Tensor MUSIC in Multidimensional Sparse Arrays Chun-Lin Liu 1 and P . - - PowerPoint PPT Presentation

▶
tensor music in multidimensional sparse arrays
SMART_READER_LITE
LIVE PREVIEW

Tensor MUSIC in Multidimensional Sparse Arrays Chun-Lin Liu 1 and P . - - PowerPoint PPT Presentation

Apr 11, 2023 281 likes •528 views

Tensor MUSIC in Multidimensional Sparse Arrays Chun-Lin Liu 1 and P . Vaidyanathan 2 . P Dept. of Electrical Engineering, MC 136-93 California Institute of Technology, Pasadena, CA 91125, USA cl.liu@caltech.edu 1 , ppvnath@systems.caltech.edu 2

slide-1
SLIDE 1

Tensor MUSIC in Multidimensional Sparse Arrays

Chun-Lin Liu1 and P . P . Vaidyanathan2

  • Dept. of Electrical Engineering, MC 136-93

California Institute of Technology, Pasadena, CA 91125, USA cl.liu@caltech.edu1, ppvnath@systems.caltech.edu2

Asilomar Conference on Signals, Systems, and Computers, 2015

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 1 / 24

slide-2
SLIDE 2

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 2 / 24

slide-3
SLIDE 3

Introduction

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 3 / 24

slide-4
SLIDE 4

Introduction Motivation

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 4 / 24

slide-5
SLIDE 5

Introduction Motivation

Harmonic Retrieval in Planar Array Processing1

  • Planar arrays

Spatial information

t

Received waveforms Temporal information Incoming plane waves

Utimate Goal Estimate source profiles (azimuth, elevation, range, Doppler, etc.) from sensor measurements efficiently and accurately.

1Harry L. Van Trees. Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. Wiley

Interscience, 2002. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 5 / 24

slide-6
SLIDE 6

Introduction Motivation

Sparse Array Processing2,3

Uniform Linear Arrays (ULAs)

ULA with N sensors and sensor separation λ/2.

  • N

λ/2 Identify at most N − 1 sources using N sensors. ✗

Linear Sparse Arrays

Nested array with N1, N2 and

  • min. separation λ/2.
  • N1

λ/2 N2 (N1 + 1)λ/2 Identify O(N2) uncorrelated sources using O(N) sensors.

✓

2Alan T Moffet. “Minimum-redundancy linear arrays”. In: IEEE Trans. Antennas Propag. 16.2 (1968), pp. 172–175. 3Piya Pal and P

. P . Vaidyanathan. “Nested Arrays: A Novel Approach to Array Processing With Enhanced Degrees of Freedom”. In: IEEE Trans. Signal Process. 58.8 (2010), pp. 4167–4181. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 6 / 24

slide-7
SLIDE 7

Introduction Motivation

Tensor Model4,5, etc.

Measurements Vector Model x = Spatial/temporal relations are mixed✗ Tensor Model X = Spatial/temporal relations are separated✓

  • 4M. Haardt, F. Roemer, and G. Del Galdo. “Higher-Order SVD-Based Subspace Estimation to Improve the Parameter

Estimation Accuracy in Multidimensional Harmonic Retrieval Problems”. In: IEEE Trans. Signal Process. 56.7 (2008),

  • pp. 3198–3213.
  • 5D. Nion and N.D. Sidiropoulos. “Tensor Algebra and Multidimensional Harmonic Retrieval in Signal Processing for MIMO

Radar”. In: IEEE Trans. Signal Process. 58.11 (2010), pp. 5693–5705. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 7 / 24

slide-8
SLIDE 8

Introduction Motivation

Main Goal of this Work

Proposed Scheme

Input Sparse arrays Tensor Models Tensor MUSIC Azimuth Elevation Doppler

  • etc. . .

Related work: ULA, tensors, and MUSIC ⇒ DOA and polarization6,7. Nested arrays, tensors, and MUSIC ⇒ azimuth, elevation, and polarization8.

6Sebastian Miron, Nicolas Le Bihan, and Jerome I Mars. “Vector-Sensor MUSIC for Polarized Seismic Sources Localization”.

In: EURASIP Journal on Advances in Signal Processing 2005.1 (2005), pp. 74–84.

  • 7M. Boizard et al. “Numerical performance of a tensor MUSIC algorithm based on HOSVD for a mixture of polarized sources”.

In: Proc. European Signal Process. Conf. 2013, pp. 1–5.

8Keyong Han and A. Nehorai. “Nested Vector-Sensor Array Processing via Tensor Modeling”. In: IEEE Trans. Signal Process.

62.10 (2014), pp. 2542–2553. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 8 / 24

slide-9
SLIDE 9

Introduction Tensors

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 9 / 24

slide-10
SLIDE 10

Introduction Tensors

Notations9

Tensor A A Outer product A ◦ B A B = Inner product A, B

  • A

, B

  • =

n-mode product A ×n U U A = A ×1 U

9Tamara G. Kolda and Brett W. Bader. “Tensor Decompositions and Applications”. In: SIAM Review 51.3 (2009), pp. 455–500.

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 10 / 24

slide-11
SLIDE 11

Introduction Tensors

Tensor Decomposition10

CANDECOMP/PARAFAC (CP) decomposition: X ≈ R

r=1 ar ◦ br ◦ cr.

High-order SVD (HOSVD): X ≈ G ×1 A ×2 B ×3 C.

10Tamara G. Kolda and Brett W. Bader. “Tensor Decompositions and Applications”. In: SIAM Review 51.3 (2009), pp. 455–500.

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 11 / 24

slide-12
SLIDE 12

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 12 / 24

slide-13
SLIDE 13

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 13 / 24

slide-14
SLIDE 14

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor

Sparse Array Processing

Physical array S Difference coarray D

Vector Model:

  • xS(k)

Estimate covariance matrix

  • RS

Auto- correlation vector

  • xD

Hermitian Toeplitz matrix11

  • R

MUSIC

  • θi,

. . .

Tensor Model (Proposed):

  • X S(k)

Estimate covariance tensor

  • RS

Auto- correlation tensor

  • X D

Coarray tensor

  • R

Tensor MUSIC

  • θi,

. . . Existing Proposed

11S.U. Pillai, et al.

“A new approach to array geometry for improved spatial spectrum estimation”. Proc. IEEE 73.10 (1985); C.-L. Liu and P . P . Vaidyanathan. “Remarks on the Spatial Smoothing Step in Coarray MUSIC”. IEEE SPL 22.9 (2015). Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 14 / 24

slide-15
SLIDE 15

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor

Some Discussions on the Coarray Tensor R

Vector model12

Rp1,p′

1 =

xDm1, p1 − p′

1 = m1.

Tensor model

Rp1,p2,...,pR,p′

1,p′ 2,...,p′ R

= X Dm1,m2,...,mR, pr − p′

r = mr,

r = 1, 2, . . . , R.

  • R avoids implementing spatial smoothing in tensors.
  • R admits the (tensor) MUSIC algorithm.

12S.U. Pillai, et al.

“A new approach to array geometry for improved spatial spectrum estimation”. Proc. IEEE 73.10 (1985); C.-L. Liu and P . P . Vaidyanathan. “Remarks on the Spatial Smoothing Step in Coarray MUSIC”. IEEE SPL 22.9 (2015). Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 15 / 24

slide-16
SLIDE 16

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 16 / 24

slide-17
SLIDE 17

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum

Tensor MUSIC

MUSIC

1 Eigen-

decomposition:

  • R =

U Λ UH.

2 Signal and noise

subspace:

  • U =
  • Us
  • Un
  • 3 MUSIC spectrum:

P(¯ θ) = 1 UH

n v(¯

θ)2 v(¯ θ): steering vectors.

Tensor MUSIC13

1 HOSVD:

  • R =

K ×1 U1 ×2 U2 · · · ×R UR ×R+1 U∗

1 ×R+2

U∗

2 · · · ×2R

U∗

R. 2 Signal and noise subspace:

  • Ur =
  • Ur,s
  • Ur,n
  • is a unitary matrix.

3 Tensor MUSIC spectrum

PHOSV D (¯ µ) = 1 V (¯ µ)×1 U1,n UH

1,n . . .×R

UR,n UH

R,n2 F

V (¯ µ): steering tensors.

  • 13M. Boizard et al. “Numerical performance of a tensor MUSIC algorithm based on HOSVD for a mixture of polarized sources”.

In: Proc. European Signal Process. Conf. 2013, pp. 1–5. Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 17 / 24

slide-18
SLIDE 18

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum

Problem with tensor MUSIC via HOSVD

Our observation: PHOSV D (¯ µ) is a separable MUSIC spectrum

PHOSV D (¯ µ) =

R

  • r=1

Pr(¯ µ(r)), Pr(¯ µ(r)) = 1 UH

r,nvU+

r (¯

µ(r))2

2

PHOSV D (¯ µ) has cross-terms

¯ µ(1) ¯ µ(2)

  • Actual

PHOSV D (¯ µ)

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 18 / 24

slide-19
SLIDE 19

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Tensor MUSIC spectrum

Proposed Tensor MUSIC spectrum via CP

CP

  • R =

D

  • ℓ=1
  • a(1)

ℓ

a(2)

ℓ

  • · · · ◦

a(R)

ℓ

a(1)∗

ℓ

a(2)∗

ℓ

  • · · · ◦

a(R)∗

ℓ

.

Signal and noise subspace

Signal subspace S = span{ a(1)

ℓ

a(2)

ℓ

  • · · · ◦

a(R)

ℓ

}D

ℓ=1,

Noise subspace N = S⊥.

Tensor MUSIC spectrum

PCP (¯ µ) = 1 projN VU+ (¯ µ) 2

F

.

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 19 / 24

slide-20
SLIDE 20

Numerical Examples

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 20 / 24

slide-21
SLIDE 21

Numerical Examples

Tensor Dimension R = 2

10 sensors (or samples) in each dimension Coprime array/sampling with M = 3 and N = 5 1000 snapshots, 0dB SNR, and D = 5 equal-power sources. PULA,HOSV D (¯ µ) PCoprime,HOSV D (¯ µ) Proposed PCoprime,CP (¯ µ) Low resolution ✗ High resolution ✓ High resolution ✓ Cross terms ✗ Cross terms ✗ No cross terms ✓

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 21 / 24

slide-22
SLIDE 22

Numerical Examples

Tensor Dimension R = 3

10 sensors (or samples) in each dimension, Coprime array/sampling with M = 3 and N = 5, 1000 snapshots, 0dB SNR, and D = 5 equal-power sources. PULA,HOSV D (¯ µ) PCoprime,HOSV D (¯ µ) Proposed PCoprime,CP (¯ µ) Low resolution ✗ High resolution ✓ High resolution ✓ Cross terms ✗ Cross terms ✗ No cross terms ✓

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 22 / 24

slide-23
SLIDE 23

Concluding Remarks

Outline

1

Introduction Motivation Tensors

2

Contribution: Tensor MUSIC in Multidimensional Sparse Arrays Coarray tensor Tensor MUSIC spectrum

3

Numerical Examples

4

Concluding Remarks

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 23 / 24

slide-24
SLIDE 24

Concluding Remarks

Concluding Remarks

Parameter estimation using

1 Sparse arrays / non-uniform sampling, 2 Tensor models, and 3 MUSIC.

Tensor MUSIC using HOSVD on R:

1 Product of MUSIC spectra 2 Cross-terms

Tensor MUSIC using CP on R:

1 No cross-terms

Liu and Vaidyanathan (Caltech) Tensor MUSIC ACSSC 2015 24 / 24