Adapitive waveform design for distributed OFDM MIMO radar system in multi-target scenario

Haitao WANG,Junpeng YU,Wenzhen YU,De BEN

Nanjing Research Institute of Electronics Technology,Nanjing 210039,China

1.Introduction

Recently,distributed Orthogonal-Frequency-Division Multiplexing(OFDM)Multiple-Input Multiple-Output(MIMO)radar system has attracted a lot of interest because this kind of radar system integrates the performance potentials of distributed MIMO radar and OFDM waveform.1,2It has been demonstrated in extensive literature that MIMO radar has more outstanding performance than the Single-Input Single-Output(SISO)counterpart.3Furthermore,compared with the collocated MIMO system,distributed MIMO radar has widely separated antennas,so it provides additional space diversity which is likely to benefit detection task.4,5

Waveform design is one of the most important issues for MIMO radar.6,7Different from phase-encoding orthogonal waveform,OFDM waveform provides the frequency diversity that may also be beneficial to radar detection.8–10What’s more,one target usually has the different Doppler frequencies for the separate antennas of distributed radar system and OFDM waveform possesses high Doppler tolerance in general,8so this kind of waveform fits distributed radar especially.The OFDM waveform used in our work is composed of a number of Linear Frequency-Modulated(LFM)signals with the same bandwidth but different sub-carriers.9,10These LFM sub-waveforms are orthogonal to each other because they are separated in frequency domain.

Adaptive Waveform Design(AWD)has become an active topic in radar research domain over the past decades.11–15This technique devotes to improving radar performance by adjusting transmitted waveform adaptively according to the priori knowledge of the environment and/or the radar system.11Many researchers have demonstrated the advantages of AWD for various radar configuration and operation tasks.Typically,the AWD approaches for several types of MIMO radars are discussed in Refs.13–15while the AWD algorithms are proposed for SISO radar systems in Refs.11,12Note that Ref.12involves the scenario of multiple targets and that the AWD issues in Refs.12,13are presented from the view of information theory.

The contribution of our work in this paper is threefold.Firstly,we regard detection and estimation of multiple targets for distributed OFDM MIMO radar system as a problem of sparse recovery and derive the sparse measurement model of this radar system.The algorithm based on Decomposed Dantzig Selectors(DDS)is applied to our sparse recovery.9,16The study in Ref.16shows that DDS is more efficient than standard Dantzig Selector(DS)if both the system matrix and the sparse vector of the measurement model have the relevant block structures.Note that this sparse measurement model is also a key foundation to apply the technique of Compressed Sensing(CS)later.

Secondly,we propose a novel AWD approach based on the Multi-Objective Optimization(MOO)for distributed OFDM MIMO radar system.One objective function we construct is to minimize the upper bound of the recovery error,and the other is to maximize the return energy of the weakest target.The former commits to reduce the recovery error and the latter devotes to improving the detection and estimation for the weak targets.These two objective functions can be solved by the result in Refs.17,18and some convex optimization approaches19respectively.Furthermore,we apply the popular Nondominated Sorting Genetic Algorithm II(NSGA-II)to solve our MOO problem in this paper.20Generally,there exist a set of optimal solutions for a MOO problem,which are called Pareto-optimal solutions.21The complex amplitudes of the transmitted OFDM waveform can be adjusted adaptively according to the optimal solutions of this MOO problem.

Thirdly,we introduce the technique of CS to the proposed scheme in this paper.The acceptable sensor performance is likely to be realized by this emerging technique when available samples are reduced dramatically.22It has a special meaning to airborne and space-borne radar systems because their capability of data transmission is restricted by wireless links.

Several numerical examples are given in this paper to show the detection and estimation improvement induced by the proposed AWD approach for distributed OFDM MIMO radar system.Their results show that,evaluated by empirical Receiver Operation Characteristics(ROC)and Root Mean Square Error(RMSE),the AWD approach based on either of two objective functions performs better than the traditional manner in which the OFDM waveform with fixed uniform amplitudes are transmitted,and that the AWD approach based on the proposed MOO technique behaves best.We also provide the numerical simulations to assess the system performance when the CS technique is introduced to the proposed scheme in this paper.Their results show that the proposed MOO-based AWD approach improves the system performance of distributed OFDM MIMO radar markedly when the available samples are reduced severally and the CS technique is applied.

The rest of the paper is organized as follows.We derived the sparse measurement model of distributed OFDM MIMO radar system in Section 2.Next,the DDS-based sparse recovery algorithm is stated brie fly in Section 3.We proposed a novel AWD approach based on the MOO in Section 4.Some numerical examples and conclusions are provided in Section 5 and Section 6 respectively.

2.Measurement model

It is assumed that the distributed OFDM MIMO radar system has MTtransmitters and MRreceivers,and that there are K point targets which move in a two-dimensional(2-D)plane.Note that our work can be generalized easily to the threedimensional(3D)case.The kth target locatesand has a velocityin a Cartesian coordinate system.The ith transmitter and the jth receiver locate at ti= ［tix，tiy］and rj= ［rjx，rjy］respectively.Each of the transmitters sends the OFDM signal at one sub-carrier.We assume that the total transmitted energyper pulse where Eiis the energy of the emitting waveform at the ith sub-carrier.

The band-pass signal arriving at the jth receiver can be written as

where aiis the complex amplitude weight of the waveform emitted by the ith transmitter.are the signal attenuation,path delay and Doppler shift respectively corresponding to the kth target between the ith transmitter and the jth receiver.Note that the signal attenuation depends on many factors,such as the signal propagation path,target scattering and system loss.fi=fc+iΔf is the ith sub-carrier of the emitting OFDM signal where fcis the operational carrier frequency.Δf=B/（MT+1）is the spacing between the adjacent sub-carriers where B is the total bandwidth of the transmitted OFDM signal.The duration time of the transmitted waveform is defined as T.

where 〈·〉is the inner product operator and c is the speed of the electromagnetic wave.andare the unit vector from the ith transmitter to the kth target and the unit vector from the kth target to the jth receiver respectively.

where eij（n）represents the additive noise,κ represents the set of the targets which contribute the energy to this match filter at the sampling index n,and Tsrepresents the sampling interval.

We define the target state vector ζ= ［px，py，vx，vy］Twhich represents the location and velocity of a target.The target state space is discretized into L grids ｛ζ（l），?l=1，2，···，L｝ which contain all the possible states.We define

What’s more,if the target state vector of the kth target is ζ（l）,we define

xijin Eq.(9)is a L×1 sparse vector which has only K nonzero elements.

We stack xij（n）,yij（n）and eij（n）into a LMR-dimensional column vector xi（n）and two MR-dimensional column vectors yi（n）and ei（n）as Eqs.(10)–(12)for every i respectively.Similarly,we exploit the matrices ψij（n）and aito form a blockdiagonal matrix ψi（n）and a diagonal matrix Aias Eqs.(13)and(14)for every i respectively.

In Eqs.(10)–(14),diag｛·｝ represents a diagonal matrix whose entries on the main diagonal are given by ｛·｝,and blkdiag｛·｝represents a block-diagonal matrix whose block matrices on the main diagonal are given by ｛·｝.Next,we stackintoaLMTMR-dimensional column vector x and two MTMR-dimensional column vectors y（n）and e（n）as Eqs.(15)–(17)respectively.Furthermore,we exploit the matricesandto form a LMTMR× MTMRblock-diagonal matrix ψ（n）and a diagonal matrix A as Eqs.(18)and(19)respectively.

The sparse vector x has KMTMRnonzero entries.

The velocities of the considered targets are far less than the speed of light,so their Doppler effect in one pulse dwelling interval is negligible and multiple pulses are needed to estimate their velocities.We stackandto form y, e and ψ as Eqs.(21)-(23)respectively.

From Eqs.(21)–(23),we obtain the sparse measurement model of distributed OFDM MIMO radar system as

The nonzero elements of x represent the signal attenuation values.

In practice,the measurement data collected by each receiver of distributed MIMO radar system are sent to a shared processor in the fusion center.The shared processor stacks the data properly and obtains the measurement vector y.In other words,the data are not processed locally by any individual receiver because such an un-centralized procedure may lead to suboptimal results.1,10,15

We derive the sparse measurement model of distributed OFDM MIMO radar system in this section.In the next section,the sparse recovery algorithm based on decomposed Dantzig selectors will be stated.

3.Sparse recovery

In this section,we describe the sparse recovery algorithm used in our work brie fly.The purpose of sparse recovery is to estimate the sparse vector x from the measurement vector y and the approach based on Dantzig selector is one of the most popular sparse recovery approaches.5,16DS regards the estimate of x as a solution of the following ?1-regularization problem:

Note that the vector x in Eq.(24)has a block structure as

Every sparse sub-vector xiin Eq.(26)has KMRnonzero entries.Furthermore,the system matrix ψ in Eq.(23)can also be rewritten as

Every NMTMR× LMRblock matrix ψiin Eq.(27)can be described as

where OMR×LMRis a MR×LMRzero matrix.The block matrices ψis are orthogonal to each other,i.e.,=0 for i1≠i2.

Based on the block structures of x and ψ,we can apply a revised vision of DS,namely decomposed DS.In this algorithm,MTsmall Dantzig selectors which correspond to MTblocks of x and ψ are run concurrently and each of the small Dantzig selectors is expressed as

4.Adaptive waveform design

In this section,we propose an AWD algorithm based on the MOO for distributed OFDM MIMO radar system.As a result,the weight vector a of the complex OFDM waveform amplitudes is adjusted to minimize the upper bound of the recovery error and to maximize the weakest target return in perfect compromise.We study two objective functions and their respective optimal solutions,and then construct the multiobjective optimization.

4.1.Minimizing upper bound of error

Lots of merits about the system matrix ψ have been proposed to evaluate the performance of sparse recovery methods.One of the most popular merits is Restricted Isometry Constant(RIC).However,it is very difficult to compute RICs for many types of system matrices.Refs.17,18propose another merit which is much easier to calculate,namely ?1-Constrained Minimal Singular Value(?1-CMSV).In our case, ?1-CMSV is defined as

where

The detailed physical explanations of the function s1（x）and?1-CMSV can be found in Refs.17,18

Considering Theory 2 in Ref.18,the measurement model Eq.(24)and the principle of DDS,we can easily prove that the sparse vector x and its estimate^x obtained by DDS satisfy the following inequality:

The above inequality is demonstrated in the supplement file owing to the limit of space.

Note that each ψiin Eq.(27)can be represented asand thus there isIn order to minimize the upper bound of the recovery error,we construct an optimization problem based on Eq.(32)as

where the vector a= ［a1，a2，···，ai，···，aMT］T.The solution of the above optimization problem is easily obtained by the Lagrange-multiplier method as

4.2.Maximizing the weakest target return

When there are multiple targets in radar scene,we shall pay more attention to the detection and estimation of weak targets.12The transmitter-receiver pairs of a distributed radar system monitor the targets from different angles and the attenuations of one target corresponding to these pairs are usually different.Every transmitter of a distributed OFDM MIMO radar system sends the signal at one sub-carrier,so we can adjust the complex amplitudes of the sub-carrier waveforms to enhance the returns of the weak targets and to improve their detection and estimation.Accordingly,we propose the second objective function as

4.3.Multi-objective optimization

Two objective functions,namely Eqs.(33)and(35),have been proposed above.Next,we propose a constrained MOO problem based on the two functions to design the waveform of distributed OFDM MIMO radar system as

As mentioned in the introduction,we solve this MOO problem by the NSGA-II20in the simulations.After the optimal solution is obtained,the fusion center of the radar system will send the waveform information to the radar transmitters and the new OFDM waveform will be radiated in the next CPI.

5.Numerical results

We provide several numerical simulations in this section to assess the performance improvement brought by the proposed AWD approach based on the constrained MOO technique.

The distributed OFDM MIMO radar system that we simulate contains three transmitters and three receivers.Their positions are identified in a Cartesian coordinate system.The transmitters are located at t1= ［1，0］km,t2= ［2，0］km and t3= ［3，0］km respectively,and the receivers are located at r1= ［0，3］km,r2= ［0，2］km and r3= ［0，1］km respectively.The radar system parameters are set as follows:carrier frequency fc=1GHz,available bandwidth B=100MHz,sub-carrier amount MT=3, sub-carrier spacing Δf=B/（MT+1）=25MHz,pulse width T=1/Δf=40ns,pulse repetition interval Tp=10μs,and pulse number in one CPI N=128.We assume that there exist three targets in the simulated scenario.The target state spaces about position and velocity are separated into 9×9 and 5×5 grids respectively.Thus,the total number L of possible target states is 2025. Furthermore, the true sparse vector x has LMTMR=18225elementsandthereareKMTMR=27 nonzero elements of them.These targets are located at p（1）= ［1.1，2.8］km,p（2）= ［0.8，2.8］km and p（3）= ［1.0，2.6］km respectively, and their velocities are given as v（1）= ［120，100］m/s, v（2）= ［110，110］m/s and v（3）=［130，130］m/s respectively.The attenuationss corresponding to all the combinations of the targets,transmitters and receivers are set as

We assume that the elements of the noise vector e come independently from the Gaussian distribution with zero mean and the variance σ2in our simulations.The Signal to Noise Ratio(SNR)for distributed OFDM MIMO radar system is defined as

In order to verify the performance of our scheme,we adopt two classical merits,namely empirical ROC and RMSE.Next,we describe brie fly how to obtain the two merits in the simulations.

As mentioned earlier,the most significant KMTMRelements of the reconstructed vector^x will be on the indices where the nonzero elements of the true sparse vector x lie if the perfect recovery is realized.We define a length-L vector~x as

Every element of~x integrates the entries of^x which correspond to all the transmitter-receiver pairs for one of the L grids in the target state space.In a similar way,a length-L vectoris structured from x.

We assume that there are nTelements of the vector~x which are considered as the target responses after a Monte Carlo run.If nDelements of these nTelements lie on the indices with true targets,the empirical probabilities of false alarm(PFA)and correct detection(PD)are calculated respectively as

where nFA=nT-nD.We obtain the RMSE value by calculating.Note that the approach by which we gain the empirical ROCs here is different from the classical one.

Four kinds of waveforms are transmitted by the radar system in our simulations and they are:

Case 1.Traditional OFDM waveform with fixed uniform weights of complex amplitudes.

Case 2.Dynamic waveform designed adaptively according to the minimization of the upper bound on the recovery error,as discussed in Section 4.1.

Case 3.Dynamic waveform designed adaptively according to the maximization of the weakest-target return,as discussed in Section 4.2.

Case 4.Dynamic waveform designed adaptively by the proposed MOO technique,as discussed in Section 4.3.

As mentioned in the introduction,we use the NSGA-II to solve the MOO problem of Eq.(36).The parameters of this algorithm are chosen as follows:mutation probability=10%,crossover probability=90%,number of generations=50 and population size=500.When the NSGA-II is carried out,we apply the relaxed constraint 0.999≤aHa≤1.001 according to the premise aHa=1 of Eq.(36).The optimal solutions and the corresponding values of two objective functions at the zeroth, fifth and fiftieth generations are shown in Figs.1 and 2 respectively.In Fig.1,we use|a1|,|a2|and|a3|as the axes of Cartesian coordinates and the constraint aHa=1 makes the solutions of every NSGA-II generation lie on the surface of the first unit octant.Note that the return energy of the weakest target is normalized by the noise level.It is shown that almost all of the solutions arrive at or come near the Pareto-front at the end of the fifth generation.We choose one of the solutions on the Pareto-front after the fiftieth generation to optimize our OFDM waveform.

Fig.1 Optimal solutions obtained by the NSGA-II.

Fig.2 Optimal values of two objective functions obtained by the NSGA-II.

The DDS approach described in Section 3 is applied for the sparse recovery in our simulations.The empirical ROCs and RMSEs corresponding to the aforementioned four kinds of waveform are shown in Figs.3 and 4 respectively.They display that three AWD approaches mentioned in Section 4 all improve the radar system performance in terms of empirical ROC and RMSE.However,only the AWD approach based on the proposed MOO realizes both the best empirical ROC and the RMSEs.

Fig.3 Empirical ROCs corresponding to four kinds of transmitted waveforms.

Next,as stated in the introduction,we try to evaluate the effect of the proposed MOO-based AWD approach when the compressed sensing technique is integrated into our scheme.Note that the measurement vector y is sparse in the space spanned by all the columns of the matrix ψ.Therefore,we

Fig.4 Estimation RMSEs corresponding to four kinds of transmitted waveforms.

can exploit the samples which are far less than the complete size of y to reconstruct the sparse vector x in accordance with the CS theory.22Every receiver of the distributed OFDM MIMO radar system projects its received data by one Gaussian sequence with length of NCSwhere NCS?NMTMRand sends the compressed data to the fusion center of the radar system.The shared processor in fusion center stacks the data from all the receivers appropriately and the measurement model with CS is expressed as

where Θ is the （NCS）× （NMTMR）sensing matrix.The entries of this matrix come form an independent Gaussian distribution.The compression level of the samples is defined as NCS/（NMTMR）× 100%.We show the advantages of the proposed AWD approach based on the MOO when the available samples are reduced severally and the CS technique is introduced.Figs.5 and 6 show that our MOO-based AWD approach gives the significant improvement in terms of both empirical ROC and estimation RMSE respectively when there are 50%of the samples.It is shown in Figs.7 and 8 that the same conclusion can be drawn even when there are only 25%of the samples.

Fig.5 Empirical ROCs with and without MOO-based adaptive waveform when 50%of samples are available.

Fig.6 Estimation RMSEs with and without MOO-based adaptive waveform when 50%of samples are available.

Fig.7 Empirical ROCs with and without MOO-based adaptive waveform when 25%of samples are available.

Fig.8 Estimation RMSEs with and without MOO-based adaptive waveform when 25%of samples are available.

6.Conclusions

We propose a novel approach of AWD based on the MOO so as to improve the system performance of distributed OFDM MIMO radar in multi-target scene.We translate the detection and estimation task of this radar system into the issue of sparse recovery and derive the sparse measurement model.The recovery algorithm based on DDS is applied according to the block structures of the sparse vector and the system matrix.Next,we construct the constrained MOO problem which minimizes the upper bound of the recovery error and maximizes the weakest target return in optimized compromise.This MOO problem is solved by the NSGA-II in our simulations.Its solution provides the optimal complex weights of the transmitted OFDM waveform amplitudes.

Several simulations are given in this paper to demonstrate the validity of the proposed AWD algorithm.Their results show that,compared with the traditional fixed uniform OFDM waveform,the MOO-based adaptive waveform leads to the significant performance improvement of the distributed OFDM MIMO radar system.We also demonstrate that our proposed AWD approach based on the MOO affords the remarkable system performance improvement when the available samples are reduced severely and the CS technique is applied.

Acknowledgement

This study was supported by the National Basic Research Program of China(No.613205212).

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