n× k datamatrix Y(i) is formed with the rows being sub-sequences of length k taken with a step-size of 1. Next, achangepoint score is assigned for the index i using the α−relative Pearson divergence, which calculates thedissimilarity in the data distribution of the adjacent matrices Y(i) and Y(i+n). Similarly, a changepoint scoreis assigned for the next index i+ 1, based on the data distributions of the matrices Y(i+ 1) and Y(i+ n+ 1)respectively. The same procedure is followed until the end of the signal, i.e., from i = 1 to i = n− k − 2.Changepoints are considered significant if their scores are larger than a numeric threshold η. The α−relativePearson divergence requires knowledge of the ratio of the probability densities of Y(i) and Y(i+ n), whichneeds to be learned from the matrices Y(i) and Y(i+n). This is done by firstly modelling the density ratio asa weighted sum of Gaussian kernels and then learning the weights in the sum by minimizing the regularizedsquared loss between the true and the estimated density ratios.Implementation details for the RuLSIF method: For the implementation, we reuse the MATLAB sourcecode [107] provided by the authors and parallelize the changepoint detection for multiple time bins, so asto reduce the total computation time. To determine the “significant” changepoints, we set η to be 80%of the maximum observed change-point score for that time-bin. Also, since we know that the signals ofinterest have a frequency span of 16 MHz (the original bandwidth is 20 MHz but out-of-band transmissionsconsume 10% on either side of the signal bandwidth), we filter out those “significant” changepoints whichfall within 12 MHz of a preceding “significant" changepoint. The width σ of the Gaussian kernels and theregularization weight λ in the loss minimization are both chosen using the same cross-validation procedureas in [107]. For the α, n, and k, the authors in [107] have recommended α = 0.1, n = 50, and k = 10 but it isunclear whether these values are suitable for the data at hand. As explained below, we run a few experimentsfor the specific example in Fig.A.1 and choose α = 0.01,n = 15, k = 3 to report the detection performance.We begin with the recommended values of α = 0.1,n = 50, k = 10 and notice that reducing the α from0.1 to 0.01 decreased the number of false positives. To illustrate the same, in Fig. A.2, we consider themid-point time-bins for the three Wi-Fi signals of interest and plot the original signal, the raw changepointscores, and the “significant" changepoints. For each time index, the reported changepoint score is the sum14010 20 30 40 50 60 70 80Time (ms)-25-20-15-10-50510152025Frequency (MHz)-140-120-100-80-60-40-20Power/frequency (dB/Hz)Three Wi-Fi signals of interestFigure A.1: Example RF capture 1 to expose the difficulty in hyperparameter tuning for the RuLSIF [107] methodof the two scores obtained by running the algorithm in the forward and reverse orders respectively. Wenotice that the number of false positives decrease as we reduce the α from 0.1 to 0.01. We therefore setα = 0.01. Next, we know that the computation time for each time bin can be reduced by decreasing n and k.We therefore try decreasing (n, k) while keeping the ratio 5 : 1 and report the resulting performance in Fig.A.3. We notice that the detection accuracy in unaffected until (n, k) = (15,3), where false positives begin toappear. We therefore set (n, k) = (15,3). In practice, if we are conducting a search for the hyperparameters,we would start with smaller (n, k) and gradually increase their value because the computation times increasewith (n, k). Since all the three Wi-Fi signals are now detected, we say that the hyperparameter tuning iscomplete for the changepoint detection step for Fig. A.1.We need to now verify if the hypeparameters chosen for the example RF capture in Fig. A.1 workwell for signal detection in other captures too. If not, we would need to tune the hyperparameters for eachcapture. Below, we present the performance of the RuLSIF method for a different example RF capture tomake obvious the need for hyperparameter tuning per capture.We study the tuning of the hyperparameters n and k, i.e., the window length and the subsequence lengthrespectively, in the RuLSIF method. Apart from the (n, k), other hyperparameters which require tuning arethe changepoint “significance" threshold η, the Pearson divergence parameter α, and the hyperparametersassociated with the multi-time-bin changepoint combining algorithm, and the signal classification algorithm.141200 400 600 800 1000406080200 400 600 800 10000510200 400 600 800 100001020200 400 600 800 1000020406080(a) Mid-point time-bin for signal at 24.88 ms200 400 600 800 100020406080200 400 600 800 10000510200 400 600 800 100001020200 400 600 800 1000020406080(b) Mid-point time-bin for signal at 45.75 ms200 400 600 800 100020406080200 400 600 800 10000510200 400 600 800 100001020200 400 600 800 1000020406080(c) Mid-point time-bin for signal at 79.54 msFigure A.2: Changepoint detection for the three WiFi signals in Fig. A.1, when (n, k) = (50,10). Reducing α from 0.1 to 0.01decreases the number of false positives.142200 400 600 800 1000406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(a) Mid-point time-bin for signal at 24.88 ms200 400 600 800 100020406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(b) Mid-point time-bin for signal at 45.75 ms200 400 600 800 100020406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(c) Mid-point time-bin for signal at 79.54 msFigure A.3: Changepoint detection for the three WiFi signals in Fig. A.1, when α = 0.01. Decreasing (n, k) from (50,10), whilekeeping the 5 : 1 ratio, does not affect the performance until (n, k) = (15,3), where false positives begin to appear (see (a)).14310 20 30 40 50 60 70 80Time (ms)-25-20-15-10-50510152025Frequency (MHz)-140-120-100-80-60-40-20Power/frequency (dB/Hz)Wi-Fi signals of interestFigure A.4: Example RF capture 2 to expose the difficulty in hyperparameter tuning for the RuLSIF [107] methodLet us study if the combination (n, k) = (15,3), which worked well for the RF capture in Fig. A.1, also workswell for the test capture in Fig. A.4. In Fig. A.5, we consider four different time-bins within the RF capturein Fig. A.4 and plot the changepoint detection performance for various (n, k). For each time-bin, please seeFig. A.4 for the frequency edges that need to be detected. The performance of the changepoint detection issummarized in Table A.1. We notice that the RuLSIF method can successfully detect the frequency edgesif the (n, k) are appropriately chosen, but making this choice is non-trivial. For example, the combination(n, k) = (15,3), which was seen to work very well for the signals in Fig. A.1, fails to detect the correctfrequency edges for the example time-bins 2, 3, and 4 in Fig. A.5. The combination (n, k) = (50,10) workswell for the example time bins 2 and 3 in Fig. A.4, but fails for the time bins 1 and 4. A different combination(n, k) = (15,3) works well for the example time bins 1 and 3, but fails for time-bins 2 and 4. In summary,we observe that it is non-trivial to find an (n, k) which works well across all four time-bins. Although onlyfour example time-bins are presented here, we notice similar behaviour for several other time-bins, i.e., thelimited generalizeability of the (n, k) is very commonplace and apparent.The above numerical examples with varying (n, k) have made obvious the need for hyperparameter tun-ing in the per-time-bin STFT analysis. The optimization procedure is however non-trivial and there are noclear guidelines currently available in this regard. At the least, we need to design a unifying loss functionand a hyperparameter update mechanism that pursues the global objective of localizing the time-frequency144200 400 600 800 100020406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(a) Example time-bin 1: 7.5521 ms to 7.5646 ms (pleasesee Fig. A.4 for the frequency edges that need to bedetected)200 400 600 800 1000406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(b) Example time-bin 2: 32.1339 ms to 32.1464 ms(please see Fig. A.4 for the frequency edges that need tobe detected)200 400 600 800 1000406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(c) Example time-bin 3: 68.969 ms to 68.9816 ms (pleasesee Fig. A.4 for the frequency edges that need to bedetected)200 400 600 800 100020406080200 400 600 800 1000020406080200 400 600 800 1000020406080200 400 600 800 1000020406080(d) Example time-bin 4: 38.2231 ms to 38.2356 ms(please see Fig. A.4 for the frequency edges that need tobe detected)Figure A.5: Changepoint detection for four example time-bins (see Fig. A.4 for the frequency edges). The edges are successfullydetected if the (n, k) is appropriately chosen but there is no single (n, k) which correctly detects the edges across all four time-bins.145Table A.1: Summary of the detection performance of the RuLSIF method for the four example time bins in Fig. A.5. None of the(n, k) perform consistently well across all the time-bins.Example time bin index (n,k) = (50,10) (n,k) = (15,3) (n,k) = (10,3)1 7 X X2 X 7 73 X 7 X4 7 7 7span of signals. This needs to be done while still being able to achieve the intermediate objectives for whichthe changepoint detection, the multi-time-bin changepoint combining, and the signal classification algo-rithms are employed. One may pursue exhaustive grid-search methods, but they become computationallyprohibitive with the increasing number of the hyperparameters and the increasing size of the search space.Unlike the per-time-bin STFT analysis, one may note that the proposed deep learning system can betrained using a single loss function and all the hyperparameters can be optimized through backpropagationto achieve the global objective of localizing the time-frequency span of the signals (c.f. [8] for details onthe loss function and the training procedure). This makes the automatic hyperparameter tuning a distinctadvantage of the proposed deep learning solution in Chapter 2.Besides, when compared to the per-time-bin STFT analysis, the proposed deep learning system is morecompact and target-oriented towards the time-frequency localization task: (i) we process a large STFTmatrix, instead of an STFT vector, at a time, (ii) we directly search for rectangular bounding boxes to obtainthe joint time-frequency span, instead of searching for changepoints on each time-bin and then combiningthe changepoints to report the time-frequency span, and (iii) we simultaneously separate out the noise andinterference using the same features that were used to detect the bounding boxes, instead of extractingseparate features for changepoint detection and noise/interference separation. These aspects, along with theautomatic hyperparameter tuning, make the proposed deep learning solution as the preferred option over theper-time-bin STFT analysis for the time-frequency localization task.146Appendix BAppendices for Chapter 3B.1 Proof of Lemma 3.1(mean and variance expressions from the GaGP method)Closed-form expression for [µ̂(GaGP)x ]l can be obtained as follows:[µ̂(GaGP)x ]l (a)= E[̂x]l ([̂x]l |˜x, P˜, p̂l)(b)= Ep̂∗l(E[̂x]l ([̂x]l |˜x, P˜, p̂∗l ))(c)= Ep̂∗l(L˜∑i=1[ψ]iφ(̂p∗l , p˜i))(d)=L˜∑i=1[ψ]i∫(α′N (̂p∗l ; p˜i,B)+γp̂∗Tl p˜i +σ2e δli)N(̂p∗l ; p̂l, Σ̂l)dp̂∗l(e)=L˜∑i=1[ψ]i{α′N (̂pl; p˜i,B+ Σ̂l)∫(N(̂p∗l ; (B−1 + Σ̂−1l )−1(B−1p˜i + Σ̂−1l p̂l), (B−1 + Σ̂−1l )−1))dp̂∗l +γp̂Tl p˜i}=L˜∑i=1α′[ψ]iN (̂pl; p˜i,B+ Σ̂l)+γ[ψ]ip̂Tl p˜i, (B.1)where (a) is obtained from (3.24), (b) from the law of iterated expectations (c) by substituting E[̂x]l ([̂x]l |˜x, P˜, p̂∗l )=∑L˜i=1 φ(̂p∗l , p˜i)[ψ]i from (3.17), (d) by substituting covariance model from (3.9), and (e) by substituting prod-uct of Gaussian terms from (B.17).Similarly, we derive a closed-form expression for [Ĉ(GaGP)x ]ll as follows:[Ĉ(GaGP)x ]ll (a)= E[̂x]l (([̂x]l |˜x, P˜, p̂l)2)− ([µ̂(GaGP)x ]l)2147(b)= Ep̂∗l(E[̂x]l (([̂x]l |˜x, P˜, p̂∗l )2))− ([µ̂(GaGP)x ]l)2(c)= Ep̂∗l([Ĉ(CGP)x ]ll + (E[̂x]l ([̂x]l |˜x, P˜, p̂∗l ))2)− ([µ̂(GaGP)x ]l)2(d)= Ep̂∗l{φ(̂p∗l , p̂∗l )−L˜∑i=1L˜∑j=1[Φ˜−1]i jφ(̂p∗l , p˜i)φ(˜pj, p̂∗l )+ (E[̂x]l ([̂x]l |˜x, P˜, p̂∗l ))2}−([µ̂(GaGP)x ]l)2(e)= Ep̂∗l(φ(̂p∗l , p̂∗l ))− ([µ̂(GaGP)x ]l)2−L˜∑i=1L˜∑j=1([Φ˜−1]i j −[ψ]i[ψ]j)Ep̂∗l(φ(̂p∗l , p˜i)φ(˜pj, p̂∗l )) (B.2)where (a) is obtained from (3.24), (b) from the law of iterated expectations, (c) from definition of co-variance in (B.18), (d) by substituting [Ĉ(CGP)x ]ll from (3.17), and (e) by substituting E[̂x]l ([̂x]l |˜x, P˜, p̂∗l ) =∑L˜i=1 φ(̂p∗l , p˜i)[ψ]i from (3.17). To proceed further, we require closed-form expressions for Ep̂∗l (φ(̂p∗l , p̂∗l ))and Ep̂∗l(φ(̂p∗l, p˜i)φ(˜pj, p̂∗l )), which we derive as follows:Ep̂∗l(φ(̂p∗l , p̂∗l ))(a)=Ep̂∗l(α′N (̂p∗l ; p̂∗l ,B)+γp̂∗Tl p̂∗l +σ2e δll)=σ2e +∫(α+γp̂∗Tl p̂∗l )N(̂p∗l ; p̂l, Σ̂l)dp̂∗l(b)=σ2e +α+γp̂Tl p̂l +γTr(Σ̂l), (B.3)where (a) is obtained by substituting covariance model from (3.9) and (b) by applying the integral ofquadratic from (B.19). Next, we haveEp̂∗l(φ(̂p∗l , p˜i)φ(˜pj, p̂∗l )) =∫(α′N (̂p∗l ; p˜i,B)+γp̂∗Tl p˜i +σ2e δli)(αN (̂p∗l ; p˜j,B)+γ(˜pj)T p̂∗l +σ2e δjl)N(̂p∗l ; p̂l, Σ̂l)dp̂∗l=I1 +I2 +I3 +I4, (B.4)where, for notational convenience, we have definedI1 =(α′)2∫N (̂p∗l ; p˜i,B)N (̂p∗l ; p˜j,B)N(̂p∗l ; p̂l, Σ̂l)dp̂∗lI2 =α′γ∫p̂∗Tl p˜iN (̂p∗l ; p˜j,B)N(̂p∗l ; p̂l, Σ̂l)dp̂∗l ,148I3 =α′γ∫N (̂p∗l ; p˜i,B)(˜pj)T p̂∗lN(̂p∗l ; p̂l, Σ̂l)dp̂∗l ,I4 =γ2∫p̂∗Tl p˜i (˜pj)T p̂∗lN(̂p∗l ; p̂l, Σ̂l)dp̂∗l (B.5)Closed-form expressions for I1, I2, I3, and I4 can be obtained as follows:I1 = (α′)2∫N (̂p∗l ; p˜i,B)N (̂p∗l ; p˜j,B)N(̂p∗l ; p̂l, Σ̂l)dp̂∗l(a)= (α′)2N (˜pi; p˜j,2B)∫N (̂p∗l ;p˜i + p˜j2,B2)N(̂p∗l ; p̂l, Σ̂l)dp̂∗l(b)= (α′)2N (˜pi; p˜j,2B)N (̂pl;p˜i + p˜j2,B2+ Σ̂l),I2 =∫α′γp̂∗Tl p˜iN (̂p∗l ; p˜j,B)N(̂p∗l ; p̂l, Σ̂l)dp̂∗l(c)= α′γ∫p̂∗Tl p˜iN (̂pl; p˜j,B+ Σ̂l)N(̂p∗l ; (B−1 + Σ̂−1l )−1(B−1p˜j + Σ̂−1l p̂l), (B−1 + Σ̂−1l )−1)dp̂∗l=α′γN (̂pl; p˜j,B+ Σ̂l){(B−1 + Σ̂−1l )−1(B−1p˜j + Σ̂−1l p̂l)}T p˜i,I3 =∫α′γN (̂p∗l ; p˜i,B)(˜pj)T p̂∗lN(̂p∗l ; p̂l, Σ̂l)dp̂∗l(d)= α′γ(˜pj)T∫p̂∗lN (̂pl; p˜i,B+ Σ̂l)N(̂p∗l ; (B−1 + Σ̂−1l )−1(B−1p˜i + Σ̂−1l p̂l), (B−1 + Σ̂−1l )−1)dp̂∗l=α′γN (̂pl; p˜i,B+ Σ̂l)(˜pj)T (B−1 + Σ̂−1l )−1(B−1p˜i + Σ̂−1l p̂l),I4 =∫γ2p̂∗Tl p˜i (˜pj)T p̂∗lN(̂p∗l ; p̂l, Σ̂l)dp̂∗l(e)= γ2p̂Tl p˜i (˜pj)T p̂l +γ2 Tr(˜pi (˜pj)T Σ̂l), (B.6)where (a)-(d) are obtained by applying product of Gaussian terms from (B.17) in each step and (e) by apply-ing the integral of quadratic from (B.19). Substituting (B.6) into (B.4) gives us a closed-form expression forEp̂∗l(φ(̂p∗l, p˜i)φ(˜pj, p̂∗l )). We then substitute (B.3) and (B.4) into (B.2) and define matrices Υ ∈ RM×L˜ and ξ ∈RL˜×L˜ such that [Υ]i = α′γN (̂pl; p˜i,B+ Σ̂l)((B−1 + Σ̂−1l )−1(B−1p˜i + Σ̂−1l p̂l)), and [ξ]i j = [Φ˜−1]i j − [ψ]i[ψ]j ,∀i, j = 1, . . ., L˜, to obtain the expression for [Ĉ(GaGP)x ]ll as given by Lemma 3.1.149B.2 Proof of Remark 3.3(similarity in the mean and variance expressions provided by the CGP and GaGP method)Let us first study the relation between [µ̂(GaGP)x ]l and [µ̂(CGP)x ]l . From (3.17), we know that [µ̂(CGP)x ]l is givenby[µ̂(CGP)x ]l =L˜∑i=1α′[ψ]iN (̂pl; p˜i,B)+γ[ψ]ip̂Tl p˜i . (B.7)In order to express the GaGP mean [µ̂(GaGP)x ]l, given by (3.25), in a similar form as (B.7), let us first expressthe Gaussian term N (̂pl; p˜i,B+ Σ̂l) in [µ̂(GaGP)x ]l (c.f. (3.25)) in terms of N (̂pl; p˜i,B) as follows:N (̂pl; p˜i,B+ Σ̂l) =exp(−12 (̂pl − p˜i)T (B+ Σ̂l)−1(̂pl − p˜i))(2pi)(M−1)/2 |B+ Σ̂l |1/2(a)=exp(− 12 (̂pl − p˜i)T (B−1 + (B+ Σ̂l)−1−B−1)(̂pl − p˜i))(2pi)(M−1)/2 |B+ Σ̂l |1/2(b)= N (̂pl; p˜i,B)λi,(B.8)where (a) is obtained by adding and subtracting B−1 and (b) by introducing a new variable λi, defined as,λi =exp(− 12 (̂pl − p˜i)T ((B+ Σ̂l)−1−B−1)(̂pl − p˜i))|I+B−1Σ̂l |1/2, (B.9)for notational convenience. Substituting (B.8) into the expression for [µ̂(GaGP)x ]l in (3.25) gives us[µ̂(GaGP)x ]l =L˜∑i=1α′[ψ]iN (̂pl; p˜i,B)λi +γ[ψ]ip̂Tl p˜i . (B.10)Observe from (B.10) and (B.7) that [µ̂(GaGP)x ]l is similar in structure as [µ̂(CGP)x ]l, but with multiplicativecorrection factors {λi} introduced by the GaGP method to the Gaussian terms in (B.7). The correctionfactors account for the stochastic nature of p̂l. Lastly, we can verify that when the test DRSS is noise-free,i.e., when Σ̂l = 0, we get λi = 1. Consequently, the expressions for [µ̂(GaGP)x ]l and [µ̂(CGP)x ]l in (B.10) and(B.7) respectively, turn out to be the exactly the same when the test DRSS is noise-free.Next, we study the relationship etween [Ĉ(GaGP)x ]ll and [Ĉ(CGP)x ]ll. From (3.17), we know that [Ĉ(CGP)x ]llis given by150[Ĉ(CGP)x ]ll (a)= α+γp̂Tl p̂l +σ2er −L˜∑i=1L˜∑j=1[(Φ˜)−1]i j{(α′)2N (̂pl; p˜i,B)N (˜pj ; p̂l,B)+α′γp̂Tl p˜iN (˜pj ; p̂l,B)+α′γ(˜pj)T p̂lN (̂pl; p˜i,B)+γ2p̂Tl p˜i (˜pj)T p̂l}(b)= α+γp̂Tl p̂l +σ2er −L˜∑i=1L˜∑j=1(α′)2[(Φ˜)−1]i j{N (˜pi; p˜j,2B)N (̂pl;p˜i + p˜j2,B2)}+α′γ[(Φ˜)−1]i j p̂Tl p˜i{N (˜pj ; p̂l,B)}+α′γ[(Φ˜)−1]i j (˜pj)T p̂l{N (̂pl; p˜i,B)}+γ2{p̂Tl p˜i (˜pj)T p̂l},(B.11)where (a) is obtained by substituting the covariance model from (3.9) into [Ĉ(CGP)x ]ll in (3.17) and (b) byapplying the product of Gaussian terms from (B.17). In order to express the GaGP variance [Ĉ(GaGP)]ll givenby (3.25) in a similar form as (B.11), we firstly simplify the Gaussian term N (̂pl; p˜i+p˜ j2 , B2 + Σ̂l) in [Ĉ(GaGP)]ll(c.f. (3.25)) by following the same procedure as in (B.8) to obtainN (̂pl;p˜i + p˜j2,B2+ Σ̂l) = N (̂pl;p˜i + p˜j2,B2)λi j, where,λi j =exp(−12 (̂pl −p˜i+p˜ j2 )T ((B2 + Σ̂l)−1−(B2 )−1)(̂pl −p˜i+p˜ j2 ))|I+ (B2 )−1Σ̂l |1/2. (B.12)In (B.12), λi j is a multiplicative correction factor introduced by GaGP. Let us also simplify the expressionsfor ([µ̂(GaGP)x ]l)2 in [Ĉ(GaGP)x ]ll (c.f. (3.25)), as follows:([µ̂(GaGP)x ]l)2 (a)=L˜∑i=1L˜∑j=1(α′[ψ]iN (̂pl; p˜i,B)λi +γ[ψ]ip̂Tl p˜i)(α′[ψ]jN (̂pl; p˜j,B)λj +γ[ψ]j (˜pj)T p̂l)(b)=L˜∑i=1L˜∑j=1[ψ]i[ψ]j{(α′)2λiλjN (˜pi; p˜j,2B)N (̂pl;p˜i + p˜j2,B2)+α′γλiN (̂pl; p˜i,B)(˜pj)T p̂l +α′γλjN (̂pl; p˜j,B)̂pTl p˜i +γ2p̂Tl p˜i (˜pj)T p̂l}, (B.13)where (a) is obtained by substituting (B.10) and (b) by applying the product of Gaussian terms from (B.17).151Next, let us simplify the expression for [Υ]i in [Ĉ(GaGP)x ]ll (c.f. (3.25) as follows[Υ]i = α′γN (̂pl; p˜i,B+ Σ̂l)((B−1 + Σ̂−1l )−1(B−1p˜i + Σ̂−1l p̂l))= α′γN (̂pl; p˜i,B+ Σ̂l)((Σ̂lB−1 + I)−1Σ̂lΣ̂−1l (Σ̂lB−1p˜i + p̂l))(a)= α′γλiN (̂pl; p˜i,B)((Σ̂lB−1 + I)−1(Σ̂lB−1p˜i + p̂l))(b)= α′γλiN (̂pl; p˜i,B)(νi +Γp̂l), (B.14)where (a) is obtained by substituting (B.8) and (b) by introducing the terms νi = (Σ̂lB−1 + I)−1Σ̂lB−1p˜i andΓ = (Σ̂lB−1 + I)−1 for notational convenience. Substituting (B.12)-(B.14) into [Ĉ(GaGP)x ]ll in (3.25) gives us[Ĉ(GaGP)x ]ll = α+γp̂Tl p̂l +σ2er +γTr(Σ̂l)−L˜∑i=1L˜∑j=1(α′)2([ξ]i jλi j+[ψ]i[ψ]jλiλj){N (˜pi; p˜j,2B)N (̂pl;p˜i + p˜j2,B2)}+α′γ([ξ]i jλj(ν j +Γp̂l))T p˜i + [ψ]i[ψ]jλj p̂Tl p˜i){N (̂pl; p˜j,B)}+α′γ([ξ]i jλi (˜pj)T (νi +Γp̂l)+ [ψ]i[ψ]jλi (˜pj)T p̂l){N (̂pl; p˜i,B)}+γ2([ξ]i j + [ψ]i[ψ]j){p̂Tl p˜i (˜pj)T p̂l}+ [ξ]i jγ2 Tr(˜pi (˜pj)T Σ̂l).(B.15)Observe from (B.11) and (B.15) that the [Ĉ(CGP)x ]ll and [Ĉ(GaGP)x ]ll expressions have a similar structure,with the GaGP introducing several additive and multiplicative correction factors to the Gaussian and innerproduct terms in [Ĉ(CGP)]ll. When Σ̂l = 0, i.e., when the test DRSS is noise-free, we can verify that λi =1, λi j = 1,νi = 0, and Γ = I, ∀ i, j,= 1, . . ., L˜. Substituting these values, along with [ξ]i j = [Φ˜−1]i j −[ψ]i[ψ]jinto (B.15), gives us an expression for [Ĉ(GaGP)x ]ll which is exactly the same as [Ĉ(CGP)]ll given in (B.11).B.3 Mathematical Formulas for Simplifications in Chapter 31. [Conditioning a joint Gaussian distribution [98] (pg. 200)] If a is aW×1 Gaussian random vector witha ∼N(u,A) and the random variables in a are partitioned into two sets aζ = [[a]1 [a]2, . . . [a]w]T ∈ Rw152and aζ ′ = [[a]w+1 [a]w+2, . . . [a]W ]T ∈ RW−w such thataζaζ ′ ∼ N©­­«uζuζ ′ª®®¬,©­­«Aζζ Aζζ ′ATζζ ′ Aζ ′ζ ′ª®®¬ , (B.16)then aζ |aζ ′ and aζ ′ |aζ are also Gaussian such thataζ |aζ ′ ∼ N(uζ +Aζζ ′A−1ζ ′ζ ′(aζ ′ −uζ ′),Aζζ −Aζζ ′A−1ζ ′ζ ′ATζζ ′),aζ ′ |aζ ∼ N(uζ ′ +ATζζ ′A−1ζζ (aζ −uζ ),Aζ ′ζ ′ −ATζζ ′A−1ζζAζζ ′).2. [Product of Gaussian expressions] Let us consider three deterministic W−dimensional vectors a, uand u0, and two W ×W positive definite matrices A and A0. The product of Gaussian expressionsN(a;u,A) and N(a;u0,A0) is then given byN(a;u,A)N(a;u0,A0) = N(u;u0,A+A0)N(a;u1,A1),where A1 = (A−1 +A−10 )−1 and u1 = A1(A−1u+A−10 u0).(B.17)3. [Covariance of a random vector] The covariance matrix A of anW−dimensional vector a has elementsgiven by[A]ww = E[a]w (([a]w)2)− (E[a]w ([a]w))2, ∀w = 1, . . .,W . (B.18)4. [Integral of a quadratic expression with respect to a Gaussian random vector] If u0 is a deterministicW ×1 vector, A0 is a deterministic positive definite matrix of size W ×W , and a is a W ×1 Gaussianrandom vector with mean u and covariance A, i.e., a ∼ N(u,A), then∫(a−u0)TA−10 (a−u0)N(a;u,A)da = (u0−u)TA−10 (u0−u)+Tr(A−10 A). (B.19)5. [Derivative of the log-determinant and inverse of a matrix] If a is a vector of unknown variables, Ais a positive definite matrix whose entries are functions of a, and ∇a(A) is the matrix of element-wisederivatives of A with respect to a, then∇a(log |A|) = Tr(A−1∇a(A)),and (B.20)153∇a(A−1) = −A−1∇a(A)A−1. (B.21)B.4 Complexity Analysis for the GP Methods and Baselines in Chapter 3We investigate the computational cost associated with the GP methods presented in Chapters 3. SectionB.4.1 presents an overview of how some frequently-occurring matrix operations in the presented GP meth-ods can be implemented in a numerically stable manner, and the computational complexity of such numeri-cally stable implementations. We make use of this overview to present a detailed analysis of the complexityof each GP method in Chapter 3.B.4.1 Numerically Stable Implementation of Matrix Operations in the GP MethodsLet A be a matrix of size W ×W , χ be the Cholesky factor of A such that A = χχT , u be a vector of sizeW ×1, and ∇aA be the matrix of element-wise derivatives of A w.r.t a.1. [A−1u]: The matrix-vector multiplication A−1u is stably calculated as χ−T (χ−1u). Cholesky decom-position of A requires O(W3) operations and the product χ−T (χ−1u) requires O(W2) operations whencomputed via forward and backward substitution.2. [Tr(A−1∇aA)]: The matrix product Tr(A−1∇aA) can be stably implemented as Tr(χ−T (χ−1∇aA)).The Cholesky factor χ of A can be obtained in O(W3) operations. After obtaining χ, the calculationof Tr(χ−T (χ−1∇aA)) requires another O(W3) operations.[log(|A|)]: The term log(|A|) can be stably calculated as 2∑Ww=1 log([χ]ww). This requires O(W3) operationsto obtain χ and O(W) operations to calculate the sum.We now present details on the computational complexity of each GP method.B.4.2 Complexity of Conventional GP MethodObserve from (3.17) that the predictive mean µ̂(CGP)x requires computation of (i) Φ˜−1x˜, which needs O(L˜3)operations to obtain the Cholesky factor of Φ˜ and O(L˜2) operations to obtain the product Φ˜−1x˜, and (ii)L˜ evaluations of the covariance function φ(̂pl, p˜i) for each test device, amounting to O(L̂ L˜M) operationsfor the L̂ test devices. In total, since we generally have L̂ << L˜ and M << L˜, calculating µ̂(CGP)x incursa complexity of O(L˜3). To calculate the predictive variance [Ĉ(CGP)x ]ll, ∀l = 1, . . ., L̂, observe from (3.17)154that we need to calculate∑L˜i=1∑L˜j=1 φ(̂pl, p˜i)[(Φ˜)−1]i jφ(˜pj, p̂l). The φ(̂pl, p˜i) terms are obtained in O(L̂M)operations. Since the matrix Φ˜ and its Cholesky factor are already known from the µ̂(CGP)x calculation,the product [(Φ˜)−1]i jφ(˜pj, p̂l) requires O(L˜ L̂M) operations to obtain the φ(˜pj, p̂l) terms and O(L˜2 L̂) toobtain the product. Once these terms are available, evaluating the sum∑L˜i=1∑L˜j=1 φ(̂pl, p˜i) [(Φ˜)−1]i jφ(˜pj, p̂l)requires another O(L˜ L̂) operations. In total, since we generally have L̂ << L˜ and M << L˜, calculating the[Ĉ(CGP)x ]ll for all the L̂ test devices incurs an additional complexity of O(L˜2 L̂).B.4.3 Complexity of NaGP MethodFrom (3.23), we know that the NaGP method requires computations of the [µ̂CGPx (s)]l and [ĈCGPx (s)]ll forS i.i.d samples p̂∗l(s), 1 ≤ s ≤ S, drawn from p̂∗l|p̂l, Σ̂l ∼ N(̂pl, Σ̂l). We therefore incur O(L˜3 + SL̂L˜M)computations to obtain µ̂NaGPx and an additional O(SL˜2 L̂) computations to obtain ĈNaGPx , respectively.B.4.4 Complexity of GaGP MethodObserve from ((3.25)) that the calculation of [µ̂(GaGP)x ]l for the L̂ devices requires one computation ofψ = Φ˜−1x˜, L̂ L˜ computations of α′N (̂pl; p˜i,B + Σ̂l) and L̂ L˜ computations of γp̂Tl p˜i. Calculating Φ˜−1x˜requires O(L˜3) operations. The L̂ L˜ computations of α′N (̂pl; p˜i,B+ Σ̂l) require a total of O(L̂ L˜M) oper-ations if the {Σ̂l} are diagonal matrices and a total of O(L̂M3 + L̂ L˜M2) operations otherwise. The L̂ L˜computations of γp̂Tlp˜i require a total of O(L̂ L˜M) operations. Therefore, considering non-diagonal {Σ̂l},L̂ << L˜, M << L˜, and L̂M2 < L˜2, the calculation of [µ̂(GaGP)x ]l for the L̂ devices incurs a total of O(L˜3)complexity. Next, observe from (3.25) that the calculation of [Ĉ(GaGP)x ]ll for the L̂ test devices requires L̂computations of Tr(Σl) and p̂Tl p̂l, one computation of ξ , L˜2 computations of N (˜pi; p˜j,2B), L̂ L˜2 computa-tions of N (̂pl; p˜i+p˜ j2 , B2 + Σ̂l), L̂ L˜ computations of {[Υ]j} , L̂ L˜2 computations of [Υ]Tj p˜i, L̂ L˜ computationsof γp̂Tlp˜i, L̂ L˜2 computations of Tr(˜pi (˜pj)T Σ̂l), and L̂ computations of the [µ̂(GaGP)x ]l values. The L̂ com-putations of Tr(Σl) and p̂Tl p̂l require O(L̂M) operations. The computation of ξ requires O(L˜3) operationsfor the Cholesky decomposition of Φ˜, O(L˜2) operations to obtain ψ, and another O(L˜2) operations to obtainthe products [ψ]i[ψ]j . The L˜2 computations of N (˜pi; p˜j,2B) require O(L˜2M) operations. If the matri-ces {Σ̂l} are diagonal, the L̂ L˜2 computations of N (̂pl; p˜i+p˜ j2 , B2 + Σ̂l) require O(L̂M3) operations to obtainthe Cholesky factors of {B2 + Σ̂l} and another O(L̂ L˜2M2) operations to obtain the Gaussian terms. Other-wise, the L̂ L˜2 computations of N (̂pl; p˜i+p˜ j2 , B2 + Σ̂l) would require O(L̂ L˜2M) operations. When computing155{[Υ]j}, we know that the Gaussian terms (c.f. (3.26)) are already available from the µ̂(GaGP)x calculations(c.f. (3.25)). Therefore, if the {Σ̂l} are non-diagonal matrices, the L̂ L˜ computations of {[Υ]j} would requireO(L̂M3) operations for Cholesky decomposition of Σ̂l and Σ̂lB+I and another O(L̂ L˜M2) operations for thematrix-vector products in [Υ]i (c.f. (3.26)). If the {Σ̂l} are diagonal matrices, the L̂ L˜ computations of [Υ]jrequire a total of O(L˜ L̂M) operations. When all the [Υ]j are available, the L̂ L˜2 computations of [Υ]Tj p˜irequire O(L˜2M) operations. The L̂ L˜ computations of γp̂Tlp˜i require O(L̂ L˜M) operations. The L̂ L˜2 compu-tations of Tr(˜pi (˜pj)T Σ̂l) require O(L̂ L˜2M2) operations if {Σ̂l} are non-diagonal and O(L̂ L˜2M) otherwise.Lastly, from our earlier discussion, calculation of µ̂(GaGP)x requires O(L˜3) operations. In total, consideringnon-diagonal {Σ̂l}, L̂ << L˜, and M << L˜, calculation of Ĉ(GaGP)x incurs complexity of O(L˜3 + L˜2 L̂M2).B.4.5 Complexity of the baselines in Chapter 3We have considered three baselines in Chapter 3: the maximum-likelihood estimator (MLE) in [55], therobust semidefinite programming estimator (RSDPE) in [61], and the k-nearest neighbours (kNN) finger-printing algorithm in [71]. In the MLE method, a non-linear least squares problem is solved iteratively usingthe Levenberg-Marquardt (LM) algorithm. Assuming K steps to convergence, the MLE method incurs a to-tal complexity of O(M3 +KM2), where M is the number of sensors deployed in the network. The numberof LM steps depends on the starting point as well as the convergence threshold. In RSDPE, a semidefiniteprogramming problem is solved iteratively using the interior-point method. This incurs a cost of O(d2M2)per iteration. The total number of iterations required for convergence in the interior-point method is boundedby O(√M ln( 1ζ )). Here, d is the number of variables to be estimated, M is the number of sensors, and ζ isthe iteration tolerance. The total complexity of the RSDPE algorithm is therefore O(d2M2.5 ln( 1ζ )). Notethat d = 2 in our work because we focus on 2D localization. The kNN algorithm requires calculation of thedistance from the test DRSS to the L̂ fingerprints in the database. It therefore incurs O(L̂) computations.156Appendix CAppendix for Chapter 4C.1 Complexity Analysis for the GP Methods and baselines in Chapter 4C.1.1 Complexity of RecGP MethodThe computational cost of RecGP method can be obtained as the sum cost of the CGP method and thereconstruction steps in (4.13), (4.17) and (4.18). The reconstruction step in (4.13) incurs O(L¯3 + L˜M2)computations because we require (i) O(L¯3) operations to obtain the right singular matrix V of P¯∗ via singularvalue decomposition, (ii) O(M2M0) operations to obtain the product V[M0](V[M0])T , (iii) O(L˜M2) operationsto obtain the product P˜V[M0](V[M0])T . Similarly, (4.17) incurs O(L˜3+ L̂M2) computations and (4.18) incursO(L¯3 + L̂M2) computations, respectively. Lastly, following the analysis for the CGP method, calculatingthe µ̂(RecGP)x and [Ĉ(RecGP)x ]ll requires an additional O(L˜3) and O(L˜2 L̂) computations, respectively.In total, to obtain µ̂(RecGP)x and Ĉ(RecGP)x , we incur O(L˜3+ L̂M2) computations when training with noise-free DRSS, and O(L˜3+ L˜M2) computations when training with noisy DRSS (assuming L¯ < L˜), respectively.C.1.2 Complexity of RecGaGP MethodThe computational cost of RecGaGP method can be obtained as the sum cost of the GaGP method and thereconstruction steps in (4.13), (4.17) and (4.18). The reconstruction step in (4.13) incurs O(L¯3+ L˜M2) com-putations because we require (i) O(L¯3) operations to obtain the right singular matrix V of P¯∗ via singularvalue decomposition, (ii) O(M2M0) operations to obtain the product V[M0](V[M0])T , (iii) O(L˜M2) computa-tions to obtain the product P˜V[M0](V[M0])T . Similarly, (4.17) incurs O(L˜3 + L̂M2) computations and (4.18)incurs O(L¯3 + L̂M2) computations, respectively. After the reconstruction step in the prediction phase, weneed to compute the residual noise covariance matrices Σ̂(rec)l (c.f. Lemma 4.1). The matrix Σ̂(rec)l can bewritten in compact form as (V[M0](V[M0])T )T Σ̂l(V[M0](V[M0])T ) . The product V[M0](V[M0])T was alreadycomputed during the reconstruction. The additional cost would therefore be O(M3) per test device, leading157to a total of O(L̂M3) computations. In total, to obtain µ̂(RecGaGP)x and Ĉ(RecGaGP)x , we incur O(L˜3 + L̂M3)computations when training with noise-free DRSS, and O(L˜3 + L˜M2 + L̂M3) computations when trainingwith noisy DRSS (assuming L¯ < L˜), respectively.Overall, we note from the above discussion that when L̂ < L˜, L¯ < L˜, and M is an order of magnitudesmaller than L˜, the location estimates from the GaGP, RecGP and RecGaGP methods can be obtained inO(L˜3) computations, which is also the case with the conventional GP method.C.1.3 Complexity of the baselines in Chapter 4The baselines in Chapter 4 include the maximum-likelihood estimator (MLE), robust semidefinite program-ming estimator (RSDPE), k-nearest neighbours (kNN), conventional GP (CGP), numerical approximationGP (NaGP), and Gaussian approximation GP (GaGP) methods. The complexity of each of these methodshas already been provided in Section B.4.158Appendix DAppendix for Chapter 5D.1 On the Inability to Pursue Weighted Least Squares for LocationEstimation in Chapter 5The optimum weight W is the inverse of the covariance Ce of the error e in (5.13) Elements of Ce are givenby[Ce]m,n = E([e]m−E([e]m))([e]n −E([e]n))T ,∀m,n,(a)= ((x̂1− x)2 + (ŷ1− y)2)2E{δ∗m2δ∗n2− δ∗m2E(δ∗n2)− δ∗n2E(δ∗m2)+E(δ∗m2)E(δ∗n2)}, (D.1)where (a) follows from substituting [e]m from (5.13) and simplifying. In order to derive a closed-formexpression for [Ce]m,n, we need to derive the terms E(δ∗m2), E(δ∗n2) and E(δ∗m2δ∗n2) in closed-form, whichwe attempt as follows. The polynomial in (5.10) has up to three roots, from which the estimate loge δ∗m isobtained through the root selection procedure presented in Section 5.2. Since the estimate loge δ∗m can beany of the three roots, we need to derive closed-form expressions for all three roots. One of the roots isgiven in closed-form by Cardano’s formula [187] asα1 = a0 +3√a1 +√a21 + (a2− a20)3 +3√a1−√a21 + (a2− a20)3,where, a0 =µγqm3σ2γ, a2 =λ2m(σ2γ + µ2γ)−σ2γq2m3σ4γ,a1 =2µ3γq3m−9qmµγ(λ2m(σ2γ + µ2γ)−σ2γq2m)−27λ2mσ2γ µγqm54σ6γ.(D.2)The remaining two roots share the same structure as α1, so the analysis below applies to them too.When the selected root for loge δ∗m is α1, the expectation E(δ∗m2) becomes E(exp(2α1)). Substituting α1from (D.2), we note that the polynomial inside the exp(.) term contains fractional-powers in the Gaussianr.v. qm because the terms inside the squared-roots are not perfect squares. To evaluate the expectation, we159therefore need to integrate over the exponential of a fraction-order polynomial in a Gaussian r.v., for whichno closed-form solution exists. We are therefore unable to derive E(δ∗m2), and subsequently Ce (c.f. (D.1))and W in closed-form. For this reason, we make the sub-optimal choice of W = I.D.2 Mathematical Simplifications to Obtain β in the CLLS EstimatorWe begin by simplifying the CLLS estimate ϑ∗(CLLS) in (5.22) as followsϑ∗CLLS=(ATA+ βH)−1(ATb−0.5βg),(a)= (I+ β(ATA)−1H)−1(ATA)−1(ATb−0.5βg),(b)= V(I+ βΛ)−1V−1(ATA)−1(ATb−0.5βg),(D.3)where (a) follows from extracting the matrix product (ATA) out of the inverse term on the right hand side(r.h.s.), and (b) from substituting the eigenvalue decomposition VΛV−1 for (ATA)−1H in order to simplifyfurther. Next, we substitute the ϑ∗CLLS from (D.3) into the constraint in (5.20) in order to obtain the optimumLagrangian parameter β∗. The quadratic and linear terms in the constraint in (5.20) can then be simplifiedas done in (D.4).(ϑ∗CLLS)THϑ∗CLLS(a)= (bTA−0.5βgT )(V(I+ βΛ)−1V−1(ATA)−1)HV(I+ βΛ)−1V−1(ATA)−1(ATb−0.5βg)(b)= (bTA−0.5βgT )V(I+ βΛ)−1V−1VΛV−1V(I+ βΛ)−1V−1(ATA)−1(ATb−0.5βg)(c)= (bTA−0.5βgT )V(I+ βΛ)−1Λ(I+ βΛ)−1V−1(ATA)−1(ATb−0.5βg),(d)= κ(I+ βΛ)−1Λ(I+ βΛ)−1θ + βη(I+ βΛ)−1Λ(I+ βΛ)−1θ+βκ(I+ βΛ)−1Λ(I+ βΛ)−1ξ + β2η(I+ βΛ)−1Λ(I+ βΛ)−1ξ, andgTϑ∗CLLS(e)= gTV(I+ βΛ)−1V−1(ATA)−1(ATb−0.5βg)( f )= −2η(I+ βΛ)−1θ −2η(I+ βΛ)−1ξ,where,κ = bTAV,θ = V−1(ATA)−1ATb,η = −0.5gTV, and ξ = −0.5V−1(ATA)−1g.(D.4)Details on the steps in (D.4) are as follows: in (a) we substitute ϑ∗CLLS from (D.3) and note that V(I+βΛ)−1V−1(ATA)−1 is a symmetric matrix (it is the inverse matrix (ATA+ βH)−1 (c.f. (D.3)), in (b) we160substitute the eigenvalue decomposition (ATA)−1H = VΛV−1, in (c) we utilize the relation V−1V = I, in (d)we introduce the auxiliary vectors κ, θ, η, and ξ , in (e) we substitute ϑ∗CLLS from (D.3), and in (f), we usethe auxiliary vectors θ, η, and ξ for the convenience of notation.Next, we substitute the (ϑ∗CLLS)THϑ∗CLLS and gTϑ∗CLLS from (D.4) into the constraint in (5.20) and sim-plify further as follows. Since the matrix (ATA)−1H is of rank 2 (c.f. H in (5.20)), we know that its eigen-value matrix Λ has only two non-zero diagonal elements. We can therefore specify Λ as diag([0;λ2;λ3]).Next, for the ease of simplification, let us also specify the vectors κ, η,θ, and ξ in (D.4), [κ1; κ2; κ3]T ,[η1; η2; η3]T , [θ1; θ2; θ3], and [ξ1; ξ2; ξ3] respectively. The equality constraint in (5.20), with the quadraticand linear terms simplified as in (D.4) and the vectors κ, η, θ, and ξ denoted as above, then simplifies intoa fifth degree polynomial in β, given by (D.5).t5β5 + t4β4 + t3β3 + t2β2 + t1β+ t0 = 0, where,t5 = −2η1ζ1(λ2λ3)2,t4 = −2η1θ1(λ2λ3)2−4η1ζ1λ2λ3(λ2 +λ3)−η3ξ3λ3λ22 −η2ξ2λ2λ23,t3 = −4η1θ1λ2λ3(λ2 +λ3)−2η1ζ1((λ2 +λ3)2 +2λ2λ3)−η3θ3λ22λ3−η2θ2λ2λ23 −2η3ξ3λ2(λ2 +λ3)−2η2ξ2λ3(λ2 +λ3)+ κ3ξ3λ22λ3 + κ2ξ2λ2λ23 −η3θ3λ22λ3−η2θ2λ2λ23,t2 = −2η1θ1((λ2 +λ3)2 +2λ2λ3)−4η1ζ1(λ2 +λ3)−2η3θ3λ2(λ2−2λ3)−2η2θ2λ3(−2λ2 +λ3)−η3ξ3(4λ2 +λ3)−η2ξ2(λ2 +4λ3)+ κ3θ3λ22λ3 + κ2θ2λ2λ23 +2κ3ξ3λ2λ3 +2κ2ξ2λ2λ3,t1 = −4η1θ1(λ2 +λ3)−2η1ζ1−2η2ξ2−2η3ξ3−η3θ3(4λ2 +λ3)−η2θ2(λ2 +4λ3)+2κ3θ3λ2λ3+2κ2θ2λ2λ3 + κ2ξ2λ2 + κ3ξ3λ3,t0 = −2η1θ1−2η2θ2−2η3θ3 + κ2θ2λ2 + κ3θ3λ3.(D.5)To solve (D.5), we use the Matlab function roots [184] and obtain up to five real roots. The following rootselection procedure is then pursued to obtain β∗. Firstly, we only consider the real roots because β is real-valued for real optimization problems. For each real root, we compute (Aϑ∗CLLS−b)T (Aϑ∗CLLS−b), which isthe objective in (5.20) when ϑ = ϑ∗CLLS. The root which yields the smallest objective value is then selectedas the optimum β∗.161D.3 Complexity Analysis for the Proposed Location Estimation Algorithmsand Baselines in Chapter 5The TLLS estimator requires (i) the solving the cubic polynomial in (5.10), which amounts to O(33) com-putations per anchor, (ii) the root selection for δm, which requires computation of | |Aϑ∗LLS − b| | for eachroot combination; this incurs O(2 ∗ 3M+1(M − 1)) computations in the worst case (3 real roots per anchor)and O(2∗9(M −1)) computations in the best case (1 real root per anchor), and (iii) the matrix operations in(5.14)-(5.18), which amounts to O(32(M −1)) operations. In total, the TLLS incurs a worst-case complexityof O(3M+1(M −1)) and best-case complexity of O(M) operations.Next, the CLLS estimator requires (i) solving the cubic polynomial in (5.10), which amounts to O(33)computations per anchor, (ii) the root selection for δm, which requires computation of | |Aϑ∗LLS − b| | foreach root combination; this incurs O(2 ∗ 3M+1(M − 1)) computations in the worst case (3 real roots peranchor) and O(2 ∗ 9(M − 1)) computations in the best case (1 real root per anchor), (iii) solving the fifthdegree polynomial in (D.5) for the Lagrangian parameter β, which amounts to O(53) computations, (iv) thematrix operations in (5.22), which amounts to O(5 ∗ 32(M − 1)) operations in the worst case (5 real rootsfrom (D.5)), and (v) the root selection for β, which requires computation of | |(Aϑ∗CLLS−b|)T (Aϑ∗CLLS−b)|for each β; this incurs a total of O(5 ∗ 32(M − 1)) computations in the worst case (5 real roots from (D.5)).In total, the CLLS incurs a worst-case complexity of O(3M+1(M − 1)) and best-case complexity of O(M)operations.Lastly, the MAP estimator requires (i) the TLLS estimate as a starting point, which incurs O(3M+1(M −1)) operations in the worst-case and O(M) operations in the best-case, and (ii) the LM update step, whichincurs O((M −1)3) operations (mainly due to inversion of Σq). Assuming K steps to convergence, the MAPestimator incurs O((3M+1(M −1)+K(M −1)3) worst-case and O(K(M −1)3) best-case complexity.Complexity of the baselines:We have considered five baseline algorithms for performance comparisons: (i) WLS-known γ, which is thetwo-step WLS proposed in [55] for known γ; (ii) RSDPE, which is the semidefinite programming solutionproposed in [54] to estimate s under imperfect γ; (iii) ReducedBiasDrss, which is the reduced bias leastsquares algorithm proposed in [57] for known γ; (iv) the ratio and numerical search method proposedin [53]; we search for γ within [µγ −2σγ, µγ +2σγ] in steps of 0.01, and (v) MLE, the maximum likelihood162method proposed in [73].The WLS-known γ and the RedBiasDRSS algorithms incur O(M3) computations, with the least squaresstep being the main contributor. The RSDPE, as reported in the complexity discussion for Chapter 3, incursO(d2M2.5 ln( 1ζ )) computations, where d = 2 is the number of unknowns to be estimated and ζ is the iterationtolerance. The PL search method incurs O(KM3) computations, where K is the number of search steps forthe path loss exponent. The MLE method requires solving a non-linear least squares problem. Whenthe LM algorithm is used, it incurs a one-time cost of O(M3) computations to invert the noise covariancematrix and additional O(M2) computations per iteration for the associated matrix multiplications. The totalcomputational cost of the MLE algorithm depends on the number of iterations to convergence, which inturn depends on our choice of the starting point. It is difficult to quantify the impact of the starting point onthe iterations to convergence in terms of the big-O notation. For an abstract comparison, we have simplyreported the runtimes for an example scenario in Section 5.4 when presenting the numerical studies.163Appendix EList of Other PublicationsPublications from my Ph.D. program that were not included in the thesis are listed below.• K. D’Souza, K. N. R. S. V. Prasad, V. K. Bhargava, “Hybrid precoding with partially connected struc-ture for mmWave massive MIMO: a parallel framework and feasibility analysis," IEEE Transactionson Wireless Communications, vol. 17, no. 12, pp. 8108-8122, Oct. 2018.• K. N. R. S. V. Prasad, E. Hossain, V. K. Bhargava, “Energy efficiency in massive MIMO-based 5Gnetworks: opportunities and challenges," IEEE Wireless Communications, vol. 24, no. 3, pp. 86-94,Jan. 2017.• K. N. R. S. V. Prasad, J. Cheng, V. K. Bhargava, “Accurate distance estimation for rss localizationwith statistical path loss exponent model," in Proc. of IEEE Global Communications Conference(Globecom), Virtual Conference, Dec. 2020, pp. 1-6.• K. N. R. S. V. Prasad and V. K. Bhargava, “A classification algorithm for blind uav detection inwideband rf systems," in Proc. of IEEE Vehicular Technology Conference (VTC Fall 2020), VirtualConference, Nov. 2020, pp. 1-7.• K. N. R. S. V. Prasad and V. K. Bhargava, “Rss-based positioning in distributed massive mimo underunknown transmit power and pathloss exponent," in Proc. of IEEE Vehicular Technology Conference(VTC Fall 2019), Honolulu, USA, Sept. 2019, pp. 1-6.• K. N. R. S. V. Prasad and V. K. Bhargava, “Energy-efficient multi-cell massive MIMO: how manyantennas should we use ?" in Proc. of IEEE International Conference on Advanced Networks andTelecommunications Systems (ANTS), Bangalore, India, Nov. 2016, pp. 1-6.• S. Mallick, R. Loodarichech, K. N. R. S. V. Prasad, and V. K. Bhargava, “Resource allocation forcooperative D2D communication,” chapter in 5G Mobile Commun., pp. 531-570, Springer, May 2017.164
Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Bhargava, Vijay K., 1948-
Machine learning algorithms for intruder signal detection and device localization in wireless radio frequency systems
Text