The Design of A Novel Sparse Array Using Two Uniform Linear Arrays considering Mutual Coupling

The sparse arrays using two uniform linear arrays have attracted considerable interest due to the capability of giving analytical expression of sensor location and owning robust direction-of-arrival (DOA) facing strong mutual coupling and sensor failure. In order to achieve the maximum consecutive virtual uniform linear array in difference coarray, in this paper, a design method of a novel sparse array using two uniform linear arrays (NSA-U2) is proposed. We first analyze the relationship between the values of displacement of two subarrays and difference coarray, and then we give the analytical expressions of the displacement and the number of consecutive lags. By discussing the selection of number of subarray sensors, the design of NSA-U2 is completed. Moreover, through choosing a proper compressed interelement spacing, NSA-U2 can be robust to mutual coupling effect. Numerical experiments prove the effectiveness and favorable performance of DOA estimation with mutual coupling.


Introduction
Direction-of-arrival (DOA) estimation is one of the key technologies in the field of passive location. It is widely used in seismic detection, military early warning, astronomical observation, radar monitoring, underwater targets positioning, and so on [1,2]. Traditionally, the researchers apply the super resolution methods, such as subspace methods, to solve DOAs mainly in uniform nonsparse arrays, such as uniform linear arrays (ULAs). In recent years, many experts and scholars have focused on nonuniform sparse arrays, whose sensor spacing can be larger than the half of wavelength of impinging signals. Minimum redundancy arrays (MRAs) [3], coprime linear arrays (CLAs) [4,5], and nested linear arrays (NLAs) [5,6] are the typical sparse arrays. Compared with ULAs, the nonuniform sparse arrays can construct the larger array aperture with the same number of sensors, which also means the higher resolution. In order to solve DOAs without any ambiguous values, spatial smoothing multiple signal classification (SS-MUSIC) [7] and direct augmentation approach (DAA) [8] are proposed, respectively. Two algorithms both try to transform the physical array to a virtual nonsparse array, where they both use the difference coarray. [9] demonstrated the effectiveness of two algorithms and concluded that the two share the same performance.
The performance of DOA estimation is mainly decided by the array aperture. While in sparse array, it is changed to be decided by the aperture of consecutive ULA belonging to the difference coarray. Although MRA has the largest virtual ULA aperture in theory, they have no closed-form expression. Thus, a structure named generalized coprime linear array (GCLA) is proposed, where NLA and CLA are the typical arrays. The sensors location of GCLA has an analytical expression, which reduces the complexity of array design. And [10][11][12][13] proves that this structure can be much more robust facing sensors failures compared with other sparse arrays. Then, to enlarge the aperture of virtual ULA under the fixed number of sensors, coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) [14,15], coprime array with multiperiod subarrays (CAMpS) [16], and shifted coprime array (SCA) [17] are proposed. CACIS and CADiS show that setting one subarray with a smaller interelement spacing can have the larger aperture of virtual ULA. CAMpS demonstrate that, in order to expand the aperture of virtual ULA, only one subarray can have a compressed interelement spacing. SCA reveals that the displacement between two subarrays is the main factor to the aperture of virtual ULA. Based on sparse array using two ULAs, although many valuable researches have been proposed, the solution of how to maximize virtual ULA aperture corresponding to different parameters has not been answered and is still deserved to study.
Moreover, besides the array aperture, the mutual coupling is another factor affecting the performance of DOA estimation. The severe mutual coupling causes the degrade of accuracy. To tackle the problem, the researchers aim to enlarge the interelement spacing and propose super nested arrays (SNAs) [18,19], the augmented nested array (ANA) [20], and the maximum interelement spacing constraint (MISC) [21]. Based on the GCLA, they divide the dense subarray into three or four sparse subarrays, which decreases the weight of interelement spacing equal to half of wavelength. However, when the mutual coupling is strong, the weight of interelement spacing bigger than half of wavelength should be considered.
Hence, in this paper, based on GCLA configuration, we propose a novel sparse array using two ULAs (NSA-U2). We first derive the analytical expression of displacement between two subarrays, which can decrease the complexity of array design, and then discuss the selection of interelement spacing of each subarray to suppress the strong mutual coupling. Moreover, the effectiveness of conclusion is demonstrated by simulation experiments. The main contribution of this paper is summarized as follows: (1) Under the fixed number of sensors and same interelement spacing, the paper gives the design of NSA-U2, which can obtain the biggest aperture of the virtual ULA compared with other arrays using two ULAs. Moreover, the paper derives the analytical expression of displacement and the aperture of the consecutive virtual ULA and gives a low-complexity array design method (2) The paper analyzes the selection of interelement spacing of each subarray, which ensures that the weight of small interelement spacing is low. When mutual coupling is strong, robust DOA estimation can be obtained The rest of this paper is organized as follows: Section II presents the model of sparse array. Section III proposes the design method of NSA-U2 in detail. Section IV analyzes the performance facing mutual coupling in NSA-U2. Section V is the simulation experiments. Section VI summarizes the paper. Throughout the paper, we make use of the following notations shown in Table 1.

Problem Formulation
Assume that there are T sensors. Based on GCLA [15], NSA-U2 is made of two ULAs, having M ′ = M − 1 and N sensors, respectively, where T = M ′ + N and M and N are coprime integers (generally assuming M < N). We set the unit interelement spacing to λ/2, where λ is the wavelength of impinging signals. So, the sensors' location set is all integers, defined as where M = M/p and M and p are integers. Here, p is the compressed factor, M, N are still coprime integers, and L is the displacement between two subarrays. Suppose that there are K far-field narrowband signals impinging on this array from fθ 1 ,⋯,θ K g with power fσ 2 1 ,⋯,σ 2 K g. Then, the received data is where the manifold matrix AðθÞ is denoted as and the steering vector a D ðθÞ can be given by where ℓ m ∈ D, m = 1, ⋯, T − 1. The signal data vector is where t = 1, ⋯, J, and J is the number of snapshots. And the noise vector is usually Gaussian random variables with zero means and variance σ 2 n . From the references mentioned above in Section I, the covariance matrix is use to construct the virtual array. Then, we can have the covariance matrix of the received data denoted as where B = A * ∘ A, e = vecðIÞ, and p are the diagonal elements of R S . Hence, the vector Z can be seen as the received data of a virtual linear array [4,15], whose location is defined as the difference coarray, given by where the difference coarray is the union of self-difference coarray D s , where and the corresponding mirrored coarray And the aperture of virtual ULAs in coarray is equivalent to the maximum number of consecutive lags in D v .
CACIS and CADiS [14,15], respectively, set L = − MðN − 1Þ and L = M + N. Comparing CACIS with CADiS, we notice that CACIS has the bigger consecutive integer range while it can still be improved, and CADiS has the bigger number of unique coarray lags while its consecutive integer range is much smaller. Hence, the value of L is a main factor to the number of consecutive lags. SCA in [17] gives a solution that when p = 1, setting L = bM/2cN − MðN − 1Þ can obtain the maximum number of consecutive lags. Then, we need to find a more general solution of L under different p. Because p = M is a special case, where nested CADiS in [?] reveals that L = N + 1 can have the maximum number of consecutive lags, we will take the 1 ≤ p < M into consideration in the following section.

Proposed Array Design Method
In order to have the maximum number of consecutive lags, defined as S v , the key of the NSA-U2 design method is to solve the value of L and the selection of M, N with the fixed T. In this section, we will, respectively, give the setting of L and M, N with the detailed proof.
3.1. The Analytical Expression of L. Based on (8), we solve the analytical expression of L in two steps. The first step is solving L to obtain the maximum number of consecutive lags in D c ∪ D − c . The second step is adjusting L when considering the effect of D s ∪ D − s to D v . In first step, in order to obtain the maximum number of consecutive lags of D c ∪ D − c , the positive and negative lags must be connected. Before solving L, we introduce the proposition proposed in [15], which tells the analytical expression of lags and holes in D c : Then, we further propose the proposition 2 based on Proposition 1: Proposition 2. D c meets the following properties: (a) The lags in D c can be divided into three parts, which are given by The holes in D c can be divided into two parts, which are given by Proof.
The holes can be divided into two parts, where D hn corresponds to the holes in D cn , and D hp corresponds to the holes in D cp . The maximum value of D hn is ð Similarly, the minimum value of D hp is M′N + L. So, any hole no less than M ′ N + L is given by where the integers a 1 , a 2 ∈ f0, 1g and a 1 , a 2 cannot equal to 0 simultaneously. Moreover, the values of a 1 , a 2 are depended on the odevity of M, N.
Proof. See Appendix A. In second step, with the theorem 3, we use D s ∪ D − s to fill the holes in D cp ∪ D − cp and obtain the detailed expression of L, which is the function of M, N. We propose the theorem 4, which states that ☐ Theorem 4. The analytical expression of L and S v are showed in Table 2.
Proof. See Appendix B.
We show an example in Figure 1, where M = 6, N = 7, and p = 2. Thus, L = −ð MN − NÞ/2 = −7 from Table 2. The physical sensors of two subarrays have one common sensor located at the Nth sensor of subarray 1 and ð M/2 + 1/2Þth sensor of subarray 2, which proves the Table 3 Journal of Sensors by D − c , which proves the Theorem 3, and the minimum value in D hp can be aligned by D s , which proves the Theorem 4. Hence, the figure demonstrates that the proposed theorems are right.
where C is the mutual coupling matrix with size T × T. Generally, in linear array, C can be a B-band symmetric Toeplitz matrix [22][23][24][25][26][27]. The element in mth row and nth column is defined as where c 0 , c 1 , ⋯, c B are coupling coefficients satisfying c 0 = 1 > c 1 > ⋯ > c B . (14) reveals that the sparse array that has a much larger interelement spacing can have weaker mutual coupling. To quantify the effect of mutual coupling, we introduce the following definitions [18][19][20][21]: Definition 6. The weight function of the virtual array D v is defined as the number of coarray lags index ℓ, which can be expressed as When |ℓ | <B, the value of wðℓÞ is smaller, and the mutual coupling is less significant.

Definition 7. The coupling leakage is defined as
where the smaller value of Le implies the weaker mutual coupling.   Applying DAA to the received data (7) of virtual sensors, we can have the MUSIC spectrum and obtain the DOA of signals. ☐   Figure 2. The figure presents that if p is small, when ℓ is small, wðℓÞ can be 1. In order to quantify the influence of mutual coupling, we assume that B = 3, c 1 = 0:3e jπ/3 , and c b = c 1 e −jπðb−1Þ/8 . Moreover, we compare NSA-U2 with the existing sparse arrays, such as NLA [6], MISC [20], ANAI-1, and ANAII-1 [20], under the same number of sen-sors. Figure 3(a) shows the relationship between S v , Le and p in NSA-U2. With the increase of p, S v increases, but Le also increases. When mutual coupling is severe, a big array aperture may not obtain the high accuracy. Figure 3(b) compares S v , Le of different kinds of sparse arrays with same T . When p = 1 and p = 2, NSA-U2 can have a small Le but also a small S v . ANAII-1 and MISC have the larger S v and also the bigger Le.
In conclusion, the selection of p needs to balance the values of S v and Le to obtain a favorable performance, where the influence to DOA estimation will be presented by simulation experiments.

Simulation Results
The definition of root mean square error (RMSE) is given as follows, Q is the number of Monte Carlo, K is the number of target signals, and b θ k,q is the DOA of the kth estimated source by the qth Monte Carlo experiment.
Simulation 9. The MUSIC spectrum in NSA-U2 under different p.
When we use DAA [?] to estimate DOAs, the maximum number of detectable signals is ðS v − 1Þ/2 in theory, which is Thus, we apply DAA and obtain the spectrum under different p, which is shown in Figure 4. The figure shows that we can estimate underdetermined signals in NSA-U2. However, with the increase of p, the influence of mutual coupling to DOA estimation is more obvious, where the peaks (blue lines) gradually deviate from the true DOAs (red lines).   Journal of Sensors Moreover, we set p = 2 and vary the Δθ from f5 ∘ , 6:67 ∘ , 2:5 ∘ , 1:67 ∘ g. The spectrum is shown in Figure 5. When the target angle difference increases, if the DOAs of signals are more close to 90 ∘ , the accuracy will decrease or even deterio-rate. The RMSE of Δθ = 5 ∘ is 0:087 ∘ , while the RMSE of Δθ = 3:75 ∘ is 0:071 ∘ . And when Δθ = 6:67 ∘ , the DOA estimation of signal from the direction higher than 60 ∘ fails. When the target angle difference decreases, it can have the higher  In this simulation, we show the RMSE performance with mutual coupling under different SNRs. The mutual coupling coefficients and the structure of NSA-U2 are the same as simulation 9, and SNR is from −30dB to 15dB with 5dB intervals. Set Q = 500, and results are shown in Figure 6(a). When SNR > −20dB, the RMSEs decrease with SNR increasing, but when SNR > −5dB, the RMSEs go to be flat due to the mutual coupling. When p > 2, Le is the main factor to affect the RMSEs, but when p ≤ 2, S v is the main factor. So, when p = 2, NSA-U2 can obtain the highest accuracy. Moreover, we compare the NSA-U2 under p = 2, with NLA, MISC, ANAI-1, and ANAII-1, where the RMSEs are shown in Figure 6(b). Considering Figure 3, this figure shows that the RMSE of NSA-U2 is lower than NLA, ANAI-1, and MISC due to the weaker mutual coupling. ANAII-1 has the lowest RMSE because it has the largest S v , but the complexity to design ANAII-1 is much higher than NSA-U2, especially when T is big.
Simulation 11. RMSE performance comparison of different snapshots.
Similar to the simulation 10, we set SNR = 10dB and vary J from f50,100,200,500,1000,2000,5000g. The RMSEs are shown in Figure 7. The simulation results show that the performance of RMSEs is improved with the increase of snapshot numbers, but when J > 500, RMSEs are gradually flatted.
Other conclusions are the same as that in simulation 10.
Simulation 12. RMSE performance comparison of different T .
Based on the setting of simulation 10, we further consider the RMSEs of NSA-U2 with different numbers of sensors defined as T under p = 2. We vary T from f8, 12, 16,20,24,28,32g. Through the conclusion in Table 2, we can have the corresponding values of S v as f33,75,115, 1 85,245,343,423g. The results are shown in Figure 8. The figure shows that the bigger T can cause the bigger S v , which means the higher accuracy. But due to the mutual coupling, the RMSEs goes to be flat when SNR > −5dB, and the gap between the RMSEs corresponding to the bigger T is not obvious. Although the difference of S v between T = 28 and T = 32 and that between T = 20 and T = 24 is big, the RMSEs between T = 28 and T = 32 and that between T = 20 and T = 24 are very close.
Simulation 13. RMSE performance comparison of different K .
In this simulation, we do the experiments about the effect of number of signals to the RMSEs. Based on the setting of simulation 10, we set θ k = 0 ∘ + ΔθðK/2 + 1/2 − kÞ, 1 ≤ k ≤ K, Δθ = 3:75 ∘ , and vary K from f3, 9, 15,19,25,33g. The results in Figure 9 show that when SNR > −5dB, with the increase of K, the RMSEs increase. If K is small, such as K = 3, the threshold, defined as the value of SNR when RMSE = 1 ∘ , is even smaller than −15dB. The threshold becomes bigger with K increasing. Moreover, the RMSEs of K = 25 and K = 33 Simulation 14. RMSE performance comparison of different |c 1 | . |c 1 | determines the strength of mutual coupling. So, in this simulation, we compare the RMSEs of different sparse arrays under different |c 1 | . The sparse arrays are the same as simulation 10. We set SNR = 10dB and J = 5000. |c 1 | varies from 0:1 to 0:8 with 0:05 intervals. The RMSEs are shown in Figure 10. For all array geometries, as |c 1 | increases, which causes more severe mutual coupling effect, the associated RMSEs increase. And NSA-U2 under p = 1 and p = 2 has the similar robust DOA estimation as MISC and ANAII-1 facing severe mutual coupling.

Conclusion
In this paper, based on GCLA, the design of sparse array using two ULAs is proposed, named NSA-U2. The paper analyzes the relationship between the displacement L between two subarrays and the number of consecutive lags S v in difference coarray. Then, the selection of the number of subarray sensors and the analytical expressions of L, S v are derived, which can obtain the maximum number of consecutive lags. At last, facing mutual coupling, the paper analyzes the balance between S v and Le in NSA-U2. Through the simulations, NSA-U2 both has a low-complexity design method and obtains the robust DOA estimation facing severe mutual coupling under proper p. Considering the advantages of NSA-U2, the design of sparse array using P ULAs can be conducted in the future research. 2) requires that L should be divisible by 2 and as big as possible. The former requirement is depended on the odevity of M, N and values of a 1 , a 2 . The latter requirement means that a 1 = 0or1 and a 2 = 0or1. When a 1 = a 2 = 0, L can never be divided by 2; so, a 1 , a 2 cannot be 0 simultaneously.
Moreover, we find that some values of L let two subarrays have one common sensor, where L meets that The possible values of some parameters are listed in Table 3. Thus, with a fixed value of T, the number of sensors of subarray 2 can be M; so, S c can be bigger as 2ðMN + LÞ − 1.