ZHAO Yongsheng,HU Dexiu,ZHAO Yongjun,and LIU Zhixin

Data and Target Engineering Institute,PLA Strategic Support Force Information Engineering University,Zhengzhou 450001,China

Abstract:Time delay and Doppler shift between the echo signal and the reference signal are two most commonly used measurements in target localization for the passive radar.Doppler rate,which can be obtained from the extended cross ambiguity function,offers an opportunity to further enhance the localization accuracy.This paper considers using the measurement Doppler rate in addition to measurements of time delay and Doppler shift to locate a moving target.A closed-form solution is developed to accurately and efficientl estimate the target position and velocity.The proposed solution establishes a pseudolinear set of equations by introducing some additional variables,imposes weighted least squares formulation to yield a rough estimate,and utilizes the function relation among the target location parameters and additional variables to improve the estimation accuracy.Theoretical covariance and Cramer-Rao lower bound(CRLB)are derived and compared,analytically indicating that the proposed solution attains the CRLB.Numerical simulations corroborate this analysis and demonstrate that the proposed solution outperforms existing methods.

Keywords:target localization,multistatic passive radar,time delay,Doppler shift,Doppler rate.

1.Introduction

Multistatic passive radar(MPR),which employs ambient signals(FM radio[1–3],digital TV[4–6],digital audio broadcast[7–9],cellphone basestation[10–13],WiMax[14,15],WiFi[16–18],satellite[19–21],non-cooperative radar signal[22],etc.)as the transmitters to detect and locate potential targets,is an attractive system for surveillance purposes.Owing to its distinct merits,such as covert operation,wide coverage,small size and hence easy to deploy,low costs of operation and maintenance,immune to directional interference,capabilities against stealth aircraft,etc.,MPR has been studied for several decades and is still a hot area of research[23,24].

The receiver of passive radar usually deploys two receiving channels:the reference channel and the echo channel[25].The reference channel,which can be implemented as a directional antenna or a digitally formed beam,points towards the transmitter to receive the original transmitted signal,i.e.,the direct path signal.The echo channel looks towards the area of interest to collect echo signals from potential targets[26].The echo signal is usually much weaker in power,hence coherent integration is needed to improve the signal-to-noise ratio(SNR)[27].One of the most typical coherent integration methods is the cross ambiguity function(CAF)[28],from which time delay and Doppler shift between the echo signal and the reference signal can be acquired by locating the correlation peak.Time delay and Doppler shift are two most commonly used measurements to estimate the target position and velocity,and copious localization methods are developed based on these two measurements[29–32].

The CAF method works well in most of the cases.However,in some situations,such as detection and localization of manoeuvring targets,the simplified assumptions made in the derivation of the CAF lead to suboptimal performance.Another situation where more complicated signal processing is required is when long time coherent integration(of the order of 1 s)is desired.For this purpose,recently some novel coherent integration methods[33–35],such as the modified CAF[34],are proposed,from which time delay,Doppler shift and Doppler rate can be obtained simultaneously.In source localization problem,the Doppler rate measurement has been used in addition to time delay and Doppler shift measurements to improve the localization accuracy[36,37].Inspired by this,for moving target localization in the multistatic passive radar,we hopefully determine the target position and velocity with a higher accuracy by jointly using the Doppler rate measurements.However,despite the fine prospects, up to now there does not exist any publication in the open literature that addresses combining measurements of time delay,Doppler shift and Doppler rate to estimate the target position and velocity in the multistatic passive radar.

Extracting the target position and velocity from measurements of time delay,Doppler shift and Doppler rate acquired at a single observation is a crucial and challenging operation,on account of the extreme nonlinearity implied in the measurement equations.Exhaustive search in the solution space seems to be a basic alternative to tackle the nonlinearity.Nevertheless,this technique usually endures a heavy computational burden because of the high dimensionality,which prohibits real-time implementation.Iterative approaches,such as Taylor series approach,Newton-Raphson approach and expectationmaximization approach,therefore,have to be envisaged,to achieve desirably accurate results with an acceptable complexity.However,iterative approaches are known to converge to the global optimal solution only if the initial solution guess is close enough to the true values of parameters.Otherwise,they may converge to a local optimum or even diverge.Therefore,there is yet a need for an alternative solution method that requires no initial guess.

Closed-form solutions are always compelling to researchers due to their advantages of independence on initial guess and acceptable computational complexity.Motivated by this, we derive in this paper a closed-from solution for moving target localization using measurements of time delay, Doppler shift and Doppler rate in the multistatic passive radar.The proposed solution follows the basic idea of two-step weighted least squares(WLS)[38],and further extends our previous work in[32]that determines the target position and velocity only using measurements of time delay and Doppler shift.It is generally composed of two WLS steps.In the first WLS step,by introducing some extra additional variables,the proposed solution first translates the time delay,Doppler shift and Doppler rate equations to a set of linear equations,from which a rough estimate of target location and additional variables is obtained by using WLS minimization.Next,in the second WLS step,by using the functional relation between the additional variables and target location,we extract another set of linear equations,from which a re fined estimate of target location is finally obtained by using WLS minimization again.To the best of our knowledge,the proposed solution is the first localization method jointly by using measurements of time delay,Doppler shift and Doppler rate in the multistatic passive radar literature.Theoretical error analysis and numerical simulations are performed and verify thevalidity and efficiency of the proposed solution.

The remaining sections of this paper are organized as follows.In Section 2,we present the measurement model and the target localization problem.In Section 3,the proposed closed-form solution for the target position and velocity is given.In Section 4,we analyze the bias,covariance as well as the Cram′er-Rao lower bound(CRLB).In Section 5,the performance of the proposed solution is evaluated via several numerical examples.Finally,in Section 6,we make some concluding remarks.

2.Problem formulation

In general,as illustrated in Fig.1,an MPR system withMtransmitters andNreceivers is deployed to locate a moving target with an unknown positionu=[x,y,z]Tand velocity˙u=[˙x,˙y,˙z]T.The position and the velocity of themth transmitter are known and denoted byand the position and the velocity of thenth receiver are known and denoted by

Fig.1 Localization geometry

Using the above notations,the range,rate and acceleration between themth transmitter and the target are respectively expressed as

Likewise,the range,rate and acceleration between the target and thenth receiver are respectively stated as

Subsequently,as defined in[34],the bistatic range,bistatic velocity and bistatic acceleration subject to themth transmitter and thenth receiver,are respectively described by

In the presence of the additive noise,the observed bistatic range,bistatic velocity and bistatic acceleration,which are directly converted from measurements of time delay,Doppler shift and Doppler rate respectively,can be stated as

where Δrm,n,Δ˙rm,nand Δ¨rm,nare the measurement noises of bistatic range,bistatic velocity and bistatic acceleration,respectively.

Next,by de fining the following vector notations:

the bistatic range,bistatic velocity and bistatic acceleration subject to theMtransmitters andNreceivers,can be compactly described by

where

Putting the three sets of measurements together forms the total measurement vectorα=[rT,,]T.Then,the corresponding true value of the measurement vector is denoted byαo=[(ro)T,()T,()T]T,and the noise vector is denoted by Δα=[ΔrT,Δ,Δ]T,which follows the Gaussian distribution with the mean zero and the covariance

GivenMtransmitters andNreceivers,there areMNbistatic range measurements,MNbistatic velocity measurements,andMNbistatic acceleration measurements.Now,the problem can be stated as,given the observationsα, find the target positionuand velocity˙uaccurately.Nevertheless,despite a clear appeal,determining the target positionuand velocity˙ufrom the observationsαis not tractable,since the target location parameters are nonlinearly related to the observations.

3.The proposed localization method

In this section,borrowing the basic framework of two-step WLS[38],we deduce a closed-from solution for the target position and velocity estimation.As mentioned above,the proposed solution generally comprises two WLS steps,from which a rough WLS solution and a re fined WLS solution are produced respectively.

3.1 The first WLS step

Begin by reformulating(10)as

Recalling the definitions ofand,squaring both sides of(17),and then simplifying,yield

where the second-order noise term has been neglected.

By taking the derivative of(18)versus time,we establish a relation between the measurement Doppler shift and the parameter target location as follows:

Further,taking the time derivative of(19)yields a relation between the measurement Doppler rate and the parameter target location as follows:

De fine an auxiliary vector as

andG1,h1and Δh1are expressed in submatrix as

where

The corresponding composite noise vector Δh1is given by

whereB1is expressed in the submatrix form as

andB,andare block diagonal matrices,i.e.,

with

From(22),an estimate ofθ1is given,based on the weighted least squares minimization,as

whereW1is the weighting matrix determined by

Troublingly,as indicated in(28),W1is a matrix related to the unknown target location.Toward this end,we turn to iteratively updating values ofW1.To be more specific,by settingW1=I3MN×3MN,we find an initial estimate ofθ1using(27).Based on this,a better weighting matrixW1can be formed by substituting the estimatedθ1into(28).Then,a more accurate estimate ofθ1is acquired by using the formed weighting matrixW1.Usually,repeating the computation ofθ1andW1one to two times is enough to achieve an acceptably accurate estimate.

Subtracting both sides of(27)by the true valueθ1=

Suppose the measurement noise inG1andB1is small enough to be neglected.Then,taking expectation on(29)and invoking the fact that E{Δα}= 03MN×1yield E{Δθ1}0(3M+6)×1,which indicates the unbiasedness over a small noise region.Accordingly,the covariance(as well as the mean square error)is given,by multiplying(29)with its transpose and taking expectation,as

3.2 The second WLS step

Observe that,in the first WLS step,to rearrange the nonlinear measurement equations into linear forms,we have introduced extra addition alvariableswhich are functions of the target position and velocity as seen in(1),(2)and(3).Following the basic idea of two-step WLS,the second WLS step exploits these functional relations to upgrade the localization accuracy.

Toward this end,begin by lettingandbe the estimates ofu,u˙,andobtained in the first WLS step,andandbe the corresponding estimation errors.First of all,the final target position and velocity estimate should maintain as close as possible to the target location values obtained in the first WLS step by minimizing the errors of the following equations:

Meanwhile,to exploit the functional relations between additional variables and target location parameters to refine the estimate,reformulate(1),(2)and(3),respectively,as

Now,the final target position and velocity estimate should also minimize the equation errors in(36),(37)and(38).Therefore,definingθ2=[uT,˙uT]Tand combining(31),(32)and(36),(37),(38),another integrated linear equation is established in the matrix form as

where

The corresponding composite noise vector Δh2is described by

whereB2is expressed in the submatrix form as

with

Invoking the WLS minimization formula yields from(39):

whereW2is the weighting matrix given by

4.Error analysis

The CRLB,which designates a fundamental lower bound on the variance for any unbiased estimators,usually serves as a benchmark for the performance of target localization algorithms.This section evaluates the performance of the proposed solution and compares it with the CRLB,under the small noise conditions.

4.1 Bias and covariance

Combining(45)and the fact thatwe obtain the estimate bias as

Assume that the noise inG2andB2is small enough to be neglected.Then,taking expectation on(47)and recalling E{Δθ1}0(3M+6)×1,result in E{Δθ2}06×1,which indicates the unbiasedness for a low noise level.The resulting covariance(as well as the mean square error)is determined,by multiplying(47)with its transpose and taking expectation,as

4.2 CRLB

To obtain the performance limit for the target localization,the CRLB is derived next.As previously mentioned,the unknown parameter vector to be estimated and the measurement vector for the CRLB evaluation areθ2=[uT,]Tandα=[rT,,]Trespectively.From the measurement noise model described in Section 2,the logarithm of the probability density function underθ2(after ignoring the constant term)is of the form

By definition,the Fisher information matrix(FIM),whose inverse yields the CRLB,is calculated as

Recalling the parametric form of the true measurement vector with respect to target location parameters given in(7),(8)and(9),the entries of submatrices are given as

form=1,2,...,M;n=1,2,...,Nand zero elsewhere.

Form(50),it follows that the CRLB can be given by

It is not difficult to show that the CRLB in(52)and the covariance in(48)are of the same form.Not only that,under the assumptions that the measurement noises are sufficiently small,we have after some algebraic manipulations

By now,it is reasonable to arrive at the conclusion that the covariance is approximately equivalent to the CRLB,implying that the CRLB is achieved by using the proposed solution for small measurement noises.

5.Numerical examples

In this section,numerical examples are performed to verify the theoretical development of the proposed solution.In the numerical examples,we consider a geolocation scenario as illustrated in Fig.1,where the positions and velocities of the transmittersT1?T5and receiversR1?R5are enumerated in Table 1.Note that in the following simulations only three transmittersT1?T3and three receiversR1?R3are employed,unless otherwise specified.The position of the aircraft target is[30 000,?30 000,1 000]Tm,and it moves with a velocity of[?500,500,50]Tm/s.

Table 1 Position and velocity of transmitters and receivers

Simulation results illustrate the performance of the proposed solution for different measurement noise levels[39],which can be obtained from the following equations:

wherecis the signal propagation speed,λis the signal wavelength,Bnis the noise bandwidth,Bsis the signal bandwidth,Tis the integration time.Assume that the MPR system employs digital video broadcasting(DVB)broadcasters as illuminators,and the corresponding parameters areλ=0.375 m,Bs=8 MHz,T=1 s andBn=20 MHz.The SNR is related to many factors including transmit power,transmitter-to-target and targetto-receiver ranges,target bistatic radar cross-section,and so on.Combining the study in[40]and[41],we let the SNR ranges from –70 dB to–20 dB to characterize the performance of the proposed solution for different SNR levels.The root mean squared error(RMSE)and bias are employed as the performance measures,which come from 1 000 independent runs for various SNR values.

5.1 Delay–Doppler-based localization versus delay–Doppler–Doppler rate-based localization

Fig.2 CRLB comparison between delay–Doppler-based localization and delay–Doppler–Doppler rate-based localization

In order to demonstrate that using the Doppler rate measurement in addition to measurements of time delay and Doppler shift provides the estimation of target position and velocity with a higher accuracy,we present and compare the CRLB of localization using delay and Doppler against localization using delay,Doppler and Doppler rate,versus various SNRs.The results are shown in Fig.2.With the increase of SNR,both delay–Doppler-based localization and delay–Doppler–Doppler rate-based localization give better localization performance.Over the whole SNR range,the CRLB of delay–Doppler–Doppler rate-based localization is remarkably below the delay–Doppler-based localization,especially for velocity estimation.The former benefit from the additional measurement Doppler rate.This signifies the necessity of taking into account the measurement Doppler rate during the design of localization algorithms.

5.2 Evaluation of different localization methods

Next,we study the localization performance of the proposed solution in terms of RMSE and bias,as SNR varies.We also compare its performance against the Taylor-series method(with the true position and velocity as the initial guess)and the delay–Doppler localization method suggested in[32].Moreover,the CRLB is also plotted as a benchmark.The results are illustrated in Fig.3 and Fig.4.As can be seen from Fig.3,the localization RMSE of delay–Doppler-based localization is apparently higher,compared to the family of delay–Doppler–Doppler ratebased localization methods.The excess RMSE for the delay–Doppler-based localization is of course due to the fact that it does not employ the Doppler rate measurements.Both the Taylor-series method and the proposed solution give the similar performance by attaining the CRLB at sufficiently high SNR conditions.Note that the localization problem here is highly nonlinear,and hence in practice,both the Taylor-series method and the proposed solution will suffer from the threshold phenomenon when the SNR is small enough.As analyzed,the Taylor-series method deviates from the CRLB and gives an inaccurate solution at SNR values below–50 dB,whereas the proposed solution still coincides with the CRLB at this level of SNR.Even at SNR values below–60 dB,the proposed solution is still significantly superior to the Taylor-series method,although its RMSE begins to digress from the CRLB.

Fig.3 Localization RMSE versus SNR

Fig.4 Localization bias versus SNR

In terms of bias,it is clear from Fig.4 that the proposed solution offers excellent localization performance compared with other methods.Note that as the SNR decreases,the biases of the three localization methods increase.Such findings are partially explained by the nonlinearity nature of the localization problem.

5.3 Effect of number of transmitters and receivers on performance of the proposed solution

It is intuitively reasonable that increasing the number of transmitters and receivers will improve the performance of localization methods.To numerically show the effect of the number of transmitters and receivers on the performance of the proposed solution,three cases with different numbers of transmitters and receivers are considered.We chooseM=3 andN=3(T1?T3,R1?R3)for the first case,M=4 andN=4(T1?T4,R1?R4)forthe second case,andM=3 andN=3(T1?T5,R1?R5)for the third case.The resulting RMSEs and biases are shown in Fig.5 and Fig.6.

Fig.5 Effect of number of transmitters and receivers on localization RMSE of the proposed solution

Not surprisingly,as displayed in Fig.5 and Fig.6,as the number of transmitters and receivers increases,the proposed solution performs better in terms of both RMSE and bias.

Fig.6 Effect of number of transmitters and receivers on localization bias of the proposed solution

On the other hand,however,as the number of transmitters and receivers increases,the improvement rate decreases rapidly. That is to say, as the number of transmitters and receivers increases,the performance curve of the proposed solution versus SNR shows a tendency to a bound curve.

5.4 Comparison of computational cost

As mentioned above,in the proposed solution,the additional measurement Doppler rate provides the estimation of target position and velocity with a higher accuracy,but meanwhile,it certainly will require a higher computational load.To numerically evaluate the computational burden of the proposed solution,in Table 2,we enumerate and compare the time cost of the proposed solution and delay–Doppler-based localization suggested in[32].It should be pointed out that,the time cost is obtained from the average time of 1 000 independent runs.Main configuration of the computer is CPU:Intel(R)Core(TM)i5-7200U@2.50 GHz;RAM:8.00G;Operating system:Windows1064bit;Software:Matlab 2018a.

Table 2 Time cost of localization methods ms

As evident in Table 2,the proposed solution costs more time than the delay–Doppler-based localization method,due to the additional measurement Doppler rate.Nevertheless,the excess time cost is not remarkable(an increase of 27%).That is to say:it is at the expense of more time cost that the proposed solution achieves a higher localization accuracy.Recalling remarkable performance improvement,the excess time cost is worthy and acceptable.

6.Conclusions

The CAF is a standard method used in passive radar,which produces time delay and Doppler shift between the echo signal and the reference signal,two most commonly used measurements in target localization.The Doppler rate,obtained from the extended CAF,makes it possible to dramatically improve the localization accuracy.This paper uses Doppler rate in addition to time delay and Doppler shift as measurements to estimate the target position and velocity.A closed-form solution is explored,by following the basic idea of two-step.The proposed closed-form solution requires no initial estimate and guarantees global convergence.Simulation results demonstrate that delay–Doppler–Doppler rate-based localization has a higher localization accuracy than delay–Doppler-based localization although the former requires higher but acceptable time cost,and the proposed solution is proved to perform better than existing methods.