# Coherent Signal Parameter Estimation by Exploiting Decomposition of Tensors

Authors: Long Liu; Ling Wang; Yuexian Wang; Jian Xie; Zhaolin Zhang
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

The problem of parameter estimation of coherent signals impinging on an array with vector sensors is considered from a new perspective by means of the decomposition of tensors. Signal parameters to be estimated include the direction of arrival (DOA) and the state of polarization. In this paper, mild deterministic conditions are used for canonical polyadic decomposition (CPD) of the tensor-based signal model; i.e., the factor matrices can be recovered, as long as the matrices satisfy the requirement that at least one is full column rank. In conjoint with the estimation of signal parameters via the algebraic method, the DOAs and polarization parameters of coherent signals can be resolved by virtue of the first and second factor matrices. Hereinto, the key innovation of the proposed approach is that the proposed approach can effectively estimate the coherent signal parameters without sacrificing the array aperture. The superiority of the proposed algorithm is shown by comparing with the algorithms based on higher order singular value decomposition (HOSVD) and Toeplitz matrix. Theoretical and numerical simulations demonstrate the effectiveness of the proposed approach.


---

## Body

## 1. Introduction

The problem of parameter estimation for coherent signals is encountered in a variety of signal processing applications including wireless communication, satellite navigation, and radar. Generally, the signal covariance matrix must be full rank for the subspace-based high-resolution estimation approaches such as multiple signal classification (MUSIC) and estimation of signal parameters via rotational invariance techniques (ESPRITs). However, there is also a fact that cannot be ignored; that is, the presence of the coherent signals can cause the rank loss of the covariance matrix. Under such circumstances, the spatial smoothing technique and its derivative algorithms [1–5] are commonly used as a conventional solution to this problem. According to previous research studies, these approaches have been proved to play an important role in eliminating the rank loss of the covariance matrix, but they are not flawless because an unignorable large aperture loss appears under these approaches as well, degrading the accuracy of parameter estimation. Apart from these approaches, the approaches of parameter estimation for coherent signals based on the Toeplitz matrix signal model have received considerable attention and developments because the rank of the Toeplitz matrix is only related to the DOA of signals and cannot be affected by the coherency between them [6–10]. In addition, in the context of underwater environment and target imaging, the coherent signal analysis methods proposed in [11, 12] are also worthy of attention. For all the approaches mentioned above, they share a common feature that the process of parameter estimation is based on matrix operations.However, the matrix can only reflect the two-dimensional characteristics of the signal. For multidimensional signals, stacking dimensions into one highly structured matrix can effectively improve the accuracy of parameter estimation. In [13], the measurement tensor has been defined, and the estimation of the subspace through higher order singular value decomposition (HOSVD) has been proposed. Similarly, the HOSVD-based algorithms have been studied to improve the performance of parameter estimation in [14–17]. However, in order to achieve the parameter estimation of coherent signals, smoothing for tensor is unavoidable in terms of the HOSVD-based algorithms, which will also lead to the sacrifice of array aperture.Recently, parameter estimation methods by exploiting the canonical polyadic decomposition (CPD) of tensor, which is a minimal decomposition into a sum of rank-1 tensors, have been studied in [18–21]. One common feature of the existing tensor CPD-based parameter estimation methods is that they are all directed to noncoherent signal models. In this paper, we will study the CPD-based approach of parameter estimation under the condition that at least two signals satisfy the coherent premise.Here, we present a signal model based on the third-order tensor, with three dimensions corresponding to the temporal, spatial, and polarized information of the signal. Furthermore, the mild deterministic conditions [22] are used for CPD of the tensor-based signal model; i.e., as long as the coherent signals comply with any one of the spatial and polarized diversities, the proposed approach can effectively estimate the parameters of the coherent signals. The parameters to be estimated mainly include the DOA and polarization parameters delivered by the signal. Computer results are reported as a function of the signal-to-noise ratio (SNR), in comparison to the advanced subspace-based approaches. In addition, the estimation accuracy of polarized parameters is also studied with respect to the different SNR values.The rest of this paper is organized as follows: Section2 derives the tensor-based signal model, followed by the proposed algorithm for extracting the signal parameters in Section 3. Meanwhile, numerical simulations are provided in Section 5. Finally, the last part of the paper will present our concluding remarks. Table 1 summarizes the algebra notations involved in this paper.Table 1
Algebra notations.
v
Scalarv
v
Vectorv
A
MatrixA
A
TensorA
ℝ
Real number field
ℂ
Complex number field
A
T
Transpose of A
A
H
Conjugate transpose of A
A
⊚ℬ
Outer product betweenA and ℬ
A
⊗
B
Kronecker product betweenA and B
A
⊙
B
Khatri–Rao product betweenA and B
A
∘
B
Hadamard product betweenA and B
r
A
Rank of the matrixA
A
F
Frobenius norm of A
v
2
2-norm ofv


## 2. Signal Model

Letθ∈0,2π and ϕ∈0,π serve as the indications of the azimuth angle and the elevation angle. Set the number of electromagnetic (EM) vector sensors in the array as L, and the components measured by an EM vector sensor are indexed as 1,…,J separately. The spatial steering vector for the array with DOA θ,ϕ is given by(1)asθ,ϕ=e−i2πf1,…,e−i2πfLT,where fl=blTϵ/λ, ϵ=−sinϕrcosθrsinϕrsinθrcosϕrT, and bl∈ℝ3 indicates the spatial location of the lth sensor, l=1,…,L. Let Ψ=θ,ϕ,γ,η indicate the spatial-polarization parameter, where γ∈0,π/2 symbolizes the polarization auxiliary angle and η∈−π,π symbolizes the polarization-phase difference. The polarization steering vector can be denoted as(2)apΨ=B−sinθcosϕcosθcosθcosϕsinθ0−sinϕcosϕcosθsinθcosϕsinθ−cosθ−sinϕ0︸Ξθ,ϕhγ,η,where hγ,η=cosγsinγeiηT and B∈ℝJ×6 is defined as the polarization selection matrix for the EM vector sensor. In particular, for the complete EM vector sensor, the polarization selection matrix is represented as B=I6×6 and B=I3×3,03×3 belongs to the three dipoles, in which I symbolizes the unit matrix. The spatial-polarization steering vector of the signal with the spatial-polarization parameter Ψ can be further given by(3)aΨ=asθ,ϕ⊗apΨ.Hereby, theR signals with the complex amplitudes srk,r=1,…,R are assumed to be received by the array. Furthermore, assume that the first R1 signals are coherent and the last R2 signals are noncoherent, but all the signals are in cofrequency, where R1+R2=R. Construct the following equation to represent the vector output of the array at the instant tk:(4)yk=∑r=1RaΨrsrk+nk,where the additive prewhitening noise nk is assumed to have a Gaussian complex circular. In order to better distinguish the multidomain diversities of the signal, we perform tensorization on (3) so that the multidomain diversities of the signal caters to the multidimensional nature of the tensor, and then the following representation is obtained:(5)A^Ψ=apΨ⊚asθ,ϕ.Furthermore, the tensor output of the consecutiveK snapshots for the array can be modeled as(6)Y=∑r=1RA^Ψr⊚sr+N=∑r=1RapΨr⊚asθr,ϕr⊚sr+N,where Y∈ℂJ×L×K and sr=sr1,…,srK. In addition, the noise tensor N∈ℂJ×L×K is yielded by tensorization with respect to the noise matrix N=n1,…,nk.


## 3. The Proposed Approach

LetT=∑r=1Rarp⊚ars⊚sr. Thereby, equation (6) can be expressed as(7)Y=T+N.Considering thatarp∈ℂJ, ars∈ℂL, and sr∈ℂK are nonzero vectors, tensor Sr=arp⊚ars⊚sr is the rank-1 tensor. If the number R of the rank-1 terms in T is minimal, then T=∑rRSr is called the CPD of T and R is called the rank of T (denoted by rT). Since the addition of noise component N, the structure of the tensor Y is significantly different from the structure of the tensor T. Herein, we are not going to talk about the structure of the tensor Y, but instead regard it as the rank R tensor and decompose it. Furthermore, the estimation of the factor matrices can be obtained.


### 3.1. Construction of Intermediate Tensor

The CPD ofY can be expressed as(8)Y=∑r=1Rar⊚br⊚cr,where ar∈ℂJ, br∈ℂL, and cr∈ℂK. Let A=a1,…,aR∈ℂJ×R, B=b1,…,bR∈ℂL×R, and C=c1,…,cR∈ℂK×R. We can also write (8) as Y=A,B,CR, where A, B, and C are called the first, second, and third factor matrix of Y, respectively. Again, we can write T as T=Ap,As,SR, where Ap=apΨ1,…,apΨR, As=asθ1,ϕ1,…,asθR,ϕR, and S=s1,…,sR. Furthermore, we call that the CPD of the tensor is unique when it is the only subject to the trivial indeterminacies, in which the factor matrices can be arbitrarily permuted and the vectors belonging to any factor matrix can be arbitrarily scaled. Obviously, the factor matrices of Y and the factor matrices of T do not satisfy the above indeterminacies due to the presence of the noise. Therefore, A, B, and C are considered as the estimated values of Ap, As, and S, respectively.Herein, we only consider the algebraic algorithm for CPD under the condition that at least one factor matrix ofT has full column rank. That is to say, in the tensor Y, at least one factor matrix contains highly collinear vectors due to the presence of noise disturbances. In order to adapt to the analysis of the coherent signal tensor model, we extend Theorem 2 and Theorem 8 in [22]. The following theorem can be obtained.Theorem 1.
LetT=Ap,As,SR and rAs=L=R, i≥0 Assume that(9)dimkerR2,iT∩S2+iℂL2+i=R.
Then,rT=R, and the CPD of T is unique. In addition, the CPD of T can be obtained algebraically. Under this premise, the proposed algorithm can achieve a unique decomposition of the tensor Y.MatrixRm,iT is defined in Appendix A. Sm+iℂLm+i⊂ℂLm+i denotes the subspace spanned by all vectors of the form x⊗⋯⊗x, where x∈ℂL is repeated m+i times.We hereby execute the decomposition of the tensor-based signal model to obtain the factor matrices, which are then used to extract the parameters of the signals. Assume thatrB=L=R. First of all, the CJmCKm×CL+m−1m matrix QmY needs to be built, and the l1,…,lm-th column of the matrix QmY can be obtained as follows:(10)QmY:,l1,…,lm=vecℱm−1Yl1,…,Ylm,where l1,…,lm∈QLm, Y1,…,YL∈ℂJ×K represents the frontal slices of the tensor Y, and QLm=l1,…,lm:1≤l1≤l2≤…≤lm≤L. An equation for the polarized compound matrix ℱm−1Y1,…,Ym can be defined as(11)ℱm−1Y1,…,Ym=∑k=1m−1m−k∑1≤i1<i2⋯<ik≤mCmYi1+Yi2+⋯+Yik,where the m-th compound matrix CmX of a given matrix X can be obtained from Appendix B.Subsequently, how to get the vectorsw^1,…,w^CRL−1 from the matrix QmY is the focus of this paper. In the absence of noise, for the tensor T, as described in [23], vectors w^1,…,w^CRL−1 form a basis of kerQmT. Combining the construction process of the matrix QmT, the vector w^i satisfying QmTw^i=0 has the following structural features:(12)w^i,1:a=w^i,1,…,w^i,aT,w^i,a+1:CL+m−1m=0,…,w^i,a+i,…,0T,where i∈1,CRL−1, a=CL+m−1m−CRL−1, and the element w^i,a+i is constant 1. Similarly, we can also write QmT as QmT=Qm1T,Qm2T, where the submatrices Qm1T and Qm2T contain the first a columns and the last CRL−1 columns of QmT, respectively. Therefore, the following equation is established:(13)Qm1Tw^i,1:a+Qm2T:,i=0,where Qm2T:,i represents the ith column of the submatrix Qm2T.However, the vectorsw^1,…,w^CRL−1 cannot be obtained by solving the basis of the matrix kerQmY due to the presence of noise. Inspired by (12) and (13) summarized in this paper, an optimization problem model is established as follows:(14)minw^i,1:aQm1Yw^i,1:a+Qm2Y:,i2.Herein, the alternating least squares (ALS) method is used to solve the optimization problem (14), and then the vectorsw^1,…,w^CRL−1 can be obtained according to (12).LetejLj=1L denotes the canonical basis of ℝL, and then ej1L⊗⋯⊗ejmLj1,…,jm∈RLm is the canonical basis of ℝLm, RLm=j1,…,jm:j1,…,jm∈1,…,L. The symmetrizer πS on the space ℝLm can be denoted as(15)πSej1L⊗…⊗ejmL=1m!∑l1,…,lm∈Pj1,…,jmel1L⊗⋯⊗elmL.In addition, the intermediate matrixG∈ℝLm×CL+m−1m needs to be built. The columns of the matrix G can be defined as follows:(16)πSej1L⊗…⊗ejmL:j1,…,jm∈QLm.Furthermore, the matrixW=Gw^1,…,w^CRL−1 can be obtained. Reshape the Lm×CRL−1 matrix W into an L×Lm−1×CRL−1 tensor W. Afterwards, we need to compute the CPD of the tensor W=F,M1,M2CRL−1 by generalized eigenvalue decomposition (GEVD). The tensor CPD procedure by GEVD is shown in [21].


### 3.2. Estimation of Factor Matrices

Once the tensorW is constructed, we can obtain the matrix unfoldings:(17)WI=W1⋮WCRL−1=FDiagm21M1T⋮FDiagm2CRL−1M1T=M2⊙FM1T,where WI∈ℂLCRL−1×Lm−1, Wk′=wi′j′k′i′,j′=1L,Lm−1 symbolizes the k′th frontal slice of W, m2i denotes the ith row of the M2, and Diag⋅ is the diagonalization operator. According to (17), the following equation can be easily derived:(18)Wk1′Wk2′†=FDiagm2k1′Diagm2k2′−1F†,where k1′≠k2′, k1′,k2′=1,…,CRL−1, and ⋅† denotes the Moore–Penrose inverse. Obviously, the columns of the factor matrix F are the eigenvectors of Wk1′Wk2′†. The corresponding eigenvalues are the first CRL−1 larger eigenvalues, which are arranged in a descending order. It is worth noting that when CRL−1>2, there are multiple combinations k1′,k2′ that can be used for EVD. In order to fully exert the statistical characteristics of (17), CH2 possible combinations are selected for EVD to obtain Fh′, where h′=1,…,CH2, H=CRL−1. Herein, we make the agreement that k1′,k2′ tuples are ordered lexicographically: the h1′th tuple k1′,k2′ is preceding the h2′th tuple k3′,k4′ if and only if k1′≤k3′ and k2′≤k4′. According to the trivial indeterminacies aforementioned for CPD uniqueness, Fh1′ and Fh2′ coincide up to column permutation and scaling, h1′≠h2′. Therefore, we need to match multiple sets of results; i.e., all the eigenvectors need to be normalized and arranged in a descending order according to the corresponding eigenvalues. The results of the matching processing are recorded as Fh′′, and the estimation of the factor matrix F can be obtained by performing the following statistical operation:(19)F=1CH2∑h′=1CH2Fh′′.Once the matrixF is obtained, we can recover the matrix B from F. First, compute R submatrices F1,…,FR of CR−1L−2 columns of F that are linearly dependent. Then, compute b1,…,bR as orthogonal complements to the submatrices aforementioned. It is worth noting that, in the presence of noise, the algorithm proposed in this paper only guarantees that brFr2≤10−15. Furthermore, the factor matrices A and C can be recovered, thanks to R1,0YB−T=A⊙C.


### 3.3. Estimation of Signal Parameters

The construction of the factor matrices clearly shows that the factor matrixA contains the spatial-polarization parameters of the signals, while the factor matrix B includes only the spatial ones of them. Putting it in other way, ar and br are the estimated counterparts of apΨr and asθr,ϕr, respectively. Thus, the following equations are going to be used to obtain the spatial DOA of the rth signal impinging on the array:(20)br1=e−i2πb1Tϵr/λ,…,brL=e−i2πbLTϵr/λ.It is obvious that the DOA of therth signal can be procured through any two equations from (20). Suppose θr,ϕrn is the representation of the nth subequations’ solution. Under such circumstances, the DOA can be estimated through the average of multiple sets’ solutions:(21)θr,ϕr=1CL2∑n=1CL2θr,ϕrn.Once the DOA is estimated, a further step is taken to extract the polarization parametersγr,ηr of the rth signal from the following formula:(22)ar=BΞθr,ϕrcosγr,sinγreiηrT.In order to simplify the decomposition of the tensor-based signal model, the value of the snapshotsK is supposed to be as small as possible when K≥R is satisfied. Therefore, we take N-segment K snapshots to improve the estimation accuracy of signal parameters. That is, the signal parameters which are estimated through decomposing over N times are going to be averaged.


## 3.1. Construction of Intermediate Tensor

The CPD ofY can be expressed as(8)Y=∑r=1Rar⊚br⊚cr,where ar∈ℂJ, br∈ℂL, and cr∈ℂK. Let A=a1,…,aR∈ℂJ×R, B=b1,…,bR∈ℂL×R, and C=c1,…,cR∈ℂK×R. We can also write (8) as Y=A,B,CR, where A, B, and C are called the first, second, and third factor matrix of Y, respectively. Again, we can write T as T=Ap,As,SR, where Ap=apΨ1,…,apΨR, As=asθ1,ϕ1,…,asθR,ϕR, and S=s1,…,sR. Furthermore, we call that the CPD of the tensor is unique when it is the only subject to the trivial indeterminacies, in which the factor matrices can be arbitrarily permuted and the vectors belonging to any factor matrix can be arbitrarily scaled. Obviously, the factor matrices of Y and the factor matrices of T do not satisfy the above indeterminacies due to the presence of the noise. Therefore, A, B, and C are considered as the estimated values of Ap, As, and S, respectively.Herein, we only consider the algebraic algorithm for CPD under the condition that at least one factor matrix ofT has full column rank. That is to say, in the tensor Y, at least one factor matrix contains highly collinear vectors due to the presence of noise disturbances. In order to adapt to the analysis of the coherent signal tensor model, we extend Theorem 2 and Theorem 8 in [22]. The following theorem can be obtained.Theorem 1.
LetT=Ap,As,SR and rAs=L=R, i≥0 Assume that(9)dimkerR2,iT∩S2+iℂL2+i=R.
Then,rT=R, and the CPD of T is unique. In addition, the CPD of T can be obtained algebraically. Under this premise, the proposed algorithm can achieve a unique decomposition of the tensor Y.MatrixRm,iT is defined in Appendix A. Sm+iℂLm+i⊂ℂLm+i denotes the subspace spanned by all vectors of the form x⊗⋯⊗x, where x∈ℂL is repeated m+i times.We hereby execute the decomposition of the tensor-based signal model to obtain the factor matrices, which are then used to extract the parameters of the signals. Assume thatrB=L=R. First of all, the CJmCKm×CL+m−1m matrix QmY needs to be built, and the l1,…,lm-th column of the matrix QmY can be obtained as follows:(10)QmY:,l1,…,lm=vecℱm−1Yl1,…,Ylm,where l1,…,lm∈QLm, Y1,…,YL∈ℂJ×K represents the frontal slices of the tensor Y, and QLm=l1,…,lm:1≤l1≤l2≤…≤lm≤L. An equation for the polarized compound matrix ℱm−1Y1,…,Ym can be defined as(11)ℱm−1Y1,…,Ym=∑k=1m−1m−k∑1≤i1<i2⋯<ik≤mCmYi1+Yi2+⋯+Yik,where the m-th compound matrix CmX of a given matrix X can be obtained from Appendix B.Subsequently, how to get the vectorsw^1,…,w^CRL−1 from the matrix QmY is the focus of this paper. In the absence of noise, for the tensor T, as described in [23], vectors w^1,…,w^CRL−1 form a basis of kerQmT. Combining the construction process of the matrix QmT, the vector w^i satisfying QmTw^i=0 has the following structural features:(12)w^i,1:a=w^i,1,…,w^i,aT,w^i,a+1:CL+m−1m=0,…,w^i,a+i,…,0T,where i∈1,CRL−1, a=CL+m−1m−CRL−1, and the element w^i,a+i is constant 1. Similarly, we can also write QmT as QmT=Qm1T,Qm2T, where the submatrices Qm1T and Qm2T contain the first a columns and the last CRL−1 columns of QmT, respectively. Therefore, the following equation is established:(13)Qm1Tw^i,1:a+Qm2T:,i=0,where Qm2T:,i represents the ith column of the submatrix Qm2T.However, the vectorsw^1,…,w^CRL−1 cannot be obtained by solving the basis of the matrix kerQmY due to the presence of noise. Inspired by (12) and (13) summarized in this paper, an optimization problem model is established as follows:(14)minw^i,1:aQm1Yw^i,1:a+Qm2Y:,i2.Herein, the alternating least squares (ALS) method is used to solve the optimization problem (14), and then the vectorsw^1,…,w^CRL−1 can be obtained according to (12).LetejLj=1L denotes the canonical basis of ℝL, and then ej1L⊗⋯⊗ejmLj1,…,jm∈RLm is the canonical basis of ℝLm, RLm=j1,…,jm:j1,…,jm∈1,…,L. The symmetrizer πS on the space ℝLm can be denoted as(15)πSej1L⊗…⊗ejmL=1m!∑l1,…,lm∈Pj1,…,jmel1L⊗⋯⊗elmL.In addition, the intermediate matrixG∈ℝLm×CL+m−1m needs to be built. The columns of the matrix G can be defined as follows:(16)πSej1L⊗…⊗ejmL:j1,…,jm∈QLm.Furthermore, the matrixW=Gw^1,…,w^CRL−1 can be obtained. Reshape the Lm×CRL−1 matrix W into an L×Lm−1×CRL−1 tensor W. Afterwards, we need to compute the CPD of the tensor W=F,M1,M2CRL−1 by generalized eigenvalue decomposition (GEVD). The tensor CPD procedure by GEVD is shown in [21].


## 3.2. Estimation of Factor Matrices

Once the tensorW is constructed, we can obtain the matrix unfoldings:(17)WI=W1⋮WCRL−1=FDiagm21M1T⋮FDiagm2CRL−1M1T=M2⊙FM1T,where WI∈ℂLCRL−1×Lm−1, Wk′=wi′j′k′i′,j′=1L,Lm−1 symbolizes the k′th frontal slice of W, m2i denotes the ith row of the M2, and Diag⋅ is the diagonalization operator. According to (17), the following equation can be easily derived:(18)Wk1′Wk2′†=FDiagm2k1′Diagm2k2′−1F†,where k1′≠k2′, k1′,k2′=1,…,CRL−1, and ⋅† denotes the Moore–Penrose inverse. Obviously, the columns of the factor matrix F are the eigenvectors of Wk1′Wk2′†. The corresponding eigenvalues are the first CRL−1 larger eigenvalues, which are arranged in a descending order. It is worth noting that when CRL−1>2, there are multiple combinations k1′,k2′ that can be used for EVD. In order to fully exert the statistical characteristics of (17), CH2 possible combinations are selected for EVD to obtain Fh′, where h′=1,…,CH2, H=CRL−1. Herein, we make the agreement that k1′,k2′ tuples are ordered lexicographically: the h1′th tuple k1′,k2′ is preceding the h2′th tuple k3′,k4′ if and only if k1′≤k3′ and k2′≤k4′. According to the trivial indeterminacies aforementioned for CPD uniqueness, Fh1′ and Fh2′ coincide up to column permutation and scaling, h1′≠h2′. Therefore, we need to match multiple sets of results; i.e., all the eigenvectors need to be normalized and arranged in a descending order according to the corresponding eigenvalues. The results of the matching processing are recorded as Fh′′, and the estimation of the factor matrix F can be obtained by performing the following statistical operation:(19)F=1CH2∑h′=1CH2Fh′′.Once the matrixF is obtained, we can recover the matrix B from F. First, compute R submatrices F1,…,FR of CR−1L−2 columns of F that are linearly dependent. Then, compute b1,…,bR as orthogonal complements to the submatrices aforementioned. It is worth noting that, in the presence of noise, the algorithm proposed in this paper only guarantees that brFr2≤10−15. Furthermore, the factor matrices A and C can be recovered, thanks to R1,0YB−T=A⊙C.


## 3.3. Estimation of Signal Parameters

The construction of the factor matrices clearly shows that the factor matrixA contains the spatial-polarization parameters of the signals, while the factor matrix B includes only the spatial ones of them. Putting it in other way, ar and br are the estimated counterparts of apΨr and asθr,ϕr, respectively. Thus, the following equations are going to be used to obtain the spatial DOA of the rth signal impinging on the array:(20)br1=e−i2πb1Tϵr/λ,…,brL=e−i2πbLTϵr/λ.It is obvious that the DOA of therth signal can be procured through any two equations from (20). Suppose θr,ϕrn is the representation of the nth subequations’ solution. Under such circumstances, the DOA can be estimated through the average of multiple sets’ solutions:(21)θr,ϕr=1CL2∑n=1CL2θr,ϕrn.Once the DOA is estimated, a further step is taken to extract the polarization parametersγr,ηr of the rth signal from the following formula:(22)ar=BΞθr,ϕrcosγr,sinγreiηrT.In order to simplify the decomposition of the tensor-based signal model, the value of the snapshotsK is supposed to be as small as possible when K≥R is satisfied. Therefore, we take N-segment K snapshots to improve the estimation accuracy of signal parameters. That is, the signal parameters which are estimated through decomposing over N times are going to be averaged.


## 4. Simulation Results

Consider a uniform linear array (ULA) with 4 complete EM vector sensors whose spacing is half wavelength. Suppose there are three signals with the polarization parametersγ1,η1=20∘,90∘, γ2,η2=45∘,90∘, and γ3,η3=60∘,90∘ impinging on the ULA from θ1,ϕ1=30∘,90∘, θ2,ϕ2=10∘,90∘, and θ3,ϕ3=60∘,90∘, respectively. s1 is independent of s2, and s3 is the replica of s2, which means that there is a coherency between s2 and s3. All of the signals have the same power. In addition, the noise component is assumed to be zero-mean additive white Gaussian noise. The root mean square error (RMSE) is defined as(23)RMSEμ=1MR∑m=1M∑r=1Rμ^rm−μr2,where M is the Monte Carlo trial number, μr denotes one of the parameters θr,ϕr,γr,ηr, and μ^rm is the estimation of μr in the mth trial.The DOA estimation of the signals atSNR=20dB with 10 independent trials is illustrated in Figure 1. As expected, our proposed algorithm succeeds in differentiating the signals without paying special attention to their coherency.Figure 1
DOA estimation of the signals with the proposed algorithm atSNR=20dB.For all the following simulations, their results derive from 300 Monte Carlo trials and have been compared with the Cramér–Rao lower bound (CRB) benchmark which is described in Appendix C.In Figure2, the proposed algorithm is compared with two advanced subspace-based algorithms in terms of the RMSE versus number of snapshots for SNR=20dB. It can be seen from Figure 2 that the DOA estimation accuracy of the proposed algorithm outperforms the other two algorithms. Especially for the case of fewer number of snapshots, the advantages of the proposed algorithm are obvious.Figure 2
DOA RMSE versus number of snapshots forSNR=20dB.Figure3 illustrates the superior performance of the proposed algorithm as compared to the other two algorithms when the snapshots number is set as 200. It is worth noting that as SNR increases, the tensor-based methods, i.e., the proposed algorithm and the HOSVD-based algorithm, present a better accuracy of the DOA estimation than the Toeplitz-based algorithm. What is more, the estimation performance of the polarization parameters is examined in Figure 4, which is conducted under the same scenarios with those in Figure 3. According to the operation process mentioned above, we know that, in the proposed algorithm, polarization parameters cannot be estimated independently without obtaining the DOA estimation because the estimation of polarization parameters can only be achieved with the predicted DOA at hand at first. When comparing Figure 4 with Figure 3, we can see clearly that the estimation accuracy of polarization parameters is inferior to that of DOA parameters, and the reason lies in the accumulation of the estimation error during these two steps. Nevertheless, the estimation accuracy of polarization parameters in the proposed algorithm is still better than that of the other two algorithms.Figure 3
DOA RMSE versus SNR forNK=200.Figure 4
Polarization parameter’s RMSE versus SNR forNK=200.


## 5. Conclusions

With the purpose of surmounting the problem of parameter estimation for coherent signals, this paper proposes a new approach which features decomposition for tensors. Specifically, factor matrices can be effectively estimated at first and then be used to extract the parameters of signals irrespective of the coherency between them. In view of this, an obvious advantage of this approach lies in its effective estimation of factor matrices even with noises present. Besides, the problem of aperture loss, which commonly appears in most traditional algorithms, can also be avoided under this approach. Meanwhile, according to the simulation results, our proposed algorithm shows a higher accuracy of parameter estimation in comparison with the state-of-the-art ones.



---

*Source: 5794791-2019-11-13.xml*