# A Novel Sparse Array Configuration with Low Coarray Redundancy for DOA Estimation in Mobile Wireless Sensor Network

Authors: Pin-Jiao Zhao; Guo-Bing Hu; Liang-Tian Wan
Journal: Mobile Information Systems
 (2021)

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## Abstract

For tacking and localizing sources in the mobile wireless sensor network, underdetermined direction of arrival (DOA) estimation with high-accuracy is a crucial issue. In this paper, a novel sparse array configuration is developed for accurate DOA estimation from the perspective of sum-difference coarray (SDCA). As compared with most of the existing sparse array configurations, the proposed array can effectively reduce the overlap between difference coarray (DCA) and sum coarray (SCA) and can achieve more consecutive degrees of freedom (DOF), more sources can be resolved accordingly. Additionally, the proposed array has hole-free DCA and SDCA. Then, the concept of coarray redundancy ratio (CRR) is introduced for evaluating the coarray overlap quantitatively and the closed-form CRR expressions of the proposed array are derived in detail. Based on the good properties of the proposed array, vectorized conjugate augmented MUSIC (VCAM) is adopted for underdetermined DOA estimation. The theoretical propositions and numerical simulations demonstrate the superior performance of the proposed array in terms of CRR, consecutive DOF, and DOA estimation accuracy.


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## Body

## 1. Introduction

In the large-scale mobile wireless communication network [1, 2], underdetermined direction of arrival (DOA) estimation [3–6] with high-accuracy is a key issue to solve for tacking and localizing sources. The optimized array configuration design is a prerequisite for accurate DOA estimation. Recently, the emerging sparse arrays break through the constraint of spatial sampling theorem by sparsely arranging the array elements and have remarkable advantages in array element layout flexibility, degrees of freedom (DOF), and virtual array aperture [7, 8]. Benefiting from the sparse arrays, more sources than the number of the physical sensors can be resolved with high-accuracy.From the view of array configuration design, coprime array (CPA) [9] and nested array (NA) [10] are the two most typical sparse arrays. In comparison to the minimum redundancy array and the minimum hole array, CPA and NA are easy to construct since their sensor locations are analytically tractable. Specifically, CPA is obtained from a pair of coprime uniform linear arrays (ULA), which can offer OMN DOF with OM+N sensors, but has holes in the virtual coarray. By contrast, NA contains two “nesting” ULAs with increasing element spacing, which has a consecutive virtual coarray with ON2 DOF from ON sensors. On the basis of these prototypes, several modifications are developed, such as generalized CPA [11], thinned CPA [12], super NA [13], and augmented NA [14], aiming at increasing consecutive DOF and expanding effective array aperture. Nevertheless, note that these arrays construct synthetic virtual arrays from the view of difference coarray (DCA), the number of consecutive DOF in the virtual array cannot be more than twice of the physical aperture. Accordingly, the number of the resolvable sources and the DOA estimation accuracy are also affected.Alternatively, several array configurations based on sum-difference coarray (SDCA) have attracted considerable interests, wherein the introduction of sum coarray (SCA) originating from multiple-input multiple-output (MIMO) system [15, 16] can bring more DOF. In [17], a novel CPA based on SDCA is presented, from which the consecutive DOF more than twice of array aperture can be obtained. Following this, a modified nested array configuration named as sum-diff NA (SdNA) is proposed in [18]. To further improve the available DOF, two kinds of improved NAs [19], termed as INAwSDCA-I and INAwSDCA-II, are developed by rearranging the translational NA. In [20], the unfold CPA configuration with SDCA is proposed. However, for the above array configurations based on SDCA, lots of overlapping virtual sensors between their DCA and SCA lead to unsatisfactory coarray utilization. In addition, there also exist holes in the SDCA. Toward this end, transformed NA (TNA) is designed in [21], which can reduce the overlap between DCA and SCA to some extent, but the degree of overlap under different number of array elements is only evaluated via simulations, lacking quantificational analysis.To tackle these problems, a novel sparse array configuration with low coarray overlap is proposed in this paper. The proposed array is constructed with two concatenated subarrays with different interelement spacings. From the perspective of virtual array, both the DCA and SDCA of the proposed array are hole-free and can provide more consecutive DOF accordingly. Moreover, the coarray redundancy ratio (CRR) of the proposed array is derived in detail to evaluate the coarray overlap quantitatively. Based on the proposed array, the vectorized conjugate augmented MUSIC (VCAM) approach can be employed for DOA estimation. Now, we briefly summarize the contributions of this work as follows:(1)
A novel array configuration is proposed to reduce the overlap between DCA and SCA(2)
The proposed array has hole-free DCA and SDCA, which has been proved via theoretical propositions(3)
More sources than twice of the number of sensors can be resolvedThe mathematical notations used throughout this paper are denoted as follows. Vectors and matrices are denoted by lowercase and uppercase bold-face letters, respectively.⋅T, ⋅H, and ⋅∗ denote transpose, conjugate transpose, and conjugate. E⋅ and ⋅ represent the statistical expectation and modulus of the internal entity, ∂fa/∂a denotes the derivation of fa with respect to a, Z denotes the nonnegative integer set, Z+ denotes the positive integer set, Zn denotes a set of consecutive positive integers ranging from 1 to n in the case of n>0, and ¢n=0 holds in the case of n=0; otherwise, ¢n denotes a set of consecutive negative integers ranging from −n to −1. The symbol ⊗ denotes the Kronecker product and ⊙ denotes the Khatri–Rao product.The remainder of this paper is organized as follows. The DOA estimation model based on SDCA is formulated in Section2. Section 3 presents the proposed low coarray redundancy array configuration in detail. Section 4 exhibits the simulation results in terms of CRR, consecutive DOF, and DOA estimation accuracy. Conclusions are drawn in Section 5. The proofs of Propositions 1 and 2 are derived in Appendix A and Appendix B.


## 2. Problem Formulation

Assume thatK far-field narrowband sources from directions θk|k=1,2,…,K impinge on an N-element nonuniform sparse antenna array. Then, the array output at time t can be represented as(1)xt=Ast+nt=∑k=1Kaθkskt+nt,where the value of K is known in advance which can be estimated via source number estimation methods [22]. st=s1t,s2t,…,sKtT is the source vector with the kth element being skt=Gkejwkt, Gk is the deterministic complex amplitude, and wk is the frequency offset. A=aθ1,aθ2,…,aθK denotes the array manifold matrix. After normalizing by the unit element spacing d, the steering vector can be expressed as aθk=e−jgπsinθk with g=g1,g2,…,gNT being the sensor location vector. nt is the Gaussian white noise vector with zero mean and variance being σn2. By collecting Ts samples, the time average function can be expressed as(2)Rxi∗xjτ=1Ts∑t=1Tsxi∗txjt+τ≈∑k=1Kejπgj−gisinθkRsk∗skτ+Rni∗njτ,where τ≠0 is the time lag, xit and xjt are respectively the ith and the jth (1≤i,j≤N) row of xt, Rsi∗sjτ=Ek2ejwkt, and Rni∗njτ=0. For the Tp pseudosnapshots, the pseudodata matrix is defined as(3)G=gxτgx∗−τ|τ=npPs,np=1,…,TP,where Ps is the pseudosampling period that satisfies the Nyquist sampling theorem. For convenience, let the 1st sensor be the reference, gxτ=Rx1∗x1τ,Rx1∗x2τ,…,Rx1∗xNτT. Then, the covariance matrix of G is calculated as(4)RG=EGGH≈1Tp∑np=1TpgnpPsgHnpPs.VectorizingRG yields(5)rG=vecRG=Bz,where z=E14,E24,…,EK4T and the kth column of B can be denoted as(6)a˜θk=aθk⊗a∗θkaθk⊗aθka∗θk⊗a∗θka∗θk⊗aθk,where the term a˜θk in (6) behaves like a virtual steering vector of a longer virtual ULA. More specifically, the union of aθk⊗a∗θk and a∗θk⊗aθk is respectively corresponding to DCA and its mirror version; aθk⊗aθk and a∗θk⊗a∗θk are respectively corresponding to SCA and its mirror version. Then, the spatial smoothing MUSIC or sparse construction approach can be performed for underdetermined DOA estimation, and the above procedure is termed as the vectorized conjugate augmented MUSIC (VCAM) [17].


## 3. Low Coarray Redundancy Array Configuration

Definition 1.
Assume2<N1≤N2, N1∈Z+, and N2∈Z+, after normalizing by d, the configuration of the proposed array with N=N1+N2 antenna elements is defined by(7)L=L1∪L2,L1=1+l−1N1,l∈ℤN1,L2=N12−N1+N2+l,l∈ℤN2.
Intuitively, an example ofN1=N2=3 is given in Figure 1, where white circles and black circles denote the sensor locations of subarray L1 and L2, respectively.
From the perspective of coarray equivalence, the proposed array, NA, and TNA are compared in Figure2 with the number of antenna elements being N = 6, where the red circles and blue triangles indicate the virtual sensors in DCA and SCA, respectively, and black crosses indicate holes. Note that the DCA, SCA, and SDCA are origin-symmetric, here the nonnegative parts of the abovementioned coarrays are selected for performance comparison, which are respectively abbreviated as n-DCA, n-SCA, and n-SDCA. The sensors in the rectangular dashed box indicate the overlapping virtual sensors between the n-DCA and the consecutive n-SCA. We can see from Figure 2 that the n-SDCA is continuous in the range of (0, 24) for the proposed array, while (0, 16) for NA and (0, 22) for TNA. Moreover, there exists only 1 overlapping virtual sensor for the proposed array, while 10 for NA and 4 for TNA.Figure 1
An example of the proposed array configuration withN1=N2=3.Figure 2
Virtual coarrays of the proposed array and NA withN=6, (a) NA, (b) TNA, and (c) the proposed array.
(a)(b)(c)Proposition 1.
Some properties of the proposed array configuration in the virtual coarray are listed as follows:(i)
The DCA of the proposed array is continuous in set ofℤ−S1∪ℤ0∪ℤS1 with S1=N12−N1+2N2−1(ii)
The SCA of the proposed array is continuous in set of¢S3−¢S2∪¢−S3−¢−S2 with S3=2N12−N1+2N2, let ε=N2−N1∈¢, and the value of S2 is given as follows:(a)
if0≤ε<2, S2=N12+2(b)
ifε=2or3, S2=N12−N1+N2+2(c)
ifε≥4, S2=N12+N1+2,N1=ε−1,N12−N1+N2+2,elseProof.
See Appendix A.Proposition 2.
The proposed array has hole-free DCA and SDCA, and the maximum consecutive DOF is4N12−N1+2N2+1.Proof.
See Appendix B. Note that for a sparse array configuration with the fixed number of antenna elements, the higher consecutive DOF means larger virtual array aperture and more accurate DOA estimation. Therefore, it is essential to look for appropriateN1 and N2 to maximize the consecutive DOF.Corollary 1.
After optimizingN1 and N2, the optimal consecutive DOF and the corresponding solutions are given as(8)DOFmax=N+12,ifN1=N2=N2,N2+8,ifN1=N−12,N2=N+12.Proof.
The optimal DOF under the constraint ofN=N1+N2 can be cast as the following optimization problem:(9)maxN1,N2DOF=4N12−N1+2N2+1s.t.N=N1+N2.
Using arithmetic mean-geometric mean (AM-GM) inequalities, the solutions are obtained from (8). According to the above analysis, the proposed array exhibits the advantages in terms of consecutive DOF and virtual array aperture. From the view of coarray efficiency, the overlapping virtual sensors between the DCA and SCA lead to coarray redundancy, which, to some extent, reduce the number of available virtual coarrays. In view of this, the concept of CRR is introduced to evaluate the coarray overlap.Definition 2.
CRR denotes the ratio of the overlapping virtual sensors between the DCA and SCA, which is given by(10)κ=2S1−S2+12S3+1.
More specifically, CRR of the proposed array configuration can be calculated as(11)if0≤ε<2,κ=4N2−2N1−44N12−N1+2N2+1,ifε=2or3,κ=2N2−44N12−N1+2N2+1,ifε≥4,κ=4N14N12−N1+2N2+1,N1=ε−1,2N2−44N12−N1+2N2+1,else.Proposition 3.
For the optimal DOF given in Corollary1, if N1=N2=N/2>2, the corresponding CRR is no more than 0.0496; if N2=N+1/2>N1=N−1/2>2, the corresponding CRR is no more than 0.1053.Proof.
IfN1=N2=N/2>2, the corresponding CRR can be simplified to κ=4N2−2N1−4/4N12−N1+2N2+1=N−4/N+12. Then, the first-order derivative of κ is taken as ∂κ/∂N=9−N/N+13. Since N is even, the maximum of κ is calculated as κmax=maxκN=8,κN=10=0.0496. If N2=N+1/2,N1=N−1/2, the corresponding CRR can be simplified to κ=4N2−2N1−4/4N12−N1+2N2+1=N−1/N2+8. Similarly, the first-order derivative of κ is taken as ∂κ/∂N=−N2+2N+8/N2+82. Note that N is odd; the maximum of κ is calculated as κmax=maxκN=7=0.1053.


## 4. Numerical Simulations

In this section, numerical simulations have been carried out to evaluate the performance of the proposed array, where SdNA [18] and TNA [21] are selected for performance comparison. In the first simulation, the CRR scatters versus ε=N2−N1 are plotted in Figure 3, and the blue asterisks indicate the case where N is even and the red squares indicate the case where N is odd. It can be observed that the CRR of the proposed array tends to increase with the increase of ε, thus the optimal CRR can be obtained from ε=0 when N is even and ε=1 when N is odd.Figure 3
The scatter plot of CRR.In the second simulation, the CRR of three array configurations withε=0 and ε=1 are respectively compared in Figures 4 and 5. The results show that the proposed array can achieve the smallest CRR compared to SdNA and TNA with the same number of antenna elements. Also, it has been concluded that the CRR of the proposed array is no more than 0.0496 when N is even and no more than 0.1053 when N is odd, which has been verified in Proposition 3.Figure 4
Comparison of CRR with different sensor number (ε=0).Figure 5
Comparison of CRR with different sensor number (ε=1).In the third simulation, we compare the consecutive DOF of three array configurations and the simulation results are depicted in Figure6. As can be seen from Figure 6, the proposed array can obtain more consecutive DOF than the other two arrays benefitting from the lower CRR. This implies that there exist few overlapping sensors between the DCA and SCA of the proposed array, thus more effective virtual sensors can be utilized for the construction of SDCA.Figure 6
Comparison of consecutive DOF.Then, the DOA estimation performance is investigated in the last simulation. We considerK = 27 narrowband sources uniformly distributed between −60° and 60° impinging on the proposed sparse array with 8 sensors, where SNR being 0 dB and Ts=Tp=200. Figure 7 depicts the MUSIC spectrum of the proposed array where the blue solid lines denote the angle estimates and the red dotted lines denote the incident sources. It can be seen that the proposed array can detect 27 impinging sources with only 8 sensors, and the estimated spectrum has sharp and clearly discernible peaks in the vicinity of the true impinging sources.Figure 7
The MUSIC spectrum of the proposed array withN = 8.Figures8 and 9 respectively depict the root mean square error (RMSE) of DOA estimation versus SNR and snapshots via 200 Monte Carlo trials, where RMSE is defined as(12)RMSE=1200K∑n=1200∑k=1Kθ∧n,k−θn,k2,with θ∧n,k being the estimate of the kth impinging source for the nth Monte Carlo trial. All the simulation conditions are the same as Figure 7, except that the snapshot number is fixed at 200 in Figure 8 and the SNR is fixed at −6 dB in Figure 9. The results show that the proposed array is superior to the other two array configurations in terms of DOA estimation accuracy, which mainly attributed to the larger consecutive DOF and the lower CRR of the proposed array.Figure 8
Comparison of DOA estimation performance (RMSE versus SNR).Figure 9
Comparison of DOA estimation performance (RMSE versus snapshots).


## 5. Conclusions

This paper presents a novel sparse array configuration for accurate DOA estimation, which has fewer coarray overlapping sensors and more consecutive DOF as compared with most of the existing sparse array configurations. From the perspective of coarray equivalence, the proposed array has hole-free DCA and SDCA. The closed-form CRR expressions of the proposed array are derived in order to evaluate the coarray overlap quantitatively. With the utilization of the proposed array, the VCAM approach is adopted for DOA estimation. The results of the theoretical analysis and the numerical simulations demonstrate the effectiveness and favorable performance of the proposed array configuration and DOA estimation performance.



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*Source: 1362640-2021-11-22.xml*