Nested Vector-Sensor Array Processing via TensorModeling 1

Keyong Han & Arye Nehorai

April 24, 2014

1This work was suppoted by the ONR Grant N000141310050.

CSSIP Lab 1

Report Documentation Page Form ApprovedOMB No. 0704-0188

Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering andmaintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, ArlingtonVA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if itdoes not display a currently valid OMB control number.

1. REPORT DATE 24 APR 2014 2. REPORT TYPE

3. DATES COVERED 00-00-2014 to 00-00-2014

4. TITLE AND SUBTITLE Nested Vector-Sensor Array Processing via Tensor Modeling (Briefing Charts)

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S) 5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Washington University in St. Louis,Preston M. Green Department ofElectrical and Systems Engineering,St. Louis,MO,63130

8. PERFORMING ORGANIZATIONREPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR’S REPORT NUMBER(S)

12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited

13. SUPPLEMENTARY NOTES

14. ABSTRACT

15. SUBJECT TERMS

16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT Same as

Report (SAR)

18. NUMBEROF PAGES

19

19a. NAME OFRESPONSIBLE PERSON

a. REPORT unclassified

b. ABSTRACT unclassified

c. THIS PAGE unclassified

Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18

Background• A vector sensor employs multiple (Nc) sensor components. For acoustic:Nc = 4; for electromagnetic: Nc = 6.

• Two strategies: matrix-based scheme (left) and tensor-based scheme (right).

Concatenation

of components

… Multiple

samples

𝑁 × 𝑁𝑐

𝑁𝑁𝑐 × 1

𝑁𝑁𝑐 × 𝑇

Multiple

samples

𝑁 × 𝑁𝑐

𝑁 ×𝑁𝑐 × 𝑇

𝑇

Concatenation

of components

… Multiple

samples

𝑁 × 𝑁𝑐

𝑁𝑁𝑐 × 1

𝑁𝑁𝑐 × 𝑇

Multiple

samples

𝑁 × 𝑁𝑐

𝑁 ×𝑁𝑐 × 𝑇

𝑇

: Sensors

: Snapshots

: Components

• Tensor-based scheme conserves the multidimensional structures of the data,and provides better performance than the matrix-based scheme.

• Our goal: Apply the nested-array strategy to vector-sensor arrays via tensormodeling.• Challenge: Multidimensional operation.

CSSIP Lab 2

Signal Model

• Consider a two-level nested linear array with N vector sensors. The output ofeach vector sensor is an Nc-dimensional vector.

• Assume K far-field sources from directions {(φk, θk), k = 1, . . . ,K}, whereφk and θk represent the azimuth and elevation angles

• Then, we obtain the tensor measurement model:

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

= +

= 𝟑 + 𝑁

𝑁𝑐 𝐾

= 𝟑 + 𝑁

𝑁𝑐 𝐾

Y (t) = A×3 x(t) +E(t), (1)

• Y (t): N ×Nc measurement matrix at time t

• A: N ×Nc ×K array manifold tensor

• x(t): K × 1 source signals

• E(t): N ×Nc measurement noise at time t—————————-

A×3 B: Mode-3 product of A ∈ CI1×I2×I3 and B ∈ CJ1×J2×I3 , defined as (A×3 B)i1i2j1j2=

∑i3ai1i2i3

bj1j2i3

CSSIP Lab 3

Signal Model (Cont.)• ai,j,k = (Ak)i,j, where Ak = dkp

Tk

• dk = [ej2πuTk r1/λ, . . . , ej2πu

Tk rN/λ]T is the phase delay vector.

• uk = [cosφkcosθk, sinφkcosθk, sinθk]T is the unit vector at the sensor pointing

towards the kth signal.

• pk is the steering vector of a single vector sensor located at the origin.

• Based on (1), we get the N ×Nc ×N ×Nc interspectral tensor:

R = E[Y ◦ Y ∗] = A×3 Rx×3A∗ + E[E ◦E∗], (2)

which is a tensor version of the covariance matrix in the scalar case.

• We apply mode-2 matricization to R, and obtain

Z , RT(2) = (A∗T(3) }A)×3 s+ σ2

e

−→I , (3)

– Z is an NcN2 ×Nc matrix

– s = [σ21, σ

22, . . . , σ

2K]T

–−→I = blkdiag(

−→1 , . . . ,

−→1 ), where

−→1 = [eT1 , e

T2 , . . . , e

TN ]T

CSSIP Lab 4

Signal Model (Cont.)

Generalized Kahtri-Rao product }

) ) )

)

j

k

𝑁𝑁𝑐

𝐾

𝑨 ) 𝑇

k

( ) )

Consider one column vector:

—————————-

A×3B: Mode-3 inner product of A ∈ CI1×I2×I3 and B ∈ CJ1×J2×I3 , defined as (A×3B)i1i2j1j2=

∑i3ai1i2i3

bj1j2i3

A } B: Generalized Khatri-Rao product of A ∈ CJ×I3 and B ∈ CI1×I2×I3 , with dimension I1J × I2 × I3, defined as

(A } B)(i1+(j−1)I1),i2,i3= aj,i3

bi1,i2,i3

A ◦B: Outer product of A ∈ CI1×I2 and B ∈ CJ1×J2 , defined as (A ◦B)i1i2j1j2= ai1i2

bj1j2

A(2): mode-2 matrix unfolding of tensor A ∈ CI1×I2×I3 , defined as (A(2))i2,(i1−1)I3+i3= (A)i1i2i3

CSSIP Lab 5

Signal Model (Cont.)

• Comparing (3) with the original signal model (1), we observe that Z in (3)behaves like a received measurement with a longer vector-sensor array whosemanifold is given by A∗T(3) }A. In (3) the equivalent source signal vector is

represented by s and the noise becomes a deterministic matrix given by σ2e

−→I .

• Looking at the structure of the tensor A∗T(3) } A, we observe that there areNc sets of horizontal slices, and they have same horizontal slices except fordifferent amplitudes. Without loss of generality we will consider to use onlythe first set.

• Note that, although we choose only one set, we use the information from allthe Nc sensor components.

CSSIP Lab 6

Signal Model (Cont.)

Dimension change of the manifold tensor using the nested-array strategy.

𝓐

𝑁

𝐾

𝑁c

c

𝑁

𝐾

𝑁c

One set

𝑁

2+ 𝑁 − 1

𝐾

𝑁c

𝓐

Manifold tensor A of theoriginal signal model.

CSSIP Lab 7

Signal Model (Cont.)

Dimension change of the manifold tensor using the nested-array strategy.

𝓐

𝑁

𝐾

𝑁c

𝑁𝑐𝑁2

𝐾

𝑁c

𝑨(3)∗𝑇 ⊚𝓐

After matricizing the

interspectral tensor

• Nc parallel sets ofhorizontal slices

• Different amplitudesbut same information

CSSIP Lab 8

Signal Model (Cont.)

Dimension change of the manifold tensor using the nested-array strategy.

𝓐

𝑁

𝐾

𝑁c

c

𝑁

𝐾

𝑁c

One set

Concentrate

on one set

• Multiple repeated horizontalslices (matrices)

CSSIP Lab 9

Signal Model (Cont.)

Dimension change of the manifold tensor using the nested-array strategy.

𝓐

𝑁

𝐾

𝑁c

c

𝑁

𝐾

𝑁c

One set

𝑁

2+ 𝑁 − 1

𝐾

𝑁c

𝓐

Remove repeated horizontal

slices and resort them

CSSIP Lab 10

Spatial Smoothing

• Spatial smoothing is used to exploit the increased DOFs by building up therank of the observation tensor.

• Similar to the scalar sensor case, we remove repeated slices and sort themaccording to virtual sensor positions:

Z = A×3 s+ σ2eE.

• Divide these 2N − 1 (N = N2/4 + N/2) virtual sensors into N partiallyoverlapping subarrays, and the lth subarray is Zl = Al ×3 s+ σ2

eEl.

• Define Rl , Zl ◦ Zl∗, and take the average of Rl over all l: T , 1

N

∑Nl=1 Rl.

• T is the N ×Nc × N ×Nc spatially smoothed interspectral tensor.

• Based on T , the nested array with N vector sensors provides N − 1 DOFs,whereas a ULA with the same number of vector sensors can provide only N−1DOFs.

CSSIP Lab 11

Higher-Order Singular Value Decomposition (HOSVD)

• HOSVD is a higher-order generalization of the matrix singular valuedecomposition (SVD) [3].

• The HOSVD of tensor T can be written as

T = K×1 U1 ×2 U2 ×3 U3 ×4 U4, (4)

where U1,U3 ∈ CN×N , and U2,U4 ∈ CNc×Nc are orthonormal matrices,provided by the SVD of the i-mode matricization of the tensor T : T (i) =

UiΛiVHi . K ∈ CN×Nc×N×Nc is the core tensor.

• Since T is an Hermitian tensor, i.e., ti1,i2,i3,i4 = t∗i3,i4,i1,i2, ∀i1, i2, i3, i4, theHOSVD of T can be written as

T = K×1 U1 ×2 U2 ×3 U∗1 ×4 U

∗2 . (5)

—————————-

[3] M. Boizard, G. Ginolhac, F. Pascal, S. Miron, and P. Forster, Numerical performance of a tensor MUSIC algorithm based on HOSVD for

a mixture of polarized sources, in EUSIPCO 2013, Marrakech, Marocco, Sep. 2013.

CSSIP Lab 12

Applications: Acoustic Vector Sensors

• For acoustic vector sensors, Nc = 4.

• The array steering matrix can be written as Ak = dkpTk , with

pk = [1,uTk ]T ,

which is the steering vector of a single vector sensor located at the origin.

CSSIP Lab 13

Applications: Electromagnetic Vector Sensors

• For EM vector sensors, Nc = 6. Here, we consider polarized signals.

• The array steering matrix can be written as Ak = dkpTk , with pk = Vkρk,

where

Vk =

−sinφk −cosφksinθkcosφk −sinφksinθk

0 cosθk−cosφksinθk sinφk−sinφksinθk −cosφk

cosθk 0

, and

ρk = [cosγk sinγkejηk]T .

• Ak is the N × Nc steering matrix of the array associated with a polarizedsignal coming from the direction (φk, θk) with polarization (γk, ηk), whereγk ∈ [0, 2π] and ηk ∈ (−π, π] are polarization parameters. Vk is the steeringmatrix of one EM vector sensor associated with the kth signal. ρk is thepolarization vector for the kth signal.

CSSIP Lab 14

Acoustic Case I: MUSIC Spectrum

Fig. 1: MUSIC spectrum using a nested acoustic vector-sensor array with 6 sensors, as a function

of elevation angle θ, T = 1000, SNR = 0dB, K = 6 sources with the same azimuth angles.

Observation: The 2-level nested array resolves all the 6 sources.

CSSIP Lab 15

EM Case II: DOA Estimation

Fig. 2: MUSIC spectrum using a ULA (left: tensor-based) and a nested array (middle: matrix-

based; right: tensor-based) with 6 EM vector sensors, as a function of azimuth φ and elevation

angles θ, T = 1000, SNR = 21.97dB, K = 2.

Observation: The proposed nested vector-sensor array strategy outperforms theULA with the same number of sensors. In addition, the tensor-based methodoutperforms the matrix-based method.

CSSIP Lab 16

EM Case III: Source Number Detection

Fig. 3: Probability of detection versus SNR using a nested array with 6 EM vector sensors and

ULAs with 6 and 12 EM vector sensors, T = 1000, K = 2.

Observation: We can see that the detection performance of all the three arraysimproves with increasing SNRs. In addition, the nested array outperforms thecorresponding ULA with same number of sensors and performs closer to the ULAwith double number of sensors.

CSSIP Lab 17

Summary

• Nested vector-sensor array processing via tensor modeling

– We established the analytical foundation of the nested vector-sensor arrayby exploiting multilinear algebra

– We constructed corresponding signal processing strategies, and verified theireffectiveness through numerical examples.

CSSIP Lab 18

Reference

• Journal Papers

1. K. Han and A. Nehorai, “Nested vector-sensor array processing via tensormodeling,” to appear in IEEE Trans. on Signal Processing.

• Conference Papers

1. K. Han and A. Nehorai, “Direction of arrival estimation using nestedvector-sensor arrays via tensor modeling,” Proc. 8th IEEE Sensor Array andMultichannel Signal Processing (SAM) Workshop, A Coruna, Spain, Jun.22-25, 2014 (invited).

CSSIP Lab 19