# DOA Estimation for Non-Gaussian Signals: Three-Level Nested Array and a Successive SS-MUSIC Algorithm

Authors: Sha Shi; Haowei Zeng; Heng Yue; Changbo Ye; Jianfeng Li
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

Direction of arrival (DOA) estimation for non-Gaussian signals using three-level nested array (THL-NA) is investigated in this paper. Motivation from larger consecutive degree of freedom (DOF) and array aperture, the THL-NA is proposed, which can take full advantages of the consecutive coarrays of TL-NA and has the closed-form expression of DOF. Specifically, firstly, the array aperture is expanded by the second order sum coarray (2-SC) of the proposed array, secondly, the nested relationship between subarrays is employed to obtain the difference coarray of 2-SC (2-DCSC), finally, a consecutive virtual array with large array aperture is obtained. Besides, a successive SS-MUSIC algorithm is proposed, which employs the spatial smoothing estimating signal parameter via rotational invariance techniques (SS-ESPRIT) algorithm and partial spectrum searching multiple signal classification (PSS-MUSIC) to obtain initial estimations and fine estimations, respectively, resulting in a better balance between computational complexity and estimation accuracy.


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## Body

## 1. Introduction

Direction of arrival (DOA) estimation, which is one of the fundamental issues in array signal processing, plays an important role in various fields, e.g., radar systems, acoustics, navigation, and wireless communications [1–3].Compared with traditional uniform linear arrays (ULAs), sparse arrays can obtain less mutual coupling and higher degrees of freedom (DOF), which can obtain DOA estimation for signals with more sources. The nested array (NA) [4] is a sparse array composed of two ULAs with different spacings, which has strong expansion capability in DOA estimation algorithms based on FOC. The coprime array (CPA) [5, 6], consisting of two ULAs with coprime interelement spacings, has less mutual coupling compared with NA. These arrays can generate sets of uniformly distributed virtual second order difference coarray (2-DC) [4], in particular, the 2-DC of NA is free from holes. However, these sparse arrays have limitations due to their array geometries. The 2-DC of CPA generates a lot of holes that decrease consecutive DOF significantly, the dense part of NA leads to more serious mutual coupling than CPA.Traditional DOA parameter estimation algorithms, such as multiple signal classification (MUSIC) algorithm [7, 8] and estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm [9, 10], mostly utilize the second order statistical characteristics of the signals. Furthermore, new algorithms have been proposed such as 2D-MUSIC and RD-MUSIC [1]. When the signals obey the Gaussian distribution, they can be described by the first order or second order statistics. However, the situation in practical applications is more complicated, and most sources to be processed are non-Gaussian signals, whose statistical characteristics can be described by fourth order cumulant (FOC) [11, 12], resulting in larger array aperture and better DOA estimation performance, as compared with second order cumulant (SOC). Considering the non-Gaussian signals, DOA estimation methods based on FOC, including MUSIC-LIKE [13, 14] and virtual-ESPRIT algorithm [15], are most exploited. The virtual coarrays utilized by vectorized FOC methods can be obtained from fourth order difference coarray (4-DC) [16] or difference coarray of sum coarray (2-DCSC) [17] operation of physical sensors. The FOC can not only suppress Gaussian white noise or color noise, but effectively expand the length of array apertures. Therefore, the estimation performance of DOA based on FOC is greatly improved.Combining the designing of sparse array and the exploiting of 4-DC or 2-DCSC, lots of array structures are proposed, such as fourth-level NA (FL-NA) [16], sparse array with fourth order difference coarray enhancement based on CPA (SAFE-CPA) [18]. FL-NA is investigated in detail as a special case of 2q-level NAs when q is equal 2 in reference [16] and SAFE-CPA constructs physical sensors structure by adding another subarray. Regarding the processing of the above sparse array structure, related scholars have proposed a series of FOC estimation algorithms based on spatial smoothing subspace methods [19, 20] such as SS-MUSIC algorithm, SS-ESPRIT algorithm, which provide a theoretical basis for the DOA estimation based on sparse arrays.In this paper, a novel sparse array structure, named three-level nested array (THL-NA), is proposed for non-Gaussian incident signals, which can take full advantages of the consecutive coarrays of TL-NA and has the closed-form expression of DOF. Firstly, the 2-SC [21] of the proposed array is obtained to expand the array aperture, and then the consecutive 2-DCSC of the proposed array is obtained by employing the nested relationship between subarrays of 2-SC. To make a better balance between computational complexity and estimation accuracy, a successive SS-MUSIC algorithm is proposed, which employs the spatial smoothing ESPRIT (SS-ESPRIT) algorithm and partial spectrum searching MUSIC (PSS-MUSIC) to obtain initial estimates and fine estimates, respectively.The three contributions of this paper are extracted as follows:(1)
From the viewpoint of constructing sum or difference coarray, the THL-NA based on the FOC is proposed to obtain large consecutive DOF and array aperture, which enhances the DOA estimation performance.(2)
From the perspective for expressing the length of the virtual array in the THL-NA, this paper derives the closed-form expressions with consecutive DOF and discusses the optimal array configuration to achieve the largest consecutive DOF.(3)
In terms of making a balance between computational complexity and estimation accuracy, a successive SS-MUSIC algorithm is proposed, which employs SS-ESPRIT algorithm and PSS-MUSIC to obtain initial angle estimations and fine angle estimations, respectively.The chapter arrangement of this paper is as follows. In Section2, we introduce 2-DC, 2-SC, 2-DCSC, 4-DC, and the properties of NA. We elaborate the array configuration of THL-NA and explain the closed-form expression in Section 3. The proposed algorithm called successive SS-MUSIC algorithm is presented in Section 4. Section 5 analyzes the performance of THL-NA and the computational complexity of the successive SS-MUSIC algorithm. Section 6 provides lots of simulations and the conclusions are drawn in Section 7.Notations. Throughout the paper, matrices are expressed by upper-case bold characters and vectors are denoted by lower-case bold characters, respectively. ⋅T, ⋅H, ⋅−1, and ⋅∗ imply the transpose, the conjugate transpose operation, inverse and complex conjugation of a vector or matrix, respectively. ⊗, ⊙, and ⊕ stand for the Kronecker product, Khatri-Rao product and Hadamard product, respectively. vec⋅ represents the vectorization operation and cum⋅ indicates the cumulant operator. angle⋅ signifies the phase operator and ⋅F shows the Frobenius norm. arcsin⋅ means the arcsine function.


## 2. Preliminaries

In this section, the 2-DC, 2-SC, 2-DCSC, 4-DC, and the properties of NA are introduced.


### 2.1. The Definitions of 2-DC, 2-SC, 4-DC, and 2-DCSC

Consider aP -sensors linear array, whose locations set can be indicated as [22](1)S=v1⋅d,v2⋅d,…,vP⋅d,where vp⋅d represents the position of the p−th sensor, p=1,2,…,P, and d is the unit spacing.Definition 1.
For the array with the set of sensor positionS in equation (1), the 2-DC set is defined as [17](2)C2−DC=Φ2−DCd,where the set of 2-DC lags(3)Φ2−DC=vp1−vp2,1≤p1,p2≤P.Definition 2.
The 2-SC set is defined as [21](4)C2−SC=Φ2−SCd,with the set of 2-SC lags(5)Φ2−SC=vp1+vp2,1≤p1,p2≤P.Definition 3.
The 4-DC set is defined as [16](6)C4−DC=Φ4−DCd,where the set of 4-DC lags(7)Φ4−DC=vp1−vp2−vp3−vp4,1≤p1,p2,p3,p4≤P.Definition 4.
The 2-DCSC set is defined as [17](8)C2−DCSC=Φ2−DCSCd,where the set of 2-DCSC lags(9)Φ2−DCSC=vp1+vp2−vp3+vp4,1≤p1,p2,p3,p4≤P.
By permutation invariance,Φ2−DCSC can be rewritten as(10)Φ2−DCSC=vp1−vp2−vp3−vp4=vp1+vp4−vp2+vp3=Φ4−DC.
for1≤p1,p2,p3,p4≤P. That is, for a particular array, the 4-DC is equivalent to the 2-DCSC(11)C4−DC=C2−DCSC.


### 2.2. The Properties of TL-NA

The configuration of TL-NA [4] has been shown in Figure 1, which contains two sparse uniform subarrays. The first subarray has M1 sensors with interelement spacing d, where d=λ/2 and λ is the wavelength, while the another subarray has M2 sensors with interelement spacing M1+1d. The position of TL-NA varies from 1d to M2M1+1d and it can be represented as(12)S=m1d,1≤m1≤M1∪m2M1+1d,1≤m2≤M2.Figure 1
Two-level nested array.Refer to Definition1, the 2-DC location set of TL-NA can be written as(13)C2−DC=−M2M1+1+1,…,−1,0,1,…,M2M1+1−1d.Refer to Definitions3 and4, the 4-DC (2-DCSC) location set of TL-NA can be expressed as [23](14)C4−DC=C2−DCSC=−2M2M1+1+2,…,−1,0,1,…,2M2M1+1−2d.From equations (13) and (14), It can be concluded that the position of the 2-DC and 2-DCSC of TL-NA are consecutive.


### 2.3. Data Model Based on FOC

AssumeK far-field narrowband uncorrelated sources impinging upon the array with locations set S from directions θ=θ1,…,θK, the received data model of the array can be expressed as [24, 25](15)xt=Ast+nt,where Aθ=aθ1,aθ2,…,aθK∈ℂP×K represents the steering matrix of array, and the steering vector at direction θk, which denotes the elevation angle of the k−th target, is given by(16)aθk=e−j2πv1dsinθk/λ,e−j2πv2dsinθk/λ,…,e−j2πvPdsinθk/λT∈ℂP×1.andst=s1t,s2t,…,sKtT∈ℂK×1, 1≤t≤L denotes non-Gaussian signal source matrix with zero mean, where L indicates the number of snapshots. And, nt is the received additive white Gaussian noise with mean zero and variance σ2.In this paper, for givenxk1,xk2,xk3∗,xk4∗, the FOC definition can be written as [26](17)C4=Cumxk1,xk2,xk3∗,xk4∗=Exk1xk2xk3∗xk4∗−Exk1xk2Exk3∗xk4∗−Exk1xk3∗Exk2xk4∗−Exk1xk4∗Exk2xk3∗.For the array, the FOC of the received signal can be [26](18)C4,X=CumX,X,X∗,X∗=EX⊗X∗X⊗X∗H−EX⊗X∗EX⊗X∗H−EXXH⊗EXXH∗.The FOC matrix of the received signal is expressed by [26](19)R4=C4,X=BθCSBHθ,where(20)Bθ=bθ1,bθ2,…,bθK=aθ1⊗a∗θ1,aθ2⊗a∗θ2,…,aθK⊗a∗θK=Aθ⊙A∗θ,CS=ES⊗SS⊗SH−ES⊗SES⊗SH−ESSH⊗ESSH∗.Bθ is the steering matrix after array manifold Aθ⇒Bθ by using the FOC. To further improve the performance of DOA estimation, we use a vectorized method [23, 26, 27] for R4 to obtain a vector, which is equivalent to the received signal from the virtual ULA.(21)z=vecR4=vecBθCsBHθ=Λθp,where(22)Λθ=b∗θ1⊗bθ1,b∗θ2⊗bθ2,…,b∗θK⊗bθK=B∗θ⊙Bθ,and p=c4,s1,c4,s2,…,c4,sKT represents the FOC matrix of source vector, and c4,sk represents the FOC of the k−th source vector.


### 2.4. Data Model with Mutual Coupling Based on FOC

Equation (16) assumes that the sensors in the array has not interference with each other. Actually, the output of each sensor is influenced by its adjacent elements owing to the presence of mutual coupling. Therefore, by introducing a mutual coupling matrix C, the received data model can be rewritten as follows [28, 29].(23)xt=CAst+nt,where the mutual coupling C can be obtained from reference [28], whose define is determined by varieties of factors involving in the distance between sensors, the operating frequency, e.g., and it can be approximated by employing the B-banded model [30–32].(24)Ci,j=0,vid−vjd>B,cvid−vjd,vid−vjd≤B,where vid,vjd∈S and 1=c0>c1>⋯>cB>cB+1=0, c1=0.3ejπ/3,cl=c1e−jl−1π/8/l, for l∈2,B, B = 100 represents the maximum spacing of sensor pairs with mutual coupling. Besides, for a given array, the total strength of the mutual coupling effect can be measured by coupling leakage as [22, 28](25)LM=C−diagCFCF.Based on the mutual coupling model in equation (24), the received signal from the virtual array in equation (21) can be reconstructed as(26)z˜=CvecΛθp,where Cvec=C⊗C∗∗⊗C⊗C∗.


## 2.1. The Definitions of 2-DC, 2-SC, 4-DC, and 2-DCSC

Consider aP -sensors linear array, whose locations set can be indicated as [22](1)S=v1⋅d,v2⋅d,…,vP⋅d,where vp⋅d represents the position of the p−th sensor, p=1,2,…,P, and d is the unit spacing.Definition 1.
For the array with the set of sensor positionS in equation (1), the 2-DC set is defined as [17](2)C2−DC=Φ2−DCd,where the set of 2-DC lags(3)Φ2−DC=vp1−vp2,1≤p1,p2≤P.Definition 2.
The 2-SC set is defined as [21](4)C2−SC=Φ2−SCd,with the set of 2-SC lags(5)Φ2−SC=vp1+vp2,1≤p1,p2≤P.Definition 3.
The 4-DC set is defined as [16](6)C4−DC=Φ4−DCd,where the set of 4-DC lags(7)Φ4−DC=vp1−vp2−vp3−vp4,1≤p1,p2,p3,p4≤P.Definition 4.
The 2-DCSC set is defined as [17](8)C2−DCSC=Φ2−DCSCd,where the set of 2-DCSC lags(9)Φ2−DCSC=vp1+vp2−vp3+vp4,1≤p1,p2,p3,p4≤P.
By permutation invariance,Φ2−DCSC can be rewritten as(10)Φ2−DCSC=vp1−vp2−vp3−vp4=vp1+vp4−vp2+vp3=Φ4−DC.
for1≤p1,p2,p3,p4≤P. That is, for a particular array, the 4-DC is equivalent to the 2-DCSC(11)C4−DC=C2−DCSC.


## 2.2. The Properties of TL-NA

The configuration of TL-NA [4] has been shown in Figure 1, which contains two sparse uniform subarrays. The first subarray has M1 sensors with interelement spacing d, where d=λ/2 and λ is the wavelength, while the another subarray has M2 sensors with interelement spacing M1+1d. The position of TL-NA varies from 1d to M2M1+1d and it can be represented as(12)S=m1d,1≤m1≤M1∪m2M1+1d,1≤m2≤M2.Figure 1
Two-level nested array.Refer to Definition1, the 2-DC location set of TL-NA can be written as(13)C2−DC=−M2M1+1+1,…,−1,0,1,…,M2M1+1−1d.Refer to Definitions3 and4, the 4-DC (2-DCSC) location set of TL-NA can be expressed as [23](14)C4−DC=C2−DCSC=−2M2M1+1+2,…,−1,0,1,…,2M2M1+1−2d.From equations (13) and (14), It can be concluded that the position of the 2-DC and 2-DCSC of TL-NA are consecutive.


## 2.3. Data Model Based on FOC

AssumeK far-field narrowband uncorrelated sources impinging upon the array with locations set S from directions θ=θ1,…,θK, the received data model of the array can be expressed as [24, 25](15)xt=Ast+nt,where Aθ=aθ1,aθ2,…,aθK∈ℂP×K represents the steering matrix of array, and the steering vector at direction θk, which denotes the elevation angle of the k−th target, is given by(16)aθk=e−j2πv1dsinθk/λ,e−j2πv2dsinθk/λ,…,e−j2πvPdsinθk/λT∈ℂP×1.andst=s1t,s2t,…,sKtT∈ℂK×1, 1≤t≤L denotes non-Gaussian signal source matrix with zero mean, where L indicates the number of snapshots. And, nt is the received additive white Gaussian noise with mean zero and variance σ2.In this paper, for givenxk1,xk2,xk3∗,xk4∗, the FOC definition can be written as [26](17)C4=Cumxk1,xk2,xk3∗,xk4∗=Exk1xk2xk3∗xk4∗−Exk1xk2Exk3∗xk4∗−Exk1xk3∗Exk2xk4∗−Exk1xk4∗Exk2xk3∗.For the array, the FOC of the received signal can be [26](18)C4,X=CumX,X,X∗,X∗=EX⊗X∗X⊗X∗H−EX⊗X∗EX⊗X∗H−EXXH⊗EXXH∗.The FOC matrix of the received signal is expressed by [26](19)R4=C4,X=BθCSBHθ,where(20)Bθ=bθ1,bθ2,…,bθK=aθ1⊗a∗θ1,aθ2⊗a∗θ2,…,aθK⊗a∗θK=Aθ⊙A∗θ,CS=ES⊗SS⊗SH−ES⊗SES⊗SH−ESSH⊗ESSH∗.Bθ is the steering matrix after array manifold Aθ⇒Bθ by using the FOC. To further improve the performance of DOA estimation, we use a vectorized method [23, 26, 27] for R4 to obtain a vector, which is equivalent to the received signal from the virtual ULA.(21)z=vecR4=vecBθCsBHθ=Λθp,where(22)Λθ=b∗θ1⊗bθ1,b∗θ2⊗bθ2,…,b∗θK⊗bθK=B∗θ⊙Bθ,and p=c4,s1,c4,s2,…,c4,sKT represents the FOC matrix of source vector, and c4,sk represents the FOC of the k−th source vector.


## 2.4. Data Model with Mutual Coupling Based on FOC

Equation (16) assumes that the sensors in the array has not interference with each other. Actually, the output of each sensor is influenced by its adjacent elements owing to the presence of mutual coupling. Therefore, by introducing a mutual coupling matrix C, the received data model can be rewritten as follows [28, 29].(23)xt=CAst+nt,where the mutual coupling C can be obtained from reference [28], whose define is determined by varieties of factors involving in the distance between sensors, the operating frequency, e.g., and it can be approximated by employing the B-banded model [30–32].(24)Ci,j=0,vid−vjd>B,cvid−vjd,vid−vjd≤B,where vid,vjd∈S and 1=c0>c1>⋯>cB>cB+1=0, c1=0.3ejπ/3,cl=c1e−jl−1π/8/l, for l∈2,B, B = 100 represents the maximum spacing of sensor pairs with mutual coupling. Besides, for a given array, the total strength of the mutual coupling effect can be measured by coupling leakage as [22, 28](25)LM=C−diagCFCF.Based on the mutual coupling model in equation (24), the received signal from the virtual array in equation (21) can be reconstructed as(26)z˜=CvecΛθp,where Cvec=C⊗C∗∗⊗C⊗C∗.


## 3. The Proposed Array

In this section, the proposed array is specifically introduced, including the position of physical sensors, the derivation of the consecutive elements part in the 2-SC and the 2-DCSC, and the properties of it. Next, we analyze the 2-SC and the 2-DCSC of the array for given parameters, and reveal the influence of different array configurations on the consecutive DOF of the array with the same total number of physical sensors.


### 3.1. Array Structure

Motivation from larger consecutive DOF and array aperture, the THL-NA is proposed, which has a concrete closed expression form with consecutive DOF. The 2-SC of the proposed array is employed to enlarge the array aperture, and then a consecutive virtual array with large array aperture is obtained by performing the difference operation on the 2-SC, whose consecutive character utilizes the nested relationship between subarrays in the 2-SC. Next, the position of physical sensors in the proposed array is given.Proposition 1.
The THL-NA is composed of three ULAs with a dense ULA and two sparse ULAs, which is shown in Figure2. The position of THL-NA sensors can be represented as(27)S1=n1d,1≤n1≤N1,S2=n2N1+1d,1≤n2≤N2,S3=n3N2+1N1+1d,1≤n3≤N3,STHL−NA=S1∪S2∪S3.Figure2 shows the distribution position of physical sensors in the proposed array. The array consists of three ULAs. The number of sensors is N1 with interelement spacing d starting from 1d in the first subarray, the number of sensors is N2 with interelement spacing N1+1d starting from N1+1d in the second subarray and the number of sensors is N3 with interelement spacing N2+1N1+1d starting from N2+1N1+1d in the third subarray. The total number of physical sensors is N and N=N1+N2+N3.Figure 2
Three-level nested array structure.


### 3.2. The Deriving of Closed-Form Expression with Consecutive Elements

The 2-SC of the proposed array ranges from2d to 2N3N2+1N1+1d, which is not a virtual consecutive array. It is worth noting that there are N3 TL-NAs and a THL-NA with the same array structure in the 2-SC. In fact, the 2-SC can be regarded as the panning of a certain position of the proposed array. For instance, the THL-NA at the last part of the 2-SC can be regarded as the number of elements position in the physical array plus the largest number of elements position, which can be understood as the panning of the initial physical array.Proposition 2.
The 2-SC can be divided intoN3+1 parts. The 2-SC position of THL-NA can be represented as(28)S1=n1d,2≤n1≤N1+1N2+1,S2=N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1,S3=2N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1,⋮SN3+1=N3N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2∪n3N2+1N1+1d,1≤n3≤N3,S2−SC=S1∪S2∪⋯∪SN3+1.According to the consecutive character of the 2-DC of the TL-NA, it can be seen that a virtual consecutive 2-DCSC is obtained by performing another difference operation on the 2-SC.Proposition 3.
The 2-DCSC position of THL-NA can be represented as(29)S2−DCSC=−2N3N2+1N1+1+2,…,−1,0,1,…,2N3N2+1N1+1−2.
Summarize the closed-form expression with the number of the consecutive lags of the array structure based on FOC as follows:(30)cDOF=22N3N2+1N1+1−2+1=4N3N2+1N1+1−3.
Next, a specific array configuration example is used to deepen understanding. Figure3 shows when the array configuration satisfies N1=3,N2=3,N3=4, the position of physical sensors in the proposed array, the position of array elements in the 2-SC and 2-DCSC. Next, the specific example is used to analyze and discuss the structure and properties of the proposed array.
The location of the physical sensors is distributed in0.5,1,1.5,2,4,6,8,16,24,32, The 2-SC of the array ranges from 2d to 2N3N2+1N1+1d namely 1 to 64 with d=0.5. According to the above analysis, the 2-SC is divided into N3+1=5 parts, and the position of coarray elements in each part can be expressed as(31)S1=n1d,2≤n1≤N1+1N2+1=1,1.5,…,8,S2=N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1=8+0.5,1,1.5,2,4,6,8,S3=2N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1=16+0.5,1,1.5,2,4,6,8,S4=3N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1=24+0.5,1,1.5,2,4,6,8,S5=4N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2∪n3N2+1N1+1d,1≤n3≤N3=32+0.5,1,1.5,2,4,6,8,16,24,32,S2−SC=S1∪S2∪⋯∪S5.There is nested relationship among subarrays circled in the black box in Figure3. The elements in the position set S2,S3,S4 construct NA structure respectively, which can be regarded as TL-NA. The last part of the 2-SC is structurally equivalent to the proposed array, which can be regarded as a THL-NA. The consecutive 2-DCSC is obtained by performing a difference operation on the 2-SC. Since the 2-DC of the TL-NA is consecutive, the 2-DC of the sets S2,S3,S4 is equivalent to the 2-DCSC of the proposed array, which is a consecutive virtual array. From Figure 3, it can be observed that the longest length of the consecutive holes is 8, and the longest length of the consecutive elements is 9. By performing the difference operation on set S5, a consecutive array is obtained. As a result, a consecutive virtual array can be obtained through the above array processing. The length of the consecutive DOF is cDOF=4N3N2+1N1+1−3=253.Figure4 shows when the array configuration satisfies N1=3,N2=3,N3=3, the position of physical sensors in the proposed array, the position of array elements in the 2-SC and 2-DCSC. According to the above analysis, the consecutive DOF is cDOF=4N3N2+1N1+1−3=189.Figure 3
The location of physical sensors, 2-SC, 2-DCSC whenN1=3,N2=3,N3=4.Figure 4
The location of physical sensors, 2-SC, 2-DCSC whenN1=3,N2=3,N3=3.


### 3.3. Optimal Array Configuration

For a fixed total number of physical sensors, there are multiple schemes about allocating sensors of each subarray in THL-NA. The problem of determining the optimal array configuration is addressed to maximize the consecutive DOF in this section. According to Proposition3, the length of consecutive virtual elements in THL-NA is 4N3N2+1N1+1−3. The consecutive DOF optimization problem can be formulated as(32)max,4N3N2N1+4N3N2+4N3N1+4N3−3,subject to,N3+N2+N1=N.In equation (32), we focus on how to configure the number of sensors of each subarray N1,N2,N3, so that the possible largest length of consecutive lags is obtained from the virtual ULA when the total number of array sensors N is certain. We define integers m and n to be the remainder and quotient of N modulo 3, where N=3n+m,0≤m≤2.Proposition 4.
One solution to the optimization problem in equation (32) is can be presented as(33)N3=N2=N1=n,ifm=0,N3=n+1,N2=N1=n,ifm=1,N3=N2=n+1,N1=n,ifm=2.According to equation (33), the corresponding consecutive DOF can be evinced as(34)c−DOF=4nn+1n+1−3,ifm=0,4n+1n+1n+1−3,ifm=1,4n+1n+2n+1−3,ifm=2,where n=N/3.Table1 shows the consecutive degree of freedom corresponding to different array configurations when the number of physical sensors is 9. Proposition 4 can be verified with specific examples in Table 1.Table 1
C-DOF of different array configurations.
N1,N2,N33, 3, 32, 3, 43, 2, 42, 4, 34, 2, 34, 3, 23, 4, 22, 2, 55, 2, 22, 5, 2c-DOF189189189177177157157177141141


## 3.1. Array Structure

Motivation from larger consecutive DOF and array aperture, the THL-NA is proposed, which has a concrete closed expression form with consecutive DOF. The 2-SC of the proposed array is employed to enlarge the array aperture, and then a consecutive virtual array with large array aperture is obtained by performing the difference operation on the 2-SC, whose consecutive character utilizes the nested relationship between subarrays in the 2-SC. Next, the position of physical sensors in the proposed array is given.Proposition 1.
The THL-NA is composed of three ULAs with a dense ULA and two sparse ULAs, which is shown in Figure2. The position of THL-NA sensors can be represented as(27)S1=n1d,1≤n1≤N1,S2=n2N1+1d,1≤n2≤N2,S3=n3N2+1N1+1d,1≤n3≤N3,STHL−NA=S1∪S2∪S3.Figure2 shows the distribution position of physical sensors in the proposed array. The array consists of three ULAs. The number of sensors is N1 with interelement spacing d starting from 1d in the first subarray, the number of sensors is N2 with interelement spacing N1+1d starting from N1+1d in the second subarray and the number of sensors is N3 with interelement spacing N2+1N1+1d starting from N2+1N1+1d in the third subarray. The total number of physical sensors is N and N=N1+N2+N3.Figure 2
Three-level nested array structure.


## 3.2. The Deriving of Closed-Form Expression with Consecutive Elements

The 2-SC of the proposed array ranges from2d to 2N3N2+1N1+1d, which is not a virtual consecutive array. It is worth noting that there are N3 TL-NAs and a THL-NA with the same array structure in the 2-SC. In fact, the 2-SC can be regarded as the panning of a certain position of the proposed array. For instance, the THL-NA at the last part of the 2-SC can be regarded as the number of elements position in the physical array plus the largest number of elements position, which can be understood as the panning of the initial physical array.Proposition 2.
The 2-SC can be divided intoN3+1 parts. The 2-SC position of THL-NA can be represented as(28)S1=n1d,2≤n1≤N1+1N2+1,S2=N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1,S3=2N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1,⋮SN3+1=N3N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2∪n3N2+1N1+1d,1≤n3≤N3,S2−SC=S1∪S2∪⋯∪SN3+1.According to the consecutive character of the 2-DC of the TL-NA, it can be seen that a virtual consecutive 2-DCSC is obtained by performing another difference operation on the 2-SC.Proposition 3.
The 2-DCSC position of THL-NA can be represented as(29)S2−DCSC=−2N3N2+1N1+1+2,…,−1,0,1,…,2N3N2+1N1+1−2.
Summarize the closed-form expression with the number of the consecutive lags of the array structure based on FOC as follows:(30)cDOF=22N3N2+1N1+1−2+1=4N3N2+1N1+1−3.
Next, a specific array configuration example is used to deepen understanding. Figure3 shows when the array configuration satisfies N1=3,N2=3,N3=4, the position of physical sensors in the proposed array, the position of array elements in the 2-SC and 2-DCSC. Next, the specific example is used to analyze and discuss the structure and properties of the proposed array.
The location of the physical sensors is distributed in0.5,1,1.5,2,4,6,8,16,24,32, The 2-SC of the array ranges from 2d to 2N3N2+1N1+1d namely 1 to 64 with d=0.5. According to the above analysis, the 2-SC is divided into N3+1=5 parts, and the position of coarray elements in each part can be expressed as(31)S1=n1d,2≤n1≤N1+1N2+1=1,1.5,…,8,S2=N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1=8+0.5,1,1.5,2,4,6,8,S3=2N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1=16+0.5,1,1.5,2,4,6,8,S4=3N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2+1=24+0.5,1,1.5,2,4,6,8,S5=4N1+1N2+1d+n1d,1≤n1≤N1∪n2N1+1d,1≤n2≤N2∪n3N2+1N1+1d,1≤n3≤N3=32+0.5,1,1.5,2,4,6,8,16,24,32,S2−SC=S1∪S2∪⋯∪S5.There is nested relationship among subarrays circled in the black box in Figure3. The elements in the position set S2,S3,S4 construct NA structure respectively, which can be regarded as TL-NA. The last part of the 2-SC is structurally equivalent to the proposed array, which can be regarded as a THL-NA. The consecutive 2-DCSC is obtained by performing a difference operation on the 2-SC. Since the 2-DC of the TL-NA is consecutive, the 2-DC of the sets S2,S3,S4 is equivalent to the 2-DCSC of the proposed array, which is a consecutive virtual array. From Figure 3, it can be observed that the longest length of the consecutive holes is 8, and the longest length of the consecutive elements is 9. By performing the difference operation on set S5, a consecutive array is obtained. As a result, a consecutive virtual array can be obtained through the above array processing. The length of the consecutive DOF is cDOF=4N3N2+1N1+1−3=253.Figure4 shows when the array configuration satisfies N1=3,N2=3,N3=3, the position of physical sensors in the proposed array, the position of array elements in the 2-SC and 2-DCSC. According to the above analysis, the consecutive DOF is cDOF=4N3N2+1N1+1−3=189.Figure 3
The location of physical sensors, 2-SC, 2-DCSC whenN1=3,N2=3,N3=4.Figure 4
The location of physical sensors, 2-SC, 2-DCSC whenN1=3,N2=3,N3=3.


## 3.3. Optimal Array Configuration

For a fixed total number of physical sensors, there are multiple schemes about allocating sensors of each subarray in THL-NA. The problem of determining the optimal array configuration is addressed to maximize the consecutive DOF in this section. According to Proposition3, the length of consecutive virtual elements in THL-NA is 4N3N2+1N1+1−3. The consecutive DOF optimization problem can be formulated as(32)max,4N3N2N1+4N3N2+4N3N1+4N3−3,subject to,N3+N2+N1=N.In equation (32), we focus on how to configure the number of sensors of each subarray N1,N2,N3, so that the possible largest length of consecutive lags is obtained from the virtual ULA when the total number of array sensors N is certain. We define integers m and n to be the remainder and quotient of N modulo 3, where N=3n+m,0≤m≤2.Proposition 4.
One solution to the optimization problem in equation (32) is can be presented as(33)N3=N2=N1=n,ifm=0,N3=n+1,N2=N1=n,ifm=1,N3=N2=n+1,N1=n,ifm=2.According to equation (33), the corresponding consecutive DOF can be evinced as(34)c−DOF=4nn+1n+1−3,ifm=0,4n+1n+1n+1−3,ifm=1,4n+1n+2n+1−3,ifm=2,where n=N/3.Table1 shows the consecutive degree of freedom corresponding to different array configurations when the number of physical sensors is 9. Proposition 4 can be verified with specific examples in Table 1.Table 1
C-DOF of different array configurations.
N1,N2,N33, 3, 32, 3, 43, 2, 42, 4, 34, 2, 34, 3, 23, 4, 22, 2, 55, 2, 22, 5, 2c-DOF189189189177177157157177141141


## 4. The Proposed Algorithm

The computational complexity of the FOC matrix and that of the conventional MUSIC method due to the global spectrum searching are expensive, the successive SS-MUSIC algorithm is presented, which employs partial spatial spectrum peak searching to drop the complexity. The SS-ESPRIT algorithm for the consecutive 2-DCSC of the THL-NA is utilized to obtain the initial estimates, which can be used to shrink the searching range of MUSIC algorithm to obtain the fine estimates.


### 4.1. Initial Estimates

From the previous discussion, there is a consecutive virtual array of the THL-NA with the range of elements−Wd,Wd, where W=2N3N2+1N1+1−2, and the length of the virtual array is T=2W+1. Data model of the part refer to Section 2.3, the steering vector of the virtual array a∗θk∈CT×1 is denoted as(35)a∗θk=e−j2π−Wsinθk/λ,…,0,…,e−j2πWsinθk/λT.According to the position of the virtual array, remove the redundant part ofz and sort z, then construct z1.(36)z1=Λ1p,where Λ1∈ℂT×K is the direction matrix through sorting it after removing the redundant parts of Λθ, and it is mathematical form can be represented by(37)Λ1=e−j2π/λ−Wdsinθ1e−j2π/λ−Wdsinθ2⋯e−j2π/λ−WdsinθK⋮⋮⋱⋮1111⋮⋮⋱⋮e−j2π/λWdsinθ1e−j2π/λ−Wdsinθ2⋯e−j2π/λWdsinθK.Under the situation, the traditional DOA algorithms, which use the eigenvalue decomposition of covariance, is invalid because of the vectorized interrelated signal. The received signal matrix is updated by intercepting the consecutive part ofz1. After obtaining the consecutive vector z1, the spatial smoothing algorithm [19, 20], which divides the ULA into several overlapping subarrays, is exploited to take the place of the invalid traditional DOA algorithms. After obtaining the sum of the covariance matrices of the divided subarrays, which have the same array structure, the spatial smoothing covariance matrix is obtained by taking the average of it. As shown in Figure 5, W+1 overlapping subarrays are obtained by dividing equally the virtual array, and each subarray incorporates W+1 elements. Where, the sensors position of the i−th subarray is(38)−i+1+nd,n=0,1,…,W.Figure 5
The spatial smoothing algorithm.The received signal matrixz1i is from row W+2−i to row 2W+2−i of z1.Construct the covariance matrix:(39)Ri=z1iz1iH.To obtain the spatial smoothing covariance matrixR, which has the same form as the signal covariance matrix based on classical subspace algorithms, we sum the covariance matrices of all W+1 subarrays and calculate the mean of it as shown in(40)R=1W+1∑i=1W+1Ri.In the following, the results of DOA estimation are obtained by using the classic ESPRIT algorithm [33]. Perform eigenvalue decomposition on R to obtain the signal subspace UN and the noise subspace US.(41)R=US∑SUSH+UN∑NUNH.The rotation invariance of array makes the matrixUS decomposed into UX∈ℂW−1×K, UY∈ℂW−1×K,corresponding to two subarrays, namely,(42)US=UXUY=UXUXΨ,Ψ=T−1ΦT.The signal subspaceUX is similar to UY, and the diagonal elements of Φ are equal to the eigenvalues of Ψ, so by calculating the eigenvalues λk of Ψ=UX+UY, where k=1,2,…,K, the diagonal elements of Φ is estimated to obtain the angle estimation value:(43)θ^kini=arcsinangleλk2πd.


### 4.2. Fine Estimates

The initial estimated anglesθ^kini are employed to shrink the range of spectrum searching and the MUSIC algorithm is utilized to perform interval processing on the spatial spectrum function, where the interval range is expressed by θ^kini−Δ,θ^kinik+Δ (Δ is a small number), and then the more accurate angle estimation parameters are obtained.The spatial spectrum function of MUSIC algorithm can be constructed as(44)PMUSIC=1aHθUNUNHaθ,where aθ is the direction vector, and aθ=1,e−j2πsinθ/λ,…,e−j2πW−1sinθ/λT∈ℂW×1. While changing the value of θ, and performing the spectrum searching in the space domain. When the denominator of the spectral function tends to 0, in where the spatial spectral function reaches a peak, the noise vector is orthogonal to the signal vector, and the signal arrival angle θ^k is equivalent as θ at this time.


### 4.3. The Advantages of the Proposed Algorithm

The main advantages of the proposed algorithm can be summarized as follows:(1)
The computational complexity of the successive SS-MUSIC algorithm is much lower than SS-MUSIC algorithm(2)
The successive SS-MUSIC algorithm owns approximately the same DOA estimation performance as the SS-MUSIC algorithm(3)
The successive SS-MUSIC algorithm makes a balance between computational complexity and estimation accuracy


## 4.1. Initial Estimates

From the previous discussion, there is a consecutive virtual array of the THL-NA with the range of elements−Wd,Wd, where W=2N3N2+1N1+1−2, and the length of the virtual array is T=2W+1. Data model of the part refer to Section 2.3, the steering vector of the virtual array a∗θk∈CT×1 is denoted as(35)a∗θk=e−j2π−Wsinθk/λ,…,0,…,e−j2πWsinθk/λT.According to the position of the virtual array, remove the redundant part ofz and sort z, then construct z1.(36)z1=Λ1p,where Λ1∈ℂT×K is the direction matrix through sorting it after removing the redundant parts of Λθ, and it is mathematical form can be represented by(37)Λ1=e−j2π/λ−Wdsinθ1e−j2π/λ−Wdsinθ2⋯e−j2π/λ−WdsinθK⋮⋮⋱⋮1111⋮⋮⋱⋮e−j2π/λWdsinθ1e−j2π/λ−Wdsinθ2⋯e−j2π/λWdsinθK.Under the situation, the traditional DOA algorithms, which use the eigenvalue decomposition of covariance, is invalid because of the vectorized interrelated signal. The received signal matrix is updated by intercepting the consecutive part ofz1. After obtaining the consecutive vector z1, the spatial smoothing algorithm [19, 20], which divides the ULA into several overlapping subarrays, is exploited to take the place of the invalid traditional DOA algorithms. After obtaining the sum of the covariance matrices of the divided subarrays, which have the same array structure, the spatial smoothing covariance matrix is obtained by taking the average of it. As shown in Figure 5, W+1 overlapping subarrays are obtained by dividing equally the virtual array, and each subarray incorporates W+1 elements. Where, the sensors position of the i−th subarray is(38)−i+1+nd,n=0,1,…,W.Figure 5
The spatial smoothing algorithm.The received signal matrixz1i is from row W+2−i to row 2W+2−i of z1.Construct the covariance matrix:(39)Ri=z1iz1iH.To obtain the spatial smoothing covariance matrixR, which has the same form as the signal covariance matrix based on classical subspace algorithms, we sum the covariance matrices of all W+1 subarrays and calculate the mean of it as shown in(40)R=1W+1∑i=1W+1Ri.In the following, the results of DOA estimation are obtained by using the classic ESPRIT algorithm [33]. Perform eigenvalue decomposition on R to obtain the signal subspace UN and the noise subspace US.(41)R=US∑SUSH+UN∑NUNH.The rotation invariance of array makes the matrixUS decomposed into UX∈ℂW−1×K, UY∈ℂW−1×K,corresponding to two subarrays, namely,(42)US=UXUY=UXUXΨ,Ψ=T−1ΦT.The signal subspaceUX is similar to UY, and the diagonal elements of Φ are equal to the eigenvalues of Ψ, so by calculating the eigenvalues λk of Ψ=UX+UY, where k=1,2,…,K, the diagonal elements of Φ is estimated to obtain the angle estimation value:(43)θ^kini=arcsinangleλk2πd.


## 4.2. Fine Estimates

The initial estimated anglesθ^kini are employed to shrink the range of spectrum searching and the MUSIC algorithm is utilized to perform interval processing on the spatial spectrum function, where the interval range is expressed by θ^kini−Δ,θ^kinik+Δ (Δ is a small number), and then the more accurate angle estimation parameters are obtained.The spatial spectrum function of MUSIC algorithm can be constructed as(44)PMUSIC=1aHθUNUNHaθ,where aθ is the direction vector, and aθ=1,e−j2πsinθ/λ,…,e−j2πW−1sinθ/λT∈ℂW×1. While changing the value of θ, and performing the spectrum searching in the space domain. When the denominator of the spectral function tends to 0, in where the spatial spectral function reaches a peak, the noise vector is orthogonal to the signal vector, and the signal arrival angle θ^k is equivalent as θ at this time.


## 4.3. The Advantages of the Proposed Algorithm

The main advantages of the proposed algorithm can be summarized as follows:(1)
The computational complexity of the successive SS-MUSIC algorithm is much lower than SS-MUSIC algorithm(2)
The successive SS-MUSIC algorithm owns approximately the same DOA estimation performance as the SS-MUSIC algorithm(3)
The successive SS-MUSIC algorithm makes a balance between computational complexity and estimation accuracy


## 5. Performance Analysis

Aiming to evaluate the performance of the proposed array geometry and the proposed algorithm, the array performance index, including the consecutive DOF, the array aperture, the SS-MUSIC spectrum, the coupling leakage e.g., and the computational complexity are compared in this section.


### 5.1. Analysis of the Proposed Array

Direction of arrival In this part, we compare the consecutive DOF of ACA, SAFE-CPA, TL-NA, FL-NA, and THL-NA, as shown in Figure6. Figure 6 demonstrates that the THL-NA has a higher consecutive DOF than other arrays. It is worth noting that when the number of array elements exceeds 12, the consecutive DOF of the THL-NA begins to be lower than that of FL-NA. From the overall trend of the image, the consecutive DOF of the THL-NA is close to but still greater than SAFE-CPA, which always surpasses that of FL-NA and TL-NA.Figure 6
c-DOF comparison versus different arrays.Table2 lists the number of sensors, the consecutive DOF and the location of different arrays with 10 sensors. The length of array aperture in the proposed array is larger than that in TL-NA, ACA, which is smaller than that in FL-NA, SAFE-CPA.Table 2
Comparison of the closed-form expression with consecutive DOF.
Arrays structureNumber of sensors (Ni,i=1,2,…,4)Consecutive DOF (Ni,i=1,2,…,4)Location (T=10)TL-NAN1+N24N2N1+1−30.5,1,1.5,2,2.5,3,6,9,12,15FL-NA∑14Ni−32∏i=14Ni−10.5,1,1.5,2,4,6,12,18,36,54ACA2N1+N2−16N1N2+2N1−2N2−10,1.5,2.5,3,4.5,5,6,7.5,10,12.5SAFE-CPA2N1+N2−1+N32M0+10,1,1.5,2,3,4.5,17.5,30.5,43.5,56.5ProposedN1+N2+N34N3N2+1N1+1−30.5,1,1.5,2,4,6,8,16,24,32(45)M0=4N1N2N3+3N1N2+2N1N3−N2N3+N1−N2+N3−1.Table3 shows the position, consecutive DOF, the weight distribution diagrams of the 2-SC, 2-DCSC, the SS-MUSIC spatial spectrum and mutual coupling coefficients of each array with 10 sensors in ACA, TL-NA, FL-NA, SAFE-CPA, THL-NA, and the consecutive DOF of the 2-DCSC with the THL-NA, SAFE-CPA, CPA, FL-NA, TL-NA can reach 253, 241, 85, 215, 117 respectively. The incident angles are evenly distributed between −50 and 50, SNR=0dB, SNAPSHOTS as L=500.Table 3
The 2-SC, 2-DCSC, c-DOF, spectrum, andL(M) comparison of different arrays.
ACATL-NAFL-NASAFE-CPATHL-NAAntennas0,1.5,2.5,3,4.5,5,6,7.5,10,12.50.5,1,1.5,2,2.5,3,6,9,12,150.5,1,1.5,2,4,6,12,18,36,540,1,1.5,2,3,4.5,17.5,30.5,43.5,56.50.5,1,1.5,2,4,6,8,16,24,322-SC locations2-DCSC locationsSpectrumC-DOF85117215241253LM0.45820.63630.46860.44090.4728The comparison of the MUSIC spatial spectrum of the array structure proposed in this paper with SAFE-CPA, CPA, FL-NA, TL-NA at the same incident angles is shown as Table3. It can be seen that the incident angle information of all signals can be effectively detected only in the spectral peak diagram of the THL-NA, while the DOA estimation based on the other four arrays fail to find target estimation and search false peaks, which fully demonstrates that the THL-NA is capable of achieving higher accuracy DOA estimation.


### 5.2. Analysis of the Proposed Algorithm

For evaluating the proposed algorithm, the computational complexity is introduced into the analysis in this part, which is compared among the SS-MUSIC algorithm, the SS-ESPRIT method and the proposed algorithm. Specifically, the total number of sensors isN, the number of signal source is K, L indicates the number of snapshots, and the total number of the virtual array is T=2W+1, where W=2N3N2+1N1+1−2.Based on the number of complex multiplications, the main complexities of the methods involve that, the calculation of FOC matrix needsOLN4. The virtual array is divided into W+1 overlapping subarrays, and each subarray contains W+1 elements. After the operation of spatial smoothing, the covariance matrix calculation of each virtual subarray needs OW+13 and eigenvalue decomposition of the covariance matrix requires OW+13, calculating Ψ needs O2K3+3K2W and the eigenvalue decomposition of Ψ requires OK3. The complexity of spectral peak searching is On1W+12W−K+3, where the number of searching n1=2KΔ/0.01, where Δ is a very small number and 0.01 is the searching accuracy.Consequently, the total complexity of the proposed algorithm is given byOLN4+2W+13+3K2W+K+n1W+12W−K+3, the total complexity of SS-ESPRIT algorithm is OLN4+2W+13+3K2W+K, the total complexity of SS-MUSIC algorithm is OLN4+2W+13+nW+12W−K+3, where n=90/0.01 stands for the peak search times over angle domain.The computational complexity of algorithms is exhibited in Table4. The histogram and the line graph are utilized to display the comparison results between the complexity of the three algorithms, as shown in Figures 7 and 8 respectively, where K=2 and L=500 in comparison versus different elements and N1=3,N2=3,N3=4 in comparison versus different snapshots. It is observed that the complexity of the proposed algorithm is significantly lower than the SS-MUSIC method but is higher than the SS-ESPRIT algorithm.Table 4
Complexity of different algorithms.
AlgorithmComputational complexityProposedOLN4+2W+13+3K2W+K+n1W+12W−K+3SS-ESPRITOLN4+2W+13+3K2W+KSS-MUSICOLN4+2W+13+nW+12W−K+3Figure 7
Complexities of different methods versus different elements.Figure 8
Complexities of different methods versus different snapshots.


## 5.1. Analysis of the Proposed Array

Direction of arrival In this part, we compare the consecutive DOF of ACA, SAFE-CPA, TL-NA, FL-NA, and THL-NA, as shown in Figure6. Figure 6 demonstrates that the THL-NA has a higher consecutive DOF than other arrays. It is worth noting that when the number of array elements exceeds 12, the consecutive DOF of the THL-NA begins to be lower than that of FL-NA. From the overall trend of the image, the consecutive DOF of the THL-NA is close to but still greater than SAFE-CPA, which always surpasses that of FL-NA and TL-NA.Figure 6
c-DOF comparison versus different arrays.Table2 lists the number of sensors, the consecutive DOF and the location of different arrays with 10 sensors. The length of array aperture in the proposed array is larger than that in TL-NA, ACA, which is smaller than that in FL-NA, SAFE-CPA.Table 2
Comparison of the closed-form expression with consecutive DOF.
Arrays structureNumber of sensors (Ni,i=1,2,…,4)Consecutive DOF (Ni,i=1,2,…,4)Location (T=10)TL-NAN1+N24N2N1+1−30.5,1,1.5,2,2.5,3,6,9,12,15FL-NA∑14Ni−32∏i=14Ni−10.5,1,1.5,2,4,6,12,18,36,54ACA2N1+N2−16N1N2+2N1−2N2−10,1.5,2.5,3,4.5,5,6,7.5,10,12.5SAFE-CPA2N1+N2−1+N32M0+10,1,1.5,2,3,4.5,17.5,30.5,43.5,56.5ProposedN1+N2+N34N3N2+1N1+1−30.5,1,1.5,2,4,6,8,16,24,32(45)M0=4N1N2N3+3N1N2+2N1N3−N2N3+N1−N2+N3−1.Table3 shows the position, consecutive DOF, the weight distribution diagrams of the 2-SC, 2-DCSC, the SS-MUSIC spatial spectrum and mutual coupling coefficients of each array with 10 sensors in ACA, TL-NA, FL-NA, SAFE-CPA, THL-NA, and the consecutive DOF of the 2-DCSC with the THL-NA, SAFE-CPA, CPA, FL-NA, TL-NA can reach 253, 241, 85, 215, 117 respectively. The incident angles are evenly distributed between −50 and 50, SNR=0dB, SNAPSHOTS as L=500.Table 3
The 2-SC, 2-DCSC, c-DOF, spectrum, andL(M) comparison of different arrays.
ACATL-NAFL-NASAFE-CPATHL-NAAntennas0,1.5,2.5,3,4.5,5,6,7.5,10,12.50.5,1,1.5,2,2.5,3,6,9,12,150.5,1,1.5,2,4,6,12,18,36,540,1,1.5,2,3,4.5,17.5,30.5,43.5,56.50.5,1,1.5,2,4,6,8,16,24,322-SC locations2-DCSC locationsSpectrumC-DOF85117215241253LM0.45820.63630.46860.44090.4728The comparison of the MUSIC spatial spectrum of the array structure proposed in this paper with SAFE-CPA, CPA, FL-NA, TL-NA at the same incident angles is shown as Table3. It can be seen that the incident angle information of all signals can be effectively detected only in the spectral peak diagram of the THL-NA, while the DOA estimation based on the other four arrays fail to find target estimation and search false peaks, which fully demonstrates that the THL-NA is capable of achieving higher accuracy DOA estimation.


## 5.2. Analysis of the Proposed Algorithm

For evaluating the proposed algorithm, the computational complexity is introduced into the analysis in this part, which is compared among the SS-MUSIC algorithm, the SS-ESPRIT method and the proposed algorithm. Specifically, the total number of sensors isN, the number of signal source is K, L indicates the number of snapshots, and the total number of the virtual array is T=2W+1, where W=2N3N2+1N1+1−2.Based on the number of complex multiplications, the main complexities of the methods involve that, the calculation of FOC matrix needsOLN4. The virtual array is divided into W+1 overlapping subarrays, and each subarray contains W+1 elements. After the operation of spatial smoothing, the covariance matrix calculation of each virtual subarray needs OW+13 and eigenvalue decomposition of the covariance matrix requires OW+13, calculating Ψ needs O2K3+3K2W and the eigenvalue decomposition of Ψ requires OK3. The complexity of spectral peak searching is On1W+12W−K+3, where the number of searching n1=2KΔ/0.01, where Δ is a very small number and 0.01 is the searching accuracy.Consequently, the total complexity of the proposed algorithm is given byOLN4+2W+13+3K2W+K+n1W+12W−K+3, the total complexity of SS-ESPRIT algorithm is OLN4+2W+13+3K2W+K, the total complexity of SS-MUSIC algorithm is OLN4+2W+13+nW+12W−K+3, where n=90/0.01 stands for the peak search times over angle domain.The computational complexity of algorithms is exhibited in Table4. The histogram and the line graph are utilized to display the comparison results between the complexity of the three algorithms, as shown in Figures 7 and 8 respectively, where K=2 and L=500 in comparison versus different elements and N1=3,N2=3,N3=4 in comparison versus different snapshots. It is observed that the complexity of the proposed algorithm is significantly lower than the SS-MUSIC method but is higher than the SS-ESPRIT algorithm.Table 4
Complexity of different algorithms.
AlgorithmComputational complexityProposedOLN4+2W+13+3K2W+K+n1W+12W−K+3SS-ESPRITOLN4+2W+13+3K2W+KSS-MUSICOLN4+2W+13+nW+12W−K+3Figure 7
Complexities of different methods versus different elements.Figure 8
Complexities of different methods versus different snapshots.


## 6. RMSE Results

In this section, the root mean square error (RMSE) via 200 Monte-Carlo trials are utilized to validate the superior performance of the proposed array and the proposed algorithm by comparing with four arrays and two algorithms, respectively. The RMSE is defined by [34, 35](46)RMSE=1K∑k=1K1200∑l=1200θ^k,l−θk2,where θk denotes the true elevation of the k−th target, θ^k,l is estimated value of θk in the l−thl=1,…,200 Monte-Carlo simulation. The non-Gaussian sources with θ=5∘,45∘ incident on the proposed array, SAFE-CPA, ACA, FL-NA, TL-NA, the total number of physical sensors is set as 10, and the consecutive DOF of the virtual coarray with the proposed array, SAFE-CPA, ACA, FL-NA, and TL-NA can reach 253, 241, 85, 215, and 117, respectively.Sections6.1–6.3 all employ the proposed successive SS-MUSIC algorithm to conduct the simulation experiments. Furthermore, Section 6.4 performs the comparison of different algorithms based on the proposed array geometry.


### 6.1. RMSE Comparison versus Snapshots

Figure9 substantiates the effectiveness of the proposed array in estimated accuracy, and manifests that the increase of sampled data leads to the improvement of estimation performance, owing to the more accurate covariance estimation.Figure 9
RMSE performance of different snapshots versus SNR.


### 6.2. RMSE Comparison of Different Array Configurations

In this simulation, we compare the RMSE results of different array configurations, involvingN1=3,N2=3,N3=3, N1=3,N2=3,N3=4 and N1=3,N2=3,N3=5, using the successive SS-MUSIC algorithm, where SNAPSHOTS = 500. Meanwhile, the RMSE results of each algorithm versus the number of snapshots are exhibited in Figure 10. As depicted in Figure 10, with the number of elements increasing, the parameter estimation performance is enhanced because of diversity gain.Figure 10
RMSE performance of different array configurations versus SNR.


### 6.3. RMSE Performance Comparison of Different Arrays

To specifically examine the DOA estimation ability of the proposed array, the experiment uses 200 independent Monte Carlo experiments to statistically compares the RMSE of the array structure proposed in this paper with SAFE-CPA, ACA, FL-NA, TL-NA at different signal-to-noise ratios and snapshots, respectively. The number of physical sensors is 10. It is obviously seen that the performance of the proposed array is better than others.Figure11 presents the RMSE comparison of DOA estimation in different arrays versus SNR, where SNR is from −15 dB to 5 dB and SNAPSHOTS = 500. With the increase of SNR, the THL-NA can reach the best DOA estimation performance compared to SAFE-CPA, ACA, FL-NA, and TL-NA.Figure 11
RMSE performance of different arrays versus SNR.Figure12 presents the RMSE comparison of DOA estimation in different arrays versus SNAPSHOTS, where SNAPSHOTS is from 200 to 1000 and SNR = −5 dB. With the increase of the SNAPSHOTS, the THL-NA can reach the best DOA estimation performance compared to SAFE-CPA, ACA, FL-NA, and TL-NA due to the largest consecutive DOF.Figure 12
RMSE performance of different arrays versus SNAPSHOTS.


### 6.4. RMSE Comparison of Different Algorithms

Figure13 gives the comparison of estimation performance with different algorithms versus SNR, where SNAPSHOTS = 500. In addition, the searching range of the successive SS-MUSIC algorithm and the SS-MUSIC method is θkini−1∘,θkini+1∘,k=1,…K and −60∘,60∘, where θkini represents the initial DOA estimates according to Section 4.3. In terms of estimated performance according to following image, the successive SS-MUSIC algorithm is close to SS-MUSIC method and far better than SS-ESPRIT algorithm.Figure 13
RMSE performance of different algorithms.Figure14 gives the comparison of estimation performance with different algorithms versus snapshots, where SNR = 0. It can be concluded from Figures 12 and 13 that the DOA estimation performance of the successive SS-MUSIC algorithm is more accurate than the SS-ESPRIT algorithm and slightly worse than the SS-MUSIC algorithm.Figure 14
RMSE performance of different algorithms.


## 6.1. RMSE Comparison versus Snapshots

Figure9 substantiates the effectiveness of the proposed array in estimated accuracy, and manifests that the increase of sampled data leads to the improvement of estimation performance, owing to the more accurate covariance estimation.Figure 9
RMSE performance of different snapshots versus SNR.


## 6.2. RMSE Comparison of Different Array Configurations

In this simulation, we compare the RMSE results of different array configurations, involvingN1=3,N2=3,N3=3, N1=3,N2=3,N3=4 and N1=3,N2=3,N3=5, using the successive SS-MUSIC algorithm, where SNAPSHOTS = 500. Meanwhile, the RMSE results of each algorithm versus the number of snapshots are exhibited in Figure 10. As depicted in Figure 10, with the number of elements increasing, the parameter estimation performance is enhanced because of diversity gain.Figure 10
RMSE performance of different array configurations versus SNR.


## 6.3. RMSE Performance Comparison of Different Arrays

To specifically examine the DOA estimation ability of the proposed array, the experiment uses 200 independent Monte Carlo experiments to statistically compares the RMSE of the array structure proposed in this paper with SAFE-CPA, ACA, FL-NA, TL-NA at different signal-to-noise ratios and snapshots, respectively. The number of physical sensors is 10. It is obviously seen that the performance of the proposed array is better than others.Figure11 presents the RMSE comparison of DOA estimation in different arrays versus SNR, where SNR is from −15 dB to 5 dB and SNAPSHOTS = 500. With the increase of SNR, the THL-NA can reach the best DOA estimation performance compared to SAFE-CPA, ACA, FL-NA, and TL-NA.Figure 11
RMSE performance of different arrays versus SNR.Figure12 presents the RMSE comparison of DOA estimation in different arrays versus SNAPSHOTS, where SNAPSHOTS is from 200 to 1000 and SNR = −5 dB. With the increase of the SNAPSHOTS, the THL-NA can reach the best DOA estimation performance compared to SAFE-CPA, ACA, FL-NA, and TL-NA due to the largest consecutive DOF.Figure 12
RMSE performance of different arrays versus SNAPSHOTS.


## 6.4. RMSE Comparison of Different Algorithms

Figure13 gives the comparison of estimation performance with different algorithms versus SNR, where SNAPSHOTS = 500. In addition, the searching range of the successive SS-MUSIC algorithm and the SS-MUSIC method is θkini−1∘,θkini+1∘,k=1,…K and −60∘,60∘, where θkini represents the initial DOA estimates according to Section 4.3. In terms of estimated performance according to following image, the successive SS-MUSIC algorithm is close to SS-MUSIC method and far better than SS-ESPRIT algorithm.Figure 13
RMSE performance of different algorithms.Figure14 gives the comparison of estimation performance with different algorithms versus snapshots, where SNR = 0. It can be concluded from Figures 12 and 13 that the DOA estimation performance of the successive SS-MUSIC algorithm is more accurate than the SS-ESPRIT algorithm and slightly worse than the SS-MUSIC algorithm.Figure 14
RMSE performance of different algorithms.


## 7. Conclusion

In this paper, the THL-NA is proposed for DOA estimation of non-Gaussian signals based on FOC, which can provide large consecutive DOF and a specific closed-form expression. The proposed array utilizes the characteristic of consecutive coarrays in TL-NA. Firstly, the array aperture is enlarged by obtaining the 2-SC of the proposed array, and then the nested relationship between subarrays in the 2-SC is employed to obtain 2-DCSC to construct a consecutive virtual array. The simulation results prove that the proposed array has higher performance than SAFE-CPA, CPA, FL-NA, TL-NA in DOA estimation. Besides, the successive SS-MUSIC method proposed in the paper has lower computational complexity than SS-MUSIC algorithm and more accurate estimate than SS- ESPRIT algorithm. In future research, sparse array design under the incidence of non-Gaussian sources based on MIMO system [36, 37] may be considered.



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*Source: 9604664-2022-04-07.xml*