An Efficient Algebraic Solution for Moving Source Localization from Quadruple Hybrid Measurements

I.   Introduction

Source localization utilizing a set of spatially distributed receivers has drawn concerns for several decades due to its numerous applicability in a variety of fields such as radar, mobile communications, emergency services, intelligent transportation, geological prospecting, indoor acoustic localization, etc.^[1-3]. Generally speaking, different sorts of measurements including time difference of arrival (TDOA)^[4], frequency difference of arrival (FDOA)^[4], angle of arrival (AOA)^[5], or their combinations, can be used to conduct source localization in a passive and uncooperative way. Each measurement defines a geometrical curve of possible source locations. The intersection of these curves with respect to the measurements produces the source location estimate. Due to the very nonlinear relationship between the mearements and the unknown source location, obtaining the intersection is far from simple.

Over the previous two decades, there has been a lot of focus on TDOA-based^[6-9], AOA-based^[10-12] and TDOA-AOA-based^[13-15] source position estimation. These algorithms^[6-15], however, cannot determines the source velocity at a single time instant. By using TDOA and/or AOA measurements obtained at different time instants, extended Kalman filter (EKF)^[16] is a viable method for determining the source's position and velocity. However, EKF’s performance relies on the correctness of the linearization in the measurement equations, and it needs to go through an initial transient period before it gives a credible source position and velocity estimate. To estimate both the position and the velocity of a moving source at a single time instant, some algorithms have been explored based on jointly using TDOA-FDOA measurements^[17-21]. And more recently, to locate the moving source with higher accuracy and a smaller number of receivers, Liu et al.^[22] offer a hybrid TDOA-FDOA-differential Doppler rate (DDR) localization approach based on the development of a pseudo-linear set of equations using nuisance parameters, from which the source position and velocity is produced using a two-stage weighted least squares (WLS) estimator. Xiong et al.^[23] propose a hybrid TDOA-FDOA-AOA localization algorithm, also using the pseudo-linearization technique and two-stage WLS estimator. The aforementioned TDOA-FDOA-based^[17-21], TDOA-FDOA-AOA-based^[23] and TDOA-FDOA-DDR-based^[22] moving source localization algorithms explicitly or implicitly follows the basic framework of TSWLS idea: in the first WLS stage, the nuisance parameters are introduced to convert the measurement equations into a pseudo-linear set of equations, from which the nuisance parameters, as well as the source position and velocity, are calculated; then in the second WLS stage, the relation between the parameters estimated in the previous WLS stage are exploited to improve the initial estimate. However, because of the two-stage processing, the algorithms are susceptible to interstage error propagation and thus to measurement noise levels. Furthermore, because the nuisance parameters, as well as the source position and velocity, must be estimated, more receivers are necessary, increasing the system’s cost and complexity.

The AOA rate is, along with TDOA, FDOA and AOA measurements, another basic radar measurement. Traditionally, the AOA rate was computed by taking many AOA measurements, which resulted in a higher inaccuracy in the AOA rate measurement. However, recent studies^{[24, 25]} in array signal processing have achieved the direct measurement the AOA of moving targets. In addition, the theoretical accuracy of direct AOA rate measurement has been explored in Refs.[26, 27], indicating that AOA rate measurement can be directly measurend with high accuracy. As a result, AOA can be deemed an accurate measurement set for use in source localization. Combining AOA rate measurement with the three other conventional radar measurements could improve localization accuracy and resilience while reducing the number of receivers necessary. Nevertheless, despite the fine prospects, to our knowledge, there is no explicit algebraic solution in the open literature for estimating source position and velocity from hybrid TDOA, FDOA, AOA, and AOA rate measurements. Motivated by these facts, in this study, we focus on the 3-D moving source localization utilizing TDOA, FDOA, AOA and AOA rate measurements, and explore an efficient closed-form solution for source position and velocity estimation. We begin our study by establishing the TDOA, FDOA, AOA and AOA rate measurement equations. Then the quadruple hybrid nonlinear measurement equations are converted into to a linear set of equations, from which the source position and velocity are produced by applying WLS minimization. Unlike the more often used two-stage WLS estimators, the proposed algorithm only has one-stage, preventing interstage error propagation and thus being less sensitive to measurement noise, and allowing for source localization with the fewest number of receivers. The Cramer-Rao lower bound (CRLB) and error covariance matrix will be deduced to analytically investigate the efficiency of the proposed algorithm. Numerical simulations will be conducted to prove the superiority of the proposed algorithm over existing algorithms.

This paper consists of six sections. Section II establishes the TDOA, FDOA, AOA and AOA rate measurement equations for moving source localization. Section III presents the deduction of the proposed localization algorithm. Section IV includes the CRLB and error covariance matrix analysis. Section V investigates the performance of the proposed algorithm through numerical simulations, and Section VI discusses conclusions.

II.   Problem Statement

Fig.1 depicts a 3-dimensional scene in which $M$ passive receivers work together to locate an unknown moving source. The receiver positions and velocities denoted by ${{\boldsymbol{ s}}}_i = [x_i, y_i, z_i]^{{\rm{ T}}}$ ($i = 1,2,\dots,M$) and ${\dot{{\boldsymbol{ s}}}}_i = [\dot{x}_i, \dot{y}_i, \dot{z}_i]^{{\rm{ T}}}$ ($i = 1,2,\dots,M$) respectively, are assumed known. The source position and velocity denoted by ${{\boldsymbol{u}}} = [x, y, z]^{{\rm{ T}}}$ and ${\dot{{\boldsymbol{u}}}} = [\dot{x}, \dot{y}, \dot{z}]^{{\rm{ T}}}$ respectively, are unknown and to be estimated.

Figure  1.  Geometry for moving source localization

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By using the aforementioned notations, the range and range rate between the source and receiver $i$ can be expressed as

where $\begin{Vmatrix} \cdot \end{Vmatrix}$ represents the Euclidean norm. By designating receiver $1$ as the reference, the TDOA and FDOA (which are usually transformed to the range-difference and range-rate-difference for ease of presentation) with respect to receiver $i$, denoted by $r_{i1}^{\rm{o}}$, is given by

In reality, the measurements are inevitably contaminated by noises and we can construct the measured TDOA and FDOA as

where $\Delta r_{i1}$ and $\Delta\dot{r}_{i1}$ denote the TDOA and FDOA measurement noise, respectively. By defining the following vectors:

we can collect the TDOA and FDOA measurement equations in Eqs.(5) and (6) for $i = 2,3,\dots,M$ in vector form as

According to the source’s spatial geometric relationship with receiver $i$, the AOA pair $(\theta _i^{{\rm{ o}}},\varphi _i^{{\rm{ o}}})$, in which $\theta _i^{{\rm{ o}}}$ and $\varphi _i^{{\rm{ o}}}$ stands for the azimuth angle and elevation angle, can be expressed as

The time derivative of Eqs.(11) and (12) gives the azimuth and elevation angle rate equations after some trigonometric manipulations as

Then considering the measurement noises, we can formulate the measured AOA and AOA rate as

where $\Delta \theta _i$ and $\Delta \varphi _i$ are measurement noises of azimuth and elevation, $\Delta \dot \theta _i$ and $\Delta \dot \varphi _i$ are measurement noises of azimuth and elevation rate. By letting

we can stack the AOA and AOA rate measurements in Eqs.(15–18) for $i = 1,2,\dots,M$ as

To put the involved TDOA, FDOA, AOA and AOA rate measurements together for easier manipulation, we define the following vectors

and the total TDOA, FDOA, AOA and AOA measurement equations can be written compactly as

where ${{\boldsymbol{ \alpha}}}$ represents the total measurement vector, ${{\boldsymbol{ \alpha}}}^{{\rm{ o}}}$ represents its true value, and $\Delta {{\boldsymbol{ \alpha}}}$ represents the corresponding measurement noise vector. Without loss of generality, $\Delta {{\boldsymbol{ \alpha}}}$ is modelled as a zero-mean Gaussian vector with covariance matrix

Till here, the task of this paper can be succinctly summarized as follows: estimate the source position ${{\boldsymbol{u}}}$ and velocity $\dot {{\boldsymbol{u}}}$ as accurately as possible, using the noisy TDOA, FDOA, AOA and AOA rate measurements contained in the total measurement vector ${{\boldsymbol{ \alpha}}}$.

III.   Localization Algorithm

Although the task is very clear, it is far from straightforward to identify the source position ${{\boldsymbol{u}}}$ and velocity $\dot {{\boldsymbol{u}}}$ from measurement vector ${{\boldsymbol{ \alpha}}}$ since the measurements and source location parameters have a high degree of nonlinearity. This section is devoted to design a closed-form algorithm for the above-mentioned localization problem by adopting parameter transformation and WLS minimization.

Firstly, to exploit the TDOA measurement equation in Eq.(5), we rearrange it as $r_{i1}+r_1^{{\rm{ o}}} = r_i^{{\rm{ o}}}+\Delta r_{i1}$, square both sides, and retain merely one order error terms, which yields

On the surface, Eq.(32) looks just like a linear equation of source position ${{\boldsymbol{u}}}$, but actually, it is pseudo-linear in terms of source position ${{\boldsymbol{u}}}$ and nuisance parameter $r_1^{{\rm{ o}}}$, because $r_1^{{\rm{ o}}}$ is nonlinearly related to ${{\boldsymbol{u}}}$ through Eq.(1). Generally, the nuisance parameter can be removed by using two-stage minimization. However, such two-stage minimization makes the estimator more sensitive to the measurement noise. For this purpose, we eliminate the nuisance parameter $r_1^{{\rm{ o}}}$ by jointly using the AOA measurements. To achieve this, define the following vector:

Obviously, we have $r_1^{{\rm{ o}}}{{\boldsymbol{ \rho}}}^{{\rm{ o}}} = {{\boldsymbol{u}}}-{{\boldsymbol{ s}}}_1$ according to the localization geometry. Multiplying $r_1^{{\rm{ o}}}{{\boldsymbol{ \rho}}}^{{\rm{ o}}} = {{\boldsymbol{u}}}-{{\boldsymbol{ s}}}_1$ with $({{\boldsymbol{ \rho}}}^{{\rm{ o}}})^{{\rm{ T}}}$, and using the fact $({{\boldsymbol{ \rho}}}^{{\rm{ o}}})^{{\rm{ T}}}{{\boldsymbol{ \rho}}}^{{\rm{ o}}} = 1$, we have

Substituting Eq.(34) into Eq.(32), leads to

By substituting $\theta _i^{{\rm{ o}}} = \theta _i-\Delta \theta _i$ and $\varphi _i^{{\rm{ o}}} = \varphi _i-\Delta \varphi _i$ into Eq.(35), and then expanding with respect to $\theta _i$ and $\varphi _i$ through first-order Taylor-series, we arrive at

where

Collecting Eq.(36) for $i = 2,3,\dots,M$ offers a set of linear equations with regard to source position and velocity vector ${{\boldsymbol{ x}}} = [{{\boldsymbol{u}}}^{{\rm{ T}}},\dot {{\boldsymbol{u}}}^{{\rm{ T}}}]^{{\rm{ T}}}$, extracted from the TDOA measurements, as

where ${{\boldsymbol{ G}}}_r$ is a $(M-1)\times 6$ matrix and ${{\boldsymbol{ h}}}_r$ is a $(M-1)\times 1$ vector, with their inner elements given by

for $i = 2,3,\dots,M$, and zero otherwise. On the right-hand side of Eq.(40), $\Delta {{\boldsymbol{ h}}}_r$ is the error vector given by

where, ${{\boldsymbol{ B}}}_{r,r}$, ${{\boldsymbol{ B}}}_{r,\theta}$, ${{\boldsymbol{ B}}}_{r,\varphi}$ is a $(M-1)\times (M-1)$, $(M-1)\times M$, $(M-1)\times M$ matrix, respectively, whose inner elements are

for $i = 2,3,\dots,M$, and zero otherwise.

Taking the time derivative of Eq.(36) yields the following linear equation derived from FDOA measurements

where

Collecting Eq.(47) for $i = 2,3,\dots,M$ together leads to

where $\dot {{\boldsymbol{ G}}}_r$ is a $(M-1)\times 6$ matrix and $\dot{{\boldsymbol{ h}}}_r$ is a $(M-1)\times 1$ vector, with their inner elements given by

for $i = 2,3,\dots,M$, and zero otherwise. On the right-side of Eq.(51), $\Delta \dot{{\boldsymbol{ h}}}_r$ is the error vector given by

where $\dot {{\boldsymbol{ B}}}_{r,r}$, $\dot {{\boldsymbol{ B}}}_{r,\theta}$, $\dot {{\boldsymbol{ B}}}_{r,\varphi}$ are a $(M-1) \times (M-1)$, $(M-$$1) \times M$, $(M-1) \times M$ matrix, respectively, whose inner elements are

for $i = 2,3,\dots,M$, and zero otherwise.

To linearize the AOA measurement equations in Eqs.(15) and (16), we rearrange them by taking tangents from their both sides and then cross-multiplying, as

Expanding Eqs.(58) and (59) with respect to $\theta_i$ and $\varphi_i$ through first-order Taylor-series yields

where

Stacking Eqs.(60) and (61) for $i = 1,2,\dots,M$, yields the two set of linear equations from azimuth and elevation angle measurements, as follow

where ${{\boldsymbol{ G}}}_{\theta}$ and ${{\boldsymbol{ G}}}_{\varphi}$ are $M \times 6$ matrices, ${{\boldsymbol{ h}}}_{\theta}$ and ${{\boldsymbol{ h}}}_{\varphi}$ are $M \times 1$ vectors, and their inner elements are

for $i = 1,2,\dots,M$, and zero otherwise. In Eqs.(67) and (68), $\Delta {{\boldsymbol{ h}}}_{\theta}$ and $\Delta {{\boldsymbol{ h}}}_{\varphi}$ are the error vectors given by

where ${{\boldsymbol{ B}}}_{\theta,\theta}$, ${{\boldsymbol{ B}}}_{\varphi,\theta}$, ${{\boldsymbol{ B}}}_{\varphi,\varphi}$ are $M \times M$ matrices with their inner elements are given by

for $i = 1,2,\dots,M$, and zero otherwise.

The time derivative of Eqs.(60) and (61) gives the linear equations with respect to the source location parameters and the AOA rate measurements as

where

In matrix form by stacking Eqs.(78) and (79) for $i = 1,2,\dots,M$,

where $\dot {{\boldsymbol{ G}}}_{\theta}$ and $\dot {{\boldsymbol{ G}}}_{\varphi}$ are $M\times 6$ matrices, $\dot {{\boldsymbol{ h}}}_{\theta}$ and $\dot {{\boldsymbol{ h}}}_{\varphi}$ are $M\times 1$ vectors, and their inner elements are

for $i = 1,2,\dots,M$, and zero otherwise. The error vectors $\Delta \dot {{\boldsymbol{ h}}}_{\theta}$ and $\Delta \dot {{\boldsymbol{ h}}}_{\varphi}$ are given by

where $\dot{{\boldsymbol{ B}}}_{\theta,\theta}, \dot{{\boldsymbol{ B}}}_{\varphi,\theta}, \dot{{\boldsymbol{ B}}}_{\varphi,\varphi}$ are $M\times M$ matrices with their inner elements given as

Now, stacking Eq.(40), Eq.(51), Eq.(67), Eq.(68), Eq.(85), Eq.(86) in a sequence offers the overall matrix equation from TDOA, FDOA, AOA and AOA rate measurements:

where

The composite error vector $\Delta {{\boldsymbol{ h}}}$ is given as

with

Then, the WLS estimate for source position and velocity vector ${{\boldsymbol{ x}}}$ to Eq.(96), can be achieved as follows

in which ${{\boldsymbol{ W}}}$ is a weighting matrix that is symmetric. The optimal weighting matrix ${{\boldsymbol{ W}}}$ for sufficiently modest measurement noises is

It must be emphasized that, as indicated in Eq.(101), the weighting matrix ${{\boldsymbol{ W}}}$ depends on the true source locations, which is definitely unknown before exiting the algorithm implementation. To solve this dilemma, we set ${{\boldsymbol{ W}}} = {{\boldsymbol{ Q}}}^{-1}$ first in Eq.(100) to generate an initial estimate for ${{\boldsymbol{ x}}}$. This initial estimate is then substituted into Eq.(101) to compute ${{\boldsymbol{ W}}}$, from which a more accurate estimate for ${{\boldsymbol{ x}}}$ can be acquired.

IV.   Performance Analysis

We will analyze the efficiency of the proposed algorithm in this section by deducing the CRLB and theoretical covariance matrix for the proposed algorithm, and focusing on their equality under some conditions.

1   Cramer-Rao lower bound

For the 3-D moving source localization using quadruple hybrid measurements, the observation vector is ${{\boldsymbol{ \alpha}}}$, and the parameter vector of interest is ${{\boldsymbol{ x}}}$. Since observation vector ${{\boldsymbol{ \alpha}}}$ is Gaussian distributed with mean ${{\boldsymbol{ \alpha}}}^{{\rm{ o}}}$ and covariance matrix ${{\boldsymbol{ Q}}}$, the logarithm of probability density function of ${{\boldsymbol{ \alpha}}}$ conditioned on ${{\boldsymbol{ x}}}$ can be written as

where $\kappa$ is a constant.

The CRLB of ${{\boldsymbol{ x}}}$ can be produced by taking the inverse of the Fisher information matrix as follows:

and according to the definition of ${{\boldsymbol{ \alpha}}}^{{\rm{ o}}}$, the inner elements of the partial derivative $\frac{\partial {{\boldsymbol{ \alpha}}}^{{\rm{ o}}}}{\partial {\boldsymbol{ x}}}$ can be further detailed in Appendix A.

2   Error covariance matrix

Subtracting both sides of Eq.(100) by the identical equation ${{\boldsymbol{ x}}} = ({{\boldsymbol{ G}}}^{{\rm{ T}}}{{\boldsymbol{ W}}}{{\boldsymbol{ G}}})^{-1}{{\boldsymbol{ G}}}^{{\rm{ T}}}{{\boldsymbol{ W}}}{{\boldsymbol{ G}}}{{\boldsymbol{ x}}}$, leads to the source location estimation error, denoted by $\Delta {{\boldsymbol{ x}}}$, as follows

Multiplying Eq.(104) by its transpose and taking expectation, leads to the covariance matrix of source location estimate $\hat {{\boldsymbol{ x}}}$ as

Comparing Eq.(105) and Eq.(103) indicates that ${{\rm{ cov}}}({{\boldsymbol{ x}}})$ and ${{\rm{ CRLB}}}({{\boldsymbol{ x}}})$ have the same structural form. To testify their equality, by substituting Eq.(101) into Eq.(105), we further rewrite ${{\rm{ cov}}}({{\boldsymbol{ x}}})$ as

where

After some straightforward mathematical manipulations, it can be shown that when the measurement noises are sufficiently small such that the noises in ${{\boldsymbol{ G}}}$ and ${{\boldsymbol{ B}}}$ can be ignored, we will have the following approximation:

which analytically testifies that

Hence, we can deduce that the error covariance matrix of the proposed algorithm approximates the CRLB given sufficiently low measurement noise conditions.

V.   Simulation Results

We use Monte Carlo simulations in this section to demonstrate the proposed algorithm’s performance and to validate the theoretical advancements. The 3-D localization geometry is simulated, where a network of $M = 8$ receivers (unless otherwise specified) whose positions and velocities are tabulated in Table 1, is deployed to localize an unknown moving source with position $[30000, 10, 0]^{{\rm{ T}}}$ m and velocity $[200, 10, 0]^{{\rm{ T}}}$ m/s.

Table  1.  Positions and velocities of the receivers

Receiver	$x_i$(m)	$y_i$(m)	$z_i$(m)	$\dot x_i$(m/s)	$\dot y_i$(m/s)	$\dot z_i$(m/s)
Rx1	0	0	0	0	0	0
Rx2	20000	0	1500	40	20	5
Rx3	20000	20000	1000	20	−20	0
Rx4	−20000	20000	1500	−20	20	1
Rx5	−20000	0	2000	−20	−20	0
Rx6	−20000	−20000	2500	12	−20	0
Rx7	0	−20000	3000	13	0	0
Rx8	20000	−2000	3500	20	40	0

 | Show Table

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According to (30), the TDOA-FDOA-AOA-AOA rate measurement vector ${{\boldsymbol{ \alpha}}}$ is simulated by adding zero-mean Gaussian random noise vector $\Delta {{\boldsymbol{ \alpha}}}$ to its true value vector ${{\boldsymbol{ \alpha}}}^{{\rm{ o}}}$. The covariance matrix of noise vector $\Delta {{\boldsymbol{ \alpha}}}$ is set as ${{\boldsymbol{ Q}}} = {{\rm{ diag}}}(\sigma_r^2{{\boldsymbol{ I}}}_{M-1},\sigma_{\dot r}^2{{\boldsymbol{ I}}}_{M-1},\sigma_{\theta}^2{{\boldsymbol{ I}}}_{M}, \sigma_{\varphi}^2{{\boldsymbol{ I}}}_{M},$$\sigma_{\dot \theta}^2{{\boldsymbol{ I}}}_{M},\sigma_{\dot \varphi}^2{{\boldsymbol{ I}}}_{M})$, where $\sigma_r$, $\sigma_{\dot r}$, $\sigma_{\theta}$, $\sigma_{\varphi}$, $\sigma_{\dot \theta}$, $\sigma_{\dot \varphi}$ represents the standard deviations of the TDOA, FDOA, AOA and AOA rate measurement noises. Note that the aforementioned standard deviations are inversely related to the square root of signal to noise ratio (SNR) as below:

where $\gamma_r$, $\gamma_{\dot r}$, $\gamma_{\theta}$, $\gamma_{\varphi}$, $\gamma_{\dot \theta}$, and $\gamma_{\dot \varphi}$ are scaling factors related to systematic parameters including receiver bandwidth, signal bandwidth and so on. Hence, by setting $\gamma_r = 5$ m, $\gamma_{\dot r} = 0.5$ m/s, $\gamma_{\theta} = \gamma_{\varphi} = 0.1$ deg, $\gamma_{\dot \theta} = \gamma_{\dot \varphi} =$$0.01$ deg/s, we can vary the TDOA-FDOA-AOA-AOA rate measurement noise level by changing the SNR values. The root mean square error (RMSE) of the position and velocity estimation, which is calculated as follows

is used to evaluate the localization performance. $N =$$5000$ is the total number of Monte Carlo simulations, $\hat {{\boldsymbol{u}}}_n$ and $\hat {\dot {{\boldsymbol{u}}}}_n$ denotes the source position and velocity estimate obtained at the nth Monte Carlo simulation. In order to objectively examine the performance of the proposed algorithm, Noroozi’s algorithm presented in Ref.[21] which locates the moving source by employing TDOA-FDOA measurements, and Xiong’s algorithm presented in Ref.[23] which locates the moving source by utilizing TDOA-FDOA-AOA measurements, are chosen as references for comparison.

1   Performance versus measurement noise level

We now inspect the localization performance of the algorithms for different measurement noise levels depending on the SNR. To achieve this, we use these noisy TDOA, FDOA, AOA and AOA rate measurements in our simulations and investigate the localization RMSE of the algorithms with respect to SNR changes.

Fig.2 presents the RMSE performance versus SNR. It shows that the proposed algorithm generally outperforms Noroozi’s algorithm and Xiong’s algorithm over the entire SNR range, both at the RMSE and CRLB level. The reason for this is due to the fact that the proposed algorithm incorporates extra AOA rate measurements apart from TDOA, FDOA and AOA measurements, thereby leading to an increase in the number of equations and an improvement in localization accuracy. Under sufficiently small measurement noise levels (${{\rm{ SNR}}}\geq-35$ dB), the proposed algorithm accomplishes the CRLB. As measurement noise level rises, The proposed algorithm’s RMSE differs from the CRLB at ${{\rm{ SNR}}}\leq-35$ dB. And the departure from the CRLB at high noise levels, known as thresholding phenomenon, is largely anticipated, since the proposed algorithm neglects the second order and higher order error terms. By contrast, even at high noise levels, the proposed algorithm still presents the smallest RMSEs among the algorithms. With additional AOA measurements, Xiong’s algorithm provides a relatively smaller RMSE compared to Noroozi’s algorithm, both at the RMSE and CRLB level. However, compared with the proposed algorithm, the localization accuracy improvement brought by the use of AOA measurements in Xiong’s algorithm is not sufficiently impressive.

Figure  2.  Performance comparison versus measurement noise level

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2   Performance versus receiver-source distance

Next, we inspect the localization performance over the receiver-source distance (defined as the distance between the source and reference receiver) using the receivers presented in Table 1. Assuming the source is initially located at position ${{\boldsymbol{u}}} = [1000, 10, 0]^{{\rm{ T}}}$ m, and is moving away from the centre of the receiver network with velocity $\dot {{\boldsymbol{u}}} = [300, 10, 0]^{{\rm{ T}}}$ m/s. The measurement noise level is kept at ${{\rm{ SNR}}} = -35$ dB. The rest of the simulation parameters are the same as they were in Section V.1.

The RMSE performance is plotted against the receiver-source distance in Fig.3. In this case, the relative performance is comparable to that seen in Fig.2, once again demonstrating that the proposed algorithm outperforms Noroozi’s algorithm and Xiong’s algorithm. It is also observed from Fig.3 that the algorithms' localization performance declines as the distance between the receiver and the source grows. This result is in line with previous findings about the effect of receiver-source geometry on localization performance. Furthermore, the results indicate the proposed algorithm achieves the CRLB performance over the moderate receiver-source distance region. When the receiver-source distance is approaching zero, the RMSE curves of the three algorithms are slightly above their respective CRLBs. This is because the relative measurement noise becomes very large when the source is close to the receiver network. Another phenomenon worthy of note is that, when the source is far from the receiver network, the proposed algorithm and Xiong’s algorithm slightly depart from their respective CRLBs. A plausible explanation is that, when the receiver-source distance to the array aperture ratio is large, the AOA measurement lines incorporated in these two algorithms become approximately parallel to each other and do not intersect each other.

Figure  3.  Performance comparison versus receiver-source distance

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3   Performance versus number of receivers

Then, we remove the receivers in the order of number, from Rx7 to Rx1, so as to investigate the effect of the number of receivers on the localization performance. The measurement noise level is kept at ${{\rm{ SNR}}} = -30$ dB. The rest of the simulation parameters are the same as they were in Section V.1.

The RMSE performance of the algorithms is compared in Fig.4 with the number of receivers ranging from 2 to 8. As expected, we observe that the proposed algorithm exhibits smaller RMSEs compared with Noroozi’s algorithm and Xiong’s algorithm for different numbers of receivers. As the number of receivers grows, the localization RMSEs of the algorithms decline, but the decline rate tends to zero. Besides, the main advantage of the proposed algorithm over existing algorithms presented in Fig.4 is that the proposed algorithm requires the least minimum number of receivers to locate a moving source. More specifically, as shown by the blue dotted CRLB curves, TDOA-FDOA-based localization method requires, notionally at least, 4 receivers to determine the source position and velocity, but Noroozi’s algorithm (marked by blue squares) requires at least 5 receivers since Noroozi’s algorithm has to estimate the source position, velocity, and two introduced nuisance parameters in the first WLS stage; from the black dotted CRLB curves, it can be seen that TDOA-FDOA-AOA-based localization method requires theoretically at least 2 receivers to determine source position and 4 receivers to determine source velocity, but owing to the introduced nuisance parameters, at least 3 receivers are required for source position estimation and 5 receivers for source velocity estimation in Xiong’s algorithm (marked by black cross). By contrast, as can be seen from the (red dotted) CRLB curves and the RMSE curves (marked by red circle), the proposed algorithm requires only at least 2 receivers to estimate the source position and velocity, since the proposed algorithm incorporates extra AOA rate measurements and does not need to introduce any nuisance parameters.

Figure  4.  Performance comparison versus number of receivers

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VI.   Conclusions

In this paper, moving source localization jointly using TDOA, FDOA, AOA and AOA rate measurements gathered from a number of spatially distributed receivers is investigated. The TDOA, FDOA, AOA and AOA rate measurement equations were firstly established according to the source’s spatial geometric relationship with the receivers. Then an efficient closed-form algorithm for 3-D moving source localization utilizing the quadruple hybrid measurements was developed, without the need for introduction of any nuisance parameters. In contrast to existing two-stage WLS estimators, the proposed algorithm, by jointly using the quadruple hybrid measurements, requires merely one-stage, allowing for source localization with the fewest receivers necessary and outperforming existing algorithms significantly. The CRLB for estimating the source position and velocity is derived given small Gaussian measurement noises, and the efficiency of the proposed algorithm is verified by attaining the CRLB performance. These theoretical developments were corroborated by numerical simulations.

Appendix A

According to the definition of ${{\boldsymbol{ \alpha}}}^{{\rm{ o}}}$, the inner elements of the partial derivative $\frac{\partial{{\boldsymbol{ \alpha}}}^{{\rm{ o}}}}{\partial{{\boldsymbol{ x}}}}$ can be expressed as

where $d_i^{{\rm{ o}}} = \sqrt{(x-x_i)^2+(y-y_i)^2}$.