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Coprime Coarray Interpolation for DOA Estimation via Nuclear Norm Minimization Chun-Lin Liu 1 . Vaidyanathan 2 Piya Pal 3 P . P 1 , 2 Dept. of Electrical Engineering, MC 136-93 California Institute of Technology, cl.liu@caltech.edu 1 ,

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SLIDE 1

Coprime Coarray Interpolation for DOA Estimation via Nuclear Norm Minimization

Chun-Lin Liu1 P . P . Vaidyanathan2 Piya Pal3

1,2Dept. of Electrical Engineering, MC 136-93

California Institute of Technology, cl.liu@caltech.edu1, ppvnath@systems.caltech.edu2

  • 3Dept. of Electrical and Computer Engineering

University of Maryland, College Park ppal@umd.edu

ISCAS 2016

Liu et al. Coprime Coarray Interpolation ISCAS 2016 1 / 19

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SLIDE 2

Outline

1

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

2

Coarray Interpolation via Nuclear Norm Minimization

3

Numerical Examples

4

Concluding Remarks

Liu et al. Coprime Coarray Interpolation ISCAS 2016 2 / 19

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SLIDE 3

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

Outline

1

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

2

Coarray Interpolation via Nuclear Norm Minimization

3

Numerical Examples

4

Concluding Remarks

Liu et al. Coprime Coarray Interpolation ISCAS 2016 3 / 19

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SLIDE 4

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

Direction-of-arrival (DOA) estimation1

θi

  • •
  • •
  • DOA Estimators

Monochromatic Uncorrelated Sources Sensor Arrays Estimated DOA θi

1Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, 2002. Liu et al. Coprime Coarray Interpolation ISCAS 2016 4 / 19

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SLIDE 5

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

ULA and sparse arrays

ULA (not sparse)

Identify at most N − 1 uncorrelated sources, given N sensors.1 Can only find fewer sources than sensors.

Sparse arrays

1 Minimum redundancy arrays2 2 Nested arrays3 3 Coprime arrays4 4 Super nested arrays5

Identify O(N 2) uncorrelated sources with O(N) physical sensors. More sources than sensors!

1Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory, 2002. 2Moffet, IEEE Trans. Antennas Propag., 1968. 3Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2010. 4Vaidyanathan and Pal, IEEE Trans. Signal Proc., 2011. 5Liu and Vaidyanathan, IEEE Trans. Signal Proc., 2016. Liu et al. Coprime Coarray Interpolation ISCAS 2016 5 / 19

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SLIDE 6

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

Coprime arrays1

The coprime array with (M, N) = 1 is the union of

1 an N-element ULA with spacing Mλ/2 and 2 a 2M-element ULA with spacing Nλ/2.

  • 3
  • 6
  • 9
  • 4
  • 8
  • 12
  • 16
  • 20
  • 0××

× × ×× ××× ××× Physical array S (M = 3, N = 4):

ULA (1) ULA (2)

  • • ••••••••••••••••••••••••••••• ••
  • ××

× × ××

−20 −14 14 20

Difference coarray D = {n1 − n2 | n1, n2 ∈ S} Holes

1Vaidyanathan and Pal, IEEE Trans. Signal Proc., 2011. Liu et al. Coprime Coarray Interpolation ISCAS 2016 6 / 19

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SLIDE 7

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

The spatial smoothing MUSIC Algorithm1

1 Sample covariance matrix:

RS = 1

K

K

k=1

xS(k) xH

S (k). 2 Sample autocorrelation function on the difference coarray:

xD.

  • • ••••••••••••••••••••••••••••• ••
  • −20

−14 14 20

× × × × × × D U

  • xD
  • xU

3 Hermitian Toeplitz matrix

R (indefinite matrix).

  • R =

  

  • xU0
  • xU−1

. . .

  • xU−14
  • xU1
  • xU0

. . .

  • xU−13

. . . . . . ... . . .

  • xU14
  • xU13

. . .

  • xU0

  

4 MUSIC on

R resolves (|U| − 1)/2 = O(N2) uncorrelated sources.

1Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2010; Liu and Vaidyanathan, IEEE Signal Proc. Letter, 2015. Liu et al. Coprime Coarray Interpolation ISCAS 2016 7 / 19

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SLIDE 8

Coarray Interpolation via Nuclear Norm Minimization

Outline

1

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

2

Coarray Interpolation via Nuclear Norm Minimization

3

Numerical Examples

4

Concluding Remarks

Liu et al. Coprime Coarray Interpolation ISCAS 2016 8 / 19

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SLIDE 9

Coarray Interpolation via Nuclear Norm Minimization

Why coarray interpolation?

  • • ••••••••••••••••••••••••••••• ••
  • −20

−14 14 20

× × × × × × D

  • xD

|D| = 3MN + M − N

  • −14

14

U

  • xU

Not all information is used. |U| = 2MN + 2M − 1

  • • ••••••••••••••••••••••••••••• ••
  • −20

−14 14 20

  • V
  • xV

All information is used. |V| = 4MN − 2N + 1

Liu et al. Coprime Coarray Interpolation ISCAS 2016 9 / 19

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SLIDE 10

Coarray Interpolation via Nuclear Norm Minimization

Previous work

1 Spatial smoothing MUSIC1: No coarray interpolation. 2 Positive-definite Toeplitz matrix completion2: Not always feasible. 3 Coarray interpolation (ICA-AI)3: Non-convex optimization. 4 Sparse support recovery techniques4: Predefined dense grid and

parameters.

5 Gridless DOA estimator via low-rank recovery5: Not used for

interpolation, but for denoising.

1Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2010; Liu and Vaidyanathan, IEEE Signal Proc. Letter, 2015. 2Abramovich, Spencer, and Gorokhov, IEEE Trans. Signal Proc., 1999. 3Friedlander and Weiss, IEEE Trans. Aero. Elec. Sys., 1992; Tuncer, Yasar, and Friedlander, Radio Science, 2007. 4Zhang, Amin, and Himed, IEEE ICASSP, 2013; Pal and Vaidyanathan, IEEE Trans. Signal Proc., 2015; 5Pal and Vaidyanathan, IEEE Signal Proc. Letter, 2014. Liu et al. Coprime Coarray Interpolation ISCAS 2016 10 / 19

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SLIDE 11

Coarray Interpolation via Nuclear Norm Minimization

The proposed method (via nuclear norm minimization)

  • R⋆

V =

arg min

  • RV∈C|V+|×|V+|

RV∗

  • s. t.
  • RV =

RH

V ,

RVn1,n2 = xDn1−n2, n1, n2 ∈ V+ = {n | n ∈ V, n ≥ 0}.

  • •
  • •
  • •
  • •

× × × × × ×

D

  • xD
  • •
  • •
  • •
  • •
  • •
  • •
  • V
  • xV ≈ autocorrelation functions
  • RV ≈ covariance matrices

Low rank, Hermitian Toeplitz

  • RV has a low-rank structure for sufficient number of snapshots.

The nuclear norm · ∗ (sum of singular values) is a convex relaxation of the matrix rank.

  • RV is Hermitian.
  • RV is a Toeplitz matrix with some known entries.

Liu et al. Coprime Coarray Interpolation ISCAS 2016 11 / 19

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SLIDE 12

Coarray Interpolation via Nuclear Norm Minimization

Advantages over the previous work

1 All the information is used. 2 Gridless. 3 Always feasible, even though

  • R⋆

V can be indefinite. 4 Convex program. 5 It is possible to resolve beyond

the limit of U.

  • •
  • •
  • •
  • •
  • •
  • •
  • V
  • xV ≈ autocorrelation functions
  • RV ≈ covariance matrices

Low rank, Hermitian Toeplitz

Coarray interpolation

  • R⋆

V =

arg min

  • RV∈C|V+|×|V+|

RV∗ subject to

  • RV =

RH

V ,

RVn1,n2 = xDn1−n2. MUSIC

  • R⋆

V =

UΛ UH,

  • U =
  • Us
  • Un
  • ,

PMUSIC(¯ θ) = 1

  • UH

n vV+(¯

θ)

  • 2

2

Liu et al. Coprime Coarray Interpolation ISCAS 2016 12 / 19

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SLIDE 13

Numerical Examples

Outline

1

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

2

Coarray Interpolation via Nuclear Norm Minimization

3

Numerical Examples

4

Concluding Remarks

Liu et al. Coprime Coarray Interpolation ISCAS 2016 13 / 19

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SLIDE 14

Numerical Examples

Simulation parameters

A coprime array with M = 3 and N = 5: (10 sensors) S = {0, 3, 5, 6, 9, 10, 12, 15, 20, 25}, |S| = 10, |S| − 1 = 9, D = {−25, −22, −20, −19, − 17, . . . , 17, 19, 20, 22, 25}, |D| = 43, (|D| − 1)/2 = 21, U = {−17, . . . , 17}, |U| = 35, (|U| − 1)/2 = 17, V = {−25, . . . , 25}, |V| = 51. S :

  • 3 •

5• 6

  • 9•

10 • 12

  • 15
  • 20
  • 25

× × × × × × × × × × × × × × × × D :

  • •• ••••••••••••••••••••••••••••••••••• •• •
  • −25

−17 17 25

× × × × × × × × U :

  • −17

17

V :

  • −25

25 Liu et al. Coprime Coarray Interpolation ISCAS 2016 14 / 19

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SLIDE 15

Numerical Examples

Simulation parameters (Cont.)

1 The maximum number of resolvable uncorrelated sources:

using U: 17, using D: 21.

2 Equal-power uncorrelated sources. 3 0 dB SNR and 500 snapshots. 4 Root-mean-squared error:

E =

  • 1

D

D

  • i=1
  • ¯

θi − ¯ θi 2 , where

{ ¯ θi}D

i=1 is the estimated normalized DOA, and

{¯ θi}D

i=1 is the true normalized DOA.

Liu et al. Coprime Coarray Interpolation ISCAS 2016 15 / 19

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SLIDE 16

Numerical Examples

Example 1: Two closely spaced sources (10 sensors)

SS-MUSIC1 Co-LASSO4 Spline3 P(¯ θ)

−0.0045 0.0045 10

−3

10

−2

10

−1

10 ¯ θ −0.0045 0.0045 10

−4

10

−3

10

−2

10

−1

10 ¯ θ −0.0045 0.0045 10

−4

10

−3

10

−2

10

−1

10 ¯ θ

E 0.00093588 0.23299 0.0019951 ICA-AI3 P .D. Toeplitz Completion2 Nuclear norm (Proposed) P(¯ θ)

−0.0045 0.0045 10

−5

10

−4

10

−3

10

−2

10

−1

10 ¯ θ −0.0045 0.0045 10

−4

10

−3

10

−2

10

−1

10 ¯ θ −0.0045 0.0045 10

−4

10

−3

10

−2

10

−1

10 ¯ θ

E 0.0046232 0.0011188 0.00073202

1-5 See page 10 for all the references

Liu et al. Coprime Coarray Interpolation ISCAS 2016 16 / 19

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SLIDE 17

Numerical Examples

Example 2: D = 19 sources (10 sensors)

Co-LASSO (E = 0.0054924) Spline (E = 0.017665)

−0.4 0.4 10

−10

10

−8

10

−6

10

−4

10

−2

10

¯ θ

−0.4 0.4 10

−3

10

−2

10

−1

10

¯ θ

ICA-AI (E = 0.018319) Nuclear norm (E = 0.003458)

−0.4 0.4 10

−6

10

−4

10

−2

10

¯ θ

−0.4 0.4 10

−5

10

−4

10

−3

10

−2

10

−1

10

¯ θ

✓

Liu et al. Coprime Coarray Interpolation ISCAS 2016 17 / 19

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SLIDE 18

Concluding Remarks

Outline

1

Introduction (DOA, Coprime Arrays, Spatial Smoothing MUSIC)

2

Coarray Interpolation via Nuclear Norm Minimization

3

Numerical Examples

4

Concluding Remarks

Liu et al. Coprime Coarray Interpolation ISCAS 2016 18 / 19

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SLIDE 19

Concluding Remarks

Concluding remarks

Coarray interpolation via nuclear norm minimization:

The correlation information on the difference coarray is fully utilized. The estimation error is reduced. More sources than (|U| − 1)/2 (the limit of SS MUSIC using xU) can be resolved.

In the future, it will be interesting to incorporate the matrix denoising idea into the interpolation approach.

Thank you!

Liu et al. Coprime Coarray Interpolation ISCAS 2016 19 / 19