A two-step weighted least-squares method for joint estimation of source and sensor locations:A general framework

Ding WANG,Jiexin YIN,*,To TANG,Ruirui LIU,Zhidong WU

aNational Digital Switching System Engineering and Technology Research Center,Zhengzhou 450002,China

bZhengzhou Institute of Information Science and Technology,Zhengzhou 450002,China

Abstract It is well known that the Two-step Weighted Least-Squares(TWLS)is a widely used method for source localization and sensor position refinement.For this reason,we propose a unified framework of the TWLS method for joint estimation of multiple disjoint sources and sensor locations in this paper.Unlike some existing works,the presented method is based on more general measurement model,and therefore it can be applied to many different localization scenarios.Besides,it does not have the initialization and local convergence problem.The closed-form expression for the covariance matrix of the proposed TWLS estimator is also derived by exploiting the first-order perturbation analysis.Moreover,the estimation accuracy of the TWLS method is shown analytically to achieve the Crame′r-Rao Bound(CRB)before the threshold effect takes place.The theoretical analysis is also performed in a common mathematical framework,rather than aiming at some specific signal metrics.Finally,two numerical experiments are performed to support the theoretical development in this paper.

KEYWORDS Crame′r-Rao Bound(CRB);Disjoint sources;General framework;Performance analysis;Sensor position refinement;Source localization;Weighted least-squares

1.Introduction

Source localization has become a very important and fundamental research topic in GPS,radar,sonar,navigation,microphone arrays, sensor networks, and wireless communications.1-6Therefore,it has attracted considerable attention over the past few years.Localization of a stationary or moving emitter is usually accomplished by using several spatially distributed sensors,which can intercept the radiated signal from the source.The positioning parameters are extracted next to locate the source.In general,the parameters used for localization cover Direction of Arrival(DOA),7-10Time of Arrival(TOA),11-17Time Difference of Arrival(TDOA),18-27Frequency of Arrival(FOA),28Frequency Difference of Arrival(FDOA),29-38Received Signal Strength(RSS),39,40Gain Ratios of Arrival(GROA),41,42etc.

Indeed,emitter localization is not a trivial task due to the nonlinear nature of the estimation problem.During the past few decades,a number of localization methods become available in the literature.Some of them are iterative algorithms that require careful initial solution guesses.Perhaps the most representative iterative method is Taylor-Series(TS)algorithm,8,22,30,33,43because it can be applicable to any localization scenario as long as there are enough measurements.The others can provide closed-form solutions,and hence they are more computationally efficient and attractive.Undoubtedly,the best-known closed-form method is Two-step Weighted Least-Squares (TWLS) method,13,15-18,24,26,29,31,34,38,40-42which works in two stages.In the first stage,the nonlinear measurement equations are transformed into a set of pseudolinear ones by introducing some auxiliary variables,so that a closed-form estimate of the unknowns can be obtained by Weighted Least-Squares(WLS)method.In the second stage,the relationship between the source location and the instrumental variables is explored to refine the source position estimate through another WLS computation.It is worthy to mention that although the TWLS method requires less computational effort than the iterative algorithms,its estimated accuracy can still achieve the Crame′r-Rao Bound(CRB)under moderate noise level.Due to the fact that the TWLS method has wide application and excellent performance,we focus on this localization method in this paper.

Generally,there are two sources of errors that can degrade the estimation accuracy of the location methods.The first is the measurement noise of the positioning parameters and the second is the uncertainties in sensor locations.It must be noted that the latter is commonly ignored by the estimator.In Refs.31,44,the source location Mean Square Error(MSE)is derived when the location estimator assumes no sensor location error but in fact they exist.The analysis result demonstrates that the sensor position uncertainties can seriously deteriorate the source location accuracy.As a consequence,the second source of error should also be taken into account in order to achieve satisfactory estimation precision.In Refs.13,16,31,34,42,several robust TWLS methods are proposed,aiming at different signal metrics,such as TOA,TDOA/FDOA and TDOA/GROA.All of them incorporate the statistical knowledge of the sensor location errors into the position estimation procedure,and therefore they are able to yield asymptotically efficient solutions for source locations.Nevertheless,it is worth pointing out that these algorithms cannot improve the position information of the sensors.The refinement of sensor locations is indeed necessary because it can achieve better localization performance and tolerate higher noise level before the threshold effect starts to occur.For this purpose,the joint estimation of the unknown source locations and the inaccurate sensor positions should be considered.In Refs.15,24,26,38,some improved TWLS methods are proposed,which can locate multiple disjoint sources and refine erroneous sensor positions simultaneously.Disjoint sources refer to the sources whose emissions do not interfere with one other,where the disjointedness can be achieved,for example,by confining the source signals to disjoint frequency bands and/or time intervals.24,34Note that the positioning parameters from different sources are subject to the same displacements in the sensor positions.Therefore,the locations of multiple sources need to be jointly estimated to achieve the optimal statistical performance.

It is noteworthy that all the above-mentioned TWLS methods are proposed for some specific signal metrics,thus leading to the lack of a general framework.For this reason,we develop a unified framework of the TWLS method that can locate multiple disjoint sources and refine the erroneous sensor positions simultaneously.The computational procedure is based on a more generic measurement model,and hence it is applicable to many different location scenarios.Compared to the iterative algorithms,the proposed method does not suffer from divergence and local convergence problem.Additionally,the compact expression for the covariance matrix of the proposed TWLS estimator is also deduced by making use of the first order perturbation analysis,which ignores the second-and higher-order error terms.Moreover,the estimated accuracy of the presented closed-form solution is proved to achieve the CRB at moderate noise level before the threshold effect occurs.Different from the previous work,our theoretical analysis is performed in a common mathematical framework,rather than aiming at some specific signal metrics.Finally,two examples are given to illustrate how to utilize the proposed TWLS method to locate the sources and refine the sensor positions.One uses the DOA/TDOA/FDOA measurements and the other is based on the DOA/TDOA/GROA parameters.The experiment results support the theoretical development in this paper.

The organization of this paper is as follows.Section 3 describes the general measurement model for source localization and formulates the problem considered.In Section 4,the TWLS closed-form solution for jointly estimating the source and sensor locations is derived.Section 5 provides the compact expression for the covariance matrix of the TWLS estimator and proves that it can reach the CRB accuracy before the threshold effect occurs.In Section 6,two examples are given to illustrate how to exploit the proposed TWLS method for source localization.Numerical results are shown in Section 7 in order to verify the theoretical development in this paper.Conclusions are offered in Section 8.The proofs of the main results are provided in the Appendixes.

2.Notational conventions

In this paper,lowercase and uppercase boldface letters are used to denote vectors and matrices,respectively.In addition,the conventions shown in Table 1 are used throughout this paper.

3.Measurement model and problem formulation

3.1.Nonlinear measurement model

We assume that there existDdisjoint sources that need to be located.In ideal situation,the location observation equation corresponding to thedth source can be expressed in a generic form as

Table 1 Mathematical notation explanation.

where zd0∈ Rl1×1is the noiseless measurement vector,l1is length of zd0;pd∈ Rl2×1denotes the position and/or velocity vector of thedth source,l2is length of pd;w ∈ Rl3×1is the system parameter vector whose elements are the sensor positions and/or velocities,l3is length of w;h（·，·）is the nonlinear function that relies on the specific localization scenario.Note that the general model Eq.(1)is refined from lots of localization measurements.It can be readily checked that all the source localization equations can be expressed as in Eq.(1)and,hence,Eq.(1)is a more general measurement model.

Obviously,if the vectors zd0and w are available,the localization problem can be considered to solve a set of nonlinear equations.However,the accurate values of these vectors are impossible to obtain in practice.First,it is assumed that the noisy version of zd0is denoted as zd.Then,we have

where φ is the noise vector that follows zero-mean Gaussian distribution with covariance matrix Φ =EφφT（ ）.Moreover,φ is uncorrelated with ｛εd｝1≤d≤D.

3.2.Pseudo-linear measurement model

For some localization scenarios,Eq.(1)can be converted into the following pseudo-linear model:

where af（zd0，w）∈ Rl1×1is the measurement vector;Bf（zd0，w）∈ Rl1×（l2+l4）is the coefficient matrix,l4is number of instrumental variables;tf，d=gf（pd，w）∈ R（l2+l4）×1is a new parameter vector that is dependent on pdand w.Besides,the vector function gf（pd，w）can be written as

where J ∈ Rl2×l3is a known and constant matrix;γ（ pd，w）∈ Rl4×1consists of all thel4instrumental variables,which are relevant with pdand w in a nonlinear manner.It is important to highlight that the reason for introducing the instrumental variables is to get the linear equations in Eq.(4).

Remark 1.Note that Eq.(4)is the first pseudo-linear equation in the TWLS method.It is a universal model because it is suitable for many signal metrics(e.g.,DOA,TDOA,TOA,FDOA,FOA,GROA,RSS,etc.),which are used for source location.In Section 6,we explain how to transform some of these nonlinear measurement equations into the pseudo-linear ones given in Eq.(4)based on two examples.

It can be readily seen that all the equations in Eq.(4)depend on the common system parameter w.Therefore,if vector w is not known accurately,we should combine these equations to perform joint estimation of all the position vectors｛ud｝1≤d≤Dand the system parameter w.This cooperative processing can result in better estimation performance compared to the case when the sources are located separately.

Gathering all theDequations in Eq.(4)together leads to

Remark 2.It is noteworthy that Eq.(10)is the second pseudolinear equation in the TWLS method.It is also a general model that can be applied to many localization problems.In Section 6,we provide two specific examples to illustrate how to obtain this pseudo-linear equation.

The problem studied here can be briefly described as follows:Given the observation vectors ｛zd｝1≤d≤Dand the noisy system parameter v, find an estimate of ｛pd｝1≤d≤D(or p)and w as accurate as possible based on the two pseudo-linear Eqs.(6)and(10).

4.Two-step weighted least-squares method for parameter estimation

The objective of this section is to present the TWLS method,which can estimate p and w simultaneously.Different from the existing approaches,the proposed method is derived in a more general framework,which is not limited to some specific signal metrics.

4.1.Weighted least-squares solution in the first stage

Before proceeding,two remarks are concluded.

Remark 4.Although Eq.(19)also provides the estimate for system parameter w,its estimation variance cannot attain the CRB(see Appendix B).Therefore,in the second stage,this estimation result needs to be further refined in order to achieve the optimal statistical performance.

4.2.Weighted least-squares solution in the second stage

In the second stage of the presented approach,an alternative WLS estimator is formed based on the vector Eq.(10),which originates from the dependence of the instrumental variables inonand w.Note that the expressions of as（·）and Bs（·）in Eq.(10)are available,but vectorcannot be accurately estimated in the first step.As a result,we need to introduce the following error vector:

As mentioned in Remark 4,the solution^wfas the estimation of w is not asymptotically efficient and,therefore,the unknown parameters in the WLS estimator deduced here should also include vector w.The corresponding optimization model is formulated as

4.3.Summary of computational procedure

At this point,it is possible to summarize the proposed algorithm step by step as follows:

Remark 7.It is clear that the algorithm described above does not require iteration or initialization and does not have the divergence problem.Table 2 summarizes the computational complexity of the proposed TWLS method in terms of the number of multiplication operations.

Although there are many computing units in the proposed method,the total amount of computation is not high because the values ofD,l1,l2,l3,andl4are small.

Table 2 Complexity of proposed TWLS method.

Remark 8.It is well known that the flip ambiguity occurs in wireless sensor networks.It may affect the reliability of the localization of the whole or a major portion of the sensor network.Hence,it is very meaningful to study how to reduce the probability of flip ambiguity occurrence.In Ref.47,a robustness criterion,called Orthogonal Projection Algorithm(OPA),to detect flip ambiguity for range-based nodes multilateration localization method is proposed.In Ref.48,a robustness criterion to detect flip ambiguities in neighborhood geometries is established,and it is also embedded in localization algorithms to enhance the reliability of the localization of the sensor network.However,it is worth noting that this paper mainly focuses on the localization for multiple disjoint emitting sources.All the observations are used for location at the same time.Therefore,the probability that flip ambiguity happens becomes small in this situation.On the other hand,there also exist some useful ways to eliminate flip ambiguity.In particular,sometimes we can get the rough area that the sources may be situated;we can choose the sensor locations that are not collinear;we can also incorporate more measurements(e.g.,DOA,FOA,FDOA,etc.)to locate the emitters.Additionally,in our future study,we intend to combine the flip ambiguity detection with the proposed TWLS method to further improve the robustness of the source localization.

Remark 9.Notice thatare the final TWLS solutions for system parameter and all source locations,respectively.Hereafter,the estimated value^wsis also denoted by^wtwlsfor notational unification.In the following section,it is proved that the TWLS solutions are able to achieve the CRB performance before the threshold effect occurs.

5.Statistical performance analysis

This section is devoted to the derivation of the explicit expression for the covariance matrix of the above TWLS solution.The first-order perturbation analysis is performed which neglects the second-and higher-order error terms.Additionally,we also prove analytically that the Mean Square Error(MSE)of the TWLS solution equals the corresponding CRB for Gaussian data model at moderate noise level.It is noteworthy that in contrast to some existing analysis method,our performance analysis is conducted in a unified mathematical framework,which is not limited to a specific measurement.

5.1.Covariance matrix of TWLS solution

The derivation starts from the functional relationship

which,combined with Eq.(34),leads to

Additionally,substituting Eq.(28)into Eq.(36)produces

Substituting Eq.(20)into Eq.(37)yields

Our proof consists of two parts.In the first part,the following result is proved.

The diagonal elements of covcan be used to predict the estimation variances of the proposed TWLS estimator.In the following subsection,we show that the covariance matrix given by Eq.(38)equals the CRB matrix.

5.2.Asymptotical efficiency of TWLS solution

In order to prove that the above TWLS solution is asymptotically efficient,it is necessary to obtain the corresponding CRB first.It follows from the results in Refs.24,34that

After substituting Eqs.(43)and(44)into Eqs.(37)and(41)holds true.□

In the second part,we obtain the desired result based on Proposition 1.

which,combined with Eq.(20),implies

Putting Eq.(46)and Proposition 1 together produces

On the other hand,inserting zd0=h（ pd，w）into Eq.(4)yields

6.Two localization examples

In this section,two specific localization examples are detailed in order to explain how to exploit the above TWLS method to locate multiple disjoint sources and improve the sensor positions simultaneously.In the first scenario,we use the DOA/TDOA/FDOA measurements for Two-Dimensional(2D)localization;in the second one,we focus on Three-Dimensional(3D)geometrical space and combine the DOA/TDOA/GROA parameters to locate the sources.

6.1.Source localization using DOA/TDOA/FDOA measurements

which completes the proof.□

Up to this point,the asymptotical efficiency of the proposed TWLS solution is proved.Moreover,it can be seen that our theoretical analysis is performed in a generic mathematical framework,and therefore it can be applied to many different localization scenarios.Nevertheless,it must be emphasized that the theoretical analysis described above is based on the first-order error approach,which ignores the second-and higher-order error terms.Therefore,when the noise variance exceeds a certain value,its estimation performance may depart abruptly from the CRB.This phenomenon is called threshold effect,which is due to the nonlinear nature of the problem.

Remark 10.In the light of the signal parameter estimation theory,45,46,49,50it can be found that the trace of E is generally inversely proportional to Signal-to-Noise Ratio(SNR).Then,it immediately follows from Eq.(59)that the trace ofNote that the vector w is not known exactly and only the noisy version of it,denoted by u,is available in practice.The TDOA and FDOA measurements with respect to the reference sensor,say sensor 1,are extracted from the received signals.Note that the TDOA and FDOA measurements are equivalent to the range difference and range difference rate measurements,respectively.In addition,the DOA parameters can also be measured for sensors 2-M.Thus,the relevant observation equations can be written as

where 2 ≤m≤M;1 ≤d≤D,θd，mis the azimuth angle;μd，mis the range difference;˙μd，mis the range difference rate.Define the following vectors:

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and then the noiseless measurement vector associated with thedth source is given by

Gathering all the measurements in a 3（M-1）D× 1 vector leads to

To employ the presented TWLS localization method,we must transform the nonlinear equations in Eq.(60)into the pseudo-linear ones by introducing some nuisance parameters.According to the first equation in Eq.(60),we have

Taking the time derivation of the third equation in Eq.(66)yields

It is straightforward to deduce from the second equation in Eq.(76)that

Remark 11.In order to simplify the problem,we just study the 2D localization in this subsection.The reason is that when the 3D scenario is considered,there will be too many unknowns that need to be estimated if the velocities for both the sources and the sensors are taken into account.However,the localization method stated above can be straightforwardly generalized to the 3D localization case because all the DOA/TDOA/FDOA measurement equations can be transformed into the pseudo-linear ones in the 3D coordination system.9,29,31

6.2.Source localization using DOA/TDOA/GROA measurements

In order to exploit the proposed TWLS method for source localization,we must transform the nonlinear equations in Eq.(78)into the pseudo-linear ones by introducing some auxiliary variables.It follows from the first equation in Eq.(78)that

Table 3 Nominal positions and velocities of sensors.

Fig.1 RMSEs of estimated position and velocity for sources versus measurement error parameter σ1for DOA/TDOA/FDOA localization.

where

It can be checked from the third equation in Eq.(78)that μd，m=||pd-wm||2-||pd-w1||2

where

Finally,from the fourth equation in Eq.(78),we arrive at

Fig.2 RMSEs of estimated position and velocity for sensor versus measurement error parameter σ1for DOA/TDOA/FDOA localization.

where

Combining Eqs.(82)-(89)yields the following pseudolinear vector equation:

where

Fig.3 RMSEs of estimated position and velocity for sources versus system error parameter σ2for DOA/TDOA/FDOA localization.

Fig.4 RMSEs of estimated position and velocity for sensor versus system error parameter σ2for DOA/TDOA/FDOA localization.

It can be readily verified from the second equation in Eq.(96)that

Remark 12.Although this subsection assumes that the sensor is static,the proposed method can be applied to the case of moving sensor if only the position parameters need to be determined.The reason is that none of the DOA/TDOA/GROA measurements depends on the velocity of the sensors

Table 4 Nominal positions of sensors.

Remark 13.Besides the two localization examples given in this section,the presented unified framework for the TWLS method can also be applied for many other signal metrics,as long as the pseudo-linear Eqs.(6)and(10)can be obtained.

7.Simulation results

Fig.5 RMSEs of estimated position for sources versus measurement error parameter σ1 for DOA/TDOA/GROA localization.

This section presents a variety of Monte Carlo simulations to reveal the behavior of the proposed TWLS method.The Root-Mean-Square-Error(RMSE)of parameter estimate is used as performance metric.Our simulation results are obtained through a total of 5000 independent trials.The proposed solution is implemented using the procedure described in Section 4.3.Its estimation accuracy is compared to the corresponding CRB given by Eq.(39)as well as the TS algorithm.Note that the TS algorithm requires initial solution guess,which is not always an easy task.For a comprehensive comparison,in the following,the TS algorithm is initialized in two ways:(A)the initial guess is chosen randomly;(B)the initial solution is equal to the true value.On the other hand,in order to show the performance gain due to the cooperative localization for multiple sources,the CRB given by Eq.(39)is also compared to the CRB for the case of locating the sources independently.

7.1.Numerical results for DOA/TDOA/FDOA source localization

In the first experiment,we fix σw（a）=5 m, σw（b）=0.03 m/s and set σDOA=0.001σ1rad, σTDOA=0.1σ1/cs,σFDOA=0.001f0σ1/cHz,where σ1ranges from 1 to 10,cis the signal propagation speed andf0is the signal carrier frequency.Note that σ1is called measurement error parameter.Fig.1(a),(c)and(e)respectively display the RMSE of position estimates for the three sources versus σ1.Fig.1(b),(d)and(f)respectively depict the RMSE of velocity estimates for the three sources versus σ1.Fig.2 plot the RMSE of location and velocity estimates for the sensors as a function of σ1.

Fig.6 RMSEs of estimated position for sensor versus measurement error parameter σ1for DOA/TDOA/GROA localization.

In the second experiment,we fix σDOA=0.005 rad,σTDOA=0.5/cm,σFDOA=0.005f0/cHz and set σw（a）= σ2m,=0.01σ2m/s,where σ2varies from 1 to 10 and it is called system error parameter.Fig.3(a),(c)and(e)respectively show the RMSE of position estimates for the three sources versus σ2.Fig.3(b),(d)and(f)respectively depict the RMSE of velocity estimates for the three sources versus σ2.Fig.4 illustrates the RMSE of location and velocity estimates for the sensors as a function of σ2.

Fig.7 RMSEs of estimated position for sources as a function of system error parameter σ2for DOA/TDOA/GROA localization.

It can be seen from Figs.1-4 that the proposed TWLS solution is able to reach the CRB accuracy given by Eq(39))under moderate noise level.Therefore,the effectiveness of the theoretical analysis in Section 5 is confirmed.Besides,the TS algorithm can also yield an asymptotically efficient solution if it is initialized with the true value.However,when the starting point of the TS algorithm is chosen randomly,its performance departs from the CRB earlier than the TWLS algorithm.In fact,it is hard to find a good initial guess for the TS algorithm because the pseudo-linear vector equation is not available in this algorithm.Additionally,we can also observe from Figs.2 and 4 that the proposed method can improve the estimation accuracy for the sensor locations and velocities in comparison to their prior RMSE.On the other hand,by comparing the two kinds of CRB,we can find that the performance improvement due to the joint localization for multiple sources is notable.Moreover,it can be found from Fig.3 that the cooperation gain increases as σ2increases.The reason stems from the fact that,as σ2increases,the correlation between the measurements related to different sources becomes larger,and hence the advantage of cooperative treatment becomes more remarkable.

7.2.Numerical results for DOA/TDOA/GROA source localization

Fig.8 RMSEs of estimated position for sensor as a function of system error parameter σ2for DOA/TDOA/GROA localization.

Fig.9 RMSEs of estimated position for sources versus measurement error parameter σ1for large scale network.

In the first experiment,we fix σw=10 m and set σDOA=0.001σ1rad, σTDOA=0.5σ1/cs, σGROA=0.005σ1,where σ1varies from 1 to 10 and it is called measurement error parameter.Fig.5 respectively display the RMSE of position estimates for the three sources versus σ1.Fig.6 illustrates the RMSE of location estimates for the sensors as a function of σ1.

In the second experiment,we fix σDOA=0.005 rad,σTDOA=5/cm, σGROA=0.005 and set σw= σ2m,where σ2ranges from 1 to 10 and it is called system error parameter.Fig.7 respectively show the RMSE of position estimates for the three sources versus σ2.Fig.8 plots the RMSE of location estimates for the sensors as a function of σ2.

It is easy to see from Figs.5-8 that the proposed TWLS method can attain the CRB accuracy computed by Eq.(39)under moderate noise level,which supports the theoretical comparison between the performance of the proposed estimator and the CRB again.Moreover,the newly presented solution still has a higher level of noise tolerance before the threshold effect occurs than the TS algorithm with random initialization.The advantage of cooperation localization is also noticeable by comparing the two kinds of CRB.On the other hand,it can also be observed from Figs.6 and8 that,compared to the prior measurements of the sensor positions,the proposed TWLS method can provide a solution with a smaller RMSE.

Finally,we would like to verify that the proposed algorithm is suitable for the wireless location system which has more sensors than the number previously used.The simulation parameters are the same as those used to produce Figs.5 and 6,except that the number and the nominal positions of the sensors are changed.The localization scenario consists of 16 receivers whose positions are tabulated in Table 5.Fig.9 respectively display the RMSE of position estimates for the three sources with respect to σ1.Fig.10 plots the RMSE of estimates position for the sensors as a function of σ1.

Fig.10 RMSEs of estimated position for sensor versus measurement error parameter σ1for large scale network.

Table 5 Nominal positions of sensors(large scale network).

Figs.9 and 10 clearly show that,for large scale network,the presented TWLS solution also can achieve the CRB accuracy under a moderate noise level.This observation is not unexpected because the proposed theoretical framework of the TWLS method is independent of the number of the sensors.

8.Conclusions

In this paper,we propose a unified framework of the TWLS method that can locate multiple disjoint sources and refine the erroneous sensor positions simultaneously.Unlike the existing localization methods,the presented computational procedure is based on a more generic measurement model,and therefore it is applicable for many different localization scenarios.Besides,the proposed method does not have the initialization and local convergence problem because it can provide a closed-form solution.Additionally,the compact expression for the covariance matrix of the proposed TWLS estimator is also deduced by exploiting the first-order perturbation analysis,which only retains the first-order error terms.Moreover,the presented closed-form solution is proved to reach the CRB accuracy at moderate noise level before the threshold effect takes place.It is noteworthy that our theoretical analysis is conducted based on a common mathematical framework,rather than aiming at some specific signal metrics.Finally,two examples are presented to explain how to utilize the proposed TWLS method to locate the sources and refine the sensor positions.One uses the DOA/TDOA/FDOA measurements and the other is based on the DOA/TDOA/GROA parameters.Numerical results show the good performance of the proposed method,and also validate the performance analysis in this paper.

Acknowledgements

This study was co-supported by the National Natural Science Foundation of China (Nos.61201381,61401513 and 61772548),the China Postdoctoral Science Foundation(No.2016M592989),the Self-Topic Foundation of Information Engineering University,China(No.2016600701),and the Outstanding Youth Foundation of Information Engineering University,China(No.2016603201).

Appendix A.

It follows from Eq.(19)that

Appendix B.

Before proceeding,it is necessary to prove the following result.

Lemma 1:.It is assumed thatX1∈ Rn×nis positive definite andX2∈ Rn×mhave column full rank.Then,it follows

Hence,Eq.(B1)holds true.□

Combining Propositions 1 and 2 leads to

where Gf，1（p，w）and Gf，2（p，w）are defined in Proposition 1.It can be seen from Eqs.(B1)and(B4)that

which implies

At this point,the proof is completed.

Appendix C.

With the use of the partitioned matrix inversion formula,it can be shown from Eq.(18)that

In addition,using the matrix inversion lemma,it can be verified that

Substituting Eq.(C2)into Eq.(C1)leads to

which completes the proof.

Appendix D.

It can be easily verified from the first equality in Eq.(7)that

Besides,according to the second equality in Eq.(7),we obtain the following two equations:

Combining Eqs.(D1)-(D4)proves Eq.(54).

Appendix E.

First,it can be checked from Eqs.(52)and(53)that

Combining Eqs.(65),(67)and(69)yields

Besides,using Eqs.(65),(67)and(69)and the second equality in Eq.(71),it can be verified that

On the other hand,it can be shown from Eqs.(25),(75)and(76)that

where

Appendix F.

First,it can be seen from Eqs.(52)and(53)that

Using Eq.(80)and the first equality in Eq.(91),it is straightforward to show that

From Eqs.(83),(85),(87)and(89),it can be deduced that where

Besides,it can be checked from Eqs.(83),(85),(87)and(89)and the second equality in Eq.(91)that

On the other hand,from Eqs.(25),(95)and(96),we have