Yongsheng ZHAO,Dexiu HU,Yongjun ZHAO,Zhixin LIU

National Digital Switching System Engineering and Technological Research Center(NDSC),Zhengzhou 450000,China

KEYWORDS Bistatic range;Bistatic range rate;Location uncertainty;MIMO radar;Target localization;Two-step weighted least squares

Abstract In this paper,the problem of moving target localization from Bistatic Range(BR)and Bistatic Range Rate(BRR)measurements in a Multiple-Input Multiple-Output(MIMO)radar system having widely separated antennas is investigated.We consider a practically motivated scenario,where the accurate knowledge of transmitter and receiver locations is not known and only the nominal values are available for processing.With the transmitter and receiver location uncertainties,which are usually neglected in MIMO radar systems by prior studies,taken into account in the measurement model,we develop a novel algebraic solution to reduce the estimation error for moving target localization.The proposed algorithm is based on the pseudolinear set of equations and two-step weighted least squares estimation.The Cramer-Rao Lower Bound(CRLB)is derived in the presence of transmitter and receiver location uncertainties.Theoretical accuracy analysis demonstrates that the proposed solution attains the CRLB,and numerical examples show that the proposed solution achieves significant performance improvement over the existing algorithms.?2019 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

In recent decades,radars deployed on multiple ground-based or airborne moving platforms1-4have drawn a great deal of attention for their applications,such as Unmanned Aerial Vehicles(UAVs)based radars5and other potential applications,6-9in civilian and military fields.With multiple sensors,the system can be implemented as a Multi-Input Multi-Output(MIMO)radar.A MIMO radar detects and locates targets of interest by utilizing multiple Transmit/Receive(TX/RX)antennas to transmit and receive signals of different waveforms.10Such systems fall into two main architectures,i.e.with colocated11and widely separated antennas.12The latter architecture has been shown to offer significant performance improvement owing to the spatial and signal diversities.In this study,we consider the target localization using the latter architecture.

Target localization is a long-lasting classic research topic and has gained much attention in academic field.In distributed MIMO radar systems,the time delay and Doppler shift are two very commonly used measurements for target localization.13,14In contrast to the definition in source localization issues,where the time delay has a direct correspondence with the Range Difference(RD)between two propagation paths and the Doppler shift has a direct correspondence with the Range Rate Difference (RRD),15in distributed MIMO radar community,the time delay corresponds to the Bistatic Range(BR)and the Doppler shift corresponds to the Bistatic Range Rate(BRR).16In the past decades,the RD-and-RRDbased source localization problem has been considerably studied,17-22but BR-and-BRR-based target localization problem is far from straightforward due to its higher nonlinearity implied in the measurement equations and relatively less studies are reported in literature.This paper focuses on the target localization problem by using BR and BRR measurements in distributed MIMO radar systems.

Over the past few years,some efficient methods have been explored to handle this challenging problem.Algebraic solution methods have always been compelling to researchers due to their advantages of independence on initial estimate and computational efficiency.Inspired by the two-stage processing method of Ho and Xu,17Du and Wei23proposed an algebraic solution for moving target localization in distributed MIMO radar systems.This method divides the measurements into groups,measurements in each group correspond to a certain transmitter,then a target location estimate is obtained from each group using Weighted Least Squares(WLS)estimation,and finally these estimates are weighted to produce a refined estimate.However,this method is not optimal if the measurement noises are correlated.Using a different way of converting nonlinear equations to linear ones, Yang and Chun24developed an improved algebraic solution,where the BR and BRR measurements are converted to the RD and RRD ones by choosing one of the transmitters or receivers as the reference.This method need not divide the measurements into groups,but theoretical analysis and numerical simulations demonstrate that the conversion from BRs and BRRs to RDs and RRDs causes a loss in the localization accuracy.More recently,Zhao et al.25developed a novel algebraic solution for moving target localization in distributed MIMO radar systems using BR and BRR measurements.This developed solution is also inspired by the 2WLS idea,17but in contrast to Refs.23,24it is shown analytically and numerically to be approximately unbiased and its variance reaches the Cramer-Rao Lower Bound(CRLB)in the general cases with correlated noise variances,because it does not need to group or convert the BR and BRR measurements to the RD and RRD ones.Nevertheless,the above studies are based on the ideal assumption that the accurate knowledge of transmitter and receiver locations is known,which is not realistic.26,27In practice,the transmitter and receiver locations are inevitably contaminated by errors to some extent,and the location errors are often nonnegligible,especially when the transmitters and/or receivers are moving,1-5such as UAVs based radars. Apparently, the existence of the transmitter and receiver location errors will deteriorate the localization performance of distributed MIMO radar systems.Therefore, transmitter and receiver location errors need to be taken into account in practical environments.

Motivated by these facts,we take into consideration the transmitter and receiver location errors in the measurement model and propose an algebraic solution for target position and velocity in distributed MIMO radar systems using BR and BRR measurements.The proposed solution follows the basic framework of the two-step weighted least squares idea17,and extends the previous study by Zhao et al.25that ignores the transmitter and receiver location errors.In contrast to recently proposed algebraic solutions,23-25the performance of the proposed solution achieves the CRLB, no matter whether there exist transmitter and receiver location errors or not.

The remainder of this paper is organized as follows.The measurement model is described in Section 2,whereas the proposed target localization algorithm is presented in Section 3.The theoretical accuracy analysis of the proposed solution is performed in Section 4.Simulation results are provided to evaluate the performance of the proposed solution in Section 5.Finally,the paper ends with conclusions in Section 6.

2.Measurement model

Consider a distributed MIMO radar on airborne or groundbased moving platforms,3as illustrated in Fig.1,which consists of M transmitters and N receivers.Let uo=[xo,yo,zo]Tandbe the position and velocity(to be determined)of the moving target,respectively.The positions and velocities of the transmitters are respectively denoted byand, whereandare the position and velocity of the mth transmitter,respectively.The positions and velocities of the receivers are respectively denoted byandwhereandare the position and velocity of the nth receiver,respectively.Note that in practice,the true valuesandare not known to the target localization processing,and only nominal valuesandare available, whereandare the nominal position and velocity of the mth transmitter and nth receiver respectively.

Fig.1 Target localization scenario.

Putting the transmitter location vectorand receiver location vectortogether leads to a 6(M+N)×1 column vector as follow:

Then the corresponding true value vector and error vector are respectively denoted by

Based on the above assumptions,the range and the range rate between the mth transmitter and the target,denoted byandrespectively,are

The range and the range rate between the nth receiver and the target,denoted byandrespectively,are

In the absence of measurement noise,the Bistatic Range(BR)and Bistatic Range Rate(BRR)corresponding to the mth transmitter and nth receiver are respectively given by

Considering the non-negligible measurement noises in practical environments,we model the BR and BRR measurements,which result from the noisy time delays and Doppler shifts,as follows:

This study aims to obtain the estimate of target position uoand velocityfrom the noisy measurement vector α together with the noisy transmitter and receiver location vector β.

3.Localization algorithm

Since directly solving the nonlinear measurement equations is a computationally difficult problem,our main strategy is to apply two-step processing.In the first step,we reorganize the nonlinear equations into a set of pseudolinear equations by introducing some proper auxiliary variables,which is a function of the target position and velocity,and then solve the pseudolinear equations using WLS estimation to obtain a rough estimate.In the second step,we exploit the constraints between the auxiliary variables and the target location to improve the estimate using WLS again.

3.1.The first WLS step

To find a rough estimate of the target position and velocity,we should firstly extract a set of pseudolinear equations from the BR and BRR measurements.To achieve this,we first rearrange Eq.(11)as

Stacking Eq.(17)for m=1,2,...,M and n=1,2,...,N,we arrive at

where

with

Taking the time derivative of Eq.(17)gives the pseudolinear equation extracted from the BRR measurements as

Stacking Eq.(27)for m=1,2,...,M and n=1,2,...,N leads to

with

Now we combine Eq.(18)and Eq.(28)to form the set of equations extracted from both BR and BRR measurements as

where

with

Now,we have achieved a set of pseudolinear equations extracted from the BR and BRR measurements.Then by minimizing,the WLS solution ofis acquired from Eq.(36)as

where W1is the weighting matrix,and according to the WLS theory,the optimal choice of W1is

It can be seen from Eq.(41)that the ideal weighting matrix for W1is a function of target position and velocity,which are to be determined.Hence,to make the problem solvable,we first set W1=I2MN×2MNand obtain an initial estimate ofthen utilize the estimated θ1to acquire a more optimal weighting matrix,from which a more accurate estimate ofwill be acquired.Repeating the above solution computation process two times is sufficient to produce a reasonable estimate.In the following simulation section,two repetitions are required because there is no obvious improvement with more repetitions.

Subtracting Eq. (40) by the factgives rise to Δθ1as

Obviously,assuming small measurement noise so that the noise in G1and D1can be ignored, we conclude from Eq.(42)that.Correspondingly,when the noise in G1and D1is small enough to be ignored,the covariance of θ1is approximated by

3.2.The second WLS step

Recall that the auxiliary variablesanddepend on target position uoand velocityas shown in Eqs.(5)and(6).Herein,using this dependency,another set of linear equation is formed to obtain a refined solution.From Eqs.(5)and(6)we acquire

Inserting the noisy quantitiesinto the right side andinto the both sides of Eq.(44)and Eq.(45)yields

where

with

where the optimal choice for weighting matrix W2is

The weighting matrix W2is a function of unknown target position and velocity.Hence,in a similar way as used in the first WLS step,we first use the estimate θ1to form W2,from which an estimate ofwill be acquired,and then utilize the estimated θ2to update W2,from which a more accurate estimate ofwill be produced.Repeating the above solution computation process two times is sufficient to produce an accurate estimate.In the following simulation section,two repetitions are needed.

It is easy to deduce that,under the small noise condition,the covariance of θ2is approximated by

4.Performance analysis

Based on Eq.(62),the CRLB for the estimation of φ is given as

It is easy to see that CRLB(φ) in Eq. (63) is a 6(M+N+1)×6(M+N+1) square matrix, where only upper left 6×6 submatrix is for the target position and velocity estimation.Thus,we rewrite CRLB(φ)in submatrix form as

in which

It is important to emphasize that,in Eq.(66),X-1is the CRLB of the target position and velocity when there is no transmitter and receiver location error,25and X-1Y(Z-YTX-1Y)-1YTX-1means the increase of CRLB arising from transmitter and receiver location error.In the subsequent numerical simulation section, the CRLB with transmitter and receiver location error and that without transmitter and receiver location error will be compared numerically to verify this analysis.

Now,inserting Eq.(59),Eq.(43)and Eq.(41)into Eq.(61)in sequence and employing matrix inversion lemma lead to

in which with

It is observed that the covariance in Eq.(67)and the CRLB in Eq.(66)are of the same form.Under assumptions that the measurement noise and transmitter and receiver location errors are sufficiently small,we can conclude,after some mathematical manipulations, that the covariance reaches the CRLB.

5.Numerical simulations

In order to evaluate the efficiency of the proposed solution,we apply it to moving target localization,and compare it with the existing methods including Refs.23-25as well as the CRLB.We consider employing a distributed MIMO radar with 4 transmitters and 4 receivers to locate a moving target with coordinates[3000,3000,3000]Tm/s and velocity[300,100,100]Tm/s.The transmitters and receivers are deployed on ground-based or airborne moving platforms,whose coordinates and velocities are listed in Table 1.The estimation performance is evaluated in terms of Root Mean Square Error(RMSE)of the position and velocity estimate,which is acquired from 1000 independent trials.The BR and BRR measurements are simulated according to Eq.(13)and Eq.(14),where the covariances of measurement noises arewhereand,with σαrepresenting the measurement noise level and Vαset to 1 in the diagonal and 0.5 elsewhere. The noisy transmitter and receiver locations are simulated by adding noises to the true values,and noises are zero-mean Gaussian with covariance matrixwhereand Qs˙=0.1Qs,with σβrepresenting transmitter and receiver location uncertainty level and

To illustrate the effects of transmitter and receiver location uncertainties on localization performance,the CRLB of target location estimation when transmitter and receiver location error exists as well as when it does not,is plotted in Figs.2 and 3.In other words,we compare the CRLB(with transmitter and receiver location uncertainty)derived here to the one(without transmitter and receiver location uncertainty)derived in Ref.25.

Fig.2 plots the CRLB along the BR and BRR measurement noise level,with the BR and BRR measurement noise level σαchanging from 10-2m to 103m and the transmitter and receiver location uncertainty level σβ=1 m.It is readily seen from Fig.2 that the CRLB with transmitter and receiver location uncertainty is above the one without transmitter and receiver location uncertainty.This implies that the existence of the transmitter and receiver location uncertainty significantly deteriorates the estimation accuracy of target position and velocity,at least in the CRLB sense.Fig.3 depicts the CRLB along transmitter and receiver location uncertainty with the transmitter and receiver location uncertainty level σβchanging from 10-2m to 103m and the BR and BRR measurementnoise level σα=10 m.Obviously,the CRLB with transmitter and receiver location uncertainty rises rapidly with increase of the transmitter and receiver location uncertainty.When the location uncertainty level reaches 10 m,the CRLB with transmitter and receiver uncertainty is tens of meters above the one without transmitter and receiver uncertainty,and this level of location uncertainty is not rare in practice.This signifies again the necessity of taking into consideration the transmitter and receiver uncertainties during the design of localization algorithms for distributed MIMO radar.

Table 1 Position and velocity of transmitters and receivers.

Fig.2 CRLB comparison at different BR and BRR measurement noise levels.

Fig.3 CRLB comparison at different transmitter and receiver location error levels.

We next examine the performance of the proposed solution in the presence of transmitter and receiver location uncertainties by comparing with the existing methods.The RMSE is used to evaluate the localization performance.As earlier,we first examine the RMSE along BR and BRR measurement noise,with the results plotted in Fig.4,and then we examine the RMSE along transmitter and receiver location uncertainty level,with the results plotted in Fig.5.

Fig.4 provides the performance comparison between the proposed solution and those without consideration of the transmitter and receiver location uncertainty,with respect to the BR and BRR measurement noises.The measurement noise level σαchanges from 10-2m to 103m and the location uncertainty level is fixed at σβ=1 m.From Fig.4,we observe that the proposed solution outperforms the existing methods and its RMSE reaches the CRLB,which implies that the proposed solution is asymptotically unbiased.The RMSE of Ref.25is smaller than that of Ref.23as well as Ref.24,which is in accordance with the conclusion drawn in Ref.25,but all these three methods cannot achieve the CRLB.This is because the proposed solution took the transmitter and receiver location uncertainties into consideration while the existing methods did not.Then it may seem unusual that the RMSE of Ref.24is below the CRLB.This is possibly because that when the measurement noise is large,Ref.24provides a biased estimate and the RMSE of a biased estimate can be smaller than the CRLB.

Fig.5 provides the performance comparison of different methods along the transmitter and receiver location uncertainty. The location uncertainty level σβchanges from 10-2m to 103m and the measurement noise level is fixed at σα=10 m.From Fig.5,we conclude that the proposed solution still has the best performance.As we expect,Ref.23and Ref.24cannot achieve the CRLB.Ref.25seems to approach the CRLB when the transmitter and receiver location error is small,but it deviates from the CRLB when the transmitter and receiver location uncertainty level σβ＞1 m.The proposed solution has smaller RMSE than the existing methods and attains the CRLB under small and moderate transmitter and receiver location uncertainty levels.For σβ≥102m,the proposed solution slightly deviates from the CRLB,but its RMSE is still below that of the existing methods,and the deviation from the CRLB at high location uncertainty levels may lie in the second-order and higher-order error terms neglected in the derivation of the proposed solution.

Fig.4 RMSE comparison at different measurement noise levels.

Fig.5 RMSE comparison at different transmitter and receiver location error levels.

Table 2 Complexity assessment of different methods.

The computational complexity is also an important index for the algorithm performance evaluation.For this purpose,we compare the computational complexity of the methods in Table 2,where Ni1(Ni1=2)and Ni2(Ni2=2)represent the number of repetitions in the first WLS step and the second WLS step,respectively.Clearly,the proposed solution endures the heaviest computational burden among the methods,approximately 3 times heavier than Ref.25.This is hardly surprising since the proposed solution took the transmitter and receiver location uncertainties into consideration while the existing methods did not.That is to say,it is at the expense of the higher computation cost that the proposed solution achieves higher localization accuracy. In consideration of remarkable performance improvement, the increased but acceptable complexity is worthy.

6.Conclusions

We have developed an algebraic solution for moving target localization using BR and BRR measurements in distributed MIMO radar systems in the presence of transmitter and receiver location uncertainties.Considering the high nonlinearity and nonconvexity of the localization problem,we borrow the basic idea of two-step weighted least squares processing to estimate target position and velocity.The proposed solution took into account the transmitter and receiver location uncertainties in the measurement model,and achieves the CRLB,no matter whether there exist transmitter and receiver location errors or not.The proposed solution does not suffer from divergence problem as in iterative methods.The proposed solution was compared with several existing methods and it showed generally better performance both theoretically and numerically.But meanwhile,the disadvantages of distributed MIMO radar compared to the conventional radar,including SNR loss,28heavy computation and so on,29should also be noted.To overcome these disadvantages,long-time integration,parallel computing technique and other signal processing techniques should be further explored for distributed MIMO radar.29,30

Acknowledgement

This research was supported by the National Natural Science Foundation of China(No.61703433).

Appendix A.

with the elements of submatrices further given by

for m=1,2,...,M and n=1,2,...,N,and zero otherwise.

In a similar manner,the partial derivative ?αo/?βocan be written as

and the elements of submatrices are given by

for m=1,2,...,M and n=1,2,...,N and zero otherwise.