Quaternion ESPRIT for Direction Finding with a Polarization Sentive - - PowerPoint PPT Presentation

Quaternion ESPRIT for Direction Finding with a Polarization Sentive Array

Xiaofeng Gong, Zhiwen Liu, Yougen Xu

2

Content

• 1. Introduction
• 2. Data model
• 3. The proposed algorithm
• 4. Simulations
• 5. Conclusion

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• 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.

4

• 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.

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• 1. Introduction

y z x

• θ

d

• ,1

,0 1 1 1 1 ,0 ,0 2 2 2 2

exp( ), , :related to angular parameters only exp( ), , :related to both angle and polarization

x x y x

E E ρ α ρ α E E ρ α ρ α = ⎧ ⎪ ⎨ = ⎪ ⎩

,0 x

E

,1 x

E

,2 x

E

,0 y

E

,1 y

E

,2 y

E

A Polarization sensitive array

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• 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.

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• 1. Introduction

x y z

1( ) x,

E t

1( ) y,

E t

2( ) x,

E t

2( ) y,

E t

3( ) x,

E t

3( ) y,

E t

1 1 2 2 3 3

( ) ( ) ( ) ( ) ( ) ( ) ( )

x, y, x, y, y, z,

E t E t E t t E t E t E t ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ x

1 1 2 2 3 3

( ) ( ) ( ) ( ) ( ) ( ) ( )

x, y, x, y, x, y,

t E t i E t E t i E t E t i E t = ⎡ ⎤ + ⋅ ⎢ ⎥ + ⋅ ⎢ ⎥ ⎢ ⎥ + ⋅ ⎣ ⎦ y

The complex long- vector model Vs. The hypercomplex model

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• 1. Introduction
• 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)

Previous works on hypercomplex based polarization sensitive array signal processing

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• 1. Introduction

However, in all the above-mentioned works dealing with DOA estimation via vector-sensors:

1.

Only the MUSIC algorithm was considered which suffers greatly from its heavy computational burden, yet the ESPRIT method with less computational efforts is left uninvestigated.

2.

Some hypercomplex

• perations for designing an ESPRIT

based algorithm, such as matrix inverse, and eigenvalue decomposition of an arbitrary square quaternion matrix, are not addressed.

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• 2. Data model
• Quaternions:

1.

Quaternions were first discovered by Hamilton in the year of 1843. A quaternion is defined on units :

2.

The algebra of quaternions is an associative division algebra, but not a commutative algebra. This implies that the quaternion product does not

• bey the commutative law. But the associative law, the distributive law
• hold. Also, the division (matrix inverse) operation could be defined.

{1, , , } i j k

q∈H

1 2 3

1 ; ; q q iq jq kq ii jj kk ij ji k jk kj i ki ik j = + + + ⎧ ⎪ = = = − ⎨ ⎪ = − = = − = = − = ⎩

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• 2. Data model
• Quaternion matrices:

1.

Quaternion matrices are matrices of quaternionic entries. The definitions of addition, product, conjugation, transpose for quaternion matrices can be naturally extended from complex matrices.

2.

A key algebraic tool is the adjoint matrix projection, which links the algebra of quaternion matrices to complex matrices:

3.

By exploiting the link of quaternion matrices and complex matrices, many operations of quaternion matrices can be realized via complex

• perations, such as matrix product and eigenvalue decomposition of a

Hermitian matrix.

4.

In this paper, we have studied the matrix inversion and eigenvalue decomposition (EVD) of an arbitrary square matrix. More details can be found in our paper.

2 2

:

M N M N

χ

× ×

→ H C

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• 2. Data model

Quaternion matrices Complex matrices Quaternion matrix The adjoint matrix of Q :

( ) 2 2

( ) ( )

j M N

Q

×

∈ χ C

1 M N

i

×

= + ∈ Q Q Q H

( ) ( )

1 ( )( ) 2

i i H M N

Q = Q Ψ χ Ψ

1 1

( ) Q

∗ ∗

⎡ ⎤ = ⎢ ⎥ − ⎣ ⎦ Q Q χ Q Q

The adjoint matrix projection

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• 2. Data model
• The quaternion model

x y z

1( ) y,

E t

1( ) x,

E t

2( ) y,

E t

2( ) x,

E t ( )

y,N

E t ( )

x,N

E t

1( ) y,N

E t

+ 1( ) x,N

E t

+

...

1 1 2 2 1 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

x, y, x, y, x,N y,N

E t i E t E t i E t t t t E t i E t

• + ⋅

⎡ ⎤ ⎢ ⎥ + ⋅ ⎢ ⎥ = = + ⎢ ⎥ ⎢ ⎥ + ⋅ ⎣ ⎦ y A s n

2 2 3 3 2 2 2 1 1

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

x, y, x, y, x,N y,N

E t i E t E t i E t t t t E t i E t

• +

+

+ ⋅ ⎡ ⎤ ⎢ ⎥ + ⋅ ⎢ ⎥ = = + ⎢ ⎥ ⎢ ⎥ + ⋅ ⎣ ⎦ y A s n

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• 2. Data model
• The quaternion model
• 1. We should note that in this quaternion model, the

coefficients of different units have different meanings:

1:The output signals of sensors parallel to the -axis :The output signals of sensors parallel to the -axis : The complex phase of signals x i y j ⎧ ⎪ ⎨ ⎪ ⎩

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• 3. The proposed algorithm
• The main idea of quaternion ESPRIT is to fulfill all the

necessary steps of ESPRIT with quaternionic matrix

• perations.

1.

Calculate the quaternion covariance matrix

2.

Signal subspace estimation via quaternionic EVD.

3.

Estimation of the shift-invariance factor using quaternionic operations, 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.

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• 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

arbitrary complex matrix with rank equal to 10. And construct a quaternion matrix by selecting any two adjacent elements of to build a quaternion entry of . Then the approximation error against the rank of the approximation matrix is plotted as follows:

10 100 q ×

∈ X H

20 100

X c

×

∈C

c

X

q

X

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• 3. The proposed algorithm

Approximation error Vs. Rank of the approximation matrix

1 2 3 4 5 6 7 8 9 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Rank Error of low-rank approximation Low-rank approximation in the quaternion case Low-rank approximation in the complex case

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• 4. Simulations
• We assume there are three far-field, narrow-band uncorrelated signals

impinging upon an eight element uniform linear array of crossed dipoles, the DOAs are , and respectively, and the polarizations are , and

• We first define the following sensor-position error:

where and denote the ideal and actual positions of the nth element, is the uniformly distributed error term, and is the model error variance

n

d

1/2 n n ε n

d d P ε = +

n

d

n

ε

ε

P

ε

d P ⋅

The actual sensor position The ideal sensor position

d

The illustration of sensor-position error

1

10 θ =

• 2

30 θ =

• 3

45 θ =

• 1

1

( , ) (22 ,30 ) γ η =

• 2

2

( , ) (33 ,45 ) γ η =

• 3

3

( , ) (44 ,60 ) γ η =

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• 4. Simulations

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 1 2 3 4 5 6 7 8 Moder error variance Overall RMSE(degree) Q-ESPRIT C-ESPRIT

Overall RMSE Vs. Sensor-position error (SNR=30dB; Number

• f snapshots=1000)

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• 4. Simulations

Overall RMSE Vs. SNR (Sensor-position error=0; Number of snapshots=10)

5 10 15 20 25 30 1 2 3 4 5 6 7 8 9 10 SNR(dB) Overall RMSE(degree) Q-ESPRIT C-ESPRIT

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• 5. Conclusion
• A novel DOA estimation algorithm, namely

quaternion-ESPRIT, is proposed based on polarization-sensitive arrays

• The proposed method outperforms the

conventional ESPRIT algorithm in the presence

• f model error, or with short data length, and

thus may be more competent in practice.

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Thank you!