Multidim Syst Sign Process DOI 10.1007/s11045-013-0239-2

Novel methods of DOA estimation based on compressed sensing Wei Zhu · Bai-Xiao Chen

Received: 17 July 2012 / Revised: 9 May 2013 / Accepted: 27 May 2013 © Springer Science+Business Media New York 2013

Abstract Making use of the sparsity of targets, three novel direction of arrival (DOA) models based on compressed sensing (CS) theory are proposed. Covariance matrix CS, interpolated array CS and beam space CS carry out compressive sampling on covariance matrix, interpolated array and beam space, respectively. High-resolution DOA estimations are obtained through reconstruction of sparse signal by convex optimization problem resolution. The proposed methods are conceptually different from subspace-based methods and provide high resolution using a uniform linear array without restricting requirements on the spatial and temporal stationary and correlation properties of the sources and the noise. Results of both simulated data and measured data show that these methods are superior to conventional DOA methods in angular estimation performance. Keywords

Compressed sensing · Direction of arrival · Array signal processing

1 Introduction Direction of arrival (DOA) estimation has been an important topic in array signal processing, which is of great interest in a variety of applications, such as radars, sonars, wireless communications and seismology. Recently, super-resolution algorithms have been conceived and developed based on their effectiveness and robustness. These approaches of DOA estimation fall into two classes: subspace-based methods and parameter estimation methods. The first group consists of various algorithms which are solely based on the decomposition of a covariance matrix whose terms consist of estimates of the correlation between the signals at the elements of an array antenna, such as the multiple signal classification (MUSIC) (Schmidt 1986) algorithm, estimation of signal parameters via rotational invariance

W. Zhu (B) · B.-X. Chen National Key Lab of Radar Signal Processing, Xidian University, Xi’an, Shaanxi 710071, China e-mail: [email protected] B.-X. Chen e-mail: [email protected]

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techniques (ESPRIT) (Roy et al. 1986), and so on. Subspace-based methods have excellent performance on high signal-to-noise ratio (SNR), long data snapshots scenarios, especially in a spatially uncorrelated sources field. However, the performance will deteriorate significantly under small numbers of snapshots and in the presence of coherent or highly correlated source signals. Since the signal coherence makes the covariance matrix rank deficient. Although the spatial smoothing (SS) technique (Pillai and Kwon 1989; Li 1992) can be used to alleviate this problem, it degrades the performance of these algorithms by introducing biases to the estimates and reducing the antenna gain and resolution. The second group is parameter estimation technique which mainly comprises a variety of maximum-likelihood (ML) (Stoica et al. 1996; Zhao et al. 2008) estimation techniques. In general, these algorithms are much less affected by the signal coherence, but are highly nonlinear and multimodal and exhibit a threshold behavior at low SNR. In addition, a novel propagator method for DOA estimation of incoherently distributed sources is analyzed in Zheng et al. (2013) and a DOA estimation method for coherent sources with a sparse acoustic vector-sensor array is proposed in Liu et al. (2013). The above algorithms are under the condition of the Nyquist/Shannon sampling theory in the spatial domain, that means the distance between adjacent sensors must less than or equal to half a wavelength of the impinging wavefronts, otherwise it may lead to grating lobes and near ambiguities in the array beampattern. These grating lobes may produce large estimation errors if potential ambiguities are not resolved. Different with Nyquist/Shannon sampling, compressed sensing theory (Donoho 2006; Candes and Tao 2006; Candas and WaKin 2008) allows reconstruction of a signal which is sparse or compressible in one basis using an optimization process from a nonadaptive linear projections onto another basis that is incoherent with the first one. Therefore, the sampling rate is dictated not by signal bandwidth but by the sparsity in the signal. The spatial spectrum can be modeled as a large discrete set of spatially distributed far-field sources with varying power levels that includes a large set of plane waves, with only a few of them being of high power. This representation allows the DOA estimation problem to be treating by the sparsity reconstruction algorithms. A l1-svd method for source localization using sparse signal reconstruction is proposed in Dmitry et al. (2005). In Cotter et al. (2005) a focal underdetermined system solver (FOCUSS) was proposed for obtaining single-measurement sparse solutions to linear inverse problems in low-noise environments in which we have access to multiple measurement vectors with sparsity structure, but is computationally complex. A weighted l1 norm penalty (Xu et al. 2012) is used based on Capon spectrum for DOA estimation problem in a sparse signal representation framework, but the choosing of the regularization parameter is a challenge. Compressive beamforming method (Gurbuz et al. 2008) show that by using random projections of the sensor data, along with a full waveform recording on one reference sensor, a sparse angle space scenario can be reconstructed. Compressive MUSIC (Kim et al. 2012) identifies the parts of support using CS, after which the remaining supports are estimated using a novel generalized MUSIC criterion that can approach the optimal l0 -bound with finite number of snapshots. A bearing estimator for sensor arrays (Gurbuz et al. 2012) uses a very small number of measurements in the form of random projections of the sensor data along with one full waveform recording at one of the sensors. The compressive sampling array (Wang et al. 2009) is proposed for array based applications, by exploiting compressive sampling in the spatial domain with random projections (or selections) of the array elements. In the MIMO radar (Yu and Poor 2010), each receive node applies compressive sampling to the received signal to obtain a small number of samples. Assuming that the targets are sparsely located in the angle-Doppler space, the CS technique can be used to reconstruct the target scene, but the receive signal waveform must be known a prior. Spatial compressive sensing (S-CS) (Bilik 2011) method reconstruct the high-resolution spatial spectrum from a

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small number of spatial measurements. S-CS exploits the array orientation diversity to form a sparse recovery problem with larger size. Motivated by the results using compressive sensing in different areas, especially encouraged by the marked advantages of DOA estimation under compressive sensing framework presented in Donoho (2006), Candes and Tao (2006), Candas and WaKin (2008), Dmitry et al. (2005), Cotter et al. (2005), Xu et al. (2012). In this paper, based on CS theory, making use of sparsity of targets, we propose covariance matrix CS (CM-CS), interpolated array CS and beam space CS three novel DOA models. This paper is organized as follows: The signal model is introduced in Sect. 2. The proposed methods are discussed in detail in Sect. 3. The simulation results and real data measurements are shown and analyzed in Sect. 4. Finally, we summarize and conclude in Sect. 5.

2 Signal model Consider a uniform linear array composed of M omnidirectional sensors, where the distance between sensors is d. Assume that K (K < M) arriving narrowband signals impinge on the array from directions θi (i = 1, 2, . . . , K ). The M × 1 array measurement vectors can be modeled as x M×1 = A M×K s K ×1 + n M×1

(1)

where A is array response matrix, A = [a(θ1 ), a(θ2 ), . . . , a(θ K )], a(θi ) is the steering vector, a(θi ) = [1, a(θi ), . . . , a M−1 (θi )]T , a(θi ) = exp(j2πd/λ sin(θi )), the superscript (·)T denotes the transpose operator, s is a vector of signals, s = [s1 , s2 , . . . , s K ], and n is a vector of additive Gaussian noise with power σn2 . In general, signals of practical interest may not be supported in space domain on a set of relatively small size. Instead, the coefficients of elements taken from signal class decay rapidly, typically like a power law. The CS based approaches discretize the bearing space into a large number N(N  M) of distinct bearing angles, where only a few of them are of high power. The signal model Eq. (1) can be approximated as a superposition of N far-filed point sources at the N distinct bearing angles and can be written as x M×1 =  M×N S N ×1 + n M×1

(2)

where x is the measurements vector with M elements,  is a M × N mapping matrix,  = [a( θ1 ), a( θ2 ), . . . , a( θ N )],  θi (i = 1, 2, . . . , N ) is the distinct bearing angle, S is the spatial K-sparse signal. Assuming that the sparsity signal remains constant during the sensing period, this work address the problem of estimating S of size N × 1 with K high-power entries from the vector of measurements x of size M × 1. Under CS framework, S can be recovered using non-adaptive linear projection onto an L × M observation matrix  that is incoherent with . The observation vector y is y L×1 =  L×M x M×1 =  L×M  M×N S N ×1 +  L×M n M×1

(3)

The K -sparse signal S can be exactly recovered by minimizing a convex function which does not assume any knowledge about the number of nonzero coordinates of S, their locations, and their amplitudes which we assume are all completely unknown. The number of observations obeys (Candas and WaKin 2008) L ≥ cK log(N /K )

(4)

where c a small constant. Then S can be reconstructed exactly with overwhelming probability.

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In CS theory, the problem consists of designing a stable observation matrix  such that the salient information in any K -sparse or compressible signal is not damaged by the dimensionality reduction from x to y and a reconstruction algorithm to recover x from observation vector y. According to Dmitry et al. (2005),  must obeys the restricted isometry property (RIP). An equivalent description of the RIP is to say that all subsets of K columns taken from  are in fact nearly orthogonal (the columns of  cannot be exactly orthogonal since we have more columns than rows).  is fixed and does not depend on the signal (Baraniuk 2007). A related condition, referred to as incoherence, requires that the rows  j of  cannot sparsely represent the columns  j of  (and vice versa). Numerous other possibilities exist for construction of CS matrices  (Donoho and Tsaig 2006). Such as random signs ensemble, uniform spherical ensemble, partial Fourier ensemble and partial Hadamard ensemble. It is pointed out that most independent and identically distributed (iid) random matrix could be selected as . However, both the RIP and incoherence can be achieved with high probability simply by selecting  as a random Gaussian matrix. Let the matrix  elements be iid random variables from a Gaussian probability density function with mean zero and variance 1/M. The random Gaussian matrix  has two useful properties: (1) The matrix  is incoherent with the basis  of delta spikes with high probability. Therefore, the sparse and compressible signal can be recovered from random Gaussian measurements; (2) The matrix  is universal in the sense that  will be iid Gaussian and thus have the RIP with high probability regardless of the choice of orthonormal basis . Because the number of observations L is much smaller than the sparsity signal length N , from the matrix theory point of view, solving under-determined equations (3) is not possible. Under the prerequisite that S is K-sparse, the problem solving under-determined equations can be transformed into minimizing l1 norm problem (Baraniuk 2007). The expression is followed as   min  S1 , s.t y =  S + n

(5)

where  S is the reconstruction of S from y. The expression (5) is a convex optimal problem that could be resolved by some algorithms as follow: basis pursuit algorithm (Chen et al. 2001), matching pursuit algorithm (Mallat and Zhang 1993), orthogonal matching pursuit algorithm (Tropp and Gilbert 2007), interior point algorithm (Kim et al. 2007) and gradient projection algorithm (Figueiredo et al. 2007).

3 The proposed methods In practice, most arrays are spacing uniform. The limited dimension of measurement vectors x due to the array size degrade the performance of conventional CS-based DOA estimation approaches. This paper proposes three novel DOA models based on CS theory to improve the accuracy of DOA estimation. 3.1 Covariance matrix CS The dimension of covariance matrix is M × M that is larger than the dimension of measurements vector that is M × 1. We introduce the idea of a sparse representation of array covariance matrix. Apply compressed sensing on covariance matrix and get covariance matrix CS (CM-CS) algorithm.

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The spatial covariance matrix R is defined by R = Rs  H + σn2 I

(6)

where Rs is the source covariance matrix and σn 2 is the noise power. While sources are uncorrelated zero-mean stationary, Rs is real and diagonal. Rs falls into the basic framework of “structured sparsity” or ”group lasso” Huang and Zhang (2010). Then, the spatial covariance matrix can be reformulated as a sparse representation Rv =  v Sv + σn2 Iv

(7)

where Rv = vec(R), vec(·) is the vectorization operator that converts the matrix to a vector by stacking the columns of a matrix into a long column vector starting from the first column,  v = [av ( θ1 ), av ( θ2 ), . . . , av ( θ N )], av ( θi ) = vec(a( θi )aH ( θi )) (i = 1, 2, . . . , K ), Sv (n) = diag(Rs ), diag(·) denotes diagonal opertator, Iv = vec(I). Observation vector y could be obtained through compressive sampling by observation matrix C which is of size L × M 2 . y = C Rv = C  v Sv + σn2 C Iv So expression (5) can be formulated as    2 min  S1 , subjectto y − C  S2 ≤ β

(8)

(9)

where  Sv is the recovered signal of Sv from y, nv = vec(n) and β is a regulation parameter which determined by the power of the measurement noise. 3.2 Interpolated array CS The interpolated array approach is based on the idea that the array manifold of a virtual array, whose element locations are chosen by the user, can be obtained by linear interpolation of the array manifold of the real array, within a limited sector. Compressed sensing could be carried out in the virtual array, named as interpolated array CS (IA-CS) method. Array interpolation is done by considering an interpolation sector  where the source locations are all assumed to be inside this sector. In general, interpolation sector  is uniformly divided into P(P  M) portions with interval θ .  = [θl , θl + θ, . . . , θr ]

(10)

where θl and θr are the left and right bound of . Let A and Av are the manifold matrices of the real array and the virtual array, respectively. BI , the mapping matrix for the conventional array interpolation, is found by the leastsquares solution. H −1 BI = Av AH  (A A )

(11)

where A = [a(θl ), a(θl + θ ), . . . , a(θr )] and Av = [av (θl ), av (θl + θ ), . . . , av (θr )]. av (θi ) is the steering vector of virtual array which has Q(Q  M) virtual sensors. The problem in finding BI is that A AH  may be ill-conditioned for certain angular sector. The Wiener solution for array interpolation improves the DOA performance for noisy observations, i.e. 2 H 2 −1 BI = σs2 Av AH  (σs A A + σn I)

(12)

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where σs2 is the source power. σs2 and σn2 could be obtained from EVD of sample covariance matrix Rx . Rx = E[xxH ]. The mapping matrix BI can be prewhitened so that the noise remains white in the output. The whiten mapping matrix TI is −1/2 H BI TI = (BH I BI )

(13)

The virtual array measurement vectors zI is zI = TI x = TI S + TI n

(14)

Observation vector y could be obtained through compressive sampling from zI by observation matrix I which is of size L × Q(L  Q). y = I zI = I TI S + I TI n Thus, expression (5) can be converted to  2   S2 ≤ β min  S1 , subjectto y − I TI 

(15)

(16)

where  S is the reconstruction of S from y and β is a regulation parameter which determined by the power of the measurement noise. The interpolated array CS can be used for sensor arrays with an arbitrary configuration. A virtual array can be obtained by virtual interpolated method from arbitrary array geometries. Then compressed sampling can be applied to the virtual array data. 3.3 Beam space CS In practice, the signal of interest usually locate at some special space that means that all source locations are inside some spatial sector. Similar to spatial domain and time domain, compressed sensing could be carried out in beam space domain, which referred to beam space CS (BS-CS) approachss. Assume that all source locations are inside the spatial sector , as described in expression (6). Consider a beam space operation denoted by its beamforming matrix BB that P beams can be formed in . √ (17) BB = [a(θl ), a(θl + θ ), . . . , a(θr )]H / M The beamforming matrix BB can be prewhitened and the whiten beamforming matrix TB is −1/2 H BB TB = (BH B BB )

(18)

The beam space measurement vectors zB is zB = TB x = TB S + TB n

(19)

Similar to Eq. (11), observation vector y could be obtained y = B zB = B TB S + B TB n where B is a observation matrix of size L × P(L  P). Therefore, expression (5) can be transformed to    2 min  S1 , subjectto y − B TB  S2 ≤ β

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(20)

(21)

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where  S is the reconstruction of S from y and β is a regulation parameter which determined by the power of the measurement noise. Spatial gain could be got through beam space transformation. Besides, beam space transformation improves the estimation accuracy and reduces the sensitivity due to the systematic err. The compressed sampling can be applied to the beam space data.

4 Simulation results and real data measurements In this section, we illustrate the performance of the proposed methods. 4.1 Simulation results Computer simulations are carried out to study the performance of the proposed methods. The number of array elements is 20. The method proposed in Roy et al. (1986), Li (1992), Zhao et al. (2008), Dmitry et al. (2005), and Bilik (2011) are referred as ESPRIT, SS-MUSIC, AP-ML, l1-svd and S-CS, respectively. They are used to compare with CM-CS, IA-CS and BS-CS described in Sect. 3. ESPRIT, SS-MUSIC and AP-ML are classical DOA estimate methods. l1-svd use sparse reconstruction with the original samples directly and S-CS form a sparse recovery problem with spatial compressive sampling. The size of the subarray is 18 in SS-MUSIC algorithm. In all CS-based approaches, observation matrix is random Gaussian matrix, the bearing space is discretized into N = 18,000 points and the interval is 0.01◦ . The minimizing l1 norm problem is resolved by orthogonal matching pursuit algorithm. In S-CS, compressive sampling length is L = 4. In the proposed methods compressive sampling length is L = 20. The interpolation sector and the spatial sector Θ = [−10◦ , −9.9◦ , . . . , 10◦ ], the interval θ is 0.1◦ . In IA-CS, 9 virtual elements are interpolated between every two elements. The total elements of virtual array is Q = 191. In BS-CS, the number of beams is P = 200. The number of snapshots is 20. We perform 500 Monte Carlo trials for each experiment. In the first simulation, we assume that d = 0.5λ and consider one source coming from 2◦ . The RMSE of the DOA estimates versus input SNR is shown in Fig. 1a. The CramerRao bound (CRB) (Stoica and Nehorai 1990) is given either. Note from the figure that with increase of SNR the RMSE of the DOA estimates is close to CRB. The figure illustrates that the estimates by the proposed methods are considerably more accurate than that by ESPRIT, l1-svd and S-CS. The second simulation considers the scenario that d = λ which means the Nyquist/Shannon sampling theory is not satisfied. Two correlated sources coming from [2◦ , −2◦ ]. Assume that all source signals are of equal power. The RMSE of the DOA estimates versus input SNR is shown in Fig. 1b. Similar to the first simulation, the result shows that our methods still have better performance than SS-MUSIC, AP-ML, l1-svd and S-CS for coherent signals. 4.2 Real data measurements A series of experiments were performed to analysis the performance of the proposed methods. 20-element uniform linear array locate on flat earth. The antenna aperture is about 18 m, and the corresponding half power beamwidth θ3dB in elevation is 4.6◦ . All elements in the array were precisely aligned to minimize errors due to misalignment. In the experimental part we use the result of SS-MUSIC as the coarse estimate θc , the interpolation

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sector  is [θc − θ3dB , θc − θ3dB ] and the interval θ is 0.1◦ . 9 virtual elements are interpolated between every two elements in IA-CS algorithm. As Fig. 2a shows, the plane fly over the route in the east from 50 to 310 km. The SNR of measured data is shown in Fig. 2b. Figure 2c and d present the processing results of the measured data, in which the lines are the true elevation angle or height recorded by GPS on the plane, symbol ’+’ and ’o’ are the results by SS-MUSIC and IA-CS algorithms, respectively. It can be concluded from the results that, between 50 and 160 km the target is at high angle, SS-MUSIC and IA-CS can obtain correct estimation of DOA. But the result of SS-MUSIC has bigger fluctuation. From 160 and 310 km the target is at low angle, under the influence of multipath propagation and beam split, SS-MUSIC cannot separate a target from its image which lying within a beamwidth. But IA-CS provides an accurate estimate of the target’s angle of arrival. According to IA-CS, the RMSE of elevation and height are 0.07◦ and 322 m, which perform better than SS-MUSIC that have elevation RMSE 0.59◦ and height RMSE 2236 m. The effectiveness of the proposed method is validated by results of real data.

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5 Conclusions In this paper, we have proposed covariance matrix CS, interpolated arrays CS and beam spaces CS for DOA estimation that is suitable for uniform linear array. The first method, covariance matrix CS algorithm is suit for uncorrelated sources. The interpolated array CS algorithm and beam space CS algorithm can be used in correlated sources environment. But an interpolation sector or a spatial sector should be known as a prior in the interpolated array CS algorithm and beam space CS algorithm. That means the performance will degrade while the sector don’t include the source DOA. This work shows that the proposed methods provide additional improvement in spatial spectrum estimation performance compared to ESPRIT, MUSIC, l1svd and spatial CS. All the advantages are obtained with a reasonable increase of computation load compare to conventional DOA methods based on CS. By using both simulated data and real data measurements, it is shown that our approaches achieve high angular resolution and outperform the other methods in single-source and multi-sources scenarios. Acknowledgments This work is supported by National Natural Science Foundation of China (61001209, 61101244), the Fundamental research Funds for the Central Universities (K5051202038) and Program for Changjiang Scholars and Innovative Research Team in University (IRT0954).

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Author Biographies Wei Zhu was born in Susong, Anhui Province, China, in September 1987. He received his B.A. and M.S. degrees from Xidian University, Xi’an, China, in July 2007 and March 2010, respectively. Now he is working towards the Eng.D. Degree in Radar signal processing from Xidian University. His advisor is Professor Baixiao Chen. His research interests include altitude measurement in digital array radar, VHF array radar, array signal processing, etc.

Bai-Xiao Chen was born in Anhui, China in 1966. In 1994 and 1997, he received the M.S. degree in circuit and system and Ph.D. in signal and information processing at Xidian University, China, respectively. He is currently a Professor of Signal and Information Processing in Key Lab for Radar Signal Processing, Xidian University. His current research interests include synthetic impulse and aperture radar (SIAR), signal processing in terminal guidance radar, and new radar system design.

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