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

Authors: Haiyun Xu; Weijia Cui; Yuxi Du; Fengtong Mei; Bin Ba
Journal: International Journal of Antennas and Propagation
 (2021)

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## Abstract

When there is coexistence of uncorrelated and coherent signals in sparse arrays, the conventional algorithms for direction-of-arrival (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. Then, 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.


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## Body

## 1. 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. The typical sparse arrays are minimum redundancy arrays (MRAs) [3], coprime linear arrays (CLAs) [4], and nested linear arrays (NLAs) [5]. Through 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 [8–10].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]. The 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. There 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. Then, 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.The rest of this paper is organized as follows. Section2 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. Throughout the paper, we make use of the notations shown in Table 1.Table 1
Key notations used in this paper.
NotationsExplanation•TTranspose•∗Conjugate•HHermitian transposegcd•Greatest common divisor operationdiag•Diagonal matrixIMIdentity matrix with sizeM×Mrank•Rank of a matrix⌊•⌋Floor integer


## 2. Signal Model

Suppose that there areK 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,d1,d2,…,dM−1 (generally assuming d1<d2<⋯<dM−1). Assume that there are P coherent signal groups, where the pth group has Lp signals. The coherent signal coming from θp,ℓ is corresponding to the ℓth multipath propagation of Spt with power σp2p=1,…,P. The signals within each group are coherent to each other and uncorrelated to those in different groups. The total number of coherent signals is Kc=∑p=1PLp. In addition, the remaining signals, Skt coming from θk with the power σk2 (k=Kc+1,…,K), are uncorrelated to each other. The number of those signals is Ku=K−Kc. Thus, the received signals is(1)Xt=∑p=1P∑ℓ=1LpaDθp,ℓβp,ℓSpt+∑k=Kc+1KaDθkSkt+Nt=AθSt+Nt,where the manifold matrix Aθ is denoted as(2)Aθ=aDθ1,1,…,aDθP,LP,aDθKc+1,…,aDθK,and the steering vector aDθk can be given by(3)aDθk=1,e−jπd1sinθk,…,e−jπdM−1sinθkT,and βp,ℓ is the complex fading coefficient of the ℓth coherent signal in the pth group. The signal data vector is(4)St=β1,1S1t,…,βP,LPSPt,SKc+1t,…,SKtT,where t=1,…,J, and J is the number of snapshots. The noise vector is usually a Gaussian random variable with zero mean and variance σn2.From (1), the covariance matrix is denoted as(5)RX=1JXXH=ARSAH+σn2IM,where RS can be written as a block-diagonal matrix given by(6)RS=R1⋱RPσKc+12⋱σK2,Rp=βp,12σp2βp,1βp,2∗σp2…βp,1βp,Lp∗σp2βp,2βp,1∗σp2βp,22σp2…βp,2βp,Lp∗σp2⋮⋮⋱⋮βp,Lpβp,1∗σp2βp,Lpβp,2∗σp2…βp,Lp2σp2.BecauserankRp=1, rankRS=P+Ku and rankRX=P+Ku<K. Thus, the conventional methods for DOA estimation fail. Spatial smoothing can let rankRX=K to satisfy the requirement of subspace methods, but it requires that the array can be divided into several same subarrays. Thus, in the next section, we will discuss the decomposition of sparse arrays.


## 3. Spatial Smoothing in Sparse Arrays

In this section, we try to divide any sparse array into several subarrays and summarize two situations. The first is that the sparse array is composed of several same sparse subarrays. The 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.


### 3.1. Sparse Arrays with Identical Sparse Subarrays

There exists a type of sparse arrays, which are made up of several same arrays. The sparse subarray can be CLAs, NLAs, MRAs, and so on. If the cascade number of array isQ, we have M=QM¯ and each stage contains M¯ physical sensors, whose location set can be Dq=dqM¯,dqM¯+1,…,dqM¯+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. The structures are shown in Figure 1(a). Thus, the received data in (1) can be rewritten as(7)X=X0⋮XQ−1=A0⋮AQ−1S+N0⋮NQ−1,where(8)Aq=aDqθ1,1,…,aDqθP,LP,aDqθKc+1,…,aDqθK.Figure 1
The structure of sparse arrays whenM=12.
(a)(b)(c)Each subarray meets the feature of rotational invariance, which can be expressed by(9)Aq=A0Φq,where(10)Φq=diage−jπΔdqsinθ1,1,…,e−jπΔdqsinθP,LP,e−jπΔdqsinθKc+1,…,e−jπΔdqsinθK,where Δdq=dqM¯−d0 is displacement between the qth subarray and first subarray and q=0,…,Q−1. Thus, the spatial smoothing covariance matrix is defined as(11)RSS=1JQ∑q=0Q−1XqXqH,=1JQA0∑q=0Q−1ΦqSSHΦqHA0H+σn2IM¯=1JQA0R¯SA0H+σn2IM¯,where Φ0=IK. In order to let rankRSS=K, we give the following theorem.Theorem 1.
IfgcdΔd1,…,ΔdQ−1=1, Q≥maxLp, and M¯≥K, rankRSS=K.Proof.
See Appendix A.Considering the requirement for setting the displacement between the subarrays [15], we have known that gcdΔd1,…,ΔdQ−1=1. So, when Q≥maxLp and M¯>K, we can estimate all DOAs θ^k by applying subspace methods [32] to RSS.


### 3.2. Sparse Arrays with ULA-Subarrays

The sparse arrays, which consist ofQ ULA-subarrays, are capable to use spatial smoothing algorithm to solve coherent signals. We first give the general model of sensors location denoted as(12)D=∪q=1QGq+mgq|0≤m≤Mq−1,where Gq is the displacement between the qth ULA-subarray and the first ULA-subarray, Mq is the sensor number of qth ULA-subarray, and gq is the interelement spacing of qth ULA-subarray.


#### 3.2.1. Sparse Arrays with Two ULA-Subarrays

Figure1(b) shows four arrays with 12 sensors using two ULAs. Thus, two subarrays, respectively, have M1 and M2 sensors. The location of subarrays can be denoted as D1=m1g1|0≤m1≤M1−1, D2=G2+m2g2|0≤m2≤M2−1, where g1,g2 are coprime integers, and generally G1=0. The parameters of arrays are defined in Table 2. Then, the received data in (1) can be rewritten as(13)X=X1X2=A1A2S+N1N2,where(14)Ai=aDiθ1,1,…,aDiθP,LP,aDiθKc+1,…,aDiθK.Table 2
Parameters of sparse arrays.
Type of sparse arrayM1,M2,g1,g2,G2CLA7,5,4,5,0CADis5,7,7,3,10SCA5,7,5,4,8NLA6,6,1,6,0Then, the spatial smoothing covariance matrix with sizeMSi×MSi of ith subarray is given by(15)RSSi=1JMi−MSi+1∑m=0Mi−MSiΓmiXiΓmiXiH,=ASiR¯SiASiH+σn2IMSi,where Γmi=0MSi×Mi−MSi−m,IMSi,0MSi×m. AS1 is given by(16)AS1=aDS1θ1,1,…,aDS1θP,LP,aDS1θKc+1,…,aDS1θK,where DS1=m1g1|0≤m1≤MS1−1. Also, we define AS2=A¯S2D, where(17)A¯S2=aDS2θ1,1,…,aDS2θP,LP,aDS2θKc+1,…,aDS2θK,where DS2=m2g2|0≤m2≤MS2−1, and(18)D=diagejπG2sinθ1,1,…,ejπG2sinθP,LP,ejπG2sinθKc+1,…,ejπG2sinθK.Moreover,(19)R¯Si=1JMi−MSi+1∑m=0Mi−MSiΨimSSHΨimH,where Ψ1=diagejπg1sinθ1,1,…,ejπg1sinθK and Ψ2=diagejπg2sinθ1,1,…,ejπg2sinθK.Based on the model above, the following statement gives the applicability of using spatial smoothing in this type of sparse array. Firstly, we introduce Theorem2 in [30] and propose Theorem 3.Theorem 2.
Letg1 and g2 denote the interelement spacing of two ULAs, respectively. Let Θk1=θk,11,…,θk,2g1−11 and Θk2=θk,12,…,θk,2g2−12 denote the estimated DOA values of the kth signal of two subarrays, respectively, where each set has multiple ambiguous values and one real value. If gcdg1,g2=1, Θk1∩Θk2=θk.Theorem 3.
Let the directions of sources have a random distribution. WhenMi−MSi+1≥maxLpp=1,…,P and MSi≥K, rankR¯SSi=K.Proof.
See Appendix B.WhenMi−MSi+1≥maxLpp=1,…,P and MSi>K, rankR¯Si=K and the requirement of using subspace methods is satisfied. Also, rankDR¯S2DH=rankR¯S2. Then, we know that g1,g2 are coprime integers, so we use the subspace method to R¯SSi and obtain the estimated values of the kth signals, defined as Θk1=θk,11,…,θk,2g1−11 and Θk2=θk,12,…,θk,2g2−12. From Theorem 2, common peak finding [30] tells that the real value θ^k=Θk1∩Θk2.


#### 3.2.2. Sparse Arrays withQ ULA-Subarrays

When it comes to the sparse arrays withQQ>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.Table 3
Parameters of sparse arrays.
ArraysDQMq,gq,GqANAI-10,1,6,13,20,27,34,4141,42,43,44,4532,1,0,7,6,6,5,1,41MISC0,1,6,14,22,30,3838,40,42,45,47,4942,1,0,5,8,63,2,38,3,2,45ANAII-10,1,1,5,9,9,16,23,30,37,4447,49,49,5052,1,0,3,4,1,6,7,9,2,2,47,2,1,49MRA0,1,3,6,6,13,20,27,34,4141,45,49,49,5052,1,0,2,3,3,6,7,6,3,4,41,2,1,49Based 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. The former is to estimate as many sources as possible, and the latter is to ensure the accuracy of estimation. Thus, we can obtain the receiving dataX1 and X2 of two chosen subarrays from X. Then, we can use (16) to calculate the spatial smoothing covariance matrix and apply common peak finding [30] to find the DOAs of coherent signals.


## 3.1. Sparse Arrays with Identical Sparse Subarrays

There exists a type of sparse arrays, which are made up of several same arrays. The sparse subarray can be CLAs, NLAs, MRAs, and so on. If the cascade number of array isQ, we have M=QM¯ and each stage contains M¯ physical sensors, whose location set can be Dq=dqM¯,dqM¯+1,…,dqM¯+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. The structures are shown in Figure 1(a). Thus, the received data in (1) can be rewritten as(7)X=X0⋮XQ−1=A0⋮AQ−1S+N0⋮NQ−1,where(8)Aq=aDqθ1,1,…,aDqθP,LP,aDqθKc+1,…,aDqθK.Figure 1
The structure of sparse arrays whenM=12.
(a)(b)(c)Each subarray meets the feature of rotational invariance, which can be expressed by(9)Aq=A0Φq,where(10)Φq=diage−jπΔdqsinθ1,1,…,e−jπΔdqsinθP,LP,e−jπΔdqsinθKc+1,…,e−jπΔdqsinθK,where Δdq=dqM¯−d0 is displacement between the qth subarray and first subarray and q=0,…,Q−1. Thus, the spatial smoothing covariance matrix is defined as(11)RSS=1JQ∑q=0Q−1XqXqH,=1JQA0∑q=0Q−1ΦqSSHΦqHA0H+σn2IM¯=1JQA0R¯SA0H+σn2IM¯,where Φ0=IK. In order to let rankRSS=K, we give the following theorem.Theorem 1.
IfgcdΔd1,…,ΔdQ−1=1, Q≥maxLp, and M¯≥K, rankRSS=K.Proof.
See Appendix A.Considering the requirement for setting the displacement between the subarrays [15], we have known that gcdΔd1,…,ΔdQ−1=1. So, when Q≥maxLp and M¯>K, we can estimate all DOAs θ^k by applying subspace methods [32] to RSS.


## 3.2. Sparse Arrays with ULA-Subarrays

The sparse arrays, which consist ofQ ULA-subarrays, are capable to use spatial smoothing algorithm to solve coherent signals. We first give the general model of sensors location denoted as(12)D=∪q=1QGq+mgq|0≤m≤Mq−1,where Gq is the displacement between the qth ULA-subarray and the first ULA-subarray, Mq is the sensor number of qth ULA-subarray, and gq is the interelement spacing of qth ULA-subarray.


### 3.2.1. Sparse Arrays with Two ULA-Subarrays

Figure1(b) shows four arrays with 12 sensors using two ULAs. Thus, two subarrays, respectively, have M1 and M2 sensors. The location of subarrays can be denoted as D1=m1g1|0≤m1≤M1−1, D2=G2+m2g2|0≤m2≤M2−1, where g1,g2 are coprime integers, and generally G1=0. The parameters of arrays are defined in Table 2. Then, the received data in (1) can be rewritten as(13)X=X1X2=A1A2S+N1N2,where(14)Ai=aDiθ1,1,…,aDiθP,LP,aDiθKc+1,…,aDiθK.Table 2
Parameters of sparse arrays.
Type of sparse arrayM1,M2,g1,g2,G2CLA7,5,4,5,0CADis5,7,7,3,10SCA5,7,5,4,8NLA6,6,1,6,0Then, the spatial smoothing covariance matrix with sizeMSi×MSi of ith subarray is given by(15)RSSi=1JMi−MSi+1∑m=0Mi−MSiΓmiXiΓmiXiH,=ASiR¯SiASiH+σn2IMSi,where Γmi=0MSi×Mi−MSi−m,IMSi,0MSi×m. AS1 is given by(16)AS1=aDS1θ1,1,…,aDS1θP,LP,aDS1θKc+1,…,aDS1θK,where DS1=m1g1|0≤m1≤MS1−1. Also, we define AS2=A¯S2D, where(17)A¯S2=aDS2θ1,1,…,aDS2θP,LP,aDS2θKc+1,…,aDS2θK,where DS2=m2g2|0≤m2≤MS2−1, and(18)D=diagejπG2sinθ1,1,…,ejπG2sinθP,LP,ejπG2sinθKc+1,…,ejπG2sinθK.Moreover,(19)R¯Si=1JMi−MSi+1∑m=0Mi−MSiΨimSSHΨimH,where Ψ1=diagejπg1sinθ1,1,…,ejπg1sinθK and Ψ2=diagejπg2sinθ1,1,…,ejπg2sinθK.Based on the model above, the following statement gives the applicability of using spatial smoothing in this type of sparse array. Firstly, we introduce Theorem2 in [30] and propose Theorem 3.Theorem 2.
Letg1 and g2 denote the interelement spacing of two ULAs, respectively. Let Θk1=θk,11,…,θk,2g1−11 and Θk2=θk,12,…,θk,2g2−12 denote the estimated DOA values of the kth signal of two subarrays, respectively, where each set has multiple ambiguous values and one real value. If gcdg1,g2=1, Θk1∩Θk2=θk.Theorem 3.
Let the directions of sources have a random distribution. WhenMi−MSi+1≥maxLpp=1,…,P and MSi≥K, rankR¯SSi=K.Proof.
See Appendix B.WhenMi−MSi+1≥maxLpp=1,…,P and MSi>K, rankR¯Si=K and the requirement of using subspace methods is satisfied. Also, rankDR¯S2DH=rankR¯S2. Then, we know that g1,g2 are coprime integers, so we use the subspace method to R¯SSi and obtain the estimated values of the kth signals, defined as Θk1=θk,11,…,θk,2g1−11 and Θk2=θk,12,…,θk,2g2−12. From Theorem 2, common peak finding [30] tells that the real value θ^k=Θk1∩Θk2.


### 3.2.2. Sparse Arrays withQ ULA-Subarrays

When it comes to the sparse arrays withQQ>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.Table 3
Parameters of sparse arrays.
ArraysDQMq,gq,GqANAI-10,1,6,13,20,27,34,4141,42,43,44,4532,1,0,7,6,6,5,1,41MISC0,1,6,14,22,30,3838,40,42,45,47,4942,1,0,5,8,63,2,38,3,2,45ANAII-10,1,1,5,9,9,16,23,30,37,4447,49,49,5052,1,0,3,4,1,6,7,9,2,2,47,2,1,49MRA0,1,3,6,6,13,20,27,34,4141,45,49,49,5052,1,0,2,3,3,6,7,6,3,4,41,2,1,49Based 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. The former is to estimate as many sources as possible, and the latter is to ensure the accuracy of estimation. Thus, we can obtain the receiving dataX1 and X2 of two chosen subarrays from X. Then, we can use (16) to calculate the spatial smoothing covariance matrix and apply common peak finding [30] to find the DOAs of coherent signals.


## 3.2.1. Sparse Arrays with Two ULA-Subarrays

Figure1(b) shows four arrays with 12 sensors using two ULAs. Thus, two subarrays, respectively, have M1 and M2 sensors. The location of subarrays can be denoted as D1=m1g1|0≤m1≤M1−1, D2=G2+m2g2|0≤m2≤M2−1, where g1,g2 are coprime integers, and generally G1=0. The parameters of arrays are defined in Table 2. Then, the received data in (1) can be rewritten as(13)X=X1X2=A1A2S+N1N2,where(14)Ai=aDiθ1,1,…,aDiθP,LP,aDiθKc+1,…,aDiθK.Table 2
Parameters of sparse arrays.
Type of sparse arrayM1,M2,g1,g2,G2CLA7,5,4,5,0CADis5,7,7,3,10SCA5,7,5,4,8NLA6,6,1,6,0Then, the spatial smoothing covariance matrix with sizeMSi×MSi of ith subarray is given by(15)RSSi=1JMi−MSi+1∑m=0Mi−MSiΓmiXiΓmiXiH,=ASiR¯SiASiH+σn2IMSi,where Γmi=0MSi×Mi−MSi−m,IMSi,0MSi×m. AS1 is given by(16)AS1=aDS1θ1,1,…,aDS1θP,LP,aDS1θKc+1,…,aDS1θK,where DS1=m1g1|0≤m1≤MS1−1. Also, we define AS2=A¯S2D, where(17)A¯S2=aDS2θ1,1,…,aDS2θP,LP,aDS2θKc+1,…,aDS2θK,where DS2=m2g2|0≤m2≤MS2−1, and(18)D=diagejπG2sinθ1,1,…,ejπG2sinθP,LP,ejπG2sinθKc+1,…,ejπG2sinθK.Moreover,(19)R¯Si=1JMi−MSi+1∑m=0Mi−MSiΨimSSHΨimH,where Ψ1=diagejπg1sinθ1,1,…,ejπg1sinθK and Ψ2=diagejπg2sinθ1,1,…,ejπg2sinθK.Based on the model above, the following statement gives the applicability of using spatial smoothing in this type of sparse array. Firstly, we introduce Theorem2 in [30] and propose Theorem 3.Theorem 2.
Letg1 and g2 denote the interelement spacing of two ULAs, respectively. Let Θk1=θk,11,…,θk,2g1−11 and Θk2=θk,12,…,θk,2g2−12 denote the estimated DOA values of the kth signal of two subarrays, respectively, where each set has multiple ambiguous values and one real value. If gcdg1,g2=1, Θk1∩Θk2=θk.Theorem 3.
Let the directions of sources have a random distribution. WhenMi−MSi+1≥maxLpp=1,…,P and MSi≥K, rankR¯SSi=K.Proof.
See Appendix B.WhenMi−MSi+1≥maxLpp=1,…,P and MSi>K, rankR¯Si=K and the requirement of using subspace methods is satisfied. Also, rankDR¯S2DH=rankR¯S2. Then, we know that g1,g2 are coprime integers, so we use the subspace method to R¯SSi and obtain the estimated values of the kth signals, defined as Θk1=θk,11,…,θk,2g1−11 and Θk2=θk,12,…,θk,2g2−12. From Theorem 2, common peak finding [30] tells that the real value θ^k=Θk1∩Θk2.


## 3.2.2. Sparse Arrays withQ ULA-Subarrays

When it comes to the sparse arrays withQQ>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.Table 3
Parameters of sparse arrays.
ArraysDQMq,gq,GqANAI-10,1,6,13,20,27,34,4141,42,43,44,4532,1,0,7,6,6,5,1,41MISC0,1,6,14,22,30,3838,40,42,45,47,4942,1,0,5,8,63,2,38,3,2,45ANAII-10,1,1,5,9,9,16,23,30,37,4447,49,49,5052,1,0,3,4,1,6,7,9,2,2,47,2,1,49MRA0,1,3,6,6,13,20,27,34,4141,45,49,49,5052,1,0,2,3,3,6,7,6,3,4,41,2,1,49Based 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. The former is to estimate as many sources as possible, and the latter is to ensure the accuracy of estimation. Thus, we can obtain the receiving dataX1 and X2 of two chosen subarrays from X. Then, we can use (16) to calculate the spatial smoothing covariance matrix and apply common peak finding [30] to find the DOAs of coherent signals.


## 4. Performance Analysis and Simulation Experiments

### 4.1. Performance Analysis

We discussmaxLp of each sparse array, that is, the maximum number of detectable coherent signals in one group. For GNSA, maxLp≤Q=M/M¯, so when Q is the maximum integer no more than ⌊M⌋, maxLp achieving maximum is equal to Q. For any sparse array consisting of Q ULAs, we arrange the number of subarray sensors M1,…,MQ in descending order and have the set M˜1,…,M˜Q, where M˜1>⋯>M˜Q. Based on the theorem that the paper proposes, we can have maxLp≤⌊M˜2/2⌋ and Kc≤M˜2−⌊M˜2/2⌋. Thus, when P=1 and Ku=0, maxLp=⌊M˜2/2⌋. Due to Q subarrays, maxM˜2=⌊M−Q+2/2⌋ achieves maximum in theory. So, the more the number of ULA-subarrays, the smaller the value of maxLp.Next, we compare the value ofmaxLp 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. maxLp becomes bigger with the increase of M. The CLA and NLA with two ULA-subarrays have the biggest value of maxLp. But the ANAII-1 and MRA with 5 or more ULA-subarrays 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 maxLp is corrected.Figure 2
maxLp in different sparse arrays with different M.Moreover, based on the papers, where the sparse arrays are proposed, the maximum array aperture of consecutive virtual ULAs defined asΩ meets that ΩMRA≥ΩANAII−1>ΩMISC>ΩANAI−1>ΩNLA>ΩCLA, and ΩGNSA is generally between ΩANAI−1 and ΩMISC. When they apply spatial smoothing, the main factor affecting the accuracy is the aperture of smoothing array, defined as Π. Then, we assume that P=1 and L1=2 and compare Π and Π-Ω-ratio γ=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.Figure 3
The comparison ofΠ and γ in different sparse arrays.
(a)(b)


### 4.2. Simulation Experiments

We use root mean square error (RMSE) to quantify the accuracy of DOA estimation, given by(20)RMSE=1FK∑f=1F∑k=1Kθ^k,f−θk2,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. The simulation conditions are shown in Table 4.Table 4
Simulation conditions for the experiments.
Simulation parametersValuesAntenna number20The subarrays for smoothing in ULA0,1,2,3,…,18,19The subarrays for smoothing in GNSA0,1,4,7,9,19,20,23,26,2876,77,80,83,85,114,115,118,121,123The subarrays for smoothing in CLA0,10,20,30,40,50,60,70,80,90,1000,11,22,33,44,55,66,77,88,99,The subarrays for smoothing in NLA0,1,2,3,4,5,6,7,8,9,100,10,20,30,40,50,60,70,80,90,100,The subarrays for smoothing in ANAI-111,22,33,44,55,66,77,88,99,110110,111,112,113,114,115,116,The subarrays for smoothing in ANAII-135,50,65,80,95,1103,11,19,27,35,The subarrays for smoothing in MRA28,43,58,73,88,103103,111,119,127,135,Monte Carlo numberF=100SNR−5,0,5,10,15Snapshot50,100,200,500,1000,2000,5000Simulation 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. Thus, assume thatP=1, Ku=0, and L1 of each sparse arrays can be seen in Figure 2. Set SNR=0dB and J=5000. The estimation values of 100 times experiments are shown in Figure 4. The figure demonstrates that the sparse arrays can use spatial smoothing to estimate DOAs of coherent signals. Moreover, in the condition of maxLp and low SNR, all estimated values are still close to the real values, which means a favorable performance.Figure 4
The estimation values of coherent signals in different sparse arrays.
(a)(b)(c)(d)(e)(f)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 setK=3, where θ1,1=25°,θ1,2=0°,θ3=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. The other three sparse arrays have the close RMSE due to their close values of Π.Figure 5
The RMSE versus SNR in different sparse arrays using spatial smoothing.Simulation 3.
RMSE performance comparison of different number of snapshots.
Similar to the simulation 2, we present the RMSEs of different sparse arrays under different snapshots withSNR=10dB. From 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.Figure 6
The RMSE versus the number of snapshots in different sparse arrays using spatial smoothing.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, whereΠ can be 1.5 times that of spatial smoothing [26]. Thus, 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. The results are shown in Figures 7 and 8. Compared with the results in simulations 2 and 3, FBSS has improved the accuracy. Thus, we can use FBSS in sparse arrays to find more coherent signals and obtain higher accuracy.Figure 7
The RMSE versus SNR in different sparse arrays using FBSS.Figure 8
The RMSE versus the number of snapshots in different sparse arrays using FBSS.


## 4.1. Performance Analysis

We discussmaxLp of each sparse array, that is, the maximum number of detectable coherent signals in one group. For GNSA, maxLp≤Q=M/M¯, so when Q is the maximum integer no more than ⌊M⌋, maxLp achieving maximum is equal to Q. For any sparse array consisting of Q ULAs, we arrange the number of subarray sensors M1,…,MQ in descending order and have the set M˜1,…,M˜Q, where M˜1>⋯>M˜Q. Based on the theorem that the paper proposes, we can have maxLp≤⌊M˜2/2⌋ and Kc≤M˜2−⌊M˜2/2⌋. Thus, when P=1 and Ku=0, maxLp=⌊M˜2/2⌋. Due to Q subarrays, maxM˜2=⌊M−Q+2/2⌋ achieves maximum in theory. So, the more the number of ULA-subarrays, the smaller the value of maxLp.Next, we compare the value ofmaxLp 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. maxLp becomes bigger with the increase of M. The CLA and NLA with two ULA-subarrays have the biggest value of maxLp. But the ANAII-1 and MRA with 5 or more ULA-subarrays 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 maxLp is corrected.Figure 2
maxLp in different sparse arrays with different M.Moreover, based on the papers, where the sparse arrays are proposed, the maximum array aperture of consecutive virtual ULAs defined asΩ meets that ΩMRA≥ΩANAII−1>ΩMISC>ΩANAI−1>ΩNLA>ΩCLA, and ΩGNSA is generally between ΩANAI−1 and ΩMISC. When they apply spatial smoothing, the main factor affecting the accuracy is the aperture of smoothing array, defined as Π. Then, we assume that P=1 and L1=2 and compare Π and Π-Ω-ratio γ=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.Figure 3
The comparison ofΠ and γ in different sparse arrays.
(a)(b)


## 4.2. Simulation Experiments

We use root mean square error (RMSE) to quantify the accuracy of DOA estimation, given by(20)RMSE=1FK∑f=1F∑k=1Kθ^k,f−θk2,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. The simulation conditions are shown in Table 4.Table 4
Simulation conditions for the experiments.
Simulation parametersValuesAntenna number20The subarrays for smoothing in ULA0,1,2,3,…,18,19The subarrays for smoothing in GNSA0,1,4,7,9,19,20,23,26,2876,77,80,83,85,114,115,118,121,123The subarrays for smoothing in CLA0,10,20,30,40,50,60,70,80,90,1000,11,22,33,44,55,66,77,88,99,The subarrays for smoothing in NLA0,1,2,3,4,5,6,7,8,9,100,10,20,30,40,50,60,70,80,90,100,The subarrays for smoothing in ANAI-111,22,33,44,55,66,77,88,99,110110,111,112,113,114,115,116,The subarrays for smoothing in ANAII-135,50,65,80,95,1103,11,19,27,35,The subarrays for smoothing in MRA28,43,58,73,88,103103,111,119,127,135,Monte Carlo numberF=100SNR−5,0,5,10,15Snapshot50,100,200,500,1000,2000,5000Simulation 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. Thus, assume thatP=1, Ku=0, and L1 of each sparse arrays can be seen in Figure 2. Set SNR=0dB and J=5000. The estimation values of 100 times experiments are shown in Figure 4. The figure demonstrates that the sparse arrays can use spatial smoothing to estimate DOAs of coherent signals. Moreover, in the condition of maxLp and low SNR, all estimated values are still close to the real values, which means a favorable performance.Figure 4
The estimation values of coherent signals in different sparse arrays.
(a)(b)(c)(d)(e)(f)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 setK=3, where θ1,1=25°,θ1,2=0°,θ3=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. The other three sparse arrays have the close RMSE due to their close values of Π.Figure 5
The RMSE versus SNR in different sparse arrays using spatial smoothing.Simulation 3.
RMSE performance comparison of different number of snapshots.
Similar to the simulation 2, we present the RMSEs of different sparse arrays under different snapshots withSNR=10dB. From 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.Figure 6
The RMSE versus the number of snapshots in different sparse arrays using spatial smoothing.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, whereΠ can be 1.5 times that of spatial smoothing [26]. Thus, 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. The results are shown in Figures 7 and 8. Compared with the results in simulations 2 and 3, FBSS has improved the accuracy. Thus, we can use FBSS in sparse arrays to find more coherent signals and obtain higher accuracy.Figure 7
The RMSE versus SNR in different sparse arrays using FBSS.Figure 8
The RMSE versus the number of snapshots in different sparse arrays using FBSS.


## 5. 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. The first type consists of several identical sparse arrays. The 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.



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*Source: 4447562-2021-09-17.xml*