# Joint Estimation of DOA and Polarization with CLD Pair Cylindrical Array Based on Quaternion Model

Authors: Guibao Wang; Haihong Tao; Lanmei Wang; Cao Zeng
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

This paper proposes a quaternion-ESPRIT algorithm for closed-form estimation of direction of arrival (DOA) and polarization, using a cylindrical conformal array configuration. First, the array steering vector and the elevation are estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Second, the azimuth is estimated. Finally, the polarization parameters are estimated using the relationships between the dipoles and the loops. These estimates are automatically matched. Simulation results show that the performance of quaternion method is obviously better than that of the long-vector method.


---

## Body

## 1. Introduction

In the last decades, thanks to advances in sensor technology, electromagnetic vector sensor (EMVS) array was widely used in source localization. The problems of estimating direction of arrival (DOA) and polarization parameters have been discussed in many articles. The first direction-finding algorithms, explicitly exploiting all six electromagnetic components, have been developed separately by Nehorai and Paldi [1, 2] and Li [3]. The cross-product-based DOA estimation algorithm was first adapted to ESPRIT by Wong and Zoltowski [4–7]. A uni-vector-sensor ESPRIT algorithm was proposed in [8]. The aforementioned six-component electromagnetic vector sensors are called complete EMVS. The incomplete EMVS antenna configurations have been extensively studied by numerous researchers, collocated loop and dipole (CLD) oriented along the z-axis of Cartesian coordinate can measure the electric-field and magnetic-field z-component [9, 10], as shown in Figure 1. It is noteworthy that the polarization parameter estimation using a CLD array is independent of the sources DOA and it requires no prior information of azimuth and elevation angles [11]. This independence can not be applied to other antenna configurations, for example two identical dipoles [12–14], two identical loops [11, 15], dipole and/or loop triad(s) [16, 17], and so forth [18, 19]. In the above-mentioned contributions, the output of each EMVS in the array can be represented as complex-valued vectors, and the data received by the EMVS array is combined into a long vector in series, which is called long-vector mode [20]. Consequently, the relevant estimators destroy locally the vector type of the signal because of the organization of the data into a large vector.Figure 1
CCCP array element geometry.More recently, the problem of estimating the DOA and polarization of the EMVS array within the algebraic system theory for quaternion and its extension has been widely researched [20–27]. Quaternion MUSIC technique was proposed based on the quaternion formalism of the two-component vector sensor array in [21]. The three-component vector sensor was expressed as a biquaternion number, then biquaternion-based MUSIC was proposed in [22]. The six-component electromagnetic vector sensor array was represented by a quad-quaternion model in [20]. The quaternion-ESPRIT algorithm for a crossed-dipole uniform linear array (ULA) was firstly proposed in [23]. The advantages of using quaternion for EMVS are that the local-vector nature of an EMVS array is preserved in multiple imaginary parts, and it could result in a more compact representation and a better estimation of signal subspace. Compared with long-vector method, the quaternion and its extension algorithms are shown to be more robust to model errors, while their computation efforts for estimating the data covariance matrices are slightly lower [20–22].Conformal antenna array, that is, array antennas with antenna elements arranged conformally on a curved surface, is widely used in a variety of airborne radar and other defense systems. The benefits include reduction of aerodynamic drag, wide angle coverage, space savings, and potential increase in available apertures [28–30]. Cylindrical conformal array is a typical conformal array.The purpose of this paper is to study the quaternion-ESPRIT algorithm for combining DOA and polarization estimation in cylindrical conformal CLD pair (CCCP) array. Compared with loop and dipole pairs oriented alongx-axis and y-axis and CDD (cocentered dipole and dipole) pairs, CLL (cocentered loop and loop) pairs oriented along x- and y-axes, respectively, the CLD pairs along the z-axis can decouple the DOA estimation from the polarization estimation [11–15]; errors of DOA and polarization herein do not cumulate. On the contrary, the related algorithms using the other pairs can not decouple the DOA estimation from the polarization estimation, and the DOA and polarization errors in numerical computation accumulate from step to step. The procedure of the proposed method is as follows: first, the array steering vector and the elevation are estimated using a quaternion eigenvalue decomposition of the data covariance matrix. Secondly, the azimuth is estimated using the relationships between direction cosine and spatial steering vector. Thirdly, the polarization parameters are estimated using the relationships between the dipoles and the loops. Hence, this proposed algorithm (1) is computationally less intensive than many open-form search methods; (2) produces closed-form solution with no extra computation needed for signal parameter estimates; (3) resolves the elevation quadrant ambiguity problem; and (4) has the advantage of parameter automatic matching and without spectral peak searching. The simulation results show that the performance of quaternion method is better than that of long-vector method.


## 2. Definition of Quaternion and Relevant Arithmetic

### 2.1. Definition of Quaternion

Quaternion, which extended imaginary numbers into a four-dimensional space, began to be developed by William Hamilton in 1843. A complex number has two components: the real and imaginary parts. The quaternion, however, has four components, that is, one real part and three imaginary parts and can be represented in Cartesian form as(1)q=a+ib+jc+kd,a,b,c,d∈R,
where a, b, c, and d are real numbers and i, j, and k are complex operators which obey the following rules:
(2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=-1.
From formulas (1) and (2), quaternion can also be expressed in the following form:
(3)q=a+ib+(c+id)j=α+βj.
The quaternion expressed in (3) is also known as the “Cayley-Dickson representation.”


### 2.2. The Relevant Arithmetic of Quaternion

The quaternion conjugate and the quaternion modulus are, respectively, given by(4)q*=a-ib-jc-kd,(5)∥q∥=a2+b2+c2+d2.
Let qi be qi=ai+ibi+jci+kdi, where ai, bi, ci, and di∈R, i=1,2; then the addition of two quaternions expressed in terms of their real and imaginary parts is given by
(6)q1+q2=(a1+a2)+i(b1+b2)+j(c1+c2)+k(d1+d2).
From the rules in (2), it is clear that multiplication is not commutative; namely, q1·q2≠q2·q1. The inner product of quaternion can be represented as
(7)〈p,q〉H=p*q.
If the inner product of quaternion meets 〈p,q〉H=0, it is said that the quaternions p and q are orthogonal.From the point of view of algebra, quaternion is an extension of the complex field. A quaternion contains more than two components than that of a complex number, so that it can contain more information in one operation. More details about quaternion can be found in [27].


## 2.1. Definition of Quaternion

Quaternion, which extended imaginary numbers into a four-dimensional space, began to be developed by William Hamilton in 1843. A complex number has two components: the real and imaginary parts. The quaternion, however, has four components, that is, one real part and three imaginary parts and can be represented in Cartesian form as(1)q=a+ib+jc+kd,a,b,c,d∈R,
where a, b, c, and d are real numbers and i, j, and k are complex operators which obey the following rules:
(2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=-1.
From formulas (1) and (2), quaternion can also be expressed in the following form:
(3)q=a+ib+(c+id)j=α+βj.
The quaternion expressed in (3) is also known as the “Cayley-Dickson representation.”


## 2.2. The Relevant Arithmetic of Quaternion

The quaternion conjugate and the quaternion modulus are, respectively, given by(4)q*=a-ib-jc-kd,(5)∥q∥=a2+b2+c2+d2.
Let qi be qi=ai+ibi+jci+kdi, where ai, bi, ci, and di∈R, i=1,2; then the addition of two quaternions expressed in terms of their real and imaginary parts is given by
(6)q1+q2=(a1+a2)+i(b1+b2)+j(c1+c2)+k(d1+d2).
From the rules in (2), it is clear that multiplication is not commutative; namely, q1·q2≠q2·q1. The inner product of quaternion can be represented as
(7)〈p,q〉H=p*q.
If the inner product of quaternion meets 〈p,q〉H=0, it is said that the quaternions p and q are orthogonal.From the point of view of algebra, quaternion is an extension of the complex field. A quaternion contains more than two components than that of a complex number, so that it can contain more information in one operation. More details about quaternion can be found in [27].


## 3. Signal and Array Models

K narrowband completely polarized electromagnetic plane wave source signals impinge upon a CCCP array, which is composed of 2N(N>K) identical CLD pairs arranged uniformly on the upper and lower circle rings of the cylinder. The distance of the two circle rings is d, which satisfies d≤0.5λmin⁡, where λmin⁡ refers to the minimal signals wavelength of the incident signals. The origin of the Cartesian coordinates is located in the center of the lower circle. Two reference CLD pairs are, respectively, placed at their own center of upper and lower circles. The radii of the upper and lower circles are both R which satisfies R≤0.5λmin⁡. The first CLD pair located on the cross point of lower circle and the positive x-axis, along the anticlockwise direction is, respectively, the first, second, …, and Nth CLD pair. Similarly, the (N+1)th CLD pair located on the cross point of upper circle and the positive x-axis along the anticlockwise direction is, respectively, the (N+1)th and (2N)th CLD pair, as shown in Figure 2.Figure 2
CCCP array geometry.For the CLD pairs, the dipoles parallel to thez-axis are referred to as the z-axis dipoles and the loops parallel to the x-y plane are referred to as the x-y plane loops, respectively, measuring the z-axis electric field components and the z-axis magnetic field components. The CLD pairs’ steering vector of the kth (1≤k≤K) unit-power electromagnetic source signal is the following 2×1 vector [3]:
(8)a=[ekzhkz]=[0-sinθksinθk0][cos⁡γksinγkejηk],
where θk∈[0,π] is the signals elevation angle measured from the positive z-axis, γk∈[0,π/2] represents the auxiliary polarization angle, and ηk∈[-π,π] symbolizes the polarization phase difference. Both the z-axis electric field ekz and the z-axis magnetic field hkz involve the same factor sinθk, so polarization estimation based on CLD pairs is independent of the sources direction of arrival and it requires no prior information of azimuth and elevation angles.Theekz and hkz can be expressed as follows with a quaternion ck:
(9)ck=ekz+ihkz=-sinθksinγkejηk+isinθkcos⁡γk.
The output of array response for the kth incident signal can be expressed as follows:
(10)xk(t)=ckq(θk,ϕk)︸a(θk,ϕk,γk,ηk)sk(t),
where ϕk∈[0,2π] is the azimuth of the kth incident signal and sk(t) is the kth incident signal. q(θk,ϕk) is the spatial steering vector constituted by the phase differences between the array elements and the origin; that is,
(11)q(θk,ϕk)=[1qLT(θk,ϕk)ej(2πdcosθk/λ)qUT(θk,ϕk)]T,
with(12)qL(θk,ϕk)=[ej(2πRsinθkcos⁡(ϕk-φ1)/λ)⋯ej(2πRsinθkcos⁡(ϕk-φN)/λ)]T(13)qU(θk,ϕk)=[ej(2π(Rsinθkcos⁡(ϕk-φ1)+dcos⁡θk)/λ)⋯ej(2π(Rsinθkcos⁡(ϕk-φN)+dcos⁡θk)/λ)]T.From (12), the phase of qL(θk,ϕk) can be expressed as
(14)arg[qL(θk,ϕk)]=2πRλ[βkαksinΔ+βkcos⁡Δ⋮αksin[(N-1)Δ]+βkcos⁡[(N-1)Δ]],
where αk=sinθksinϕk, βk=sinθkcos⁡ϕk, and Δ=2π/N.The received data collected by the CCCP array at timet can be represented as
(15)X(t)=AS(t)+N(t),
where X(t), S(t), N(t), and A are the 2N+2 received data, the K uncorrelated incident signals, the zero-mean additive complex Gaussian noise, and the steering vector matrix of K incident signals, respectively.The lower and upper circle subarray steering vectorsA1 and A2 are constructed by the first N+1 rows and the last N+1 rows of vector A. Their relationship can be expressed as
(16)A2=A1Φ,
where
(17)Φ=[ej(2πd/λ)cos⁡θ1⋱ej(2πd/λ)cos⁡θK].


## 4. Quaternion-ESPRIT Algorithm

The covariance matrix of received dataX(t) is given by
(18)Rx=E[XXH]=ARsAH+σ2I,
with E[·] symbolizing the statistical mean (·)H denoting the complex conjugate transpose, σ2 indicating the white noise power and Rs=E[s(t)sH(t)] representing the source covariance matrix. Let Es be the (2N+2)×K signal subspace matrix composed of the K eigenvectors corresponding to the K largest eigenvalues of Rx and let En denote the noise subspace composed of the remaining 2N+2-K eigenvectors of Rx. According to the subspace theory, there exists K×K nonsingular matrix T, and the signal subspace can be expressed explicitly as
(19)Es=AT,E1 and E2 are constructed directly using the first N+1 rows and the last N+1 rows of matrix Es. According to the definition of signal subspace, the relationship of matrices E1 and E2 can be showed explicitly as
(20)E1=A1T,E2=A2T=A1ΦT.
The following expression can be obtained by crunching matrix operation:
(21)E1#E2T-1=T-1Φ,
where E1#=(E1HE1)-1E1H.LetΓ=E1#E2=(E1HE1)-1E1HE2; then (21) can be rewritten as
(22)ΓT-1=T-1Φ.Equation (22) implies that the estimation of Φ is a matrix whose diagonal elements are composed of the K largest eigenvalues of matrix Γ and the full-rank matrix T-1 is composed of the K eigenvectors corresponding to the K largest eigenvalues of matrix Γ. The estimations of A1, A2, and A can be obtained:
(23)A^1=E1T-1,A^2=E2T-1,A^=EsT-1.


### 4.1. The Estimations of DOA

From the expression ofΦ^, the estimation of elevation angle can be given as
(24)θ^k=cos⁡-1[λ2πdarg(Φ^kk)].The elevation obtained by the cosine function in formula (24) can eliminate quadrant ambiguity; the values of elevation can be taken in range from 0 to π.The estimation of signal spatial steering vectorq^D(θk,ϕk) is obtained from A1:
(25)q^D(θk,ϕk)=A1(:,k)A1(:,1)=[1q^L(θk,ϕk)].
From formulas (14) and (25), the following expression can be gotten as follows:
(26)Ω=arg[q^L(θk,ϕk)]=W·[α^kβ^k],
where
(27)W=2πRλ[01sinΔcos⁡Δ⋮⋮sin[(N-1)Δ]cos⁡[(N-1)Δ]].
According to (26), the following expression can be obtained:
(28)[α^kβ^k]=W#Ω,
where W#=(WHW)-1WH.From formulas (24) and (28), the estimates of azimuth are given:
(29)ϕ^k=arctan(α^kβ^k),β^k≥0,ϕ^k=π+arctan(α^kβ^k),β^k<0.


### 4.2. The Estimations of Polarization

According to formulas (10) and (15), the matrix A can be expressed as another form:
(30)A=Ae+Ahj,
where Ae and Ah are, respectively, the dipole and loop subarray steering vectors, which can be reconstructed from the real and three imaginary parts of A. From (9), (10), and (15), it can be seen that
(31)Ae=AhΨ,
where
(32)Ψ=[-tanγ1ejη1⋱-tanγKejηK]=Ah#Ae,
with Ah#=(AhHAh)-1AhH is pseudoinverse matrix of Ah.According to (32), the polarization parameters estimation are presented as
(33)γk=tan-1(|Ψkk|),ηk=arg(-Ψkk).


## 4.1. The Estimations of DOA

From the expression ofΦ^, the estimation of elevation angle can be given as
(24)θ^k=cos⁡-1[λ2πdarg(Φ^kk)].The elevation obtained by the cosine function in formula (24) can eliminate quadrant ambiguity; the values of elevation can be taken in range from 0 to π.The estimation of signal spatial steering vectorq^D(θk,ϕk) is obtained from A1:
(25)q^D(θk,ϕk)=A1(:,k)A1(:,1)=[1q^L(θk,ϕk)].
From formulas (14) and (25), the following expression can be gotten as follows:
(26)Ω=arg[q^L(θk,ϕk)]=W·[α^kβ^k],
where
(27)W=2πRλ[01sinΔcos⁡Δ⋮⋮sin[(N-1)Δ]cos⁡[(N-1)Δ]].
According to (26), the following expression can be obtained:
(28)[α^kβ^k]=W#Ω,
where W#=(WHW)-1WH.From formulas (24) and (28), the estimates of azimuth are given:
(29)ϕ^k=arctan(α^kβ^k),β^k≥0,ϕ^k=π+arctan(α^kβ^k),β^k<0.


## 4.2. The Estimations of Polarization

According to formulas (10) and (15), the matrix A can be expressed as another form:
(30)A=Ae+Ahj,
where Ae and Ah are, respectively, the dipole and loop subarray steering vectors, which can be reconstructed from the real and three imaginary parts of A. From (9), (10), and (15), it can be seen that
(31)Ae=AhΨ,
where
(32)Ψ=[-tanγ1ejη1⋱-tanγKejηK]=Ah#Ae,
with Ah#=(AhHAh)-1AhH is pseudoinverse matrix of Ah.According to (32), the polarization parameters estimation are presented as
(33)γk=tan-1(|Ψkk|),ηk=arg(-Ψkk).


## 5. Simulation Results

In this section, some simulations are conducted to evaluate the performances on DOA and polarization estimation by the proposed method. Two uncorrelated equal-powered signals with parameters(θ1,ϕ1,γ1,η1)=(72∘,85∘,30∘,120∘) and (θ2,ϕ2,γ2,η2)=(30∘,43∘,67∘,80∘) impinge upon the CCCP array with N=6 CLD pairs sensors. The circle radius of CCCP is 0.5λmin⁡. The distance of the two circle rings is 0.5λmin⁡. The signal-to-noise ratio (SNR) is from 0 to 45 dB; 1024 snapshots are used in each of the 500 independent Monte Carlo simulation experiments. The results are shown in Figures 3, 4, 5, and 6.Figure 3
Standard deviation of azimuth versus SNR.Figure 4
Standard deviation of elevation versus SNR.Figure 5
Standard deviation of PPD versus SNR.Figure 6
Standard deviation of APA versus SNR.The solid line with star and circular data points in Figures3–6 indicates the standard deviations of azimuth, elevation, polarization phase difference (PPD), and auxiliary polarization angle (APA), respectively. The results are estimated by the long-vector and the proposed quaternion methods, at various signal-to-noise ratio (SNR) levels. The proposed quaternion procedure is better than the long vector. The estimation precision at 0 dB based on the quaternion model has improved to be larger than 0.24° for azimuth, 1.29° for elevation, 0.31° for PPD, and 0.17° for APA, compared with that of the long-vector method. Moreover, the standard deviations of azimuth, elevation, PPD, and APA are reduced evidently as the SNR increases using the quaternion method. The enhanced performance is rooted in the special data model of quaternion, which can provide a better signal subspace approximation than the long-vector methods.


## 6. Conclusion

A quaternion-ESPRIT algorithm for estimating DOA and polarization using CCCP array has been studied in this paper. The novel method can decouple the DOA estimation from the polarization estimation; the DOA and polarization errors herein do not cumulate. The proposed algorithm overcomes quadrant ambiguity in elevation angle when CCCP array is used for wide range elevation angle (more than 90 degrees) estimation. The performance of the proposed method is superior to the long-vector method because the quaternion matrix operations can maintain the vectorial property of the vector sensor and provide a better signal subspace approximation than the long-vector methods.



---

*Source: 806853-2014-05-15.xml*