# Increasing Utilization of Redundant Virtual Array for DOA Estimation Based on Coprime Array

Authors: Chenghong Zhan; Guoping Hu; Zixin Zhang; Ziang Feng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, we initiated a method to estimate the direction of arrival (DOA) of far-field, narrowband, and incoherent targets using coprime array. First, we proposed a coprime array structure and analysed the distribution of difference coarray (DCA). The degrees of freedom (DOF) of the proposed coprime array became clearer by referring to the DCA conception. However, previous algorithm only uses the continuous virtual array, which causes the virtual array elements in the repeated position being abandoned. Therefore, the paper analyses the distribution of virtual array based on DCA conception and averages the receiving signal on these redundant virtual array elements to increase the utilization of receiving data. As a result, the algorithm has high precision in parameter estimation. Simulation results have shown the superiority of the proposed algorithm.


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## Body

## 1. Introduction

Because sparse arrays can overcome the limitation of interelement spacing, direction of arrival (DOA) estimation based on it has attracted great attention. Compared to the uniform linear arrays (ULAs), sparse arrays, such as minimum redundancy array (MRA) [1], nested array (NA) [2, 3], and coprime array (CPA) [4], have great advantages in improving estimation accuracy, enhancing angle resolution, and reducing physical cost [5–8]. Larger array element spacing expands the virtual aperture, which leads to the rise in the number of detectable sources and angle resolution. In addition, it can also effectively suppress the mutual coupling effect of array elements and the coherence of received noise. Thus, improving the estimation, low redundancy can effectively reduce wastage of resources and accelerate operation [9–11].Coprime arrays have more DOFs than ULAs and have no mutual coupling problems compared with nested arrays [12]. Furthermore, it has closed form expression of freedom extension degree and systematic array structure design. Therefore, DOA estimation based on coprime arrays has become a research hotspot. Most algorithms with coprime arrays derive extended virtual array and operate the corresponding signals to estimate DOA [13–16]. In [17], the paper proposes a modified Toeplitz matrix reconstruction method using a random nonzero row of the covariance matrix in the virtual uniform linear array. Besides, a full rank covariance matrix is obtained by restructuring a Toeplitz matrix. In [18], the author recovered the Toeplitz matrix by solving a nuclear norm minimization problem and decohered the signal. In addition, the discrete virtual array elements are utilized by interpolating virtual elements. However, the virtual arrays in repeated position were abandoned. In [16], a generalized coprime array configuration for DOA estimation is proposed. Based on generalized sum and difference coarray (GSDC) concept, the proposed configuration enlarges the interelement space, and hence, the DOFs of the arrays are increased. Besides, there are a large number of virtual arrays in the same position which are not utilized. On the other hand, in [19], an algorithm with low computational complexity and high estimation accuracy was initiated. The method first used estimates signal parameters via rotational invariance technique (ESPRIT) algorithm. Then, the search scope was narrowed down based on the initial results. Finally, more accurate estimation results were obtained by multiple signal classification (MUSIC). Similarly, in order to reduce the computational cost, the authors of [20] propose a successful signal subspace fitting (SSF) method from a computationally efficient perspective. All of the algorithms above focused either on improving the array structure to obtain higher DOFs and larger virtual array aperture or on reducing the complexity by avoiding spatial spectrum search and so on. Obviously, the redundant virtual elements in the same position were not utilized, and their data are wasted.In this paper, we take the coprime array with the difference coarray (DCA) as an example to illustrate the redundant virtual array elements distribution and then average the virtual elements in the repeated position to increase the utilization efficiency of received signal data. Consequently, the snapshots of received signal have increased, and the DOA estimation performance of targets is improved noticeably.


## 2. Coprime Array Signal Model

As shown in Figure1, a coprime array structure consists of two uniform linear subarrays. Subarray 1 has N sensors with an interelement spacing N. The other subarray has 2M−1 sensors with interelement spacing M. Mand N are a pair of coprime integers M<N. We assume the unit of interelement spacing is d, where d=λ/2 is half-wavelength.Figure 1
Coprime array MIMO radar structure.According to Figure1, sensors’ location is expressed as follows:(1)S=Mnd,0≤n≤N−1∪Nmd,0≤m≤2M−1.We assume narrowband and uncorrelated sources in the far-field from distinct angles impinging on the array with sensors. The output of the received signal after the matched filter can be denoted as(2)Xt=ASt+Nt.In the above equation,St=s1t,s2t,…,sKtT, where skt is the reflection coefficient of the kth target. N(t) is the complex-valued additive white Gaussian noise matrix with mean value 0 and variance σ2. Ais the steering vector which can be denoted as(3)A=aθ1,aθ2,…,aθK,where aθk can be expressed as(4)aθk=1,e−jπsinθk,e−jπMsinθk,…,e−jπN−1Msinθk,e−jπNsinθk,…,e−jπ2M−1NsinθkT.Then, the covariance matrix can be given as(5)R=EXtXHt=ARSAH+σ2IN×N,where RS=EStSHt represents the target covariance matrix. E⋅ denotes mathematical expectation; ⋅H denotes conjugate transpose; σ2 denotes the energy of the noise; and IN×N is N×N dimensional identity matrix. Because the signal is incoherent, RS=EStSHt can be given as(6)RS=EStSHt=diagσ12,σ22,…,σK2,where the σk2 denotes the power of the signal.In practice, the signal covariance matrix is often estimated withL snapshots.(7)R^=1L∑t=1LXtXHt.R^ needs to be vectorized to obtain the virtual array elements. Consequently, the vectorizing covariance matrix of signal can be expressed as(8)r=vecR^=A∗∘Ap+σ2vecIN×N=Bp+σ2vecIN×N,where vec⋅ denotes vectorizing the covariance matrix, ⋅∗ denotes conjugate operation, and ∘ denotes Khatri–Rao product. The virtual array elements are formed from steering vector B=A∗∘A. Then, combining formulae (3) and (8), the DAC conception can be expressed as(9)B=A∗∘A=a∗θ1⊗aθ1,a∗θ2⊗aθ2,…,a∗θK⊗aθK,where ⊗ denotes the Kronecker product. Thus, vectorizing the covariance matrix R^ is an effective method for obtaining the virtual array and extending the array’s aperture.We express the virtual array elements asSCPA, and then it can be represented as(10)SCPA=dn−dn˜,dn,dn˜∈PCPA=Ss∪Sc,Ss=±Mn,n=0,…,N−1∪±Nm,m=0,…,2M−1,Sc=±Mn−Nm,m=0,1,…,2M−1,n=0,1,…,N−1,where Ss denotes the self-difference set and Sc denotes the cross-difference set.FromSs⊂Sc, formula (10) can be simplified as(11)SCPA=±Mn−Nm,m=0,1,…,2M−1,n=0,1,…,N−1.From formula (11), DOFs of the virtual array is OMN that only needs 2M+N−1 physical sensors. It is obvious that the virtual array elements are discontinuous. For example, when M=2,N=3, the continuous range of virtual elements is −7,7. Next, the distribution and application of virtual redundant elements are discussed.


## 3. Improved Algorithm for Coprime Array

In this section, we will compare traditional algorithms based on difference coarray and reuse the redundant virtual elements.


### 3.1. Review of Traditional Algorithm

In equations (8) and (9), the virtual elements formed by the vectorization of the signal covariance matrix are reordered, and redundant virtual elements are deleted. Then, a new steering vector of the single snapshot can be acquired and is marked as signal B^, which is expressed as(12)B^=e−jπNMaxsinθ1e−jπNMaxsinθ2⋯e−jπNMaxsinθK⋮⋮⋱⋮e−jπsinθ1e−jπsinθ2⋯e−jπsinθK11⋯1ejπsinθ1ejπsinθ2⋱ejπsinθK⋮⋮⋮⋮ejπNMaxsinθ1ejπNMaxsinθ2⋯ejπNMaxsinθK,where NMax=MN+M−1 denotes the maximum position of extended continuous virtual elements. Correspondingly, the range of the virtual array elements is −NMax,NMax.Figure2 shows the distribution of virtual array elements when M=2andN=3.Figure 2
Distribution of virtual array.In Figure2, there are many redundant virtual array elements in the repeated position.In the previous algorithm, redundant virtual elements in duplicate positions are deleted and reordered. In addition, the Toeplitz matrix method is used to eliminate the single snapshot effect and obtain a new signal covariance matrix.The new single snapshot signal is marked asr0. Based on formulae (8) and (12), r0 can be expressed as(13)r0=B^p+σn2e⟶,where r0 denotes that a virtual ULA with continuous sensors in the range of −MN+M−1,MN+M−1. Based on the observation vector of the virtual ULA r0, the covariance matrix of the virtual array can be expressed as(14)Rv=r0r0H.Because the new vectorr0 is a single snapshot signal, the rank of virtual array covariance is one. Thus, estimating the DOA by traditional algorithm is inappropriate. However, the Toeplitz matrix method with the symmetric structure of the steering vector can solve the coherence problem.We assume thatr0i−NMax≤i≤NMax denotes i+NMax row of the vector r0. Obviously, r0i=r0−i∗, and a Toeplitz matrix can be constructed as(15)R˜1=r00r0−1⋯r0−NMaxr01r00⋯r0−NMax+1⋮⋮⋱⋮r0NMaxr0NMax−1⋯r00.In this way, the traditional subspace methods can be employed on the covariance matrixR1 for DOA estimation based on the coprime array.In order to making full use of redundant virtual array elements and increase the utilization of received signal data, data of redundant elements are compiled into a new received signal. Section3.2 will show how the algorithm works.


### 3.2. Improved Algorithm Based on Reusing the Redundant Virtual Array Elements

In order to effectively utilize the received signal information in the redundant virtual array element, we take the average redundant elements in the repeated positions. Moreover, we replace the virtual elements in the original positions with the averaging redundant elements. The process is as follows:(16)−7,…,−3,−3,−3︸averaging values,−2,−2,−2︸averaging values−1,−1︸averaging values,0,0,0,0,0,0︸averaging values,1,1︸averaging values,2,2,2︸averaging values,3,3,3︸averaging values,…,7.From equation (16), the redundant virtual array elements are fully utilized. Then, we can get a new observation vector r¯0, which is denoted as(17)r¯0=B^¯p¯+σn2e⟶.In equation (17), the new covariance matrix R¯ is reconstructed by restructuring the Toeplitz matrix method. Then, Root-MUSIC algorithm for DOA estimation is introduced with covariance matrix R¯. Firstly, we define a polynomial as follows:(18)fz=eiHpz,i=K+1,…,NMax,where ei is NMax−K smaller eigenvectors of the covariance matrix R¯ and pz=1,z,…,zNMaxT. Because the signal subspace and noise subspace are orthogonal, equation (18) can be expressed as(19)fz=pHzUNUNHpz.Considering the conjugate ofz existing in fz, formula (19) can be modified into(20)fz=zNMax−1pTz−1UNUNHpz.Therefore, the DOAs of the targets can be expressed by the following formula:(21)θi=arcsinλ2πd0argzi,i=1,…,K.


### 3.3. Algorithm Steps

In order to better illuminate the algorithm ideas, we add detailed algorithm steps in Algorithm1.Algorithm 1: Algorithm steps.
Input: received signal XtOutput: θ^k,k=1,2,…,KStep 1: after the matched filter, received signal Xtwill be obtained.Step 2: according to the formula R=EXtXHt=ARSAH+σ2IN×N, the received signal covariance matrix is obtained.Step 3: the covariance matrix R is vectorized to obtain the vector matrix r.Step 4: the virtual array elements in the vector matrix r are sorted according to the position size.Step 5: averaging the virtual array elements in the same position.Step 6: new vector r¯ will be obtained.Step 7: making r¯ become a Toeplitz matrix to overcome the single snapshot effect, then a new covariance matrix R˜1 will be obtained.Step 8: using Root-MUSIC algorithm to estimate DOA of targets.


## 3.1. Review of Traditional Algorithm

In equations (8) and (9), the virtual elements formed by the vectorization of the signal covariance matrix are reordered, and redundant virtual elements are deleted. Then, a new steering vector of the single snapshot can be acquired and is marked as signal B^, which is expressed as(12)B^=e−jπNMaxsinθ1e−jπNMaxsinθ2⋯e−jπNMaxsinθK⋮⋮⋱⋮e−jπsinθ1e−jπsinθ2⋯e−jπsinθK11⋯1ejπsinθ1ejπsinθ2⋱ejπsinθK⋮⋮⋮⋮ejπNMaxsinθ1ejπNMaxsinθ2⋯ejπNMaxsinθK,where NMax=MN+M−1 denotes the maximum position of extended continuous virtual elements. Correspondingly, the range of the virtual array elements is −NMax,NMax.Figure2 shows the distribution of virtual array elements when M=2andN=3.Figure 2
Distribution of virtual array.In Figure2, there are many redundant virtual array elements in the repeated position.In the previous algorithm, redundant virtual elements in duplicate positions are deleted and reordered. In addition, the Toeplitz matrix method is used to eliminate the single snapshot effect and obtain a new signal covariance matrix.The new single snapshot signal is marked asr0. Based on formulae (8) and (12), r0 can be expressed as(13)r0=B^p+σn2e⟶,where r0 denotes that a virtual ULA with continuous sensors in the range of −MN+M−1,MN+M−1. Based on the observation vector of the virtual ULA r0, the covariance matrix of the virtual array can be expressed as(14)Rv=r0r0H.Because the new vectorr0 is a single snapshot signal, the rank of virtual array covariance is one. Thus, estimating the DOA by traditional algorithm is inappropriate. However, the Toeplitz matrix method with the symmetric structure of the steering vector can solve the coherence problem.We assume thatr0i−NMax≤i≤NMax denotes i+NMax row of the vector r0. Obviously, r0i=r0−i∗, and a Toeplitz matrix can be constructed as(15)R˜1=r00r0−1⋯r0−NMaxr01r00⋯r0−NMax+1⋮⋮⋱⋮r0NMaxr0NMax−1⋯r00.In this way, the traditional subspace methods can be employed on the covariance matrixR1 for DOA estimation based on the coprime array.In order to making full use of redundant virtual array elements and increase the utilization of received signal data, data of redundant elements are compiled into a new received signal. Section3.2 will show how the algorithm works.


## 3.2. Improved Algorithm Based on Reusing the Redundant Virtual Array Elements

In order to effectively utilize the received signal information in the redundant virtual array element, we take the average redundant elements in the repeated positions. Moreover, we replace the virtual elements in the original positions with the averaging redundant elements. The process is as follows:(16)−7,…,−3,−3,−3︸averaging values,−2,−2,−2︸averaging values−1,−1︸averaging values,0,0,0,0,0,0︸averaging values,1,1︸averaging values,2,2,2︸averaging values,3,3,3︸averaging values,…,7.From equation (16), the redundant virtual array elements are fully utilized. Then, we can get a new observation vector r¯0, which is denoted as(17)r¯0=B^¯p¯+σn2e⟶.In equation (17), the new covariance matrix R¯ is reconstructed by restructuring the Toeplitz matrix method. Then, Root-MUSIC algorithm for DOA estimation is introduced with covariance matrix R¯. Firstly, we define a polynomial as follows:(18)fz=eiHpz,i=K+1,…,NMax,where ei is NMax−K smaller eigenvectors of the covariance matrix R¯ and pz=1,z,…,zNMaxT. Because the signal subspace and noise subspace are orthogonal, equation (18) can be expressed as(19)fz=pHzUNUNHpz.Considering the conjugate ofz existing in fz, formula (19) can be modified into(20)fz=zNMax−1pTz−1UNUNHpz.Therefore, the DOAs of the targets can be expressed by the following formula:(21)θi=arcsinλ2πd0argzi,i=1,…,K.


## 3.3. Algorithm Steps

In order to better illuminate the algorithm ideas, we add detailed algorithm steps in Algorithm1.Algorithm 1: Algorithm steps.
Input: received signal XtOutput: θ^k,k=1,2,…,KStep 1: after the matched filter, received signal Xtwill be obtained.Step 2: according to the formula R=EXtXHt=ARSAH+σ2IN×N, the received signal covariance matrix is obtained.Step 3: the covariance matrix R is vectorized to obtain the vector matrix r.Step 4: the virtual array elements in the vector matrix r are sorted according to the position size.Step 5: averaging the virtual array elements in the same position.Step 6: new vector r¯ will be obtained.Step 7: making r¯ become a Toeplitz matrix to overcome the single snapshot effect, then a new covariance matrix R˜1 will be obtained.Step 8: using Root-MUSIC algorithm to estimate DOA of targets.


## 4. Simulation Results

In this section, in the coprime array withM=2 and N=3, we assume that both subarrays start from zero. Obviously, the total physical sensors are 2M+N−1=6.We compared the performance of improved algorithm with traditional algorithm based on coprime array and ULA with six M=6 physical sensors.Simulation will show advantages of reusing the redundant virtual array elements in terms of root means square error and spatial spectral resolution.


### 4.1. Simulation of Spatial Spectral Resolution

In the simulation, a pair of coprime integers are assumed asM=2andN=3. Similarly, the uniform linear array has M=6 physical sensors. Considering three equal power targets impinging on the coprime array from the directions in −20∘,0∘,20∘ with SNR=0dB, the snapshots is L=500. The following formula helps achieve the spatial spectrum:(22)fMUSICθ=1aHθUn2,where aθ denotes the steering vector of the signal and Un consists of eigenvectors corresponding to NMax−K eigenvalues.Figure3 shows the resolution performance of the three algorithms. It can be seen that the spectrum peak of the algorithm with redundant virtual array element (red solid line) is sharper, and the side lobe is lower. Thus, the red line represents the best performance. Similarly, the algorithm resolution with conventional CPA structure is better than the ULAs referring to the higher array aperture.Figure 3
Comparison of resolution performance.


### 4.2. Relationship between SNR and RMSE

We assume three incoherent signals with equal power impinging on the arrays from the directionsθ1=10∘, θ2=20∘, andθ3=30∘. Then, the SNR varies from −10dB to 10dB with an interval of 2dB. Meanwhile, the number of snapshots is L=500 , which is the same with that of Monte-Carlo trials. The noise is assumed to be Gaussian white noise with a mean value of 0 and variance of σ2. In the next all simulations, we will use the Root-MUSIC algorithm to calculate the DOAs of the targets. The formula for calculating root mean square error is defined as(23)RMSE=1LK∑i=1500∑k=1Kθ^ki−θk2,where K denotes the number of the targets, while L denotes the snapshots of the received signal i=1,2,…,K.As shown in Figure4, under the same SNR condition, the root mean square error of the proposed algorithm is the smallest. With the increase of SNR, the algorithm obtains higher estimation accuracy and smaller root mean square error value.Figure 4
Relationship between SNR and RMSE.


### 4.3. Relationship between Snapshots and RMSE

Based on the same assumption in partB, the SNR is 0dB. The number of snapshots L varies from 100 to 1000 with an interval of 100. Moreover, the number of Monte-Carlo trials is 500. The noise is assumed to be Gaussian white noise with a mean value of 0 and variance of σ2.From Figure5, with the increase of snapshots, the root means square error plummets. Under the same number of snapshots, the proposed algorithm has better performance.Figure 5
Relationship between snapshots and RMSE.


## 4.1. Simulation of Spatial Spectral Resolution

In the simulation, a pair of coprime integers are assumed asM=2andN=3. Similarly, the uniform linear array has M=6 physical sensors. Considering three equal power targets impinging on the coprime array from the directions in −20∘,0∘,20∘ with SNR=0dB, the snapshots is L=500. The following formula helps achieve the spatial spectrum:(22)fMUSICθ=1aHθUn2,where aθ denotes the steering vector of the signal and Un consists of eigenvectors corresponding to NMax−K eigenvalues.Figure3 shows the resolution performance of the three algorithms. It can be seen that the spectrum peak of the algorithm with redundant virtual array element (red solid line) is sharper, and the side lobe is lower. Thus, the red line represents the best performance. Similarly, the algorithm resolution with conventional CPA structure is better than the ULAs referring to the higher array aperture.Figure 3
Comparison of resolution performance.


## 4.2. Relationship between SNR and RMSE

We assume three incoherent signals with equal power impinging on the arrays from the directionsθ1=10∘, θ2=20∘, andθ3=30∘. Then, the SNR varies from −10dB to 10dB with an interval of 2dB. Meanwhile, the number of snapshots is L=500 , which is the same with that of Monte-Carlo trials. The noise is assumed to be Gaussian white noise with a mean value of 0 and variance of σ2. In the next all simulations, we will use the Root-MUSIC algorithm to calculate the DOAs of the targets. The formula for calculating root mean square error is defined as(23)RMSE=1LK∑i=1500∑k=1Kθ^ki−θk2,where K denotes the number of the targets, while L denotes the snapshots of the received signal i=1,2,…,K.As shown in Figure4, under the same SNR condition, the root mean square error of the proposed algorithm is the smallest. With the increase of SNR, the algorithm obtains higher estimation accuracy and smaller root mean square error value.Figure 4
Relationship between SNR and RMSE.


## 4.3. Relationship between Snapshots and RMSE

Based on the same assumption in partB, the SNR is 0dB. The number of snapshots L varies from 100 to 1000 with an interval of 100. Moreover, the number of Monte-Carlo trials is 500. The noise is assumed to be Gaussian white noise with a mean value of 0 and variance of σ2.From Figure5, with the increase of snapshots, the root means square error plummets. Under the same number of snapshots, the proposed algorithm has better performance.Figure 5
Relationship between snapshots and RMSE.


## 5. Conclusions

In this paper, a new algorithm via reusing the redundant virtual array elements is proposed for DOA estimation of uncoherent targets. This method averages the virtual array elements in the repeated position. In this way, a new covariance matrix of signal is constructed, which increases the utilization of the received signal data. In addition, Root-MUSIC algorithm is employed to estimate the DOAs of these targets. Unlike the existing algorithm, the improved algorithm makes full use of the information in the redundant virtual array elements. Consequently, the proposed algorithm has greatly improved the resolution performance and reduced estimation error. Furthermore, the algorithm can also be introduced to the MARs and NAs based on the conception of difference coarray and sum-difference coarray. Simulation results show that our algorithm provides better performance for uncoherent signals than conventional methods.



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