A Two-Step Compressed Sensing Approach for Single-Snapshot DOA Estimation of Closely Spaced Signals

Hindawi Mathematical Problems in Engineering Volume 2021, Article ID 1491097, 11 pages https://doi.org/10.1155/2021/1491097

Research Article A Two-Step Compressed Sensing Approach for Single-Snapshot DOA Estimation of Closely Spaced Signals

Shuang Wei , Yanhua Long , Rui Liu, and Ying Su

College of Information, Mechanical and Electrical Eng, Shanghai Normal University, Shanghai, China

Correspondence should be addressed to Yanhua Long; [email protected]

Received 22 June 2020; Revised 12 December 2020; Accepted 13 January 2021; Published 2 February 2021

Academic Editor: Ramon Sancibrian

Copyright © 2021 Shuang Wei et al. 'is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Single-snapshot direction-of-arrival (DOA) estimation plays an important role in dynamic target detection and tracking applications. Because a single-snapshot signal provides few information for statistics calculation, recently compressed sensing (CS) theory is applied to solve single-snapshot DOA estimation, instead of the traditional DOA methods based on statistics. However, when the unknown sources are closely located, the spatial signals are highly correlated, and its overcomplete dictionary is made up of dense grids, which leads to a serious decrease in the estimation accuracy of the CS-based algorithm. In order to solve this problem, this paper proposed a two-step compressed sensing-based algorithm for the single-snapshot DOA estimation of closely spaced signals. 'e overcomplete dictionaries with coarse and reﬁned grids are used in the two steps, respectively. 'e measurement matrix is constructed by using a very sparse projection scheme based on chaotic sequences because chaotic sequences have determinism and pseudo-randomness property. Such measurement matrix is mainly proposed for compressing the overcomplete dictionary in preestimation step, while it is well designed by choosing the steering vectors of true DOA in the accurate estimation step, in which the neighborhood information around the true DOAs partly solved in the previous step will be used. Monte Carlo simulation results demonstrate that the proposed algorithm can perform better than other existing single-snapshot DOA estimation methods. Especially, it can work well to solve the issues caused by closely spaced signals and single snapshot.

1. Introduction much attention, which is an active topic widely used in automotive radar/sonar applications like driver assistance sys- Direction-of-arrival (DOA) estimation plays an important tems [7]. Moreover, Ha¨cker and Yang [7] also investigated the role for target/source localization, which is widely used in performance of traditional DOA estimators using a single many fields including radar, sonar, speech, communication, snapshot, such as Bartlett beamformer, MUSIC, deterministic and medical diagnosis [1–4]. Traditional high-resolution maximum likelihood, stochastic maximum likelihood, and DOA estimation algorithms use the statistics of observed weighted subspace fitting. It was shown that these algorithms signals to improve the performance efficiency, such as cannot work at all or cannot show a superior performance as multiple signal classification (MUSIC) algorithm and esti- expected, especially for multitarget complex scenarios in- mation method of signal parameters via rotational invariance cluding correlated targets under low SNR situation. techniques (ESPRIT) [5, 6]. 'ese algorithms require to re- In order to improve the accuracy of single-snapshot DOA ceive the signals observed in a period of time. However, in estimation, the latest research studies on this topic mainly dynamic target detection and tracking systems, only single used the compressed sensing theory [8, 9] because in CS- snapshot or a small number of snapshots are available for based algorithms, the signals can be reconstructed in spatial DOA estimation. In this case, the statistics information is not by an overcomplete dictionary, whose sampling interval could accurate, and thus traditional algorithms degrade dramati- be smaller than the Nyquist limit [10]. However, considering cally. 'erefore, single-snapshot DOA estimation attracts that the unknown angles are closely spaced, an overcomplete 2 Mathematical Problems in Engineering dictionary is composed of a uniform sampling grid, so that the alternate minimization methods were proposed to design on-grid points should be densely distributed to cover all the the optimal measurement matrix in terms of reducing the exactly steering vectors to reduce the gap between the real coherence between the atoms of such matrix [25–29]. 'e DOA and its nearest grid points. 'at means vectors in- optimization target is to minimize the difference between the volving true DOAs in the overcomplete dictionary will be Gram matrix of the equivalent dictionary and the identity high dimensional, and computational cost will be high in matrix [30]. 'e equivalent dictionary is defined as the sequence. Furthermore, the columns in the matrix with high product of the measurement matrix and the overcomplete correlation will inevitably degrade the estimation perfor- dictionary. 'us, in this paper, a new double-structure mance. 'erefore, the design of overcomplete dictionary measurement matrix is constructed by combining a part of becomes the key step in the CS-based single-snapshot DOA the unit matrix and the chaotic measurement matrix because estimation algorithms for closely spaced signals, which is also the chaotic measurement matrix has been proved to satisfy the main content of this paper. the RIP [22, 31, 32] and has better properties than Gaussian Many researchers focus on the design of overcomplete matrix [33]. In addition, the very sparse random projection dictionary to improve the accuracy of CS-based estimators. (VSRP) method is used to thin the chaotic matrix, in order to Some existing CS-based estimators have designed iterative reduce the memory cost, especially when the measurement schemes to obtain the optimal overcomplete dictionary [11–13]. number is large. Some researchers designed the approach based on off-grid 'e main contribution of this paper is to develop a novel model which can achieve a smaller MSE than those methods approach for single-snapshot DOA estimation of closely based on on-grid model. Among them, some off-grid methods spaced signals. Its contributions are twofold. require the covariance matrix of multiple observed samples, such as the sparse asymptotic minimum variance (SAMV) We proposed a two-step compressed sensing approach method in [14]. However, in some dynamic applications, the to improve the estimation accuracy by decreasing high statistics of multiple previous snapshots are inconsistent with correlation in the overcomplete dictionary. In the first the characteristics of the current single snapshot. Some step, the search space with coarse grids is defined, so methods have been developed based on Bayesian compressed that high correlation of the vectors will be reduced to a sensing framework to model the off-grids in the overcomplete low level and high computation cost will be avoided. In dictionary, such as the off-grid sparse Bayesian inference the second step, an updated search space with refined (OGSBI) method in [15, 16]. Some DOA estimation methods grids is designed, which is adaptively determined by the use the total least-squares to reduce the error of the over- solution obtained from the first step. Subsequently, a complete dictionary, such as [17]. One drawback of these smaller dictionary set defined around the true DOAs methods is its slow speed in the case of a dense sampling grid. lies in the preconceived results, which can improve the When the estimated DOA is densely distributed, the iterative single-snapshot DOA estimation efficiency. algorithms based on the probabilistic model need to spend a 'e proposed CS-based method used a double-structure large amount of computational load to maintain the estimation measurement matrix for closely spaced DOA estimation. accuracy. In order to minimize the difference between its Gram In order to improve the estimation accuracy, this paper matrix and the identity matrix, a part of the unit matrix proposed a two-step method to construct the overcomplete and a chaotic-based measurement matrix are combined dictionary. In the first step, the search space with coarse grids to construct this double-structure measurement matrix. will be used, which is useful to determine a narrow optimal 'e chaotic-based measurement matrix is built by lo- range of the second step with a reduced computational cost. gistic mapping chaotic sequence and very sparse random 'us, high correlation of the vectors will be reduced to a low projection method, where very sparse projection scheme level, and the key neighborhood information around the true is used to thin the chaotic matrix. Experimental results DOAs lies in the preconceived results [15]. In the following show that it significantly improves the estimation ac- step, the previous result will be used to reconstruct the curacy of CS-based method and outperforms those overcomplete dictionary with a smaller size, which is useful to obtained using the general Gaussian random matrix. reduce the memory cost. 'en, an updated search space with refined grids will be used, so that a more accurate DOAs es- 'e remainder of the paper is organized as follows. timation can be expected with high probability in the final step. Section 2 describes a single-snapshot DOA estimation Besides, in order to obtain the sparse angles from a model using CS theory. Section 3 presents the proposed single-snapshot signal, a measurement matrix is required to double-structure measurement matrix designed by cha- sample the observed signals without loss of information [18]. otic sequence and very sparse random projection method. 'e measurement matrix has to meet the restricted isometry 'e proposed two-step CS-based method for DOA esti- property (RIP) to ensure the completeness of useful information is then presented in Section 4 by using the pro- mation [19–22]. Generally, Gaussian random matrix and posed measurement matrix. In Section 5, some numerical Bernoulli matrix are used to construct the measurement experiments are given by using dense spatial source sig- matrix because it is found [23, 24] that they can perform nals, and the results are analyzed from the perspective of better with small measurement number than other matrices. different measurement numbers, SNRs, and sparse de- However, their randomness and high memory cost are two grees. Conclusions and future works are provided in issues in application. In order to solve this issue, some Section 6. Mathematical Problems in Engineering 3

2. One-Step DOA Estimation of a its randomness and high memory cost are two issues that Single Snapshot need to be solved in application. 'us, we proposed a double-structure measurement matrix in Section 3. Consider K narrow-band signals from the unknown di- 'e sparse characteristic of y shown in equation (6) can rections θi(i � 1, 2, ... ,K) incidenting to uniform linear be considered as a constraint condition [39], so that the array (ULA) composed of N sensors with interelement vector y in equations (5) and (6) can be solved by a l1 norm spacing d; then, the observed output at a single snapshot optimization problem as [37] CN×1 x ∈ is expressed as [34] min ‖y�‖ 1 ( ) x � As + w, (1) 7 s.t. ‖z − ΦΨy�‖2 < ε, where s ∈ CK×1 is the signal vector, w ∈ CN×1 is the additive N×K where y� denotes the estimated sparse solution and ε is a small Gaussian white noise, and A ∈ C is the array manifold parameter. 'us, the estimated DOAs of the spatial signals matrix, which is given as [35, 36] can be decided by the nonzero position of the sparse vector y after such vector y is solved according to the measured vector A ��aθ1 �, aθ2 �, ... , aθK ��, (2) T z. In summary of the above, a one-step CS-based algorithm − j(2π/λ)d sin θ − j(2π/λ)(N− 1)d sin θ where a(θi) � � 1 e i ... e i � for a single snapshot is shown in the following (Algorithm 1). with i � 1, 2, ... ,K is the steering vector and λ is the wavelength. Consider the angular search range of interest Θ in- 3. ProposedMeasurementMatrix Based onVery volving from θb to θe, which is defined by Sparse Chaotic Sequences T �� , (3) Θ θ1 θ2 ... θL Considering a chaotic measurement matrix constructed by logistic chaotic sequences, the mathematical expression of with the candidate angles defined by θ � θ + (j − j b mapping equation is [24] 1/L) × (θe − θb)(j � 1, 2, ... ,L). L is the number of the uniform-distributed candidate angles, and it has relation xg+1 � μxg�1 − xg �, n � 0, 1, 2 ... , G, (8) L ≫ K. 'us, the overcomplete array manifold matrix N×L th Ψ ∈ C composed of the steering vectors associating with where xg ∈ (0, 1) represents the g value of chaotic sequence each angle orientation θj is expressed as and g denotes the iteration number. In the case of μ � 4, the values of x can traverse the entire area of 0 to 1, and each 1 g Ψ � √�� a �θ �, a �θ �, ... , a �θ ��, (4) point of the sequences has the property of pseudo-randomness L 1 2 L [31, 33]. So, the proposed method defines μ � 4. 'e initial T − j2π(d sin θ /λ) − j2π(N− 1)(d sin θ /λ) value is set as x0 � 0.256. where a(θj) � � 1 e j ... e j � In order to improve the stability, after generating ∈ CN×1 is the steering vector corresponding to each angle x0, x1, ... , xG, the first t items are discarded to form a new orientation θ . So, equation (1) can be rewritten as [37] j sequence xt, xt+1, ... , xG. 'e chaotic sequence is sampled x � Ψy + w, (5) by the equal interval of v, that is,

T L×1 zσ � xt+σv, (9) where y � � y1 y2 ... yL � ∈ C denotes the candidate signal component corresponding to all L possible DOAs. where v denotes the sampling rate of the sequence x. 'e Because only true(unknown) DOAs have the larger com- pseudorandom sequence z0, z1, ... , z(G−t)/v is then obtained. ponents than the other angles, y can be considered a In our experiments, G is set to 3000 × 3000, v is set to 3, and t K-sparse vector. 'at is, y is a sparse vector to be solved, is set to 1000. Because Yu and Barbot et al. [24] have proved which has only K larger components and the rest L − K is that a matrix constructed by a chaotic sequence column by close to zero. M×N column is sufficient to satisfy RIP with high probability, the Based on CS theory, a measurement matrix Φ ∈ C is first M × M values are selected to generate a chaotic matrix designed to measure the original output x, and a measured Γ ∈ RM×M with M being the measurement number: signal z ∈ CM×1 is obtained as [37] z0 zM ... zM×(M−1) ⎢⎡ ⎥⎤ z � Φx � Φ(Ψy + w), (6) ⎢ z z ... z ⎥ ⎢ 1 M+1 M×(M−1)+1 ⎥ Γ �⎢ ⎥. (10) ⎢ ⎥ where M is the measurement numbers and M < N. ⎣⎢ ...... ⎦⎥ According to CS theory, as long as the measurement number zM−1 z2M−1 ... zM×M−1 M satisfied the following equation, i.e., M ≥ K × log2(N/K), the vector y can be obtained accurately with a big probability When the number of measurements M is large, the [38]. In equation (6), the measurement matrix Φ is com- memory cost of chaotic matrix Γ shown in equation (10) monly constructed by random sequences which can let the would increase greatly. In order to compress Γ without the columns of the equivalent dictionary have low correlations loss of useful information, very sparse random projection with each other, such as Gaussian random matrix. However, (VSRP) method is applied to the chaotic measurement 4 Mathematical Problems in Engineering

Input: 'e observed single-snapshot signal x ∈ CN×1; 'e measurement matrix Φ ∈ CM×N; Output: 'e DOA estimator, [ϕ1, ϕ2, ... , ϕK]; (1) Divide the initial search space Θ into L parts to construct the overcomplete dictionary matrix Ψ ∈ CN×L. (2) Generate the measured signal z � Φx and the sensing matrix T � ΦΨ. (3) Calculate the DOA estimator [ϕ1, ϕ2, ... , ϕK] using the OMP method in [40]. (4) return [ϕ1, ϕ2, ... , ϕK].

ALGORITHM 1: 'e one-step CS-based algorithm. matrix [32], and it is also used to reduce the data recorded in measurement matrix Φ ∈ RM×N is designed by combining a M×(N− M) the memory [41] and make the measurement matrix easy to part of unit diagonal matrix Φ3 ∈ R with Φ2, that is, implement [42]. Φ ��Φ Φ �. (13) In fact, very sparse projection is the generalized form of 3 2 sparse projection and still satisfies the sparse projection In order to verify the advantage of the proposed mea- distribution as surement matrix, the memory storage cost of the proposed √� ⎧⎪ 1 matrix and Gaussian matrix is compared in Table 1. It is seen ⎪ S , with prob. , ⎪ 2S that the proposed matrix occupies a much smaller memory ⎪ ⎪ storage than Gaussian matrix of the same scale. Since the ⎪ ⎨ √� 1 proposed measurement matrix is sparse, its sampling rate is Φ1(i, j) � ⎪ − S , with prob. , (11) much lower than the Gaussian matrix under the same ⎪ 2S ⎪ conditions. 'us, it provides convenience for dealing with ⎪ ⎪ high-dimensional signals, saving memory storage and fa- ⎪ 1 ⎩ 0, with prob.1 − , cilitating hardware storage and implementation. Based on S these, the measurement matrix Φ ∈ RM×N shown in equation (13) is applied in this paper. where in the case of 1 ≤ S ≤ 3 every element in Φ1 meets the sparse distribution, while in the case of S ≫ 3 every element in Φ1 meets the very sparse distribution [43]. Inspired by 4. Two-Step DOA Estimation of a this, we propose a very sparse projection scheme Φ2 to Single Snapshot improve the chaotic measurement matrix as follows: A novel two-step approach for DOA estimation of single- ⎧⎪ √�� 1 ⎪ M , √�� > Γ(i, j) ≥ 0, snapshot signal will be presented and some notations in the ⎪ M ⎪ previous section will be employed in this section. ⎪ ⎪ ⎨⎪ √�� √2�� √1�� 4.1. %e First Step: DOA Preestimation. In the preestimation Φ2(i, j) �⎪ − M , > Γ(i, j) ≥ , (12) ⎪ M M L ⎪ step, the search space of interest Θ is divided into 1 parts by ⎪ c c ⎪ the interval π/ 1, where 1 is a predefined step factor and ⎪ 2 L < L. 'us, the parameter L can be expressed as ⎩⎪ 0, 1 > Γ(i, j) ≥ √�� . 1 1 M θb − θe M×M L1 � INT � � + 1, (14) Here, Φ2 ∈ R is the matrix obtained by thinning the π/c1 � chaotic matrix Γ, and Φ2 has similar characteristics with Φ1. Since very sparse matrix has been proved to meet the RIP where θb − θe denotes the whole angular range in Θ and INT(F) denotes the largest integer that does not exceed the [32, 43], Φ2 satisfies the RIP theory as well. According to the CS theory, a measurement matrix value F. composed of two different structural matrices is superior to a 'en, the overcomplete orthogonal dictionary with a matrix of one single structure [38]. According to the RIP rough division is used to construct the sparse transform N×L1 condition of very sparse projection matrix, the correlation matrix Ψ1 ∈ C , that is, between very sparse projection vectors and the unit vectors 1 ... 1 ⎡⎢ ⎤⎥ can be controlled in a restricted range with high probability. ⎢ − j2π(d/λ)sin φ − j2π(d/λ)sin φ ⎥ ⎢ 1 L1 ⎥ ⎢ e ... e ⎥ 'us, the combination of the unit matrix and very sparse Ψ � ⎢ ⎥. 1 ⎢ ⎥ projection matrix inherits the property of each single matrix. ⎣⎢ ...... ⎦⎥ Since both the unit diagonal matrix and very sparse pro- − j(N− 1)2π(d/λ)sin φ − j(N− 1)2π(d/λ)sin φ e 1 ... e L1 jection matrix satisfy RIP, the combination of these matrices also satisfies RIP. 'erefore, a new double-structure (15) Mathematical Problems in Engineering 5

Table 1: Memory cost of the proposed matrix and Gaussian matrix. 1 ... 1 ⎢⎡ ⎥⎤ ⎢ − j2π(d/λ)sin ϕ1 − j2π(d/λ)sin ϕL ⎥ MN Proposed matrix (bytes) Gaussian matrix (bytes) ⎢ e ... e 2 ⎥ ⎢ ⎥ 10 40 1128 3200 Ψ2 � ⎢ ⎥. ⎢ ...... ⎥ 20 40 3000 6400 ⎣⎢ ⎦⎥ − j(N− 1)2π(d/λ)sin ϕ − j(N− 1)2π(d/λ)sin ϕ 30 40 4520 9600 e 1 ... e L2 40 40 6600 12800 (21)

'us, the signal can be expressed as Next, the signal can be expressed based on the new sparse base matrix as

x � Ψ1y1 + w, (16) x � Ψ2y2 + w, (22)

L1×1 where y ∈ C denotes the sparse representation of the L2×1 1 where y2 ∈ C denotes the sparse representation of the estimated angle in a rough manner and φ � θ + j b accurate estimated angle and ϕj � min(Θ) + (j − 1/L2) × th th (j − 1/L1) × (θe − θb)(j � 1, 2, ... L1) denotes the j angle span(Θ)(j � 1, 2, ... L2) denotes the j angle in the search in the search space Θ. space Θ� . M1×1 M2×1 Finally, the observed signals s1 ∈ C are obtained by Finally, the observed signals s2 ∈ C are obtained by projecting the signal x into the double-structure measure- projecting the signal x into the double-structure measurement matrix ΦI, that is, ment matrix ΦII, that is,

s2 � ΦIIx � ΦIIΨ2y2 + w �, (23) s1 � ΦIx � ΦIΨ1y1 + w �, (17) M2×N where ΦII ∈ R is constructed by equation (13). Here, M1×N where ΦI ∈ R is constructed by using equation (13). M2 denotes the measurement number in the second step. Here, M1 denotes the measurement number in the first step. Based on the new sparse representation of spatial signal, the After the sparse representation of spatial signal, the accurate DOA estimator [ϕ1, ϕ2, ... , ϕK] is obtained using th rough DOA estimator [φ1, φ2, ... , φK] is obtained using the the OMP method, where ϕk is the k accurate solution. th OMP method. Here, φk is the k angle solution in the first 'e flowchart of the proposed two-step DOA estimation step and K is the sparse degree. In general, the sparse degree algorithm is shown in Algorithm 2. is the same as the number of actual angles. 5. Experiment Simulations and Discussion 4.2. %e Second Step: Accurate DOA Estimation. In the In the simulation experiments, we investigate the feasibility � second step, the search space of interest is narrowed as Θ, and accuracy of the proposed method for the single-snapshot which is generated adaptively according to the rough esti- DOA estimation, in terms of different measurement numbers, mator [φ1, φ2, ... , φK], which is SNRs, and sparse degrees, respectively. 'e proposed method is compared with the general CS-based single-snapshot DOA � � � � Θ ��Θ1, Θ2, ... , ΘK�, (18) estimation method, which is called the single-step Gaussian matrix method. In order to verify the effectiveness of the with measurement matrix we used, it is also compared with the � general Gaussian measurement matrix in the two-step Θi � φi − Δφi, φi + Δφi�, i � 1, 2, ... , K, (19) scheme, which is called the two-step Gaussian matrix method. 'e Monte Carlo method is adopted to evaluate the esti- where Θ� is a neighborhood area with the center φ and i i mation performance. Root mean square error (RMSE) is used radius Δφi. � to evaluate the accuracy of DOA estimation, which is de- Similar to the first step, the search space of interest Θ is scribed as below. divided into L2 parts by the interval π/c2, where c2 is a �� predefined step factor. 'us, the parameter L2 can be K CNT expressed as 1 1 2 RMSE � � � �ϕk,cnt − θk � , (24) K CNT � span(Θ� ) k�1 cnt 1 L � INT� � + 1. (20) 2 c π/ 2 where CNT is the number of Monte Carlo loops, K is the th sparse degree, ϕk,cnt is the estimated value of k angle in the th th Here, span means the difference between the maximum cnt Monte Carlo loop, and θk is the actual value of the k and minimum of the search space of interest. Note that the angle. 'us, the RMSE value is measured by degree. predefined step factor is larger than that in the first step, i.e., 'e parameters are defined as follows: the number of c2 > c1, because the divided parts of the second step are array antenna N � 40, d � λ/2, the sparse degree K � 4, smaller than those of the first step. and SNR � 15. Note that though the parameter K is as- � 'en, the search space Θ is subdivided into L2 parts to sumed to be known, it is not required to be exactly the N×L2 construct overcomplete matrix Ψ2 ∈ R , that is, same as the number of sources in practice. 'e parameters 6 Mathematical Problems in Engineering

Input: 'e observed single-snapshot signal x ∈ CN×1; M1×N A new double-structure measurement matrix for preestimation, ΦI ∈ R ; M2×N A new double-structure measurement matrix for the second step, ΦII ∈ R ; Output: 'e DOA estimator, [ϕ1, ϕ2, ... , ϕK]; N×L1 (1) Divide the initial search space Θ into L1 (L1 < L) parts to construct the overcomplete dictionary matrix Ψ1 ∈ C . (2) Generate the measured signal z1 � ΦIx and the sensing matrix T1 � ΦIΨ1. (3) Calculate the DOA rough estimator [φ1, φ2, ... , φK] using the OMP method [40]. � � � � � (4) Narrow the search space as Θ � [Θ1, Θ2, ... , ΘK] based on the rough estimator, where Θi � (φi − Δφi, φi + Δφi)i � 1, 2, ... ,K. � N×L2 (5) Divide the search space Θ into L2 parts to construct the overcomplete dictionary matrix Ψ2 ∈ C . (6) Generate the measured signal z2 � ΦIIx and the sensing matrix T2 � ΦIIΨ2. (7) Calculate the DOA accurate estimator [ϕ1, ϕ2, ... , ϕK] using the OMP method in [40]. (8) return [ϕ1, ϕ2, ... , ϕK].

ALGORITHM 2: 'e proposed two-step CS-based algorithm. in the two-step CS-based method are defined as follows: 0 the search space of interest Θ is [−90°, 90°], the step factor c � for preestimation 1 30, and the step factor for accurate –5 estimation. c2 � 100 'e 100 Monte Carlo loops are re- ° alized. As we found, two closely spaced DOAs, i.e., 1 and –10 2°, are generally estimated by mistake as one DOA by MUSIC algorithm due to high correlation between two –15 closely spaced sources, as shown in Figure 1. Here, the correlation coefficient between two closely spaced DOAs is –20 0.9407. To verify the reliability of the proposed method in Power (dB) solving such estimation problem of highly correlated –25 signals between each pair of sources, the actual angle values used in this experiment are randomly generated as –30 ° ° ° ° ° θ � � −20 1 2 35 �. 'e radius Δφi is chosen as 5 . In Figure 2, the relationship between measurement –35 –40 –30 –20 –10 0 10 20 30 40 number M2 and RMSE is presented when M1 � 21 DOA (degrees) (Figure 2(a)) and when M1 � 25 (Figure 2(b)). It is seen that RMSE in Figure 2(b) is lower than that in Figure 2(a) because True DOAs when M1 increased, estimation accuracy of the first step was MUSIC spectral improved, which enhanced the accuracy of the final accurate Figure 1: DOA estimation results using MUSIC algorithm. estimation. When the measurement numbers M1 and M2 are increased, the RMSE of the proposed method is reduced. ° Such error mainly comes from the estimation process of 1 are defined as M1 � 21, M2 � 35. It is seen that the estimated and 2°. 'us, for the dense spatial signal estimation in a DOAs of the proposed method can distinguish the dense linear array system, the proposed algorithm can use a limited signals emitted from 1° and 2° with a higher accuracy. In number of measurements to accurately estimate spatial order to evaluate the estimation accuracy more clearly, signals that are spaced closer than the general resolution. Figure 4 indicates the relationship between M2 and RMSE Figure 3 shows the RMSE performance with different under different sparse degree conditions. In Figure 4(a), SNRs. Here, the parameters of N and K are defined the same as K � 4, the actual angle values θ � � −20° 1° 2° 35° �. In the experiment mentioned above. 'e parameters of different Figure 4(b), K � 5, the actual angle values CS-based methods are defined as follows. 'e measurement θ � � −40° −20° 1° 2° 35° �. It is shown that the proposed number in each step is defined as 21. 'e search space of two-step method performs better than others under different interest Θ is [−90°, 90°]. In the two-step CS-based methods, degrees of sparsity. the step factor for preestimation c1 � 30 and the step factor for accurate estimation c2 � 100. In the one-step CS-based method, the step factor is defined as 100. It is seen that the 5.1. Comparison between One-Step and Two-Step DOA proposed method asymptotically follows CRB and performs Estimation. Among the above experiments, the single-step better than other methods. Even when the SNR is lower than CS approach and two-step CS approach are compared to 0 dB, the proposed method can obtain a lower RMSE. analyze the effectiveness of two-step scheme. According to Table 2 shows the average estimation results of 100 Figure 2, it is observed that the RMSE results obtained by Monte Carlo trials when the sparse degrees are set as two-step estimation methods are smaller than those of k � 3, 4, 5, 6, respectively. Here, the measurement numbers single-step estimation method under the same measurement Mathematical Problems in Engineering 7

30 30

25 25

20 20

15 15 RMSE (deg) RMSE RMSE (deg) RMSE 10 10

5 5

0 0 8 11 14 17 20 23 26 29 32 35 38 8 111417202326 29 32 35 38 M M Measurement number 2 Measurement number 2 Single-step Gaussian matrix Single-step Gaussian matrix Two-step Gaussian matrix Two-step Gaussian matrix Proposed two-step approach Proposed two-step approach (a) (b) Figure 2: Relationship between RMSE and M2 when (a) M1 � 21 and (b) M1 � 25.

RMSE (deg) 10

0 –1 2 5 8 11 14 17 20 23 26 29 SNR (dB) Single-step Gaussian matrix Proposed two-step Two-step Gaussian matrix approach CRLB Figure 3: Relationship between RMSE and SNR. number condition. Figure 3 shows that under the same SNR proposed double-structure matrix is better than that of condition, the two-step estimation methods are superior to Gaussian matrix, which verifies the reliability of the pro- the single-step estimation method. For the cases with dif- posed double-structure measurement matrix. Considering ferent degrees of sparsity, Table 2 and Figure 4 indicate that that the estimation results of the first step determined the compared with the single-step method, the two-step esti- search space of the second step in the CS-based approach, mation methods can reduce the estimated error caused by different measurement matrices in the first step would affect the dense signals emitted from 1° and 2°. the final estimation results. 'us, we investigate the performance of different measurement matrices in the first step for dense DOA estimation. In the experiment, two closely 5.2. Discussion about Measurement Matrix Construction. spaced DOAs are considered for estimation, i.e., Additionally, Figures 2–4 present the comparison results of θ � �1° 2° �. 'e proposed measurement matrix and the two-step CS approaches with different measurement Gaussian matrix are used in the first step, respectively, and matrices. It is found that estimation accuracy of the the same second step is used as in the proposed algorithm. 8 Mathematical Problems in Engineering

Table 2: Comparison of average DOA estimation results using various methods. Estimated angles Sparsity True angles Proposed 2-step Gaussian 1-step 1° 0.6° 0.5910° −1.7° k � 3 2° 2.4° 2.544° 12.4° 35° 34.8° 34.935° 36.6° −20° −21° −20.478° −22.347° 1° 0° −12.501° −1.287° k � 4 2° 1.8° 2.295° 10.674° 35° 34.8° 34.854° 35.514° −40° −39° −38.82° −42.228° −20° −19.2° −22.422° −22.896° k � 5 1° 0° −13.113° −2.799° 2° 1.8° 2.928° 11.16° 35° 34.8° 34.89° 36.423° −40° −39° −38.91° 35.514° −20° −19.2° −22.395° −22.797° 1° 0° −12.591° −2.628° k � 6 2° 1.8° 3.102° 11.583° 35° 34.8° 34.899° 37.449° 70° 70.8° 69.999° 70.299°

18 18

16 16

14 14

12 12

10 10

8 8 RMSE (deg) RMSE RMSE (deg) RMSE 6 6

4 4

2 2

0 0 15 20 25 30 35 40 15 20 25 30 35 40 M M 2 2 Single-step Gaussian matrix Single-step Gaussian matrix Two-step Gaussian matrix Two-step Gaussian matrix Proposed two-step approach Proposed two-step approach (a) (b) Figure 4: Relationship between RMSE and M2 when (a) k � 4 and (b) k � 5.

'e comparison results are plotted in Figure 5. It is found 5.3. Comparison with Other Single-Snapshot DOA Estimation that the proposed algorithm can obtain a better RMSE than Methods for Dense DOA Estimation. Finally, the proposed Gaussian matrix for dense DOA estimation. 'e reason is method is compared with other existing single-snapshot that in the first CS-based procedure, the proposed mea- DOA estimation methods. Considering that the latest surement matrix can obtain a better preestimator with single-snapshot DOA estimation methods are mainly higher accuracy than Gaussian matrix. 'e better pre- designed based on the CS theory, especially some off-grid estimator is helpful to construct a better search range Θ� for CS-based methods might solve the problem of dense pa- the second step and then generate a better accurate esti- rameter estimation. So, we compared the proposed method mator. 'erefore, it is efficient to apply the double-structure with two off-grid CS-based approaches, that is, the OGSBI measurement matrix to two-step CS-based approach to method [15] and the SAMV method [14]. In this experi- improve the estimation accuracy. ment, N � 40, a single-snapshot signal is observed from Mathematical Problems in Engineering 9

7 Carlo loop but three diﬀerent methods in the same ex- ample are recorded. 'e results show that the average 6 execution times are around 3 seconds, 0.01 seconds, and 0.3 seconds when using OGSBI, SAMV, and the proposed 5 methods, respectively. 'ough SAMV could achieve the solution with a lower time cost, its estimation accuracy is 4 poor. Both the OGSBI method and the proposed method require a certain time cost to deal with grid updating 3 process. However, compared to OGSBI, the proposed RMSE (deg) method costs less time. 2

1 6. Conclusion In this paper, a two-step algorithm based on CS theory is 0 10 15 20 25 30 35 40 proposed to solve the single-snapshot dense DOA estima- M 2 tion problem. 'e features of the proposed method include the following: (1) the search spaces with coarse and refined Gaussian matrix for first step Proposed matrix for first step grids are used in the two estimations, respectively, so that the size of the overcomplete matrices is reduced to a low level; Figure 5: Comparison of two different measurement matrices in (2) a new double-structure measurement matrix is firstly M the first step in terms of 2 and RMSE with two densely distributed presented using very sparse projection scheme and chaotic signals. sequence matrix; and (3) the performance of the proposed algorithm is quite good demonstrated by numerical examples. 25 It is shown that the proposed double-structure mea- 20 surement matrix costs less memory than pure chaotic 15 matrix, which performs better than single-step and two-step 10 Gaussian method, and it can work well with smaller mea- 5 surement numbers and lower SNR. It is also shown that the 0 proposed method performs better for single-snapshot DOA –5 estimation of high correlation signals, when compared with – Power (dB) 10 some existing off-grid DOA estimation methods (OGSBI –15 – and SAMV). It should be noted that the number of array 20 elements N should be not smaller than the measurement –25 number M and the value of M should meet the requirement –30 –40 –30 –20 –10 0 10 20 30 40 of the CS theory, i.e., M ≥ K × log2(N/K) in our proposed Direction of arrival (deg) method. True DOAs SAMV In our future work, we will further analyze the perfor- OGSBI Proposed approach mance of the combined methods of SVD and the proposed two-step CS algorithm for the low-altitude target dynamic Figure 6: Comparison results of OGSBI and SAMV with the detection, which is the key to automotive assistance driver proposed approach. system and so on. four angles θ � � −20° 1° 2° 35° �. In order to make the Data Availability comparative experiment fair, the grid spacing of OGSBI and SAMV is chosen as 2°. 'e number of Monte Carlo 'e data used to support the findings of this study are in- loops is 50. Figure 6 shows one simulation trail of the cluded within the article. Monte Carlo experiments to compare the estimated DOAs of three methods. It is seen that OGSBI and SAMV fail to Conflicts of Interest distinguish two peaks when estimating two dense DOAs. In other words, in this experiment, the proposed method 'e authors declare that they have no conflicts of interest. can successfully estimate four DOAs, but the OGSBI and SAMV methods can only obtain three DOAs because two dense DOAs are misunderstood as a single DOA. In brief, Acknowledgments there is usually a problem of high computational com- 'is study was supported by the National Natural Science plexity while existing estimation approaches achieve Foundation of China under grant nos. 61401145 and higher accuracy for dense DOA estimation. In our cal- 61701306 and the Natural Science Foundation of Shanghai culations, the average time costs using the same Monte under grant no. 19ZR1437600. 10 Mathematical Problems in Engineering

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