# Joint Position and Velocity Estimation of a Moving Target in Multistatic Radar by Bistatic Range, TDOA, and Doppler Shifts.

1. Introduction

Single parameter can be used to estimate the position or velocity in some cases, but the accuracy needs to be improved. Several localization algorithms have been researched using two types of measurements, such as BR and Doppler shifts [15], BR and bearing measurements [16], and ellipse and hyperbola [17]. Also, there is an estimation algorithm using time delay, Doppler shifts, and AOA measurements [18]. Using two or three kinds of estimated parameters improves the position or velocity accuracy than only one estimated parameter.

In this paper, we present a joint position and velocity estimation algorithm, which uses Doppler shifts, BR, and TDOA, to estimate the position and velocity of a moving target. This algorithm is optimized based on two-step weighted least squares minimisations (WLS) [9, 18]. In the first step, a set of pseudolinear BR, TDOA, and Doppler shift equations are established by introducing nuisance parameters, rewriting pseudolinear equations to a linear equation that can utilize the calculation formula of WLS, and then the initial estimation of position and velocity is obtained. In the second step, the error term of the first-stage solution is estimated by using the relationship between initial estimated terms and nuisance variables. Similarly, error terms are calculated by WLS, and then the final estimation of position and velocity is obtained through subtracting the error terms from initial values. Simulation results show that it has a better localization and velocity accuracy not only than the method of ellipse and hyperbola but also than BR and Doppler shifts.

We shall use the common convention that bold upper and lower case letters denote matrix and column vector. [I.sub.m], [0.sub.m], and [0.sub.m,n] represent an m x m identity matrix, a zero vector with the length m, and m x n zero matrix, respectively. The notations [A.sup.T] and [A.sup.-1] denote transpose and inverse of matrix A. The [l.sub.2] norm is denoted by [parallel] * [parallel]. The symbol [cross product] stands for Kronecker products, and [a.sup.0] denotes the true value of the variable [alpha].

2. Problem Formulation for the Multistatic Radar System

We consider the localization of a moving target in a three-dimensional (3D) scenario using the multistatic radar system network with M transmitters and N receivers, whose positions are denoted by [t.sub.i] = [([x.sup.t.sub.i], [y.sup.t.sub.i], [z.sup.t.sub.i]).sup.T], i = 1,2, ..., M and [s.sub.j] = [([x.sup.s.sub.j], [y.sup.s.sub.j], [z.sup.s.sub.j]).sup.T], j = 1, 2, ..., N. The target position [u.sup.0] = [(x, y, z).sup.T], and the target velocity [v.sup.0] = [([v.sub.x], [v.sub.y], [v.sub.z]).sup.T]. The geometry structure of the multistatic radar system is shown in Figure 1.

Each transmitter radiates a signal of different frequency, and all receivers observe the signal through two paths, one is the direct propagation from the transmitter and the other is the indirect reflection of a moving target. Signals from the two paths for each transmitter-receiver pair provide differential delay and Doppler shift measurements, where the latter comes from the target movement. The BRs [r.sub.i] + [r.sub.j] are obtained based on the time synchronization of transmitters and receivers. TDOA is the time difference of signals reflected from the target to spatially separated receivers; it can be obtained due to the time synchronization of receivers. We will take advantage of these readily available parameters, BR, TDOA, and Doppler shift, to estimate the position and velocity of a moving target.

The range between the direct and the indirect signal from transmitter i to receiver j is given by

[r.sup.0.sub.ij] = c[tau] = [r.sub.i] + [r.sub.j] - [d.sub.ij], (1)

where c is the speed of signal transmission, t is the differential delay between the direct and the indirect signal transmission, and [r.sub.j] = [parallel] [u.sup.0] - [s.sub.j] [parallel], [r.sub.i] = [parallel] [u.sup.0] - [t.sub,i], and [d.sub.ij] = [parallel] [t.sub.i] - [s.sub.j] [parallel]; therefore, the BRs of the multistatic radar system are [r.sup.0.sub.ij] + [d.sub.ij] = c[tau] + [d.sub.ij].

Collecting the range measurements gives

[mathematical expression not reproducible] (2)

where [mathematical expression not reproducible].

It is assumed that receiver [s.sub.1] is the reference receiver. After multiplying TDOA by signal propagation speed, the range difference between the receivers and [s.sub.1] can be expressed as [r.sup.0.sub.ij] = [r.sub.j] - [r.sub.1], j = 2,3, ..., N and the measured version of [r.sup.0.sub.j1] is [r.sub.j1] = [r.sup.0.sub.j1] + [DELTA][r..sub.j1], where [DELTA][r.sub.j1] is the additive noise. We can get the N - 1 hyperbolic measurements in vector form as

[mathematical expression not reproducible] (3)

Doppler shift is defined as the rate of the change of total path length of the signal to the time because transmitters and receivers are stationary, and the true Doppler shift is

[mathematical expression not reproducible] (4)

where [mathematical expression not reproducible] denotes a unit vector from [t.sub.i] to [u.sup.0] and [f.sub.ci] is the carrier frequency of transmitter [t.sub.i]; the measured [f.sup.0.sub.ij] is [f.sub.ij] = [f.sup.0.sub.ij] + [DELTA][f.sub.ij], where [DELTA][f.sub.ij] is the additive noise. Collecting the Doppler measurements gets

[mathematical expression not reproducible] (5)

where [mathematical expression not reproducible].

Defining m = [[[r.sup.T], [[??].sup.T]. [f.sup.T]].sup.T], m is a (2MN + N - 1) x 1 measurement vector, the error vector [DELTA]m = [[DELTA][r.sup.T], [DELTA][[??].sup.T]. [DELTA][f.sup.T]].sup.T] is assumed to be zero-mean Gaussian with covariance matrix [Q.sub.m] = diag([Q.sub.r], [Q.sub.[??]], [Q.sub.f]), where [Q.sub.r], [Q.sub.[??]], and [Q.sub.f] are covariance matrices of r, [??], and f. We will use BR, TDOA, and Doppler shifts to estimate the position and velocity of a moving target.

The proposed algorithm is optimized by two steps. First, get the initial estimates through establishing a series of pseudolinear equations which introduce nuisance parameters. Second, get the final estimates by exploring the estimation error and estimating the amount of correction to refine the solution.

2.1. First Step. We calculate the initial position and velocity by introducing nuisance parameters and establishing a set of pseudolinear equations about BR, TDOA, and Doppler shift.

Rearranging [r.sup.0.sub.ij] = [r.sub.i] + [r.sub.j] - [r.sub.ij], substituting [r.sub.ij] = [r.sup.0.sub.ij] + [DELTA][r.sub.ij], and squaring both sides yield the elliptic measurement equations. It can be obtained that

[mathematical expression not reproducible] (6)

Note that BR and ellipse localization will be used interchangeably in this paper.

Similarly, rearranging [r.sup.0.sub.j1] = [r.sub.j] - [r.sub.1], substituting [r.sub.j1] = [r.sup.0.sub.j1] + [DELTA][r.sub.j1], and squaring both sides yield the hyperbolic measurement equations. It can be expressed as

[mathematical expression not reproducible] (7)

Note that TDOA and hyperbola will be used interchangeably in this paper.

Then, rearranging (4) and [f.sub.ij] = [f.sup.0.sub.ij] + [DELTA][f.sub.ij], it can be expressed as

[mathematical expression not reproducible] (8)

In (6)-(8), the second-order noise terms have been ignored, and the unknown vector in the first stage can be [mathematical expression not reproducible], where [mathematical expression not reproducible] which contains 2M + 7 nuisance variables. Equations (6)-(8) can be rewritten in the matrix form as follows:

[B.sub.1][DELTA]m = [h.sub.1] - [G.sub.1][[phi].sub.0]. (9)

This is a typical equation that can be solved using WLS, where [mathematical expression not reproducible];

[mathematical expression not reproducible] (10)

The first-step WLS solution is [19, 20]

[mathematical expression not reproducible] (11)

where [W.sub.1] = [B.sup.-T.sub.1] [Q.sup.-1] [B.sup.-1.sub.1] is the weighting matrix. Since the variable [B.sub.1] depends on the estimated position and velocity, however, the initial estimations are unknown in the beginning; hence, we can first set [W.sub.1] = [Q.sup.-1.sub.m] to produce an initial solution, then use the initial solution to recalculate [B.sub.1], and finally expected [W.sub.1] can be obtained. When the noise is small, the covariance matrix of [[phi].sub.1] is approximately equal to cov([[phi].sub.1]) [approximately equal to] [([G.sup.T.sub.1][W.sub.1][G.sub.1]).sup.-1].

2.2. Second Step. The error terms are estimated by using the relationship between the solution of the first stage and nuisance variables.

We denote the first-stage estimation as [mathematical expression not reproducible]

Rearranging the parameters of [[phi].sub.1] and [DELTA][[phi].sub.1], we get the equations about the nuisance variables and the estimation values from the first step:

[mathematical expression not reproducible] (12)

[mathematical expression not reproducible] (13)

[mathematical expression not reproducible] (14)

Stacking (12)-(14) and putting them together yield the matrix form equation (15) which can be used to obtain estimation error terms by WLS:

[mathematical expression not reproducible] (15)

where

[mathematical expression not reproducible] (16)

The second-step WLS solution is

[mathematical expression not reproducible] (17)

where [W.sub.2] = [B.sup.-T.sub.2] cov ([[phi].sub.1])[B.sup.-1.sub.2]. [DELTA]u = [[phi].sub.2] (1 : 3) and [DELTA]v = [[phi].sub.2] (4 : 6). Under small noise conditions, we can get

cov([[phi].sub.2]) [approximately equal to] [([G.sup.T.sub.2][W.sub.2][G.sub.2]).sup.-1], (18)

and [mathematical expression not reproducible] can be obtained from (11).

Finally, the estimations of position and velocity are

[mathematical expression not reproducible]. (19)

3. Performance Analysis

The CRLB for estimating the unknown [??] = [[[u.sup.T], [v.sup.T]].sup.T] is equal to the inverse of the Fisher information matrix (FIM):

CRLB ([??]) = [([[nabla].sup.T] [Q.sup.-1.sub.m] [nabla]).sup.-1], (20)

where [nabla] = [[partial derivative][??]/[partial derivative]u, [partial derivative][??]/[partial derivative]9v].

The covariance matrix of [??] is obtained from (21) as [1]:

[mathematical expression not reproducible] (21)

where [G.sub.3] = [B.sup.-1.sub.1][G.sub.1] [B.sup.-1.sub.2][G.sub.2].

It can be obtained that [G.sub.3] [approximately equal to] [nabla] when [mathematical expression not reproducible]. The calculation of this follows similar procedures as in [2, 3, 15, 20-23].

Consequently, we conclude from (20) and (21) that

cov ([??]) [approximately equal to] CRLB ([??]), (22)

when those small noise conditions are satisfied.

4. Simulation Results

The simulation scenario is as follows. A moving target is at [u.sup.0] = [[1000, 1500, 3000].sup.T] m and its velocity is [v.sup.0] = [120, 150, 100]T m/s. The multistatic radar system has T = 7 transmitters and R = 8 receivers. The transmitters are at [t.sub.1] = [[3000, 1000, 0]sup.T] m, [t.sub.2] = [[1000, 1500, 1000].sup.T] m, [t.sub.3] = [[3000, 5000, 2000].sup.T] m, [t.sub.4] = [[3500, 2000,1000].sup.T] m, [t.sub.5] = [[3500,2000, 0].sup.T] m, [t.sub.6] = [[0, 0, 0].sup.T] m, and [t.sub.7] = [[-1000, 3000, 0].sup.T] m. The receivers' position are at [s.sub.1] = [[0, 5000, 0].sup.T] m, [s.sub.2] = [[5000, 0, 1000].sup.T] m, [s.sub.3] = [[-5000, 0, 1500].sup.T] m, [s.sub.4] = [[0, - 5000, 1000].sup.T] m, [s.sub.5] = [[0, 0, 1000].sup.T] m, [s.sub.6] = [[0, 1000, 1000].sup.T] m, [s.sub.7] = [[0, - 2000, 0].sup.T] m, and [s.sub.8] = [[-5000, 2000, 0].sup.T] m. The signal propagation speed c is 300000000 m/s. The carrier frequencies of the transmitted signals are [f.sub.ci] = [20,19, 12,26,28,16,8] MHz. The covariance matrix [Q.sub.m] of BR, TDOA, and Doppler shift measurements is [Q.sub.m] = [[sigma].sup.2] blkdiag([I.sub.M], [I.sub.N-1], [10.sup.5] [I.sub.M]), [sigma] represents the level of noise. The localization and velocity accuracy are assessed via root-mean-square error (RMSE). The results in Figures 2-8 are obtained from Monte Carlo simulations of 5,000 ensemble runs. We use CRLB as a benchmark for performance evaluation.

We compared performance of the proposed method with the CRLB and two kinds of algorithms, one is the hybrid of ellipse and hyperbola algorithm and the other is the algorithm of joint BR and Doppler shifts. The comparison algorithms were shown to outperform the same kind of algorithms. Simulation results are summarised in Figures 2-8, in which the curve marked with circle corresponds to the hybrid of ellipse, hyperbola, and Doppler shifts (AE + H + Doppler), and curves marked with triangle and plus correspond to the joint ellipse and Doppler shifts (AE + Doppler) and the joint ellipse and hyperbola (AE + H), respectively. The dotted line represents the CRLB under the corresponding setting.

In this paper, two kinds of modes are simulated, one is one transmitter and multiple receivers and the other is multiple transmitters and multiple receivers. The results show that the proposed method is able to reach the CRLB when the measurement error is small. As is shown in Figures 2-5, under the condition of changing the number of transmitters or receivers, the proposed method has a better performance in the positioning compared to the hybrid of elliptic and hyperbolic method and the joint method of BR and Doppler shifts. Also, with the increase of a, the position RMSE of the proposed method is obviously lower than that of the hybrid of elliptic and hyperbolic method and the joint method of BR and Doppler shifts. Then, in Figures 6-8, the velocity RMSE is also analysed, and with the increase of a, its gap between the proposed method and the hybrid of BR and Doppler shifts method becomes larger. The advantage of estimation performance is more obvious with the increase of measurement error. Moreover, when the number of receivers or transmitters is changed, the proposed algorithm has much better velocity accuracy than the joint method of BR and Doppler shifts. In Figures 3-5, 7, and 8, CRLB is close to zero when the number of transmitters and receivers is large. It is illustrated that the proposed algorithm has much better performance not only in the positioning but also in the velocity. Moreover, the estimation performance is affected by the number of transmitters and receivers, and estimation accuracy is improved when the number of transmitters and receivers is increased.

5. Conclusion

In this paper, we investigated the location and velocity of a moving target in the multistatic radar system by combining BR, TDOA, and Doppler shifts. This nonlinear estimation problem is solved by introducing nuisance variables and using two-step WLS to refine the estimate. It is found that the proposed method can obtain the CRLB accuracy over a small error region in the case of Gaussian noise. Simulation results have also shown its effectiveness. The proposed algorithm in this paper enriches the positioning methods of moving targets in the multistatic radar system.

https://doi.org/10.1155/2019/4943872

Data Availability

MATLAB codes used in this study are available from the first author upon request (lcfylj@163.com).

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (no.61671333), the Natural Science Foundation of Hubei Province (2014CFA093), the Fundamental Research Funds for the Central Universities (2042019K50264, 2042019gf0013), and the Fundamental Research Funds for the Wuhan Maritime Communication Research Institute (2017J-13).

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Lijuan Yang [ID], Huotao Gao [ID], Boya Li, Yunkun Yang, and Guobao Ru [ID]

Electronic Information School, Wuhan University, Wuhan 430072, China

Correspondence should be addressed to Huotao Gao; gaoght863@163.com

Received 5 June 2019; Revised 25 September 2019; Accepted 6 November 2019; Published 23 November 2019

Caption: Figure 1: Geometry for multistatic radar.

Caption: Figure 2: Comparison of the position RMSE of the proposed method with AE + Doppler, AE + H, and CRLB when T = 1, R = 8. The accuracy is lower as the noise power increases.

Caption: Figure 3: Comparison of the position RMSE of the proposed method with AE + Doppler, AE + H, and CRLB when T = 3, R = 8. The CRLB is close to zero, but it is going up as the noise power increases.

Caption: Figure 4: Comparison of the position RMSE of the proposed method with AE + Doppler, AE + H, and CRLB when T = 7, R = 8. The CRLB is close to zero, but it is going up as the noise power increases.

Caption: Figure 5: Comparison of the position RMSE of the proposed method with AE + Doppler, AE + H, and CRLB when T = 3, R = 7. The CRLB is close to zero, but it is going up as the noise power increases.

Caption: Figure 6: Comparison of the velocity RMSE of the proposed method with AE +Doppler and CRLB when T = 1, R = 8. The accuracy is lower as the noise power increases.

Caption: Figure 7: Comparison of the velocity RMSE of the proposed method with AE + Doppler and CRLB when T = 3, R = 8. The CRLB is close to zero, but it is going up as the noise power increases.

Caption: Figure 8: Comparison of the velocity RMSE of the proposed method with AE + Doppler and CRLB when T = 3, R = 7. The CRLB is close to zero, but it is going up as the noise power increases.