# A Novel Designed Sparse Array for Noncircular Sources with High Degree of Freedom

Authors: Yan-kui Zhang; Hai-yun Xu; Da-ming Wang; Bin Ba; Si-yao Li
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

The existing coprime array is mainly applicable to circular sources, while the virtual array degree of freedom (DOF) for noncircular sources is enhanced limitedly. In order to perfect the array DOF and the direction of arrival (DOA) estimation accuracy, a high degree of freedom sparse array design method for noncircular sources is put forward. Firstly, the method takes the advantages of the characteristic of the noncircular sources to expand the array manifold and then explores and solves the location distribution of the physical array sensors on the basis of the virtual array model with the help of the searching approach. The array configuration can obtain the longest continuous virtual array. The comparisons between the proposed array configuration and the common array configurations are advanced. The simulation experiments show that the sparse array presented in this paper can effectively increase the continuous virtual array aperture of noncircular sources, improve the array DOF and DOA estimation accuracy, and achieve the purpose of better estimation of multiple DOAs in underdetermined conditions.


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## Body

## 1. Introduction

Direction of arrival (DOA) estimation is one of the main research topics in array signal processing. It is widely used in radar, sonar, radio astronomy, and so on [1, 2]. The traditional DOA estimation algorithms are mainly based on the uniform array, and the M array sensors can achieve at most M-1 effective estimation of the azimuths [3, 4]. The number of target sources needed to be located is sometimes greater than the number of array sensors, that is, underdetermined condition. In order to improve the array degree of freedom (DOF) and realize the effective estimation of multiple sources in undetermined condition, in recent years, many sparse array structures have been proposed by experts and scholars. With the new sparse arrays, the apertures of arrays can be effectively extended, and the array DOFs and estimation accuracy could be improved [5, 6].Typical sparse arrays include the minimum redundancy array (MRA), the minimum hole array (MHA), the nested array, and the coprime array [7]. MRAs and MHAs are the ideal array configurations for circular sources; the optimal virtual array can be achieved [8, 9]. The nested array consists of a nonsparse uniform array and a sparse uniform array, which ensures a higher array DOF and continuous virtual array [10, 11]. The coprime array is composed of two uniform sparse arrays with larger array sensor spacing and a high array DOF, and the nested array is a special type of coprime array [12, 13]. The general structure of the coprime array and the solutions to basic DOA estimation are given in [14, 15]. The authors of [16, 17] propose improved coprime arrays and use the spatial smoothing algorithm for the virtual array to improve the estimation precision. In [18, 19], a coprime array with displaced subarrays (CADiS) structure is exploited. This coprime array configuration greatly expands the array aperture, improves the array DOF, and increases estimation accuracy. On the basis of the CADiS configuration, [20] summarizes the general configuration of coprime array with multiperiod subarrays (CAMpS) and proves that the CADiS is a special form of CAMpS. A kind of shifted coprime array (SCA) is proposed in [21], which further improves the virtual array aperture of CADiS configuration. The sparse array configurations mentioned above are mainly focused on the DOA estimation of circular sources, and there is no open research on the sparse configurations designed for noncircular sources.A noncircular source is a special transmission signal. Compared with a circular source, a noncircular source has more useful information. One of the most important features is that the ellipse covariance of noncircular sources is not zero. Therefore, this characteristic can be used to improve the array DOF and the DOA estimation accuracy. A DOA estimation method based on symmetric shift invariance array for noncircular sources is presented in [22], which enhances the array DOF by using the noncircular characteristic. Some DOA estimation methods for noncircular sources are listed in document [23–27]. These methods extend the array aperture by using the noncircular characteristics of sources and advance the array DOF and estimation accuracy.The DOA estimation methods mentioned above for the noncircular sources are all based on the array configuration of circular sources, and the array DOF and estimation accuracy are improved by using the characteristic that the ellipse covariance is not zero, but the use of this characteristic is not sufficient. In order to further increase the array DOF of noncircular sources and improve the accuracy of DOA estimation, a novel high DOF sparse array design approach for noncircular sources is presented in this paper. In this configuration, the virtual array model is constructed by using the characteristics of noncircular sources. Then the maximum continuous virtual array structure of noncircular sources is given by search method on the basis of difference. At last, the typical DOA estimation methods under this model are given. The validity of the method is proven by simulation experiments.The main contributions of this paper are as follows.( 1 ) A novel designed sparse array for noncircular sources (SANC) with high DOF is proposed, which effectively extends the continuous virtual array aperture and improves the DOA estimation accuracy.( 2 ) The design method and flow chart of the SANC configuration are given in this paper. The comparison between the array configuration proposed and the common array configurations is given, and typical DOA estimation methods under this array configurations are introduced.( 3 ) A virtual array model is constructed, and the Cramer-Rao lower bound (CRLB) is derived under the proposed model, effectively proving that the proposed array configuration not only can realize the effective estimation in overdetermined conditions but also is suitable for underdetermined conditions.The remainder of this paper is arranged as follows. Section2 introduces the typical coprime array model. Section 3 describes the SANC configuration. The typical DOA estimation methods under this array configuration are given in Section 4. The performance analysis and simulation experiments are presented in Section 5. Section 6 is the conclusion of this paper.


## 2. Coprime Array Model

### 2.1. Notation Conventions

Notations used in this paper are as follows:(i)
IN represents N-dimensional unit array.(ii)
(·)∗, (·)T, and (·)H, respectively, represent conjugate, transposition, and conjugate transpose.(iii)
·m stands for the mth element of this vector.(iv)
⊙ represents Hadamard product.(v)
⋃ stands for the intersection.(vi)
vec· is the operation of “vectorization,” which means turning a matrix into a vector by columns.(vii)
⊗ is the Kronecker product.(viii)
E(·) denotes the mathematical expectation.(ix)
·0, ·1, and ·22, respectively, represent the 0-norm, 1-norm, and 2-norm.(x)
Re· and Im· denotes the real part and the imaginary part.


### 2.2. Coprime Array Model

We assume that there areD stationary narrowband targets transmitting plane waves to the measuring array. The direction-of-arrivals is θ=θ1,θ2,⋯θD and the corresponding source powers are p=σ12,σ22,⋯σD2. The measuring array is a linear array with M sensors. k is the snapshot index, k∈1,2⋯K; the kth receiving snapshot can be expressed as(1)xk=∑i=1Daθisik=Ask+nkwhere x(k) is the receiving signal vector, s(k) is the transmission signal vector, A is the array manifold, n(k) is the additive Gaussian white noise vector with a power of σn2, and a(θi) is the steering vector:(2)aθi=1,e-j2πd1sin⁡θi/λ,e-j2πd2sin⁡θi/λ,⋯e-j2πdM-1sin⁡θi/λTdm represents the distance between the mth antenna and the first antenna in the array configuration, dm∈P, where P is the location set of the physical sensors. If the array is a coprime array, then(3)dm∈0,M1d⋯M1M2-1d⋃M2d⋯M2M1-1dThe geometric configuration of the coprime array is shown in Figure 1, where d is the fundamental sensor spacing, d=λ/2, and λ is the wavelength of the transmitting signal. M1 and M2 are coprime integers, and M=M1+M2-1 is the total number of sensors. The covariance matrix of the received signal is expressed as(4)RXX=ExkxkH=∑i=1Dσi2aθiaθiH+σn2I=ARssAH+σn2IRss is the covariance matrix of the transmitted signal. In practice, the statistical mean is used to replace the mathematical mean; that is,(5)RXX=1K∑k=1KxkxkHExukxv∗k is the uth line and vth column of the covariance matrix RXX, which can be expressed as(6)Ru,v=e-j2πsin⁡θi/λdu-dvσi2where du∈P and dv∈P. We define the difference set as(7)D=du-du∣0≤u,v≤M-1The difference set is the basis of the virtual array, and the elements in the difference set represent the relative position of the virtual array. The location distribution of the physical elements and the virtual array are shown in Figure 2 for M1=4 and M2=5. Obviously, the independent difference is symmetric on the center of origin.Figure 1
Geometry of coprime array.Figure 2
Position distribution of the physical array and virtual array (M1=4 and M2=5).


## 2.1. Notation Conventions

Notations used in this paper are as follows:(i)
IN represents N-dimensional unit array.(ii)
(·)∗, (·)T, and (·)H, respectively, represent conjugate, transposition, and conjugate transpose.(iii)
·m stands for the mth element of this vector.(iv)
⊙ represents Hadamard product.(v)
⋃ stands for the intersection.(vi)
vec· is the operation of “vectorization,” which means turning a matrix into a vector by columns.(vii)
⊗ is the Kronecker product.(viii)
E(·) denotes the mathematical expectation.(ix)
·0, ·1, and ·22, respectively, represent the 0-norm, 1-norm, and 2-norm.(x)
Re· and Im· denotes the real part and the imaginary part.


## 2.2. Coprime Array Model

We assume that there areD stationary narrowband targets transmitting plane waves to the measuring array. The direction-of-arrivals is θ=θ1,θ2,⋯θD and the corresponding source powers are p=σ12,σ22,⋯σD2. The measuring array is a linear array with M sensors. k is the snapshot index, k∈1,2⋯K; the kth receiving snapshot can be expressed as(1)xk=∑i=1Daθisik=Ask+nkwhere x(k) is the receiving signal vector, s(k) is the transmission signal vector, A is the array manifold, n(k) is the additive Gaussian white noise vector with a power of σn2, and a(θi) is the steering vector:(2)aθi=1,e-j2πd1sin⁡θi/λ,e-j2πd2sin⁡θi/λ,⋯e-j2πdM-1sin⁡θi/λTdm represents the distance between the mth antenna and the first antenna in the array configuration, dm∈P, where P is the location set of the physical sensors. If the array is a coprime array, then(3)dm∈0,M1d⋯M1M2-1d⋃M2d⋯M2M1-1dThe geometric configuration of the coprime array is shown in Figure 1, where d is the fundamental sensor spacing, d=λ/2, and λ is the wavelength of the transmitting signal. M1 and M2 are coprime integers, and M=M1+M2-1 is the total number of sensors. The covariance matrix of the received signal is expressed as(4)RXX=ExkxkH=∑i=1Dσi2aθiaθiH+σn2I=ARssAH+σn2IRss is the covariance matrix of the transmitted signal. In practice, the statistical mean is used to replace the mathematical mean; that is,(5)RXX=1K∑k=1KxkxkHExukxv∗k is the uth line and vth column of the covariance matrix RXX, which can be expressed as(6)Ru,v=e-j2πsin⁡θi/λdu-dvσi2where du∈P and dv∈P. We define the difference set as(7)D=du-du∣0≤u,v≤M-1The difference set is the basis of the virtual array, and the elements in the difference set represent the relative position of the virtual array. The location distribution of the physical elements and the virtual array are shown in Figure 2 for M1=4 and M2=5. Obviously, the independent difference is symmetric on the center of origin.Figure 1
Geometry of coprime array.Figure 2
Position distribution of the physical array and virtual array (M1=4 and M2=5).


## 3. Sparse Array for Noncircular Sources (SANC) with High Degree of Freedom

### 3.1. SANC Configuration Design

The circular sources are the universal assumption of narrow band sources. Its definition comes from the first- and second-order statistical characteristics of the sources. When the ellipse covariance of the sources is zero, the sources are called circular sources. When the ellipse covariance is not zero, the sources are noncircular sources, and the common noncircular sources are BPSK, AM, ASK, and PAM. The constellation diagram of circular sources consists of in-phase components and quadrature components. Compared with circular sources, noncircular sources have more useful information, so the DOA estimation accuracy of noncircular sources can be improved by using this characteristic.A notable feature of noncircular sources is that the elliptical covariance of sources is not zero.(8)R′XX=ExkxkT≠0With the noncircular characteristics of the sources, the aperture of the array can be expanded.(9)yk=xkx∗k=AA∗sk+nkn∗kThe equivalent array manifold is B=AA∗=b(θ1)⋯,b(θD); b(θi)=a(θi)a∗(θi); and the extended covariance matrix is(10)Ryy=EykykH=∑i=1Dσi2bθibθiH+σn2I=BRssBH+σn2IBecause bθibθiH extends the virtual array aperture, according to formula (6), the virtual array manifold of the extended array for noncircular sources can be obtained; that is,(11)DNC=du-dv,du+dv,-du-dv∣0≤u,v≤M-1Generally, to achieve the maximum array DOF with the spatial smoothing method, the virtual array aperture needs to reach the maximum; that is, the number of continuous virtual array sensors reaches the maximum. In order to achieve the maximum number of continuous virtual array sensors, exhaustion method is used to search the optimal distribution of physical sensors. The search steps are divided into four steps as follows.Step 1.
Based on the number of physical elementsM, we set the number of continuous virtual array sensors 2L+1; that is, the virtual array sensors are continuous in the range -L,L, because the virtual array is symmetry about origin of coordinates, so the distribution range of the physical sensor is set 0,2L (the physical array sensor distribution can also be taken in a wider range, but the search is more complex), the position of the first physical element is 0, and take arbitrary M-1 physical positions in the range 1,2L.Step 2.
According to the physical sensors position taken in Step1, the extended difference set of the noncircular sources is constructed by formula (11). If the number of continuous virtual elements is less than 2L+1, the Step 1 is repeated. If all the possible physical array sensors are taken out and the length of the continuous virtual array sensors is all less than 2L+1, then Step 4 is executed; if there is a physical array distribution, which makes the number of continuous virtual elements greater than or equal to 2L+1, then perform Step 3.Step 3.
LetL=L+1; return to execute Step 1.Step 4.
LetL=L-1; the position distribution of the physical sensors is obtained when the continuous virtual array length is equal to 2L+1 in Step 2, which is the optimal physical sensors distribution of SANC. The flow chart of this method is shown in Figure 3.Figure 3
Flow chart of the design method for SANC configuration.


### 3.2. Comparison of Different Array Structures

The SANC with high DOF is designed to increase the continuous virtual aperture effectively. In order to show the aperture advantage of the proposed array configuration, this section firstly gives the physical sensors distribution of SANC when the number of array elements is 2 to 10, as shown in Table1.Table 1
The physical array sensors distribution of SANC.
M d i 2 L + 1 2 0,1 5 3 0,2 , 3 13 4 0,2 , 4,5 21 5 0,1 , 2,8 , 13 33 6 0,4 , 8,10,11,13 49 7 0,5 , 7,12,13,15,16 65 8 0,6 , 12,15,16,17,19,20 81 9 0,6 , 12,18,21,22,23,25,26 105 10 0,6 , 12,18,24,27,28,29,31,32 129The distribution of physical sensors and virtual array sensors of SANC and five typical sparse arrays with 8 array elements are shown in Figure4. To fully exploit the advantages of the CADiS array, the nested CADiS is used in the comparison. In the comparison, the distribution of physical array sensors is {0,6,12,15,16,17,19,20}. From Figure 4, we can see that the virtual aperture of the proposed array is the largest and the number of continuous virtual sensors is the largest. To clearly present the structural advantages of SANC, Table 2 lists the maximum number of continuous virtual array elements and the DOF for the case of a physical array of 8 sensors; the DOF here is an example for space smoothing algorithm. According to the data comparison in Table 2, the proposed array configuration effectively improves the aperture of the virtual array and increases the array DOF.Table 2
Comparison of maximum continuous virtual array aperture and DOF for typical array configurations.
Array configurations d i 2 L + 1 DOF Uniform array 0,1,2,3,4,5,6,7 29 15 Nest array 0,1 , 2,3 , 4,8 , 12,16 41 21 Coprime array 0,4 , 5,8 , 10,12,15,16 57 29 Nested CADiS 0,1 , 2,3 , 8,12,16,20 49 25 MRA 0,1 , 2,5 , 8,11,12,13 53 27 Proposed (SANC) 0,6 , 12,15,16,17,19,20 81 41Figure 4
Distribution of the physical sensors and the virtual elements of typical array configurations: (a) uniform array, (b) nested array, (c) coprime array, (d) nested CADiS array, (e) MRA, and (f) SANC array.


## 3.1. SANC Configuration Design

The circular sources are the universal assumption of narrow band sources. Its definition comes from the first- and second-order statistical characteristics of the sources. When the ellipse covariance of the sources is zero, the sources are called circular sources. When the ellipse covariance is not zero, the sources are noncircular sources, and the common noncircular sources are BPSK, AM, ASK, and PAM. The constellation diagram of circular sources consists of in-phase components and quadrature components. Compared with circular sources, noncircular sources have more useful information, so the DOA estimation accuracy of noncircular sources can be improved by using this characteristic.A notable feature of noncircular sources is that the elliptical covariance of sources is not zero.(8)R′XX=ExkxkT≠0With the noncircular characteristics of the sources, the aperture of the array can be expanded.(9)yk=xkx∗k=AA∗sk+nkn∗kThe equivalent array manifold is B=AA∗=b(θ1)⋯,b(θD); b(θi)=a(θi)a∗(θi); and the extended covariance matrix is(10)Ryy=EykykH=∑i=1Dσi2bθibθiH+σn2I=BRssBH+σn2IBecause bθibθiH extends the virtual array aperture, according to formula (6), the virtual array manifold of the extended array for noncircular sources can be obtained; that is,(11)DNC=du-dv,du+dv,-du-dv∣0≤u,v≤M-1Generally, to achieve the maximum array DOF with the spatial smoothing method, the virtual array aperture needs to reach the maximum; that is, the number of continuous virtual array sensors reaches the maximum. In order to achieve the maximum number of continuous virtual array sensors, exhaustion method is used to search the optimal distribution of physical sensors. The search steps are divided into four steps as follows.Step 1.
Based on the number of physical elementsM, we set the number of continuous virtual array sensors 2L+1; that is, the virtual array sensors are continuous in the range -L,L, because the virtual array is symmetry about origin of coordinates, so the distribution range of the physical sensor is set 0,2L (the physical array sensor distribution can also be taken in a wider range, but the search is more complex), the position of the first physical element is 0, and take arbitrary M-1 physical positions in the range 1,2L.Step 2.
According to the physical sensors position taken in Step1, the extended difference set of the noncircular sources is constructed by formula (11). If the number of continuous virtual elements is less than 2L+1, the Step 1 is repeated. If all the possible physical array sensors are taken out and the length of the continuous virtual array sensors is all less than 2L+1, then Step 4 is executed; if there is a physical array distribution, which makes the number of continuous virtual elements greater than or equal to 2L+1, then perform Step 3.Step 3.
LetL=L+1; return to execute Step 1.Step 4.
LetL=L-1; the position distribution of the physical sensors is obtained when the continuous virtual array length is equal to 2L+1 in Step 2, which is the optimal physical sensors distribution of SANC. The flow chart of this method is shown in Figure 3.Figure 3
Flow chart of the design method for SANC configuration.


## 3.2. Comparison of Different Array Structures

The SANC with high DOF is designed to increase the continuous virtual aperture effectively. In order to show the aperture advantage of the proposed array configuration, this section firstly gives the physical sensors distribution of SANC when the number of array elements is 2 to 10, as shown in Table1.Table 1
The physical array sensors distribution of SANC.
M d i 2 L + 1 2 0,1 5 3 0,2 , 3 13 4 0,2 , 4,5 21 5 0,1 , 2,8 , 13 33 6 0,4 , 8,10,11,13 49 7 0,5 , 7,12,13,15,16 65 8 0,6 , 12,15,16,17,19,20 81 9 0,6 , 12,18,21,22,23,25,26 105 10 0,6 , 12,18,24,27,28,29,31,32 129The distribution of physical sensors and virtual array sensors of SANC and five typical sparse arrays with 8 array elements are shown in Figure4. To fully exploit the advantages of the CADiS array, the nested CADiS is used in the comparison. In the comparison, the distribution of physical array sensors is {0,6,12,15,16,17,19,20}. From Figure 4, we can see that the virtual aperture of the proposed array is the largest and the number of continuous virtual sensors is the largest. To clearly present the structural advantages of SANC, Table 2 lists the maximum number of continuous virtual array elements and the DOF for the case of a physical array of 8 sensors; the DOF here is an example for space smoothing algorithm. According to the data comparison in Table 2, the proposed array configuration effectively improves the aperture of the virtual array and increases the array DOF.Table 2
Comparison of maximum continuous virtual array aperture and DOF for typical array configurations.
Array configurations d i 2 L + 1 DOF Uniform array 0,1,2,3,4,5,6,7 29 15 Nest array 0,1 , 2,3 , 4,8 , 12,16 41 21 Coprime array 0,4 , 5,8 , 10,12,15,16 57 29 Nested CADiS 0,1 , 2,3 , 8,12,16,20 49 25 MRA 0,1 , 2,5 , 8,11,12,13 53 27 Proposed (SANC) 0,6 , 12,15,16,17,19,20 81 41Figure 4
Distribution of the physical sensors and the virtual elements of typical array configurations: (a) uniform array, (b) nested array, (c) coprime array, (d) nested CADiS array, (e) MRA, and (f) SANC array.


## 4. DOA Estimation Methods

There are many typical DOA estimation methods, such as subspace-based methods, compressed sensing methods, and high-order statistics methods. Here we briefly introduce the common subspace-based method and compressed sensing method.


### 4.1. Spatial Smoothing MUSIC Algorithm

In order to make full use of the maximum continuous virtual array of SANC above, the received source covariance matrix is vectorized as follows:(12)z=vecRyy=vec∑i=1Dσi2bθibθiH+σn2I~=B~θ1,θ2,⋯θDp+σn2I~where(13)B~θ1,⋯θD=b∗θ1⊗bθ1,b∗θ2⊗bθ2,⋯b∗θD⊗bθD(14)I~=vecI2MB~ is the extended virtual array manifold. If the length of the maximum continuous virtual array is 2L+1, we take the maximum continuous virtual element response part in B~ to reconstruct a 2L+1×D-dimensional uniform array. So(15)z⌢=B⌢p+σn2I⌢We apply spatial smoothing to z⌢, where z⌢ has a conjugate symmetric distribution. Therefore, the length of the smooth segment is L+1. The position of the ith smooth subarray is z⌢i, and the noise vector is σn2e⌢i; for the virtual array,(16)-i+1+nd,n=0,1,⋯,L(17)z⌢i=B⌢ip+σn2I⌢i=B⌢1Φi-1p+σn2I⌢iwhere(18)Φ=e-jπsin⁡θ1e-jπsin⁡θ2⋱e-jπsin⁡θD(19)B⌢1=11⋯1b⌢θ1b⌢θ2⋯b⌢θD⋮⋮⋮⋮b⌢θ1Lb⌢θ2L⋯b⌢θDLb⌢θi=1,b⌢θi,⋯,b⌢θiLT; b⌢θi=e-jπsin⁡θi. The weighted mean for all of the covariance matrices of the smooth subarray is(20)R=1L+1∑i=1L+1z⌢iz⌢iHEigenvalue decomposition (EVD) is applied to R as follows:(21)R=UsΣsUsH+UnΣnUnHBecause the signal subspace and the noise subspace are orthogonal, the steering vector and the signal subspace belong to the same vector space, so the spatial spectral expression of the DOAs is(22)P=1b⌢HθUnUnHb⌢θThe root-MUSIC method or spectral peak search method can be used to solve all the direction-of-arrivals.


### 4.2. Compressed Sensing Approach

Other typical DOA estimation methods are based on compressed sensing approaches. Since the DOAs fall on the finite discrete angles, the spatial signal spectra are sparse. Therefore, sparsity-based techniques can be applied to DOA estimation to improve the array DOF. There are many typical compressed sensing methods [28–30]; LASSO is one of the representative compressed sensing methods. Firstly, a complete set Π=θ¯1,θ¯2,⋯θ¯Γ is constructed, which is called a redundant dictionary. Π contains all the possible direction-of-arrivals, and Γ≫D. Based on this, a compressed sensing model is build as follows:(23)z=B~θ¯1,θ¯2,⋯θ¯Γp¯+σn2I~B~θ¯1,θ¯2,⋯θ¯Γ is the sensing matrix; p¯=p¯1,⋯,p¯Γ is the sparse source power. When θ¯i is the true direction of the arrival, that is, θ¯i=θi, then p¯i=σi2 and p¯i=0 except for the true direction of the arrival. Construct optimization cost function based on this sparse model:(24)minp¯p¯0s.t.z-B~θ¯1,θ¯2,⋯θ¯Γp¯-σn2I~<εε is positive and small enough. Unconstrained transformation and convex transformation are applied to the preceding expression, and the following expressions are obtained:(25)p^=minp¯12z-B~θ¯1,θ¯2,⋯θ¯Γp¯2+ηp¯1According to the convex optimization knowledge, the sparse source power vector p^ can be obtained, and the angles in the redundant dictionary corresponding to the nonzero elements of the sparse signal power vector are DOAs.


## 4.1. Spatial Smoothing MUSIC Algorithm

In order to make full use of the maximum continuous virtual array of SANC above, the received source covariance matrix is vectorized as follows:(12)z=vecRyy=vec∑i=1Dσi2bθibθiH+σn2I~=B~θ1,θ2,⋯θDp+σn2I~where(13)B~θ1,⋯θD=b∗θ1⊗bθ1,b∗θ2⊗bθ2,⋯b∗θD⊗bθD(14)I~=vecI2MB~ is the extended virtual array manifold. If the length of the maximum continuous virtual array is 2L+1, we take the maximum continuous virtual element response part in B~ to reconstruct a 2L+1×D-dimensional uniform array. So(15)z⌢=B⌢p+σn2I⌢We apply spatial smoothing to z⌢, where z⌢ has a conjugate symmetric distribution. Therefore, the length of the smooth segment is L+1. The position of the ith smooth subarray is z⌢i, and the noise vector is σn2e⌢i; for the virtual array,(16)-i+1+nd,n=0,1,⋯,L(17)z⌢i=B⌢ip+σn2I⌢i=B⌢1Φi-1p+σn2I⌢iwhere(18)Φ=e-jπsin⁡θ1e-jπsin⁡θ2⋱e-jπsin⁡θD(19)B⌢1=11⋯1b⌢θ1b⌢θ2⋯b⌢θD⋮⋮⋮⋮b⌢θ1Lb⌢θ2L⋯b⌢θDLb⌢θi=1,b⌢θi,⋯,b⌢θiLT; b⌢θi=e-jπsin⁡θi. The weighted mean for all of the covariance matrices of the smooth subarray is(20)R=1L+1∑i=1L+1z⌢iz⌢iHEigenvalue decomposition (EVD) is applied to R as follows:(21)R=UsΣsUsH+UnΣnUnHBecause the signal subspace and the noise subspace are orthogonal, the steering vector and the signal subspace belong to the same vector space, so the spatial spectral expression of the DOAs is(22)P=1b⌢HθUnUnHb⌢θThe root-MUSIC method or spectral peak search method can be used to solve all the direction-of-arrivals.


## 4.2. Compressed Sensing Approach

Other typical DOA estimation methods are based on compressed sensing approaches. Since the DOAs fall on the finite discrete angles, the spatial signal spectra are sparse. Therefore, sparsity-based techniques can be applied to DOA estimation to improve the array DOF. There are many typical compressed sensing methods [28–30]; LASSO is one of the representative compressed sensing methods. Firstly, a complete set Π=θ¯1,θ¯2,⋯θ¯Γ is constructed, which is called a redundant dictionary. Π contains all the possible direction-of-arrivals, and Γ≫D. Based on this, a compressed sensing model is build as follows:(23)z=B~θ¯1,θ¯2,⋯θ¯Γp¯+σn2I~B~θ¯1,θ¯2,⋯θ¯Γ is the sensing matrix; p¯=p¯1,⋯,p¯Γ is the sparse source power. When θ¯i is the true direction of the arrival, that is, θ¯i=θi, then p¯i=σi2 and p¯i=0 except for the true direction of the arrival. Construct optimization cost function based on this sparse model:(24)minp¯p¯0s.t.z-B~θ¯1,θ¯2,⋯θ¯Γp¯-σn2I~<εε is positive and small enough. Unconstrained transformation and convex transformation are applied to the preceding expression, and the following expressions are obtained:(25)p^=minp¯12z-B~θ¯1,θ¯2,⋯θ¯Γp¯2+ηp¯1According to the convex optimization knowledge, the sparse source power vector p^ can be obtained, and the angles in the redundant dictionary corresponding to the nonzero elements of the sparse signal power vector are DOAs.


## 5. Performance Analysis and Simulation Experiments

### 5.1. Cramer-Rao Lower Bound

The Cramer-rao lower bound is the lower bound of the unbiased estimation of variance, which is used to measure the deviation of the DOAs estimation. Based on [31–33], the CRLB for SANC is given. It is assumed that the noise follows the complex Gauss distribution. According to formula (9), the probability density function of the received signal is as follows:(26)Py1,⋯,yK=12π2MKσn2/22MKe-1/σ2∑k=1Kyk-BskHyk-BskThe logarithmic likelihood function is obtained by taking the logarithm of formula (26).(27)Ly1,⋯,yK=-2MKln2π-2MKlnσn22-1σn2∑k=1Kyk-BskHyk-BskDefine s¯k=Resk, s~k=Imsk, and the Fisher information matrix is Ω=EψψT-1, where(28)ψT=∂L∂σn2s¯T1s~T1⋯s¯TKs~TKθTThe CRLB expression for the SANC configuration can be obtained from the Fisher information matrix as(29)CRLBθ=σn22∑k=1KReFHkDHPB⊥DFk-1where Fk=diagsk, D=∂BH/∂θ1,∂BH/∂θ2,⋯,∂BH/∂θD, and PB⊥=I-PB=I-BBHB-1BH.


### 5.2. Simulation Results

In order to verify the DOA estimation performance of the proposed array configuration, the simulation experiments of SANC in underdetermined condition are given. The root mean square error (RMSE) and the CRLB with the different SNRs and the different snapshots are also simulated, respectively. The compared array configurations are the typical array configurations under the same physical array sensor number, such as the uniform array, the coprime array, the nested array, the MRA array, and the CADiS array. To maximize the aperture advantages of CADiS array, the nested CADiS array is used in the simulation. The root mean square error (RMSE) represents the magnitude of the angle estimation bias, and its mathematical expression is as follows:(30)RMSE=1KmD∑m=1Km∑i=1Dθ^im-θi2where Km is the number of Monte Carlo simulations, D is the number of source targets, and θ^im is the ith source DOA in the mth Monte Carlo simulation. The simulation conditions are listed in Table 3.Table 3
Simulation conditions for the experiments.
Simulation parameters Value Antenna number 8 Sensor locations of SANC 0,6 , 12,15,16,17,19,20 Sensor locations of uniform array 0,1 , 2,3 , 4,5 , 6,7 Sensor locations of nested array 0,1 , 2,3 , 4,8 , 12,16 Sensor locations of coprime array 0,3 , 5,6 , 9,10,12,15 Sensor locations of nested CADiS 0,1 , 2,3 , 8,12,16,20 Sensor locations of MRA 0,1 , 2,5 , 8,11,12,13 Source number D = 5 DOAs - 4 0 ∘ , - 2 0 ∘ , 0 ∘ , 2 0 ∘ , 4 0 ∘ Snapshot number K = 200 Carrier frequency f = 2.1 G H z Speed of light c = 3 × 1 0 8 m / s Monte Carlo times K m = 200 Signal-to-noise ratio (SNR) S N R = 0 ~ 20 d BSimulation 1.
DOA estimation performance of SANC configuration in underdetermined condition.
One of the most important features of the sparse array is that it can estimate DOAs in underdetermined condition. SANC array configuration retains this advantage. To measure the DOA estimation performance of SANC in underdetermined conditions, we conducted Monte Carlo simulations with the source number of 29 and SNRs of 0dB and 20 dB. The normalized spatial spectrums for the SANC configuration are presented in Figure5. The simulation results show that the array configuration proposed in this paper can all effectively estimate the DOAs no matter in a high SNR condition or a low SNR condition, and the higher SNR is the sharper spectrum peak is.Figure 5
Normalized spatial spectrum results for a source number of 29: (a) SNR = 0dB; (b) SNR = 20dB.
(a) (b)Simulation 2.
RMSE and CRLB comparison of uniform array, coprime array, nested array, nested CADiS array, MRA array, and SANC for different SNRs.
Figure6(a) shows the results of the RMSE and CRLB performance of the different configurations. Figure 6(b) presents the results of the RMSE of the different DOA estimation methods, including MUSIC, the spatial smoothing MUSIC (ss-MUSIC), root-MUSIC, LASSO, and the four-order cumulant (FOC) methods. The simulation results show that the proposed configuration obviously improves the DOA estimation accuracy, and the CRLB is effectively reduced. Under this SANC configuration, the estimation performances of spatial smoothing MUSIC and root-MUSIC are basically the same. The performance of the spatial smoothing MUSIC, root-MUSIC, LASSO, and MUSIC methods is higher than the four-order cumulant method.Figure 6
DOA estimation comparison under different SNRs: (a) RMSE for different array configurations. (b) RMSE for different DOA estimation methods.
(a) (b)Simulation 3.
RMSE comparison of uniform array, coprime array, nested array, nested CADiS array, MRA array, and SANC for different snapshot numbers.
The number of snapshots is an important factor for the DOA estimation accuracy. The Monte Carlo experiments simulate the RMSE of the DOA estimations with different snapshot numbers at SNRs of 0dB and 20dB. The simulation results in Figure7 show that as the number of snapshots increases, the RMSE decreases. For each snapshot number, the RMSE of SANC is lower than that of the other configurations no matter in a low SNR or a high SNR.Figure 7
RMSE performance for different snapshot numbers: (a) SNR of 0dB; (b) SNR of 20dB.
(a) (b)


## 5.1. Cramer-Rao Lower Bound

The Cramer-rao lower bound is the lower bound of the unbiased estimation of variance, which is used to measure the deviation of the DOAs estimation. Based on [31–33], the CRLB for SANC is given. It is assumed that the noise follows the complex Gauss distribution. According to formula (9), the probability density function of the received signal is as follows:(26)Py1,⋯,yK=12π2MKσn2/22MKe-1/σ2∑k=1Kyk-BskHyk-BskThe logarithmic likelihood function is obtained by taking the logarithm of formula (26).(27)Ly1,⋯,yK=-2MKln2π-2MKlnσn22-1σn2∑k=1Kyk-BskHyk-BskDefine s¯k=Resk, s~k=Imsk, and the Fisher information matrix is Ω=EψψT-1, where(28)ψT=∂L∂σn2s¯T1s~T1⋯s¯TKs~TKθTThe CRLB expression for the SANC configuration can be obtained from the Fisher information matrix as(29)CRLBθ=σn22∑k=1KReFHkDHPB⊥DFk-1where Fk=diagsk, D=∂BH/∂θ1,∂BH/∂θ2,⋯,∂BH/∂θD, and PB⊥=I-PB=I-BBHB-1BH.


## 5.2. Simulation Results

In order to verify the DOA estimation performance of the proposed array configuration, the simulation experiments of SANC in underdetermined condition are given. The root mean square error (RMSE) and the CRLB with the different SNRs and the different snapshots are also simulated, respectively. The compared array configurations are the typical array configurations under the same physical array sensor number, such as the uniform array, the coprime array, the nested array, the MRA array, and the CADiS array. To maximize the aperture advantages of CADiS array, the nested CADiS array is used in the simulation. The root mean square error (RMSE) represents the magnitude of the angle estimation bias, and its mathematical expression is as follows:(30)RMSE=1KmD∑m=1Km∑i=1Dθ^im-θi2where Km is the number of Monte Carlo simulations, D is the number of source targets, and θ^im is the ith source DOA in the mth Monte Carlo simulation. The simulation conditions are listed in Table 3.Table 3
Simulation conditions for the experiments.
Simulation parameters Value Antenna number 8 Sensor locations of SANC 0,6 , 12,15,16,17,19,20 Sensor locations of uniform array 0,1 , 2,3 , 4,5 , 6,7 Sensor locations of nested array 0,1 , 2,3 , 4,8 , 12,16 Sensor locations of coprime array 0,3 , 5,6 , 9,10,12,15 Sensor locations of nested CADiS 0,1 , 2,3 , 8,12,16,20 Sensor locations of MRA 0,1 , 2,5 , 8,11,12,13 Source number D = 5 DOAs - 4 0 ∘ , - 2 0 ∘ , 0 ∘ , 2 0 ∘ , 4 0 ∘ Snapshot number K = 200 Carrier frequency f = 2.1 G H z Speed of light c = 3 × 1 0 8 m / s Monte Carlo times K m = 200 Signal-to-noise ratio (SNR) S N R = 0 ~ 20 d BSimulation 1.
DOA estimation performance of SANC configuration in underdetermined condition.
One of the most important features of the sparse array is that it can estimate DOAs in underdetermined condition. SANC array configuration retains this advantage. To measure the DOA estimation performance of SANC in underdetermined conditions, we conducted Monte Carlo simulations with the source number of 29 and SNRs of 0dB and 20 dB. The normalized spatial spectrums for the SANC configuration are presented in Figure5. The simulation results show that the array configuration proposed in this paper can all effectively estimate the DOAs no matter in a high SNR condition or a low SNR condition, and the higher SNR is the sharper spectrum peak is.Figure 5
Normalized spatial spectrum results for a source number of 29: (a) SNR = 0dB; (b) SNR = 20dB.
(a) (b)Simulation 2.
RMSE and CRLB comparison of uniform array, coprime array, nested array, nested CADiS array, MRA array, and SANC for different SNRs.
Figure6(a) shows the results of the RMSE and CRLB performance of the different configurations. Figure 6(b) presents the results of the RMSE of the different DOA estimation methods, including MUSIC, the spatial smoothing MUSIC (ss-MUSIC), root-MUSIC, LASSO, and the four-order cumulant (FOC) methods. The simulation results show that the proposed configuration obviously improves the DOA estimation accuracy, and the CRLB is effectively reduced. Under this SANC configuration, the estimation performances of spatial smoothing MUSIC and root-MUSIC are basically the same. The performance of the spatial smoothing MUSIC, root-MUSIC, LASSO, and MUSIC methods is higher than the four-order cumulant method.Figure 6
DOA estimation comparison under different SNRs: (a) RMSE for different array configurations. (b) RMSE for different DOA estimation methods.
(a) (b)Simulation 3.
RMSE comparison of uniform array, coprime array, nested array, nested CADiS array, MRA array, and SANC for different snapshot numbers.
The number of snapshots is an important factor for the DOA estimation accuracy. The Monte Carlo experiments simulate the RMSE of the DOA estimations with different snapshot numbers at SNRs of 0dB and 20dB. The simulation results in Figure7 show that as the number of snapshots increases, the RMSE decreases. For each snapshot number, the RMSE of SANC is lower than that of the other configurations no matter in a low SNR or a high SNR.Figure 7
RMSE performance for different snapshot numbers: (a) SNR of 0dB; (b) SNR of 20dB.
(a) (b)


## 6. Conclusion

In order to solve the problem of low DOF and smaller continuous virtual aperture in the existing sparse array for noncircular sources, a high degree of freedom sparse array design method suitable for noncircular sources is presented. The method uses the difference set to establish the virtual array model and then expands the virtual array aperture with the characteristic of noncircular sources. The physical sensors distribution of the maximum continuous virtual array length is given by searching method. SANC gives the optimal array design for noncircular sources, and the comparisons and analysis between the proposed array configuration and the typical array configurations are also given. The simulation experiments show that, compared with the uniform array, the coprime array, the nested array, the MRA array, and the CADiS array under the same physical sensor number, the sparse array proposed for noncircular sources is significantly improved in the array aperture, array DOF, and estimation precision.



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*Source: 1264715-2019-01-29.xml*