Tóm tắt Luận án Research on a solution for improvement of radio direction finding

unit matrix with size of M – 2p. Let set 4 [ ( )], then the equation (2.9) can be re- written as follows: (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 5 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: 1. UCA: - Antenna elements are evenly spaced in a circle. - Number of elements per array: 10. - Antenna element type: Isotropic. 6 - Distance between antenna elements: λ/2. 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. 3. Radiation sources: - Number of radiation sources: 2. - Signal to noise ratio SNR for both radio sources: -5dB. - Arrival of angles (elevation, azimuth): [(25 o , 70 o ), (80 o , 310 o )] and [(25 o , 70 o ), (25 o , 310 o )]. - Number of signal samples: L =1000. 4. 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 (25 o , 70 o ) and (80 o , 310 o ) in Gaussian white noise condition Figure 2.5: Results of direction finding for azimuth angles of two radio sources (25 o , 70 o ) and (80 o , 310 o ) 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 7 azimuth under Gaussian white noise. In terms of accuracy, the proposed solution has small direction errors in accordance with two incident waves which are (0,02 o ; 0,04 o ) and (0,01 o ; 0,01 o ). 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 (0,07 o ; 0,06 o ) and (0,16 o ; 1,01 o ). Figure 2.6: Results of direction finding for elevation angles of two radio sources (25 o , 70 o ) and (80 o , 310 o ) in nonlinear noise condition Figure 2.7: Results of direction finding for azimuth angles of two radio sources (25 o , 70 o ) and (80 o , 310 o ) 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 8 5dB 8 25 33 25,3 32,91 10dB 6 25 31 25,06 30,97 Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ -5dB 10 70 80 69,88 79,98 0dB 5 70 75 70,04 75,97 5dB 3 70 73 70 73 10dB 2 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 7 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 2 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 (6 o , 2 o ) and (7 o , 2 o ) 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 (25 o , 70 o ) and (80 o , 310 o ) according to the number of signal samples 9 Figure 2.11: The dependence of RMSE on azimuth angle of two radio sources (25 o , 70 o ) and (80 o , 310 o ) according to the number of signal samples Figures 2.12 and 2.13 represent the RMSE direction errors corresponding to the angle of elevation and azimuth according to SNR. Figure 2.12: The dependence of RMSE on elevation angle of two radio sources (25 o , 70 o ) and (80 o , 310 o ) versus SNR Figure 2.13: The dependence of RMSE on azimuth angle of two radio sources (25 o , 70 o ) and (80 o , 310 o ) versus SNR In terms of computational complexity, the MUSIC algorithm contains 2[M 2 L + O(M 3 )] multiplications. Where 2M 2 L multiplications are used to compute the covariance matrix and 2O(M 3 ) multiplications are used to derive the eigenvalues [9]. Meanwhile, the proposed method only requires 2[2p(M - p)L + 2O(4p 3 )] multiplications. 2.4. Proposed 1D direction finding solution for uncorrelated radiation sources using ULA antenna array 10 2.4.1. Modeling and proposal of solution Let consider p radio sources s(t) of the same wavelength λ: ( ) [ ( ) ( ) ( )] with relative incident angles ϕi (i = 1, , p). In case of one signal sample, the signal vector x and y at time t are represented as follows: ( ) [ ( ) ( ) ( ) ] ( ) ( ) ( ) (2.31) ( ) [ ( ) ( ) ( ) ] ( ) ( ) ( ) (2.32) Where: ( ) ( ) ( ) ( ) (2.33) ( ) [ ( ) ( ) ( ) ( ) ( )( ) ( )] (2.34) ( ) ( ) ( ) ( ) (2.35) ( ) ( ) ( ) ( ) (2.36) ( ) [ ( ) ( ) ( ) ( ) ( )( ) ( )] (2.37) ( ) ( ) ( ) ( ) (2.38) Where: ( ), ( ) is the direction matrix of size (M + 1) x p of p radio sources; ( ), ( ) of size (M + 1) x 1 is a white Gaussian noise vector; ( ), ( ) is the direction vector of radio source i (i = 1, , p). The signal vector ̃( ) can be constructed from vector y(t) (by eliminating the last row) and the signal vector x(t): ̃( ) [ ( ) ( ) ] [ ( ) ( ) ( ) ( ) ( ) ] (2.39) It can be seen that the signal vector ̃( ) has the size (2M + 1) x 1 so the associated Toepliz matrix ̃ ( ) of the size (M +1) x (M + 1) can be constructed as the following: 11 ̃ ( ) [ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ] (2.40) The covariance matrix can be obtained as follows: ̃ ( ) ̃ ( ) ( ) ( ) (2.51) Where: ̃( ) ̃( ) is the covariance matrix of the radio 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: 1. ULA: - Antenna elements are evenly spaced in a straight line. - Number of elements per array: 8. - Antenna element type: Isotropic. - Distance between antenna elements: λ/2. 2. Radiation sources: - Number of radiation sources: 3. - Signal to noise ratio SNR for both radio sources: 0dB. - Azimuth angles: (100 o , 120 o , 140 o ) in case of uncorrelated radio sources and (60 o , 75 o , 95 o ) in case of correlated radio sources. - Number of signal samples: L = 1 with the proposed solution, L = 5 with TLS and ESPRIT algorithm in case of uncorrelated radio sources, L = 1 with TLS and Matrix Pencil algorithm in case of correlated radio sources 3. 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 100 o , 120 o and 140 o . In terms of accuracy, this solution has very small direction errors relatively at 0,04 o ; 0,01 o and 0,07 o . Figure 2.18: Direction results for three uncorrelated radiation sources with incident angles [100 o , 120 o , 140 o ] Figure 2.19 shows the simulation results when three totally correlated radio sources arriving at the antenna array in the directions of 60 o , 75 o and 95 o . Figure 2.19: Direction results for three totally correlated radiation sources with incident angles [60 o , 75 o , 95 o ] Similar to the case of uncorrelated radiation sources, the proposed method has successfully found all three incident waves with direction errors 0,16 o ; 0,2 o and 0,04 o 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 Resolution (deg) Real angle (deg) Obtained angle (deg) Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ -5dB 9 100 109 99,54 109,29 0dB 8 100 108 99,84 108,18 13 5dB 6 100 106 102,02 105,83 10dB 5 100 105 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 8 65 73 65,1 72,83 5dB 7 65 72 65,06 71,91 10dB 5 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 9 o for uncorrelated radiation sources and 10 o for correlated radiation sources. Under this simulation conditions, at SNR equal to 10 dB, the resolution of proposed method is derived at 5 o . 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 11- element 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 55 o and 70 o . Figure 2.28 and 2.29 shows the simulation results of RMSE versus SNR for three totally correlated radio sources with incident angles of 45 o , 60 o and 75 o . Figure 2.26: The dependence of RMSE on SNR for L = 1 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 = 1 signal sample of proposed method Figure 2.29: The dependence of RMSE on SNR for L = 1 signal sample of Matrix Pencil and TLS algorithms [76] 2.5. Conclusion for Chapter Chapter 2 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 1 on the diagonal and 0 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) As Nz is a symmetric Toeplitz matrix hence Nz T , Nz H and JNzJ are symmetric Toeplitz matrices and JNz T J = (JNzJ) T = Nz hence the following is obtained: Nz = JNz T J = 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: 1. ULA on x and z axis: 17 - Antenna elements are evenly spaced in a straight line. - Number of elements per array: 9. - Antenna element type: Isotropic. - Distance between antenna elements: λ/2. 2. Radiation sources: - Number of radiation sources: 6. - Signal to noise ratio SNR for both radio sources: -15dB. - Arrival of angles (elevation, azimuth): [(12 o , 10 o ), (65 o , 65 o ), (20 o , 85 o ), (75 o , 30 o ), (125 o , 90 o ), (95 o , 150 o )]. - Number of signal samples: L =10 with the proposed solution and L =200 with MUSIC algorithm. 3. 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 [(12 o , 10 o ), (65 o , 65 o ), (20 o , 85 o ), (75 o , 30 o ), (125 o , 90 o ), (95 o , 150 o )] using proposed method for 10 signal samples. Figure 3.7: Results of elevation angles for six totally correlated radiation sources with incident angles [(12 o , 10 o ), (65 o , 65 o ), (20 o , 85 o ), (75 o , 30 o ), (125 o , 90 o ), (95 o , 150 o )] 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 (0,5 o ). 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 y z x ... θ ϕ M M -M 1 1 1 s(t) O dd ... -M Figure 3.9: Model of ULA-ULA antenna array with symmetric phase center Figure 3.9 represents an antenna array consisting of 2 symmetric ULA antenna array of N = 2M + 1 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 1. ( ̂ ) ( )( ) ∑ ( { ( ̂ ) ( ̂ )}) ( )( ) (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: 1. 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. 2. Radiation sources: - Number of radiation sources: 3. - Signal to noise ratio SNR for both radio sources: 10dB. - Arrival of angles (elevation, azimuth): [(17 o , -43 o ), (45 o , 25 o ), (5 o , 60 o )]. - Number of signal samples: L =1000. 3. 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 , - 43 o ), (45 o , 25 o ) and (5 o , 60 o ). 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 [17 o , 45 o , 5 o ] and [-43 o , 25 o , 60 o ]. 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 [(17 o , -43 o ), (45 o , 25 o ), (5 o , 60 o )], 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 [(17 o , -43 o ), (45 o , 25 o ), (5 o , 60 o )], 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 9 17 26 17,03 25,58 10dB 8 17 25 16,71 24,65 Δϕ ϕ1 ϕ2 ϕ1’ ϕ2’ 0dB 11 25 36 25,81 35,16 21 5dB 9 25 34 25,37 33,47 10dB 7 25 32 25,17 31,59 Under this simulated condition, the proposed solution is capable of distinguishing two adjacent angles of incidence (8 o , 7 o ) 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 , -43 o ), (45 o , 25 o ) and (5 o , 60 o ) have large errors respectively at (0,69 o ; 0,63 o ), (0,36 o ; 0,23 o ) and (0.28 o , 0,31 o ). When the SNR increases to 10dB, RSME has a very small value (less than 0,1 o ) 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 [(17 o , -43 o ), (45 o , 25 o ), (5 o , 60 o )], 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