Considering the computational complexity and redundancy of traditional array signal arrival angle (DOA) estimation algorithms, the compressed sensing ...

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Novel methods of DOA estimation based on compressed sensing

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Off-Grid DOA Estimation Via Real-Valued Sparse Bayesian Method in Compressed Sensing

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A new DOA estimation algorithm based on compressed sensing Zhang Yong1 · Zhang Li-Yi1 · Han Jian-Feng1 · Ban Zhe1 · Yang Yi1 Received: 18 July 2017 / Revised: 4 January 2018 / Accepted: 6 January 2018 © Springer Science+Business Media, LLC, part of Springer Nature 2018

Abstract Considering the computational complexity and redundancy of traditional array signal arrival angle (DOA) estimation algorithms, the compressed sensing technology was used to improve the real-time and accurate performance of the DOA estimation algorithm, in which, the space sparse signals were reconstructed from the array data by means of array manifold matrix. Compared with the classical MUSIC algorithm, the compressed sensing DOA estimation method could effectively improve the direction finding accuracy and angle resolution with low SNR and snapshot deficiency. Moreover, the proposed algorithm could achieve the coherent signal estimation correctly, and the simulation results show that its performance was superior to that of traditional algorithm. Keywords DOA estimation · Compress sensing · Sparse signal

1 Introduction The estimation of direction-of-arrival (DOA) was one of the hotspots and key technologies in the array signal processing research field, which were mainly based on time domain spectrum estimation and spatial filtering. The DOA estimation has been widely used in many fields, such as target monitoring and localization [1,2]. The classical methods were mainly implemented by MUSIC [3,4] and ESPRIT [5,6] algorithm, in which the estimation of the target arrival angle was achieved mainly based on the probability and statistical characteristics of the received array signal with mathematical decomposition methods. MUSIC algorithm and ESPRIT algorithm could achieve high angular resolution for the DOA estimation, but they were only suitable for the uncorrelated signals or weak correlated signals [7]. Although, the subspace fitting algorithm could be directly applied for the coherent signals and easily to be implemented, but the computation burden of the methods was very heavy and which was not suitable for practical engineering application. Some scholars further studied the MUSIC algorithm for reducing the amount of calculation, such as the root MUSIC algorithm and some

B

Zhang Li-Yi [email protected] Zhang Yong [email protected]

1

School of Information Engineering, Tianjin University of Commerce, Tianjin 300134, China

improved root MUSIC algorithm [4,8], but the algorithms were still relatively complicated with low estimation accuracy. In recent years, being a new signal processing theory, the compressed sensing (CS) [9,10] has attracted wide attention of scholars and been applied in pattern recognition, graphic image processing, wireless communications and other fields. As for the compressed or sparse signal processing problems, the signals could be sampled below the Nyquist rate to reduce redundant data processing, transmission and storage pressure, which could provide a new solution for the DOA estimation in the future. Some Scholars have carried out a preliminary study on the application of compressed sensing technology to DOA estimation, and achieved corresponding results [11–16]. Malioutov was the first scholar who applied the sparsity idea to the array signal DOA estimation problem [11]. The main idea of the method was to establish a sparse signal reconstruction model based on discretization space angles firstly, and then using the two order cone programming (SOC) optimization method for the DOA estimation solving, finally the high resolution DOA estimation was achieved with the spatial mesh size adjusting. Gribonval [12] reconstructed the sparse point space model based on the random projection of the sensor array measurements and the complete waveform was obtained by reference array sensor nodes, then the ultimate goal of the source numbers and DOA estimation would be achieved. In [13], the compressed sensing tech-

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nology was used for the target source estimation of MIMO radar echo sparse signals, and compared with the traditional method, the compressed sensing method could effectively obtain more target information to improve the resolution performance. In [14], the compressed sensing technology was firstly used to achieved the compressed sampling of the received signals with the sensor array, and the DOA estimation of high resolution array based on MUSIC algorithm was achieved with the sampling points of sensor array reducing. Lee studied the Narrowband signal DOA estimation of snapshot data and wideband signal DOA estimation problem with measurement vector model based on the compressed sensing technology [15]. In [16], the influence between the estimation performance and sparse model selection was analysed in the compressed sensing DOA estimation. As for the compressed sensing DOA estimation, the first problem which should be solved was the corresponding method for signal sparse representation and proper sparse model reconstruction. The sparse space representation method often determined the performance of the proposed model reconstruction. At present, the research of the establishment of sparse model has not been systematically expounded for compressed sensing DOA estimation, in which the spatial equal angle thinning method was usually used. In this paper, we proposed a new CS based DOA estimation method, in which, the sinusoidal space division method and space angle division method was used for the signal sparse representation and sparse model reconstruction. Simulation results show that the realization of the sinusoidal space division and model reconstruction method was more superior to the traditional space angle division method, and it could achieve fast and accurate DOA estimation with higher reconstruction performance than the traditional method.

2 Compressed sensing DOA estimation model The principle of CS: If the signal is compressible or sparse, the high-dimensional signal could be projected to the low dimensional signal with the measurement matrix, which is not related to the sparse transform, and then reconstruct the original signal from a few projections through solving the convex optimization problem. It usually consists of three core parts: sparse signal representation, measurement matrix design and signal reconstruction algorithm. In this paper, the DOA estimation problem of compressed sensing was mainly studied based on uniform linear sensor array (ULA). The signal vector s = [ s1 s2 . . . s N ]T were assumed inthe whole spatial domain (− 90◦ ∼ 90◦ ), and θ1 θ2 . . . θ N mean the incident angles in turn and correlate θi with si in a one-to-one correspondence, and all possible sources were exist in the N signals, as shown in Fig. 1:

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θi τ d

d

d

d

Fig. 1 Uniform linear sensor array

The mathematical model of signals receiving by sensor array could be described as: y = As + n

(1)

where, A was the array flow pattern matrix A ∈ M×N (M << N ), which could be defined as: A = [a(θ1 ), a(θ2 ), · · · ⎡ 1 ⎢ e-j2π d sinλ θ1 ⎢ =⎢ ⎢ .. ⎣. (M−1)d sin θ1 λ e-j2π

, a(θ N )] 1 d sin θ2 e-j2π λ .. .

(M−1)d sin θ2 λ e-j2π

··· 1 d sin θ N · · · e-j2π λ ..

. . ..

· · · e-j2π

(M−1)d sin θ N λ

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(2) where, d was the distance between the two adjacent array elements, λ was the wavelength of light, M was the number of observed signals. Generally, the number of received signals s = [ s1 s2 . . . s N ]T was far greater than the number of real target signals. Suppose that there only K nonzero values in received signals s, and we could say that the signals s were with K -sparsity. The array flow pattern matrix A ∈ M×N (M << N )was determined by the given spatial sparseness θ1 θ2 . . . θ N , so the matrix A would be not depended on the direction of the K real target signals any more. In the compressed sensing DOA estimation method, the signal s = [ s1 s2 . . . s N ]T would be reconstructed through y with the array flow pattern matrix A, and the K nonzero value signals in s = [ s1 s2 . . . s N ]T were the real target signals, which could be achieved with the one-to-one correspondence between θi and si in the compressed sensing DOA estimation of the target signals.

3 The sparse reconstruction of compressed sensing 3.1 The constraint condition of sparse reconstruction The theory of compressive sensing could be essentially described as an underdetermined problem to find the sparse

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solution of the linear model b = As given by the measurement matrix A ∈ M×N (M << N ) and the observed value vector b ∈ M . And the vector s ∈ N was signals with unknown sparse need to be reconstructed, which could be usually resolved as a l0 norm problem: sˆ = argmin sl0

s.t. As = b

(3)

where sl0 was the number of nonzero elements in the signal vector s. The solution of Eq. (3) could be describe as nonconvex optimization problem, in which, if sl0 ≤ K , the vector s could be called K -sparsity signals and it was also a NP-Hard problem for the l0 norm solving. Generally, it can be transformed into a l1 convex optimization problem: sˆ = argmin sl1

s.t.As = b

⎡

max

1≤i≤ j≤N

ai , a j

1 ⎢ e–jπ sin θ1 ⎢ A=⎢ ⎢ .. ⎣. e-jπ(M−1) sin θ1

1 e-jπ sin θ2 .. . e-jπ(M−1) sin θ2

··· ··· .. .

1 e-jπ sin θ N .. . · · · e-jπ(M−1) sin θ N

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(7) (5)

The RIP constraint was the first condition of the compressed sensing problem. However, as for the real applications, it was difficult to design a RIP satisfied measurement matrix directly. And it was very difficult to verify whether the condition was satisfied RIP for a given corresponding measurement matrix. Therefore, most scholars established a non-RIP theory based on the theory of matrix coherence, which was easy to implement and apply and had become a breakthrough point of practical application of compressive sensing theory. Sometime, it was much easier to analyze and calculate the coherence coefficients of a matrix than RIP analysis than the Restricted Isometry Constant analytical calculation, which using for RIP description. The coherence coefficient of a matrix A ∈ M×N (M << N )was defined as follows: μ=

As for the DOA estimation problem in this paper, the reconstruction of signals s through y based on the theory of compressed sensing was mainly depended on the correlation or orthogonality between the columns in A ∈ M×N (M << N ), which was usually related to spatial grid partition. The current space grid partition methods included two kinds: the spatial equal angle division and space equal sine division. In the equal angle division, the ◦ whole airspacearea (− 90◦ ∼ 90 ) wouldπ be divided into equal intervals θ1 θ2 . . . θ N and θi = − 2 + (i − 1) Nπ−1 , i = 1, 2, . . . , N . At the same time, let d = λ/2to simplify the analysis, then:

(4)

The precondition of Eq. (4) was that the measurement matrix A should satisfy the Restricted Isometry Property (RIP), it mean that if 0 < δ < 1, any K -sparsity signal s should be satisfied the following equation: (1 − δ) s22 ≤ As22 ≤ (1 + δ) s22

3.2 The spatial meshing method for compressed sensing

(6)

Given the constraint condition that the measurement matrix A ∈ M×N (M << N ) could be decomposed into two orthogonal matrices and μ < 2K1−1 , the above l1 convex optimal problem could be reconstructed exactly for any K sparsity signals [17]. Gribonval and Nielsen generalized the orthogonal constraints of the measurement matrix to a general situation case [18]. In [19], it is pointed out that the best criterion for solving the l1 convex optimization problem with arbitrary K -sparsity s signals was μ < 2K1−1 , and it could also be applied to the robustness of l1 convex optimization problems.

The space equal sine division is that sinθi = −1 + (i − 1) N2-1 i = 1, 2, . . . , N , so: ⎡

1 ⎢ jπ ⎢e ⎢ A = ⎢. ⎢ .. ⎣ ej(M−1)π

··· 1 -jπ −1+(i−1) N2-1 ··· e . . .. . .

-jπ(M−1) −1+(i−1) N2-1 ··· e

⎤

··· 1 · · · e-jπ ..

⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. . ..

· · · e-j(M−1)π

(8) Through the above correlation theory analysis, we could see that the two division methods both have the RIP property and satisfied the necessary conditions of signal reconstruction, and the more significant of the RIP, the better performance of signal reconstruction.

3.3 DOA estimation algorithm based on CS reconstruction algorithm According to the signal reconstruction theory of CS, the greedy algorithm was widely used with the less computation and complexity. In this paper, the greedy algorithm was adopted to solve the problem of signal reconstruction. The DOA estimation process based on orthogonal matching pursuit (OMP) algorithm was as follows and shown in Fig. 2: Step1: Assume the sparsity of y, A, s were known, initialized residuals r0 = y, indexed sets 0 = ∅, the number of iterations i = 0.

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Fig. 3 Estimation spectrum of MUSIC algorithm

Step6: To determine whether the iteration stop condition is satisfied, if not satisfied, going to Step2. If else, iterating stop; Step7: Approximately reconstructing the signal s = [ s1 s2 . . . s N ]T Step8: According to the one-to-one correspondence between θi and si , DOA estimation could be obtained.

4 Simulation and discussion 4.1 Simulation analysis of classical algorithms Firstly, the DOA estimation of non-coherent signals and coherent signals based on MUSIC algorithm and ESPRIT algorithm were analyzed through Matlab simulations [20]. 4.1.1 DOA estimation of non-coherent signals

Fig. 2 The OMP based DOA estimation process diagram

Assumed the non-coherent signals respectively with − 40◦ , 40◦ and 60◦ angel incident to the 8 element ULA, and the sub-array element number is 7, SNR is 10 dB, the sampling number 100. The simulation results are shown in Figs. 3 and 4. From the angle corresponding to the spectral peak in Fig. 3 and the DOA estimation value in Fig. 4, it could be seen that both MUSIC and TLS-ESPRIT algorithms can achieve effective DOA estimation for noncoherent signals. 4.1.2 DOA estimation of coherent signals

Step2: Find the most closely related column to the residu

als in the matrix A, that was λi = argmax j=1,..., N a j ,r n−1

Step3: Update the index set i = i−1 ∪ {λi } and set Ai = Ai−1 ∪ aλi −1 T Ai y Step4: Calculating signals si = AiT Ai Step5: Update residual ri = y − Ai si

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It is assumed that two coherent signals incident to ULA with − 45◦ and 60◦ respectively. The elements number is 8, the number of sub-array elements is 7, the sampling number is 1024, the signal-to-noise ratio is 20 dB, and the simulation results of the two algorithms were shown in Figs. 5 and 6.

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Fig. 4 DOA estimation of TLS-ESPRIT method

Fig. 7 Estimation spectrum of MUSIC algorithm

From the angle corresponding to the spectral peak in Fig. 5 and the DOA estimation in Fig. 6, it could be seen that neither of the two algorithms can achieve the DOA estimation of the coherent signal source. This is because the coherent signal will cause the rank loss of the covariance matrix of the received data of the array, and it would result in the mutual penetration of the signal and the noise subspace, Eventually cause the algorithm to fail. 4.1.3 DOA estimation of MUSIC and ESPRIT algorithm with different SNR

Fig. 5 Estimation spectrum of MUSIC algorithm

Fig. 6 DOA estimation of TLS-ESPRIT method

Assume signal source incident to the 8 element ULA with − 20◦ , 20◦ and 25◦ angel respectively, the sampling number is 100, the simulation results of MUSIC with − 20, 0 and 20 dB SNR was shown in Fig. 7; the signal source using ESPRIT algorithm to get 100 experimental RMSE results with 20◦ in the − 10 dB ∼ 10 dB SNR was shown in Fig. 8. Figures 7 and 8 show that the estimation accuracy and resolution of the subspace decomposition class algorithm were improved with the increase of signal-to-noise ratio, and the estimation performance of the ESPRIT algorithm was close to that achieved by the two methods of LS and TLS. The same experiment shows that the estimation accuracy of MUSIC and ESPRIT algorithms will be improved with the increase of the number of array parameters and snapshots in a certain range. Assume signal source incident to the 8 element ULA with 30◦ angel, sub-array element number is 7, the number of snapshots is 1024 points, the signal-to-noise ratio from 0 dB to 35 dB with 5 dB interval increased, the simulation results was shown in Fig. 9, the RMSE of the MUSIC algorithm in the same SNR is lower than the ESPRIT algorithm, so the MUSIC algorithm is better than ESPRIT in estimation accuracy.

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Fig. 10 Signal reconstruction with equal angle division

Fig. 8 RMSE of ESPRIT algorithm varies with SNR

4.2 Simulation analysis of CS In this paper, the simulation experiments of the DOA estimation based on CS theory was as following: 4.2.1 DOA estimation based on OMP algorithm with space equal angles divination

Fig. 9 Comparison of DOA and TLS methods for MUSIC estimation of RMSE

The running time of the two algorithms with different signal to noise ratios was shown in Table 1. It could be seen that the MUSIC algorithm had a longer running time cycle than the ESPRIT algorithm with different SNR, the reason is that the spectrum search should be used in the whole airspace for the MUSIC algorithm, while the ESPRIT algorithm avoids this process to estimate the DOA eigenvalue directly.

Assume the source signal incident on the ULA with to − 60◦ , − 30◦ , 10◦ , 40◦ and 60◦ angel respectively, the DOA estimation simulation experiment was fulfilled based on the OMP algorithm with single sampling, the reconstruction results was shown in Fig. 10 and the DOA estimation results was shown in Fig. 11. As shown in Fig. 10, the reconstructed signal was more different from the original signal and the reconstruction error is larger. This is because that the observation matrix can not satisfy the RIP property very well with the space equal angles divination. As shown in Figure 11, the estimated DOA values were − 61.1◦ , − 27.7◦ , − 22.3◦ , 2.3◦ and 60◦ according to the one-to-one correspondence between the reconstructed signal and its incident angle. It can be seen that the accurate estimation is only achieved in the direction of 60◦ , and the estimation results had a great obvious deviation. Therefore, the experimental results show that the CS theory can not be used to estimate DOA exactly and accurately under equal angles divination.

Table 1 Comparison of operation time between MUSIC and TLS SNR (dB)

0

5

10

15

20

25

30

35

MUSIC (s)

4.675505

4.645961

4.478536

4.645797

4.612681

4.555253

4.498401

4.530326

ESPRIT (s)

0.203792

0.211400

0.189630

0.194083

0.207170

0.201223

0.199145

0.194483

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Fig. 11 DOA estimation results with equal angle division

Fig. 13 DOA estimation with equal sinusoidal division

Fig. 12 Signal reconstruction with equal sinusoidal division

Fig. 14 Curve of RMSE with SNR variation in DOA estimation based on CS

4.2.2 DOA estimation based on OMP algorithm with space equal sinusoidal divination The experimental conditions were the same as that of the space grids with equal angles. The result of reconstruction was shown in Fig. 12, and the result of DOA estimation was shown in Fig. 13. It can be seen from Fig. 12 that the reconstructed signal basically coincides with the original signal, so the performance of reconstruction with space equal sine divination mode is better than that in the equal angle mode. The experimental results fully illustrate that the array flow pattern matrix with space equal sine divination mode has more significant RIP property and is more suitable for sparse signal reconstruction in the DOA estimation problem. As can be seen from Fig. 13, the DOA estimation results − 59.9◦ , − 30◦ , 10.1◦ , 40◦ and 60.1◦ obtained by the reconstructed signal were obviously close to the real inci-

dence angle. That is to say the DOA estimation with space equal sine divination mode could get more accurate performance. 4.2.3 Simulation of DOA estimation based on OMP algorithm with different SNR Assume source signal incident to the ULA with the 40◦ angel, the OMP algorithm is used to reconstruct the signal and estimate DOA with space equal sine divination and single sampling mode, SNR were from − 10 to 30 dB with the 5 dB interval increased, the RMSE of 100 DOA estimation experiments was shown in Fig. 14. It can be seen that the performance of DOA estimation based on OMP algorithm will be better improved with the increase of SNR. But when SNR increases to a certain value, the

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5 Conclusion The DOA estimation method based on compressed sensing theory was mainly studied in this paper, As for the traditional MUSIC algorithm only applies to the non-coherent signal DOA estimation, the compressed sensing DOA estimation method could not only realize coherent signal DOA estimation, but also it could achieve an accurate DOA estimation with single snapshot conditions. The simulation results show that the proposed method would be better than the MUSIC algorithm in the estimation accuracy and real-time performance, and the sine grid space division model was super to the angle space division, and its estimation accuracy could be improved with the signal-to-noise ratio increasing. Fig. 15 Comparison of CS and MUSIC methods

Table 2 Computing time of DOA estimation based on MUSIC and CS algorithm Algorithm

MUSIC

CS

Run Time (s)

0.3254

0.0709

Acknowledgements The authors wish to thank for the financial support of Natural Science Foundation of China (61573253, 61271321), Tianjin Natural Science Foundation (16JCYBJC16400), Tianjin Enterprise Science and Technology Project of Special Correspondent (17JCTPJC54700), Tianjin Science and Technology Project (16YFZCGX 00360,16ZXZNGX00080), National Training Programs of Innovation and Entrepreneurship for Undergraduates (201610069007, 20171006 9023). The corresponding author is Professor Zhang Liyi.

References estimation performance will not be significantly improved more.

4.2.4 DOA estimation comparative analysis between CS and MUSIC algorithm Assume source signal incident to the ULA with − 30◦ , 10◦ , 50◦ and 60◦ angel, the DOA estimation were finished based on the OMP algorithm and MUSIC algorithm in a single sampling mode, the simulation results of DOA spectrum estimation was shown in Fig. 15, and the comparison of operation time was shown in Table 2. It can be seen from Fig. 15, the CS DOA estimation has an accurate performance, as for the MUSIC estimation algorithm, there were some high sidelobe in the DOA spectrum amplitude, which would affect the accuracy of the DOA estimations. So in order to improve the MUSIC DOA estimation performance, it should be need to increase the number of sampling. Therefore, the DOA estimation problem using CS theory could overcome the shortcoming of the traditional algorithm that requires a large amount of sampled data. Table 2 was the comparison of the run times of the two algorithms. It can be seen that the computation time of DOA estimation using CS theory is less than that of MUSIC algorithm, and it would reduce the computational complexity and be superior to the traditional method in real-time DOA estimation.

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Han Jian-Feng is an associate professor in Tianjin University of Commerce, China. His research interests include Dangerous Goods Logistics Monitoring and Control.

Ban Zhe is an undergrads in Tianjin University of Commerce, China.

Zhang Yong is an associate professor in Tianjin University of Commerce, China. His research interests include Sensor networks and Distributed Estimation.

Yang Yi is an undergrads in Tianjin University of Commerce, China.

Zhang Li-Yi is a professor and Ph.D. candidate supervisor in Tianjin University of Commerce, China. His research interests include Signal detection and information processing.

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