WEN Jinfang,YI Jianxin,WAN Xianrong,GONG Ziping,and SHEN Ji

School of Electronic Information,Wuhan University,Wuhan 430072,China

Abstract: This paper considers multi-frequency passive radar and develops a multi-frequency joint direction of arrival (DOA)estimation algorithm to improve estimation accuracy and resolution.The developed algorithm exploits the sparsity of targets in the spatial domain.Specifically,we first extract the required frequency channel data and acquire the snapshot data through a series of preprocessing such as clutter suppression,coherent integration,beamforming,and constant false alarm rate (CFAR)detection.Then,based on the framework of sparse Bayesian learning,the target’s DOA is estimated by jointly extracting the multi-frequency data via evidence maximization.Simulation results show that the developed algorithm has better estimation accuracy and resolution than other existing multi-frequency DOA estimation algorithms,especially under the scenarios of low signal-to-noise ratio (SNR) and small snapshots.Furthermore,the effectiveness is verified by the field experimental data of a multifrequency FM-based passive radar.

Keywords: multi-frequency passive radar,DOA estimation,sparse Bayesian learning,small snapshot,low signal-to-noise ratio(SNR).

1.Introduction

Passive radars detect and track targets through a third-party illuminator of opportunity (IO) such as radio,television,mobile communications,and navigation signals.It has a lot of unique characteristics,like excellent low-altitude coverage,good concealment,and immunity to anti-radiation missiles [1-3].For the target localization in the bistatic passive radar,the direction of arrival (DOA) is a key parameter.However,when the aperture of the antenna array is small,the theoretical performance of DOA estimation is relatively poor in the case of low signal-tonoise ratio (SNR) and small snapshots.Therefore,it is still a challenge and an important topic to improve the DOA estimation in passive radars.

Note that there may be multiple frequency channels available in a single IO in practice.It is possible to jointly utilize the multiple frequency channels to improve DOA estimation.Nevertheless,few studies are published on this topic up to now.In [4],an initial attempt was made to estimate the DOA by collecting multiple frequency modulated (FM) channels data from the IO.As only two surveillance antennas were considered,the DOA was estimated by averaging the phase difference over multiple frequency channels.In [5],the maximum-likelihood (ML)approach was employed to estimate the DOA of the passive radar using data from multiple frquency channels and confirm the feasibility with real data.However,the ML algorithm implemented by maximizing the joint probability density function of the signal is only applicable to the case of a single target.In [6],an algorithm for estimating multiple target DOAs was proposed using the ML method with multiple FM channel data and derive the estimated Cramer-Rao bound (CRB).The work in [6] derives the theoretical performance whereas the developed algorithm therein is difficult to be implemented in practice.The DOA estimation methods mentioned above can be collectively regarded as the averaged multi-frequency(MF) method [7].It should be noted that the estimation performance of the averaged MF method may be seriously affected by a poor frequency channel.In addition,the focused MF algorithm [8] is another class of the MF DOA estimation method.The basic idea is to use a focused steering vector to approximately represent that of all the frequency channels.Then,the DOA is estimated as if there is only one frequency channel.Apparently,there is information loss due to the approximation used in the focused MF algorithm.

To solve the above-mentioned problems,we establish a complete model and introduce the sparse representation(SR) method for the DOA estimation using multiple frequency channels.In fact,the SR method that makes use of the sparsity of the target in the spatial domain provides a new idea for the DOA estimation.In the SR framework,typical DOA estimation approaches include convex relaxation [9],“ ?pregularized least square” [10],and probabilistic Bayesian DOA estimation algorithms[11,12].In recent years,SR techniques have been extended to passive radar DOA estimation.In [13,14],the SR technique was applied to passive radar to accomplish superresolution estimation of multi-target DOA with a single snapshot data.Compared with [13],Zuo et al.[14] considered the influence of array error on the DOA estimation.In [15],a joint angle-range-Doppler estimation method was presented,with one of the key features being that the snapshot data for DOA estimation is derived from range-Doppler (RD) maps generated by the SR method.Since RD map sidelobe generated by the SR method is low,this method can improve the detection of weak targets under the strong target mask and increase the estimation accuracy of trajectory.To improve accuracy,two transmitters are utilized together to sparsely estimate DOA in [16].However,it is extremely difficult to associate the target’s RD information from multiple transmitters in practice.Therefore,we only consider passive radar configurations with multiple frequency channels for a single transmitter in this paper.All in all,these researches cannot be applied directly to the scenario of multiple frequency channels considered in this paper.

The main contributions of this paper can be summarized as follows:

(i) A multi-frequency joint sparse Bayesian learning(MFJSBL) algorithm is developed to improve the DOA estimation accuracy of passive radar.To the best of the author’s knowledge,the DOA estimation of passive radar targets using the joint sparse Bayesian learning (SBL)method with multiple channel data has rarely been studied in the open literature,but it is useful in practical applications.

(ii) As combining data from multiple channels is analogous to increasing the number of snapshots,the estimation accuracy of the MFJSBL algorithm improves as the number of frequency channels increases.In addition,the MFJSBL algorithm uses the same hyperparameter constraints for the coefficient matrices of all channels,which improves the fitting probability of the algorithm.Therefore,compared with the averaged and focused MF algorithms,the developed MFJSBL algorithm has higher estimation accuracy and resolution.

(iii) The effectiveness of the MFJSBL algorithm is demonstrated by the flight data acquired by an FM-based passive radar developed at Wuhan University.In the experiment,data from two frequency channels of one transmitter are extracted to estimate the DOA.Compared with the SBL algorithm using only one frequency channel data and the averaged SBL algorithm using two frequency channels data,the MFJSBL algorithm using two frequency channels data yields a higher DOA estimation accuracy.

The remainder of this paper is organized as follows: In Section 2,we present an MF DOA estimation model for passive radar.Section 3 elaborates on the implementation of the MFJSBL algorithm in detail.Sections 4 and 5,respectively,test the performance of the developed algorithm through simulation data and real data,followed by conclusions in Section 6.

2.Model and process flow

In this section,we present the signal model,the DOA estimation model,and the entire processing scheme of sparse DOA estimation for multi-frequency passive radar.

2.1 Signal model

The geometry of wideband passive radar usingNcarrier signalsfrom the same transmitter is shown in Fig.1.The letters “Tx” and “Rx” stand for the transmitter and the receiver,respectively.

Fig.1 Geometry of wideband passive radar

The passive radar system involves two types of receiving antennas: reference and surveillance antennas.The surveillance antenna beam steers towards the direction to be surveyed,while the reference antenna points to the transmitter to receive IO signals.For the sake of simplicity,we assume that the reference signalxref(t) is exactly the direct-path signald(t).Therefore,the received signal of themth (m=1,2,···,M) surveillance antenna on thenth (n=1,2,···,N)frequency channel can be modeled as

The target signal strength is still lower than the noise after removing the clutter,thus the target SNR must be enhanced further through the coherent integration.The RD map generated by the cross-correlation [17] between themth surveillance signal of thenth frequency channel and the corresponding reference signal can be written as

is the noise after coherent integration.As shown in (4),the steering vectors of targets are preserved by the coherent integration.As a result,the target complex amplitude after coherent integration can be used as snapshots to estimate the DOA.Assemble (4) into a matrix form that can be written as

2.2 DOA estimation model

Considering that the targets in the detection scenario are sparse,we can utilize the SR technology to estimate the target DOA.The whole detection angle is discretized intogrid points,i.e.,θ=[θ1,θ2,···,θi,···,]，with each grid point representing a potential DOA.The DOA estimation model can then be built using discretized sparse MF technology as

2.3 Processing scheme

Fig.2 describes the processing scheme of the MF DOA estimation for passive radar,where the abbreviations“Ref.”,“Surv.”,“Fre.”,and “Chan.” stand for “Reference”,“Surveillance”,“Frequency”,and “Channel”,respectively.To begin with,we extract the channel data corresponding to the desired carrier frequency from the reference and surveillance signals.The second step is to filter out direct-path and multipath clutter from the surveillance signal using the generalized subband cancellation algorithm [18].The algorithm utilizes subband signal processing and tailored clutter subspace construction to solve practical non-ideal factors such as carrier frequency offset,sampling frequency offset,and so on.The third step is to improve the target SNR further by crosscorrelation coherent integration of the reference and surveillance signals.This process of generating an RD map is also known as matched filtering.For ease of detection,the ordered statistical constant false alarm rate (OS-CFAR)threshold is applied to the data after beamforming [19].Once the targets are detected,the snapshots required for DOA estimation are collected from the RD map before beamforming.Then,the DOAs are estimated jointly using snapshots from multiple frequencies.Finally,the target can be located in theX-Yplane according to the DOA and bistatic range.

Fig.2 Processing scheme of MF DOA estimation for passive radar

3.MFJSBL algorithm

In this section,we derive the MFJSBL algorithm and analyze its computational complexity.This method can be viewed as a subset of wideband DOA estimation.Although the passive radar receives wideband signals,data from different frequency channels may come from different transmitters.Associating target RD information from different transmitters is,in fact,a quite difficult task.As a result,for the MF DOA estimation method,we only analyze frequency channel data from the same transmitter.

3.1 Sparse Bayesian formulation

In the SBL treatment [20,21],we assume that the observations,the noise,and each element in the coefficient matrix corresponding to each channel are independent of each other.Each column of the coefficient matrix obeys complex Gaussian distribution with a zero-mean covariance matrix Γ=diag(γ),whereγ=[γ1,γ2,···,γi,···,γK～](γi≥0),and diag(γ) is a diagonal matrix with vectorγ as its diagonal elements.The variance vector γ is termed as hyperparameter prior,and it controls the sparsity of the model.The MFJSBL algorithm assumes that the hyperparameter prior for all frequency channels is governed by the same statistical distribution.The shared variance vector is equivalent to imposing the same sparsity constraint on all coefficient matrices,which can increase the fitting probability of the algorithm.Hence,we have

The observations of thenth frequency channel,under an assumption of complex Gaussian white noise,follow the complex Gaussian probability distribution with mean ΦnWnand covarianceIM,whereis the noise variance of thenth frequency channel data andIMis the identity matrix of sizeM×M.Therefore,the likelihood of observations from multiple frequency channels can be regarded as the cascade of the likelihood of the observations of the single-channel,denoted as

3.2 Target power and noise estimation

The evidencep(;γ,β) is the marginal distribution of the data and can be expressed as the product of the likelihood and the prior integrated over the coefficient matrix:

where tr(·) and det(·) denote the trace and the determinant of the matrix,respectively.The logarithm of (11) can be written as

Let (16) be equal to zero,the update of γiis

The algorithm combines the observations from multiple frequency channels with corresponding over-complete dictionaries to obtain the update criterion of the hyperparameter vector.The hyperparameters for the coefficient matrices corresponding to multiple dictionaries are the same.We only need to setN=1 when one frequency channel data is used.It is discovered that the hyperparameter update equation estimated jointly from multiple channel data is not a linear superposition of the estimated hyperparameters from a single channel data.The algorithm is more stable than averaging the estimates over multiple channel data because of the joint fitting of the covariance of multiple channels.

Like γi,the noise variance vector β can also be obtained by finding that the derivative of evidence relative to the noise variance equals zero.Then,the updating equation for the noise varianceof thenth channel is

For the comparative illustration in Section 4,we present typical averaged MF algorithms and focused MF algorithms,including averaged maximum a posterior-based SBL (AMAPSBL) algorithm,averaged expectation maximization-based SBL (AEMSBL) algorithm,focused maximum a posterior-based SBL (FMAPSBL) algorithm,and focused expectation maximization-based SBL (FEMSBL)algorithm.The AMAPSBL algorithm uses the MAPSBL[23] algorithm to estimate DOA for each channel data,then the hyperparameter update of thenth channel data follows

The noise estimation of the AMAPSBL algorithm is the same as that of the MFJSBL algorithm.The AEMSBL algorithm utilizes the expectation maximum SBL(EMSBL) [22] algorithm to estimate DOA for each channel data,then the hyperparameters and noise variance of thenth channel data are expressed as

3.3 Algorithm summary

To achieve the predetermined convergence rate,the algorithm iteratively updates the hyperparameters and noise variance.The convergence rate of the algorithm is defined as

where ‖·‖1is ?1norm.The algorithm is terminated if the iteration reaches the maximum numberNmaxor the convergence rate ε is less than the user-defined tolerance εmin.The hyperparameter corresponds to the power of the target.The estimated DOA can be obtained,once we get the estimated value of the hyperparameter.The MFJSBL algorithm is summarized in Algorithm 1.

3.4 Computational complexity analysis

4.Simulation results

In this section,we present several simulation results to illustrate the performance of the MFJSBL algorithm.

4.1 Simulation setup

The simulation takes the FM-based passive radar to test the performance of the MFJSBL algorithm.The surveillance antenna adopts a seven-element array,and the received echo parameters are displayed in Table 1,where CNR is an abbreviation for clutter to noise ratio.For a more intuitive analysis of the algorithm performance,the root mean square error (RMSE) is used as the measured metric and defined as

Table 1 Parameters of the simulation

4.2 Performance analysis

Simulation 1 aims to reveal the DOA estimation performance of the MFJSBL algorithm under different FM channel numbers.When the frequency channel number is 1,3,and 5,the corresponding set of carrier frequencies(in MHz) are {97.4},{97.4,101.8,102.6},and {97.4,98.6,101.8,102.6,103.7},respectively.We investigate the effects of the number of targets,the number of snapshots,and the array form on DOA estimation accuracy.Assume that the surveillance antenna is a uniform linear array(ULA) with an interval of 1.5 m and the number of snapshots is 15.The simulation discretizes the detection angle with a step of 0.2°.

When Target 1 from Table 1 is added to the simulation scenario,the result is shown in Fig.3(a).The RMSE of the DOA decreases as the integrated target SNR increases,regardless of the number of the frequency channel.When Target 1 and Target 2,or Target 1 and Target 3 from Table 1 are added to the simulation scenario,the result is shown in Fig.3(b).The RMSE with the target SNR for two targets follows the same trend as the RMSE with SNR for a single target.With the same SNR,the RMSE of two near-interval targets is higher than that of two farinterval targets.Meanwhile,we investigate the effect of the number of snapshots on estimation accuracy.The RMSE estimation results for Target 1 with a single snapshot are shown in Fig.3(c).To obtain an RMSE that is close to that of snapshot number 15,the integrated target SNR under a single snapshot must be increased.To test the influence of array form on the DOA,Fig.3(d)presents the estimation results of the azimuth angle of Target 1 at an elevation angle of 0° using a single snapshot of the uniform circular array (UCA) antenna.The array radius of the UCA antenna is 1.5 m,and the number of array elements is the same as that of ULA.The integrated SNR of Target 1 must be increased to achieve roughly the same estimation accuracy as the ULA(Fig.3(c)).Overall,the estimation accuracy increases with the number of frequency channels for the different number of targets,snapshots,and array forms,validating the correctness of the MFSBL algorithm.

Fig.3 RMSE of DOA estimation versus integrated target SNR for n=1,3,and 5

4.3 Comparison with existing MF algorithms

Simulation 2 compares the MFJSBL algorithm with the AMAPSBL algorithm,the AEMSBL algorithm,the FMAPSBL algorithm,and the FEMSBL algorithm.Moreover,CRB [6,28] is also utilized as a benchmark to evaluate the performance of these algorithms.Consider a scenario in which a ULA composed of seven sensors is used to receive target echoes.Three frequency channel data in the received wideband signal are selected for estimating the target DOA.

Fig.4 shows the RMSE of various DOA estimation methods versus integrated target SNR via 1 000 Monte-Carlo runs.Fig.4(a) considers the case of one target and 15 snapshots of each channel.Fig.4(b) considers the case of two far-interval targets and the single snapshot of each channel data.From Fig.4,we can see that in both cases,the developed MFJSBL algorithm has the lowest RMSE among the five DOA estimation methods and is the closest to the CRB.The main reason for the superior performance of the developed algorithm over the AMPASBL algorithm is that,as seen in (15),the developed MFJSBL algorithm combines all channel data to maximize the joint evidence and more data points make the update more stable for small perturbations.However,according to (19) and (22),the AMPASBL algorithm processes the data for each channel separately and then averages them.The averaged algorithm does not overcome the errors present in the individual estimation.Additionally,the AMAPSBL algorithm slightly outperforms the AEMSBL algorithm,which may be related to the utilization of a priori information of the target number.The FEMSBL and FMAPSBL algorithms have the highest RMSE due to information loss caused by approximation.Fig.5 presents the average number of iterations and average computation time of various DOA estimation methods versus integrated target SNR via 1 000 Monte-Carlo runs.The simulation assumes one target in the scene,single snapshot data for each frequency channel,the angular detection range varies from -60° to 60° at an interval of 1°.It can be seen from Fig.5 that both the average number of iterations and the average computation time of various DOA estimation methods decrease as the integrated target SNR increases.For a fixed target SNR,the developed MFJSBL algorithm has the smallest number of iterations,i.e.,the fastest convergence speed,among the five MF algorithms.The AMAPSBL algorithm exhibits faster convergence than the AEMSBL algorithm.The FMAPSBL algorithm spends the shortest computational time because the focused MF algorithm converts the data of multiple frequency channels into a single frequency to estimate the target DOA.

Fig.4 RMSE of different DOA estimation methods

Fig.5 Average number of iterations and average computation time for various DOA estimation methods (ULA,L=1,θ=0°)

Besides,the resolution performance of the MFJSBL algorithm is validated fu rther.Assume that the DOAs of two targets are θ1=0°,θ2=0°+Δθ,where Δθ changes from 5°to 20°with a step of 3°.Fig.6 shows the RMSE of the DOA estimation versus two targets angular separation via 1 000 Monte-Carlo runs,where the target SNR is 15 dB and the number of snapshots is 15.The developed MFJSBL algorithm has the best estimation accuracy in each angular separation.The estimation accuracy of the AEMSBL algorithm for two targets is only second to that of the MFJSBL algorithm,but its performance is unstable when the angle interval between two targets is 5.The performance of FEMSBL and FMAPSBL algorithms is inferior to that of the AEMSBL algorithm.

Fig.6 RMSE of DOA estimation versus two targets angular separation

4.4 Single snapshot multi-target DOA estimation

Simulation 3 verifies the performance of the MFJSBL algorithm in estimating multi-targets (such as aircraft formation and UAV colony) using a single snapshot of data.Assume that the three targets in Table 1 are in the same RD,that is,the targets are located at the bistatic range of 100 km and Doppler frequency of 50 Hz.For compactness,we adjust the DOA of Target 3 to θ=-20°.The time required for coherent integration of data is 1 s.The normalized power spectrum of DOA estimated by the MFJSBL algorithm with a single snapshot is shown in Fig.7.It can be observed that the conventional multiple signal classification (MUSIC) [29] algorithm fails in the case of a single snapshot data and multiple targets.The DOAs estimated by the MFJSBL algorithm are consistent with the true values of the three targets indicated by the red circles.This result confirms the feasibility of the algorithm to estimate group targets in a single snapshot.

Fig.7 Normalized power spectrum with a single snapshot data

5.Experimental results

In this section,we present the results of applying the MFJSBL algorithm to the data collected by the very high frequency (VHF) passive radar developed at Wuhan University.The experimental scenario and radar system are identical to those described in [30].Specifically,a real data acquisition campaign for aircraft targets has been conducted in August 2018,Deyang,China.The reference and the surveillance antennas in the experiments are a highly directional Yagi antenna and a seven-element UCA with a radius of 1.5 m,respectively.The unit of the UCA is a half-wave dipole antenna.

Fig.8 depicts the signal spectrum of one of the surveillance channels acquired by the wideband VHF passive radar.As can be observed,the system can collect FM audio broadcasts from 95 MHz to 105 MHz.This is consistent with the 100 MHz operating center frequency and 10 MHz acquisition bandwidth of the passive radar,indicating that the system is operating normally.Then,a multichannel analog-to-digital converter (ADC) and downsampling are used to obtain the corresponding baseband signals.Using a digital filter with a baseband signal sampling rate of 200 kHz,two frequency channels of 97 MHz and 102.6 MHz data from the Chengdu Mengzhuiwan television tower are extracted from the received broadband data.The data integration time is set to 2 s.The true value of DOA for civil aircraft is confirmed by the automatic dependent surveillance-broadcast (ADS-B) records obtained during the measurement campaign.The target DOA of the measured data refers to the azimuth angle and the elevation angle is assumed to be 0°.

Fig.8 Signal spectrum acquired by wideband passive radar

The results of the data analysis for the field experiment are shown in Fig.9.The blue dashed and orange solid lines in Fig.9(a) and Fig.9(d) give the SNRs of Target 1 and Target 2 after the coherent integration of the 97 MHz and 102.6 MHz channel data in surveillance antenna 2,respectively.Other surveillance antennas’ target SNR is similar to that of surveillance antenna 2.It is important to note that the target SNR is low during some periods,but target detection is applied to the data after beamforming.Beamforming can increase the output SNR of the target,allowing us to detect it.The orange triangular line and the blue square line in Fig.9(b) and Fig.9(e) are the DOA errors of Target 1 and Target 2 estimated by the MAPSBL algorithm using individually the 97 MHz and 102.6 MHz channel data,respectively.The MFJSBL algorithm,as shown in Fig.9(c) and Fig.9(f),can obtain a relatively small DOA error fluctuation range when compared to the SBL algorithm for single-frequency channel data and the AMAPSBL algorithm for two frequency channel data.The real data results validate the developed algorithm’s effectiveness,i.e.,the joint fitting covariance makes the estimation results more stable in the presence of small disturbances.To simplify the analysis,we do not list the estimation results of other MF algorithms for real data.In addition,it should be pointed out that the deviation of DOA estimation error to negative error is caused by system error.

Fig.9 Experiment results

Table 2 summarizes the RMSE statistical results of the DOA estimation.The results show that the MFJSBL algorithm outperforms the MAPSBL algorithm for singlefrequency channel data and the AMAPSBL algorithm for joint two-frequency channel data in terms of DOA estimation error.There are two possible explanations for this.One reason is that combining data from multiple channels can be seen as increasing the number of equivalent snapshots.The other is that the hyperparametric constraint strategy is used to jointly fit the DOA,which makes the algorithm performance more stable in the presence of small disturbances.The AMAPSBL algorithm may be adversely affected by poor estimation performance at specific instants in the frequency channel data.In addition,we can see that the DOA estimation result from real data is relatively poor.The reason for this is that the target SNR of the real data remains low in comparison to the target SNR required to achieve the estimation accuracy shown in Fig.3(d).In summary,the real data results are consistent with the simulation analysis,confirming the correctness of the MFJSBL algorithm.

Table 2 RMSEs of the experimental DOA estimation results (°)

6.Conclusions

In this paper,we address the problem of DOA estimation in passive radar.An MFJSBL algorithm is developed to estimate the DOA for passive radar by exploiting frequency diversity and sparsity of targets in detection scenarios.The results show that the estimation accuracy of DOA increases with the increase of the number of frequency channels.The MFJSBL algorithm outperforms both averaged and focused MF processing in terms of estimation accuracy and resolution performance.Besides,the superiority of the algorithm is supported by data from aircraft targets detected by FM-based passive radar.Since this paper only estimates the DOA of the target on the discrete grid points,off-grid/gridless technology can be introduced in future work to enhance the DOA estimation accuracy.In addition,obtaining snapshot data for DOA estimation from RD maps generated by the SR technique to improve trajectory estimation accuracy is another research topic in future work.