DOA estimation method for multipath targets based on TR MIMO radar

To reduce the large errors for direction of arrival (DOA) estimation in the multipath condition, a DOA estimation method for multipath targets based on time reversal (TR) multi-input multi-output (MIMO) radar is proposed. First, the proposed algorithm combines the focusing characteristic of TR technology with the waveform diversity and symmetry characteristic of MIMO radar, obtaining the virtual sub-arrays of echo signal. Then, the authors multiplex the rows and columns and apply forward–backward spatial smoothing algorithm to remove the coherence of the virtual sub-arrays, effectively improving the DOA estimation accuracy for multipath targets. Simulation results verify the validity of the proposed method.


Introduction
Target angle tracking under low-altitude environment [1,2] is a difficult problem in modern radar.The main reason is that the received signals include direct signal from target and ground (sea) multipath reflection signal, and the elevation angle is too small, resulting in a sharp decline in the accuracy of radar angle estimation.With the generation of the multi-input multi-output (MIMO) radar system, it adopts radar antenna diversity and an orthogonal waveform technique, greatly improving the performance of radar by using virtual sub-arrays of the echo signal.Compared with those of ordinary phased array radar, MIMO radar has unique performance advantages in anti-interference ability, direction finding accuracy, anti-radiation missile capability, targets detection performance, coverage range and so on [3][4][5][6].Therefore, MIMO radar is applied to the multipath environment, effectively improving the estimation accuracy of the radar system.Time reversal (TR) is originated in optical research, achieving energy focusing of optical signal by phase conjugation.Nowadays, the application of TR technology in electromagnetic field has aroused much interest of scholars [7][8][9][10][11].In [7], on the basis of linear superposition principle of electromagnetic fields, the TR mirror is used to generate microwave spatial fields with arbitrary patterns, which can be applied to wireless communications, imaging and so on.The method in [8] utilises TR technique to generate given 3-D field intensity distribution.The model of TR-based target detection is proposed in [9,10], and simulation results verify that the proposed method outperforms the traditional methods.Wu et al. [11] applied the TR algorithm in ground-penetrating radar multitarget imaging, improving the efficiency of imaging.
In recent years, a lot of domestic and foreign scholars have thoroughly studied the MIMO radar angle estimation problems.In [12,13], the proposed method uses the TR method to direction of arrival (DOA) estimation in MIMO radar, improving Cramer Rao Bounds (CRBs) and estimation accuracy.Mohammad et al. [14] utilised TR and compressive sensing technology to MIMO radar to create a new way for joint estimation of DOA, direction of departure (DOD) and Doppler information on potential targets in multipath environment.However, these methods are only suitable in free space and will worse in multipath environment due to the coherency of signals.In [15], the proposed method of low-altitude target angle estimation uses the waveform diversity of MIMO radar to obtain virtual arrays, increasing the detection aperture, overcoming the influence of coherence by adopting maximum likelihood algorithm and improving the accuracy of low-altitude target estimation.However, the accuracy is still inadequate compared with other high-resolution algorithms.The method in [16] adopts shared transmitting and receiving antennas of MIMO to detect low-altitude targets.The symmetry of virtual sub-arrays is utilised to perform row column reuse, and the forward-backward spatial smoothing (FBSS) technique is used to removes coherence of targets, so as to improve the accuracy of angle estimation.However, this method is only applicable to DOA estimation of single target under low altitude.A DOA estimation method for multipath targets based on compressive sensing and TR MIMO radar is proposed in [17].The method focuses multipath scattering energy by the TR technique and uses the compressive sensing algorithm to estimate the DOAs, greatly improving the accuracy of DOA estimation.However, the information of the received signal has not been fully utilised.In other words, the high-resolution DOA estimation in multipath environment has not been well settled yet.
Aiming at the difficulties of DOA estimation in low-altitude environment, in this paper, a novel DOA estimation method for multipath targets based on TR MIMO radar is proposed.The method makes full use of the focusing characteristic of TR method to concentrate the energy of multipath signal, performs rows and columns reuse of virtual sub-arrays and adopts the FBSS algorithm to remove coherence of targets.Compared with that of the above methods, the proposed method can effectively improve the estimation accuracy of DOAs of multi-targets under multipath environment.

MIMO radar signal model in multipath environment
As shown in Fig. 1, the monostatic MIMO radar adopts shared transmitting and receiving antennas in this paper.It can be seen that we need to consider four paths when using MIMO radar under multipath condition.
Consider a MIMO radar consisting of mutual transmitting and receiving arrays equipped with a uniform linear array (ULA), having M antennas separated by half a wavelength.The centre of the radar is h a from the ground.Assume K far-field uncorrelated targets, locating at height H.In addition, the DOAs for direct and reflected signals of kth target is denoted by θ dk and θ rk k = 1, 2, …, K , respectively.Here, the signal matrix of transmitter can be written as The signals transmitted by M array elements are orthogonal to each other due to the waveform diversity of MIMO radar, also expressed as ∫ s i (t)s j (t)dt = σ s 2 δ i j , where σ s 2 is the transmitting power.When i = j, δ i j = 1; otherwise, δ i j = 0.
Thus, the signals of targets can be given by where ε k = ρ k e j2π ΔR k /λ is the ground reflection coefficient.2π ΔR k /λ represents the phase difference depending on the path difference of direct and reflected signals, which is denoted as ΔR k .λ is the wavelength and ρ k is a complex coefficient.
sinθ rk ] T and ( ⋅ ) T denotes transpose.The steering matrix in (2) can be represented as Suppose the matrix of ground reflection coefficient is The received signal matrix of MIMO can be given by where α = diag α 1 , …, α K is the radar cross-section fading coefficient; following Gaussian distribution of zero mean and covariance σ 2 , n(t) is an additive Gaussian white noise vector.

TR MIMO radar signal model in multipath environment
According to the DOA estimation method of TR MIMO in [18], we can perform the following operation: First, the signal matrix of ( 5) is conjugated and reversed in time.Then, it is normalised and transmitted again.Therefore, the retransmitted signal is τX * −t .The received signal matrix of TR MIMO radar can be expressed as where η = diag α 1 2 , …, α K 2 .( ⋅ ) * and ( ⋅ ) H denote conjugation and conjugate transpose, respectively.W(t) and V(t) are the observation noise in the TR stage and the accumulated noise from both the conventional and TR stages, respectively.Similar to the noise n t , the noise W(t) is also temporally and spatially complex white Gaussian processes with zero mean and variance σ 2 .As described in [12], the accumulated noise V(t) can be proved to be approximately white noise due to the focusing performance of TR technology.

Proposed DOA estimation method
According to [18], to obtain virtual sub-arrays of TR MIMO radar, after matched filtering, the received signal matrix is transposed to where V is an M × M dimensional noise matrix whose elements obey the Gaussian distribution.
Due to the translation invariance of each column in Y, we can extract each column of Y as the received signal of single sub-array.Thus, ith column of Y can be expressed as = ω H e jπ(i − 1)sinθ d1 , e jπ(i − 1)sinθ r1 , …, e jπ(i − 1)sinθ dK , e jπ(i − 1)sinθ rK T (10) where i = 1, 2, …, M, V ci is the ith column of noise matrix V.
The traditional smoothing algorithm divides the array into subarrays.When the smoothing algorithm is applied to virtual subarrays in this paper, the numbers of smoothing sub-arrays and virtual array elements satisfy Q 0 = M − M 0 + 1, where the number of smoothing sub-arrays is Q 0 and the number of sub-array elements is M 0 .
The output signal of qth sub-array can be written as where A 0 is the first M 0 rows of A and V ci_q is the vector comprising the qth to the (q + M 0 − 1)th elements of V ci D = diag e − jθ d1 , e − jθ r1 , …, e − jθ dK , e − jθ rK (12) Then, the covariance matrix of qth sub-array is where , and E V ci_q V ci_q H is the covariance matrix of noise vector.
Assuming J M 0 is an M 0 × M 0 dimensional transformation matrix, we have where only the elements on the back diagonal line are 1; the rest are 0. Therefore, the reverse covariance matrix of Y ci_q can be written as The forward spatial smoothing (FSS) algorithm can be expressed as The FBSS algorithm can be given by Each row of virtual matrix Y, which is denoted as Y rk , has similar characteristics as Y ci .Dividing the whole array into sub-arrays along the row, the output signal of qth sub-array is Y rk_q .The spatial smoothing procedure is similar to that in (9)-- (13).The covariance matrix of Y rk_q is R rk_q .As the deduction of R rk_q is similar to R ci_q , we have omitted the detail in the body and described the detailed deduction in the Appendix.
The reverse covariance matrix of R rk_q can be written The FSS algorithm can be given by The FBSS algorithm can be given by Then, we calculate the mean value of M covariance matrices (R cFB i or R rFB k ) as According to (21) or ( 22), we can perform eigenvalue decomposition to get noise and signal subspace, and then, methods such as the multiple signal classification (MUSIC) algorithm can be applied to estimate the DOAs of low-altitude targets with TR MIMO radar.
To make full use of the data information of TR MIMO radar virtual matrix, after rows and columns reuse, we have It can be seen from ( 23) that compared with former methods, the performance of the proposed method is affected by R ci_q and R rk_q .
According to ( 13) and ( 26) (in the Appendix), the signal parts of R ci_q and R rk_q (the first items on the right of the equal sign) are completely equal and originate from samples Y(l), so it cannot improve the estimation accuracy of the algorithm.Assume that only one equal element due to the kth element of V ci_q and the ith element of V rk_q being the same.The proposed method can improve the DOA estimation performance of radar due to a more accurate estimation of noise covariance matrix (the second items on the right of the equal sign in ( 13) and ( 26)).

Algorithm complexity
The computational complexity of the proposed method mainly includes calculating the covariance matrix, calculating the eigenvalue decomposition and seeking the spectrum peak.From the aspect of algorithm, TR technology would not increase the complexity and compared with that of FSS, FBSS also does not increase the computation in this paper.Therefore, the increase of complexity is mainly focused on the row and column reuse of matrices.The complexity of the proposed method can be expressed as where L denotes the number of snapshots, n is the total search times, M 0 denotes the number of smoothing sub-array elements and M is the elements number of the whole array.Assume L = 200, M = 20, n = 2000, Fig. 2 describes the computation complexity of both algorithms with different number of sub-array elements.It can be seen that the proposed algorithm sacrifices a certain complexity and the increase of complexity is less in the case of a small sub-array element, compared with the former method of TR MIMO.

Simulation results
In the simulation, we consider a MIMO radar consisting of mutual transmitting and receiving arrays equipped with ULA, having M = 10 antennas separated by half a wavelength.The array elements transmit mutually orthogonal signals, and the snapshots are L = 200.The step size of the spectrum peak searching is 0.01°, and the estimation performance is examined by 300 Monte Carlo trials.When we perform spatial smoothing of TR MIMO, the number of sub-arrays is p = 5.Assume there are two targets whose direct signal and reflected signal angles are θ d1 = 6°, θ r1 = − 6°a nd θ d2 = 2°, θ r2 = − 2°, respectively, with reflection coefficients ε 1 = 0.8e j 160°/180°π and ε 2 = 0.6e j 160°/180°π .Simulation 1: We compare the spatial spectrum of the proposed method and that of the method without rows and columns reuse.The SNR is 10 dB.From Fig. 3, both algorithms can accurately estimate the DOAs, but the spectrum peaks of TR MIMO with rows and columns reuse are higher, so the proposed method has a better performance.Simulation 2: We examine the estimation performance by the RMSEs versus SNR in this simulation, where both FSS and FBSS are employed.Fig. 3 shows the RMSEs with FSS, and Fig. 4 describes the RMSEs with FBSS.As shown in Figs. 4 and 5, the proposed method outperforms the other methods.It is because that TR MIMO has good focusing characteristic and makes full use of multi-path signal energy.Compared with TR MIMO without rows and columns reuse, the estimation accuracy of the proposed method is higher under the low SNR conditions.It is because the proportion of noise to the signal is larger (it is the same that rows and columns reuse can improve the estimation accuracy of noise signal discussed earlier).It can be seen from Figs. 4 and 5 that FBSS has a better estimation performance than FSS due to more sub-arrays.Simulation 3: We examine the estimation performance by the RMSEs versus the number of snapshots in this simulation.We compare the performance of DOA estimation methods using FSS and FBSS when SNR is 0 dB.The estimation performance of the proposed method improves with more number of snapshots in Figs. 6 and 7. Compared with conventional MIMO radar, TR MIMO still has good performance under the small number of snapshots.The accuracy of MIMO with rows and columns reuse is better than that of TR MIMO, because the performance of rows and columns reuse is stronger than focusing when the array elements are eight.

Conclusion
In this paper, by applying TR technology to MIMO radar with colocated transmitters and receivers, and combining with the idea of sample reuse, an effective DOA estimation method for multipath targets based on TR MIMO radar is proposed.The algorithm can realise DOAs estimation of multi-targets and improve estimation accuracy under low altitude by making full use of the focusing characteristic of the TR method and the waveform diversity of MIMO radar.
When performing spatial smoothing, we divide the sub-array in the same way as Y ci in the former part of this paper.The output signal of qth sub-array can be given by where V rk_q is the vector consisted by the qth to the (q + M 0 − 1)th elements of V rk .Therefore, where

Fig. 1
Fig. 1 Received signal model of MIMO radar in low-altitude environment