# Highly Accurate Multi-Invariance ESPRIT for DOA Estimation with a Sparse Array

Authors: Chen Gu; Hong Hong; Yusheng Li; Xiaohua Zhu; Jin He
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This paper proposes a multi-invariance ESPRIT-based method for estimation of 2D direction (MIMED) of multiple non-Gaussian monochromatic signals using cumulants. In the MIMED, we consider an array geometry containing sparseL-shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor. Firstly, we define a set of cumulant matrices to construct two matrix blocks with multi-invariance property. Then, we develop a multi-invariance ESPRIT-based algorithm with aperture extension using the defined matrix blocks to estimate two-dimensional directions of the signals. The MIMED can provide highly accurate and unambiguous direction estimates by extending the array element spacing beyond a half-wavelength. Finally,we present several simulation results to demonstrate the superiority of the MIMED.


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## Body

## 1. Introduction

Estimation of direction-of-arrival (DOA) and polarization parameters using an array of diversely polarized electromagnetic vector sensors is very important in many applications, such as radar and wireless communications. In the past two decades, many electromagnetic vector sensor direction finding algorithms have been proposed [1–8]. Among them, ESPRIT-based algorithms [2–4] are highly popular due to their low computations and high estimation accuracy. Generally, the ESPRIT-based algorithm exploits the spatial invariance between two identical subarrays to offer direction estimation.Multi-invariance ESPRIT (MI-ESPRIT), first proposed in [9], is an extension of conventional ESPRIT by exploiting the inherent multiple invariances embedded in the array structure. With exploitation of the special multi-invariance structure, MI-ESPRIT can offer better estimation performance than original version of ESPRIT [9].In recent years, MI-ESPRIT algorithm has been applied to solve direction finding problems, where the impinging signals can be MC-CDMA [10], Noncircular [11], and narrowband chirp [12]. However, all the MI-ESPRIT-based algorithms mentioned above are developed using scalar arrays, with array intersensor spacing being not beyond a half-wavelength to guarantee unique and unambiguous direction estimates.In this paper, we apply the idea of MI-ESPRIT to sparse electromagnetic vector sensor arrays to achieve aperture extension for highly accurate direction estimation. We present a multi-invariance ESPRIT-based method for estimation of two-dimensional direction (MIMED) of multiple non-Gaussian signals using cumulants. In the proposed MIMED, we consider an array geometry that contains sparseL-shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor (dipole). Firstly, a set of cumulant matrices is defined to form two matrix blocks that possess the multi-invariance property. Then, a multi-invariance ESPRIT-based algorithm with aperture extension using the defined matrix blocks to estimate two-dimensional directions of the signals is developed. The MIMED can provide highly accurate and unambiguous angle estimates by exploitingthe fact that the spatial phase factor is larger than a half-wavelength.Note also that the MIMED algorithm is related to the algorithm presented in [13] in the sense that both algorithms define the cumulant matrices to estimate the directions of signals. However, since the sensors used in the MIMED algorithm are different from those in [13], the algorithmic derivation and performance bounds are different from those in [13]. For example, (i) [13] is derived based on a scalar sensor array, whereas the MEMID algorithm is derived using a vector sensor array; (ii) the algorithm in [13] defines cumulant matrices to formulate the tensor model, whereas the MIMED algorithm defines cumulant matrices to form the multi-invariance ESPRIT model.Notation. Throughout the paper, superscripts T, H, ∗, and †, respectively, represent the transpose, conjugate transpose, complex conjugate, and pseudoinverse, and ⊗ denotes the Kronecker-product operator. ⌈x⌉ denotes the smallest integer not less than x, ⌊x⌋ represents the largest integer not greater than x, and arg{z} signifies the principal argument of the complex number z between -π and π.


## 2. Signal Model

ConsiderM narrowband completely polarized source signals, parameterized by {θ1,ϕ1,γ1,η1}, θ2,ϕ2,γ2,η2,…,{θM,ϕM,γM,ηM},where0≤θm<π,0≤ϕm<2π,0≤γm<π/2,and-π≤ηm<π, respectively, denote the elevation angle, azimuth angle, orientation angle, and ellipticity angle of themth source, impinging upon an array, which is composed of L¯=2L+1 sparse L-shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor, as shown in Figure 1. The location of the lth (l=1,…,L+1) sensor lied on the x-axis is ((l-1)Dx,0), and the location of the kth (k=L+2,…,L¯) sensor lied on the y-axis is (0,(k-L-1)Dy)), where Dx≫λ and Dy≫λ are the intersensor spacing. Each electromagnetic vector sensor consists of six spatially orthogonal but colocated components: three electric dipoles plus three magnetic loops. The electromagnetic vector sensor response for the mth source signal produces the following 6×1 steering vector [2]:(1)cm=cθm,ϕm,γm,ηm=c1θm,ϕm,γm,ηmc2θm,ϕm,γm,ηmc3θm,ϕm,γm,ηmc4θm,ϕm,γm,ηmc5θm,ϕm,γm,ηmc6θm,ϕm,γm,ηm=defexθm,ϕm,γm,ηmeyθm,ϕm,γm,ηmezθm,γm,ηmhxθm,ϕm,γm,ηmhyθm,ϕm,γm,ηmhzθm,γm=sin⁡γmcos⁡θmcos⁡ϕmejηm-cos⁡γmsin⁡ϕmsin⁡γmcos⁡θmsin⁡ϕmejηm+cos⁡γmcos⁡ϕm-sin⁡γmsin⁡θmejηm-cos⁡γmcos⁡θmcos⁡ϕm-sin⁡γmsin⁡ϕmejηm-cos⁡γmcos⁡θmsin⁡ϕm+sin⁡γmcos⁡ϕmejηmcos⁡γmsin⁡θmwhere the first three entries (em=[ex,m,ey,m,ez,m]T) and the last three entries (hm=[hx,m,hy,m,hz,m]T) represent the electric and the magnetic field vectors, respectively. Note that the array manifold of electromagnetic vector sensors does notcontain the angle-related phase factor. This fact is pivotal to the MIMED in improving the estimation accuracy by extending the array aperture beyond that limited by the spatial Nyquist sampling theorem. Also, note that the Frobenius-norm of the normalized Poynting vector is independent of the parameters of the signal and is equal to unity. That is, the electric field vector em and the magnetic field vector hm are orthogonal to each other and to the source signal’s normalized Poynting vector pm, i.e., [2](2)pm=defem×hmH=defumvmwm=defsin⁡θmcos⁡ϕmsin⁡θmsin⁡ϕmcos⁡θmwhere um, vm, and wm denote the direction cosines of the x-axis, y-axis, and z-axis.Figure 1
Array configuration for the MIMED:L¯=2L+1 sparse L-shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor.Then, the6(L+1)×1 data vectors measured by the first tothe(L+1)th vector sensors at time t can be expressed as(3)zxt=∑m=1Mqxθm,ϕm⊗cmsmt+nxt=Axst+nxtwhere sm(t) denotes the phasor representation of the mth signal, s(t)=[s1(t),…,sM(t)]T denotes the signal vector, Ax=[qx(θ1,ϕ1)⊗c1,…,qx(θM,ϕM)⊗cM], with qx(θm,ϕm)=[1,ej2π/λDxum,…,ej2π/λ(L-1)Dxum]T, and nx(t) represent the 6(L+1)×1 additive noise vector. Likewise, the 6L×1 output vector measured bythe(L+2)th to L¯th vector sensors can be expressed as(4)zyt=∑m=1Mqyθm,ϕm⊗cmsmt+nyt=Ayst+nytwhere Ay=[qy(θ1,ϕ1)⊗c1,…,qy(θM,ϕM)⊗cM], with qy(θm,ϕm)=[ej2π/λDyvm,…,ej2π/λ(L-1)Dyvm]T, and ny(t) represent the 6L×1 additive noise vector.Next, we assume the single polarized scalar sensor is placed arbitrarily at the location(dx,dy). Then, the data collected by the single polarized scalar sensor can be represented as(5)z0t=∑m=1Mq0θm,ϕmsmt+n0twhere q0(θm,ϕm)=ej2π/λ(dxum+dyvm) and n0(t) is additive noise.With a total ofN snapshots taken at {tn:n=1,…,N}, the problem is to determine the azimuth-elevation directions {θm,ϕm,m=1,…,K} from these snapshots. For beamforming purposes, it may be also useful to subsequently estimate the corresponding polarization parameters {γm,ηm,m=1,…,K}. We will present the MIMED algorithm to solve the above-mentioned problems in Section 3, under the following assumptions: (i) the parameters (θ1,ϕ1,γ1,η1),…,(θM,ϕM,γM,ηM) are distinct with each other, and the array steering vector matrix is of full column rank; (ii) the impinging signals are zero-mean and stationary, mutually independent, and non-Gaussian, having nonzero fourth-order cumulants; (iii) the noise is zero-mean, complex Gaussian, and possibly spatially correlated.


## 3. Algorithm Development

### 3.1. Formulation of the Cumulant Matrices

In this subsection, we define2L+1 cumulant matrices that can be linked to multi-invariance model for direction estimation. In forming the cumulant matrices, the single polarized scalar sensor is used as a reference sensor. Let zi,ı(t), zj,ȷ(t), and zp,κ(t) be the data measured by the ıth, ȷth, and κth (ı,ȷ,κ=1,…,6) vector sensor element from the L-shaped vector sensors, and let the fourth-order cumulants of z0(t), zi,ı(t), zj,ȷ(t), and zp,κ(t) be cum⁡(z0(t),zi,ı∗(t),zj,ȷ(t),zp,κ∗(t)). Using the assumptions made in Section 2 and the cumulant properties in [14], we have (6)cum⁡z0t,zi,ı∗t,zj,ȷt,zp,κ∗t=∑m=1Mζmcı∗θm,ϕm,γm,ηmej2π/λx0,i,j,pum+y0,i,j,pvmwhere i,j,p∈{1,…,2L+1} and (7)x0,i,j,p=dx-xi+xj-xpy0,i,j,p=dy-yi+yj-ypwith (xi,yi) being the location of the ith vector sensor, and(8)ζm=cum⁡smt,sm∗t,smt,sm∗t=Esmt4Then, denoting the 6L¯×1 vector z(t)=[zxT(t),zyT(t)]T, we define the L¯6L¯×6L¯ cumulant matrices as(9)Rl=∑k=16cum⁡z0t,zl,k∗t,zt,zHt,l=1,…,L¯where zl,k(t) is the kth row of zl(t). After some mathematical computations, we can obtain(10)R1=AGAH(11)Rl=AΦul-1GAH,l=2,…,L+1(12)Rl¯=AΦvl¯-L-1GAH,l¯=L+2,…,2L+1where(13)G=diag⁡ζ1∑k=16ck∗θ1ej2π/λdxu1+dyv1,…,ζM∑k=16ck∗θMej2π/λdxuM+dyvM(14)Φu=diag⁡ej2π/λDxu1,…,ej2π/λDxuM(15)Φv=diag⁡ej2π/λDyv1,…,ej2π/λDyvMwith θm=(θm,ϕm,γm,ηm).Next, we construct two6(L+1)L¯ data blocks using the defined cumulant matrices as(16)Rx=R1R2⋮RL+1,Ry=R1RL+2⋮R2L+1With the foregoing definitions, we will show in next subsection that Rx and Ry are of multi-invariance properties, which can be exploited to estimate the directions of signals using MI-ESPRIT algorithm.


### 3.2. MI-ESPRIT Algorithm for Direction Estimation

It is easily to verify that the blockRx is of the form(17)Rx=R1R2⋮RL+1=AAΦu⋮AΦuLGAHThe first L¯ rows of Rx can be considered as measurement of signal GAH by physical vector sensors, while the ith L¯ rows ((i-1)L¯+1 to iL¯ rows) of Rx can be viewed as measurement of signal GAH by virtual vector sensors,which are virtually placed by shifting the physical sensors along x-axis with distance (i-1)Dx. Then, for all i=2,…,L+1, altogether L different virtual sensor groups can be formed. Each virtual sensor group has its own spatial invariance, so that L distinct spatial invariances can be provided [9]. Therefore, the matrix (17) is of multi-invariance characteristic [9] which can be exploited for use of MI-ESPRIT for estimating directions. The signal subspace matrix Es,x, which contains the eigenvectors associated with M largest eigenvalues of RxRxH, can be represented as(18)Es,x=AAΦu⋮AΦuLTwhere T is an M×M full rank matrix.LetEs,x,1 and Es,x,2, respectively, be the first 6LL¯ rows and the last 6LL¯ rows of Es,x; we have(19)Es,x,2=Es,x,1T-1ΦuTFrom (19), we can infer that, with the estimation of E^s,x,1 and E^s,x,2, the diagonal elements of Φu can be estimated from eigenvalues of E^s,x,1†E^s,x,2 and the matrix T can be estimated from the eigenvectors of E^s,x,1†E^s,x,2, i.e.,(20)E^s,x,1†E^s,x,2=T^-1Φ^uT^where {[Φ^u]m,m=ej2π/λDxu^m,m=1,2,…,M}. Since Dx>λ/2, a set of ambiguous direction cosine estimates um that satisfy (20) can be obtained.These estimates are expressed as [15](21)u^mnx=μm+nxλDx(22)Dxλ-0.5-μm≤nx≤Dλ0.5-μm(23)μm=arg⁡Φ^um,m2πDx/λSimilarly, we can obtain a set of cyclically ambiguous direction cosine estimatesvm from the data block Ry. These estimates are expressed as [15](24)v^mny=νm+nyλDy(25)Dyλ-0.5-νm≤ny≤Dyλ0.5-νm(26)νm=arg⁡Φ^vm,m2πDy/λwhere {[Φ^v]m,m=ej2π/λDyv^m,m=1,2,…,M}.The above low-variance but cyclically ambiguous direction cosine estimates can be disambiguated by using the set of high-variance but unambiguous direction cosine estimates extracted from the estimation of electromagnetic vector sensor array manifolds. This extraction may be accomplished by decoupling the matrixA^ as(27)C^0=∑l=1L¯A^l∑l=1L¯A^lFwhere A^l denotes the ((l-1)×6+1)th to (l×6)th elements of A^. According to (18), A^ can be estimated as the first 6L¯ rows of E^s,xT^-1. Denoting the mth column of C^0 as gc^m, the high-variance but unambiguous direction cosine estimates of the mth signal can be directly calculated by the vector cross product between the normalized electric field vector and the magnetic field vector, i.e.,(28)p^m=u^mrefv^mrefw^mref=e^me^m×h^mHh^mwhere e^m and h^m, respectively, correspond to the first three and the last three columns of gc^m.Finally, the high-variance but unambiguous direction cosine estimates(u^mref,v^mref),m=1,…,M would serve as reference direction cosine estimates to resolve the cyclically ambiguous direction cosines estimates (u^m(nx),v^m(ny)),m=1,…,M. The disambiguated x-axis and y-axis direction cosines, u^m and v^m, are found from u^m(nx) and v^m(ny) when the value of u^mnx-u^mref and v^mny-v^mref are minimized. Mathematically, the disambiguated x-axis direction cosine estimates u^m are given by(29)u^m=μm+nxoλDwhere nxo is estimated as(30)nxo=argminnx⁡u^mref-μm-nxλDAnalogously, the disambiguated y-axis direction cosine estimates v^m are(31)v^m=νm+nyoλDwhere nyo is estimated as(32)nyo=argminny⁡v^mref-νm-nyλDNote that since u^mref is paired with v^mref, the estimated u^m and v^m are then automatically paired without any additional processing.


### 3.3. Implementation of the MIMED Algorithm

The implementation of the proposed MIMED algorithm is summarized as follows:(S1)
Estimate the cumulant matricesR^l, l=1,…,L¯, from the data samples.(S2)
Form the matricesR^x and R^y using (16); then perform the eigenvalue decomposition to R^xR^xH and R^yR^yH to construct the signal subspaces E^s,x and E^s,y, whose columns correspond to the eigenvectors associated with M largest eigenvalues of E^s,x and E^s,y.(S3)
LetE^s,x,1 and E^s,x,2, respectively, be the first 6LL¯ rows and the last 6LL¯ rows of E^s,x, E^s,y,1, and E^s,y,2, respectively, be the first 6LL¯ rows and the last 6LL¯ rows of E^s,y. Perform eigenvalue decomposition of E^s,x,1†E^s,x,2 and E^s,y,1†E^s,y,2.(S4)
Estimate the ambiguous direction cosine estimatesu^m(nx) and v^m(ny) from (21) to (23) and (24) to (26).(S5)
EstimateC^0 using (27).(S6)
Compute reference direction cosine estimatesu^mref and v^mref using (28).(S7)
Obtain the disambiguated direction cosine estimatesu^m and v^m using (29) and (31).


## 3.1. Formulation of the Cumulant Matrices

In this subsection, we define2L+1 cumulant matrices that can be linked to multi-invariance model for direction estimation. In forming the cumulant matrices, the single polarized scalar sensor is used as a reference sensor. Let zi,ı(t), zj,ȷ(t), and zp,κ(t) be the data measured by the ıth, ȷth, and κth (ı,ȷ,κ=1,…,6) vector sensor element from the L-shaped vector sensors, and let the fourth-order cumulants of z0(t), zi,ı(t), zj,ȷ(t), and zp,κ(t) be cum⁡(z0(t),zi,ı∗(t),zj,ȷ(t),zp,κ∗(t)). Using the assumptions made in Section 2 and the cumulant properties in [14], we have (6)cum⁡z0t,zi,ı∗t,zj,ȷt,zp,κ∗t=∑m=1Mζmcı∗θm,ϕm,γm,ηmej2π/λx0,i,j,pum+y0,i,j,pvmwhere i,j,p∈{1,…,2L+1} and (7)x0,i,j,p=dx-xi+xj-xpy0,i,j,p=dy-yi+yj-ypwith (xi,yi) being the location of the ith vector sensor, and(8)ζm=cum⁡smt,sm∗t,smt,sm∗t=Esmt4Then, denoting the 6L¯×1 vector z(t)=[zxT(t),zyT(t)]T, we define the L¯6L¯×6L¯ cumulant matrices as(9)Rl=∑k=16cum⁡z0t,zl,k∗t,zt,zHt,l=1,…,L¯where zl,k(t) is the kth row of zl(t). After some mathematical computations, we can obtain(10)R1=AGAH(11)Rl=AΦul-1GAH,l=2,…,L+1(12)Rl¯=AΦvl¯-L-1GAH,l¯=L+2,…,2L+1where(13)G=diag⁡ζ1∑k=16ck∗θ1ej2π/λdxu1+dyv1,…,ζM∑k=16ck∗θMej2π/λdxuM+dyvM(14)Φu=diag⁡ej2π/λDxu1,…,ej2π/λDxuM(15)Φv=diag⁡ej2π/λDyv1,…,ej2π/λDyvMwith θm=(θm,ϕm,γm,ηm).Next, we construct two6(L+1)L¯ data blocks using the defined cumulant matrices as(16)Rx=R1R2⋮RL+1,Ry=R1RL+2⋮R2L+1With the foregoing definitions, we will show in next subsection that Rx and Ry are of multi-invariance properties, which can be exploited to estimate the directions of signals using MI-ESPRIT algorithm.


## 3.2. MI-ESPRIT Algorithm for Direction Estimation

It is easily to verify that the blockRx is of the form(17)Rx=R1R2⋮RL+1=AAΦu⋮AΦuLGAHThe first L¯ rows of Rx can be considered as measurement of signal GAH by physical vector sensors, while the ith L¯ rows ((i-1)L¯+1 to iL¯ rows) of Rx can be viewed as measurement of signal GAH by virtual vector sensors,which are virtually placed by shifting the physical sensors along x-axis with distance (i-1)Dx. Then, for all i=2,…,L+1, altogether L different virtual sensor groups can be formed. Each virtual sensor group has its own spatial invariance, so that L distinct spatial invariances can be provided [9]. Therefore, the matrix (17) is of multi-invariance characteristic [9] which can be exploited for use of MI-ESPRIT for estimating directions. The signal subspace matrix Es,x, which contains the eigenvectors associated with M largest eigenvalues of RxRxH, can be represented as(18)Es,x=AAΦu⋮AΦuLTwhere T is an M×M full rank matrix.LetEs,x,1 and Es,x,2, respectively, be the first 6LL¯ rows and the last 6LL¯ rows of Es,x; we have(19)Es,x,2=Es,x,1T-1ΦuTFrom (19), we can infer that, with the estimation of E^s,x,1 and E^s,x,2, the diagonal elements of Φu can be estimated from eigenvalues of E^s,x,1†E^s,x,2 and the matrix T can be estimated from the eigenvectors of E^s,x,1†E^s,x,2, i.e.,(20)E^s,x,1†E^s,x,2=T^-1Φ^uT^where {[Φ^u]m,m=ej2π/λDxu^m,m=1,2,…,M}. Since Dx>λ/2, a set of ambiguous direction cosine estimates um that satisfy (20) can be obtained.These estimates are expressed as [15](21)u^mnx=μm+nxλDx(22)Dxλ-0.5-μm≤nx≤Dλ0.5-μm(23)μm=arg⁡Φ^um,m2πDx/λSimilarly, we can obtain a set of cyclically ambiguous direction cosine estimatesvm from the data block Ry. These estimates are expressed as [15](24)v^mny=νm+nyλDy(25)Dyλ-0.5-νm≤ny≤Dyλ0.5-νm(26)νm=arg⁡Φ^vm,m2πDy/λwhere {[Φ^v]m,m=ej2π/λDyv^m,m=1,2,…,M}.The above low-variance but cyclically ambiguous direction cosine estimates can be disambiguated by using the set of high-variance but unambiguous direction cosine estimates extracted from the estimation of electromagnetic vector sensor array manifolds. This extraction may be accomplished by decoupling the matrixA^ as(27)C^0=∑l=1L¯A^l∑l=1L¯A^lFwhere A^l denotes the ((l-1)×6+1)th to (l×6)th elements of A^. According to (18), A^ can be estimated as the first 6L¯ rows of E^s,xT^-1. Denoting the mth column of C^0 as gc^m, the high-variance but unambiguous direction cosine estimates of the mth signal can be directly calculated by the vector cross product between the normalized electric field vector and the magnetic field vector, i.e.,(28)p^m=u^mrefv^mrefw^mref=e^me^m×h^mHh^mwhere e^m and h^m, respectively, correspond to the first three and the last three columns of gc^m.Finally, the high-variance but unambiguous direction cosine estimates(u^mref,v^mref),m=1,…,M would serve as reference direction cosine estimates to resolve the cyclically ambiguous direction cosines estimates (u^m(nx),v^m(ny)),m=1,…,M. The disambiguated x-axis and y-axis direction cosines, u^m and v^m, are found from u^m(nx) and v^m(ny) when the value of u^mnx-u^mref and v^mny-v^mref are minimized. Mathematically, the disambiguated x-axis direction cosine estimates u^m are given by(29)u^m=μm+nxoλDwhere nxo is estimated as(30)nxo=argminnx⁡u^mref-μm-nxλDAnalogously, the disambiguated y-axis direction cosine estimates v^m are(31)v^m=νm+nyoλDwhere nyo is estimated as(32)nyo=argminny⁡v^mref-νm-nyλDNote that since u^mref is paired with v^mref, the estimated u^m and v^m are then automatically paired without any additional processing.


## 3.3. Implementation of the MIMED Algorithm

The implementation of the proposed MIMED algorithm is summarized as follows:(S1)
Estimate the cumulant matricesR^l, l=1,…,L¯, from the data samples.(S2)
Form the matricesR^x and R^y using (16); then perform the eigenvalue decomposition to R^xR^xH and R^yR^yH to construct the signal subspaces E^s,x and E^s,y, whose columns correspond to the eigenvectors associated with M largest eigenvalues of E^s,x and E^s,y.(S3)
LetE^s,x,1 and E^s,x,2, respectively, be the first 6LL¯ rows and the last 6LL¯ rows of E^s,x, E^s,y,1, and E^s,y,2, respectively, be the first 6LL¯ rows and the last 6LL¯ rows of E^s,y. Perform eigenvalue decomposition of E^s,x,1†E^s,x,2 and E^s,y,1†E^s,y,2.(S4)
Estimate the ambiguous direction cosine estimatesu^m(nx) and v^m(ny) from (21) to (23) and (24) to (26).(S5)
EstimateC^0 using (27).(S6)
Compute reference direction cosine estimatesu^mref and v^mref using (28).(S7)
Obtain the disambiguated direction cosine estimatesu^m and v^m using (29) and (31).


## 4. Simulation Results

Simulation results are presented to demonstrate the efficacy of the MIMED. The array configuration in Figure1 with L¯=15 elements is used. Two narrowband uncorrelated monochromatic signals with identical power impinge on the array. The signal direction cosines u1=0.3, v1=0.2, u2=0.43, and v2=0.35 are simulated. The first signal is left-circularly polarized and the second right-circularly polarized. The snapshots used are N=200 snapshots. 500 independent Monte Carlo trials are performed. Further the additive white noise is assumed to be complex Gaussian. In all the simulations, the performance metric used is the root mean squared error (RMSE) of the first signal.We first assess the performance of the MIMED for various values ofDx=Dy=D, the intersensor spacing. Figure 2 shows, on a log-log scale, the RMSEs of the reference direction estimates (θ^1ref,ϕ^1ref) and the disambiguated direction estimates (θ^1,ϕ^1) as a function of D, varying from 1λ to 81λ, λ being the wavelength. The SNR for each of the signals is set to 20dB. It is seen that the RMSE of the disambiguated direction estimates decrease linearly as the intersensor spacing increases from 1λ up to 25λ. The performance of the reference direction estimates keeps almost unchanged as the sensor spacing increases from 1λ to 81λ. For D>36λ, the disambiguated direction estimates exhibit almost the same statistical errors as the reference direction estimates. This behavior is similar to that in [4] and can be explained as follows. Referring to (29) and (31), the estimates u1(nx) and v1(ny) suffer ambiguities of some unknown integer multiples of the grid size λ/D. As the intersensor spacing D increases, the grid sizes shrink relative to the variances of the reference estimates u^1ref and v^1ref. Therefore, it becomes increasingly probable that u^1ref and v^1ref would identify the wrong grid point. As the intersensor spacing continues to increase, the grid misidentification will become the dominant error, and the disambiguated estimates u^1 and v^1 eventually have the same error statistics as the reference estimates u^1ref and v^1ref. These imply that θ^1 and ϕ^1 would have the same error statistics as θ^1ref and ϕ^1ref. Also note that the performance of the MIMED algorithm is very close to the CRB.Figure 2
RMSEs of the direction cosine estimates versus intersensor spacing, varying from1λ to 81λ. SNR = 20 dB. Two monochromatic signals with digital frequencies f1=0.1,f2=0.4 and direction cosines u1=0.3 and v1=0.2 and u2=0.43 and v2=0.35 impinge upon the array.In the second example, we compare the RMSEs of the MIMED with the cumulant-based ESPRIT algorithm proposed by Liu and Mendel [16] and the second-order statistics based ESPRIT-based algorithm proposed by Zoltowski and Wong [4]. For the algorithm in [16], we assume 96-element arbitrarily-spaced array configuration that contains three presets guiding sensors located at (0,0),(λ/2,0), and (0,λ/2). For the algorithm in [4], we use a 16-element square-shaped electromagnetic vector sensor array. For these two algorithms, the estimated sets of directions are assumed to have been correctly paired. For the MIMED and the one in [4], we set D=5λ. Figure 3 shows the RMSEs of the three algorithms as a function of the SNR, varying from 0 dB to 40 dB. It is seen from the figure that the MIMED has a performance that is better than those of the other two algorithms for a wide range of the SNR (SNR >10dB).Figure 3
RMSEs of the direction cosine estimates versus SNR. Two monochromatic signals with digital frequenciesf1=0.1,f2=0.4 and direction cosines u1=0.3 and v1=0.2 and u2=0.43 and v2=0.35 impinge upon the array.In the third example, we compare the RMSEs of the algorithms as a function of the number of snapshots. The simulation conditions are similar to those in the second example, except that the SNR is set at20 dB, and the number of snapshots is varying from 10 to 2000. The RMSEs of the three algorithms are shown in Figure 4, where the CRBs are also plotted for comparison. It is seen from the figure that the RMSEs of the MIMED algorithm are lower than those of the other two algorithms and are close to the CRBs.Figure 4
RMSEs of the direction cosine estimates versus snapshot number. Two monochromatic signals with digital frequenciesf1=0.1,f2=0.4 and direction cosines u1=0.3 and v1=0.2 and u2=0.43 and v2=0.35 impinge upon the array.


## 5. Conclusions

We have presented a MIMED method for finding directions of non-Gaussian signals using cumulants. By using an array containing sparseL-shaped diversely polarized vector sensors plus an arbitrarily-placed single polarized scalar sensor, we have defined a set of cumulant matrices to construct two matrix blocks with multi-invariance property. We have then developed a MI-ESPRIT-based algorithm using the defined matrix blocks to estimate two-dimensional directions of the signals. The MIMED extends the array aperture by being able to space the sensors much farther apart than a half-wavelength.



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*Source: 5325817-2019-02-18.xml*