The Analysis of Using Spatial Smoothing for DOA Estimation of Coherent Signals in Sparse Arrays

When there is coexistence of uncorrelated and coherent signals in sparse arrays, the conventional algorithms for direction-ofarrival (DOA) estimation using difference coarray fail. In order to solve the problems, this paper analyzes the feasibility of using spatial smoothing in sparse arrays. Firstly, we summarize the two types of sparse arrays, one consisting of identical sparse subarrays and the other consisting of several uniform linear subarrays. ,en, we give the feasibility analysis and the processes of applying spatial smoothing. At last, we discuss the performance of the number of detectable coherent signals in different sparse arrays. Numerical experiments prove the conclusions proposed by the paper.


Introduction
Direction of arrival (DOA) is the one of the key parameters of the wireless positioning technique. It has been widely used in the fifth generation communication systems, military early warning, radar monitoring, sonar targets positioning, and so on [1,2]. Traditionally, the super-resolution DOA estimation methods, such as subspace methods, mainly use uniform nonsparse arrays like uniform linear arrays (ULAs). In recent years, many experts and scholars focus on nonuniform sparse arrays, which can provide larger array aperture with the same number of sensors. e typical sparse arrays are minimum redundancy arrays (MRAs) [3], coprime linear arrays (CLAs) [4], and nested linear arrays (NLAs) [5].
rough transforming the sparse array to a virtual ULA based on difference coarray, spatial smoothing multiple signal classification (SS-MUSIC) [6] and direct augmentation approach (DAA) [7] are proposed to solve DOAs. Moreover, compressed sensing can be directly applied to difference coarray to estimate DOAs [8][9][10].
Based on the DOA estimation methods mentioned above, many researchers proposed the improved array design methods. One design thought is based on the model of CLA. Coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) [11,12], coprime array with multiperiod subarrays (CAMpS) [13], shifted coprime array (SCA) [14], generalized nested array (GNA) [15], and novel sparse arrays with two uniform arrays (NSA-U2) [16] were proposed, where they all consist of two ULA-subarrays and have larger array aperture than that of CLA. CACIS and CADiS show that setting one subarray with a smaller interelement spacing can have larger aperture of virtual ULA. CAMpS demonstrates 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. Although GNA has the same degree of freedom as NLA, it owns a sparser array structure. NSA-U2 presents the solution to have the maximum degree of freedom for sparse arrays with two uniform arrays. In order to further improve the array aperture, the sparse arrays with multiple ULA-subarrays are proposed. Super-nested arrays (SNAs) [17,18], the augmented nested array (ANA) [19], and the maximum interelement spacing constraint (MISC) [20] divide the dense subarray of CLA into several sparse ULA-subarrays. Another design thought is combining several identical sparse arrays, such as nested MRA (NMRA) [21,22], generalized nested subarray (GNSA) [23], and displaced multistage cascade subarrays (MSC-DiSA) [24]. e subarrays can be any sparse array and the displacement between the subarrays depends on sensors' location of another chosen sparse array. Although the aperture is smaller, the latter design method is less complex than that of the former method. Unfortunately, the structure design of sparse arrays and DOA estimation algorithms are both based on the assumption that the impinging signals are uncorrelated to each other. But there exist coherent signals in real environment, such as in multipath channel. Many decoherence methods, such as spatial smoothing [25], forward/backward spatial smoothing (FBSS) [26], and Toeplitz reconstruction [27], are only applicable to the uniform structure arrays. Hence, the DOA estimation of coherent signals in sparse arrays has been a focus of interest. ere have been some algorithms only for CLAs. Signal separation and Toeplitz reconstruction (SSTR) [28] and spatial smoothing using fourth-order cumulant (SS-FOC) [29] are all utilized in the uniform sparse subarrays and combined with common peak finding [30] to resolve the real values. en, a method [31] combining spatial smoothing and matrix completion theory was proposed, which was applied to the data of physical sensors, but it also has strict restriction about array structure.
In order to solve coherent signals, we aim to give the process of using spatial smoothing in sparse arrays. We first summarize that the sparse array can be seen as two types: one is using several identical sparse arrays and the other consists of several ULA-subarrays. We then, respectively, apply the spatial smoothing and analyze the feasibility of decoherence methods. Next, we discuss the performance of the existing sparse arrays about the number of detectable coherent signals. At last, the simulation experiments are presented to prove the effectiveness of the proposed method. e rest of this paper is organized as follows. Section 2 presents the model of received data including coherent and uncorrelated signals. Section 3 introduces the spatial smoothing process in sparse arrays. Section 4 gives the performance analysis and simulation experiments. Section 5 summarizes the paper. roughout the paper, we make use of the notations shown in Table 1.

Signal Model
Suppose that there are K far-field narrow-band signals impinging on a sparse array with M sensors. Define the unit interelement spacing as λ/2, where λ is the wavelength of signals, and a integer set corresponding to the sensors location is given by D � 0, d 1 , d 2 , . . . , d M−1 (generally assuming d 1 < d 2 < · · · < d M−1 ). Assume that there are P coherent signal groups, where the pth group has L p signals. e coherent signal coming from θ p,ℓ is corresponding to the ℓth multipath propagation of S p (t) with power σ 2 p (p � 1, . . . , P). e signals within each group are coherent to each other and uncorrelated to those in different groups. e total number of coherent signals is K c � P p�1 L p . In addition, the remaining signals, S k (t) coming from θ k with the power σ 2 k (k � K c + 1, . . . , K), are uncorrelated to each other. e number of those signals is K u � K − K c . us, the received signals is where the manifold matrix A(θ) is denoted as and the steering vector a D (θ k ) can be given by and β p,ℓ is the complex fading coefficient of the ℓth coherent signal in the pth group. e signal data vector is where t � 1, . . . , J, and J is the number of snapshots. e noise vector is usually a Gaussian random variable with zero mean and variance σ 2 n . From (1), the covariance matrix is denoted as where R S can be written as a block-diagonal matrix given by Because rank(R p ) � 1, rank(R S ) � P + K u and rank us, the conventional methods for Table 1: Key notations used in this paper.
International Journal of Antennas and Propagation DOA estimation fail. Spatial smoothing can let rank(R X ) � K to satisfy the requirement of subspace methods, but it requires that the array can be divided into several same subarrays. us, in the next section, we will discuss the decomposition of sparse arrays.

Spatial Smoothing in Sparse Arrays
In this section, we try to divide any sparse array into several subarrays and summarize two situations. e first is that the sparse array is composed of several same sparse subarrays.
e second is that the sparse array can be divided into several uniform sparse linear arrays, where although the structure of ULA is different, the process of spatial smoothing is applied to each ULA, and the ULAs with different interelement spacings are applied to remove the ambiguous values.

Sparse Arrays with Identical Sparse Subarrays.
ere exists a type of sparse arrays, which are made up of several same arrays. e sparse subarray can be CLAs, NLAs, MRAs, and so on. If the cascade number of array is Q, we have M � QM and each stage contains M physical sensors, whose location set can be D q � d qM , d qM+1 , . . . , d qM+M−1 . MSC-DiSA and GNSA, which are two typical arrays, have different structure rules, but they are the same when the subarray is MRA with four sensors and the displacement is the function to the location of MRA with three sensors. e structures are shown in Figure 1(a). us, the received data in (1) can be rewritten as where Each subarray meets the feature of rotational invariance, which can be expressed by where where Δd q � d qM − d 0 is displacement between the qth subarray and first subarray and q � 0, . . . , Q − 1. us, the spatial smoothing covariance matrix is defined as where Φ 0 � I K . In order to let rank(R SS ) � K, we give the following theorem.
Proof. See Appendix A.

□
Considering the requirement for setting the displacement between the subarrays [15], we have known that gcd(Δd 1 , . . . , Δd Q−1 ) � 1. So, when Q ≥ max(L p ) and M > K, we can estimate all DOAs θ k by applying subspace methods [32] to R SS .

Sparse Arrays with ULA-Subarrays.
e sparse arrays, which consist of Q ULA-subarrays, are capable to use spatial smoothing algorithm to solve coherent signals. We first give the general model of sensors location denoted as where G q is the displacement between the qth ULA-subarray and the first ULA-subarray, M q is the sensor number of qth ULA-subarray, and g q is the interelement spacing of qth ULA-subarray.

Sparse Arrays with
Two ULA-Subarrays. Figure 1(b) shows four arrays with 12 sensors using two ULAs. us, two subarrays, respectively, have M 1 and M 2 sensors. e location of subarrays can be denoted as where g 1 , g 2 are coprime integers, and generally G 1 � 0. e parameters of arrays are defined in Table 2. en, the received data in (1) can be rewritten as where International Journal of Antennas and Propagation en, the spatial smoothing covariance matrix with size M S i × M S i of ith subarray is given by
and M S i > K, rank(R S i ) � K and the requirement of using subspace methods is satisfied. Also, rank(DR S 2 D H ) � rank(R S 2 ). en, we know that g 1 , g 2 are coprime integers, so we use the subspace method to R SS i and obtain the estimated values of the kth signals, defined as k,2g 2 −1 . From eorem 2, common peak finding [30] tells that the real value θ k � Θ (1) k ∩ Θ (2) k .

Sparse Arrays with Q ULA-Subarrays.
When it comes to the sparse arrays with Q(Q > 2) ULAs-subarrays, we need to point out that the sensor location of existing sparse arrays satisfies (13). We just give the examples about ANAI-1, MISC, ANAII-1, and MRA with M � 12 in Table 3.
Based on the analysis in sparse arrays with two subarrays, we just need to select two subarrays, whose interelement spacings are coprime integers, to solve the DOA estimation. Besides the requirement for setting the inter-element spacing, another two criteria for subarray parameters selection are the number of sensors that are as large as possible and the subarray apertures that are as large as possible. e former is to estimate as many sources as possible, and the latter is to ensure the accuracy of estimation. us, we can obtain the receiving data X 1 and X 2 of two chosen subarrays from X. en, we can use (16) to calculate the spatial smoothing covariance matrix and apply common peak finding [30] to find the DOAs of coherent signals.

Performance Analysis.
We discuss max(L p ) of each sparse array, that is, the maximum number of detectable coherent signals in one group. us, when P � 1 and K u � 0, max(L p ) � M 2 /2 . Due to Q subarrays, max(M 2 ) � ⌊(M − Q + 2)/2⌋ achieves maximum in theory. So, the more the number of ULA-subarrays, the smaller the value of max(L p ).
Next, we compare the value of max(L p ) of GNSA with that of CLA, NLA, ANAI-1, MISC, ANAII-1, and MRA. We vary M from 8 to 20 with 2 intervals, and the results are shown in Figure 2. max(L p ) becomes bigger with the increase of M. e CLA and NLA with two ULA-subarrays have the biggest value of max(L p ). But the ANAII-1 and MRA with 5 or more ULAsubarrays have the smallest value, and only when M > 12, they can use spatial smoothing to estimate coherent signals. Hence, with a fixed number of sensors, the conclusion that the less subarrays can have the bigger max(L p ) is corrected.
Moreover, based on the papers, where the sparse arrays are proposed, the maximum array aperture of consecutive virtual ULAs defined as Ω meets that and Ω GNSA is generally between Ω ANAI−1 and Ω MISC . When they apply spatial smoothing, the main factor affecting the accuracy is the aperture of smoothing array, defined as Π.
en, we assume that P � 1 and L 1 � 2 and compare Π and Π-Ω-ratio c � Π/Ω of different sparse arrays. Figure 3 presents that the more number of ULA-subarrays can achieve bigger Ω, but for spatial smoothing, it is opposite that the less number of ULA-subarrays can achieve bigger Π.

Simulation Experiments.
We use root mean square error (RMSE) to quantify the accuracy of DOA estimation, given by where F is the Monte Carlo number, K is the number of target signals, and θ k,f is the DOA of the kth estimated source by the fth Monte Carlo experiment. e simulation conditions are shown in Table 4.
Simulation 1. feasibility of estimate coherent signals in different sparse arrays.
In first simulation, we show the feasibility of estimate coherent signals with maximum number in one group. us, assume that P � 1, K u � 0, and L 1 of each sparse arrays can be seen in Figure 2. Set SNR � 0dB and J � 5000. e estimation values of 100 times experiments are shown in Figure 4. e figure demonstrates that the sparse arrays can use spatial smoothing to estimate DOAs of coherent signals. Moreover, in the condition of max(L p ) and low SNR, all estimated values are still close to the real values, which means a favorable performance.

Simulation 2. RMSE performance comparison of different SNRs.
In this simulation, we compare the RMSEs of different sparse arrays and ULAs, when there are both uncorrelated and coherent signals. We set K � 3, where θ 1,1 � 25°, θ 1,2 � 0°, θ 3 � 15°, and J � 5000. Because ANAII-1 and MRA cannot estimate 3 signals after spatial smoothing, we only compare the other arrays, and the results are shown in Figure 5. Obviously, the RMSEs decrease with the increase of SNR. GNSA has the highest RMSE due to the smallest Π. ULA has the second smallest Π as 19, so the RMSE of it is just lower than that of GNSA. e other three sparse arrays have the close RMSE due to their close values of Π.      Figure 6, the RMSEs decrease with the increase of snapshot numbers, but when J > 500, the downtrend of RMSEs become slow. Other conclusions are the same as those in simulation 2.

International Journal of Antennas and Propagation
Simulation 4. RMSE performance comparison using FBSS.
In this simulation, we use FBSS to replace spatial smoothing and do simulations 2 and 3 again. FBSS can be seen as an improvement method of spatial smoothing, where Π can be 1.5 times that of spatial smoothing [26]. us, we do not need to present the analysis in Part 1 of this section about using FBSS. But we should note that FBSS is not applicable to GNSA because it needs that the subarrays have uniform structure, where we still use spatial smoothing in GNSA. e results are shown in Figures 7 and 8. Compared with the results in simulations 2 and 3, FBSS has improved  International Journal of Antennas and Propagation the accuracy. us, we can use FBSS in sparse arrays to find more coherent signals and obtain higher accuracy.

Conclusions
In this paper, the DOA estimation methods using spatial smoothing for coherent signals in sparse arrays are proposed. We divide the sparse arrays into two parts. e first type consists of several identical sparse arrays. e second type can be decomposed of several ULA-subarrays. In view of subarrays, spatial smoothing can be applied in sparse arrays. Based on the analysis of the maximum number of detectable coherent signals in one group, the sparse arrays with less subarrays are capable to estimate more signals and own bigger smoothing array aperture. Also, the simulation experiments prove that CLA and NLA have better performance than other arrays.

A
Proof of eorem 1. e matrix R S can be rewritten as where R S is a block-diagonal matrix and (20) can be denoted as If rank(R p ) � L p , rank(R S ) � K. us, we rewrite R p as Based on (25), it can also be simplified to give with C p denoting the Hermitian square root of R p . Because rank(R p ) � rank(V p ), we need to prove that rank(V p ) � L p . We take column permutations to V p , which cannot change the rank of a matrix, and have rank V p � rank where c i,j represents element in the ith row and jth column of C p and ] is expressed as   International Journal of Antennas and Propagation 9 every row of C cannot be all zeros. e matrix [] 1 , · · · , ] L p ] can be seen as the manifold matrix of a sparse array and 0, Δd 1 , . . . , Δd Q−1 are denoted as the location of sensors. Hence, we introduce the theorem in [32,33], which tells that when gcd(Δd 1 , Δd 2 , . . . , Δd Q−1 ) � 1, the array manifold ] k is invertible. Invertibility means that if θ 1 ≠ θ 2 , then ] 1 ≠ ] 2 . en, we can obtain the conclusion that if Q ≥ L p , the rank of [] 1 , . . . , ] L p ] is L p . So, we prove that if gcd(Δd 1 , Δd 2 , . . . , Δd Q−1 ) � 1 and Q ≥ max(L p ), rank(V p ) � L p , and rank(R p ) � L p for any p, then rank(R S ) � K. Hence, when M ≥ K, rank(R SS ) � K.

B
Proof of eorem 3. In this situation, R S i is also a blockdiagonal matrix, given by and R p,i is changed to with C p,i denoting the Hermitian square root of R p,i . We just need to prove that rank(V p,i ) � L p because rank(R p,i ) � rank(V p,i ). Considering the analysis in Appendix A, we should prove that each row of C p,i has at least one nonzero element and vectors ] 1,i , . . . , ] L p ,i are linearly independent, where ν k,i � 1, e jπg i sin θ k , . . . , e jπg i sin θ k . (

B.4)
It is easy to obtain that each row of C p,i has at least one nonzero element. [] 1,i , . . . , ] L p ,i ] is a Vandermonde matrix and 0, g i , . . . , g i (M i − M S i ) are denoted as the location of sensors. Because gcd(g i , . . . , g i (M i − M S i )) � g i ≥ 1, ] k is not invertible. For example, if g i � 2, θ 1,1 � 30°, and θ 1,2 � −30°, then ] 1,i � ] 2,i and rank(R p,i ) � 1, while if θ 1,2 � −30.1°, rank(R p,i ) � 2. In general, assuming that g i � α, if sin θ k − sin θ l ≠ 2πn α , (B.5) for any integer n, ] k ≠ ] l and rank(R p,i ) � L p . Considering that the directions of sources generally have a random distribution in real environment, the parameters (sin θ k , sin θ l ) satisfy (33) with probability one. In other words, if g i > 1 and M i − M S i + 1 ≥ max(L p ), rank(R p,i ) � L p and rank(R S i ) � K in this situation. Hence, when M S i ≥ K, rank(R SS i ) � K.
Data Availability e data used in this article are provided by our simulations and the data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.