Quaternion ESPRIT for Direction Finding with a Polarization Sentive Array Xiaofeng Gong, Zhiwen Liu, Yougen Xu
Content 1. Introduction 2. Data model 3. The proposed algorithm 4. Simulations 5. Conclusion 2
1. Introduction Direction of arrival (DOA) estimation is of significant � importance in radar, sonar, wireless communication and so on. The main idea of DOA estimation with sensor arrays, is � to exploit the amplitude and phase relationship between signals recorded on different sensors to obtain the angle estimates. The subspace methods such as MUSIC and ESPRIT � are among the most popular DOA estimation methods. 3
1. Introduction Methods based on scalar arrays only exploit the spatial � information which is related to the angular parameters. Methods based on polarization sensitive arrays make � use of both spatial information and polarization information, and have been proven to offer better performance. 4
1. Introduction z A Polarization θ sensitive array E E E y y y y ,0 ,1 ,2 o ������������� d E E E x x x ,0 ,1 ,2 = ⎧ E E ρ α ρ α exp( ), , :related to angular parameters only ⎪ x x ,1 ,0 1 1 1 1 ⎨ = x ⎪ E E ρ α ρ α exp( ), , :related to both angle and polarization ⎩ y x ,0 ,0 2 2 2 2 5
1. Introduction However, most of the existing methods arrange the � polarization sensitive array signals into a complex long- vector that somehow destroys the vector nature of incident signals. Thus, in the recent several years people started to use � hypercomplex algebras, such as quaternions and biquaternions, to model and analyze polarization sensitive array signals. The hypercomplex methods are proved to occupy less � memory resource and offer better estimation of subspaces. 6
1. Introduction z The complex long- vector model y E t E t E t 1 ( ) 2 ( ) 3 ( ) y, y, y, E t E t E t 1 ( ) 2 ( ) 3 ( ) x, Vs. x, x, x The hypercomplex model ⎡ ⎤ E t ( ) = y t x, 1 ( ) ⎢ ⎥ E t ( ) ⎢ ⎥ y, ⎡ ⎤ 1 + ⋅ E t i E t ( ) ( ) ⎢ ⎥ x, y, E t ⎢ 1 1 ⎥ ( ) x, = ⎢ + ⋅ x t 2 ⎥ E t i E t ( ) ⎢ ( ) ( ) ⎥ x, y, E t 2 2 ( ) ⎢ ⎥ ⎢ ⎥ y, + ⋅ 2 E t i E t ( ) ( ) ⎣ ⎦ ⎢ ⎥ E t x, y, 3 3 ( ) y, ⎢ 3 ⎥ ⎢ E t ⎥ ⎣ ( ) ⎦ z, 3 7
1. Introduction Previous works on hypercomplex based polarization sensitive array signal processing N. Le Bihan, and J. Mars, “Singular value decomposition of � quaternion matrices: a new tool for vector-sensor signal processing” ( On polarized real signal separation recorded on tripoles ) S. Miron, N. Le Bihan and J. Mars, “Quaternion-MUSIC for vector- � sensor array processing” ( On direction finding based on two-component vector-sensor arrays ) � N. Le Bihan, S. Miron and J. Mars, “MUSIC algorithm for vector- sensors array using biquaternions” ( On direction finding based on three-component vector-sensor arrays ) 8
1. Introduction However, in all the above-mentioned works dealing with DOA estimation via vector-sensors: Only the MUSIC algorithm was considered which suffers 1. greatly from its heavy computational burden, yet the ESPRIT method with less computational efforts is left uninvestigated. Some hypercomplex operations for designing an ESPRIT 2. based algorithm, such as matrix inverse, and eigenvalue decomposition of an arbitrary square quaternion matrix, are not addressed. 9
2. Data model Quaternions: � Quaternions were first discovered by Hamilton in the year of 1843. A 1. q ∈ H i j k {1, , , } quaternion is defined on units : = + + + ⎧ q q iq jq kq 0 1 2 3 ⎪ = = = − ii jj kk ⎨ 1 ⎪ = − = = − = = − = ij ji k jk kj i ki ik j ⎩ ; ; The algebra of quaternions is an associative division algebra, but not a 2. commutative algebra. This implies that the quaternion product does not obey the commutative law. But the associative law, the distributive law hold. Also, the division (matrix inverse) operation could be defined. 10
2. Data model Quaternion matrices: � Quaternion matrices are matrices of quaternionic entries. The 1. definitions of addition, product, conjugation, transpose for quaternion matrices can be naturally extended from complex matrices. H × → C × M N M N χ 2 2 A key algebraic tool is the adjoint matrix projection, : 2. which links the algebra of quaternion matrices to complex matrices: By exploiting the link of quaternion matrices and complex matrices, 3. many operations of quaternion matrices can be realized via complex operations, such as matrix product and eigenvalue decomposition of a Hermitian matrix. In this paper, we have studied the matrix inversion and eigenvalue 4. decomposition (EVD) of an arbitrary square matrix. More details can be found in our paper. 11
2. Data model Complex Quaternion 1 = i i H Q Ψ χ Q Ψ ( ) ( ) ( )( ) matrices matrices M N 2 ⎡ ⎤ ∗ Q Q = ⎢ χ Q 0 1 ⎥ ( ) ∗ − Q Q ⎣ ⎦ 1 0 × ∈ C j M N × χ Q = + ∈ H M N The adjoint matrix of Q : ( ) 2 2 Q Q i Q ( ) ( ) Quaternion matrix 0 1 The adjoint matrix projection 12
2. Data model The quaternion model � z + ⋅ ⎡ E t i E t ⎤ ( ) ( ) x, y, 2 2 ⎢ ⎥ + ⋅ E t i E t ( ) ( ) ⎢ ⎥ x, y, = = + 3 3 y t A s t n t ( ) ( ) ( ) ⎢ ⎥ � 2 2 2 ⎢ ⎥ y + ⋅ E t i E t ( ) ( ) ⎣ ⎦ + + x,N y,N 1 1 ... x E t E t E t E t ( ) 1 ( ) 2 ( ) 1 ( ) x,N + x, x, x,N E t E t E t E t ( ) 1 ( ) 2 ( ) 1 ( ) y,N + y, y, y,N + ⋅ ⎡ ⎤ E t i E t ( ) ( ) x, y, 1 1 ⎢ ⎥ + ⋅ E t i E t ( ) ( ) ⎢ ⎥ x, y, = 2 2 = + y t A s t n t ( ) ( ) ( ) ⎢ ⎥ � 1 1 1 ⎢ ⎥ + ⋅ E t i E t ( ) ( ) 13 ⎣ ⎦ x,N y,N
2. Data model The quaternion model � 1. We should note that in this quaternion model, the coefficients of different units have different meanings: ⎧ x 1:The output signals of sensors parallel to the -axis ⎪ i y ⎨ :The output signals of sensors parallel to the -axis ⎪ j ⎩ : The complex phase of signals 14
3. The proposed algorithm The main idea of quaternion ESPRIT is to fulfill all the � necessary steps of ESPRIT with quaternionic matrix operations. Calculate the quaternion covariance matrix 1. Signal subspace estimation via quaternionic EVD. 2. Estimation of the shift-invariance factor using quaternionic operations, 3. such as the matrix inverse and quaternionic EVD of a non-Hermitian square matrix. All the above-mentioned steps are similar to the complex � based ESPRIT, and thus are not further addressed here. More details can be found in our paper. 15
3. The proposed algorithm The main advantage of quaternion ESPRIT, is that the low-rank � approximation of a quaternion matrix via quaternion EVD is more accurate than its complex based counterpart, so that the signal subspace may be more accurately estimated. The following test illustrates the conclusion above. We consider an � × ∈ C X c 20 100 arbitrary complex matrix with rank equal to 10. And ∈ H × X 10 100 construct a quaternion matrix by selecting any two adjacent q X X elements of to build a quaternion entry of . Then the c q approximation error against the rank of the approximation matrix is plotted as follows: 16
3. The proposed algorithm 0.8 Low-rank approximation in the quaternion case Approximation error Low-rank approximation in the complex case 0.7 Vs. 0.6 Rank of the approximation Error of low-rank approximation matrix 0.5 0.4 0.3 0.2 0.1 17 0 1 2 3 4 5 6 7 8 9 10 Rank
4. Simulations We assume there are three far-field, narrow-band uncorrelated signals � impinging upon an eight element uniform linear array of crossed dipoles, θ = θ = θ = o o o the DOAs are , and respectively, and the 30 45 10 2 3 1 = = = γ η o o o o γ η o o γ η polarizations are , ( , ) (22 ,30 ) ( , ) (33 ,45 ) and ( , ) (44 ,60 ) 1 1 2 2 3 3 = + d d P ε 1/2 We first define the following sensor-position error: � n n ε n d ε where and denote the ideal and actual positions of the n th element, d n n n P is the uniformly distributed error term, and is the model error variance ε The actual sensor position ⋅ d P d ε The ideal sensor position The illustration of sensor-position error 18
4. Simulations 8 Overall RMSE Vs. Sensor-position error 7 (SNR=30dB; Number 6 of snapshots=1000) Overall RMSE(degree) 5 4 3 Q-ESPRIT C-ESPRIT 2 1 0 19 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 Moder error variance
4. Simulations 10 Overall RMSE Vs. SNR 9 (Sensor-position error=0; 8 Number of snapshots=10) 7 Overall RMSE(degree) 6 Q-ESPRIT 5 C-ESPRIT 4 3 2 1 0 5 10 15 20 25 30 20 SNR(dB)
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