Direction of Arrival Estimation With Co-prime Arrays Via Compressed Sensing Methods Tianyi Jia, Haiyan Wang, Xiaohong Shen, Xingchen Liu School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, China Email:
[email protected] Abstract—In this paper, we consider the problem of direction of arrival (DOA) estimation with co-prime arrays via compressed sensing methods. A sparse signal recovery model based on the framework of co-prime array is presented. We propose to use OMP algorithm to efficiently implement the optimization procedure, which have a lower computational cost. The sparse recovery model of DOA estimation obeys the isotropy property and incoherence property. Therefore, by exploiting the RIPless theory in compressed sensing, we develop the upper bound of degrees of freedom (DOF) of the proposed model. The results establish a basic relationship between upper bound of DOF, the number of samplers and the probability of recovery. Numerical examples show the superiority of the proposed method in detection performance and estimation accuracy compared with the existing spatial smoothing MUSIC algorithm using co-prime arrays.
I. I NTRODUCTION Direction of arrival (DOA) estimation using sensor arrays finds broad applications in sonar, radar, and wireless communications. Conventional research on DOA estimation using array processing focuses primarily on uniform linear arrays (ULA). While conventional DOA estimation methods resolve fewer sources than the number of array sensors, detecting a higher number of sources is usually of tremendous interest. Specifically, by implementing a ULA with N sensors, the degrees of freedom (DOF) that can be resolved are only N −1. In fact, the non-uniform linear arrays are introduced into the field of array signal processing to increase the DOF and hence can resolve more sources than the physical sensors. The familiar existing non-uniform linear arrays called minimum redundancy arrays (MRA) have been used earlier in [1], [2] to increase the DOF by constructing an augmented covariance matrix. However, there are several shortcomings with MRA. First of all, the augmented covariance matrix is not positive semidefinite matrix for finite number of snapshots. Secondly the optimum design of MRA relys on complicated algorithms or computer simulations. Thirdly it is difficult to provide closed form expression for the array geometry and achievable DOF for a given number of physical sensors. Therefore, the above mentioned shortcomings restrict MRA to applications. In recent years, a new geometric structure of non-uniform linear arrays called co-prime arrays have been proposed to increase the DOF of the array [3]–[5]. The structure can 1 This work was supported by National Natural Science Foundation of China (61571367, 61571365, 61401364).
978-1-4673-9724-7/16/$31.00 ©2016 IEEE
Fig. 1. Geometry of Co-prime Arrays
achieve O(M N ) DOF with O(N + M ) physical sensors. Furthermore, it is possible to provide closed-form expressions for the sensor locations and the exact DOF obtainable from the proposed arrays as a function of the total number of sensors. The increased DOF provided by the co-prime array can be used for DOA estimation. The co-prime arrays consist of two ULA with N and 2M sensors respectively, as shown in Fig.1. The N sensors of one ULA locate in M nd, 0 ≤ n ≤ N − 1, and the 2M sensors of the other ULA locate in N md, 0 ≤ m ≤ 2M − 1. Because the first sensor is collocated, there are a total of N + 2M − 1 sensors. Note that N and M are co-prime numbers. The co-prime array can behave like a large virtual array which has 2M N − 1 sensors located at nd, −M N ≤ n ≤ M N . The spatial smoothing MUSIC algorithm with co-prime arrays was proposed in [6] to detect N M targets with N + 2M − 1 sensors. However, because the application of spatial smoothing leads to a significant reduction of the obtained virtual array, the corresponding DOF of detecting is decreased. Sparse recovery methods in compressed sensing was introduced in [7] to overcome the above difficulty. They focus on compressed sensing methods considering off-grid targets to co-prime arrays and increase the DOF. However, their work do not include a theoretical analysis about the upper bound on the DOF of detecting. The early papers [8]–[10] in compressed sensing triggered a massive amount of research on sparse signal recovery. This theory states that a discrete signal x ∈ RN can be reconstructed from a random measurement matrix A provided that x is sparse. Formally, suppose x has at most s nonzero terms which we do not know the locations and we are given the value of its measurements y = Ax, where A ∈ CM ×N . Then the theory asserts that one can recovery x if the matrix A and signal x satisfy the restricted isometry property (RIP).
However, it is extremely difficult to verify the RIP of a sparse problem, and so the restricted isometry machinery does not directly apply in the setting. A probabilistic and RIPless theory of compressed sensing was introduced in [11]. This paper present a simple and very general theory of compressive sensing which does not require the RIP to hold for the sparsity in question. The goal of this paper is to explore how to utilize the compressed sensing methods for DOA estimation with coprime arrays. Furthermore, we will focus on analysing the upper bound of DOF of the proposed method based on the RIPless theory. The main contribution of our work is to combine the DOA estimation of compressed sensing and the RIPless theory for theoretical analyses. We start with a brief introduction to the signal model of co-prime arrays and spatial smoothing MUSIC algorithm in section II. In section III, we describe the sparse signal recovery model of DOA estimation with co-prime arrays and present the upper bound of DOF in details based on RIPless theory. In section IV, the performance of the mentioned two methods is explored using simulated experiments and the proposed upper bound of DOF is utilized for analyzing the property of DOA estimation. Finally, conclusions are made in section V. II. C O - PRIME A RRAY AND T HE MUSIC A LGORITHM A. Signal Model of Co-prime Arrays Suppose that a linear array with N + 2M − 1 sensors, such as the one in Fig.1, observes T snapshots of spatial signals with independent Gaussian distributions. We assume there are K spatial narrowband signals sk (t) located at ϕ1 , ϕ2 , . . . , ϕK , arriving at the ith sensors, after being corrupted by additive noise ni (t), resulting in sensor outputs xi (t). Let x(t) = [x1 (t), x2 (t), . . . , xN +2M −1 (t)] and the signal received by the array at time t can be expressed as [7] x(t) =
K
a(ϕk )sk (t) + n(t) = As(t) + n(t)
(1)
where n(t) is an i.i.d Gaussian noise. The matrix A is socalled array manifold matrix, whose (i, k)th element contains the delay and gain information from the kth source to ith sensor. The columns a(ϕk ) of A are called steering vectors. Let λ denotes the wavelength of the signal and li denotes the location of the ith sensors. Specifically, a(ϕk ) can be expressed as 2π
2π
(2)
We assume that sk (t) follows an independent Gaussian distribution with variance σk2 . Because the source signals are uncorrelated form each other and also from noise, the correlation matrix of the received data x(t) can then be expressed as Rxx = E[x(t)x(t)H ] =
K k=1
σk2 a(ϕk )a(ϕk )H + σ 2 I
where σ 2 is the noise variance. Then, we vectorize Rxx in column and use kronecker product to get the following vector z
= =
vec(Rxx ) K 1n σk2 a(ϕk )a(ϕk )H ] + σ 2 ˆ vec[
=
1n Φ(ϕ1 , ϕ2 , . . . , ϕK )p + σ 2 ˆ
k=1
(3)
(4)
where Φ(ϕ1 , ϕ2 , . . . , ϕK ) = [a(ϕ1 )∗ ⊗ a(ϕ1 ), . . . , a(ϕK )∗ ⊗ 2 a(ϕK )], p = [σ12 , σ22 , . . . , σK ] and ˆ 1n = [e1 , e2 , . . . , eN +2M −1 ] with ei being a column vector of all zero terms except a 1 at the ith term. Based on the property of kronecker, We can see that Φ(ϕ1 , ϕ2 , . . . , ϕK ) is a matrix of size (N + 2M − 1)2 × K from equation (4). Moreover, z in (4) behaves like a received signal of an array whose manifold is given by Φ. It was shown in [6] that if N and M are co-prime numbers, a new array manifold matrix A of a longer virtual ULA with 2M N + 1 sensors located at all integer multiples of d from −M N d to M N d exists in a subset of rows of Φ. Next we construct A by Φ. We aim to construct a new matrix A of size (2M N +1)×K from Φ where we have extracted precisely those rows from Φ which correspond to the 2M N + 1 successive differences. Then we sort the rows so that the (n, k)th element of 2π A is ei λ nd sin(ϕk ) , n = −M N, −M N + 1, . . . , M N, k = 1, 2, . . . , K. We use the same rule to remove the corresponding rows from the observation vector z and sort them to get a new vector y. Now A and y satisfy y = Ap + σ 2 eˆ
k=1
a(ϕk ) = [ei λ l1 sin(ϕk ) , . . . , ei λ lN +2M −1 sin(ϕk ) ]
Fig. 2. Spacial Smoothing
(5)
where eˆ is a vector produced by ˆ 1n in equation (4) with the same operation above. The equation (5) implies a single snapshot created by a ULA with with 2M N + 1 sensors. B. Spatial Smoothing MUSIC Algorithm The key of spatial smoothing MUSIC algorithm is to construct a larger size positive semidefinite matrix from original correlation matrix Rxx . Considering A is nearly identical to the manifold of an ULA with 2M N + 1 sensors located from −M N d to M N d. We divide this array into M N + 1 overlapping subarrays, each with M N + 1 elements, as shown in Fig.2. The ith subarray corresponds to the ith to (i+M N )th rows of A. We denote the observation of ith subarray as yi = Ai p + σ 2 eˆi
(6)
where Ai is manifold matrix of size (M N + 1) × K of the ith subarray. Obviously, the correlation matrix of the the received
data yi can be estimated by Ri = yi yi . At last, we take the average of Ri over all i and get the spatially smoothed matrix Rss . Then Rss =
M N +1 M N +1 1 1 Ri = yi yi M N + 1 i=1 M N + 1 i=1
(7)
The theorem in [6] proved that the matrix Rss has the same form as the correlation matrix of the signal received by a longer ULA consisting of M N + 1 sensors. Thus upto M N targets can be detected by applying MUSIC algorithm. III. D IRECTION OF A RRIVAL E STIMATION V IA C OMPRESSED S ENSING M ETHODS A. Sparse Signal Recovery Model
B. Upper Bound of DOF
The equation (5) can be viewed as a sparse signal recovery problem from the perspective of compressed sensing. We first discretize the range of interest as a grid θ1 , θ2 , . . . , θD , D K which is a sampling grid of all source locations of interest. Then we construct the observation matrix A(θ1 , θ2 , . . . , θD ) composed of steering vectors corresponding to each potential source locations as its columns: A
= =
[a(θ1 ), a(θ2 ), . . . , a(θD )] ⎞ ⎛ −i 2πd M N sin(θ ) 2πd 1 . . . e−i λ M N sin(θD ) e λ ⎟ ⎜ .. .. ⎠ (8) ⎝ . . ei
2πd λ MN
sin(θ1 )
...
ei
2πd λ MN
sin(θD )
In this framework A is known and does not depend on the real source locations ϕk , and consequently the model in equation (5) can be transformed into y = A(θ1 , θ2 , . . . , θD )p + σ 2 eˆ
min p0
s.t. y = Ap + σ 2 eˆ
(10)
The · 0 called l0 norm in equation denotes the number of nonzero terms. Considering the optimization problem of l0 norm is a NP hard problem, we transform the model into a convex optimization problem of l1 norm. min p1
p∈RD
s.t. y = Ap + σ 2 eˆ
Specifically written as . min p1 =
p∈RD
s.t. yn =
D i=1
pi e i
2πd λ
D i=1
|pi |
sin(θi )(n−M N −1)
+ σ 2 eˆn
The upper bound of DOF with sparse signal recovery can be obtained via RIPless theory in compressed sensing. The RIPless theory presented in [11] which proved that if the probability distribution F obeys a simple incoherence property and an isotropy property, one can faithfully recover approximately sparse signals from the linear measurements. The distribution F include all standard models e.g., random measurement matrix, Gaussian, frequency measurements. In the problem of DOA estimation, the row vectors bn of observation matrix A in equation (8) are considered to be selected independently at random from a probability distribution F , which can be iid denoted as bn ∼ F . Before we introduce the incoherent sampling theorem in RIPless theory, we need to know the two critical concepts. 1) Isotropy property: We say that a distribution F obeys the isotropy property if
(9)
where y is a received vector calculated by the sensor outputs x(t) of a co-prime array. If the kth target is located at θi , we have pi = 0 otherwise pi = 0. Therefore, the vector p is a sparse vector because of the spatial sparseness of the signal sources. In equation (9), if y and A is given, we can obtain the sparse signal recovery model via compressed sensing stated as p∈RD
greedy algorithm in compressed sensing. Classical algorithms of convex programming have BP [12], LASSO [13] and FOCUSS [14]. However, there are two defects with most existing algorithms of convex programming. Besides the large computation times, the user parameters are hard to decide and easily affect the performance. The greedy algorithm [15]– [17], such as orthogonal matching pursuit (OMP) [17] is faster and easier to implement if the number of targets is given. In particular, OMP is especially efficient when the signal is highly sparse, which correspond to the spatial sparseness of the targets. Therefore, it is an attractive alternative to convex programming for sparse signal recovery problems of DOA estimation.
(11)
where n = 1, 2, . . . , 2M N + 1. The sparse signal recovery problem can be either solved by convex programming or
E(bn bH n ) = I,
bn ∼F
(12)
Suppose sin(θi ), i = 1, 2, . . . , D are independently sampled from a uniform distribution U (−1, 1). Then the distribution F of the problem in DOA estimation obeys the isotropy property. Next we will give the details. Assume we have the components iid bn ∼ F and define φi = sin(θi ). We have bn = [ei
2πd λ (n−M N )φ1 )
, . . . , ei
2πd λ (n−M N )φD )
]
where n = 0, 1, . . . , 2M N and
0.5 −1 ≤ φi ≤ 1 P (φi ) = 0 otherwise Take ξ = φi − φj (i = j) and we function of ξ as ⎧ ⎨ 0.25(2 − ξ) 0.25(2 + ξ) P (ξ) = ⎩ 0
get the distribution density 0≤ξ≤2 −2 ≤ ξ < 0 otherwise
Typically, we pick d = λ/2 and then we have +∞ iπ(n−M N )ξ E(e ) = P (ξ)eiπ(n−M N )ξ dξ =
0
−∞
And
⎛ iπ(n−M N )(φ −φ ) 1 1 e ⎜ .. E(bn bH n)=E⎝ . eiπ(n−M N )(φD −φ1
... ...
⎞ eiπ(n−M N )(φ1 −φD ) ⎟ .. ⎠ . eiπ(n−M N )(φD −φD
Now, combining these with above we have E(bn bH n)=I 2) Incoherence property: We take the coherence parameter μ(F ) to be the smallest number such that with bn = [b(1), b(2), . . . , b(D)]∼F max |b(i)|2 ≤ μ(F )
1≤i≤D
(13)
(a)
A simple calculation shows that μ(F ) = 1 since |b(i)|2 = 1 for all i with bn = [eiπ(n−M N )φ1 , . . . , eiπ(n−M N )φD ]. To obtain the upper bound, the noise term in equation (11) is erased and we now consider the noiseless l1 mininization program. min p1 s.t. y = Ap (14) p∈RD
Based on the incoherence sampling theorem in [11], we present a corollary about the upper bound of DOF. Corollary III.1. [Upper Bound of DOF] Let p be a fixed but otherwise arbitrary K-sparse vector in RD and pick any scalar β > 0. Then with probability at least 1 − 5/D − e−β , p is the unique minimizer to (14) provided that K≤
2M N + 1 Cβ log D
(15)
(b) Fig. 3. Spacial spectra with the two methods of 15 sources. (a) Spatial smoothing MUSIC algorithm; (b) DOA estimation with OMP algorithm.
More precisely, Cβ may be chosen as C0 (1 + β) for positive numerical constant C0 . This corollary is about a basic relationship between the number of sources K, the number of grid D and the number of samplers 2M N + 1. The upper bound on sparsity level lead to stable DOA estimation. IV. E XPERIMENTAL R ESULTS In this section, we conducted several experiments to test the performance of the mentioned two methods (spatial smoothing MUSIC algorithm and OMP algorithm). Consider a co-prime array with 10 physical sensors, which means that we pick N = 5, M = 3 in Fig.1. Specifically, the positions of all the sensors are located at [0, 3, 5, 6, 9, 10, 12, 15, 20, 25]d, with d taken as half of the wavelength. The frequencies of all signals are f = 1000Hz and the sampling frequency is fs = 8192Hz. A. Detection Performance In this example, we consider several narrowband sources, the directions of which distribute uniformly in the region from −60◦ to 60◦ . We take grid from −60◦ to 60◦ with step size 0.5◦ , and so D = 241. The correlation matrix is estimated using 500 snapshots and the SNR is set to be 0 dB. Fig.3 shows that both the two methods perform well when the number of target is 15. The red dashed lines in figures
Fig. 4. DOA estimation of 16 sources
are the underlying DOA and the solid lines mean estimations of DOA. As we can see, all the DOAs are clearly identified. However, the MUSIC can estimate at most M N = 15 sources because the choice of M and N . When there are more than 15 sources in different directions, the MUSIC algorithm will not work and the OMP algorithm will show its superiority in this case. Fig.4 shows that by implementing the OMP algorithm, we
developed the upper bound of DOF of the proposed methods. We started with a framework of co-prime array and presented the existing MUSIC algorithm implementing on the signal model. The sparse signal recovery model based on the framework of co-prime array was obtained and we demonstrated the upper bound of DOF in details with the RIPless theory in compressed sensing. An efficient optimization procedure using OMP algorithm was proposed, which have a lower computational cost. Finally, numerical examples showed the superiority of the proposed method in detection performance and estimation accuracy. Future work can extend the work here to a general situation of DOA estimation. R EFERENCES Fig. 5. Plots of the probability of recovery versus upper bound of DOF for C0 = 0.2 and C0 = 0.25
Fig. 6. Estimation accuracy of one source
can detect 16 targets which the spatial smoothing MUSIC algorithm fail to detect. Thus we can see that the OMP algorithm can detect more targets than spatial smoothing MUSIC algorithm. In the simulation, we find the probability of recovery decrease as the number of sources increase, which verifies the corollary III.1. Based on the corollary, we obtain Fig.5 which shows the relation between the probability of recovery and the upper bound of DOF. This result means the DOA estimation is not stable when the number of sources increase. B. Estimation Accuarcy In the last experiment, we test DOA estimation accuracy of both methods. Now we randomly generate one source in the region from −90◦ to 90◦ . For every testing point we run 5000 iterations of Monte Carlo simulations. The error in coordinateaxis Y is denoted in degrees. Fig.6 shows that the OMP algorithm has a better performance than the spatial smoothing MUSIC algorithm when the SNR is ranged from −20 dB to 10 dB. In addition, the gap is widening as SNR increase. V. C ONCLUSION In this paper, we explored the compressed sensing methods for DOA estimation in the signal model of co-prime array and
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