Beam resource scheduling for a VHF-MIMO radar network in low-angle tracking

Howei ZHANG, Weijin LIU, Tiyong FEI

aAir and Missile Defense College, Air Force Engineering University, Xi’an 710051, China

bAir Force Early Warning Academy, Wuhan 410039, China

KEYWORDSConvex optimization;Low-angle tracking;MIMO radar;PC-CRLB;Resource allocation

AbstractThe low-angle tracking in multipath interference is a challenging problem for the Very High Frequency (VHF) radar.The colocated Multi-Input Multi-Output (MIMO) technique can remedy such a defect.In this paper, a Joint Beam-Target Assignment and Power Allocation(JBTAPA) strategy is proposed for the VHF-MIMO radar network tracking low-angle targets.The core of the JBTAPA strategy is to improve the worst tracking accuracy among multiple targets by assigning appropriate beams to targets and allocating the power resource in each beam using the feedback information in the tracking cycle.Taking into account the transmit multipath and receive multipath,we derive the Crame′r-Rao Lower Bound(CRLB)on angle estimate,which is then incorporated in the Predicted Conditional CRLB (PC-CRLB).A more accurate and consistent lower bound is provided as the optimization metric since the PC-CRLB is based on the most recently realized measurements.A two-stage-based technique is proposed to solve the JBTAPA problem,which is originally NP-hard.Simulation results verify the effectiveness and efficiency of the proposed method.The results also imply that the target reflectivity plays one of the important roles in resource allocation.

1.Introduction

1.1.Background and motivation

The Very High Frequency (VHF) radar has received much attention in recent years, owing to its superior potential in detecting stealth targets and anti-antiradiation missiles.1-6However,due to the wide beamwidth,it suffers from multipath interference in low-angle target tracking seriously.2The multipath interference usually originates from the signal reflected by the ground/sea surface and is composed of the specular and diffuse components.7,8For the case of smooth surfaces, the specular reflection is the main influence on the elevation angle measurement.Traditional methods are not able to distinguish the direct signal and reflected signal in time,Doppler,and spatial domains,due to the high correlation between them,leading to the continuous fluctuation of the measurement error and degraded tracking performance.Recently, the colocated Multi-Input Multi-Output (MIMO) radar9is used to overcome the above difficulty.It transmits multiple uncorrelated waveforms and forms the focused receive beams by the digital beamforming technique.The superior Direction of Arrival(DOA)estimation performance is achievable owing to the virtual aperture extension, compared with traditional phased array radars.6,10Furthermore,the longer dwell time on targets improves the Doppler resolution, which is very effective in detecting slow targets.Thus, the VHF-MIMO radar is on a promising way to military applications.A typical scenario is to incorporate several VHF-MIMO radars to form a network system for multi-target tracking,because a VHF-MIMO radar network has obvious advantages over the monostatic radar,such as spatial and multiplexing diversity gains as well as immunity to one radar failure3,11-13.

Note that there are two finite resources for a VHF-MIMO radar network in multi-target tracking, namely, (A) the maximum number of available beams of each radar, and (B) the finite power budget of each radar.Thus,it is of critical importance to efficiently utilize the limited resources to improve the overall tracking accuracy.Though this problem has been partly addressed in Ref.12and Ref.,13the fixed beamwidth is used to characterize the Crame′r-Rao Lower Bound (CRLB)on the angle estimate.It was shown therein that the radar is insensitive to the target angle spread.However, this does not hold when faced with multipath interference, where both the direct signal and the reflected signal are within the same beam.More importantly, the beam allocation algorithms in Ref.12and Ref.13cannot guarantee each target being allocated at least one beam, which easily leads to the track loss.Thus,new system models and solution strategies need to be explored for resource allocation in a VHF-MIMO radar network tracking low-angle targets.

1.2.State-of-the-art algorithms

Many methods have been proposed to separate the target signal from its multipath imagines, mainly including the enhanced monopulse techniques and array signal processing methods.The modified monopulse techniques,e.g.,double null,7,14symmetrical beam pattern,5,14and complex angle15,16methods,attempt to mitigate the multipath effects by redesigning beam patterns.However, a more complex system and strict requirements restrict their applications.The array signal processing methods can be roughly divided into two categories, i.e., the nonparametric and parametric estimation methods.The first category includes the Multiple Signal Classification (MUSIC)algorithm17and the minimum norm method,18which identify the DOA of signals by searching peaks in the spatial spectrum.However, they are impractical due to the need for multiple snapshots of data.Moreover,since the direct and multipath signals are highly correlated, nonparametric estimation methods usually show poor performance.19Though spatial smoothing techniques20-22are proposed to alleviate such problems, the array aperture is sacrificed, which is unacceptable for VHF radars.To tackle this issue, the General MUSIC (GMUSIC)algorithm23and iterative adaption algorithms24,25are proposed so that the separation between the coherent signal arrival angles only involves a two-dimensional search.The second category of techniques attempts to fit the received data to a parametric model and usually adopts the Maximum Likelihood(ML)principle.26-28These methods offer better performance since they are immune to the coherency between signal sources,and more importantly, they can work with one snapshot.However, the high computational burden is inevitable due to the nonlinear processing.To lower the computational complexity,some work tries to reduce the number of unknown parameters using the prior information,29-31by modeling the uncertainty of the steering vector,32and by employing the eigendecomposition and Alternate Projection (AP) algorithm33,34.

In the resource-aware design of radar systems, the Power Allocation (PA) is a classical topic.Intuitively speaking, most vehicle-equipped radars have finite power budgets, and they must be efficiently utilized.The existing work on this topic can be classified into two categories.The first category is based on radar configuration, where the total power consumption is reduced by choosing a small subset of local radars, and the greedy search-based algorithm35and the convex relaxation based algorithm36are representative solving methods.The second category tries to achieve the optimal PA to all the local radars.37-40The typical solutions are the game theory37and the convex optimization-based methods.39There are also some combinations of the two categories41-45.

Though previous studies have made seminal contributions to multipath suppression and resource allocation, some issues still need to be addressed.

(1) The complex ground surface condition is always faced with in the low-angle tracking,resulting in degraded performance of the ideal symmetric reflection model.26,28-32Thus, it is necessary to consider the ground reflection coefficient while estimating the target elevation angle.

(2) The estimation performance improvement is based on signal processing level-based techniques,5–7,14–34but a systemic view is absent, i.e., resource allocation.The multipath effect must be taken into account in lowangle tracking.However, it is neglected in the existing resource allocation works,12,13,35–45where a fixed halfpower beamwidth is used for the angle CRLB.Actually,angle CRLB must be characterized by specific signal models and signal processing methods.

(3) The work12,13,37,39,40,44uses the Posterior CRLB(PCRLB)46as the optimization metric, but the PCRLB is derived by taking the expectation with respect to (w.r.t.)the target state and measurement data from the initial time to the current time.In this case, the most recently achieved measurements are averaged out, leading to an offline bound.

Therefore, it is of interest to investigate the joint beam and power allocation problem for low-angle target tracking in the VHF-MIMO radar network while incorporating the above issues.

1.3.Main contributions

In the tracking recursive cycle, much prior target information can be utilized.We adopt the cognitive idea47to form the radar network as a closed-loop system, where the target estimate feedback from the previous period flows into the resource allocation strategy to enhance the estimate accuracy in the next period.The main contributions of this paper are summarized as follows.

(1) The global Predicted Conditional CRLB (PC-CRLB)48considering multipath and ground reflectivity is derived,based on which the optimization model for resource allocation in low-angle tracking is established.The CRLB on elevation angle is formulated considering the effects of transmit multipath, receive multipath, and ground reflectivity, which is then embedded in the PC-CRLB.The optimization model is established as minimizing the worst-case PC-CRLB under the constraints of beam direction and power budget,which is referred to as the Joint Beam-Target Assignment and Power Allocation(JBTAPA) problem.

(2) An efficient two-stage-based solution for solving such a nonconvex optimization problem is proposed.By introducing an auxiliary variable,we first determine the normalized contributions of each beam to the targets,based on which a sequence of Beam-Target Assignment (BTA) alternatives are obtained.Then, the optimal JBTAPA is achieved by solving the corresponding PA problems.Simulation results confirm the low complexity but high precision of the proposed technique,compared with the state-of-the-art algorithms12,13.

(3) A cognitive framework for the VHF-MIMO radar network tracking low-angle targets is established.In our framework, first, the Alternate Projection-Maximum Likelihood(AP-ML)34estimator and the Square-root Cubature Kalman Probability Data Association Filter(SCK-PDAF)49,50are used to provide the cyclical parameter estimation in each local radar.Then, the states and the estimate error covariance matrices are fused and predicted for the calculation of global PC-CRLB.Next, the optimization model is formulated and solved by the two-stage-based solution technique.Finally,the results are sent to local radars to guide the JBTAPA at the next round of probing period.Overall,our tracking system can be regarded as the reaction using the prior target information to mitigate multipath and false alarm effects, where the cognition idea is implied.

The rest of this paper is organized as follows.The system model is established in Section 2.Section 3 introduces the low-angle target tracking framework.Section 4 derives the PC-CRLB for low-angle tracking, based on which the optimization model is formulated and solved by a proposed twostage-based technique.Section 5 presents the simulation results and analysis.Section 6 concludes the paper.

2.System model

Consider that a Three-Dimensional (3D) colocated VHFMIMO radar network is operated in the distributed tracking,where the communication burden of the fusion center can be evidently reduced.13,51The radar network is responsible for tracking Q widely separated and point targets, where Q has been known from the radar search phase,and their trajectories have been initialized.12,13Each radar is equipped with nrsubarrays arranged in one row on the y-z plane,and each subarray has N × M antennas perpendicular to the horizontal plane,where N is in the azimuth dimension, and M is in the vertical dimension.The antennas are spaced with half of the waveform length, i.e., λ/2.Each subarray transmits one beam, the transmit signals between any two subarrays are uncorrelated, and each subarray is responsible for processing its own transmit signal.The ith radar is located at (xRi,yRi,zRi), for i = 1,2,...,I, and zRiis the height of the central antenna.An AP-ML estimator and an SCK-PDAF are used in each local radar to send the state estimates and covariance matrices to the fusion center.According to the Rayleigh criterion, the diffuse scattering can be absorbed into the Gaussian noise in the VHF-MIMO radar.52It is noticeable that the radar index i and the target index q will be always omitted in this paper, unless doing so causes confusion.

2.1.Signal model

In the vertical dimension, the transmit multipath and receive multipath signals are considered simultaneously,53i.e., directdirect, direct-reflected, reflected-direct, and reflected-reflected.An illustration is shown in Fig.1.

In the following,the central subarray is utilized as an example, and the receive signal vectors of other subarrays can be handled accordingly.The receive signal vector of the central subarray can be expressed as

In Eq.(1), ˉq=qe-j2πcΔτ/λis the integration of ground complex reflection coefficient q and the phase shift brought by the time-delay difference Δτ between single-trip direct path and reflected path, with

Fig.1 Illustration of signal propagation in presence of multipath effects.

where Γ(θs) is the Fresnel reflection coefficient, qs(θs) is the scattering coefficient from the rough surface, and D(θs) is the divergence factor.Since the three factors are functions of θsand can be modelled empirically, we omit their derivations for brevity.More details can be found in Refs.55-58.

Remark 1.In the scenario of radar and target being close, e.g.,within or around one hundred kilometers, and the target being low, e.g., within one kilometer, the plane reflection model is appropriate.However, when their distance is longer, e.g.,thousands of kilometers, the radar is elevated, and the target is higher than one kilometer, the earth’s curvature must be taken into account.In this paper, we focus on the first scenario.The second scenario can be found in Ref.22and Ref.33.

Remark 2.Some strict conditions, e.g., the target elevation angle being within one half-power beamwidth,59have been incorporated in our simulations to support the multipath scenario.However, to establish a highly accurate multipath signal propagation model,more factors should be taken into account,such as the effects of the lower atmosphere and multibounce.However,since this paper concentrates on the resource allocation strategy for low-angle tracking, the above factors may be a little trivial and could be ignored.

2.2.Target motion model

2.3.Measurement model

When tracking multiple targets with measurement origin uncertainty, the ith radar may receive multiple measurements at one time, and it is unknown which one is from the target or just a false alarm.Thus, the measurement set is denoted as

3.Overall procedure of closed-loop feedback low-angle tracking framework

In this section,we will introduce our low-angle target tracking framework based on the resource-aware design.In the colocated VHF-MIMO radar network, each local radar is equipped with a SCK-PDAF to provide the state means and covariance matrices.Then, these data are sent to the fusion center and fused by the Covariance Intersection (CI) rule for the global tracking results.51Next, the global state means and covariance matrices are predicted and used to calculate the PC-CRLB, based on which the optimization model is established and solved by the proposed two-stage-based solver.Finally, the JBTAPA results are used to guide the beam scheduling and power allocation in each local radar.The steps are shown below.

3.1.Prediction in fusion center

which is easily obtained from the target motion models.Eq.(16) and Eq.(17) will be used for the calculation of PCCRLB and the implementation of the JBTAPA strategy, to be shown in Section 4.

3.2.Multi-target tracking in local radar

After obtaining the JBTAPA results, each colocated VHFMIMO radar turns its beam directions and tunes the transmit power in each beam.Then, the target information is extracted from receive echoes by the AP-ML estimator.Next, the extracted data are used for the target state estimation via the SCK-PDAF, as shown in the following.

3.2.1.AP-ML estimator

At the kth tracking interval,we estimate the azimuth angle φk,the real and imaginary parts of target complex gain αdRkand αdIk,the target elevation angle θdk,and the real and imaginary parts of integrated complex reflection coefficient ˉqRkand ˉqIkby the ML-type estimator.The ML estimator shows good performance in single snapshot,and can approach the CRLB asymptotically.However, the six-dimensional search in the traditional ML estimator may be computationally expensive.To solve this problem efficiently,the AP-ML estimator,which integrates the ML and AP operations, is utilized.

According to the receive signal model(1),the ML function can be formulated as62

where tr(.)is the trace operation,rEk（l）is the receive signal vector of the lth snapshot at the kth tracking interval,and L is the number of snapshots.

In applications,a six-dimensional search is computationally expensive.The AP operation is employed to reduce the computational burden.The AP operation transforms a multidimensional optimization problem into a set of one-dimensional optimization problems, and much computational complexity is saved34.

We define the parameter vector to be estimated as

The detailed steps of the AP-ML estimator are shown in Algorithm 1.Note that the AP-ML estimator is also used for the target distance estimation, and it is equivalent to the ML estimator at that time.

Input: the iteration limit Litmax, the preset value ε0Initialization:Obtain Θ （0）■k = ^φ （0 ）k ;^α （0）dRk;^α （0）dIk;^θ（0）dk;^ˉq（0）Rk;^ˉq（0）Ik■T,and set lit= 1, J = 6, O1= 0, O2= Inf.while |O1-O2|>ε0&& lit< Litmaxfor j = 1:J Calculate ^θ（lit）j by solving Eq.(22)if j==1 Obtain O1=||Θ（lit）k ||2elseif j==J Obtain O2=||Θ（lit）k ||2end if end for end while Output:Θ （lit）■k = ^φ （lit）k ;^α （lit）dRk;^α （lit）dIk;^θ（lit）dk ;^ˉq（lit）Rk;^ˉq（lit）■T Ik.

3.2.2.SCK-PDAf

Since the detailed steps of SCKF and PDAF are mature, we just give the indicated procedure considering the space limitation.

3.3.Fusion results using CI rule

Next,all the validated measurements are utilized to update the posterior PDF of the qth target in the ith local radar

Thus,consistent fusion results irrespective of unknown correlations among local radars are obtained.In addition, both the communication bandwidth and the computational complexity are reduced compared with the centralized fusion framework12,37–39,40,41,43,45.

Remark 4.The intractability in low-angle tracking mainly originates from the multipath effects, the unknown ground reflection coefficient, and the time-varying false alarms.60The AP-ML estimator and the SCK-PDAF are adopted to tackle these issues in this paper.To achieve better state estimate accuracy, other signal processing level-based techniques6and data processing level-based techniques65are able to be applied.However, as aforementioned, this paper is devoted to improving the low-angle tracking performance by the resource allocation strategy, and the devotion of the above-mentioned techniques is out of this paper’s scope.

4.JBTAPA strategy

Mathematically speaking,the JBTAPA problem can be formulated as an optimization model,which is aimed at achieving the best overall tracking accuracy subject to the constraints of available beams and power budget in a colocated VHFMIMO radar network.Here, we first introduce the BTA vector ukand PA vector Pk

4.1.PC-CRLB for low-angle tracking

In resource-aware design concept, many works12,13,36,39,40,44adopt the PCRLB as the optimization criterion, since it provides a lower bound for any unbiased estimator.However,the derivation of PCRLB is expected over the estimated state and the measurements from the initial time to the current time.In this case, the useful measurement information, which has been actually achieved by time k-1, is averaged out, leading to an offline bound.Thus, we prefer to use the PC-CRLB,48which is based on the most recently achieved measurement.The following inequality always holds for the PC-CRLB45

Note that the false alarms are still uniformly distributed in this gate,and the probability that one measurement is from the target is

The expectation operator conditioned on Zq;1:k-1in Eq.(38)could be dropped, and a clearer expression is.

Remark 5.A more rigorous name forin Eq.(38) is the generalized CRLB.This is because the true target parameters are unknown in applications,and we have to substitute the estimated parameters into it.However, there is a large gap between the estimate errors and the CRLB in low-SNR case,resulting in the mismatch between the true CRLB and the generalized CRLB.67This point will be further discussed in Section 5.1.

Remark 6.It is demonstrated in Ref.45that the PC-CRLB provides a tighter bound than the standard PCRLB.This is also applicable to the derived PC-CRLB for low-angle target tracking, which will be confirmed by the following simulations.

4.2.Optimization model formulation

In multi-target tracking, we intend to ensure that the tracking error of each target is not too large.Thus, the objective function is established as

The optimization model is then formulated as

where the first constraint means that the available beams of each radar are restricted by nr, the second constraint means that at least one beam should be assigned to the qth target to avoid the track loss, meanwhile, the number of beams cannot exceed nc,the third constraint means that total power budget in each radar is finite, and in the last constraint, Pminand Pmaxare used to bound the allocated power to each target.

The problem described in Eq.(46)is a mixed integer nonlinear optimization problem, which is known as NP-hard.Specifically, its intractability is from the following aspects:(A) the BTA variable is in the binary form, i.e., either 0 or 1, and the PA variable is in the continuous form; (B) the two variables are both coupled in the constraints and objective function;(C)after fixing the BTA variable,the optimization problem w.r.t.the PA is still nonconvex.A general solution is partitioning the two variables and solving them separately.Specifically, we can use the Exhaustive Search(ES) method to obtain the BTA results, and then use the modified convex optimization tools68to solve the PA problem.However, its time consumption is unaffordable for the real-time demand in radar systems.An efficient solution technique is proposed in the following.

4.3.Two-stage-based solution

We refer to the proposed technique for solving Eq.(46)as twostage-based algorithm.In the first stage, we use an intermediate variable δi;q;k=ui;q;kPi;q;kto reformulate the optimization problem (46) as.

Though the problem in Eq.(48) is nonlinear and nonconvex, it can be solved by the Modified Gradient Projection(MGP) method.68Intuitively speaking, the solution of Eq.(48) describes the beam contributions to each target, which provides the basis for a BTA algorithm designation.

In the second stage, we first use the BTA algorithm, as shown in Algorithm 2,to obtain a set of alternative beam allocation results.The main mechanism of our algorithm is choosing the most promising element in the solution to Eq.(48) as the assigned beam.69Since only the maximum available beams of radar i and allocated beams to target q are predefined, i.e.,ncand nr,we set a null set uk,optsetto load all the alternatives.In Step 3, the largest element in the normalized δkis sequentially selected while satisfying the constraints in Eq.(48).Specially,after one element being selected, the corresponding column is set to be 0I×1,which ensures that at least one beam is assigned to one target.The output of Step 3 is indeed a feasible solution of uk, where all the constraints in Eq.(48) have been satisfied.However, a further search of possible solutions needs to be conducted, as shown in Step 5.

Step 1.Initialize δ0= 0I×Q, uk= 0I×Q, uk,optset= [] (null set).Step 2.Acquire the solution δkof the Eq.(48) by the MGP method,δk= reshape(δk,I,Q), normalize δi,q,k= δi,q,k/Pi,totaland set δ0= δk, where the operation reshape(A,a,b) reshapes the matrix A as an a × b matrix.Step 3.while δ0≠0I×QFind the index [r,c] = arg max{δk}Set uk(r,c) = 1 If sum(uk,1) ≤nc× 11×Qand sum(uk,2) ≤nr× 1I×1δ0(:,c) = 0I×1, δk(r,c) = 0 else δ0(r,c) = 0, uk(r,c) = 0 end if end while Step 4.uk,optset= [uk,optset, reshape(uk,I × Q,1)].Step 5.for j = 1: I × Q Find the index [rj,cj] = arg max{δk}Set uk(rj,cj) = 1 If sum(uk,1) ≤nc× 11×Qand sum(uk,2) ≤nr× 1I×1uk,optset= [uk,optset, reshape(uk,I × Q,1)]else uk(rj,cj) = 0 end if end for Step 6.Output the optimal beam-target assignment set uk,optset.

After assigning the beams to the targets, the next problem for Eq.(46) is how to optimally allocate the power resource of each radar to each target.

where J is the number of BTA alternative solutions stored in uk,optset.In Eq.(49), the optimal PA is obtained by enumerating all possible uk,jand choosing Pk,jwith the lowest F(Pk,j).Similar to Eq.(48), Eq.(49) can be solved by the MGP method.Thus, the optimal BTA vector uk,optand PA vector Pk,optcan be obtained by the proposed two-stage-based algorithm, where all possible uk,jare solved in the first stage and the corresponding Pk,jare solved in the second stage.In the algorithm applications, we have the following remarks.

Remark 7.The convergence of the proposed two-stage-based solution technique can be guaranteed.The MGP method is the main solver in our technique, and its convergence and performance have been verified in Ref.39and Ref.45In addition,all the alternatives are enumerated and compared in Eq.(49) to output the final solution.Such a scheme can guarantee the optimality of Eq.(49).

Remark 9.The difficulty of solving Eq.(46)lies in designing an efficient BTA algorithm, since the PA can be obtained via modified convex optimization methods after the BTA matrix being determined.In Ref.,12a heuristic rounding technique is put forward.However, it obtains only one BTA matrix with the maximum available beams in radars(nrI)or the maximum beams can be allocated to the targets (ncQ).Its optimality/sub-optimality cannot be guaranteed.This is because if the couple of BTA with worse geometry relationship or longer distance is added,it naturally dilutes the limited power resource and results in inefficient utilization42.

Remark 11.From the optimization model (46), we can see that the resource allocation results are affected by many factors,e.g.,the variation of target reflectivity,the target position(which contains distance, elevation angle and azimuth angle), and the ground complex reflection coefficient.70However, we cannot list all the effects on resource allocation results due to the space limitation.In the simulations,we mainly investigate the influence of target reflectivity, which is a salient characteristic in low-angle tracking33.

5.Simulation results and analysis

5.1.Effectiveness and efficiency of JBTAPA strategy

The VHF-MIMO radar positions and target initial positions are randomly placed in a 200 km × 200 km area.The target initial heights are random values drawn from the interval of[0.3,0.8] km.The target velocities and accelerations are elaborately initialized to guarantee the validity of the multipath signal model.Specifically, the target velocities are toward radars and the accelerations are not very high to support the Rayleigh criterion and the elevation angle being within one half-power beamwidth during simulations.When the initialized parameters do not support the two conditions, the values will be regenerated.The maximum number of beams generated by each radar is nr= 3, which also implies that the number of subarrays in each radar is nr= 3, and the maximum number of allocated beams to each target is nc= 3.The target RCSs w.r.t.radars at each frame are set as 1.A rough reflective surface with a fluctuant standard deviation of 0.1 m is assumed.The height of the reflective point (within 1 m) is unknown for the estimators,the relative dielectric constant is 4,and conductivity is 1×10-5S/m.55-58The tracking interval is Ts=2 s,and K = 30 frames of data are utilized to support each simulation.

Fig.2 Comparison between generalized CRLB and true CRLB in different SNR cases.

First, we show the difference between the generalized CRLB and the true CRLB in different SNR scenarios, which is accomplished by controlling the noise power from 103.5× 10-11W to 1 × 10-11W.In this case, we set Q = 1.

Fig.2 show the comparison between generalized CRLB and true CRLB on distance, azimuth angle, and elevation angle,respectively.We can see that there is a large gap between the two items in the low SNR case, whereas they are very close when the SNR is higher than 5 dB.This finding verifies the effectiveness of AP-ML estimator and provides the criterion for our optimization model.In the following simulations, we set the SNR to be larger than 5 dB to support the validation of the established optimization model.

Then, we focus on the verification of the effectiveness and efficiency of the JBTAPA strategy.The fixed BTA with uniform PA allocation (FBTAUPA) strategy, the Joint Beam Selection and Power Allocation (JBSPA) strategy,12and the Joint Beam and Power Scheduling (JBPS) strategy13are adopted as the benchmarks.In the FBTAUPA strategy, each local radar uses Q beams to illuminate all the targets, and the power is uniformly allocated in each beam.Hence, this scheme ignores the physical limitations of nr= 3.Additionally, considering that the proposed JBTAPA strategy involves the problem relaxation procedure and may not always offer the optimal solutions, the integration of the ES and the MGP(ES-MGP) method is introduced to analyze the performance loss.The ES-MGP method enumerates all possible BTA matrices,and at the same time,solves the PA problem via the MGP method.Then, the alternative solution with the minimum objective function is chosen as the optimal solution.The posterior conditional-CRLB (PC-CRLB) (not the predicted) and the corresponding Root Mean Square Error (RMSE) are adopted as the optimization criterion, and the later item is defined as.

Fig.3 Performance comparison among different strategies withI = 3.

Fig.3 compares the estimate accuracies achieved by the five strategies with the different number of targets and radars averaged over K frames.The number of radars is I=4.It is noticeable that, for a specific Q, the performance tendencies of each strategy with the different number of radars (I = 2 - 6) are accordant, and we just present the representative results due to space limitation.This is because only the total power budget is improved with the increase of I,which results in higher SNR.Additionally,the data of the ES-MGP algorithm when Q>6 are unavailable due to its extremely large time consumption,which will be shown in Table 1.We can see that when the number of targets is small,e.g.,Q=2,the estimate accuracies provided by the five strategies are close.However, the performance differences become evident as the number of targets increases.This is due to the fact that more targets bring more degrees of freedom in resource allocation.In addition,we can see that the PC-CRLB, which is closer to the RMSE than the standard PCRLB, provides a favorable alternative for resource allocation.As to the five strategies, the FBTAUPA strategy shows the worst tracking accuracy due to failing to consider the geometric relationship of targets w.r.t.local radars.The JBSPA strategy outperforms the FBTAUPA strategy.However, as stated in Remark 9, the heuristic rounding technique only obtains a feasible solution but cannot guarantee its optimality.Thus,it shows higher estimate errors than the other three strategies.The JBPS strategy provides higher RMSE than the proposed strategy, and the estimate precision of the proposed JBTAPA strategy and the ES-MGP method are very close, which confirms the effectiveness of the JBTAPA strategy.

The runtime comparison of the four strategies is shown in Table 1.Combining the results in Table 1 and Fig.3, we can see that the proposed strategy needs shorter runtime while providing roughly the same solutions as the ES-MGP.Though the JBPS strategy sometimes offers the competitive estimate accuracy, its longer runtime should be highlighted.This is because the BTA matrix in the JBPS strategy shrinks based on the evaluation of objective function in an iterative manner.In each iteration, all the elements in the BTA matrix must be enumerated.In contrast, in our strategy, the importance of each element in the BTA matrix has been determined in the first stage by one-time calculation.Thus,much computational burden is saved.

5.2.Influence factors on JBTAPA strategy

Fig.4 Square root RCSs in α2case.

Table 1 Runtime comparison among different strategies.

Table 2 Initial parameters of targets.

Fig.5 Worst cases of RMSEs, PC-CRLBs and PCRLBs achieved by different strategies in Case 1.

Fig.6 Resource allocation results achieved by JBTAPA strategy in case 1.

Fig.7 Worst cases of RMSEs, PC-CRLBs and PCRLBs achieved by different strategies in Case 2.

In this subsection,we focus on the influence factors on the proposed JBTAPA strategy.A colocated MIMO radar network with I = 4 and targets being tracked with Q = 4 are preset.The coordinates of radars are (-2 × 104,1.4 × 105,5), (0,0,5),(8 × 104, -2 × 104,5), and (4 × 104,1.5 × 105,5), for i = 1,2, 3 and 4.The initial parameters of targets are shown in Table 2.

To reveal influence factors on the JBTAPA strategy, we consider two kinds of target RCS model:

α2: the square root RCSs of target 1 and 2 are shown in Fig.4, and others are set the same as α1.

The following two cases are simulated.

5.2.1.Case 1: α1

In this case, the target RCSs w.r.t.each radar are assumed to be the same.Thus, the resource allocation results are mainly affected by the angle spread and the distance of targets.

The worst cases of RMSEs, PC-CRLBs and PCRLBs achieved by the four strategies are given in Fig.5.We can see that the RMSEs approach the PC-CRLBs as the simulation processes, which verifies the correctness of the established optimization model.In addition, the JBTAPA strategy results in the lowest estimate error, verifying its effectiveness.

Fig.6 presents the resource allocation results in Case 1,where the block with the mazarine color denotes that ui,q,k= Pi,q,k= 0, and other colors denote the normalized PA ratio to the target from each radar:

5.2.2.Case 2: α2

We then consider the effects of the target RCS on resource allocation.The deployment of local radars and targets is the same as that in Case 1.The worst cases of RMSEs, PCCRLBs as well as PCRLBs of the four strategies are depicted in Fig.7, where the proposed strategy still achieves the lowest estimate error.

Fig.8 Resource allocation results achieved by JBTAPA strategy in Case 2.

The JBTAPA results are depicted in Fig.8.Comparing Fig.8(a) with Fig.6(a), we can see that radar 3 no longer focuses on target 1 due to the reduction of target RCS.Hence,radars 1 and 2 are employed to track target 1 to maintain the system performance.In addition, the power contribution of radar 2 to target 1 is evidently improved since the target reflectivity becomes better compared with Case 1.A similar phenomenon can be seen by comparing Fig.8(b) with Fig.6(b),where more power resource from radar 3 is allocated to target 2, owing to the target reflectivity improvement.This can be interpreted that,in the JBTAPA strategy,the radar with better observing conditions is often designated to track the corresponding targets,for the efficient utilization of power resource.

6.Conclusions

A JBTAPA problem was established for the colocated VHFMIMO radar network tracking low-angle targets.The APML estimator and the SCK-PDAF were adopted in each local radar to mitigate multipath and false alarm effects.To provide a stable estimate,we applied the CI fusion rule to a distributed fusion architecture and derived the PC-CRLB as the optimization criterion.A two-stage-based technique was proposed for solving this problem considering that the JBTAPA optimization model is NP-hard.The simulation results show effectiveness and efficiency.In addition, the results also imply that the target reflectivity is one of the main factors that influence the resource allocation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Nature Science Foundation of China (No.62001506).

Appendix A.Derivation of CRLB of measurement error

According to Eq.(1) and Eq.(20), the receive signal vector of whole subarray is rewritten as

and the interested parameters are the azimuth angle φ,the real part of target complex gain αdR, the imaginary part αdI,elevation angle θd,the real part of the integrated complex reflection coefficient ˉqR, and the imaginary part ˉqI.The partial derivatives of rE(l) w.r.t.φ are: