Joint DOA and polarization estimation for unequal ... - IEEE Xplore

Journal of Systems Engineering and Electronics Vol. 27, No. 3, June 2016, pp.501 − 513

Joint DOA and polarization estimation for unequal power sources based on reconstructed noise subspace Yong Han1, Qingyuan Fang1,2,* , Fenggang Yan1 , Ming Jin1, and Xiaolin Qiao1 1. School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin 150001, China; 2. School of Information Science and Technology, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Abstract : In most literature about joint direction of arrival (DOA) and polarization estimation, the case that sources possess different power levels is seldom discussed. However, this case exists widely in practical applications, especially in passive radar systems. In this paper, we propose a joint DOA and polarization estimation method for unequal power sources based on the reconstructed noise subspace. The invariance property of noise subspace (IPNS) to power of sources has been proved an effective method to estimate DOA of unequal power sources. We develop the IPNS method for joint DOA and polarization estimation based on a dual polarized array. Moreover, we propose an improved IPNS method based on the reconstructed noise subspace, which has higher resolution probability than the IPNS method. It is theoretically proved that the IPNS to power of sources is still valid when the eigenvalues of the noise subspace are changed artificially. Simulation results show that the resolution probability of the proposed method is enhanced compared with the methods based on the IPNS and the polarimetric multiple signal classification (MUSIC) method. Meanwhile, the proposed method has approximately the same estimation accuracy as the IPNS method for the weak source. Keywords: invariance property of noise subspace (IPNS), joint DOA and polarization estimation, multiple signal classification (MUSIC), reconstruction of noise subspace, unequal power sources. DOI: 10.1109/JSEE.2016.00053

1. Introduction Joint direction of arrival (DOA) and polarization estimation based on dual polarized antenna arrays has been extensively investigated during the past three decades Manuscript received December 05, 2014. *Corresponding author. This work was supported by the National Natural Science Foundation of China (61501142), the China Postdoctoral Science Foundation (2015M571414), the Fundamental Research Funds for the Central Universities (HIT.NSRIF.2016102), Shandong Provincial Natural Science Foundation (ZR2014FQ003), and the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF 2013130; HIT(WH)XBQD 201022).

[1−5] as polarization diversity is another key factor in improving the performance of sources localization, target recognition and interference suppression in radar, communication and sonar systems. The most representative joint DOA and polarization estimation methods are based on the so-called subspace algorithms, e.g., the multiple signal classification (MUSIC) method and the ESPRIT method [6−11]. However, the sources with power difference, which are very common in practical applications, are not studied in most algorithms for DOA and polarization estimation. In multiple input multiple output (MIMO) systems, the random spatial distribution of users and shadowing conditions leads to unequal power reception in cellular MIMO networks, depressing the network capacity [12]. Especially in electronic warfare environment, the radar decoy is often used as the electronic jamming to disturb the detection of the passive radar based on the array signal processing technology. The working frequencies of the radar decoy and the real target are generally close to each other, and the radar decoy usually radiates stronger electromagnetic waves than the real target. Thus, the power of the electromagnetic waves received by passive radar is usually unequal, and the estimation of the real target (usually the weak source) in the presence of the strong interference is critical for increasing the performance of passive radar. However, the performance of passive radar for the weak source is seriously depressed when confronted with the strong interference. Various beamformers for strong interferences suppression are proposed to maximize the signal of interest (SOI) when the DOA and polarization of the interferences are known [13−15]. However, when the sources are noncooperative, just like in passive radar systems, the DOA and polarization parameters of all sources should be estimated without prior information. The DOA estimation of sources in the presence of an intermittent jammer in the slow frequency-hopped spread spectrum system was discussed in [16]. Two models were derived depending on whether the jammer was deterministic or random, and

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Journal of Systems Engineering and Electronics Vol.27, No. 3, June 2016

the corresponding Cramér–Rao bound (CRB) was studied. In [17], the co-channel interference was reduced after interference rejection through subarray beamforming, and the DOA estimation of targets was enhanced. However, this method can only provide the DOA information of the SOI rather than all the signals including the interferences. In [18], the DOAs of both weak and strong signals were obtained by searching with a new steering vector. The performance of this method depends on the corresponding eigenvector accuracy of the sources which is limited to the eigenvalue decomposition (EVD). However, the accuracy of the eigenvectors with respect to the weak sources is seriously affected by the strong sources when the sample covariance matrix is applied for EVD in practical applications. In [19], a signal subspace scaled MUSIC (SSMUSIC) algorithm was proved valid in estimating the DOA and the power of each source. The SSMUSIC algorithm outperforms the MUSIC algorithm in resolving the unequal power and the correlated sources. In [20], the invariance property of noise subspace (IPNS) to power of sources was proposed to estimate the DOA information of the unequal power sources. The eigenvalues of noise subspace remain invariant while the powers of sources are increased by introducing an auxiliary source. In [21] the eigenvalues of the noise subspace were reconstructed by the diagonal loading based on the IPNS method (IPNSDL method), however, the changing of the signal noise subspace is still limited. The estimation of the DOA is focused in the aforementioned literature. However, the issue of joint DOA and polarization estimation for unequal power sources remains to be investigated. In this paper, we propose a joint DOA and polarization estimation method for unequal power sources with reconstructed noise subspace based on the IPNS method. The eigenvalues of the noise subspace are increased artificially to obtain a new antenna array output covariance matrix. We theoretically prove that the IPNS to power of sources is still valid for the new covariance matrix with reconstructed noise subspace. Apparently, the proportion of the signal to noise ratio in the new covariance matrix is decreased compared with the original antenna array output covariance matrix. However, the simulation results illustrate that the reconstructed noise subspace has favorable effect on the resolution probability when we use the IPNS method to do joint DOA and polarization estimation. Meanwhile, the proposed method has almost the same estimation accuracy as the methods based on the IPNS for the weak source, and shows approximate performances with the polarimetric MUSIC algorithm. This paper is organized as follows. Section 2 discusses the signal model based on the dual polarized antenna array for joint DOA and polarization estimation. Section 3 proposes the improved IPNS method with reconstructed noise subspace for joint DOA and polarization estimation

of the unequal power sources. Section 4 provides the simulation results to evaluate the performance of the proposed method compared with those based on the IPNS method and the polarimetric MUSIC algorithm. Finally, the conclusion is presented in Section 5.

2. Data model Given that a transverse electromagnetic (TEM) wave propagates in the free space, the electric field of this wave [22, 23] can be expressed by E(r , t ) = ( Eθ eθ + Eϕ eϕ )e j(wt− kr +φ ) = ( E x e x + E y e y + E z e z )e j(wt− kr +φ )

(1)

where eθ and eϕ are the respective unit vectors along the elevation and azimuth directions in the spherical coordinate system, and Eθ and Eϕ are the corresponding electric field components. E x , E y and E z are the respective electric field components along the x, y and z axes in the Cartesian coordinate system. k = −k0 er is the propagation vector, where k0 = 2π / λ0 is the wave number, λ0 is the free-space wavelength for this TEM wave, and er is the unit vector that points outward along the radial direction. r is the spatial position vector that points from the coordinate origin to the observation point, and φ is the initial phase of the TEM wave. When the signal has arbitrary polarization, the electric field amplitude of this signal can be presented [24] as ⎡sin γ e jη ⎤ ⎡ Eθ ⎤ (2) ⎥ ⎢ ⎥ = E0 ⎢ ⎣ Eϕ ⎦ ⎣ cos γ ⎦ where E0 is the signal amplitude, γ ( 0 ≤ γ ≤ π / 2 ) denotes the polarization angle and η ( 0 ≤ η ≤ 2π ) represents the polarization phase difference. We consider a case in which the dual polarized antenna array (i.e. crossed dipoles), as shown in Fig.1, is used to receive the TEM wave in free space.

Fig. 1

Uniform linear array configuration

Given that the crossed dipoles have no response to the electric field component along the z-axis, then the output

Yong Han et al.: Joint DOA and polarization estimation for unequal power sources based on reconstructed noise subspace

of the crossed dipoles (located at r ) z (r , t ) maintains only the electric field components along the x- and y-axes. These components can be written as ⎡ x (r , t ) ⎤ ⎡ E x ⎤ j(wt− kr +φ ) z (r , t ) = ⎢ (3) ⎥ = ⎢ ⎥e ⎣ y (r , t ) ⎦ ⎣ E y ⎦ where x(r , t ) and y (r , t ) are the output of the crossed dipoles along the x- and y-axes, respectively. The transformation expression for the unit vectors from a spherical coordinate system to a Cartesian coordinate system is given by ⎡⎣ er eθ eϕ ⎤⎦ = ⎡⎣ e x e y e z ⎤⎦ ⋅ ⎡sin θ cos ϕ cos θ cos ϕ − sin ϕ ⎤ ⎢sin θ sin ϕ cos θ sin ϕ cos ϕ ⎥ ⎢ ⎥ ⎢⎣ cos θ 0 ⎥⎦ − sin θ

(4)

where θ and ϕ are the respective elevation and azimuth angles in the spherical coordinate system. For simplicity, we assume that the TEM wave is in the yz-plane ( ϕ = 90 ). Thus the electric field amplitudes of the TEM wave along x- and y-axes can be written as ⎡ E x ⎤ ⎡ cos θ cos ϕ − sin ϕ ⎤ ⎡ Eθ ⎤ ⎢ ⎥=⎢ ⎢ ⎥= cos ϕ ⎥⎦ ⎣ Eϕ ⎦ ⎣ E y ⎦ ⎣ cos θ sin ϕ ⎡ − cos γ ⎤ E0 ⎢ = E0 u(θ , γ ,η ) jη ⎥ ⎣ cos θ sin γ e ⎦

where u(θ , γ ,η ) = ⎡⎣ − cos γ

(5) jη ⎤ T

cos θ sin γ e ⎦

and

(i)

T

denotes the transpose. According to (3) to (5), the output of the crossed dipoles located at γ can be written as z (r , t ) = E0 u(θ , γ ,η )e j(wt − kr +φ ) = u(θ , γ ,η ) s (t )e− jkr

(6)

q(θ ) = e

j

2 πd

λ

sin θi

503

(9)

and nk (t ) denotes the received noise components at the kth crossed dipoles nk (t ) = [nx (t ), n y (t )]T .

(10)

The output of the ULA can be written in a matrix form as follows: Z (t ) = AS (t ) + N (t ) (11) where Z (t ) , S (t ) and N (t ) are the column vectors that contain the ULA output, sources and noise, respectively, i.e., ⎡ z1 (t ) ⎤ ⎡ s1 (t ) ⎤ ⎢ z (t ) ⎥ ⎢ ⎥ ⎥ , S (t ) = ⎢ s2 (t ) ⎥ , Z (t ) = ⎢ 2 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ z M (t ) ⎦ ⎣ sP (t ) ⎦ ⎡ n1 (t ) ⎤ ⎢ n (t ) ⎥ ⎥. N (t ) = ⎢ 2 ⎢ ⎥ ⎢ ⎥ ⎣ nM (t ) ⎦

(12)

A denotes the antenna array manifold matrix A = [ a (θ1 , γ 1 ,η1 ) a (θ 2 , γ 2 ,η 2 ) a (θ P , γ P ,η P ) ] , (13)

the column vector a (θi , γ i ,ηi ) of which is presented [24] by a (θi , γ i ,ηi )= ⎡⎣1 q(θi )

T

q (θi ) M −1 ⎤⎦ ⊗

u(θi , γ i ,ηi ) (14) where ⊗ represents the Kronecker product. The problem addressed in this study is the joint estimation of the DOA and polarization parameters of all the sources, as denoted by Ω = {(θi , γ i ,ηi ) | i = 1, 2,… , P} ,

from a finite number of snapshots of Z (t ) which are taken at time t j ( j = 1, 2,… , L) . For the unique parameter

where s (t ) denotes j(wt +φ )

s (t ) = E0 e . (7) We then consider a 2M-element uniform linear array (ULA) consisting of M pairs of crossed dipoles placed along the y-axis with inter-element spacing d as shown in Fig.1. The received signal from each dipole is to be processed separately by the estimator. We assume that P narrow-band sources impinge on the ULA from the direction (θi , ϕi ) , i = 1, 2,… , P. For simplicity, all the sources

are assumed to be in the yz-plane, that is, ϕi = 90 , i = 1, 2,… , P. According to (6), the output of the kth ( k = 1, 2,… , M ) crossed dipoles can be written as P

zk (t ) = ∑ u(θi , γ i ,ηi )si (t )q k −1 (θi ) + nk (t ) i =1

where q(θi ) is the space phase factor

(8)

estimation, the following assumptions are made: (i) The number of sources is predetermined and is less than the number of array elements, that is, P < M . (ii) The inter-element spacing between each two pair of crossed dipoles d satisfies d ≤ λ / 2 to avoid the ambiguity problem of DOA and polarization estimation. (iii) The P sources are incoherent (independent or correlated) zero-mean stationary processes. The covariance matrix of the sources is written as Rs = E[ S (t ) S (t ) H ] (15) where E[i] denotes the expectation operator, (i) H represents the conjugate transpose, and the rank of Rs is P. (iv) The additive noise components received at each crossed dipoles, which are independent of the sources, are stationary zero-mean complex white Gaussian processes

504


with the covariance matrix Rn = E[ N (t ) N (t ) H ] = σ n2 I ,

(16)

2

where σ is the noise power received at each dipole and I is a 2M × 2 M identity matrix. According to the aforementioned assumptions, the covariance matrix of the antenna array output is expressed as R = E ⎡⎣ Z (t ) Z (t ) H ⎤⎦ = ARs AH + σ n2 I . (17) In practical situations, R is unavailable and is usually replaced by a maximum likelihood estimate Rˆ that can be obtained as follows: 1 L Rˆ = ∑ Z (t j )Z (t j ) H . (18) L j =1 We will discuss the joint estimation of the DOA and polarization of the sources with large power level differences on the basis of this sample covariance matrix.

3. Joint DOA and polarization estimation using IPNS with reconstructed noise subspace 3.1 Subspace decomposition The EVD of R yields R = U Σ U H =U S Σ S U SH + U N Σ N U NH

(19)

where U ∈ C2 M ×2 M and Σ ∈ C2 M ×2 M are the eigenvectors of R , and Σ = diag ( λ1 , λ2 ,… , λ2 M ) contains the corresponding eigenvalues that are arranged in a decreasing order as [25] λ1 ≥ λ2 ≥ λP ≥ λP +1 = λP + 2 = λ2 M = σ n2 . (20)

Σ S comprises the first P dominant eigenvalues as Σ S = diag(λ1 , λ2 ,… , λP ) (21) and Σ N is composed of the remaining 2M − P small eigenvalues, the values of which are equal to σ n2 , as given by Σ N = diag(λP +1 , λP + 2 ,… , λ2 M ) . (22) U S ∈ C2M × P and U N ∈ C2 M ×(2 M − P ) are the eigenvectors

that correspond to Σ S and Σ N , respectively. The space spanned by U S ( span(U S ) ) and that by U N ( span(U N ) ) are called the signal and noise sub-

spaces, respectively, that is span(U S ) ⊥ span(U N ) . Meanwhile, the space spanned by the array manifold matrix A ( span( A) ) is equal to the signal subspace which indicates that

span( A) = span(U S )

and

span( A) ⊥

span(U N ). Actually, each column of A that contains the DOA and polarimetric information with respect to one source is in the signal subspace and is orthogonal to the noise subspace. The polarimetric MUSIC algorithm util-

izes this property to estimate the DOA and polarization of sources. In most applications, the sources have different power levels especially for passive radar systems in which the jamming usually possesses a significantly higher power level than the actual targets. However, in the presence of a power difference between sources, the performance of the polarimetric MUSIC algorithm degrades distinctly when the sample covariance matrix in (18) is used for estimation instead of the ideal covariance matrix.

3.2

Joint DOA and polarization estimation

based on IPNS to power of sources It was certified in [20] that the noise subspace keeps invariant to the power of sources. That means if the direction and polarization parameters of sources are fixed and only the power levels of sources are changed, the noise subspace eigenvalues of the covariance matrix R will remain invariant. The power level change of one exact source can be simulated by introducing an auxiliary signal (a virtual signal that does not really exist and is only used for mathematical realization) that has the same DOA and polarization parameters with that exact source. Hence, assume that the auxiliary signal impinges on the array from direction θ ′ with polarization angle γ ′ and polarization phase difference η ′. The auxiliary signal is defined as independent from all sources and the noise. A new covariance matrix D based on the array output covariance matrix R is given as D R + ρ a2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H (23) where ρ a2 is a positive constant scalar denoting the power of auxiliary signals and a (θ ′, γ ′,η ′) is the corresponding array response vector. Assume that all the impinging sources (independent or correlated) belong to the set Ω = {(θi , γ i ,ηi ) | 1 ≤ i ≤ P} . Thus, when the auxiliary signal coincides with one of the sources that is (θ ′, γ ′,η ′) ∈ Ω, the introducing of the auxiliary signal in (23) is equivalent to the power increasing of that exact sources. The EVD of D yields D =U Σ U H = U S Σ S U SH + U N Σ N U SH (24) where the diagonal matrix Σ consists of the eigenvalues of D and can be written as Σ = diag(λ1 , λ2 ,… , λ2 M ) . (25) From [20], we know that the noise subspace eigenvalues of D are the same with the noise subspace eigenvalues of R when the auxiliary signal coincides with one of the sources. Now, we extend this conclusion for joint DOA and polarization estimation of unequal power sources based on a dual polarized array. Moreover, the case of correlated sources is also discussed. We concisely


rewrite the IPNS to the power of sources as follows. Theorem 1 If and only if the auxiliary signal coincides with one of the sources (θ ′, γ ′,η ′) ∈ Ω , the noise subspace eigenvalues of R will remain invariant to the power of sources that is Σ N = Σ N . Proof Assume the auxiliary coincides with one of the sources, i.e. the ith source, that is (θ ′, γ ′,η ′) = (θi , γ i ,ηi ) . According to (23), the matrix D becomes

D = R+

ρ a2 a (θi , γ i ,ηi )a (θi , γ i ,ηi ) H

.

(26)

(27)

Replace the matrix R in (26) with expression in (17), we get D = ARS AH + σ n2 I + ρv2 a (θi , γ i ,ηi )a (θi , γ i ,ηi ) H = ARS AH + σ n2 I + ρv2 Awi wi T AH = ARS AH + σ n2 I (28)

where 2 ⎪⎧ R (m, n) + ρv , m = n = i RS (m, n) = ⎨ S . ⎪⎩ RS (m, n), otherwise

(29)

and has no extra effect on others. Even if the sources are correlated that Rs ≠ diag( ρ12 , ρ 22 ,… , ρ P2 ) , comparison of (28) to (17) reveals that the auxiliary signal only affects the sources covariance matrix. Hence, the EVD of D yields D =U Σ U H = U S Σ S U SH + U N Σ N U SH , where the eigenvalues of D in the decreasing order are = λ2 M = σ n2 (30)

Σ N = diag

(λP+1 , λP + 2

,… , λ2 M ) . From (20) and (29), it is clear that Σ N = Σ N .

Thus, we conclude that the noise subspace eigenvalues keep invariant if (θ ′, γ ′,η ′) ∈ Ω. The certification reveals that if the DOA and polarization parameters of auxiliary signals coincide with those of the sources, then the auxiliary signal has no influence on the noise subspace of the array output covariance matrix R . However, when the auxiliary signal is different from any one of sources (θ ′, γ ′,η ′) ∉ Ω , the expression in (28) becomes D = ARS AH + ρv2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H + σ n2 I = ⎡ RS [ A a (θ ′, γ ′,η ′)] ⎢ ⎣⎢ 0

are λ1 ≥ λ2 ≥

= λ2 M = σ n2 ,

≥ λP +1 ≥ λP + 2 = λP + 3 =

where Σ S = diag(λ1 , λ2 ,… , λP +1 ) and Σ N = diag(λP + 2 , (2 M − P − 1) diagonal matrix which is different from

0 ⎤ H ⎥ [ A a (θ ′, γ ′,η ′)] + ρv2 ⎦⎥

Thus we conclude that Σ N ≠ Σ N if (θ ′, γ ′,η ′) ∉ Ω . According to Theorem 1, the DOA and polarization parameters can be jointly estimated by judging whether the noise subspace eigenvalues are changed after introducing the auxiliary signal for each set of (θ ′, γ ′,η ′) . Hence, the spectrum can be expressed as G (θ ′, γ ′,η ′) =

From (17), (19) and (22), it is clear that the noise power (variance) determines the noise subspace eigenvalues. According to (28) and (29), it is evident that the auxiliary signal just affects the ith element in the diagonal of Rs

and Σ S = diag(λ1 , λ2 ,… , λP ),

It is implicit that the auxiliary signal affects the array manifold matrix and the sources covariance matrix at the same time. The eigenvalues of D in the decreasing order

Σ N in (22) (the (2 M − P) × (2 M − P ) diagonal matrix).

w = [01×(i −1) 1 01×(2 M −i ) ]T .

≥ λP ≥ λP +1 = λP + 2 =

(31)

λP + 3 ,… , λ2 M ). It is clear that Σ N is a (2 M − P − 1) ×

We define a selective vector w as

λ1 ≥ λ2 ≥

σ n2 I = A′RD′ S A′H + σ n2 I .

505

1 2M

∑

i = P +1

.

(32)

(λi − λi )

From (32), the estimations for DOA and polarization parameters of sources are P sets of (θ ′, γ ′,η ′) that make G (θ ′, γ ′,η ′) reach P maximums.

3.3 Improved IPNS method with reconstructed noise subspace for joint DOA and polarization estimation From (32), it is evident that if the eigenvalues of the noise subspace with respect to R are changed, the performance of the IPNS method will change. However, it is found from computer simulations that the resolution probability of the IPNS method is enhanced when the eigenvalues of the noise subspace are increased. The increasing of the resolution probability for the proposed method can be interpreted from the physical property of signal and noise subspaces. By increasing the eigenvalues of the noise subspace, the energy of the signal subspace that is leaked into the noise subspace is reduced through another EVD after the auxiliary source is introduced. Thus, the resolution probability of the proposed method is increased compared with the IPNS method when applying more accurate eigenvalues of the noise subspace in estimating the DOA and polarization information of sources. The IPNS to the power of sources based on the covariance matrix with reconstructed noise subspace will be proved in two steps. Firstly, a new matrix R is defined to denote the covariance matrix with reconstructed noise subspace as

506


⎡ Σ − σ r2 I R = U Σ U H =[U S U N ] ⎢ S ⎢⎣ where U S and U N are the signal and

⎤ H ⎥ [U S U N ] (33) 0 ⎥⎦ noise eigenvectors

of R as shown in (19), respectively. Σ S − σ r2 I and 0 are the new eigenvalues corresponding to the signal and noise subspace. It should be noticed that σ r2 satisfies σ r2 < λP in order to assure the eigenvalues of the signal subspace will be larger than the eigenvalues in noise subspace after the reconstruction of noise subspace. We will certify that the IPNS is still valid when R is used as the array output covariance matrix instead of R. Suppose that the auxiliary signal impinges on the array just as defined in Section 3.2, and the new matrix D is defined as

It is known that span(U S ) = span( A) , hence (38) is equal to zero. From (36) we get that U N is the eigenvectors u2 M ] and the matrix of D where U N = [uP +1 uP + 2 corresponding eigenvalues are zeros. To see that more clear, we give a more simple form as (39) DU N = U N 02 M ×(2 M − P ) .

Thus it is evident according to (39) and (33) that Σ N = 0. Otherwise, if the auxiliary signal is different from any one of the sources, we will prove that D contains 2M − P − 1 zero eigenvalues and P + 1 non-zero eigenvalues when (θ ′, γ ′,η ′) ∉ Ω. The steering vector a (θ ′, γ ′,η ′) can be expressed by the linear superposition of U S and U N as a (θ ′, γ ′,η ′) = [U S U N ][ cS c N ]

T

D

R + ρv2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H .

(34)

The EVD of D yields ⎡Σ S ⎤ H D = U Σ U H =[U S U N ] ⎢ ⎥ [U S U N ] ⎢⎣ Σ N ⎥⎦

(35)

(40)

where cS and c N are the unknown coefficients. Let v1 = U N c N , then (40) can be written as a (θ ′, γ ′,η ′) = U S cS + v1. It is obvious that a (θ ′, γ ′,η ′) ∈ span (U S , v1 ) .

Make the other 2M − P − 1 eigenvectors in U N orthog-

where U S and U N are the respective eigenvectors of the

onal to v1 through Schmidt orthogonalization, then we get

signal and noise subspaces of D , and Σ S and Σ N are the corresponding eigenvalues matrices. The eigenvalues of the noise subspace with respect to D remain invariant to those of R as the auxiliary signal coincides with the sources, although the noise subspace eigenvalues of R are assigned artificially. This property can be concisely rewritten as follows. Theorem 2 If and only if the auxiliary signal coincides with one of the sources (θ ′, γ ′,η ′) ∈ Ω, the noise subspace eigenvalues of the constructed covariance matrix R will remain invariant to the power of sources, that is Σ N = 0. Proof Assume the auxiliary coincides with any one of the incident signals, i.e. the ith source (θ ′, γ ′,η ′) =

v1 ⊥ vi (i = 2,3,… , 2 M − P ), where vi is the new eigenvector obtained via Schmidt orthogonalization. Let V = [ v2 v3 v 2 M − P ], it is obvious that span (U S , v1 ) ⊥

(θi , γ i ,ηi ). From (34), we get

DU N

( R + ρ a(θ , γ ,η )a(θ , γ ,η ) ) ⋅ 2 v

i

i

i

i

i

i

H

U N = 02 M ×(2 M − P )

(36)

where RU N

⎡ Σ − σ r2 I = [U S U N ] ⎢ S ⎣⎢

⎡ Σ − σ r2 I [U S U N ] ⎢ S ⎣⎢

(

)

DV = R + ρv2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H V =

02 M ×(2 M − P −1) .

(41)

From (41) it can be seen that D contains 2M − P − 1 zero eigenvalues whose corresponding eigenvectors are V . Multiplying the matrix D by v1 , we get

(

)

Dv1 = R + ρv2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H v1 =

ρv2 (U S cS + v1 )(U S cS + v1 ) H v1 = ρv2 (U S cS + v1 ) . If cS = 0, we get Dv1 = ρv2 v1 which means value corresponding to v1 is ρv2 > 0. Otherwise,

(42)

the eigenif cS ≠ 0,

we get Dv1 ≠ hv1 where h ∈ R , which means v1 is not the eigenvector of D. Assume v s is the eigenvector of

⎤ H ⎥ [U S U N ] U N = 0 ⎦⎥

⎤ T ⎥ [0 I ] = 02 M ×(2 M − P ) 0 ⎦⎥

span (V ) . According to (34), we have

D except for the eigenvectors V = [ v2 v3

v2 M − P ].

Then v s can be written as (37)

and

ρv2 a (θi , γ i ,ηi )a (θi , γ i ,ηi ) H U N = 02 M ×(2 M − P ) . (38)

P

v s = ∑ ρ k uk + ρ P +1v1 .

(43)

k =1

Assuming λs is eigenvalues of D whose corresponding eigenvector is v s , we get


Dv s = vs λs =

( R + ρ a(θ ′, γ ′,η ′)a(θ ′, γ ′,η ′) ) v . 2 v

H

s

(44)

Multiplying (44) by v1H , we have v1H Dv s = v1H Rv s +

ρv2 v1H a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H v s = ρv2 v1H (U S cS + v1 )(U S cS + v1 ) H v s = ρv2 (U S cS + U N c N ) H v s = λs ρ P +1 .

(45)

According to (44) and (45), we have

P + 2,… , 2 M , where P is the number of sources. (ii) The eigenvalues of Rˆ are reconstructed by

ρv2 uiH a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H v s = λi ρi + ρv2 cSi (U S cS + v1 ) H vs = (46)

where cSi is the ith element of cS . From (46), it is clear that λi ρi = λs ( ρi − ρ P +1cSi ) , i ∈ [1, 2, …, P] . (47) If the eigenvalue λs = 0, we get ρi = 0 and v s = ρ P +1v1 because the eigenvalues λi ≠ 0(i ∈ [1, 2, …, P]) . So the eigenvalue corresponding to eigenvector v s is 0, which contradicts the conclusion derived from (42) that the eigenvalue corresponding to v1 is ρv2 > 0. So we conclude that the eigenvalue corresponding to eigenvector v s is λs > 0 because D is a positive definite matrix.

When the auxiliary signal (θ ′, γ ′,η ′) ∉ Ω , it can be obtained from the above certification that ⎧⎪λ i = 0, i = P + 2, P + 3,… , 2 M ⎨ ⎪⎩λ i > 0i , i = P + 1 which illustrates Σ N = 0. Secondly, we define a new matrix R as ⎡Σ S ⎤ R = R + σ r2 I = [U S U N ] ⎢ [U S U N ]H 2 ⎥ σ r I ⎥⎦ ⎢⎣

Until now, it has been proved that the reconstructed covariance matrix with assigned noise subspace eigenvalues σ r2 I can also be applied in the IPNS method for joint DOA and polarization estimation. Hence, the spectrum of the proposed method has the same form as that of the IPNS method shown in (32). The procedure of the improved IPNS method is summarized as follows: (i) The sample covariance matrix Rˆ in (18) is computed, and the eigenvalues of Rˆ are obtained, which in a descending order are given by λˆ1 ≥ λˆ2 ≥ ≥ λˆ2 M . The noise subspace eigenvalues are denoted by λˆ , i = P + 1, i

uiH Dv s = uiH v s λs = uiH Rv s +

λi ρi + λs ρ P +1cSi = λs ρi

⎧λˆ = λˆ , i = 1, 2, … , P i ⎪ i (51) ⎨ˆ ⎪λi = λˆP +1 + λˆP / 2, i = P + 1, P + 2, …, 2M ⎩ where the assigned value of the noise subspace eigenvalues is optimized through computer simulations. A new antenna array output covariance matrix Rˆ is constructed as shown in (49). (iii) The auxiliary signal is assumed to impinge on the antenna array from (θ ′, γ ′,η ′) , then the new covariance matrix Dˆ in (50) is formulated, where the idea recons

(

)

tructed covariance matrix R is replaced by Rˆ . Let the positive constant scalar ρv2 be equal to ρv2 = tr ( R) / 2 M ( tr (i) denotes the matrix race) [20], then the eigenvalues of Dˆ are obtained via EVD which are arranged in a descending order as λˆ1 ≥ λˆ2 ≥

(48)

≥ λˆP ≥ λˆP+1 ≥ ≥ λˆ2 M . (iv) For each set of (θ ′, γ ′,η ′) , the noise subspace ei-

ˆ genvalues of Rˆ λ i (i = P + 1, P + 2,… , 2 M ) are compared

with those of

Dˆ λˆ i (i = P + 1, P + 2,… , 2 M ) according

to G (θ ′, γ ′,η ′) , which is given as (49)

G (θ ′, γ ′,η ′) =

are σ r2 I . Replace

R whose noise subspace eigenvalues in (34) with R in (49), the new matrix D is defined as D

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R + ρv2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H =

R + σ r2 I + ρv2 a (θ ′, γ ′,η ′)a (θ ′, γ ′,η ′) H = D + σ r2 I . (50)

From Theorem 2, the noise subspace eigenvalues of D can be obtained Σ N = 0. Thus, it is obvious in (50) that the noise subspace eigenvalues of D are σ r2 I , which are the same with those of R when the auxiliary signal coincides with the sources [21].

1 2M

∑

i = P +1

ˆ (λˆ i − λ i )

.

(52)

(v) The joint estimation of the DOA and polarization parameters of the actual sources comprises the P sets of (θ ′, γ ′,η ′) that enables G (θ ′, γ ′,η ′) to reach P maximums.

4. Simulations A ULA that consists of M = 9 pairs of crossed dipoles that are spaced half wavelength is used to certify the performance of the proposed method. Two independent sources with unequal powers are assumed to impinge on

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the ULA whose initial phases and polarization phase differences are both set at φ = 0° and η = 20° , respectively. We define the power of strong source to that of weak source ratio as the strong source to the weak source ratio (SWR), and the signal-to-noise ratio (SNR) is assumed as 10 lg σ 2 dB, where σ 2 denotes the signal power. The noise that is received by the crossed dipoles is an additive Gaussian process. Computer simulations with 200 independent Monte Carlo trials are conducted to assess the resolution probability and performance of the proposed method. In addition, the IPNS method, the polarimetric MUSIC algorithm and the CRB [26] are also applied for comparison.

4.1

Spectrums of the DOA and polarization

estimation In this simulation, we consider two unequal power sources that impinge on the antenna array with SWR=20 dB. The SNR of the weak source is 5 dB and a number of L = 200 samples are employed. The spectrums that are obtained in one Monte Carlo trial through different methods are provided in Fig.2, where the strong source is located at (θ1 , γ 1 )=(3°, 30°) , SWR=20 dB and the weak source is located at (θ 2 , γ 2 ) = (7°,35°) . Evidently, the two sources are estimated correctly by using the proposed method, the IPNS method and the IPNSDL method as shown in Fig.2 (d), Fig.2 (b) and Fig.2 (c), whereas only strong source is available when the polarimetric MUSIC algorithm is used, as shown in Fig.2 (a). Hence, the resolution abilities are different for different methods when a power difference exists between sources. From the comparison of Fig.2 (b), Fig.2 (c) and Fig.2 (d), the spectrum for the weak source obtained through the proposed method is more apparent than that from the IPNS method and the method proposed in [21]. The proposed method exhibits higher resolution probability for unequal power sources than the other two methods.

Fig.2

4.2

Spectrums of the DOA and polarization estimation

Resolution probability

To see the resolution probability of the proposed method more clearly, we conduct several experiments to evaluate the effectiveness of estimating the weak sources in the presence of strong sources. In the following simulations,


two unequal power sources are supposed to impinge on the antenna array with a power difference. The resolution probabilities of different methods are obtained for several scenarios to certify the resolution capability of the proposed method in resolving unequal power sources, where a number of L = 200 samples are used in the following simulations. Suppose the directions θˆ and θˆ are the DOA 1

2

estimations for the two sources, whose true directions are θ1 and θ 2 , so that one Monte Carlo trial is regarded as a successful estimation when the estimation results satisfy ⎧ ˆ θ1 − θ 2 ⎪ θ1 − θ1 ≤ ⎪ 2 . ⎨ ⎪ θˆ − θ ≤ θ1 − θ 2 ⎪⎩ 2 2 2

(53)

To see the influence of the power difference on the resolution probability, the SNR of the weak source is fixed as the SWR increases. The resolution probabilities for different methods as the SWR varies over a range from 0 dB to 40 dB are shown in Fig. 3. It is obvious that the resolution probability of the polarimetric MUSIC algorithm suffers from degradation as the SWR increases, whereas the resolution probabilities of other methods are not affected by the increasing SWR. The polarimetric SSMUSIC algorithm also exhibits better resolution probability compared with the MUSIC algorithm. However, the resolution probability of the proposed method is the highest in all methods as the SWR increases. Moreover, the proposed method also shows better resolution probability for equal power sources which are obtained from Fig. 3 as SWR = 0 dB. In Fig. 3, the strong source is located at (θ1 , γ 1 ) = (3 ,30 ) and the weak source is located at (θ 2 , γ 2 ) = (7 ,35 ) .

509

The resolution probabilities demonstrated in Fig. 4 as the SNR of the weak source varies over a range from −5 dB to 15 dB evidently shows that the proposed method has better resolution capability than the polarimetric MUSIC algorithm, as well as the other three methods, especially when the SNR of the weak source is low. Noting that the proposed method is effective in dealing with unequal power sources, the resolution probability is enhanced as the power of the strong source increases, whereas it deteriorates for the MUSIC algorithm. In Fig. 4, the strong source is located at (θ1 , γ 1 ) = (3 ,30 ) and the weak source is at (θ 2 , γ 2 ) = (7 ,35 ) with SNR=5 dB.

Fig. 4 Resolution probability against the SNR of the weak source

To examine the resolution probability for adjacent sources with power difference, the direction and polarization angles of the strong source are fixed as the weak source varies. The resolution probabilities versus the angular separation and polarimetric separation between two sources are shown in Fig. 5 and Fig. 6 respectively, where the corresponding angular separation varies over a range from 1 degree to 7 degree and the polarimetric separation varies over a range from 10 degree to 80 degree. It indicates in Fig. 5 where the strong source is located at (θ1 , γ 1 ) = (3 ,30 ) and the weak source is at (θ 2 , γ 2 ) = (θ1 + Δ,35 ) with SNR=5 dB that the proposed method has better resolution probability than the other three methods in resolving adjacent sources with equal and unequal power. In addition, the same conclusion can also be derived from Fig. 6 where the strong source is located at (θ1 , γ 1 ) = (3 ,30 ) and the weak source located

Fig. 3

Resolution probability against the SWR

at (θ 2 , γ 2 ) = (6 , γ 1 + Δ ) with SNR=5 dB.

510


the strong source is located at (θ1 , γ 1 ) = (3o ,30o ) with SWR=20 dB and the weak source is at (θ 2 , γ 2 ) = (7o ,35o ), respectively. It can be seen in Fig. 7 and Fig. 8 that the RMSEs of the DOA and polarization estimation for the IPNS algorithm, the IPNSDL algorithm and the proposed method are approximately the same, while the RMSEs of the polarimetric MUSIC algorithm are slightly lower than the other methods and the polarimetric SSMUSIC algorithm has the lowest estimation accuracy in all the methods.

Fig. 5 tion

Resolution probability against the angular separa-

Fig. 7 RMSE of the DOA for the strong source versus the SNR of the weak source

Fig. 6 Resolution probability against the polarimetric separation

4.3

Estimation accuracy of the DOA and

polarization To examine the estimation accuracy of the proposed method, two unequal power sources are assumed to impinge on the antenna array with SWR=20 dB, where the root mean square errors (RMSE) are obtained as functions of the SNR with respect to the weak source. The RMSEs of the proposed method are compared with the IPNS algorithm, the IPNSDL algorithm, the polarimetric MUSIC algorithm and the polarimetric SSMUSIC algorithm, as well as the CRB. The RMSEs of DOA estimation and polarization angle estimation for the strong source are shown in Fig. 7 where the strong source is located at (θ1 , γ 1 ) = (3o ,30o ) with SWR=20 dB and the weak source is at (θ 2 , γ 2 ) = (7o ,35o ) , and Fig. 8 where

Fig. 8 RMSE of the polarization angle for the strong source versus the SNR of the weak source

The RMSEs of DOA estimation and polarization angle estimation for the weak source are shown in Fig. 9 where the strong source is located at (θ1 , γ 1 ) = (3o ,30o ) with SWR=20 dB and Fig. 10 where the strong source is located at (θ1 , γ 1 ) = (3o ,30o ) with SWR=20 dB and the weak source is at (θ 2 , γ 2 ) = (7o ,35o ) , respectively. Fig. 9 and Fig. 10 indicate that the RMSEs of the proposed method are the same with the IPNS algorithm and the


IPNSDL algorithm. In addition, the RMSEs of the proposed method are lower than the polarimetric SSMUSIC algorithm and the polarimetric MUSIC algorithm as the SNR of the weak source is low. In electronic warfare environment, the weak source is usually the real target. The estimation accuracy of methods for the weak source is more important, especially when the SNR of the weak source is low, which will affect the detection accuracy of passive radar.

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20 dB and the weak source is at (θ 2 , γ 2 ) = (7o ,35o ) .

Fig. 11 RMSE of the DOA for the weak source versus the number of sensors

Fig. 9 RMSE of the DOA for the weak source versus the SNR of the weak source

Fig. 12 RMSE of the polarization angle for the weak source versus the number of sensors

Fig. 10 RMSE of the polarization angle for the weak source versus the SNR of the weak source

To examine the estimation accuracy of the proposed method versus different number of sensors, two unequal power sources are assumed to impinge on the antenna array with SWR=20 dB. The weak source is usually the real target in electronic warfare environment, so the estimation accuracy of the weak source determines the performance of passive radar. Hence the RMSEs DOA and polarization angle estimation for weak source are concerned, and are exhibited in Fig. 11 where the strong source is located at (θ1 , γ 1 ) = (3o ,30o ) with SWR=20 dB and the weak source is at (θ 2 , γ 2 ) = (7o ,35o ) , and Fig. 12 where the strong source is located at (θ1 , γ 1 ) = (3o ,30o ) with SWR=

It is evident that the RMSEs of the proposed method is approximately the same with the IPNS algorithm, the IPNSDL algorithm and the polarimetric SSMUSIC algorithm, which is lower than the polarimetric MUSIC algorithm for the DOA estimation as well as the polarization angle estimation when the number of sensors is small. Thus it is concluded that the proposed method has approximately the same estimation accuracy with the IPNS algorithm, from which the proposed method is derived.

5. Conclusions In this study, the improved IPNS method based on the covariance matrix with reconstructed noise subspace is proposed to estimate the DOA and polarization parameters for unequal power sources. The IPNS to the power of sources is theoretically proved valid even the noise subspace of the covariance matrix is changed. As shown in the simulation results, the resolution probability of the proposed method is higher than the IPNS algorithm, the IPNSDL algorithm, the polarimetric SSMUSIC algorithm and the polarimetric MUSIC algorithm. Moreover, the proposed method also shows a higher resolution prob-

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ability, when confronted with equal power sources. In addition, the proposed method exhibits approximately the same estimation accuracy with the methods based on the IPNS method. Compared with the polarimetric MUSIC algorithm and the polarimetric SSMUSIC algorithm, the methods based on the IPNS algorithm need an EVD in each direction when searching for the spatial and polarimetric spectrum peaks.

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Biographies Yong Han was born in 1976. He received his Ph.D. degree from Harbin Institute of Technology, Harbin, China, in 2010. He is now a lecturer in School of Information and Electrical Engineering in Harbin Institute of Technology, Weihai, China. His research interests include array signal processing and polarization sensitive array signal processing. E-mail: [email protected] Qingyuan Fang was born in 1987. She received her M.S. degree and Ph.D. degree from Harbin Institute of Technology in 2010 and 2015, respectively. Now, she is a lecturer in School of Information Science and Technology in Shijiazhuang Tiedao University, Shijiazhuang, China. Her research interests include anti-jamming technology in passive radar and antenna technology. E-mail: [email protected]

Yong Han et al.: Joint DOA and polarization estimation for unequal power sources based on reconstructed noise subspace Fenggang Yan was born in 1983. He received his B.E., M.S., and Ph.D. degrees in information and communication engineering from Xi’an Jiaotong University, Xi’an, the Graduate School of Chinese Science of Academic, Beijing, and Harbin Institute of Technology (HIT), Harbin, in 2005, 2008, and 2013, respectively. Since October 2013, he has been a teacher with the Department of Electronics Information Engineering, HIT, Weihai, China. His current research interests include array signal processing and statistical performance analysis. E-mail: [email protected] Ming Jin was born in 1968. He received his M.S. and Ph.D. degrees in information and communicationengineering from Harbin Institute of Technology, Harbin, China, in 1998, and 2004, respectively. His current interests are in the areas of array signal processing, parallel signal processing, and radar polarimetry. E-mail: [email protected]

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Xiaolin Qiao was born in 1948. He received his B.E., M.S., and Ph.D. degrees in information and communication engineering from Harbin Institute of Technology, Harbin, China, in 1976, 1983, and 1991, respectively. His research interests are in the areas of signal processing, wireless communication, special radar, parallel signal processing, and radar polarimetry. E-mail: [email protected]