An Approximate Maximum Likelihood Algorithm for Target Localization in Multistatic Passive Radar

Transcription

1 Chinese Journal of Electronics Vol.28, No.1, Jan An Approximate Maximum Likelihood Algorithm for Target Localization in Multistatic Passive Radar WANG Jun, QIN Zhaotao, GAO Fei and WEI Shaoming (School of Electronics and Information Engineering, Beihang University, Beijing , China) Abstract This paper addresses the problem of target localization using Bistatic range (BR) measurements in a distributed multistatic passive radar system. The rangebased positioning technique employs multiple transmitterreceiver pairs, which provide separate BR measurements. Based on the Maximum likelihood (ML) function, an efficient algebraic Approximate maximum likelihood (AML) algorithm for single target localization is proposed. The closed-form AML solution has neither initial condition requirements nor convergence difficulty. Simulations are included to compare its performance to that of the Cramer- Rao lower bound (CRLB) and the Two-step Weighted least squares (TS-WLS) algorithm. The proposed method is shown to be able to achieve the CRLB accuracy under Gaussian measurement noise. It is more robust to noise than the TS-WLS method, and presents relative insensitivity to target-sensor geometry. Key words Bistatic range (BR) measurement, Multistatic passive radar, Target localization, Approximate maximum likelihood (AML) algorithm, Maximum likelihood (ML) function. I. Introduction Target localization is an important problem in multistatic passive radar systems [1,2], and has drawn considerable attentions in recent years [3,4]. It has been utilized in many applications such as surveillance and tracking. Multistatic radar system consists of multiple transmitter-receiver pairs, each of which can provide separate positioning parameters for target localization. These positioning parameters can be classified into different types, mainly including Bistatic range (BR) [4 10], Doppler shift [11,12], and angle of arrival [13,14]. In this study, we focus on the BR-based methods, which are the most common owing to their high accuracy and simplicity. The locus of points of the constant BR measurement is an ellipsoid on which the target lies, with the associated transmitter and receiver as its foci. The intersection of the ellipsoids from multiple transmitter-receiver pairs yields the estimated target position. The localization problem is potentially challenging due to the nonlinear relationship between the desired target position and measured parameters. A lot of elliptic localization methods have been developed. Generally speaking, target localization can be implemented either using direct or indirect form. In direct methods, such as in Ref.[15], the target position is directly estimated from the received signals. Although the mentioned method is asymptotically optimum, it is computationally expensive due to its multi-dimensional grid search. In indirect methods, e.g., Refs.[4 10], the target location is obtained by two steps: calculating measurements and then estimating the target position. The localization technique employed in Ref.[4] is based on the Gauss-Newton iteration method, and requires a proper initial guess, which is often infeasible in practical situations. Various algebraic closed-form methods have been proposed in order to warrant global convergence and reduce complexity. In Ref.[5], two simple solutions for elliptic localization were developed, namely the Spherical-interpolation (SI) and Spherical-intersection (SX) algorithms, but they generally cannot reach the Cramer-Rao lower bound (CRLB) accuracy. Ref.[6] introduced a closed-form solution by converting the elliptic equations to hyperbolic ones, from which the target position was obtained using the least squares approach. In Ref.[7], the target-locating problem was studied using a Weighted least squares (WLS) estimator under conditions of different noise levels. Recently, based on the introduction of nuisance parameters, a Two-step Weighted least squares (TS- WLS) algorithm was proposed in Refs.[8 10], which Manuscript Received Sept. 12, 2017; Accepted Dec. 4, This work is supported by the National Natural Science Foundation of China (No , No , No ), the Foundation of ATR Key Laboratory (No ) Chinese Institute of Electronics. DOI: /cje

2 196 Chinese Journal of Electronics 2019 approximates the Maximum likelihood (ML) estimator when BR measurement noise is small. Nevertheless, the accuracy of the obtained location estimates using the TS- WLS method may be undesirable when the noise is large. Furthermore, the TS-WLS method suffers from geometry sensitivity, and cannot provide efficient performance for some geometries, as shown by simulations in Section IV. In this paper, a novel BR-based method for locating a single target using multistatic passive radars is developed. The proposed estimator is motivated by the previous Refs.[16 18] that are employed in the radiating source localization systems [19 26]. Starting from the ML function, it is first changed into a pseudo-linear equation with respect to the unknown target position parameters, whose coefficients are also dependent on the unknowns. Subsequently, with an initial value of target position, the Approximate maximum likelihood (AML) algorithm solves the pseudo-linear equation for the new target position and updates their coefficients. After several updates, the AML checks the ML cost function with the target position from each update, and chooses the minimum as the solution. Note that the AML solver in this paper uses a weighted least squares solution as the initial guess of the source position. Moreover, the AML takes the solution from the simplified linear equation, which ignores the weighting matrix, as an alternative target position besides the aforementioned several updates. According to simulation results, the proposed method is shown to be an approximately efficient estimator, whose localization accuracy is able to reach the CRLB even at a sufficiently high Gaussian noise level before the threshold effect occurs. Additionally, in contrast to the existing TS-WLS algorithm, the developed estimator not only performs better in terms of noise robustness, but also presents a lower sensitivity to targetsensor geometry, which make it useful in many practical field environments [18]. The rest of this paper is organized as follows. Section II formulates the BR-based localization problem. The closed-form algorithm is developed in Section III. In Section IV, simulations are included to evaluate the performance of the proposed method. Finally, conclusions are drawn in Section V. II. Target Localization Problem Here, we shall consider the scenario of one transmitter and multiple receivers, which has been widely used in UWB network localization [27,28] and radio communications [29]. This paper will focus on finding an individual target position in Three-Dimensional (3-D) space using a multistatic passive radar system shown in Fig.1. The positions of the transmitter and receivers are known exactly. It is assumed that receivers are present, whose positions are denoted by s i = [x s i, ys i, zs i ]T, i = 1, 2,, M, where superscript T stands for the transpose. The transmitter is placed at location t = [x t, y t, z t ] T. The unknown position of the single target is denoted by u = [x, y, z] T. Fig. 1. Target localization scenario in multistatic passive radar The signal radiated from the transmitter is detected [30,31] and observed by the receivers from direct propagation and from indirect reflection of the target. For each transmitter-receiver pair, signals from the two paths provide a differential delay measurement. The time delay from the transmitter to the corresponding receiver is then obtained by adding the measured differential delay and the known direct propagation delay. We aim to estimate the target position from the multiple time delays, each of which is sum of transmitter-to-target and target-toreceiver delays. The distance between the target and the transmitter is r t = u t (1) where represents the l 2 norm. Similarly, the distance between the target and the ith receiver is r s i = u s i, i = 1, 2,, M (2) The true time delay between the transmitter and the ith receiver multiplied by the wave propagation speed is known as the elliptic bistatic range, which is denoted by R i = r t + r s i (3) The vector of the true BRs corresponding to the M receivers can be written as R = [R 1, R 2,, R M ] T. The measured version of the time delay formed by the transmitter and the ith receiver is expressed as τ i. Collecting all the observed time delays gives τ = [τ 1, τ 2,, τ M ] T. It is straightforward to obtain the observed version of BR measurements as follows: = c τ = [ 1, 2,, M ] T (4)

3 An Approximate Maximum Likelihood Algorithm for Target Localization in Multistatic Passive Radar 197 where c is the constant signal propagation speed, and i = c τ i is the BR measurement from transmitter to the ith receiver. The measurement noise vector of is assumed to be Gaussian with zero mean and the covariance matrix Q. The problem of target localization is estimating u from the noisy BR measurement vector as accurately as possible, while maintaining a manageable complexity. It is difficult to solve directly, as is nonlinear and nonconvex with respect to the unknown target position u. In the next section, we will develop an efficient closed-form algorithm for this problem, which can approximate an ML estimator. III. AML for BR Based Localization The Probability density function (PDF) [16,21,32] of conditioned on u is given by { f ( /u) = (2π) M 2 (det Q) 1 2 exp J } (5) 2 where operator det denotes the determinant of matrix, and J = [ R (u)] T Q 1 [ R (u)] (6) where R (u) represents the BR vector function of u. The ML estimate is the u that minimizes the cost function J. Setting the gradient of J with respect to u equal to zero, the ML equation can be given as follows: J u = 2 R(u)T u Q 1 [ R (u)] = 0 (7) where R (u)/ u is the partial derivative of BR vector function with respect to u, and is a M 3 matrix. Its ith row can be denoted by R i (u)/ u = ρ T u,t + ρ T u,s i, where ρ a,b = (a b)/ a b is a unit vector from b to a. Via simple manipulations, Eq.(7) can be written as where W ã = 0 (8) W = R(u)T u Q 1 (9) ã = R (u) = [ ] T R 1 (u) 1 R M (u) M (10) The entries of vector [ R 1 (u) 1 R M (u) M ] T in Eq.(10) can be transformed into R i (u) i = r s i + r t i = rs i 2 (r t i ) 2 r s i rt + i (11) Now, substituting Eq.(11) into Eq.(10) changes it to ã = Λa (12) where ( ) 1 1 Λ = diag r1 s rt + 1 rm s rt + M (13) [ ] a = r s 1 2 (r t 1 ) 2 r s M 2 (r t M ) 2 T Let Φ = W Λ; thus, Eq.(8) becomes Expanding the elements in a as (14) Φa = 0 (15) r s2 i (r t i ) 2 =2(x t x s i )x + 2(y t yi s )y + 2(z t z s i )z 2 i k t + k s i + 2r t i (16) and substituting them into Eq.(14) results in the matrix equation a = 2Du v (17) where and x t x s 1 y t y1 s z t z s 1 D =... (18) x t x s M yt ym s zt zm s v = v 1 2r t (19) v 1 = [ k t k s 1 2 M + k t k s M] T (20) ki s = x s2 i + y s2 i + z s 2 i (21) k t = x t2 + y t2 + z t2 (22) Then, the exact ML equation can be obtained by inserting Eq.(17) into Eq.(15) 2ΦDu = Φ(v 1 2r t ) (23) Although this is a linear equation in u, the weighting matrix Φ contains elements of the unknown u, and an AML approach is necessary. The first step is to obtain the initial u to update the values of Φ. Thus, substituting Φ into Eq.(23) produces the updated u in terms of r t as u = 1 2 (ΦD) 1 Φ(v 1 2r t ) (24) Putting Eq.(24) into Eq.(1), a quadratic equation in r t is obtained. Next, the Root selection routine (RSR) is used to choose the best root as follows. If only one root is positive, it is the correct value. If both roots are positive, it will select the one that gives a smaller J in Eq.(6). If both roots are negative or imaginary, it will take the absolute values of the real parts. Subsequently,

4 198 Chinese Journal of Electronics 2019 substituting the selective r t into Eq.(24) yields the target position u. AML uses the updated u to calculate Φ again, and the same steps for calculating and selecting r t are repeated to obtain new values of u. Iterating the above procedure q times produces q values of J. The AML solver selects the solution that gives the minimum J. In the simulation, q = 5. How to obtain the initial value u is an issue that has not been covered thus far. Ignoring the weighting matrix Φ, i.e., setting Φ as an identity matrix, simplifies Eq.(23) to 2Du = v 1 2r t (25) The weighted least square solution to this linear equation is given by u = 1 2 (DT Q 1 D) 1 D T Q 1 (v 1 2r t ) (26) This solution determines u in terms of r t, and then following the procedure after Eq.(24), produces the initial value. This is a common method in Refs.[16 18], which is actually the SX algorithm [5,20] whose solution is the same as Eq.(26). However, SX algorithm is quite sensitive to target-sensor geometry, causing the initial value to deviate from the true solution in some poor geometries. Under these circumstances, AML may fail to converge to an accurate target position. In the case of our study, another weighted least square solution inspired by Ref.[21] is utilized to provide a better initial value. θ = ( G T Q 1 G ) 1 G T Q 1 v 1 (27) where θ = [ u T, r t] T, G = 2 [D, ]. It then takes u from θ as the initial value of the AML solver directly, without solving the quadratic equation. The new matrix G is composed of the sub-matrix and vector which have already occurred above. Therefore, this method provides the initial u without greatly increasing the computational complexity. There is an additional case added to values of J before they are compared to generate the final solution. Although the SX method no longer provides an initial value in our study, it is still meaningful to operate for a special J when the weighting matrix Φ is ignored in the exact ML equation. The purpose of this operation is to improve the noise robustness and reasonability of the proposed method. As a result, the AML solver selects the final solution that gives the minimum J from the q + 1 values of J. IV. Simulation Results This section presents a series of simulations performed to assess the performance of the proposed method and validate the theoretical developments. We first examine the effect of increasing the number of receivers on the performance of the AML method. Second, we draw a localization performance comparison between the proposed method and other estimators. Finally, we conduct an analysis of the sensitivity to target-sensor geometry between the AML and TS-WLS methods. This section presents a series of simulations performed to assess the performance of the proposed method and validate the theoretical developments. We first examine the effect of increasing the number of receivers on the performance of the AML method. Second, we draw a localization performance comparison between the proposed method and other estimators. Finally, we conduct an analysis of the sensitivity to target-sensor geometry between the AML and TS-WLS methods. A total of 10,000 geometries are created randomly, where the target, transmitter, and receivers are within a cube with an edge length of 500m. Their Cartesian coordinates are independently uniformly distributed. The BR measurement estimates are generated by adding the zero mean Gaussian noise to the true values. The measurement covariance matrix is δd 2I M, where I M is the M M identity matrix and δd 2 is the variance of the measurement noise. The localization accuracy is evaluated according to the Root mean square error (RMSE) of the target location estimates. The number of ensemble runs is 5,000 for each of the random geometries. 1. Experiment 1 This experiment examines the effect of increasing the number of receivers on the performance of the proposed method. The simulations have been carried out with M= 3, 4, 5, 6, 7, 8 receivers to locate a 3D target. Fig.2 shows the average performance of the six positioning cases over 100 geometries selected randomly from the 10,000 geometries as δd 2 increases. The target fails to be located with substantial position errors for the case of three receivers. When there are more than three receivers, better position accuracy will be achieved with increasing number of receivers. Interestingly, the location performance is improved significantly from four receivers Fig. 2. Effect of increasing the number of receivers on localization performance

5 An Approximate Maximum Likelihood Algorithm for Target Localization in Multistatic Passive Radar 199 to five receivers, whereas the improvement gradually becomes small from five receivers to more receivers. The localization performance will be improved as the number of receivers increases. However, the passive location system will become more complicated as well. In order to achieve a compromise between positioning accuracy and system complexity, the number of receivers should be reasonably determined in the practical environment. In this paper, an array of five receivers will be employed in the subsequent simulation experiments. 2. Experiment 2 This experiment compares the proposed AML algorithm and the SX [5] and TS-WLS [8] methods, with the CRLB as a benchmark. The localization geometry is shown in Fig.3, where the target and the transmitter are fixed at (440, 170, 180)m and (340, 340, 330)m, respectively. The positions of the five receivers are tabulated in Table 1. Table 1. Positions of receivers Receiver s1 s2 s3 s4 s5 x(m) y(m) z(m) The threshold effect of the proposed method occurs at a noise power that is about 10dB later than that of the TS-WLS method. Another study is carried out, in which the localization results are averaged over 100 geometries chosen randomly from the 10,000 geometries. The geometries for all three positioning approaches are same. As can be seen from Fig.5, the AML method has the best performance whereas the SX method has the worst. It is worth noting that the SX method is more sensitive to geometry than the other two methods. This is why we do not use the SX method to provide the initial value for the proposed method. The TS-WLS method behaves close to AML when the noise level is less than about 6dB. However, the AML solver performs slightly better than the TS-WLS method in relatively high noise levels. Fig. 3. Localization geometry The accuracy of position estimates of the three methods is reported in Fig.4 by varying the measurement noise variance. The performance of the SX method is suboptimal, clearly not attaining CRLB accuracy. The TS- WLS method departs from the CRLB precipitously at a noise power of about 10dB, while the AML estimator begins to deviate from CRLB at a noise power of about 0dB. Fig. 4. The performance of the AML method with the SX and TS-WLS methods over a certain geometry vs. measurement error Fig. 5. The average performance of the AML method with the SX and TS-WLS methods over 100 geometries vs. measurement error The above two simulations in Experiment 2 demonstrate that the proposed method has better localization performance than the SX and TS-WLS methods, especially in terms of noise robustness. This appealing feature benefits the cases with large BR measurement errors. 3. Experiment 3 This experiment conducts an analysis of sensitivity to target-sensor geometry between the proposed method and the TS-WLS method. At the noise level of 0dB, the localization performance of the AML together with the TS-WLS method and the corresponding CRLB in Fig.6 are illustrated separately for 50 geometries obtained by sampling at regular intervals among the 10,000 geometries. It appears that the AML method is always better than the TS-WLS method. The positioning accuracy of the AML solver can reach CRLB accuracy over almost every geometry except for index 24. However, the TS-WLS technique exceeds CRLB occasionally with considerable margin for seven unfavorable geometries that have large localization errors, e.g., at geometry indices 8 and 22. According to statistical analysis, the mean value of excess values calculated from AML positioning accuracy subtracting CRLB accuracy for all geometries

6 200 Chinese Journal of Electronics 2019 is dB, whereas that of the TS-WLS method is dB. Furthermore, the maximum of the excess values for the TS-WLS method is dB, which is much larger than that of AML, 1.787dB. Fig. 6. The performance of the AML and TS-WLS methods in a sample of 50 selected geometries when noise level is 0dB V. Conclusions In this paper, an AML method for determining the position of a target based on BR measurements in multistatic passive radar is developed. The derivation of the AML algorithm is based on the ML function. The obtained method is closed-form without requiring an initial guess. The number of the receivers has an effect on the performance of the proposed method. The more receivers we use, the better position accuracy will be achieved. The proposed estimator is shown to be able to reach CRLB accuracy under Gaussian measurement noise. It exhibits better robustness to increased noise levels over the SX and TS-WLS methods. This feature is appealing for both moderate and high BR measurement errors in practice. Moreover, the AML solver is relatively insensitive to geometry, and is superior to the TS-WLS estimator. References Fig. 7. The performance of the AML and TS-WLS methods in a sample of 50 selected geometries when noise level is 10dB When the measurement noise level rises to 10dB, the localization performance over another 50 geometries obtained in the same way as those at the 0dB noise level case are given in Fig.7. Compared with Fig.6, the localization accuracy becomes worse overall due to the higher noise level. Though AML is not able to attain CRLB at geometry indices 38 and 48, it still behaves better than TS-WLS at the same indices. The AML solver has only two geometries at which the RMSE of the position estimates go beyond CRLB, whereas the TS-WLS method has twelve such geometries. The mean value of excess values over CRLB for the AML approach is dB, whereas that of the TS-WLS method is dB. The maximum of the excess values of the TS- WLS technique is 9.976dB, and that of the AML method is 3.84dB. These observations and statistical data indicate that the developed method performs significantly better than the TS-WLS method in terms of localization sensitivity to target-sensor geometry. The relative insensitivity of the AML method to physical geometry is an improvement over the popular TS-WLS algorithm. [1] E. Hanle, Survey of bistatic and multistatic radar, Proc. of the Institution of Electrical Engineers, Vol.133, No.7, pp , [2] A. Farina, Fundamentals of multisite radar systems: Multistatic radars and multiradar systems, IEEE Aerospace and Electronic Systems Magazine, Vol.16, No.4, pp.44 44, [3] S. Gogineni, M. Rangaswamy, B.D. Rigling, et al., Cramerrao bounds for UMTS-based passive multistatic radar, IEEE Transactions on Signal Processing, Vol.62, No.1, pp , [4] L.Y. Rui and K.C. Ho, Elliptic localization: Performance study and optimum receiver placement, IEEE Transactions on Signal Processing, Vol.62, No.18, pp , [5] M. Malanowski and K. Kulpa, Two methods for target localization in multistatic passive radar, IEEE Transactions on Aerospace and Electronic Systems, Vol.48, No.1, pp , [6] M. Dianat, M.R. Taban, J. Dianat, et al., Target localization using least squares estimation for MIMO radars with widely separated antennas, IEEE Transactions on Aerospace and Electronic Systems, Vol.49, No.4, pp , [7] A. Noroozi and M.A. Sebt, Weighted least squares target location estimation in multi-transmitter multi-receiver passive radar using bistatic range measurements, IET Radar, Sonar and Navigation, Vol.10, No.6, pp , [8] M. Einemo and H.C. So, Weighted least squares algorithm for target localization in distributed MIMO radar, Signal Processing, Vol.115, pp , [9] R. Amiri, F. Behnia and H. Zamani, Asymptotically efficient target localization from bistatic range measurements in distributed MIMO radars, IEEE Signal Processing Letters, Vol.24, No.3, pp , [10] R. Amiri and F. Behnia, An efficient weighted least squares estimator for elliptic localization in distributed MIMO radars, IEEE Signal Processing Letters, Vol.24, No.6, pp , [11] Y. Liu, L. Yang and K.C. Ho, Moving target localization in multistatic sonar by differential delays and doppler shifts,

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Ho, Efficient closed-form estimators for multistatic sonar localization, IEEE Transactions on Aerospace and Electronic System, Vol.51, No.1, pp , WANG Jun was born in He received the B.S. degree from the Northwestern Polytechnical University, X- ian, China, in 1995 and the M.S. and Ph.D. degrees from Beihang University (BUAA), Beijing, China, in 1998 and 2001, respectively. He is currently a professor with the School of Electronic and Information Engineering, BUAA. His main research interests are signal processing, DSP/FPGA real time architecture, target recognition and tracking. ( wangj203@buaa.edu.cn) QIN Zhaotao was born in He received the B.S. degree in electronic and information engineering from Civil Aviation University of China (CAUC), Tianjin, China, in He is currently working toward the Ph.D. degree in electronic and information engineering from Beihang University (BUAA), Beijing, China. His current research interests include radar signal processing and passive localization. ( qzt2012@buaa.edu.cn) GAO Fei (corresponding author) was born in He received the B.S. and M.S. degree from the Xian Petroleum Institute, Xian, China, in 1996 and 1999, respectively, and the Ph.D. degrees from Beihang University (BUAA), Beijing, China, in He is currently an associate professor with the School of Electronic and Information Engineering, BUAA. He is interested in radar signal processing, moving target detection and image processing. ( feigao2000@163.com) WEI Shaoming was born in He received the B.S. and Ph.D. degree in electronic and information engineering from Beihang University (BUAA), Beijing, China, in 2007 and 2013, respectively. He is currently an experimentalist of the School of Electronic and Information Engineering, BUAA. His research interests include high-resolution radar signal processing, 3-D reconstruction, and multi-target tracking. ( shaoming.wei@buaa.edu.cn)