TDOA and FDOA Hybrid Positioning of Mobile Radiation Source with Receiver Position Errors

Abstract

High-precision position awareness is essential to ubiquitous wireless networks, which can provide real-time position information and abundant status information for numerous practical applications. However, It is a challenge to obtain accurate position estimation utilizing traditional onefold parameter estimation, especially for the accurate position estimation of moving radiation source in the presence of receiver position errors. In this work, we developed an Improved Taylor Series estimation method in three-dimensional positioning scene, in which time difference of arrival (TDOA) and frequency difference of arrival (FDOA) are used to estimate the position and velocity of the target, and the position of the receiver is iteratively updated to reduce the influence of the receiver position errors. The closed-form expressions of Cramer–Rao low bound (CRLB) based on joint TDOA and FDOA positioning with receiver position errors are derived. In the simulation, CRLB with and without receiver position errors are evaluated to illustrate the influence of the receiver position errors on the positioning performance. Theory analysis and simulation results show that the proposed algorithm has lower complexity, smaller RMSE and better positioning performance than multidimensional scaling (W-MDS) algorithm, constrained total least squares algorithm and two-step weighted least squares algorithm for both near-field and far-field emitters.

Introduction

Nowadays, determining the position of an unknown source plays an indispensable role in target tracking, navigation, intelligent transportation, emergency rescue, environmental monitoring, public safety, and industrial control [1, 2]. Location awareness in wireless sensor networks (WSN) has received significant attention over the last two decades, and spawned numerous unforeseen applications [3, 4]. Whether in the civil field or military field, utilizing WSN to obtain the location of the target is a prerequisite for many other activities [5, 6]. The research of high-precision positioning techniques have practical needs and significance. Generally, radio positioning can be classified into active and passive two basic categories [7]. As the most typical representative of active positioning, global navigation satellite systems (GNSS) [8, 9–10] such as GPS, GLONASS, Galileo and Beidou have obtained great success and have widely penetrated into our daily life [11, 12, 13, 14–15]. Compared with active positioning, passive positioning has been widely concerned and studied because of its advantages in the aspects of no extra signal transmission, low energy consumption, easy hiding and no need to rely on complex infrastructure [16].

In terms of emitter positioning, multi-station passive positioning can be divided into two categories: direct position determination (DPD) [17] and two-step positioning method [18, 19] based on positioning parameters estimation. The direct positioning method uses all the data of the signal to directly estimate the position of the target source. And it needs to search the target emitter in 2D or 3D range, which has high computational complexity [20]. Therefore, the DPD method is difficult to meet the needs of the scene with high real-time positioning requirements. The two-step positioning method first estimates the position parameters from the signals received by the receiver, and then calculates the emitter position information according to the relationship between the position parameters and the target position. Typical position parameters include angle of arrival (AOA), time of arrival (TOA), time difference of arrival (TDOA), frequency difference of arrival (FDOA) and received signal strength (RSS) [19, 21, 22, 23, 24–25]. Compared with DPD method, the two-step positioning method can greatly reduce the computational complexity of the algorithm and save the deployment cost of the system, so it is more widely used in passive positioning.

TOA and TDOA positioning models have limitations, and can only be used to estimate the location of the target [26]. In order to estimate the velocity of the target, FDOA is applied to the localization algorithm, so that TDOA and FDOA can be jointly used to estimate the position and velocity of the target. TDOA localisation has been studied extensively, FDOA less so. This is mainly due to the fact that TDOA level surfaces are well known to be hyperbolic, whereas FDOA level curves and surfaces are much more complex. The work of [27] shows examples of the FDOA level curves and surfaces, and shows that they simplify dramatically in the far field.

The location problem based on TDOA and FDOA can be transformed into a highly nonlinear ML estimation problem, which is difficult to solve directly [28]. Ho et al. proposed the weighted least squares (WLS) algorithm [29], which is a non-iterative algorithm. The principle is to introduce two redundant variables to linearize the TDOA and FDOA equations. W-MDS [30] solves the joint localization of TDOA and FDOA based on the optimization of the cost function of scalar product matrix correlation in the classical MDS framework, which has strong robustness when the measurement noise is large. Ali et al. [31] presented an improved version of WLS to determine the position and velocity of a moving source using TDOA and FDOA, where a closed-form solution is obtained from the minimization of the WLS criterion in each stage. The problem in [32] first reformulated based on the robust least squares criterion and then perform semidefinite relaxation (SDR) to obtain a convex semidefinite programming problem. In [33], an efficient algorithm is developed to find the location and velocity of a moving source, which employs an iterative reweighted least square (IRLS) approach with a varying weighting matrix. A joint TDOA/FDOA location algorithm is proposed in [34], which solves the problem that WLS methods have a large positioning error when the number of observation stations is not over-determined. This algorithm is suitable for well-posed conditions, and gets rid of the dependence on the constraints of Earth’s surface. In general, these joint TDOA and FDOA method do not have local convergence issue and they are able to achieve good performance when the position of the receiver is in the ideal situation. Nonetheless, these methods may not have satisfactory performance when the receiver position errors are taken into account.

However, in the actual scene, the receiving terminal may be arranged on the mobile facilities such as cars, planes, ships, etc., and its position has great uncertainty. The uncertainty of the receiver position will seriously affect the localization performance of the target. Therefore, the authors in [35, 36] explored some feasible tactics to improve the source position estimation accuracy when the receivers have position errors. [37] proposed a two-step weighted least squares (TSWLS) algorithm to solve the problem of joint location using TDOA / FDOA when the receiver position error exists. But this method will seriously deviate from CRLB [38] when the measurement error is large. Yu et al. [39] proposed a constrained total least squares (CTLS) algorithm to estimate the position and velocity of the moving emitter when the sensor position is uncertain. The Lagrange multiplier technique was used to constrain the solution by using the relationship between the known intermediate variables and the emitter coordinates. CRLB can be achieved only when the measurement noise and sensor positioning error are small enough. Mao et al. [40] proposed a modified TSWLS method that takes the sensor location errors into account, which focuses on the location problem of moving sources in the presence of random sensor location errors. This method first utilizes the weighted spherical interpolation approach to obtain an initial estimation, and then substitute it into the original criterion to get a final estimation. The work in [41] shows the optimal sensor deployment and velocity configuration of UAV swarms mounted with TDOA and FDOA based sensors. Wang et al. [42] presented an iterative constrained weighted least squares (ICWLS) estimator for multiple-target joint localization using TDOA/FDOA measurements from both target sources and unmanned aerial vehicles (UAV) calibration emitters. This method can improve the TDOA/FDOA localization accuracy, but it is heavily reliant on the help of UAV calibration emitters. All the above methods apply bias reduction based on pseudo linear LS formulation, and can only reach the CRLB when the measurement errors and receiver position errors are small enough. This paper focuses on iteratively updating the receiver position to reduce its impact on positioning errors in cases of large receiver errors and achieve CRLB in the presence of receiver position errors.

In this paper, a robust TDOA and FDOA hybrid positioning approach of mobile radiation source is proposed. And the method takes into account the position error of the receiver, the position of the receiver is iteratively updated to reduce the influence of positioning error. The CRLB of the proposed joint TDOA and FDOA positioning method is deduced when the position error exists in the receiver. In the simulation, the difference of the CRLB in the cases of the receiver with or without position error is considered to illustrate the influence of the position error on the localization performance. Then we compare the performance of the proposed algorithm with other algorithms in the case of near-field sources and far-field sources, respectively. Theory anlysis and simulation results demonstrate that the proposed algorithm has smaller RMSE and better performance than TSWLS, CTLS and W-MDS for both near-field and far-field emitters.We summarize the contributions of this work in following:

1. An Improved Taylor Series position estimation method for three-dimensional moving radiation source localization. in which a hybrid TDOA and FDOA is used to estimate the position and velocity of the target.

2. A two-step iterative method based on positioning error correction is proposed to estimate the initial position and velocity of the target and amend the error.

3. The RMSE analysis show that the solutions of the Improved Taylor Series method is able to achieve the CRLB accuracy in the case of receiver position errors are considered.

4. Theoretical analysis and simulation results show that the algorithm proposed in this paper has lower complexity, smaller RMSE and better performance compared to other algorithms, whether in the near-field or far-field.

This paper is organized as follows: In Sect. 2, we discuss TDOA and FDOA measurement model and foundations of mathematics. In Sect. 3 we present an improved joint TDOA and FDOA mobile source localization algorithm and technological process. In Sect. 4, the Cramer–Rao lower bound of the proposed algorithm is derived. In Sect. 5, the simulation results are presented, and the performance of the algorithm is analyzed and compared. Finally, the conclusion of this paper is given in Sect. 6.

Measurement Model

Let us consider the three-dimensional localization of a radiation source on an unmanned aerial vehicle (UAV), with K receivers distributed randomly around the radiation sources. The system model of the joint TDOA and FDOA is shown in Fig. 1. Wherein receiver stations ${\mathbf{p}}_i$ and ${\mathbf{p}}_j$, $i,j=1,2,\dots ,K$ constitute WSN, which is intended to estimate the position and velocity of the target emitter simultaneously.

The real location of the receiving station is represented by ${\mathbf{p}}_i^0={\left[x_i^0,,,y_i^0,,,z_i^0\right]}^T$, the real position of the target is represented by ${\mathbf{q}}^0={\left[x^0,,,y^0,,,z^0\right]}^T$, and the real speed of the target is ${\dot{\mathbf{q}}}^0={\left[{\dot{x}}^0,,,{\dot{y}}^0,,,{\dot{z}}^0\right]}^T$.The symbol ${\left(·\right)}^0$ represents the real value of the variable, and the symbol ${\left(·\right)}^T$ represents the transpose of the matrix or vector. It is assumed that the emitter signal propagates in line of sight between the target and each receiving station.

[See PDF for image]

Fig. 1

System model of joint TDOA and FDOA positioning

The true distance between the radiation source and the i-th receiving station can be expressed as

1

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill r_i^0=& {∥{\mathbf{q}}^0,-,{\mathbf{p}}_i^0∥}_2\hfill \\ \hfill =& \sqrt{{\left({\mathbf{q}}^0,-,{\mathbf{p}}_i^0\right)}^T\left({\mathbf{q}}^0,-,{\mathbf{p}}_i^0\right)},i=1,2,\dots ,K\hfill \end{array}\end{array}$$where the symbol ${∥·∥}_2$ represents the 2-norm of a matrix or vector.

Taking the first receiver ${\mathbf{p}}_1^0$ as the reference receiver, the real distance difference between the i-th receiver and the reference receiver can be calculated as

2

$$\begin{array}{c}\hfill \begin{array}{c}\hfill r_{i1}^0=r_i^0-r_1^0=c{\tau }_{i1}^0,i=2,3,\dots ,K\end{array}\end{array}$$where $c=3\times {10}^8$ m/s is the propagation speed of electromagnetic wave signal; ${\tau }_{i1}^0$ is the real propagation time difference between the ith receiving station and the reference receiving station, i.e. TDOA value. Calculating the derivative of Eq. (2) with respect to time, one can get that the rate of distance change (i.e. velocity) is

3

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\dot{r}}_i^0=\frac{{\left({\mathbf{q}}^0,-,{\mathbf{p}}_i^0\right)}^T\left({\dot{\mathbf{q}}}^0,-,{\dot{\mathbf{p}}}_i^0\right)}{r_i^0}\end{array}\end{array}$$The real speed difference between the ith receiving station and the reference receiving station is

4

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\dot{r}}_{i1}^0={\dot{r}}_i^0-{\dot{r}}_1^0=\frac{c}{f_0}{f}_{i1}^0\end{array}\end{array}$$where $f_0$ is the carrier frequency of the signal, and $f_{i1}^0$ is the real value of the frequency difference between the electromagnetic wave signal arriving at the ith receiving station and the signal arriving at the reference receiving station, that is, the FDOA value. Based on TDOA and FDOA, the problem of joint localization of moving target is to determine the position vector and velocity vector of moving target by using the measured value of TDOA and FDOA. TDOA measurement can be equivalent to range difference of arrival (RDOA) [43]. The measurement value of distance difference is expressed as: $r_{i1}=r_{i1}^0+c{n}_{t,i1},i=2,3,\dots ,K$, where $n_{t,i1}$ is the measurement error of TDOA, and we assume that it obeys the Gaussian distribution with the mean value of 0. The vector form is written as $\tilde{\mathbf{r}}={\left[r_{21},,,r_{31},,,\dots ,,,r_{K1}\right]}^T={\mathbf{r}}^0+{\mathbf{n}}_{\text{TDOA}}$, where ${\mathbf{r}}^0={\left[r_{21}^0,,,r_{31}^0,,,\dots ,,,r_{K1}^0\right]}^T$, ${\mathbf{n}}_{\text{TDOA}}={\left[c,n_{t,21},,,c,n_{t,31},,,\dots ,,,c,n_{t,K1}\right]}^T$ and ${\mathbf{n}}_{\text{TDOA}}$ obey the Gaussian distribution with mean value of 0 and variance of ${\sigma }_T^2$.

Similarly, the FDOA measurement value can be equivalent to the measured value of the distance difference change rate. The measured value of range difference change rate is expressed as ${\dot{r}}_{i1}={\dot{r}}_{i1}^0+\frac{c}{f_0}{n}_{f,i1}$, $i=2,3,\dots ,K$, where $n_{f,i1}$ is the measurement error of FDOA, assuming that it follows the Gaussian distribution with mean value of 0 and is not related to the measurement error of TDOA. It can be written in vector form $\widetilde{\dot{\mathbf{r}}}={\left[{\dot{r}}_{21},,,{\dot{r}}_{31},,,\dots ,,,{\dot{r}}_{K1}\right]}^T={\dot{\mathbf{r}}}^0+{\mathbf{n}}_{\text{FDOA}}$, $\widetilde{\dot{\mathbf{r}}}$ obeys the Gaussian distribution with mean value of 0 and variance of ${\mathbf{n}}_{\text{FDOA}}$. where ${\dot{\mathbf{r}}}^0={\left[{\dot{r}}_{21}^0,,,{\dot{r}}_{31}^0,,,\dots ,,,{\dot{r}}_{K1}^0\right]}^T$, ${\mathbf{n}}_{\text{FDOA}}={\left[\frac{c}{f_0},n_{f,21},,,\frac{c}{f_0},n_{f,31},,,\dots ,,,\frac{c}{f_0},n_{f,K1}\right]}^T$. At the same time, considering the possible measurement error between the position of the receiving station and its real position, then $\mathbf{p}={\mathbf{p}}^0+\mathrm{\Delta }\mathbf{p}$, the receiver position error $\mathrm{\Delta }\mathbf{p}$ is assumed to be a Gaussian variable with mean value of 0 and variance of ${\sigma }_p^2$.

The Proposed Iteration Positioning Algorithm

Considering that there are errors in the position coordinates of the receiving station, the position coordinates of the receiving stations are jointly estimated while the position and velocity of the emitter are estimated. Let $\mathit{\eta }={\left[{\mathbf{q}}^{0T},,,{\dot{\mathit{q}}}^{0T},,,{\mathbf{p}}_1^{0T},,,{\mathbf{p}}_2^{0T},,,.,.,.,,,{\mathbf{p}}_K^{0T}\right]}^T$. The vector $\mathit{\eta }$ contains the unknown real position and velocity vector of the emitter and the unknown real coordinate vectors of the receivers. The vector $\mathit{\eta }$ is estimated by the corresponding measurements. The measurement error vector can be expressed as

5

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{\Gamma }=\mathbf{\Theta }-\mathit{g}\left(\mathit{\eta }\right)\end{array}\end{array}$$where $\mathbf{\Theta }={\left[{\tilde{\mathbf{r}}}^T,,,{\widetilde{\dot{\mathbf{r}}}}^T,,,{\mathbf{p}}_1^T,,,\dots ,,,{\mathbf{p}}_K^T\right]}^T$ is composed of distance difference measurement value $\tilde{\mathbf{r}}$, distance difference change rate measurement value $\widetilde{\dot{\mathbf{r}}}$ and receiving station position measurement value ${\mathbf{p}}_i$, and $\mathrm{\Gamma }={\left[{\mathbf{n}}^T,,,{\dot{\mathbf{n}}}^T,,,{\mathrm{\Delta }\mathbf{p}}_1^T,,,\dots ,,,{\mathrm{\Delta }\mathbf{p}}_K^T\right]}^T$. Assume that the mean is zero and the covariance matrix is $\mathbf{Q}$

6

$$\begin{array}{c}\hfill \mathbf{Q}=\left[\begin{array}{ccc}{\mathbf{Q}}_t& {\mathbf{O}}_{\left(K,-,1\right)\times \left(K,-,1\right)}& {\mathbf{O}}_{\left(K,-,1\right)\times \left(3,K\right)}\\ {\mathbf{O}}_{\left(K,-,1\right)\times \left(K,-,1\right)}& {\mathbf{Q}}_f& {\mathbf{O}}_{\left(K,-,1\right)\times \left(3,K\right)}\\ {\mathbf{O}}_{\left(3,K\right)\times \left(K,-,1\right)}& {\mathbf{O}}_{\left(3,K\right)\times \left(K,-,1\right)}& {\mathbf{Q}}_p\end{array}\right]\end{array}$$Where ${\mathbf{Q}}_t$ represents the covariance matrix of TDOA measurement error vector, ${\mathbf{Q}}_f$ is the covariance matrix of FDOA measurement error vector, ${\mathbf{Q}}_p$ indicates the covariance matrix of receiver position measurement error vector, ${\mathbf{Q}}_{m\times n}$ denotes $m\times n$ dimensional zero matrix.

The first $K-1$ elements of the column vector $\mathbf{g}\left(\mathit{\eta }\right)$ are determined by the RDOA relationship according to Eq. (2), the Kth to $(2K-2)$th elements are determined by the FDOA relationship according to Eq.  (4), and the $(2K-1)$th to $(5K-2)$th elements are determined by the position coordinate vector of the receiving station. In next subsection, we present an Improved Taylor Series method to reduce the bias according to Eq. (5).

Improved Taylor Series Method

We first construct the cost function of measurement error vector according to Eq. (5)

7

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{J}\left(\mathit{\eta }\right)={\left(\mathbf{\Theta },-,\mathbf{g}\left(\mathit{\eta }\right)\right)}^T{\mathbf{Q}}^{-1}\left(\mathbf{\Theta },-,\mathbf{g}\left(\mathit{\eta }\right)\right)\end{array}\end{array}$$The corresponding WLS optimization model is

8

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathit{\eta }=argmin{\left(\mathbf{\Theta },-,\mathbf{g}\left(\mathit{\eta }\right)\right)}^T{\mathbf{Q}}^{-1}\left(\mathbf{\Theta },-,\mathbf{g}\left(\mathit{\eta }\right)\right)\end{array}\end{array}$$As $\mathbf{g}\left(·\right)$ is usually a nonlinear function, the Eq. (8) can be solved by iterative algorithm based on Taylor series expansion. ${\mathit{\eta }}_a$ is the initial estimation of $\mathit{\eta }$, and $\mathbf{g}\left(\mathit{\eta }\right)$ can be expanded as its first-order Taylor series at ${\mathit{\eta }}_a$, so that from Eq.  (5) we can get

9

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{\Theta }\approx \mathbf{g}\left(\mathit{\eta }\right){|}_{\mathit{\eta }={\mathit{\eta }}_a}+F_{{\mathit{\eta }}_a}\left(\mathit{\eta },-,{\mathit{\eta }}_{\mathit{a}}\right)+\mathbf{\Gamma }\end{array}\end{array}$$where $\mathbf{F}$ is the Jacobian matrix of $\mathbf{g}\left(\mathit{\eta }\right)$ at ${\mathit{\eta }}_{\mathit{a}}$ in the following form

10

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{F}}_{{\mathit{\eta }}_a}={\left(\left[\frac{\partial \mathbf{g}(\mathit{\eta })}{\partial \mathbf{q}},,\frac{\partial \mathbf{g}(\mathit{\eta })}{\partial \dot{\mathit{q}}},,\frac{\partial \mathbf{g}(\mathit{\eta })}{\partial \mathbf{p}}\right]\right|}_{\mathit{\eta }={\mathit{\eta }}_a}\end{array}\end{array}$$Let $\mathbf{r}={\left[r_{21,a},,,\dots ,,,r_{K1,a}\right]}^T$, $\dot{\mathit{r}}={\left[{\dot{r}}_{21,a},,,\dots ,,,{\dot{r}}_{K1,a}\right]}^T$, $\mathbf{p}={\left[{\mathbf{p}}_1^T,,,\dots ,,,{\mathbf{p}}_K^T\right]}^T$, then ${\mathbf{F}}_{{\mathit{\eta }}_a}$ is

11

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{F}}_{{\mathit{\eta }}_a}={\left(\left[\begin{array}{ccc}\frac{\partial \mathbf{r}}{\partial \mathbf{q}}\hfill & \frac{\partial \mathbf{r}}{\partial \dot{\mathit{q}}}& \frac{\partial \mathbf{r}}{\partial \mathbf{p}}\\ \frac{\partial \dot{\mathit{r}}}{\partial \mathbf{q}}\hfill & \frac{\partial \dot{\mathit{r}}}{\partial \dot{\mathit{q}}}& \frac{\partial \dot{\mathit{r}}}{\partial \mathbf{p}}\\ \frac{\partial \mathbf{p}}{\partial \mathbf{q}}\hfill & \frac{\partial \mathbf{p}}{\partial \dot{\mathit{q}}}& \frac{\partial \mathbf{p}}{\partial \mathbf{p}}\end{array}\right]\right|}_{\mathit{\eta }={\mathit{\eta }}_a}\end{array}\end{array}$$The element of above-mentioned matrix can be determined as follows

12

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\left(\frac{\partial \mathbf{r}}{\partial \mathbf{q}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}={\left(\frac{\partial \dot{\mathit{r}}}{\partial \dot{\mathit{q}}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}=\left[\begin{array}{c}\frac{{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{2,a}\right)}^T}{r_{2,a}}-\frac{{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T}{r_{1,a}}\hfill \\ ⋮\hfill \\ \frac{{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{K,a}\right)}^T}{r_{K,a}}-\frac{{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T}{r_{1,a}}\hfill \end{array}\right]\end{array}\end{array}$$

13

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left(\frac{\partial \dot{\mathit{r}}}{\partial \mathbf{q}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}=\hfill \\ \hfill & \left[\begin{array}{c}\frac{{\dot{\mathit{q}}}_a^T}{r_{2,a}}-\frac{{\dot{r}}_{2,a}}{r_{2,a}^2}{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{2,a}\right)}^T-\left[\frac{{\dot{\mathit{q}}}_a^T}{r_{1,a}},-,\frac{{\dot{r}}_{1,a}}{r_{1,a}^2},{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T\right]\hfill \\ ⋮\hfill \\ \frac{{\dot{\mathit{q}}}_a^T}{r_{K,a}}-\frac{{\dot{r}}_{K,a}}{r_{K,a}^2}{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{K,a}\right)}^T-\left[\frac{{\dot{\mathit{q}}}_a^T}{r_{1,a}},-,\frac{{\dot{r}}_{1,a}}{r_{1,a}^2},{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T\right]\hfill \end{array}\right]\hfill \end{array}\end{array}$$

14

$$\begin{array}{c}\hfill \begin{array}{cc}& {\left(\frac{\partial \mathbf{r}}{\partial \mathbf{p}}\right|}_{\mathit{\eta }\mathbf{=}{\mathit{\eta }}_a}=\hfill \\ \hfill & \left[\begin{array}{cccc}\frac{{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T}{r_{1,a}}\hfill & \frac{-{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{2,a}\right)}^T}{r_{2,a}}\hfill & \cdots \hfill & {\mathbf{0}}_{1\times 3}\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \frac{{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T}{r_{1,a}}\hfill & {\mathbf{0}}_{1\times 3}\hfill & \cdots \hfill & \frac{-{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{K,a}\right)}^T}{r_{K,a}}\hfill \end{array}\right]\hfill \end{array}\end{array}$$

15

$$\begin{array}{c}\hfill {\left(\frac{\partial \dot{\mathit{r}}}{\partial \mathbf{p}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}=\left[\begin{array}{cccc}\frac{{\dot{\mathit{q}}}_a^T}{r_{1,a}}-\frac{{\dot{r}}_{1,a}}{r_{1,a}^2}{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T\hfill & \frac{-{\dot{\mathit{q}}}_a^T}{r_{2,a}}+\frac{{\dot{r}}_{2,a}}{r_{2,a}^2}{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{2,a}\right)}^T\hfill & \cdots \hfill & {\mathbf{0}}_{1\times 3}\hfill \\ ⋮\hfill & ⋮\hfill & \ddots \hfill & ⋮\hfill \\ \frac{{\dot{\mathit{q}}}_a^T}{r_{1,a}}-\frac{{\dot{r}}_{1,a}}{r_{1,a}^2}{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{1,a}\right)}^T\hfill & {\mathbf{0}}_{1\times 3}\hfill & \cdots \hfill & \frac{-{\dot{\mathit{q}}}_a^T}{r_{K,a}}+\frac{{\dot{r}}_{K,a}}{r_{K,a}^2}{\left({\mathbf{q}}_a,-,{\mathbf{p}}_{K,a}\right)}^T\hfill \end{array}\right]\end{array}$$

16

$$\begin{array}{cc}\hfill & {\left(\frac{\partial \mathbf{r}}{\partial \dot{\mathit{q}}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}={\mathbf{O}}_{(K-1)\times 3}\hfill \end{array}$$

17

$$\begin{array}{cc}\hfill & {\left(\frac{\partial \mathbf{p}}{\partial \mathbf{q}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}={\left(\frac{\partial \mathbf{p}}{\partial \dot{\mathit{q}}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}={\mathbf{O}}_{3K\times 3}\hfill \end{array}$$

18

$$\begin{array}{cc}\hfill & {\left(\frac{\partial \mathbf{p}}{\partial \mathbf{p}}\right|}_{\mathit{\eta }={\mathit{\eta }}_a}={\mathbf{I}}_{3K\times 3K}\hfill \end{array}$$From Eqs. (9) and (8), it’s not difficult to find the linear WLS optimization model of the result of the second iteration

19

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\eta }}_a^{\left(l,+,1\right)}\text{=arg min}{\left({\mathbf{F}}_{{\mathit{\eta }}_a^{\left(l\right)}},\left(\mathit{\eta },-,{\mathit{\eta }}_a^{\left(l\right)}\right),-,\left(\mathbf{\Theta },-,\mathbf{g},\left({\mathit{\eta }}_a^{\left(l\right)}\right)\right)\right)}^T\\ \hfill ·{\mathbf{Q}}^{-1}\left({\mathbf{F}}_{{\mathit{\eta }}_a^{\left(l\right)}},\left(\mathit{\eta },-,{\mathit{\eta }}_a^{\left(l\right)}\right),-,\left(\mathbf{\Theta },-,\mathbf{g},\left({\mathit{\eta }}_a^{\left(l\right)}\right)\right)\right)\end{array}\end{array}$$Equation (19) is a quadratic optimization problem about the objective function, so there is an optimal closed form solution

20

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathit{\eta }}_a^{\left(l,+,1\right)}=& {\mathit{\eta }}_a^{\left(l\right)}\hfill \\ \hfill & +{\left[{{\mathbf{F}}_{{\mathit{\eta }}_a^{\left(l\right)}}}^T,{\mathbf{Q}}^{-1},{\mathbf{F}}_{{\eta }_a^{\left(l\right)}}\right]}^{-1}{\mathbf{F}}_{{\mathit{\eta }}_a^{\left(l\right)}}^T{\mathbf{Q}}^{-1}\left(\mathbf{\Theta },-,\mathbf{g},\left({\mathit{\eta }}_a^{\left(l\right)}\right)\right),\hfill \\ \hfill & l=0,1,2,\dots \hfill \end{array}\end{array}$$when $∥{\mathit{\eta }}_a^{\left(l,+,1\right)},-,{\mathit{\eta }}_a^{\left(l\right)}∥<\epsilon$ is satisfied, the iteration is terminated and the convergence value ${\mathit{\eta }}_a^{\left(l\right)}$ is denoted as ${\mathit{\eta }}_b$. Where $0<\epsilon \ll 1$ is the iteration termination condition set in advance. At the end of the iteration, the first three elements of ${\mathit{\eta }}_b$ constitute the position vector of the target, and the fourth to sixth elements constitute the velocity vector of the target.

The main characteristic of Taylor series method is that it needs to provide an initial value of the emitter state, and the accuracy of the initial estimation will affect the localization performance. If the initial estimate is accurate, the iterative will converge very fast, and the performance of the positioning method will be better. And if not, the iteration will be caught in a numerical global search, making it difficult to converge and producing significant bias. To further reduce the bias, we present a two-step iterative method based on positioning error correction to estimate the initial position and velocity of the target and amend the error.

Two Step iterative Method for Bias Correction

By introducing Eq. (1) into Eq. (2), we can get

21

$$\begin{array}{c}\hfill \begin{array}{c}\hfill r_{i1}^0+\sqrt{{\left({\mathbf{q}}^0,-,{\mathbf{p}}_1^0\right)}^T\left({\mathbf{q}}^0,-,{\mathbf{p}}_1^0\right)}=\sqrt{{\left({\mathbf{q}}^0,-,{\mathbf{p}}_i^0\right)}^T\left({\mathbf{q}}^0,-,{\mathbf{p}}_i^0\right)}\end{array}\end{array}$$By squaring Eq. (21), we can get

22

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{p}}_i^T{\mathbf{p}}_i-{\mathbf{p}}_1{\mathbf{p}}_1-2{\left({\mathbf{p}}_i,-,{\mathbf{p}}_1\right)}^T{\mathbf{q}}^0=r_{i1}^0_{}^2+2r_{i1}^0{r}_1^0\end{array}\end{array}$$The derivative of Eq. (22) with respect to time can be expressed as

23

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\dot{\mathit{p}}}_i^T{\mathbf{p}}_i-{\dot{\mathit{p}}}_1^T{\mathbf{p}}_1-{\left({\dot{\mathit{p}}}_i,-,{\dot{\mathit{p}}}_1\right)}^T{\mathbf{q}}^0-{\left({\mathbf{p}}_i,-,{\mathbf{p}}_1\right)}^T{\dot{\mathit{q}}}^0\\ \hfill ={\dot{r}}_{i1}^0{r}_{i1}^0+{\dot{r}}_{i1}^0{r}_1^0+r_{i1}^0{\dot{r}}_1^0\end{array}\end{array}$$The proposed method is to solve the nonlinear problem, which has two steps. The first step is using weighted least squares estimation to estimate the position and velocity of the radiation source. After solving the pseudo linear equations, the Improved Taylor Series is used to amend the initial parameters of position and velocity of the radiation source.

Step 1: Weighted Least Squares Estimation.

The auxiliary variable ${\mathit{\theta }}_1^0={\left[{\mathbf{q}}^{0T},,,r_1^0,,,{\dot{\mathit{q}}}^{0T},,,{\dot{r}}_1^0\right]}^T$ is defined. Assuming that ${\mathbf{q}}^{0T},r_1^0,{\dot{\mathit{q}}}^{0T},{\dot{r}}_1^0$ is uncorrelated, Eqs. (22) and (23) can be written in matrix form

24

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\xi }}_{\mathbf{1}}={\mathbf{h}}_{\mathbf{1}}-{\mathbf{G}}_{\mathbf{1}}{\theta }_{\mathbf{1}}^{\mathbf{0}}={\mathbf{B}}_1{\mathbf{n}}_1\end{array}\end{array}$$where,

25

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{h}}_1=\left[\begin{array}{c}{r}_{21}^2-{\mathbf{p}}_2^T{\mathbf{p}}_2+{\mathbf{p}}_1^T{\mathbf{p}}_1\hfill \\ ⋮\hfill \\ r_{K1}^2-{\mathbf{p}}_K^T{\mathbf{p}}_K+{\mathbf{p}}_1^T{\mathbf{p}}_1\hfill \\ 2\left({\dot{r}}_{21},r_{21},-,{\dot{\mathit{p}}}_2^T,{\mathbf{p}}_2,+,{\dot{\mathit{p}}}_1^T,{\mathbf{p}}_1\right)\hfill \\ ⋮\hfill \\ 2\left({\dot{r}}_{21},r_{21},-,{\dot{\mathit{p}}}_K^T,{\mathbf{p}}_K,+,{\dot{\mathit{p}}}_1^T,{\mathbf{p}}_1\right)\hfill \end{array}\right]\end{array}\end{array}$$

26

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{G}}_1=-2\left[\begin{array}{cccc}{\left({\mathbf{p}}_2,-,{\mathbf{p}}_1\right)}^T\hfill & r_{21}\hfill & {\mathbf{0}}^T\hfill & 0\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ {\left({\mathbf{p}}_K,-,{\mathbf{p}}_1\right)}^T\hfill & r_{K1}\hfill & {\mathbf{0}}^T\hfill & 0\hfill \\ {\left({\dot{\mathit{p}}}_2,-,{\dot{\mathit{p}}}_1\right)}^T\hfill & {\dot{r}}_{21}\hfill & {\left({\mathbf{p}}_2,-,{\mathbf{p}}_1\right)}^T\hfill & r_{21}\hfill \\ ⋮\hfill & ⋮\hfill & ⋮\hfill & ⋮\hfill \\ {\left({\dot{\mathit{p}}}_K,-,{\dot{\mathit{p}}}_1\right)}^T\hfill & {\dot{r}}_{K1}\hfill & {\left({\mathbf{p}}_K,-,{\mathbf{p}}_1\right)}^T\hfill & r_{K1}\hfill \end{array}\right]\end{array}\end{array}$$

27

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{B}}_1=\left[\begin{array}{cc}\mathbf{B}\hfill & \mathbf{O}\hfill \\ \dot{\mathit{B}}\hfill & \mathbf{B}\hfill \end{array}\right]\hfill \\ \hfill & \mathbf{B}=2diag\left\{r_2^0,,,r_3^0,,,\dots ,,,r_K^0\right\}\hfill \\ \hfill & \dot{\mathit{B}}=2diag\left\{{\dot{r}}_2^0,,,{\dot{r}}_3^0,,,\dots ,,,{\dot{r}}_K^0\right\}\hfill \\ \hfill \end{array}\end{array}$$

28

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{n}}_1={\left[\begin{array}{cc}{\mathbf{n}}_t^T\hfill & {\mathbf{n}}_f^T\end{array}\right]}^T\end{array}\end{array}$$According to Eq. (24), the weighted least squares estimation of ${\mathit{\theta }}_1^0$ can be obtained

29

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\theta }}_1={\left({\mathbf{G}}_1^T,{\mathbf{W}}_1,{\mathbf{G}}_1\right)}^{-1}{\mathbf{G}}_1^T{\mathbf{W}}_1{\mathbf{h}}_1\end{array}\end{array}$$In the above formula, ${\mathbf{W}}_1$ is defined as

30

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{W}}_1& ={\left({\mathbf{B}}_1^T,{\mathbf{Q}}_1,{\mathbf{B}}_1\right)}^{-1}\hfill \end{array}\end{array}$$

31

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{Q}}_1& =\left[\begin{array}{cc}{\mathbf{Q}}_t& \mathbf{O}\\ \mathbf{O}& {\mathbf{Q}}_f\\ \end{array}\right]\hfill \end{array}\end{array}$$The covariance matrix of ${\mathit{\theta }}_1$ estimation error is

32

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathbf{cov}\left({\mathit{\theta }}_1\right)={\left({\mathbf{G}}_1^T,{\mathbf{W}}_1,{\mathbf{G}}_1\right)}^{-1}\end{array}\end{array}$$Step 2: Positioning Error Correction.

Next, we estimate the errors $\mathrm{\Delta }\mathbf{q}$ and $\mathrm{\Delta }\dot{\mathit{q}}$ of the emitter position and velocity obtained in the first weighted least square estimation, and calculate the final estimation of the emitter position and velocity after correction. Using the first-order Taylor series expansion, $r_1^0$ and ${\dot{r}}_1^0$ can be expanded at $\widehat{\mathit{q}}$ and $\widehat{\dot{\mathit{q}}}$ as

33

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\widehat{r}}_1={∥{\mathbf{q}}^0,-,{\mathbf{p}}_1∥}_2+\mathrm{\Delta }{r}_1\approx {∥\widehat{\mathit{q}},-,{\mathbf{p}}_1∥}_2-{\rho }_{\widehat{\mathit{q}},{\mathbf{p}}_1}^T\mathrm{\Delta }\mathbf{q}+\mathrm{\Delta }{r}_1\end{array}\end{array}$$

34

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\widehat{\dot{r}}}_1=\frac{{\left({\dot{\mathit{q}}}^0,-,{\dot{\mathit{p}}}_1\right)}^T\left({\mathbf{q}}^0,-,{\mathbf{p}}_1\right)}{r_1^0}+\mathrm{\Delta }{\dot{r}}_1\\ \hfill & \approx \frac{{\left(\widehat{\dot{\mathit{q}}},-,{\dot{\mathit{p}}}_1\right)}^T\left(\widehat{\mathit{q}},-,{\mathbf{p}}_1\right)}{{\widehat{r}}_1}-\mathbf{A}\mathrm{\Delta }\mathbf{q}-\mathbf{B}\mathrm{\Delta }\dot{\mathit{q}}+\mathrm{\Delta }{\dot{r}}_1\hfill \end{array}\end{array}$$Where ${\rho }_{\widehat{\mathit{q}},{\mathbf{p}}_1}^T=\frac{{\left(\widehat{\mathit{q}},-,{\mathbf{p}}_1\right)}^T}{{∥\widehat{\mathit{q}},-,{\mathbf{p}}_1∥}_2}$, $\mathbf{A}$ and $\mathbf{B}$ are the gradient matrices of the first-order Taylor series expansion term of ${\dot{r}}_1^0$ at $\widehat{\mathit{q}}$ and $\widehat{\dot{\mathit{q}}}$ respectively. The expansion terms of Taylor series of second-order and above are ignored. $\mathbf{A}$ and $\mathbf{B}$ can be expressed as

35

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathbf{A}& =\frac{{\left(\widehat{\dot{\mathit{q}}},-,{\dot{\mathit{p}}}_1\right)}^T{∥\widehat{\mathit{q}},-,{\mathbf{p}}_1∥}_2-{\left(\widehat{\dot{\mathit{q}}},-,{\dot{\mathit{p}}}_1\right)}^T\left(\widehat{\mathit{q}},-,{\mathbf{p}}_1\right){\rho }_{\widehat{\mathit{q}},{\mathbf{p}}_1}^T}{{∥\widehat{\mathit{q}},-,{\mathbf{p}}_1∥}_2^2}\hfill \end{array}\end{array}$$

36

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathbf{B}& =\frac{{\left(\widehat{\mathit{q}},-,{\mathbf{p}}_1\right)}^T}{{∥\widehat{\mathit{q}},-,{\mathbf{p}}_1∥}_2}={\rho }_{\widehat{\mathit{q}},{\mathbf{p}}_1}^T\hfill \end{array}\end{array}$$According to the (33) and (34), it can be concluded that

37

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\xi }}_2={\mathbf{h}}_2-{\mathbf{G}}_2{\mathit{\theta }}_2^0={\mathbf{B}}_2\mathrm{\Delta }{\mathit{\theta }}_1\end{array}\end{array}$$where

38

$$\begin{array}{c}\hfill \begin{array}{cc}& {\mathbf{h}}_2=\hfill \\ \hfill & {\left[\begin{array}{cccc}{\mathbf{0}}_{1\times 3}& {\widehat{r}}_1-{∥\widehat{\mathit{q}},-,{\mathbf{p}}_1∥}_2& {\mathbf{0}}_{1\times 3}& {\widehat{\dot{r}}}_1-\frac{{\left(\widehat{\dot{\mathit{q}}},-,{\dot{\mathit{p}}}_1\right)}^T\left(\widehat{\mathit{q}},-,{\mathbf{p}}_1\right)}{{\widehat{r}}_1}\\ \end{array}\right]}^T\hfill \end{array}\end{array}$$

39

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{G}}_2=\left[\begin{array}{cc}{\mathbf{I}}_{3\times 3}& {\mathbf{0}}_{3\times 3}\\ -{\mathit{\rho }}_{\widehat{\mathit{q}},{\mathbf{p}}_1}^T& {\mathbf{0}}_{1\times 3}\\ {\mathbf{0}}_{3\times 3}& {\mathbf{I}}_{3\times 3}\\ -\mathbf{A}& -\mathbf{B}\\ \end{array}\right]\end{array}\end{array}$$

40

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\theta }}_2^0=\left[\begin{array}{c}\mathrm{\Delta }\mathbf{q}\\ \mathrm{\Delta }\dot{\mathit{q}}\\ \end{array}\right]\end{array}\end{array}$$

41

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{B}}_2=\left[\begin{array}{cc}{\mathbf{B}}_{21}& {\mathbf{O}}_{4\times 4}\\ {\mathbf{O}}_{4\times 4}& {\mathbf{B}}_{21}\\ \end{array}\right]\end{array}\end{array}$$

42

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{B}}_{21}=\left[\begin{array}{cc}-{\mathbf{I}}_{3\times 3}& {\mathbf{0}}_{3\times 1}\\ {\mathbf{0}}_{1\times 3}& 1\\ \end{array}\right]\end{array}\end{array}$$From Eq. (37), the weighted least square solution of the second step parameter ${\mathit{\theta }}_2^0$ is obtained as

43

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\theta }}_2={\left({\mathbf{G}}_2^T,{\mathbf{W}}_2,{\mathbf{G}}_2\right)}^{-1}{\mathbf{G}}_2^T{\mathbf{W}}_2{\mathbf{h}}_2\end{array}\end{array}$$Where

44

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{W}}_2={\left({\mathbf{B}}_2,\text{cov},\left({\mathit{\theta }}_1\right),{\mathbf{B}}_2^T\right)}^{-1}\end{array}\end{array}$$Use Eq. (43) to correct ${\mathit{\theta }}_1$ and update the estimated value of emitter position and velocity information

45

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathit{\theta }}_3={\left[{\mathit{\theta }}_1,\left(1,:,3\right),{\mathit{\theta }}_1,\left(5,:,7\right)\right]}^T-{\mathit{\theta }}_2\end{array}\end{array}$$The value of ${\mathit{\theta }}_3$ is used as the initial estimation of the position and velocity of the emitter in the Improved Taylor Series method. In the second step, Taylor series expansion is used to obtain the linear least mean square estimation of the estimation error in the first step. Compared with the second step (including square operation) of TSWLS method, the positioning error is smaller, which can provide more accurate initial value, and ensure the convergence speed of the Improved Taylor Series joint estimation algorithm.

In conclusion, the method proposed in this paper is shown in Tables 1 and 2

Table 1. Summary of algorithm steps

Step 1	Construct the cost function of measurement error vector and use WSL to get $\mathit{\eta }$ as shown in (8)
Step 2	Define ${\mathit{\eta }}_{\mathit{a}}$ as the initial estimation of $\mathit{\eta }$ use first-order Taylor series expansion to expand $g(\mathit{\eta })$ then rewrite the vector (5) as (9)
Step 3	Obtain $l+1$ iteration results (19) via linear WLS optimization on (8) and (9) then use quadratic optimization to find ${{\mathit{\eta }}_{\mathit{a}}}^l$

Table 2. Steps of finding initial estimation ${\mathit{\eta }}_{\mathit{a}}$

Step 1	Rewrite (22) and (23) into matrix form as (24), and get the WLS estimation (29)
Step 2	Use first-order Taylor series expansion to expand $r_1^0$ and ${\dot{r}}_1^0$  at $\widehat{\mathit{q}}$ and $\widehat{\dot{\mathit{q}}}$ as (33) and (34), conclude (37) and find the WLS solution (43). Finally, use (43) to correct (29) to get the initial estimate ${\mathit{\theta }}_3$ as shown in (45)

The CRLB of the Proposed Approach

We first define the position and velocity vector ${\mathit{\theta }}_{\text{TFDOA}}={[{\mathbf{q}}^T,{\dot{\mathit{q}}}^T]}^T$, TDOA and FDOA measurement error vector $\mathrm{\Delta }{\mathit{\alpha }}_{\text{TFDOA}}=[{\mathbf{n}}_{\text{TDOA}}^T,{\mathbf{n}}_{\text{FDOA}}^T]$. The position error vector of the receiving station $\mathrm{\Delta }{\mathit{\beta }}_{\text{TFDOA}}=\mathrm{\Delta }\mathbf{p}$, $\mathrm{\Delta }\mathbf{p}={[\mathrm{\Delta }{\mathbf{p}}_1^T,\mathrm{\Delta }{\mathbf{p}}_2^T,\dots ,\mathrm{\Delta }{\mathbf{p}}_3^T]}^T$. $\mathrm{\Delta }{\mathit{\beta }}_{\text{TFDOA}}$ is a Gaussian random vector with zero mean, and the covariance matrix is ${\mathbf{Q}}_p$. When the receivers have position errors, the CRLB [29] for hybrid TDOA and FDOA estimation is

46

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \begin{array}{c}\text{CRLB}\left({\mathit{\theta }}_{\text{TFDOA}}\right)={\mathbf{G}}_{{\theta }_{\text{TFDOA}}}^{-1}+{\mathbf{G}}_{{\theta }_{\text{TFDOA}}}^{-1}{\mathbf{G}}_{\theta {\beta }_{\text{TFDOA}}}\hfill \\ \times {\left({\mathbf{G}}_{{\beta }_{\text{TFDOA}}},-,{\mathbf{G}}_{{\beta }_{\text{TFDOA}}}^T,{\mathbf{G}}_{{\theta }_{\text{TFDOA}}}^{-1},{\mathbf{G}}_{{\beta }_{\text{TFDOA}}}\right)}^{-1}\hfill \\ \times {\mathbf{G}}_{{\beta }_{\text{TFDOA}}}^T{\mathbf{G}}_{{\theta }_{\text{TFDOA}}}^{-1}\hfill \end{array}\end{array}\end{array}$$where

47

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{G}}_{{\theta }_{\text{TFDOA}}}={\left(\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\theta }}_{\text{TFDOA}}}\right)}^T{\mathbf{Q}}_{\alpha }^{-1}\left(\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\theta }}_{\text{TFDOA}}}\right)\end{array}\end{array}$$

48

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{G}}_{\theta {\beta }_{\text{TFDOA}}}={\left(\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\theta }}_{\text{TFDOA}}}\right)}^T{\mathbf{Q}}_{\alpha }^{-1}\left(\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\beta }}_{\text{TFDOA}}}\right)\end{array}\end{array}$$

49

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{G}}_{{\beta }_{\text{TFDOA}}}={\left(\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\beta }}_{\text{TFDOA}}}\right)}^{\text{T}}{\mathbf{Q}}_{\alpha }^{-1}\left(\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\beta }}_{\text{TFDOA}}}\right)+{\mathbf{Q}}_{\beta }^{-1}\end{array}\end{array}$$and

50

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\theta }}_{\text{TFDOA}}}=\left[\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \mathbf{q}},,,\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \dot{\mathit{q}}}\right]\end{array}\end{array}$$

51

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial {\mathit{\beta }}_{\text{TFDOA}}}=\left[\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \mathbf{p}},,,\frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \dot{\mathit{p}}}\right]\end{array}\end{array}$$where

52

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \mathbf{q}}=\left[\begin{array}{c}{\mathbf{C}}_{\text{TFDOA}1}\hfill \\ {\mathbf{C}}_{\text{TFDOA}2}\hfill \end{array}\right]{\mathbf{E}}_{\text{TFDOA}}\end{array}\end{array}$$

53

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{\partial {\mathit{\alpha }}_{\mathit{TFDOA}}}{\partial \dot{\mathit{q}}}=\left[\begin{array}{c}{\mathbf{0}}_{(K-1)\times 3}\hfill \\ {\mathbf{C}}_{\text{TFDOA}1}{\mathbf{E}}_{\text{TFDOA}}\hfill \end{array}\right]\end{array}\end{array}$$

54

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \mathbf{p}}=-\left[\begin{array}{c}{\mathbf{C}}_{\text{TFDOA}1}\hfill \\ {\mathbf{C}}_{\text{TFDOA}2}\hfill \end{array}\right]\end{array}\end{array}$$

55

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \frac{\partial {\mathit{\alpha }}_{\text{TFDOA}}}{\partial \dot{\mathit{p}}}=\left[\begin{array}{c}{\mathbf{0}}_{(K-1)\times 3K}\hfill \\ -{\mathbf{C}}_{\text{TFDOA}1}\hfill \end{array}\right]\end{array}\end{array}$$The ith row elements of ${\mathbf{C}}_{\text{TFDOA}1}and{\mathbf{C}}_{\text{TFDOA}2}$ are

56

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{C}}_{\text{TFDOA}1}(i,:)=\left[-,{{\overline{\mathit{f}}}^{\mathit{T}}}_1,,,{\mathbf{0}}_{3(i-1)\times 1}^T,,,{{\overline{\mathit{f}}}^{\mathit{T}}}_{i+1},,,{\mathbf{0}}_{3(K-i-1)\times 1}^T\right]\end{array}\end{array}$$

57

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{C}}_{\text{TFDOA}2}(i,:)=\left[-,{{\overline{\mathit{g}}}^{\mathit{T}}}_1,,,{\mathbf{0}}_{3(i-1)\times 1}^T,,,{{\overline{\mathit{g}}}^{\mathit{T}}}_{i+1},,,{\mathbf{0}}_{3(K-i-1)\times 1}^T\right]\end{array}\end{array}$$Where,

58

$$\begin{array}{cc}\hfill {\overline{\mathit{f}}}_i=& \frac{\left(\mathbf{q},-,{\mathbf{p}}_i\right)}{r_i}\hfill \end{array}$$

59

$$\begin{array}{cc}\hfill {\overline{\mathit{g}}}_i=& \frac{\left(\mathbf{q},-,{\mathbf{p}}_i\right)}{r_i}-\frac{{\dot{r}}_i}{r_i^2}\left(\mathbf{q},-,{\mathbf{p}}_i\right)\hfill \end{array}$$It should be specially pointed out that CRLB without position error of receiving station is

60

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \text{CRLB}\left({\mathit{\theta }}_{\text{TFDOA}}\right)={\mathbf{G}}_{{\theta }_{\text{TFDOA}}}^{-1}\end{array}\end{array}$$

Simulation and Performance Analysis

Simulation Analysis

Simulation 1: Comparison of CRLB with and Without Position Error of Receiver Station

The influence of the receiver position errors on joint TDOA and FDOA positioning accuracy is studied by comparing the results of Eqs.  (46) and (60). We considered six receiver stations with the real locations listed in Table 3.

Table 3. Real locations of six receiver stations

Targets	x (m)	y (m)	z (m)
1	200	1000	180
2	420	160	120
3	300	500	50
4	100	200	500
5	− 200	− 100	− 100
6	300	− 200	− 300

The receiver position error vector is a Gaussian random variable with the mean value of zero, and the receiver position error covariance matrix ${\mathbf{Q}}_p$ which can be set to

61

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{Q}}_p=& {\sigma }_p^{2}diag([1,1,1,2,2,2,10,10,10,40,40,40,20,20,20,\hfill \\ \hfill & 3,3,3])\hfill \end{array}\end{array}$$The covariance matrix of TDOA measurement error is ${\mathbf{Q}}_t={\sigma }_T^2\mathrm{\Lambda }$ and the covariance matrix of FDOA is ${\mathbf{Q}}_f={\sigma }_F^2\mathrm{\Lambda }$. $\mathrm{\Lambda }$ is a $M-1$ matrix whose main diagonal element is 1 and other elements are 0.5. Assume that the position of near-field emitter is $\left(136,,,354,,,305\right)\text{m}$, the position of far-field emitter is $\left(2000,,,1000,,,3000\right)\text{m}$, and the velocity is $\left(-,10,,,15,,,30\right)\text{m/s}$.

Figures 2 and 3 show the CRLB for the position and velocity estimates of the near-field and far-field radiation sources when the variance of the position error of the receiver changes. When the receiving station does not have position error, ${\sigma }_T^2={\sigma }_F^2={10}^{-6}$, and ${\sigma }_p^2=0$. CRLB of the near-field and far-field radiation source positions calculated according to Eq. (60) are $-$39.6 dB and $-$8.3 dB respectively, and CRLB estimated by speed are $-$30.5 dB and $-$0.5 dB respectively. It can be seen from Figs. 2 and 3 that when the position error of the receiving station is very small, CRLB estimated by the position and velocity of the near-field radiation source and far-field radiation source are almost the same as those without error. With the increase of the receiver position error, the CRLB of radiation source position and velocity estimation increases. Therefore, the influence of the receiver position error must be considered when using TDOA and FDOA to locate the radiation source.

[See PDF for image]

Fig. 2

CRLB of near-field and far-field emitter position estimation in the presence of receiver position error

[See PDF for image]

Fig. 3

CRLB of near-field and far-field emitter velocity estimation in the presence of receiver position error

Simulation 2: Comparison of Performance Between the Proposed Algorithm and Other Existing Algorithms for Near-Field Radiation Source Positioning

This subsection will discuss the performance of the proposed algorithm for near-field emitters positioning. RMSE of position estimation of W-MDS algorithm [30], CTLS algorithm [39], and TSWLS algorithm [37] are compared with the CRLB of Eq. (46) of the proposed method. Let the near-field radiation source is located at $\left(136,,,354,,,305\right)\text{m}$ and the velocity is $\left(-,10,,,15,,,30\right)\text{m/s}$. In the simulation experiments, the covariance matrix ${\mathbf{Q}}_t={\sigma }_T^2\mathrm{\Lambda }$ of receiver position error is the same as Eq. (61), and ${\sigma }_p^2={10}^{-5}$. The covariance matrix of TDOA measurement error is set to ${\mathbf{Q}}_t={\sigma }_T^2\mathrm{\Lambda }$, and the covariance matrix of FDOA is set to ${\mathbf{Q}}_f=0.1{\mathbf{Q}}_t$. All the results are the average of 5000 Monte Carlo simulations.

As shown in Fig. 4, when ${\sigma }_T^2<-10$ dB, the curves of proposed algorithm and the other three algorithms can reach or approach to the CRLB of position estimation. With the increase of TDOA measurement error, the curves of W-MDS algorithm, CTLS algorithm, and TSWLS algorithm deviate from the CRLB in different degrees. When the measurement error is 30 dB, their RMSE value of position estimation differ from the CRLB by 4.05 dB, 6.3 dB and 7.5 dB, respectively. The proposed algorithm can still approach the CRLB, so it has better performance for the near-field emitter positioning.

[See PDF for image]

Fig. 4

Performance comparison of different algorithms for emitter coordinate estimation is $\left(136,,,354,,,305\right)\text{m}$

From the result in Fig. 5, we can find that the proposed algorithm and other three algorithms can reach or approach the CRLB of speed estimation when ${\sigma }_T^2<-15$ dB. However, with the increase of TDOA measurement error, the curves of W-MDS algorithm, CTLS algorithm, and TSWLS algorithm deviate from the CRLB in different degrees. When the measurement error is 30 dB, the RMSE differences between the speed estimation and the CRLB are 4.13 dB, 4.52 dB and 5.5 dB, respectively. However, the proposed algorithm can still approach the CRLB and has better performance for the near-field emitter velocity estimation.

[See PDF for image]

Fig. 5

Performance comparison of different algorithms when the emitter coordinate is $\left(136,,,354,,,305\right)\text{m}$

Simulation 3: Performance Comparison Between the Proposed Algorithm and Other Existing Algorithms for Far-Field Radiation Source Positioning

The performance of the proposed algorithm for far-field emitters positioning will be evaluated in this subsection. The far-field emitter is located at $\left(2000,,,1000,,,3000\right)\text{m}$ and the velocity is $\left(-,10,,,15,,,30\right)\text{m/s}$. The covariance matrix ${\mathbf{Q}}_p$ of receiver position error is the same as Eq. (61), and ${\sigma }_p^2={10}^{-5}$. The covariance matrix of TDOA measurement error is set to ${\mathbf{Q}}_t={\sigma }_T^2\mathrm{\Lambda }$, and the covariance matrix of FDOA is ${\mathbf{Q}}_f=0.1{\mathbf{Q}}_t$. All the results are the average of 5000 Monte Carlo simulations.

As shown in Fig. 6, when ${\sigma }_T^2<-20$ dB, the proposed algorithm and the other three algorithms can reach or approach the CRLB of position estimation, but with the increase of ${\sigma }_T^2$, the curves of W-MDS algorithm, CTLS algorithm and TSWLS algorithm all deviate from the CRLB in different degree. When the measurement error is 30 dB, the differences between RMSE of position estimation and the CRLB are 3.1 dB, 4.07 dB and 6.2 dB, respectively. However, the proposed algorithm is still close to the CRLB in ${\sigma }_T^2=15$ dB, thus it has better performance in determining the location of far-field emitter.

[See PDF for image]

Fig. 6

Performance comparison of different algorithms for position estimation when the emitter coordinate is $\left(2000,,,1000,,,3000\right)\text{m}$

RMSE of the proposed algorithm and the other three algorithms are shown in Fig. 7. Obviously, the RMSE of the proposed algorithm is smaller than those of other three algorithms, and it is close to the CRLB. The proposed method has better performance for the estimation of the velocity of far-field radiation source.

[See PDF for image]

Fig. 7

Performance comparison of different algorithms for estimating speed when the emitter coordinate is $\left(2000,,,1000,,,3000\right)\text{m}$

It can be seen from Figs. 4, 5, 6 and 7, the RMSE value of emitter location and velocity estimation utilizing the proposed algorithm is smaller than those of W-MDS algorithm, CTLS algorithm and TSWLS algorithm. When the measurement error increases, the proposed algorithm still approximate to the CRLB with better performance.

Complexity Analysis

Assuming that L is the maximum number of iterations in the proposed Improved Taylor Series method, the computational complexity of this algorithm is O(LK), where K denotes the number of receivers.

The average processing times of the methods in the simulation are recorded by MATLAB R2020a on a computer with Intel Core i7-9700 CPU and 16 GB RAM. The average experimental time for the Improved Taylor series method is 3.3299 s, compared to 12.4042 s for the TSWLS method. In contrast, the CTLS and W-MDS algorithms take even longer, with the average processing time for experiment approaching 10 times that of the Improved Taylor Series method, increasing by an order of magnitude. Among these four methods, the Improved Taylor Series method requires the least amount of time for a single experiment. In addition, we found that the TSWLS and W-MDS methods need more time when fixing noise power and increasing receiver position error level. The reason for this is that a large error in the receiver positions requires more iterations before convergence.

Conclusion

To improve the accuracy of emitter position estimation in the case of location errors at receivers, we propose an Improved Taylor Series method with hybrid TDOA and FDOA estimation, in which the joint iterative estimation of the receiver position are carried out to reduce the influence of the receiver position errors on the positioning accuracy. The positioning error correction is used to achieve more accurate estimation, which can ensure the convergence and speed up convergence to improve the target positioning accuracy. We also derived CRLB of joint TDOA and FDOA positioning with receiver position errors in closed form expression. The comparison simulations of the performance of the proposed algorithm with and without the receiver position errors are implemented to illustrate the influence of the receiver position error on the positioning performance. Theory analysis and simulation results show that the proposed algorithm has lower complexity and smaller RMSE than W-MDS algorithm, CTLS algorithm, and TSWLS algorithm for both near-field and far-field emitters and thus has a better location estimation performance.

Acknowledgements

The authors would like to thank the editors and the anonymous reviewers.

Author Contributions

YZ and FH designed the method and wrote the main part of the manuscript, HZ designed the experiments and contributed to the writing of the manuscript, HY conducted the complexity analysis of the algorithm, ZD and ZX made grammar modifications to the manuscript. All authors reviewed the manuscript.

Funding

This work was supported in part by the Natural Science Foundation of Guangdong Province under Grant 2022A1515011975, in part by Shenzhen Fundamental Research Program under grant JCYJ20220530141017040, in part by the Natural Science Foundation of Shandong Province under grants ZR2022LZH005, in part by the Science and Technology Program in Qingdao under Grants 22-3-7-CSPZ-2-nsh.

Data availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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