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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
A. Xenaki (SIO/DTU) Compressive beamforming, JASA 2014 UA 2014 3 / 16
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
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
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)
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
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)
RMVSBL-EMCSExhaustCBFMusic
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)
RMVSBL-EMCSExhaustCBFMusic
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.
References 1. D. Malioutov, M. Cetin, A.S. Willsky. A sparse signal reconstruction
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.
References 8. 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.
9. P. Gerstoft, A. Xenaki, C.F. Mecklenbräuker. Multiple and single snapshot compressive beamforming. J. Acoust. Soc. Am., 138(4):2003– 2014, 2015.
10. P.Stoica, P.Babu. Spice and likes:Two hyperparameter-free methods for sparse-parameter estimation. Signal Proc., 92(7):1580–1590, 2012.
11. D. P. Wipf, B. D. Rao. Sparse Bayesian learning for basis selection. IEEE Trans. Signal Proc, 52(8):2153–2164, 2004.
12. Z. Zhang, B.D. Rao. Sparse signal recovery with temporally correlated source vectors using sparse bayesian learning. IEEE J Sel. Topics Signal Proc.,, 5(5):912–926, 2011.
13. M.E. Tipping. Sparse Bayesian learning and the relevance vector machine. J. Machine Learning Research, 1:211–244, 2001.
References 14. J.F. Böhme. Source-parameter estimation by approximate maximum
likelihood and nonlinear regression. IEEE J. Oc. Eng.,10(3):206–212,1985. 15. J.F. Böhme. Estimation of spectral parameters of correlated signals in
wavefields. Signal Processing, 11:329–337, 1986. 16. 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.
17. P. Stoica, A. Nehorai. On the concentrated stochastic likelihood function in array processing. Circuits Syst. Signal Proc., 14(5):669– 674, 1995.
18. Z.-M. Liu, Z.-T. Huang, Y.-Y. Zhou. An efficient maximum likelihood method for direction-of-arrival estimation via sparse bayesian learning. IEEE Trans. Wireless Comm., 11(10):1–11, Oct. 2012.
19. R. Tibshirani. Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B, 58(1):267–288, 1996.
20. A.P. Dempster, N.M. Laird, D.B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B, pp. 1–38, 1977.
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
A. Xenaki (SIO/DTU) Compressive beamforming, JASA 2014 UA 2014 15 / 16
Horizontal towed array
Xenaki & Gerstoft 2015
Random arrays work well with CS
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)
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 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