Multi Snapshot Sparse Bayesian Learning for Direction of ... · Multi Snapshot Sparse Bayesian Learning for Direction of Arrival Estimation ... Y = A X + N Y A X N Multi Snapshot

Multi Snapshot Sparse Bayesian Learning for Direction of Arrival Estimation

Christoph Mecklenbräuker and Peter Gerstoft

Plan MAP+Lasso path (3 slides) Performance simulation (6 slides) Three acoustic data sets (4 slides)

IEEE SPL 23(10):1469–1473, Oct. 2016

Y = A X + N

Y A X N

Multi Snapshot Sparse Bayesian Learning for DOA P. Gerstoft, C.F. Mecklenbräuker, A. Xenaki, S. Nannuru

CS approach to geophysical data analysis

CS of Earthquakes

Yao, GRL 2011, PNAS 2013

Sequential CS

Mecklenbrauker, TSP 2013

a) Sequential h0=0.5

5 10 15 20 25 30 35 40 45 500

45

90

135

180

Time

DOA

(deg

)

b) Sequential h0=0.05

5 10 15 20 25 30 35 40 45 500

45

90

135

180

10

15

20

25

30

35

40

CS beamforming

Xenaki, JASA 2014, 2015Gerstoft JASA 2015

CS fathometer

Yardim, JASA 2014

CS Sound speed estimation

Bianco, JASA 2016 Gemba, JASA 2016

CS matched field

Linear Time-Variant MIMO Channel

28.10.2008 Page 3

h(t,τ) x(t)

y(t)

y(t) = h(t, t − τ )x(τ )dτ0

∞

∫For MIMO: h(.,.) is a matrix. For fading channel: : h(.,.) is a random process

+ Noise + Interference

Multipath wave propagation

13.06.17 Page 4

- 5/32 -

•  Road crossing

Emergency vehicle warning, intersection collision warning, pre-crash sensing warning.

• General LOS obstructions

Hazardous location notification • Merging lanes

Wrong way driving warning, co-operative merging assistance

Safety Critical Scenarios (I)

- 6/32 -

• Traffic congestion

Traffic condition warning • In-tunnel

Emergency electronic brake lights, slow vehicle warning, lane change assistance, co-operative forward collision warning

Safety Critical Scenarios (II)

- 7/32 -

!  mobile stations (MS) are moving, base station (BS) is fixed

!  time-variant multi-path propagation !  interference

Communication Scenarios

Cellular

!  transmitter and receiver are mobile !  safety critical scenarios

vtx

vrx

vsct

d~100m vtx

Vrx=0

d = 100m ... 10km

BS

MS

Vehicular

- 8/32 -

Multipath Propagation Multipath Propagation: SISO Case

⌘0e

j2⇡f0t�(t � ⌧0)

scatterer

user

scatterer

⌘1e

j2⇡f1t�(t � ⌧1)

v

receiver⌘2e

j2⇡f2t�(t � ⌧2)

v velocity` path

⌘` attenuation⌧` time delayf` Doppler shiftL

0 number of paths�` angle of arrivalfC carrier frequencyc0 speed of light

Time-variant channel impulse response

h

p

(t, ⌧) =L

0�1X

`=0⌘`ej2⇡f`t�(⌧ � ⌧`) , f` =

v cos�`fCc0

Thomas Zemen, Nicolai Czink March 17, 2011 8 / 42

13.06.17 Page 5

Relevant Scatterers

© FTW - 19 -

Power delay profile (PDP)

General LOS obstruction in highway Main MPC contributions -  LOS

-  Other cars

-  Trucks

LOS

Other cars

Trucks

Scenario: General LOS obstruction - highway

Vehicular Channel @ 5.9 GHz measured with 240 MHz BW (non-sparse least-squares estimate)

Laura Bernadó et al.

13.06.17 Page 6

Scenario: In-tunnel

© FTW - 21 -

Power delay profile (PDP)

In-tunnel Main MPC contributions -  LOS

-  Other cars

-  Trucks

-  Buildings -  Big metallic structures (traffic

signs, ventilation systems…)

-  Ceiling and walls Rx enters the tunnel

Other cars

LOS

Ventilation system

Shield Ceiling

Relevant Scatterers Vehicular Channel @ 5.9 GHz measured with 240 MHz BW (non-sparse least-squares estimate)

Laura Bernadó et al.

Direction of Arrival (DOA) estimation with arrays

Beamfroming vs Compressive sensing

y = AM⇥Nx+n,M < N

n 2 CM , SNR = 20log10kAxk2knk2 , knk2 ✏

Beamformingsimplified l2-norm minimization (AAH = IM )

x̂ = A

Hy = A

HAx+A

Hn

Compressive sensingl1-norm minimization

x̂ = argminx2CN

kxk1 s.t. kAx� yk2 ✏

ULA M = 8, d� = 1

2 , [✓1, ✓2] = [0, 5]�, SNR = 20 dB

−90 −60 −30 0 30 60 90

−20

−10

0

θ [◦]

P[dB

remax

]

−90 −60 −30 0 30 60 90

−20

−10

0

θ [◦]

P[dB

remax

]

A. Xenaki (SIO/DTU) Compressive beamforming, JASA 2014 UA 2014 7 / 16

CBF : x̂ = AH y CS: min Ax − y2

2+µ x

1( )

N>>M

High resolution No sidelobes

DOA estimation with sensor arrays

-90o 90o

0o

-45o 45o

�1 �2

x1

x2

p1(r,t) = x1 ej(�t-k1r)

k1

k2

p2(r,t) = x2 ej(�t-k2r)

r

�

�

x 2 C, ✓ 2 [�90�, 90�]

k = �2⇡

�sin ✓, �:wavelength

ym =X

n

xnej 2⇡� rm sin ✓n

m 2 [1, · · · ,M]: sensorn 2 [1, · · · ,N]: look direction

y = AM⇥Nx

y = [y1, · · · , yM ]T , x = [x1, · · · , xN ]T

A = [a1, · · · , aN ]

an =1pM

[e j2⇡� r1 sin ✓n , · · · , e j

2⇡� rM sin ✓n ]T


Conventional beamformer Compressive Sensing

Maximum A Posteriori (MAP) estimation

Likelihood (noise complex Gaussian) p(y | x)∝ exp −Ax − y

2

2

σ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Prior (Laplacian) p(x)∝ exp −x

1

ν

⎛

⎝⎜

⎞

⎠⎟

Bayes rule p(x|y)∝ p(y|x)p(x)∝ exp −Ax − y

2

2

σ 2 −x

1

ν

⎛

⎝⎜⎜

⎞

⎠⎟⎟

MAP

solutions. The choice of the (unconstrained) LASSO for-mulation (8) over the constrained formulation (7) allowsthe sparse reconstruction method to be interpreted in astatistical Bayesian setting, where the unknowns x andthe observations y are both treated as stochastic (ran-dom) processes, by imposing a prior distribution on thesolution vector x which promotes sparsity14–16.

The Bayes theorem32 connects the posterior distribu-tion p(x|y), of the model parameters x conditioned onthe data y, with the data likelihood p(y|x), the prior dis-tribution of the model parameters p(x) and the marginaldistribution of the data p(y),

p(x|y) = p(y|x)p(x)p(y)

. (9)

From the Bayes rule (9), the maximum a posteriori(MAP) estimate is,

x̂MAP = argmaxx

ln p(x|y)

= argmaxx

[ln p(y|x) + ln p(x)]

= argminx

[� ln p(y|x)� ln p(x)] ,

(10)

where the marginal distribution of the data p(y) is omit-ted since it is independent of the model x.

Based on a complex Gaussian noise model with i.i.d.real and imaginary parts, n ⇠ CN (0,�2

I), the likelihoodof the data is also complex Gaussian distributed p(y|x) ⇠N (Ax,�

2I),

p(y|x) / e

� ky�Axk22�

2. (11)

Assuming that the coe�cients of the solution vector x

have i.i.d. Laplace (i.e., double exponential) priors33,

p(x) /NY

i=1

e

(� |xi

|⌫

) = e

(� kxk1⌫

), (12)

the LASSO estimate (8) can be interpreted as the maxi-mum a posteriori (MAP)estimate,

x̂MAP = argminx

⇥

ky �Axk22 + µkxk1⇤

= x̂LASSO(µ),

(13)where µ = �

2/⌫. The Laplace prior distribution encour-

ages sparse solutions with many zero components since itconcentrates more mass near 0 than in the tails. There-fore, the model selected by the LASSO optimization al-gorithm has the highest posterior probability under theBayesian framework.

V. REGULARIZATION PARAMETER SELECTION

The choice of the regularization parameter µ in (8),also called LASSO shrinkage parameter, is crucial as itcontrols the balance between the degree of sparsity ofthe estimated solution and the data fit determining thequality of the reconstruction.

For large µ, the solution is very sparse (with small `1-norm) but the data fit is poor. As µ decreases towardszero, the data fit is gradually improved since the cor-responding solutions become less sparse. Note that forµ = 0 the solution (8) becomes the unconstrained leastsquares solution.

A. The LASSO path

As the regularization parameter µ evolves from 1 to 0,the LASSO solution (8) changes continuously followinga piecewise smooth trajectory referred to as the solutionpath or the LASSO path18,19,34. In this section, we showthat the singularity points in the LASSO path are as-sociated with a change in the degree of sparsity of thesolution and can be used to indicate a proper value forµ.We obtain the full solution path using convex optimiza-

tion to solve (8) iteratively for di↵erent values of µ. Weuse the cvx toolbox for disciplined convex optimizationwhich is available in the Matlab environment. It usesinterior point solvers to obtain the global solution of awell-defined optimization problem17,28,29.

Let L(x, µ) denote the objective function in (8),

L(x, µ) = ky �Axk22 + µkxk1. (14)

The value x̂ minimizing (14) is found by di↵erentiation,

g(µ) = infx2CN

L(x, µ),

@

x

L(x, µ) = 2AH (Ax� y) + µ@

x

kxk1,(15)

where the subdi↵erential operator @x

is a generalizationof the partial di↵erential operator for functions that arenot di↵erentiable everywhere (Ref.29 p.338). The sub-gradient for the `1-norm is the set of vectors defined as,

@

x

kxk1 =�

s : ksk1 1, sHx = kxk1

, (16)

which implies,

si =xi

|xi

| , xi 6= 0|si| < 1, xi = 0,

(17)

i.e., for every active element xi 6= 0 of the vector x 2CN , the corresponding element of the subgradient is aunit vector in the direction of xi. For every null elementxi = 0 the corresponding element of the subgradient hasamplitude less than unity. Thus, the amplitude of thesubgradient is uniformly bounded by unity, ksk1 1.Denote,

r = 2AH (y �Ax̂) , (18)

the beamformed residual vector for the estimated solu-tion x̂. The minimum (15) is attained if,

0 2 @

x

L(x, µ) ) r 2 µ@

x

kxk1. (19)

Then, from (17) and (19), the coe�cients ri =2aHi (y �Ax̂) of the beamformed residual vector r 2 CN

have amplitude such that,

|ri| = µ, x̂i 6= 0|ri| < µ, x̂i = 0,

(20)

Compressive beamforming 3

µ

CS: LASSO for Multiple Snapshots

Y = [y1,…, yL ]∈CNxL Data

A = [a1,!,aM ]∈CNxL Sensing matrix

X = [x1,…,xL ]∈CMxL Source amplitudesData Fit

Y -AX2

2

which has the least squares solutionX =AH (AAH )−1Y ≈ AHY→ a new solution for every snapshot.Convetional beamforming

x(θm ) = 1LamH (YYH )am

One source magnitude for all snapshots.

Row sparsity constraint

X21= xn

l2

n=1

N

∑ with xnl2 = xnl

l=1

L

∑2

CS

X̂ =X∈CMxL

argmin Y - AX2

2+µ X

21

Snapshots5 10 15 20 25 30 35 40 45 50

DO

A (

de

gre

es)

-80

-60

-40

-20

0

20

40

60

80

0

2

4

6

8

10

12

14

16

18

The complex amplitude of X is allowed to vary across snapshots, but the sparsity pattern is assumed to be constant across snapshots

Problem with Degrees of Freedom

•  As the number of snapshots (=observations) increases, so does the number of unknown complex source amplitudes

•  PROBLEM: LASSO for multiple snapshots estimates the realizations of

the random complex source amplitudes.

•  However, we would be satisfied if we just estimated their power

γm = E{ |xml|2 } •  Note that γm does not depend on snapshot index l.

The problem revisited

Multiple snapshots: l=1,...,L

Likelihood function (observations conditioned on source amplitudes):

Sparsity promoted by Gaussian prior?



−2 −1.5 −1 −0.5 0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

a = 0.125a = 0.25a = 0.5

Proceeding with Bayes rule

is

Evidence

To determine the hyperparameters γ1, γ2, ..., γM, and σ2 the evidence is maximized. The evidence is the product of the likelihood and the prior integrated over the complex source signals.

Maximizing the Evidence: Covariance Fitting



= 0

We assume independent sources: covariance is diagonal


= 0

Exploiting Jaffer’s necessary condition

A.G. Jaffer. Maximum likelihood direction finding of stochastic sources: A separable solution. In IEEE Int. Conf. on Acoust., Speech, and Sig. Proc. (ICASSP-88), vol. 5, pp. 2893–2896, 1988.

Sparse Bayesian Learning Algorithm



D.P. Wipf, B.D. Rao. An empirical Bayesian strategy for solving the 1000 simultaneous sparse approximation problem. IEEE Trans. Signal Proc.,55(7):3704–3716, 2007.

Example Scenario

In the simulation, we consider an array with N = 20 antenna elements and intersensor spacing d = λ/2. The DOAs for plane wave arrivals are assumed to be on a fine angular grid [−90:0.5:90]◦and L = 50 snapshots are observed. The CS solution is found using LASSO extended to multiple measurement vectors (multiple snapshots) 3 sources at DOAs [−3, 2, 75] degrees with magnitudes [4, 13, 10].

Example Scenario

N = 20 elements

Source 1 DOA = -3° Magnitude = 4

Source 2 DOA = +2° Magnitude = 13

Source 3 DOA = +75° Magnitude = 10

P [d

B]

-10-505

10152025a)

0

50

100CS SNR=0 RMSE:1.2 9.7 0.49b)

0

50

100CBF SNR=0 RMSE:16 18 0.5

DOA [◦]-90 -60 -30 0 30 60 90

Bin

coun

t

0

50

100Exhaust SNR=0 RMSE:0.66 0.15 0.5

Array SNR (dB)-5 0 5 10 15 20 25 30

DOA

RMSE

[◦]

0

5

10

15c)

CSCBFMVDRexhaust

Multiple snapshots:

SNR=0dB

CS CBF Exhaustive

Spectra Histogram

DOA

Bin

cou

nt

50 snapshots 100 MCMC at each SNR

For M=20, λ/2 spacing, N=361, K=3, L=50, 100 MCMC simulations Compressive sensing (CS) •  min(||Y-AX||22+µ||X||1) •  20 equations with 361 unknowns! Conventional beamforming (CBF),Very fast Exhaustive search •  N*…(N-K+1)/K! possibilities •  K=2: 64,000 possibilities. •  K=3: 7,700,000 possibilities

P [

dB

]

-10

0

10

20a)

0

20

40CS SNR=0 RMSE:1.2 9.8 0.49b)

0

20

40CBF SNR=0 RMSE:16 20 0.51

DOA [◦]-90 -60 -30 0 30 60 90

Bin

co

un

t

0

20

40MVDR SNR=0 RMSE:20 17 0.56

Array SNR (dB)-5 0 5 10 15 20 25 30

DOA

RMSE

[◦]

0

2

4

6

8

10c)

CSExhaustCBFMVDRMusic

Multiple snapshot: Three sources

µ0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

|x̂i|

0

0.5

1 (a)

-20

-10

0 (b)

0

1

2(c)

-20

-10

0

x̂CS[dB

remax

]

(d)

0

1

2

|r|

(e)

-20

-10

0 (f)

0

0.5

1

(g)

θ [◦]-10 0 10 20 30

-20

-10

0 (h)

θ [◦]-90 -45 0 45 90

0

0.25

0.5 (i)

FIG. 5. (Color online) The LASSO path with the configura-tion as in Fig. 4 but for the refined angular grid [�90:1:90]�

which resolves basis mismatch but introduces increased am-biguity in the solution. (b)–(c) µ = 1.68, (d)–(e) µ = 0.94,(f)–(g) µ = 0.8 and (h)–(i) µ = 0.32.

active, while other noisy components become active con-tributing to the estimated solution for µ 0.26 (omittedin Fig. 5).

Consequently, improved precision from angular grid re-finement comes at the expense of increased basis coher-ence which causes bias in the estimates (cf. the sourceat 21� instead of the true DOA at 20� in Fig. 5(h)) andnoisy artifacts depending on the noise realization (cf. thespurious source at �4� in Fig. 5(h)).

VII. DOA ESTIMATION ERROR EVALUATION

If the source DOAs are well separated with not toodi↵erent magnitude, the DOA estimation for multiplesources using CBF and CS turns out to behave simi-larly. They di↵er, however, in their behavior whenevertwo sources are closely spaced. The same applies forMVDR under the additional assumptions of incoherentarrivals and su�cient number of snapshots, L � M . Thedetails are of course scenario dependent.

For the purpose of a quantitative performance eval-uation with synthetic data, the estimated, b

✓k, and thetrue, ✓truek , DOAs are paired with each other such thatthe root mean squared DOA error is minimized in eachsingle realization. After this pairing, the ensemble rootmean squared error is computed,

RMSE =

vuutE

"1

K

KX

k=1

(b✓k � ✓

truek )2

#. (31)

CBF su↵ers from low-resolution and the e↵ect of side-lobes for both single and multiple data snapshots, thusthe simple peak search used here is too simple. These

problems are reduced in MVDR for multiple snapshotsand they do not arise with CS,In the following simulation study, we consider an array

with M = 20 elements and intersensor spacing d = �/2.The DOAs for plane wave arrivals are assumed to be ona fine angular grid [�90�:0.5�:90�]. The regularizationparameter µ is chosen to correspond to the Kth largestpeak of the residual in Eq. (27) using the procedure in SecV.A. Note that panel c in Figs. 6–9 shows the simulationresults versus array SNR defined in Eq. (4).

A. Single Snapshot

In the first scenario, we consider a single snapshot casewith additive noise with K = 2 well-separated DOAs at[2, 75]� with magnitudes [13, 10], see Fig. 6. A thirdweak source is included in the second scenario very closeto the first source: Thus, K = 3 and the source DOAsare [�3, 2, 75]� with magnitudes [4, 13, 10], see Fig. 7.The synthetic data is generated according to Eq. (2).

For the first scenario, the CS diagrams in Fig. 6a showDOA estimation with small variance but indicate a biastowards endfire, as for the true DOA 75� the CS estimateis 76�. Towards endfire the main beam becomes broaderand absorbs more noise power. The CBF spectra Fig. 6aare characterized by a high sidelobe level but for the twowell-separated similar-magnitude sources this is a minorproblem. Nevertheless, this indicates in which way theCBF performance is fragile.

Using Monte Carlo simulations, we repeat the CS in-version for 1000 realizations of the noise in Fig. 6b. TheRMSE increases towards the endfire directions. This is tobe expected as the main beam becomes wider and thisresults in a lower DOA resolution4. Since the sourcesare well-separated in this scenario, CS and CBF performsimilarly with respect to RMSE.

Repeating the Monte Carlo simulations at severalSNRs gives the RMSE performance of CS and CBF inFig. 6b. Their performance is about the same since theDOAs are well-separated.

In the second scenario, the CBF cannot resolve thetwo closely spaced sources with DOAs [�3, 2]�. Theyare less than a beamwidth apart as indicated in Fig. 7a.Sidelobes cause a few DOA estimation errors at �65�

in the CBF histogram Fig. 7b. Since CS obtains highresolution even for a single snapshot, it performs muchbetter than CBF Fig. 7c.

B. Multiple Snapshot

In the multiple snapshot scenarios, the CBF andMVDR use the data sample covariance matrix Eq. (18)whereas CS works directly on the observations X

Eq. (16). The sample covariance matrix is formed byaveraging L synthetic data snapshots. A source’s magni-tude is considered invariant from snapshot to snapshot.The source’s phases are independent realizations sampledfrom a uniform distribution on [0, 2⇡).

Compressive beamforming 7Array SNR

2 10

CS=Ex for closely spaced sources

SNR=0dB

CS CBF MVDR

Spectra Histogram

DOA B

in c

ount

R

MSE

DO

A Er

ror

50 snapshots 100 MCMC at each SNR

P [d

B]

-10-505

10152025a) SNR=5

0100200300400500

CS SNR=5 RMSE:18 14 6.2b)

0100200300400500

CBF SNR=5 RMSE:22 14 1.8

DOA [◦]-90 -60 -30 0 30 60 90

Bin

coun

t

0100200300400500

Exhaust SNR=5 RMSE:17 15 13

Array SNR (dB)-5 0 5 10 15 20 25 30

DOA

RMSE

[◦]

0

5

10

15c)

CSCBFExhaust

Single snapshot: Three sources

µ0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

|x̂i|

0

0.5

1 (a)

-20

-10

0 (b)

0

1

2(c)

-20

-10

0

x̂CS[dB

remax

]

(d)

0

1

2

|r|

(e)

-20

-10

0 (f)

0

0.5

1

(g)

θ [◦]-10 0 10 20 30

-20

-10

0 (h)

θ [◦]-90 -45 0 45 90

0

0.25

0.5 (i)









RMSE =

vuutE

"1

K

KX

k=1

(b✓k � ✓

truek )2

#. (31)




A. Single Snapshot










Compressive beamforming 7Array SNR

Spectra Histogram

12

CS=Ex for closely spaced sources

1000 MCMC simulations at each SNR

CS CBF Exhaustive

DOA B

in c

ount

R

MSE

DO

A Er

ror

Example Results

0

10

20

P [d

B]

a)

0

20

40 CS SNR=0 RMSE:1.2 9.8 0.49b)

0

20

40 CBF SNR=0 RMSE:16 20 0.51

-90 -60 -30 0 30 60 90DOA [◦]

0

20

40

Bin

coun

t SBL SNR=0 RMSE:0.64 0 0.66

-5 0 5 10 15 20Array SNR (dB)

0

2

4

6

8

10

DOA

RMSE

[◦]

c)

RMVSBL-EMCSExhaustCBFMusic

Example RMSE Performance

0

10

20

P [d

B]

a)

0

20

40 CS SNR=0 RMSE:1.2 9.8 0.49b)

0

20

40 CBF SNR=0 RMSE:16 20 0.51

-90 -60 -30 0 30 60 90DOA [◦]

0

20

40

Bin

coun

t SBL SNR=0 RMSE:0.64 0 0.66

-5 0 5 10 15 20Array SNR (dB)

0

2

4

6

8

10DOA

RMSE

[◦]

c)


RVM-ML SBL-EM LASSO Exhaust CBF MUSIC

Example RMSE Performance

0

10

20

P [d

B]

a)

0

20

40 CS SNR=0 RMSE:1.2 9.8 0.49b)

0

20

40 CBF SNR=0 RMSE:16 20 0.51

-90 -60 -30 0 30 60 90DOA [◦]

0

20

40

Bin

coun

t SBL SNR=0 RMSE:0.64 0 0.66

-5 0 5 10 15 20Array SNR (dB)

0

2

4

6

8

10DOA

RMSE

[◦]

c)


RVM-ML SBL-EM LASSO Exhaust CBF MUSIC

Example CPU Time

1 10 100 1000Snapshots

0.15

1

10

100

1000C

PU

tim

e (

s) RVMRVM1LASSO

1 10 100 1000Snapshots

0

0.5

1

1.5

2

DO

A R

MS

E [°]

RVMRVM1LASSORVM-ML

RVM-ML1 LASSO

Conclusions

•  Sparse Bayesian Learning for complex valued array data using evidence maximization.

•  In examples it is ~ 50% faster than the SBL Expectation Maximization (SBL-EM) approach.

•  For multiple measurement vectors (snapshots) with stationary sources the benefit of RVM-ML is pronounced.

–  For each DOA it uses the hyperparameter γm as a proxy, with computational effort independent of no. of snapshots.

–  Increasing no. of snapshots improves the RMSE. –  The RMSE performance of RVM and exhaustive search are equal in this example

Full story: P. Gerstoft, C.F. Mecklenbräuker, A. Xenaki, S. Nannuru: Multi Snapshot Sparse Bayesian Learning for DOA, IEEE Signal Processing Letters, vol. 23, no. 10, pp. 1469–1473, Oct. 2016.

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perspective for source localization with sensor arrays. IEEE Trans. Signal Process., 53(8):3010–3022, 2005.

2.  A. Xenaki, P. Gerstoft, and K. Mosegaard. Compressive beamforming. J. Acoust. Soc. Am., 136(1):260–271, 2014.

3.  A. Xenaki, P. Gerstoft. Grid-free compressive beamforming. J. Acoust. Soc. Am., 137:1923–1935, 2015.

4.  H.L. Van Trees. Optimum Array Processing, chapter 1–10. Wiley- Interscience, New York, 2002.

5.  G.F. Edelmann, C.F. Gaumond. Beamforming using compressive sensing. J. Acoust. Soc. Am., 130(4):232–237, 2011.

6.  C.F. Mecklenbräuker, P. Gerstoft, A. Panahi, and M. Viberg. Sequential Bayesian sparse signal reconstruction using array data. IEEE Trans. Signal Process., 61(24):6344–6354, 2013.

7.  S. Fortunati, R. Grasso, F. Gini, M. S. Greco, and K. LePage. Single- snapshot DOA estimation by using compressed sensing. EURASIP J. Adv. Signal Process., 120(1):1–17, 2014.

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Multiple snapshots

Y = [y1,…, yL ]∈CMxL Data

A = [a1,!,aM ]∈CMxN Sensing matrix

X = [x1,…, xL ]∈CNxL Source amplitudesData Fit

Y - AX2

2

which has the least squares solutionX = AH (AAH )−1Y ≈ AHY→ a new solution for every snapshot.

Conventional beamforming

x(θm ) = 1LamH (YYH )am

⇒One source magnitude for all snapshots.

SwellEx 96 data

θ[◦]

-40

-20

0

20

40(a) f =112Hz (b) f =130Hz (c) f =148Hz

θ[◦]

-40

-20

0

20

40(d) f =166Hz (e) f =201Hz (f) f =235Hz

P [dB re max]-20 -10 0

θ[◦]

-40

-20

0

20

40(g) f =283Hz

P [dB re max]-20 -10 0

(h) f =338Hz

P [dB re max]-20 -10 0

(i) f =388Hz

80 snapshots 64 elements CBF low resolution MVDR fails due to coherent arrivals CS high resolution

Gerstoft et al. 2015

LRAD data Exeprimental data

array

G = |AHA|

CBF

reweighed l1

M=64,

d� = 1

4

M=16,

d� = 1

M=16,

random


Horizontal towed array

Xenaki & Gerstoft 2015

Random arrays work well with CS

HF97 experiment 3100 Hz Moored source and receiver 800 snapshots~68s CDB (color) CS (dots)

Some benefits of CS •  Beamforming:

–  Works for single snapshot, coherent arrivals –  For multiple snapshot perform as well as MVDR (or better..)

•  Don’t need to sample at double the Nyquist rate •  Many signals are sparse, but are solved as non-sparse

–  Beamforming, Fourier transform, Layered structure

•  Inverse methods are inherently sparse: The simplest way to describe the data (Occams razor)

Lots of low-hanging fruits… Further work: - More on tracking (Easy to incorporate a motion model) -  Instead of beamforming apply it to MFP -  Geoacoustic inversion -  Real data

-  No need for source deconvolution

Single snapshot: Two well-separated sources

P [dB

]

-10

-5

0

5

10

15

20

25a) SNR=5

0

500

1000CS SNR=5 RMSE:0.4 3.7b)

DOA [◦]-90 -60 -30 0 30 60 90

Bin

count

0

500

1000CBF SNR=5 RMSE:0.38 1.8

Array SNR (dB)-5 0 5 10 15 20 25 30

DOA

RMSE

[◦]

0

5

10

15c)

CSCBF

P [d

B]

-10-505

10152025a) SNR=5

0

500

1000CS SNR=5 RMSE:0.4 3.7b) CS SNR=5 RMSE:0.4 3.7b)

0

500

1000CBF SNR=5 RMSE:0.38 1.8CBF SNR=5 RMSE:0.38 1.8

DOA [◦]-90 -60 -30 0 30 60 90

Bin

coun

t

0

500

1000Exhaust SNR=5 RMSE:0.4 1.9

Array SNR (dB)-5 0 5 10 15 20 25 30

DOA

RMSE

[◦]

0

5

10

15c)c)

CSCBFExhaust

1000 MCMC simulations at each SNR

µ0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

|x̂i|

0

0.5

1 (a)

-20

-10

0 (b)

0

1

2(c)

-20

-10

0

x̂CS[dB

remax

]

(d)

0

1

2

|r|

(e)

-20

-10

0 (f)

0

0.5

1

(g)

θ [◦]-10 0 10 20 30

-20

-10

0 (h)

θ [◦]-90 -45 0 45 90

0

0.25

0.5 (i)









RMSE =

vuutE

"1

K

KX

k=1

(b✓k � ✓

truek )2

#. (31)




A. Single Snapshot










Compressive beamforming 7

Array SNR CS=CBF=Ex for well-separated sources

CS CBF Exhaustive

Spectra Histogram

DOA B

in c

ount

R

MSE

DO

A Er

ror

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Multi Snapshot Sparse Bayesian Learning for Direction of ... · Multi Snapshot Sparse Bayesian Learning for Direction of Arrival Estimation ... Y = A X + N Y A X N Multi Snapshot