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MINISTRY OF EDUCATION AND TRAINING MINISTRY OF DEFENCE ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY NGUYEN TUAN MINH RESEARCH ON A SOLUTION FOR IMPROVEMENT OF RADIO DIRECTION FINDING Specialization: Electronic engineering Code: 9520203 SUMMARY OF TECHNICAL DOCTORAL THESIS HA NOI – 2020 This Thesis was completed at: Academy of Military Science and Technology, Ministry of Defense Scientific Supervisors: Dr Le Thanh Hai Dr Nguyen Trong Luu Reviewer 1: Prof Dr Bach Gia Duong Reviewer 2: Assoc Prof Dr Nguyen Huy Hoang Reviewer 3: Assoc Prof Dr Bui Ngoc My This thesis will be defended in front of Doctor Examining Committee held at Academy of Military Science and Technology in time hour … date … month … year 2020 This thesis may be found at: - Library of Academy of Military Science and Technology - The Vietnam National Library INTRODUCTION Necessity: Direction finding of radiation sources plays an important role in many areas in life especially for Security - Defense [1], [2], [5], [6], which has been applied in many fields such as: Radar, radio monitoring, frequency management, search and rescue, aerospace, optimization in communications, reconnaissance, electronic warfare and many other areas Although there have been many proposed solutions to improve the quality of radio direction finding, there are still some limitations that need to be addressed such as: Accuracy, resolution, computational complexity, signal to noise ratio (SNR), noise and interference affects Therefore, the improvement of radio direction finding of radiation sources also has issues that need further attention and study This is a research direction of scientific, practical and urgent significance to serve the fields of Economics - Society as well as Security - Defense Therefore, in this thesis, the following research topic is selected: “Research on a solution for improvement of radio direction finding” Objectives of the study: To study and propose direction finding solutions to solve the limitations of accuracy, resolution, computational complexity as well as the influence of color noise and uncertainty in the information of the number of radio sources Content of study: Study on improved PM algorithm, signal models for ULA-UCA and ULA antenna array with application in fast-track problem solving; Study on the influence of color noise and signal models applied to L-shape antenna and orthogonal ULA-ULA antenna with symmetric phase center in solving 2D problem of correlated radio radiation sources Study subjects: Antenna array structures, signal processing algorithms, solutions for combining antenna array construction and signal processing algorithms Methodology: To study theory, signal models, application of mathematical tools, simulation and assessments using computers Scientific significance: The thesis has contributed a number of highquality solutions on direction finding of radio radiation sources The content presented in the thesis can be a useful reference for research and teaching in academics, schools and research institutions Practical significance: The thesis has proposed some solutions for direction finding of radio sources in accordance with the current research trend in the world These proposals can be applied for design and testing of systems of direction finding and positioning of radio radiation sources for National Defense, Security and National Economy for early detection, monitoring and accurate positioning of targets in real time CHAPTER 1: OVERVIEW ON DIRECTION FINDING OF RADIO SOURCES 1.1 Introduction on radio direction finding There are many ways to classify radio direction finding systems (referred to as radiation sources), but most of them commonly rely on system structure and signal processing methods [1] The evaluation of advantages - disadvantages of each conventional directing finding algorithm is relied on the following basic criteria: Accuracy, resolution, operational speed, capability of finding direction in multi-path environment, sensitivity and noise elimination The quality of each direction finding algorithm is affected by a number of basic factors as follows [1]: Number of antenna elements, number of signal samples, SNR, distances between antenna elements (d), correlation between signals and other factors (heterogeneity of amplitude and phase on antenna elements, coupling errors and position deviation of antenna elements) To meet the goal of improving the quality of radio direction finding, it can be divided into three basic study directions as follows: - To study the construction of a model of antenna array structure - To study, propose, improve and apply several signal processing algorithms - To study the combination of building antenna structure and signal processing algorithm These are the main study directions in this thesis 1.2 Overview of research literature The studies mainly focus on resolving existing limitations as the follows: - Problems about accuracy, resolution and computational complexity - Effects of nonlinear noise, color noise and uncertainty of the information about the number of radio sources - The use of direction finding outcomes for finding positions of radio sources Accordingly, the thesis will focus on solving specific problems as follows: - The study and propose a high-precision direction finding solution with low computational complexity and capacity of operating in small SNR conditions - To study and propose solutions for direction finding in conditions affected by some types of color noise and to solve the 2D direction finding problem under uncertain information about radio sources - To study and propose positioning solutions with highly practical applications based on the direction finding results 1.3 General direction finding problem 1.4 Several typical direction finding algorithms 1.5 Several positioning methods based on direction finding results 1.6 Research questions 1.7 Chapter conclusion On the basis of studying general theory of radio direction finding, Chapter sets out the issues to be studied and methods to evaluate the results for the next chapters of the thesis CHAPTER 2: PROPOSAL OF DIRECTION FINDING METHOD USING IMPROVED PM ALGORITHM 2.1 Chapter Introduction 2.2 Improved PM Algorithm Assuming that the number of radiation sources (p) is known and the number of antennas (M) should meet the condition: M ≥ 2p + 2, direction vector A can be decomposed into the following form [12]: (2.2) Where: The matrix and of size p x p, and the matrix of size (M – 2p) x p The partly cross-correlated matrices can be constructed as follows [12]: ( )( ) ( ) ( ) (2.3) ( )( ) ( ) ( ) (2.4) ( )( ) ( ) ( ) (2.5) ) is obtained from row i to row j of the matrix Where: ( )( ( ), ( ) ( ) is the covariance matrix of signals Because radiation sources are independent, both R and , are reversible matrices, hence: ( ) (2.6) By similar transformation: ( ) (2.7) By adding both sides of the equations (2.6) and (2.7): (2.8) The equation (2.8) has an equivalent form as follows: [ (2.9) ( )] Where: ( ) is an unit matrix with size of M – 2p Let set [ written as follows: ( ) ], then the equation (2.9) can be re- (2.10) When p signals are attached to the direction of incident wave θi then: ( ) (2.11) Where: ( ) denotes the direction vector corresponding to Similar to MUSIC algorithm, from the equation (2.11), the signal power spectrum obtained ( ) has the following form: ( ) (2.12) ( ) ( ) From (2.9), (2.10), the determination of does not require any methods for development of eigenvalues hence the computational complexity can be significantly reduced Moreover, the information about covariance matrix form of noise is not involved in , so it can be used in case of non-linear noise This is the basis for the thesis to propose direction finding solutions with low computational complexity suitable for applications with small SNR 2.3 Proposed direction finding solution for uncorrelated radiation sources using ULA-UCA antenna array 2.3.1 Modeling and proposal of solution Figure 2.1: ULA-UCA antenna array model ULA-UCA antenna model shown in Figure 2.1 is a combination of the ULA and UCA antenna arrays in which a ULA antenna array is placed vertically at the center of a UCA antenna array The output value of ULA direction finder at time t is expressed as follows [9]: ( ) (2.13) Where: is the transpose of the array weights and ( ) is the signal receive vector We assume p uncorrelated signal sources [s1(t) s2(t) … sp(t)] simultaneously arrive at the antenna array on the elevation angles (θ1, θ2, …, θp) The signal vector ( ) can be presented as follows: ( ) ( ) ( ) ( ) (2.14) The direction vector at θi (i = 1, …, p) can be represented as follows: ( )( ( ) ( ) ( ) ( ) ) ( ) ( ) [ ] (2.15) For UCA direction finder, the output signal at time t has the form [9]: ( ) (2.18) Where: is the transpose of the circular array weights vector and ( ) is the signal receive vector ( ) ( ) (̂ ) ( ) (2.19) Where: ̂ is the pre-defined elevation angle and (̂ ) [ (̂ ) (̂ ) (̂ )] (2.20) (̂ ) ( ) (̂ ) ( ) (̂ ) ( ) (̂ ) * + (2.21) By applying improved PM algorithm, the signal spectrum power can be obtained as following: - For ULA antenna array: ( ) (2.24) ( ) ( ) - For UCA antenna, the signal power spectrum of the source i (i = 1, …, p) is: (̂ ) (2.25) (̂ ) (̂ ) According to equation (2.25), the azimuth is calculated separately for each signal power spectrum corresponding to the found elevation angle This is necessary to avoid confusion in pairing up the elevation angle and the azimuth and should be applied in the next 2D direction finding solutions in the thesis 2.3.2 Simulation and result evaluation In order to evaluate the performance of proposed solution, we conduct simulations according to the algorithm flowcharts under some selected simulation conditions as follows: UCA: - Antenna elements are evenly spaced in a circle - Number of elements per array: 10 - Antenna element type: Isotropic - Distance between antenna elements: λ/2 ULA: - Antenna elements are evenly spaced in a straight line - Number of elements per array: 10 - Antenna element type: Isotropic - Distance between antenna elements: λ/2 Radiation sources: - Number of radiation sources: - Signal to noise ratio SNR for both radio sources: -5dB - Arrival of angles (elevation, azimuth): [(25o, 70o), (80o, 310o)] and [(25o, 70o), (25o, 310o)] - Number of signal samples: L =1000 Noise: White Gaussian noise and non-uniform noise The first simulation is to evaluate the accuracy, resolution of the proposed solution, traditional PM and MUSIC algorithms with the same simulation conditions in Gaussian white noise environment and nonlinear noise Figure 2.4: Results of direction finding for elevation angles of two radio sources (25o, 70o) and (80o, 310o) in Gaussian white noise condition Figure 2.5: Results of direction finding for azimuth angles of two radio sources (25o, 70o) and (80o, 310o) in Gaussian white noise condition Figures 2.4 and 2.5 show the average signal spectrum power obtained after 1000 Monte Carlo trials for finding elevation angle and azimuth under Gaussian white noise In terms of accuracy, the proposed solution has small direction errors in accordance with two incident waves which are (0,02o; 0,04o) and (0,01o; 0,01o) Within the research scope, this thesis assumes nonlinear noise on ULA-UCA antenna array with covariance matrix as follows: (2.26) (2.27) The results are shown in Figures 2.6 and 2.7 with the corresponding errors (0,07o; 0,06o) and (0,16o; 1,01o) Figure 2.6: Results of direction finding for elevation angles of two radio sources (25o, 70o) and (80o, 310o) in nonlinear noise condition Figure 2.7: Results of direction finding for azimuth angles of two radio sources (25o, 70o) and (80o, 310o) in nonlinear noise condition The resolutions Δθ and Δϕ obtained for elevation and azimuth angles with each SNR are presented in Tables 2.1 and 2.2 Table 2.1 Direction resolution obtained by proposed solution for ULA-UCA antenna array under white Gaussian noise SNR Resolution (deg) Real angle (deg) Obtained angle (deg) Δθ θ1 θ2 θ1 ’ θ2 ’ -5dB 15 25 40 25,28 40,01 0dB 11 25 36 25,52 35,59 5dB 10dB 25 33 25,3 32,91 25 31 25,06 30,97 Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ -5dB 10 70 80 69,88 79,98 0dB 70 75 70,04 75,97 5dB 70 73 70 73 10dB 70 72 70,01 71,99 Table 2.2 Direction resolution obtained by proposed solution for ULA-UCA antenna array under nonlinear noise SNR Resolution(deg) Real angle (deg) Obtained angle (deg) Δθ θ1 θ2 θ1 ’ θ2 ’ -5dB 31 25 56 24,02 56,49 0dB 26 25 51 24,09 51,49 5dB 10 25 35 24,54 35,34 10dB 25 32 24,92 32,06 Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ -5dB 35 70 105 70,06 105 0dB 34 70 104 70,96 103,68 5dB 13 70 83 69,05 83,14 10dB 70 72 69,38 72,26 The data in Tables 2.1 and 2.2 show that resolutions of the proposed solution depends greatly on SNR Under this simulation condition, the proposed solution has resolution of (6o, 2o) and (7o, 2o) corresponding to Gaussian white noise and nonlinear noise at SNR of 10dB Figures 2.10 and 2.11 represent the RMSE dependence corresponding to the angle of elevation and azimuth according to the number of signal samples Figure 2.10: The dependence of RMSE on elevation angle of two radio sources (25o, 70o) and (80o, 310o) according to the number of signal samples 11 ̃ ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ] (2.40) ( ) ( ) ( ) The covariance matrix can be obtained as follows: ( )̃ ( ) ( ) ( ) (2.51) ̃ ̃( ) ̃( ) is the covariance matrix of the radio Where: source By mapping the signal vector into the Hermitian Toeplitz form, it is possible to determine a maximum of (M – 1) correlated radio sources because the radio sources then become uncorrelated [39] On the other hand, since the condition (1) must be satisfied, the number of incident angles can be determined to be at most equal to (M – 2)/2p To separate the signal space from noise space, the improved PM algorithm is used as described in section 2.2 Then, the signal power spectrum is determined as follows: (2.52) ( ) ( ) 2.4.2 Simulation and result evaluation Conditions of simulation: ULA: - Antenna elements are evenly spaced in a straight line - Number of elements per array: - Antenna element type: Isotropic - Distance between antenna elements: λ/2 Radiation sources: - Number of radiation sources: - Signal to noise ratio SNR for both radio sources: 0dB - Azimuth angles: (100o, 120o, 140o) in case of uncorrelated radio sources and (60o, 75o, 95o) in case of correlated radio sources - Number of signal samples: L = with the proposed solution, L = with TLS and ESPRIT algorithm in case of uncorrelated radio sources, L = with TLS and Matrix Pencil algorithm in case of correlated radio sources Noise: White Gaussian noise First, the ability of proposed solution is evaluated in case the radio sources are uncorrelated and completely correlated The simulation was performed using 1000 Monte Carlo's trials Figure 2.18 shows the simulation results when three uncorrelated radio sources arriving at the 12 antenna array in the directions of 100o, 120o and 140o In terms of accuracy, this solution has very small direction errors relatively at 0,04 o; 0,01o and 0,07o Figure 2.18: Direction results for three uncorrelated radiation sources with incident angles [100o, 120o, 140o] Figure 2.19 shows the simulation results when three totally correlated radio sources arriving at the antenna array in the directions of 60o, 75o and 95o Figure 2.19: Direction results for three totally correlated radiation sources with incident angles [60o, 75o, 95o] Similar to the case of uncorrelated radiation sources, the proposed method has successfully found all three incident waves with direction errors 0,16o; 0,2o and 0,04o respectively It is realized that although only one single signal sample and small SNR (0dB) was used, the proposed solution could still successfully determine the incident wave directions for uncorrelated and correlated radiation sources at high accuracy The resolution Δϕ obtained for each SNR is summarized in Tables 2.3 and 2.4 Table 2.3 Resolution in finding direction for uncorrelated radio sources using ULA antenna array SNR -5dB 0dB Resolution (deg) Δϕ Real angle (deg) ϕ1 ϕ2 100 109 100 108 Obtained angle (deg) ϕ1’ ϕ2’ 99,54 109,29 99,84 108,18 13 5dB 100 106 102,02 105,83 10dB 100 105 99,91 104,93 Table 2.4 Resolution in finding direction for correlated radio sources using ULA antenna array SNR Resolution (deg) Real angle (deg) Obtained angle (deg) Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ -5dB 10 65 75 65 75,01 0dB 65 73 65,1 72,83 5dB 65 72 65,06 71,91 10dB 65 70 65,3 69,84 From the data of Tables 2.3 and 2.4, it can be seen that the resolution in finding directions of correlated and uncorrelated sources show small difference Although for small SNR (-5dB), the solution is still capable to distinguish two adjacent angles with incident angles of o for uncorrelated radiation sources and 10o for correlated radiation sources Under this simulation conditions, at SNR equal to 10 dB, the resolution of proposed method is derived at 5o For better view on the improvement of accuracy, in next simulation, the quality of proposed method is evaluated in pairs with ESPIRIT, Matrix Pencil and TLS algorithms The simulation is done using 11element ULA array (equally spacing of λ/2) Figures 2.26 and 2.27 represent the RMSE inversely proportional to SNR in case of two uncorrelated radio sources with incident angles of 55o and 70o Figure 2.28 and 2.29 shows the simulation results of RMSE versus SNR for three totally correlated radio sources with incident angles of 45o, 60o and 75o Figure 2.26: The dependence of RMSE on SNR for L = signal sample of proposed method 14 Figure 2.27: The dependence of RMSE on SNR for L = 10 signal sample of ESPRIT and TLS algorithms [76] Figure 2.28: The dependence of RMSE on SNR for L = signal sample of proposed method Figure 2.29: The dependence of RMSE on SNR for L = signal sample of Matrix Pencil and TLS algorithms [76] 2.5 Conclusion for Chapter Chapter has presented the improved PM algorithm in detail and then proposed 1D and 2D direction finding solutions with low computational complexity at small SNR The simulation results have 15 demonstrated the good performance of these solutions under the assumed conditions CHAPTER 3: PROPOSAL OF 2D DIRECTION FINDING SOLUTION UNDER COLOR NOISE AND UNCERTAINTY OF INFORMATION ABOUT NUMBER OF RADIO SOURCES, AND A POSITIONING SOLUTION BASED ON DIRECTION FINDING 3.1 Chapter Introduction 3.2 Proposal of 2D direction finding using a L-shape antenna array under symmetric Toeplitz colored noise 3.2.1 Symmetric Toeplitz colored noise The two types of noise of the most interest are spherical isotropic noise (three-dimensional noise field) and cylindrical isotropic noise (two-dimensional noise field) This phenomenon occurs when the noise field around the antenna array is a set of symmetric distribution points [54] Although these two types of noise are less common in practice, they can be roughly assumed in the case of antenna elements arranged on a two-dimensional plane [73] It is realized that the correlation function for these two types of noise is in the form sine(X)/X and ( ) respectively Therefore, the covariance matrix of noise will have the symmetric Toeplitz form with the correlation coefficient equal to one on the diagonal and the other correlation coefficients have smaller values when moving away from the diagonal [54] 3.2.2 Modeling and proposal of solution Let consider an L-shaped antenna array consisting of two ULA antenna arrays perpendicular to each other at the origin O (common reference element) as shown in Figure 3.3 Each antenna array consists of M elements arranged evenly (d) by half wavelength (λ) Figure 3.3: L-shape antenna array model 16 Let J be the conversion matrix with values of on the diagonal and in the remaining ( (3.8) ) Set ̃ Then the covariance matrix obtained ̃ has the following form: ̃ ̃̃ ( ) ( ) (3.9) By subtracting Rz with ̃ a new covariance matrix is derived as ̃ ( ) ( ) ( ) ( ) (3.10) T H As Nz is a symmetric Toeplitz matrix hence Nz , Nz and JNzJ are symmetric Toeplitz matrices and JNzTJ = (JNzJ)T = Nz hence the following is obtained: Nz = JNzTJ = JNz*J (3.11) During this event: ( ) ( ) ( ) ( ) (3.12) It can be seen that the covariance matrix of noise Nz in the equation (3.12) has been totally eliminated and has full order when the incident radio sources are correlated in pair [75] On the other hand, there are p eigenvalues not equal to zero hence the number of antenna elements M should only satisfy the condition M > p For radio sources with correlation in pairs, the number of elements should only satisfy M < 2p Therefore, the number of elements used should be p < M < 2p instead of using M > 2p as in other methods By applying PM algorithm with respect to to find the incident angle based on the power spectrum ( ) ( ) (3.13) ( ) ( ) Where: is the noise matrix with size of M x (M – p) derived using PM algorithm For ULA array on x axis, the power spectrum ( ̂ ) can be determined using the following equation: (̂ ) (3.18) ̂ (̂ ) (̂ ) Similar to , is the noise matrix with size of M x (M – p) 3.2.3 Simulation and evaluation of results Conditions of simulation: ULA on x and z axis: 17 - Antenna elements are evenly spaced in a straight line - Number of elements per array: - Antenna element type: Isotropic - Distance between antenna elements: λ/2 Radiation sources: - Number of radiation sources: - Signal to noise ratio SNR for both radio sources: -15dB - Arrival of angles (elevation, azimuth): [(12o, 10o), (65o, 65o), (20o, 85o), (75o, 30o), (125o, 90o), (95o, 150o)] - Number of signal samples: L =10 with the proposed solution and L =200 with MUSIC algorithm Noise: The covariance matrix of noise has a Toeplitz symmetric form To evaluate the quality of proposed method, in this thesis, colored noise is assumed with the covariance matrix commensurate with antenna arrays on z axis and x axis as in equations (3.19) and (3.20) This assumption does not void the generality as the noise component in (3.12) and (3.17) has been totally eliminated Nz = Toeplitz([1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6]) (3.19) Nx = Toeplitz([1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2]) (3.20) Figure 3.5: Results of elevation angles for six totally correlated radiation sources with incident angles [(12o, 10o), (65o, 65o), (20o, 85o), (75o, 30o), (125o, 90o), (95o, 150o)] using proposed method for 10 signal samples Figure 3.7: Results of elevation angles for six totally correlated radiation sources with incident angles [(12o, 10o), (65o, 65o), (20o, 85o), (75o, 30o), (125o, 90o), (95o, 150o)] using proposed method for 10 signal samples 18 By observation of results in Figures 3.5 and 3.7, although the proposed method requires only L = 10 samples but it successfully found all six incident angles with almost absolute accuracy and high resolution (0,5o) With very low noise background, this solution also allows applications with SNR less than -15dB 3.3 Proposed method for finding direction using ULA-ULA antenna array with symmetric phase center and uncertain priori information about the number of radiation sources 3.3.1 Modeling and proposal of solution z M θ -M s(t) 1 y O ϕ M d d x -M Figure 3.9: Model of ULA-ULA antenna array with symmetric phase center Figure 3.9 represents an antenna array consisting of symmetric ULA antenna array of N = 2M + elements symmetrically arranged across x axis and z axis respectively Step 1: Finding elevation angle θi based on ULA antenna array located on z axis [92] The signal power spectrum has the following form: ( ) (3.42) ( )( ) ( )( ) ∑ ( { ( ) ( )}) From equation (3.42), we can determine the elevation angles ̂ commensurate with the peaks of signal spectrum ( ) Step 2: Finding azimuth angle ϕ based on ULA antenna array located on x axis with the angles ̂ obtained in step (̂ ) (3.44) ( )( ) ̂ ̂ ( )( ) ∑ ( { ( ) ( )}) The noise covariance matrix at element (k, l) can be determined as follows: | | ( ) ( ) (3.45) 19 Where: is the level of noise power, is the correlation coefficient of noise Large value relates to the large correlation and vice versa for the case of white noise 3.3.2 Simulation and evaluation of results Conditions of simulation: ULA on x and z axis: - Antenna elements are evenly spaced in a straight line, symmetric through the coordinate axis - Number of elements per array: 15 - Antenna element type: Isotropic - Distance between antenna elements: λ/2 Radiation sources: - Number of radiation sources: - Signal to noise ratio SNR for both radio sources: 10dB - Arrival of angles (elevation, azimuth): [(17o, -43o), (45o, 25o), (5o, 60o)] - Number of signal samples: L =1000 Noise: White noise and colored noise with correlation coefficient of In the first simulation, the thesis evaluates the accuracy of the proposed solution with the assumption that there are three radiation sources arriving at the antenna array with corresponding angles of (17 o, 43o), (45o, 25o) and (5o, 60o) The simulation is performed with 1000 Monte Carlo attempts and the selected SNR value of 10dB for all three radiation sources Figures 3.11(a) and 3.12(a) show results in the case of radiation sources affected by white noise Meanwhile, Figures 3.11 (b) and 3.12 (b) show the results when affected by the correlated colored noise with the correlation coefficient It is noticed that in Figures 3.11 and 3.12, three observable signal power peaks are clearly visible around the corners [17o, 45o, 5o] and [-43o, 25o, 60o] The difference in the cases of colored noise and white noise is negligible 20 (a) Affected by white noise (b) Affected by the correlated colored noise Figure 3.11: Results of finding elevation angles for three radiation sources with respective incident angles [(17o, -43o), (45o, 25o), (5o, 60o)], whereas the second and third sources are fully correlated (a) Affected by white noise (b) Affected by the correlated colored noise Figure 3.12: Results of finding azimuth for three radiation sources with respective incident angles [(17o, -43o), (45o, 25o), (5o, 60o)], whereas the second and third sources are fully correlated The resolution of the proposed solution under the condition of correlated colored noise is shown in Table 3.1 Table 3.1 Direction resolution obtained by proposed solution for ULA-UCA antenna array under correlated color noise SNR Resolution (deg) Real angle (deg) Obtained angle (deg) Δθ θ1 θ2 θ1 ’ θ2 ’ 0dB 14 17 31 17,81 31,15 5dB 17 26 17,03 25,58 10dB 17 25 16,71 24,65 Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ 0dB 11 25 36 25,81 35,16 21 5dB 25 34 25,37 33,47 10dB 25 32 25,17 31,59 Under this simulated condition, the proposed solution is capable of distinguishing two adjacent angles of incidence (8o, 7o) at SNR of 10dB Finally, the accuracy of the proposed solution is assessed through the dependence of RMSE versus SNR and the number of signal samples used after 1000 independent tests Figures 3.15 and 3.16 represent RMSE values inversely proportional to SNR in the case of finding elevation angle and azimuth angle respectively The number of signal samples used is 1000 At SNR = 0dB, the results of elevation angles of the radiation sources (17 o, -43o), (45o, 25o) and (5o, 60o) have large errors respectively at (0,69o; 0,63o), (0,36o; 0,23o) and (0.28o, 0,31o) When the SNR increases to 10dB, RSME has a very small value (less than 0,1o) for all three radiation sources Figure 3.15: The dependence of the RMSE of elevation angle according to SNR of three radiation sources with the corresponding incident angles [(17o, -43o), (45o, 25o), (5o, 60o)], in which the second and third radiation sources are fully correlated in the condition of correlated color noise Figure 3.16: The dependence of the RMSE of azimuth angle according to SNR of three radiation sources with the corresponding incident angles [(17o, -43o), (45o, 25o), (5o, 60o)], in which the second and third radiation sources are fully correlated in the condition of correlated color noise 22 Figure 3.17: The dependence of the RMSE of elevation angle according to SNR of three radiation sources with the corresponding incident angles [(17o, -43o), (45o, 25o), (5o, 60o)], in which the second and third radiation sources are fully correlated in the condition of correlated color noise Figure 3.18: The dependence of the RMSE of elevation angle according to SNR of three radiation sources with the corresponding incident angles [(17o, -43o), (45o, 25o), (5o, 60o)], in which the second and third radiation sources are fully correlated in the condition of correlated color noise Figures 3.17 and 3.18 represent RMSE dependency on number of samples used in the case of finding elevation angle and azimuth angle respectively The SNR values for all three radiation sources are selected by 10dB It is noticed that although only 100 signals were used, the largest RMSE was only 0,23o for the elevation angle and 0,15o for the azimuth When increasing the number of signal samples to 700, the RMSE for all three radiation sources is unchanged (the same value at zero) 3.4 Proposal of positioning model based on direction finding result and priori topographical information Figure 3.28 shows a general model of the principle of positioning using 3D terrain The combined information of receiver coordinates, the azimuth of incoming signals and 3D terrain model will allow the reflection of objects to be determined The the information related to reflective objects can be extracted such as: Coordinates the reflection point and azimuth of the reflected surface normal This process can be 23 done automatically using Ray-Tracer algorithm [89] or manually manipulated on Google Earth software 3.5 Conclusion for Chapter Chapter has presented two 2D direction finding solutions of correlated radiation sources in the conditions of color noise The simulation results show that this proposed solution works well in the assumptions as set out herein In the last part of Chapter 3, the thesis proposes a positioning model using direction finding results and priori terrain information in the context of only NLOS signals CONCLUSION Research results The thesis has studied a number of solutions to solve two problems: Problems about accuracy, resolution and computational complexity: The thesis has studied the use of improved PM algorithm applied to ULA-UCA and ULA antenna structure The results show that this combination brings following advantages: Low computational complexity, high accuracy even at small SNRs The minimum number of elements required on each antenna array is at least M = 2p + With the ULA-UCA structure with M = 10, the solution using the improved PM algorithm is able to successfully identify two incoming directions with only 50 signal samples and SNR = -10dB The simulation process has proven that this solution improves accuracy and resolution compared to the use of traditional PM and MUSIC algorithms (especially in case of nonlinear noise) For the ULA antenna array, the re-construction of signal vector of Toeplitz format before the application of improved PM algorithm allows to find directions of correlated radiation sources with only one signal pattern Compared to ESPRIT, Matrix Pencil and TLS algorithms, this solution has better accuracy The effect of nonlinear noise, color noise and uncertainty of the information about the number of radio sources: The thesis has studied the influence of color noise and built signal models based on orthogonal L-shape antenna and ULA-ULA arrays with symmetric phase center For the L-shape antenna array, the the covariance matrix is rebuilt based on the symmetry of Toeplitz color noise before using the PM algorithm This solution is able to find directions of correlated radiation sources in pairs at very small SNR (-15dB) and only need to use a few signal samples (10 samples) For smaller SNR, this solution still has high accuracy and resolution because the noise component has been 24 completely removed from the signal power spectrum Meanwhile, the use of orthogonal ULA-ULA antenna array with symmetric phase center in combination with signal models based on quaternary invariant matrices allows to find directions of correlated radiation sources affected by correlated color noise Compared to FBSS method, this solution does not need to know the priori information on the number of radiation sources Besides, the thesis has studied and proposed a positioning solution based on the results of direction finding and topographic information The analysis and simulation results show that this solution can be applied in practice with further research and development New contributions of thesis  2D direction finding solution has been proposed for uncorrelated radiation sources using ULA-UCA antenna array together with fast 1D direction finding solution for correlated radiation sources using ULA antenna array and improved PM algorithm  A 2D direction finding solution has been proposed for correlated radiation sources using L-shape antenna array under the condition of symmetric Toeplitz colored noise and orthogonal ULA-ULA antenna array with symmetric phase center in the context of uncertain priori information about the number of radiation sources Further development of thesis Study of direction finding solutions taking into account the influence of the movement speed of radiation sources in space, And study of 2D direction finding solutions of broadband radiation sources in the context of color noise SICENTIFIC PUBLICATIONS Nguyen Tuan Minh, Le Thanh Hai, Nguyen Trong Luu, Tran Cong Thin (2018), “Method of direction–of–arrival estimator for signal sources not on the same reference reflective surface based on several antenna arrays”, Journal of Military Science and Technology, code: ISSN: 1859-1043, No FEE, pp 144 – 153 Nguyen Tuan Minh, Le Thanh Hai, Nguyen Trong Luu (2018), “A method of derection–of-arrival estimator for coherent radio-frequency sources with elevation angle and azimuth angle when unknown number of arrival sources”, Journal of Military Science and Technology, code: ISSN: 1859-1043, No 57, pp 26 – 36 Nguyen Tuan Minh, Le Thanh Hai, Nguyen Trong Luu (2019), “A fast method of directing the arrival estimation using ULA antenna array”, Journal of Military Science and Technology, code: ISSN: 18591043, No 59, pp 48-57 Nguyen Tuan Minh, Le Thanh Hai, Nguyen Trong Luu (2019), “Proposal of a method 2D-DOA for correlated sources using L shape antenna array”, Journal of Science and Technology – Ha Noi University Of Industry, code: ISSN: 1859-3585, No.50, pp 41-44 Nguyen Tuan Minh, Le Thanh Hai, Nguyen Trong Luu (2019), “The positioning method based on the derection of arrival mesured and apriori topographic information”, Journal of Science and Technology – Ha Noi University Of Industry, code: ISSN: 1859-3585, No 51, pp 9-12 ... ) is an unit matrix with size of M – 2p Let set [ written as follows: ( ) ], then the equation (2.9) can be re- (2.10) When p signals are attached to the direction of incident wave θi then: (... of correlated radiation sources with only one signal pattern Compared to ESPRIT, Matrix Pencil and TLS algorithms, this solution has better accuracy The effect of nonlinear noise, color noise... improvement of radio direction finding of radiation sources also has issues that need further attention and study This is a research direction of scientific, practical and urgent significance