# A Mutiparameter Joint Estimation Algorithm for Dual-Polarized Cylindrical Conformal Array in the Presence of Mutual Coupling

Authors: Chao Liu; Feiyang Zhou
Journal: International Journal of Antennas and Propagation
 (2022)

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## Abstract

In this article, we proposed a multiparameter joint estimation algorithm based on the dual-polarized cylindrical conformal array (DCCA) in the presence of mutual coupling. Using the characteristic of the dual-polarized cylindrical conformal array, 2D direction-of-arrival (DOA) estimation can be divided into 1D estimations of elevation and azimuth. Sensors on the boundary of the DCCA are set as auxiliary sensors to eliminate the influence of mutual coupling. Then, elevation can be estimated by the generalized eigenvalues utilizing signal subspace eigenvectors (GEESE). After that, polarization sensitivity can be eliminated by projection transformation and the proposed dual-polarized forward-backward smoothing algorithm. Consequently, a dual-polarized spatial spectrum can be developed to estimate the azimuth based on the estimated elevation. Furthermore, the angles of the signals can be reestimated to improve the accuracy of DOA estimation. Simulation results confirm the effectiveness of the proposed algorithm.


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## Body

## 1. Introduction

Conformal arrays are widely used in aircraft, missiles, unmanned aerial vehicles, and other fields for the advantages of saving space, good aerodynamic characteristics, reducing radar cross-section, and increasing angular coverage [1]. However, due to the limited layout space of the conformal array, the electromagnetic coupling between the array elements becomes more serious [2]. And this effect will significantly degrade the performance of most existing DOA estimation methods. Common methods to eliminate the influence of mutual coupling are the iterative autocorrection method and auxiliary array element method [3]. In [4], the array is firstly divided into several subarrays, and then the MUSIC algorithm is used to realize the DOA estimation algorithm of the conformal array. However, the mutual coupling and polarization sensitivity of the array are not considered. In [5, 6], an iterative self-correction algorithm is proposed, which gradually obtains the optimal solution of angle and mutual coupling through loop iteration. This kind of multidimensional nonlinear solution algorithm has a large amount of calculation and is easy to fall into local optimum. In [7], an effective DOA estimation algorithm for the conformal array is proposed in the presence of mutual coupling. This algorithm reconstructs the steering vector of the subarrays and constructs a segmented space spectrum for DOA estimation. However, the influence of polarization sensitivity is not considered. In [8], a method is proposed to realize the joint estimation of mutual coupling and angle using an auxiliary element. However, the auxiliary element must be accurate and error-free, which is difficult to realize in many occasions.Each polarized signal can be decomposed into two orthogonal components, and they can be modeled as a pair of coherent signals [9]. Thus, elimination of polarization sensitivity can also be seen as a process of decoherence. Common decoherence methods include spatial smoothing algorithm, signal feature vector method, and difference method [10]. Most of these methods are based on a specific array structure without universal applicability. In [11, 12], a DOA and polarization estimation algorithm for arbitrary array configurations is proposed. The algorithm separates the polarization sensitivity and angle information through the decoherence method to achieve DOA estimation. But the algorithm is not suitable for the scenario where mutual coupling and polarization sensitivity coexist. In [13], a blind DOA estimation algorithm is proposed for the conformal array. The algorithm uses a virtual transformation to avoid the effect of the polarization and the inconsistency of element pattern. Then, ESPRIT can be used to realize the DOA estimation. However, this method is based on the fourth-order cumulants of the array output, which brings a lot of calculation.In DOA estimation of the conformal array, mutual coupling and polarization sensitivity still face great challenges. Scholars have proposed some DOA estimation algorithms for conformal array considering only polarization sensitivity or mutual coupling. These algorithms can achieve effective DOA estimation when only considering mutual coupling or polarization sensitivity, but they are no longer applicable with both of them. However, in the DOA of conformal array, mutual coupling and polarization sensitivity are inevitable factors. In view of this, we propose a mutiparameter joint estimation algorithm to solve this problem.In this article, we use a method of stepped estimation to estimate the angle and polarization parameters. The 2D DOA estimation can be divided into two 1D estimations of elevation and azimuth according to the characteristic of DCCA. The influence of mutual coupling can be eliminated by setting the sensors on the boundary of the DCCA as auxiliary sensors. Elevation can be firstly estimated with the GEESE algorithm. Then, polarization sensitivity can be eliminated by projection transformation and dual-polarized forward-backward smoothing algorithm. After eliminating the influence of polarization sensitivity, we construct a spatial spectrum to estimate the azimuth. Consequently, polarization parameters can be estimated based on DOA estimation results. Since the polarization parameters have been obtained, the angles of the signals are reestimated to improve the accuracy of DOA estimation. Cramer-Rao bound (CRB) is also derived as a comparison standard of the estimation performance. Finally, Monte Carlo simulations are carried out to demonstrate the excellent performance of the proposed method.


## 2. Signal Model

Suppose a narrowband far-field signal with arbitrary polarization impinges on the array from directionθ,ϕ, where θ and ϕ denote the elevation and azimuth, respectively. Consider a dual-polarized cylindrical conformal array (DCCA) with M×N elements. The cylindrical array is spatially composed of M layers, and each layer is a ring array with N elements uniformly arranged. The distance between adjacent elements is dxy, and the distance between adjacent layers is dz, as shown in Figure 1.Figure 1
Conformal array coordinate diagram.


### 2.1. Dual-Polarized Cylindrical Conformal Array

Considering the polarization inconsistency of the array elements, the polarization signal need to be decomposed and projected to the local coordinate system of each element to calculate the array response. Without considering the noise, the array output can be expressed as(1)xt=Gθ,ϕTθ,ϕaθ,ϕ⊗I2sθtsϕt,Gθ,ϕ=gθ1′00…00gϕ1′0…000⋱00⋮⋮0gθMN′000…0gϕMN′Tθ,ϕ=T1θ,ϕ0…00T2θ,ϕ…0⋮⋮⋱⋮00…TMNθ,ϕaθ,ϕ=a1θ,ϕ,…,aMNθ,ϕT,where gθn′ and gϕn′n=1,2,…,MN are patterns of θ polarization and ϕ polarization of the n th array element in the local coordinate system, respectively. Tnθ,ϕ represents transformation matrix from global coordinate system to local coordinate system of the n th array element [14]. anθ,ϕ=ejkrnv, where k=2π/λ is the wavenumber, λ is the wavelength of the signal, rn=xn,yn,zn is the position vector, andv=sinθcosϕ,sinθsinϕ,cosθT is the unit direction vector. “⋅T” indicates the transpose of the matrix. I2 represents a 2×2 identity matrix. “⊗” represents the Kronecker product. Thus, the steering vector a˜θ,ϕ of the DCCA is shown as follows:(2)a˜θ,ϕ=Gθ,ϕTθ,ϕaθ,ϕ⊗I2=a˜θθ,ϕ,a˜ϕθ,ϕ.Assume that there areK sources sit, i=1,2,…,K in the space. They impinge on the array at certain angles θ1,ϕ1,θ2,ϕ2,…,θK,ϕK, where θi and ϕi are elevation and azimuth of the i th signal. The equivalent array manifold of the array can be expressed as(3)A˜=a˜θθ1,ϕ1,a˜ϕθ1,ϕ1,…,a˜θθK,ϕK,a˜ϕθK,ϕK.Each polarized signal can be decomposed into two orthogonal polarization signal components, which can be regarded as two coherent components with the same direction but different polarization.(4)siθtsiϕt=sinγiejηicosγisit,where γi and ηi represent the auxiliary polarization angle and polarization phase difference of the i th signal. Define the matrix P containing all polarization parameter information.(5)P=pγ1,η10…00pγ2,η2…0⋮⋮⋱⋮00…pγK,ηK,where pγi,ηi=sinγiejηi,cosγiT.Thus, the array output can be rewritten as(6)xt=A˜Pst+nt,where(7)st=s1t,…,sKtTnt=n1θt,n1ϕt,…,nMNθt,nMNϕtT,st represents the K×1 vectors of the incident signals. nt represents the 2MN×1 zero-mean white Gaussian noise vector with variants σn2.


### 2.2. Dual-Polarized Cylindrical Conformal Array with Mutual Coupling

Considering the influence of mutual coupling on the array, the array manifold should be corrected. Assume that each element in the cylindrical array can only receive the electromagnetic coupling interference generated by copolarization component of adjacent elements.cx, cz, and cxz are the mutual coupling coefficients between elements, as shown in Figure 2 [15].Figure 2
Schematic diagram of mutual coupling of DCCA.According to the spatial structure of the cylindrical array, the mutual coupling matrix of the array can be expressed as follows:(8)C=C1C20⋯0C2C1C2⋯0⋮⋮⋱⋮⋮0⋯C2C1C20⋯0C2C12MN×2MNC1=Toeplitz1,0,cx,0,…,0,cx,0C2=Toeplitzcz,0,cxz,0,…,0,cxz,0.Considering the effect of mutual coupling, (9) can be rewritten as(9)xt=CA˜Pst+nt.Thus, the covariance matrix of the received signal can be expressed as(10)Rx=ExtxHt=CA˜R˜sA˜HCH+σn2Ix,where R˜s=PRsPH. Rs is a K×K matrix, representing the covariance matrix of the signal source. σn2 is the noise power and Ix is the 2MN×2MN identity matrix.


## 2.1. Dual-Polarized Cylindrical Conformal Array

Considering the polarization inconsistency of the array elements, the polarization signal need to be decomposed and projected to the local coordinate system of each element to calculate the array response. Without considering the noise, the array output can be expressed as(1)xt=Gθ,ϕTθ,ϕaθ,ϕ⊗I2sθtsϕt,Gθ,ϕ=gθ1′00…00gϕ1′0…000⋱00⋮⋮0gθMN′000…0gϕMN′Tθ,ϕ=T1θ,ϕ0…00T2θ,ϕ…0⋮⋮⋱⋮00…TMNθ,ϕaθ,ϕ=a1θ,ϕ,…,aMNθ,ϕT,where gθn′ and gϕn′n=1,2,…,MN are patterns of θ polarization and ϕ polarization of the n th array element in the local coordinate system, respectively. Tnθ,ϕ represents transformation matrix from global coordinate system to local coordinate system of the n th array element [14]. anθ,ϕ=ejkrnv, where k=2π/λ is the wavenumber, λ is the wavelength of the signal, rn=xn,yn,zn is the position vector, andv=sinθcosϕ,sinθsinϕ,cosθT is the unit direction vector. “⋅T” indicates the transpose of the matrix. I2 represents a 2×2 identity matrix. “⊗” represents the Kronecker product. Thus, the steering vector a˜θ,ϕ of the DCCA is shown as follows:(2)a˜θ,ϕ=Gθ,ϕTθ,ϕaθ,ϕ⊗I2=a˜θθ,ϕ,a˜ϕθ,ϕ.Assume that there areK sources sit, i=1,2,…,K in the space. They impinge on the array at certain angles θ1,ϕ1,θ2,ϕ2,…,θK,ϕK, where θi and ϕi are elevation and azimuth of the i th signal. The equivalent array manifold of the array can be expressed as(3)A˜=a˜θθ1,ϕ1,a˜ϕθ1,ϕ1,…,a˜θθK,ϕK,a˜ϕθK,ϕK.Each polarized signal can be decomposed into two orthogonal polarization signal components, which can be regarded as two coherent components with the same direction but different polarization.(4)siθtsiϕt=sinγiejηicosγisit,where γi and ηi represent the auxiliary polarization angle and polarization phase difference of the i th signal. Define the matrix P containing all polarization parameter information.(5)P=pγ1,η10…00pγ2,η2…0⋮⋮⋱⋮00…pγK,ηK,where pγi,ηi=sinγiejηi,cosγiT.Thus, the array output can be rewritten as(6)xt=A˜Pst+nt,where(7)st=s1t,…,sKtTnt=n1θt,n1ϕt,…,nMNθt,nMNϕtT,st represents the K×1 vectors of the incident signals. nt represents the 2MN×1 zero-mean white Gaussian noise vector with variants σn2.


## 2.2. Dual-Polarized Cylindrical Conformal Array with Mutual Coupling

Considering the influence of mutual coupling on the array, the array manifold should be corrected. Assume that each element in the cylindrical array can only receive the electromagnetic coupling interference generated by copolarization component of adjacent elements.cx, cz, and cxz are the mutual coupling coefficients between elements, as shown in Figure 2 [15].Figure 2
Schematic diagram of mutual coupling of DCCA.According to the spatial structure of the cylindrical array, the mutual coupling matrix of the array can be expressed as follows:(8)C=C1C20⋯0C2C1C2⋯0⋮⋮⋱⋮⋮0⋯C2C1C20⋯0C2C12MN×2MNC1=Toeplitz1,0,cx,0,…,0,cx,0C2=Toeplitzcz,0,cxz,0,…,0,cxz,0.Considering the effect of mutual coupling, (9) can be rewritten as(9)xt=CA˜Pst+nt.Thus, the covariance matrix of the received signal can be expressed as(10)Rx=ExtxHt=CA˜R˜sA˜HCH+σn2Ix,where R˜s=PRsPH. Rs is a K×K matrix, representing the covariance matrix of the signal source. σn2 is the noise power and Ix is the 2MN×2MN identity matrix.


## 3. Multiparameter Joint Estimation Algorithm

Simultaneous estimation of multiple parameters in the array brings a large amount of calculation, which is almost impossible to realize. In this section, a stepped multiparameter joint estimation algorithm is proposed based on the dual-polarized cylindrical conformal array in the presence of mutual coupling. The 2D DOA estimation can be divided into two 1D DOA estimations. Elevation can firstly be estimated with the GEESE algorithm proposed in this section. Then, using the estimated elevation result, azimuth and polarization parameters can be estimated through the constructed space spectrum. Finally, the angles of the signals can be reestimated to improve the accuracy of DOA estimation.


### 3.1. Elevation Estimation

The cylindrical conformal array can be regarded as a linear array composed of multiple uniform circular arrays. Therefore, the dual-polarized cylindrical conformal array has some properties of a uniform linear array, which provides a theoretical basis for proposing the GEESE algorithm. Then, sensors on the boundary of the DCCA are set as auxiliary sensors to eliminate the influence of mutual coupling. Afterwards, elevation can be estimated with the GEESE algorithm.Considering the structural characteristics of the array, the steering vector of the dual-polarized cylindrical conformal array is rewritten as follows:(11)a˜θ,ϕ=Gθ,ϕTθ,ϕazθ⊗axyθ,ϕ⊗I2,where azθ is the steering vector of the uniform linear subarray.(12)azθ=1,βz,βz2,…,βzM−1Tβzθ=expj2πλdzcosθ.axyθ,ϕ is the steering vector of the uniform circular subarray.(13)axyθ,ϕ=βxy1,…,βxyNTβxyjθ,ϕ=exp−j2πλxjsinθcosϕ+yjsinθsinϕ,where j=1,2,…,N. According to the characteristics of Kronecker product, (17) is rewritten as(14)a˜θ,ϕ=Gθ,ϕTθ,ϕazθ⊗axyθ,ϕ⊗I2=azθ⊗Gθ,ϕTθ,ϕaxyθ,ϕ⊗I2=azθ⊗a˜xyθ,ϕ,where a˜xyθ,ϕ=Gθ,ϕTθ,ϕaxyθ,ϕ⊗I2.Therefore, the steering vector of the dual-polarized cylindrical conformal array can be defined as the Kronecker product of a uniform linear array and a dual-polarized uniform circular conformal array. Then, considering the existence of mutual coupling, the true steering vector is as follows:(15)Ca˜θ,ϕ=Cazθ⊗a˜xyθ,ϕ.The premise of the GEESE algorithm is that the two subarrays divided by the array have similar displacement vectors in space. In order to meet the above condition, it is necessary to discard the top and bottom layer elements of the dual-polarized cylindrical array. Then, divide two subarrays with similar spatial structures from the remaining elements, as shown in Figure3. Define two 2M−3N×2MN selection matrices P1 and P2 to extract the subarrays.where 0 represents 2N×2N zero matrix. I represents the 2N×2N identity matrix. The output vector of the two subarrays can be expressed by the following equations.(17)x1t=P1xt=P1CA˜Pst+P1nt,x2t=P2xt=P2CA˜Pst+P2nt.Figure 3
Spatial structure of subarray.(16)P1=0I0⋯00000I⋯000⋮⋮⋮⋮⋱⋮⋮000⋯I00P2=00I0⋯00000I⋯00⋮⋮⋮⋮⋮⋮⋮000⋯0I0,Thus, the true steering vectors of the two subarrays can be expressed asP1Ca˜θ,ϕ and P2Ca˜θ,ϕ. The steering vectors in the θ polarization and ϕ polarization directions, respectively (22), can be expressed as follows:(18)a˜θ,ϕ=azθ⊗a˜xyθθ,ϕ,a˜xyϕθ,ϕ=a˜θθ,ϕ,a˜ϕθ,ϕ,where a˜xyθθ,ϕ and a˜xyϕθ,ϕ are 2N×1 steering vectors and a˜θθ,ϕ and a˜ϕθ,ϕ are 2MN×1 steering vector matrices. Taking the θ polarization direction as an example, the true steering vectors of the two subarrays P1Ca˜θ and P2Ca˜θ are as follows:(19)P1Ca˜θ=0I0⋯00000I⋯000⋮⋮⋮⋯⋮⋮⋮000⋯I00C1C20⋯0C2C1C2⋯0⋮⋮⋱⋮⋮0⋯C2C1C20⋯0C2C1.a˜xyθβza˜xyθ⋮βzM−1a˜xyθ=C2a˜xyθ+C1βza˜xyθ+C2βz2a˜xyθC2βza˜xyθ+C1βz2a˜xyθ+C2βz3a˜xyθ⋮C2βzM−4a˜xyθ+C1βzM−3a˜xyθ+C2βzM−2a˜xyθ,(20)P1Ca˜θ=0I0⋯00000I⋯000⋮⋮⋮⋯⋮⋮⋮000⋯I00C1C20⋯0C2C1C2⋯0⋮⋮⋱⋮⋮0⋯C2C1C20⋯0C2C1.a˜xyθβza˜xyθ⋮βzM−1a˜xyθ=C2a˜xyθ+C1βza˜xyθ+C2βz2a˜xyθC2βza˜xyθ+C1βz2a˜xyθ+C2βz3a˜xyθ⋮C2βzM−4a˜xyθ+C1βzM−3a˜xyθ+C2βzM−2a˜xyθ,From (28)–(29), the following equation can be obtained:(21)P2Ca˜θ=P1Ca˜θβz.Theϕ polarization component has similar results, and the following equation can be obtained:(22)P2Ca˜ϕ=P1Ca˜ϕβzwhen there areK sources in the space. Moreover, the following equation can be obtained:(23)P2CA˜P=P1CA˜Φ⊗I2P=P1CA˜PΦ,where Φ=diagβz1,βz2,…,βzK is a K×K matrix containing the elevation information but not mixed with other information. βzi is the simplified representation of βzθi. a˜θ and a˜xyθ are simplified representations of a˜θθ,ϕ and a˜xyθθ,ϕ, respectively. Thus, the following equation can be obtained:(24)x1t=P1xt=P1CA˜Pst+P1nt,x2t=P2xt=P1CA˜PΦst+P2nt.Then, the GEESE algorithm can be used to estimate the elevations. The signal subspaceUs can be obtained through the eigenvalue decomposition of the covariance matrix Rx. Define Us1=P1Us, Us2=P2Us. The signal subspace spanned by the eigenvectors corresponding to the eigenvalues is the same as the subspace spanned by the array manifold matrix [16]. Thus, there is a K×K full rank matrix T that satisfied the following equation:(25)Us1Us2=A¯TA¯ΦT,where A¯=P1CA˜P. νi=ν1,ν2,…,νK is defined as the generalized eigenvalue of the matrix pair Us1HUs1,Us1HUs2.(26)Us1HUs1−νiUs1HUs2=THA¯HA¯IK−νiΦT=0i=1,2,…,K.Since the matrixT and array manifold A¯ have the same rank K, it can be obtained from formula (26):(27)IK−νiΦ=0.Thus, the eigenvalues of matrix Φ are equal to νi. The elevation estimator can be estimated.(28)θi=cos−1λ2πdzargνi,where “argx” represents the phase of x.


### 3.2. Azimuth and Polarization Parameter Estimation

Each polarized signal is regarded as two orthogonal polarized components. The two components can be treated as a pair of coherent signals that have the same direction but different polarization. The traditional decoherence methods are only suitable for planar arrays but not suitable for conformal arrays. Therefore, the dual-polarized cylindrical conformal array must be projected into a rectangular array. After using the projection transformation and spatial smoothing algorithm, the dual-polarized cylindrical conformal array signal can make decoherence.The projection operator satisfiesPA∣UCA˜=Av and PA∣UUn=0 [17]. Av is the array manifold of the virtualized dual-polarized rectangular array, which will be used as the signal subspace of the oblique projection. Un is the noise subspace after eigendecomposition of the covariance matrix of equation (16). The orthogonal projection matrix PU⊥ of the noise subspace Un is shown as follows:(29)PU⊥=IU−PU,PU=UnUnHUn−1UnH,where IU is a 2MN×2MN identity matrix.Then, the oblique projection matrixPA|U is expressed as follows:(30)PA∣U=AvCA˜HPU⊥CA˜−1CA˜HPU⊥.From (16), the projected covariance matrix is written as follows:(31)R=PA∣UHRxPA∣U=PA∣UHCA˜R˜sPA∣UHCA˜H+σn2PA∣UHPA∣U.For the existence of the projection matrix, the noise will no longer be white. In order to reconstruct white noise, the new projection matrix is defined as follows:(32)P˜A∣U=PA∣UHPA∣U−1/2PA∣UH.Thus, the projected covariance matrix is rewritten as follows:(33)R˜=P˜A∣URxP˜A∣UH.Due to the particularity of a dual-polarized conformal array, smoothing should be carried out between copolarization components. Through the following forward and backward smoothing, a new covariance matrix can be constructed as follows [18]:(34)Rfb=R˜+DR˜∗D2D=JD00000JD00000⋱00000JD00000JD,where D=IM⊗JD is the dual-polarized forward and backward smoothing matrix. Matrix D ensures that the copolarized components are smoothed, and the cross-polarized components do not affect each other. JD=JN⊗I2 is a 2N×2N matrix. I2 is a 2×2 identity matrix, JN is a N×N inverse identity diagonal matrix, and IM is a M×M identity matrix. “⋅∗ ” represents complex conjugate of the matrix. The proof process is shown in Appendix A.Through the above smoothing, the rank of covariance matrix is restored fromK to 2K, and then Rfb is expressed as follows:(35)Rfb=UsfbΛsUsfb+UnfbΛnUnfb.Usfb is a 2MN×2K matrix and Unfb is a 2MN×2MN−2K matrix. Λs and Λn are the diagonal matrices composed of corresponding eigenvalues.According to the principle of subspace orthogonality, the steering vector of the decoherent array signal is orthogonal to the noise subspace. The following equation is satisfied:(36)UnfbHP˜A|UCa˜θi,ϕi=0i=1,2,…,K.Since we have already got the estimated values of elevation in Subsection3.1, we can define Qθθ^i,ϕ and Qϕθ^i,ϕ to construct the spatial estimation spectrum:(37)Qθθ^i,ϕ=UnfbHP˜A|UCa˜θθ^i,ϕ2,Qϕθ^i,ϕ=UnfbHP˜A|UCa˜ϕθ^i,ϕ2.When the scanning angle is equal to the actual arrival angle of the signal, at least one of the above two functions is zero. Therefore, the following spectrum can be constructed to estimate the azimuth:(38)Pθ^i,ϕ=1minQθθ^i,ϕ,Qϕθ^i,ϕ.The azimuth can be estimated by searching the highest peak of the spatial spectrum.Consequently, the polarization parametersγ and η can be estimated using the obtained DOA estimation result θ^i,ϕ^i. The steering vector of the array considering the polarization sensitivity is expressed as follows:(39)a˜θ^i,ϕ^i,γ,η=a˜θ^i,ϕ^ipγ,η.The polarization parameters can be estimated through the following equation:(40)γ^i,η^i=argmaxγ,η1UnHCa˜θ^i,ϕ^i,γ,η2,where Un is the noise subspace of Rx. Each set of polarization parameters corresponds to the angle estimation.Using the estimated polarization parametersγ^i,η^i, the angles of the signals can be reestimated to improve the accuracy of DOA estimation. The steering vector is rewritten as follows:(41)a˜θ,ϕ,γ^i,η^i=a˜θ,ϕpγ^i,η^i.Then, a spatial spectrum can be constructed according to the orthogonality of the subspace to complete the DOA reestimation:(42)Pθ,ϕ=∑i=1K1UnHCa˜θ,ϕ,γ^i,η^i2.From (42), the reestimation of DOA can be achieved.


## 3.1. Elevation Estimation

The cylindrical conformal array can be regarded as a linear array composed of multiple uniform circular arrays. Therefore, the dual-polarized cylindrical conformal array has some properties of a uniform linear array, which provides a theoretical basis for proposing the GEESE algorithm. Then, sensors on the boundary of the DCCA are set as auxiliary sensors to eliminate the influence of mutual coupling. Afterwards, elevation can be estimated with the GEESE algorithm.Considering the structural characteristics of the array, the steering vector of the dual-polarized cylindrical conformal array is rewritten as follows:(11)a˜θ,ϕ=Gθ,ϕTθ,ϕazθ⊗axyθ,ϕ⊗I2,where azθ is the steering vector of the uniform linear subarray.(12)azθ=1,βz,βz2,…,βzM−1Tβzθ=expj2πλdzcosθ.axyθ,ϕ is the steering vector of the uniform circular subarray.(13)axyθ,ϕ=βxy1,…,βxyNTβxyjθ,ϕ=exp−j2πλxjsinθcosϕ+yjsinθsinϕ,where j=1,2,…,N. According to the characteristics of Kronecker product, (17) is rewritten as(14)a˜θ,ϕ=Gθ,ϕTθ,ϕazθ⊗axyθ,ϕ⊗I2=azθ⊗Gθ,ϕTθ,ϕaxyθ,ϕ⊗I2=azθ⊗a˜xyθ,ϕ,where a˜xyθ,ϕ=Gθ,ϕTθ,ϕaxyθ,ϕ⊗I2.Therefore, the steering vector of the dual-polarized cylindrical conformal array can be defined as the Kronecker product of a uniform linear array and a dual-polarized uniform circular conformal array. Then, considering the existence of mutual coupling, the true steering vector is as follows:(15)Ca˜θ,ϕ=Cazθ⊗a˜xyθ,ϕ.The premise of the GEESE algorithm is that the two subarrays divided by the array have similar displacement vectors in space. In order to meet the above condition, it is necessary to discard the top and bottom layer elements of the dual-polarized cylindrical array. Then, divide two subarrays with similar spatial structures from the remaining elements, as shown in Figure3. Define two 2M−3N×2MN selection matrices P1 and P2 to extract the subarrays.where 0 represents 2N×2N zero matrix. I represents the 2N×2N identity matrix. The output vector of the two subarrays can be expressed by the following equations.(17)x1t=P1xt=P1CA˜Pst+P1nt,x2t=P2xt=P2CA˜Pst+P2nt.Figure 3
Spatial structure of subarray.(16)P1=0I0⋯00000I⋯000⋮⋮⋮⋮⋱⋮⋮000⋯I00P2=00I0⋯00000I⋯00⋮⋮⋮⋮⋮⋮⋮000⋯0I0,Thus, the true steering vectors of the two subarrays can be expressed asP1Ca˜θ,ϕ and P2Ca˜θ,ϕ. The steering vectors in the θ polarization and ϕ polarization directions, respectively (22), can be expressed as follows:(18)a˜θ,ϕ=azθ⊗a˜xyθθ,ϕ,a˜xyϕθ,ϕ=a˜θθ,ϕ,a˜ϕθ,ϕ,where a˜xyθθ,ϕ and a˜xyϕθ,ϕ are 2N×1 steering vectors and a˜θθ,ϕ and a˜ϕθ,ϕ are 2MN×1 steering vector matrices. Taking the θ polarization direction as an example, the true steering vectors of the two subarrays P1Ca˜θ and P2Ca˜θ are as follows:(19)P1Ca˜θ=0I0⋯00000I⋯000⋮⋮⋮⋯⋮⋮⋮000⋯I00C1C20⋯0C2C1C2⋯0⋮⋮⋱⋮⋮0⋯C2C1C20⋯0C2C1.a˜xyθβza˜xyθ⋮βzM−1a˜xyθ=C2a˜xyθ+C1βza˜xyθ+C2βz2a˜xyθC2βza˜xyθ+C1βz2a˜xyθ+C2βz3a˜xyθ⋮C2βzM−4a˜xyθ+C1βzM−3a˜xyθ+C2βzM−2a˜xyθ,(20)P1Ca˜θ=0I0⋯00000I⋯000⋮⋮⋮⋯⋮⋮⋮000⋯I00C1C20⋯0C2C1C2⋯0⋮⋮⋱⋮⋮0⋯C2C1C20⋯0C2C1.a˜xyθβza˜xyθ⋮βzM−1a˜xyθ=C2a˜xyθ+C1βza˜xyθ+C2βz2a˜xyθC2βza˜xyθ+C1βz2a˜xyθ+C2βz3a˜xyθ⋮C2βzM−4a˜xyθ+C1βzM−3a˜xyθ+C2βzM−2a˜xyθ,From (28)–(29), the following equation can be obtained:(21)P2Ca˜θ=P1Ca˜θβz.Theϕ polarization component has similar results, and the following equation can be obtained:(22)P2Ca˜ϕ=P1Ca˜ϕβzwhen there areK sources in the space. Moreover, the following equation can be obtained:(23)P2CA˜P=P1CA˜Φ⊗I2P=P1CA˜PΦ,where Φ=diagβz1,βz2,…,βzK is a K×K matrix containing the elevation information but not mixed with other information. βzi is the simplified representation of βzθi. a˜θ and a˜xyθ are simplified representations of a˜θθ,ϕ and a˜xyθθ,ϕ, respectively. Thus, the following equation can be obtained:(24)x1t=P1xt=P1CA˜Pst+P1nt,x2t=P2xt=P1CA˜PΦst+P2nt.Then, the GEESE algorithm can be used to estimate the elevations. The signal subspaceUs can be obtained through the eigenvalue decomposition of the covariance matrix Rx. Define Us1=P1Us, Us2=P2Us. The signal subspace spanned by the eigenvectors corresponding to the eigenvalues is the same as the subspace spanned by the array manifold matrix [16]. Thus, there is a K×K full rank matrix T that satisfied the following equation:(25)Us1Us2=A¯TA¯ΦT,where A¯=P1CA˜P. νi=ν1,ν2,…,νK is defined as the generalized eigenvalue of the matrix pair Us1HUs1,Us1HUs2.(26)Us1HUs1−νiUs1HUs2=THA¯HA¯IK−νiΦT=0i=1,2,…,K.Since the matrixT and array manifold A¯ have the same rank K, it can be obtained from formula (26):(27)IK−νiΦ=0.Thus, the eigenvalues of matrix Φ are equal to νi. The elevation estimator can be estimated.(28)θi=cos−1λ2πdzargνi,where “argx” represents the phase of x.


## 3.2. Azimuth and Polarization Parameter Estimation

Each polarized signal is regarded as two orthogonal polarized components. The two components can be treated as a pair of coherent signals that have the same direction but different polarization. The traditional decoherence methods are only suitable for planar arrays but not suitable for conformal arrays. Therefore, the dual-polarized cylindrical conformal array must be projected into a rectangular array. After using the projection transformation and spatial smoothing algorithm, the dual-polarized cylindrical conformal array signal can make decoherence.The projection operator satisfiesPA∣UCA˜=Av and PA∣UUn=0 [17]. Av is the array manifold of the virtualized dual-polarized rectangular array, which will be used as the signal subspace of the oblique projection. Un is the noise subspace after eigendecomposition of the covariance matrix of equation (16). The orthogonal projection matrix PU⊥ of the noise subspace Un is shown as follows:(29)PU⊥=IU−PU,PU=UnUnHUn−1UnH,where IU is a 2MN×2MN identity matrix.Then, the oblique projection matrixPA|U is expressed as follows:(30)PA∣U=AvCA˜HPU⊥CA˜−1CA˜HPU⊥.From (16), the projected covariance matrix is written as follows:(31)R=PA∣UHRxPA∣U=PA∣UHCA˜R˜sPA∣UHCA˜H+σn2PA∣UHPA∣U.For the existence of the projection matrix, the noise will no longer be white. In order to reconstruct white noise, the new projection matrix is defined as follows:(32)P˜A∣U=PA∣UHPA∣U−1/2PA∣UH.Thus, the projected covariance matrix is rewritten as follows:(33)R˜=P˜A∣URxP˜A∣UH.Due to the particularity of a dual-polarized conformal array, smoothing should be carried out between copolarization components. Through the following forward and backward smoothing, a new covariance matrix can be constructed as follows [18]:(34)Rfb=R˜+DR˜∗D2D=JD00000JD00000⋱00000JD00000JD,where D=IM⊗JD is the dual-polarized forward and backward smoothing matrix. Matrix D ensures that the copolarized components are smoothed, and the cross-polarized components do not affect each other. JD=JN⊗I2 is a 2N×2N matrix. I2 is a 2×2 identity matrix, JN is a N×N inverse identity diagonal matrix, and IM is a M×M identity matrix. “⋅∗ ” represents complex conjugate of the matrix. The proof process is shown in Appendix A.Through the above smoothing, the rank of covariance matrix is restored fromK to 2K, and then Rfb is expressed as follows:(35)Rfb=UsfbΛsUsfb+UnfbΛnUnfb.Usfb is a 2MN×2K matrix and Unfb is a 2MN×2MN−2K matrix. Λs and Λn are the diagonal matrices composed of corresponding eigenvalues.According to the principle of subspace orthogonality, the steering vector of the decoherent array signal is orthogonal to the noise subspace. The following equation is satisfied:(36)UnfbHP˜A|UCa˜θi,ϕi=0i=1,2,…,K.Since we have already got the estimated values of elevation in Subsection3.1, we can define Qθθ^i,ϕ and Qϕθ^i,ϕ to construct the spatial estimation spectrum:(37)Qθθ^i,ϕ=UnfbHP˜A|UCa˜θθ^i,ϕ2,Qϕθ^i,ϕ=UnfbHP˜A|UCa˜ϕθ^i,ϕ2.When the scanning angle is equal to the actual arrival angle of the signal, at least one of the above two functions is zero. Therefore, the following spectrum can be constructed to estimate the azimuth:(38)Pθ^i,ϕ=1minQθθ^i,ϕ,Qϕθ^i,ϕ.The azimuth can be estimated by searching the highest peak of the spatial spectrum.Consequently, the polarization parametersγ and η can be estimated using the obtained DOA estimation result θ^i,ϕ^i. The steering vector of the array considering the polarization sensitivity is expressed as follows:(39)a˜θ^i,ϕ^i,γ,η=a˜θ^i,ϕ^ipγ,η.The polarization parameters can be estimated through the following equation:(40)γ^i,η^i=argmaxγ,η1UnHCa˜θ^i,ϕ^i,γ,η2,where Un is the noise subspace of Rx. Each set of polarization parameters corresponds to the angle estimation.Using the estimated polarization parametersγ^i,η^i, the angles of the signals can be reestimated to improve the accuracy of DOA estimation. The steering vector is rewritten as follows:(41)a˜θ,ϕ,γ^i,η^i=a˜θ,ϕpγ^i,η^i.Then, a spatial spectrum can be constructed according to the orthogonality of the subspace to complete the DOA reestimation:(42)Pθ,ϕ=∑i=1K1UnHCa˜θ,ϕ,γ^i,η^i2.From (42), the reestimation of DOA can be achieved.


## 4. Simulation Results

In the simulation experiment, a dual-polarized cylindrical conformal array with10×15 elements is considered. The array elements adopt the dual-polarized patch antennas. The spacing between the adjacent elements is half the wavelength. The patterns of the antenna elements are as follows [19]:(43)gθθ′,ϕ′=J2πdsinθλ′−J0πdsinθ′λ,⋅cosϕ′−jsinϕ′,0≤θ′≤π2,gϕθ′,ϕ′=J2πdsinθλ′+J0πdsinθ′λ,⋅cosθ′sinϕ′−jcosϕ′,0≤θ′≤π2,gθθ′,ϕ′=gϕθ′,ϕ′=0,θ′>π2,where J0 and J2 are the zero-order and second-order Bessel functions of the first kind, respectively.In the first simulation, we assume there are three independent far-field sources from directions35°,−30°45°,15°65°,40° in the space impinging on the array. The polarization parameters of the three sources are 20°,60°30°,45°60°,30°, respectively. We set the scan ranges of the elevation and azimuth to 0°,140° and −70°,70°, respectively. And the scan step size is set to 0.1°. The mutual coupling coefficients are set to cx=0.5+0.2i, cz=0.4+0.1i, and cxz=0.2−0.2i, respectively. The signal-to-noise (SNR) is fixed to 10 dB. And the number of the snapshots is fixed to 200.The elevation initial estimation result is shown in Figure4(a). And the spectra of the three azimuth initial estimation are shown in Figures 4(b)–4(d), respectively. Figures 5–7 plot the spectral peaks corresponding to the polarization parameter estimation results. Then, the spatial spectrum obtained by (42) is shown in Figure 8. From the figure, it can be seen that the spectrum peaks coincide with the real signal DOAs. In order to reduce the amount of calculation, we set the scan step size to 2° during the angle refined estimation, so there will be an angle estimation deviation of about 1°. Secondary refine scanning can be carried out subsequently with a step of 0.1° to acquire higher accuracy results. From the simulation results, we can see the proposed multiparameter joint estimation algorithm can realize the estimation of the elevation, azimuth, and polarization parameters, which demonstrate the effectiveness of the proposed algorithm.Figure 4
The DOA initial estimation.
(a)(b)(c)(d)Figure 5
Polarization estimation withθ,ϕ=30°,−40°.Figure 6
Polarization estimation withθ,ϕ=45°,15°.Figure 7
Polarization estimation withθ,ϕ=65°,40°.Figure 8
DOA refined estimation result.Furthermore, Monte Carlo experiments are carried to verify the estimation accuracy of the proposed algorithm. The root mean square error (RMSE) of the parameter estimation is calculated through 100 Monte Carlo experimentsMc=100 and compared with the CRB lower bound. The derivation of CRB can refer to in Appendix B. The RMSE of DOA and polarization parameters can be defined as follows:(44)RMSEθ=1McK∑m=1Mc∑i=1Kθi−θ^i,m212RMSEϕ=1McK∑m=1Mc∑i=1Kϕi−ϕ^i,m212RMSEγ=1McK∑m=1Mc∑i=1Kγi−γ^i,m212RMSEη=1McK∑m=1Mc∑i=1Kηi−η^i,m212.In the second simulation, the SNR changes from 0 dB to 20 dB with the step of 2 dB. The other conditions are consistent with the first simulation. Figure9 plots the RMSE curves of initial and refined angle estimation with the SNR increases. And the performance of refined angle estimation is close to the CRB. Figure 10 plots the RMSE curves of polarization parameters for SNR from 0 dB to 20 dB with the step of 2 dB.Figure 9
RMSE of DOA with different SNR.Figure 10
RMSE of polarization parameters with different SNR.In the third simulation, the snapshots change from 100 to 1000 with a step of 100. And the SNR is fixed to 5 dB. The other conditions are consistent with the first simulation. Figure11 plots the RMSE curves of initial and refined angle estimation with the snapshots increases. Figure 12 plots the RMSE curves of polarization parameters for snapshots from 100 to 1000 with the step of 100.Figure 11
RMSE of DOA with different snapshots.Figure 12
RMSE of polarization parameters with different snapshots.In Figures9 and 11, the DOA estimation performance gets better and better with the increase of SNR or snapshots. Then, the RMSE curves have a downward trend. With the increase of SNR and snapshots, the DOA estimation RMSE quickly decreases and approaches the Cramer-Rao lower bound.In Figures10 and 12, the RMSE values of the polarization parameters estimation gradually decrease regardless of the increase in the SNR or the snapshots. Compared with DOA estimation, the accuracy of polarization parameter estimation is relatively low because the polarization parameter estimation only has two receiving data in the polarization parameter estimation. But in the spatial domain, there are 10×15 elements for DOA estimation, so it is reasonable that the performance of DOA estimation is higher than that of polarization parameter estimation. The RMSE curves of the polarization parameters approach the CRB lower bound curve with the increase of the number of snapshots or SNR, which indicates that the polarization parameter estimation also has good performance.


## 5. Conclusion

A multiparameter joint estimation algorithm has been proposed based on the dual-polarized cylindrical conformal array in the presence of mutual coupling. The 2D DOA estimation is firstly divided into the 1D estimation of elevation and azimuth. By setting the elements on the boundary of the DCCA as auxiliary sensors, the effect of mutual coupling is proved to be eliminated, and elevation can firstly be estimated by the GEESE algorithm. Then, polarization sensitivity is eliminated by oblique projection transformation and the dual-polarized forward-backward smoothing technique. After eliminating the influence of polarization sensitivity, we construct a spatial spectrum to estimate the azimuth. After that, a spectrum in the polarization domain can be constructed based on the obtained DOA estimation. Furthermore, the angles of the signals are reestimated to improve the accuracy of DOA estimation by using all the output of the array. Finally, Monte Carlo simulations are carried out to demonstrate the excellent performance of the algorithm.



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*Source: 7321449-2022-02-08.xml*