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Wideband Direction-of-Arrival (DOA) EstimationMethods for Unattended Acoustic Sensors
Nicholas Roseveare
Department of Electrical and Computer EngineeringColorado State University
Thesis Defense, September 28, 2007
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Introduction Outline
Outline of Presentation
1 IntroductionOutlinePrevious WorkResearch Objectives
2 Wideband DOA EstimationSignal ModelReview of Wideband DOA Estimation Algorithms
3 General Source Error ModelsNon-Ideal Source ModelsGeneral Source Error Coherence
4 Robust Wideband DOA Estimation Methods5 Conclusions and Future Work
Conclusions and SummarySuggestions for Future Work
(5 min)
(10 min)
(10 min)
(10 min)(5 min)
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Introduction Previous Work
Background on DOA Estimation using UGS
Unattended ground sensors (UGS) have application in battlefield surveillanceand situation awareness:
• They are rugged, reliable, and can be left in the field for a long time afterdeployment
• They can be used to capture acoustic signatures of a variety of sourcesin different types of terrain (MOUT, etc.)
• The acoustic information may then be used to spatially locate and tracksources such as ground vehicles, airborne targets, or even personnel
Generally, high performance DOA estimation can separate multiple closelyspaced sources. Complications to this arise in acoustic arrays due to:
• Variability and nonstationarity of source acoustic signatures
• Signal attenuation and fading effects as a function of range and Doppler
• Coherence loss due to environmental conditions and wind effects
• Near-field, multipath, or other non-plane wave effects
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Introduction Previous Work
Review of Literature
Direction-of-Arrival (DOA) estimation
• Wideband DOA estimation and tracking with acoustic arrays[Pham] Benchmark of wideband DOA estimation algorithms[1]
[Azimi] Localization of multiple wideband sources and the use ofdistributed arrays to combat sensor location and non-uniformfading error[2, 3]
[Damarla],[Hohil] Tracking, counting, and classifying vehicles[4, 5]
• Narrowband DOA estimation algorithms with application toacoustic arrays[Capon] Minimum Power Distortionless Response (MPDR)[6]
[Schmidt] MUltiple SIgnal Classification (MUSIC) method[7]
[Viberg] Weighted Subspace Fitting (WSF) method[8]
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Introduction Previous Work
Review of Literature
• Wideband frequency combining methods[Krolik] Coherent focusing with Steered Covariance Matrices[9, 10]
[Pham],[Azimi] Wideband incoherent averaging methods[11, 2]
[Kaveh],[Di Claudio] Coherent MUSIC and WSF[12, 13]
• Non-idealized array source models for DOA estimation[Swindlehurst] Models for array geometry and calibration errors[14]
[Asztély] Multipath model for local scattering[15, 16]
[Valaee] Models for spatially coherent or incoherent sources[17]
[Meng],[Scharf] Array source model for partialincoherence[18, 19, 20]
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Introduction Research Objectives
Objectives of this Research
• To benchmark and illustrate deficiencies in existing widebandDOA estimation algorithms
• To develop a better understanding of non-ideal array signal models
• This understanding will help appropriately modify thecomputationally simple Capon to make it robust to the errors inour acoustic data sets
• To benchmark and show the performance improvements of thedeveloped algorithms
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Wideband DOA Estimation
Outline
1 IntroductionOutlinePrevious WorkResearch Objectives
2 Wideband DOA EstimationSignal ModelReview of Wideband DOA Estimation Algorithms
3 General Source Error ModelsNon-Ideal Source ModelsGeneral Source Error Coherence
4 Robust Wideband DOA Estimation Methods5 Conclusions and Future Work
Conclusions and SummarySuggestions for Future Work
(5 min)
(10 min)
(10 min)
(10 min)(5 min)
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Wideband DOA Estimation Signal Model
Wideband Signal Model
Consider d far-field sources observed by L sensors in an arbitrarynoise wavefield for frequency bin fj and sample k
x(fj , k) = A(fj ,φ)s(fj , k) + n(fj , k) =
d∑
i=1
a(fj , φi)si (fj , k) + n(fj , k) (1)
A(fj ,φ) = [a(fj , φ1), . . . , a(fj , φd )] array manifold of steering vectorss(fj , k) = [s1(f , k), . . . , sd (f , k)]T vector of sourcesφ = [φ1, . . . , φd ] vector of source directionsn(fj , k) noise vector
The sample covariance matrix for frequency fj is given as
Rxx(fj ) =K
∑
k=1
x(fj , k)xH(fj , k) = A(fj ,θ)Rs(fj)AH(fj ,θ) + Rn(fj)
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
Basic DOA Estimation: Beamforming
• The most basic DOA estimation method is the beamformer whichis given by the inner product of the array output vector and aweight vector as
y(fj , k , θ) = wH(fj , θ)x(fj , k), (2)
where the weight vector w(fj , θ) steers the beam response of thearray to observation angle, θ.
• The quadratic power spectrum becomes
p(fj , θ) = wH(fj , θ)Rxx(fj )w(fj , θ), (3)
from which the peaks are used as DOA estimates.
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
Capon Beamforming
Minimizes the overall received power while requiring the signal ofinterest (SOI) to be received at unit power, i.e.
minw(fj ,θ)
wH(fj , θ)Rxxw(fj , θ) s.t . wH(fj , θ)a(f,θ) = 1 (4)
This results in the optimal beamformer weights
w∗(fj , θ) =R−1
xx (fj)a(fj , θ)
aH(fj , θ)R−1xx (fj)a(fj , θ)
, (5)
this beamformer yields the power spectrum
pCapon(fj , θ) =1
aH(fj , θ)R−1xx (fj)a(fj , θ)
. (6)
incoherent averaging across frequency using the geometric mean is
PG(θ) =J
∏
j=1
pCapon(fj , θ) =J
∏
j=1
1
aH(fj , θ)R−1xx (fj)a(fj , θ)
.
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
Review MUSIC, WSF, and STCM
Idea DrawbacksMUSIC
WSF
STCM
Exploits orthogonality between signal andnoise subspaces, DOA estimate is mindistance between steering vector andnoise eigenvectors
Fits data to a search array manifoldmatrix in least-squares sense, DOAestimates found from minimum of errorbetween fitted and actual array response
Uses unitary transforms to focuses thefrequency spectra from multiplenarrowband bins; applied in tandem witha narrowband algorithm
Sensitive to choice ofnoise subspace andrequires SVD
Requires SVD and anintense multi-dimensionalsearch over sets ofangles
Focusing reduces theresolution capabilities ofwhichever algorithm it isused with
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
Results on Baseline and Distributed Array Data Sets
The baseline array data:
• Textron R© five-elementwagon-wheel ADAS type array
• A variety of military vehicletypes provided acousticsources
• Samples were collected at1024Hz for the uncalibratedtime series
• Phase/gain calibration for50 − 250Hz and 50% overlapsliding Hamming windowproduced 2048 samples perobservation period for thecalibrated data
The distributed array data:
• Fifteen wireless CrossbowTelos R© sensor nodes
• Two types of mid-sized movingtrucks were the acousticsources
• Samples were collected at1024Hz; only 876 samples perobservation period
• No calibration; additional errordue to sensor positionuncertainty (up to .2m) due toGPS measurement error
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
DOA Estimation Results - Baseline Run 1
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
50
100
150
Time (seconds)D
OA
(de
gree
s)
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−50
0
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Time (seconds)
DO
A (
degr
ees)
Arithmetic Capon Geometric Capon Harmonic Capon
0 50 100 150 200 250 300 350 400
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Time (seconds)
DO
A (
degr
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0 50 100 150 200 250 300 350 400
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DO
A (
degr
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0 50 100 150 200 250 300 350 400
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0
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Time (seconds)
DO
A (
degr
ees)
STCM Geometric MUSIC WSFThe markers ‘∗’, ‘△’, and ‘×’ or ‘×’, correspond to the DOAs obtained from the first, second, and third peaks of the spectrum.
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
DOA Estimation Results - Baseline Run 4
0 50 100 150 200
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Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200
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0
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OA
(de
gree
s)
0 50 100 150 200
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DO
A (
degr
ees)
Arithmetic Capon Geometric Capon Harmonic Capon
0 50 100 150 200
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DO
A (
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STCM Geometric MUSIC WSF
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
Error Statistics for Run 4
−5 0 5 10 150
0.1
0.2
0.3
0.4
0.5
Error (degrees)
No.
Err
ors
in N
orm
aliz
ed B
ins
Distribution of Error
Geometric CaponArithmetic CaponHarmonic CaponGeometric MUSICSTCMWSF
Geo. Capon Ari. Capon STCM Har. Capon Geo. MUSIC WSF
µe 2.8619◦ 2.9778◦ 3.3340◦ 2.7903◦ ∗2.5633◦ 2.6842◦
σ2e 5.0469 4.2967 8.1961 4.9874 6.8435 ∗ 3.0665
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
DOA Estimation Results - Distributed Run 2
0 20 40 60 80 100 120
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A (
degr
ees)
0 20 40 60 80 100 120
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OA
(de
gree
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0 20 40 60 80 100 120
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A (
degr
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Arithmetic Capon Geometric Capon Harmonic Capon
0 20 40 60 80 100 120
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A (
degr
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A (
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STCM Geometric MUSIC WSF
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Wideband DOA Estimation Review of Wideband DOA Estimation Algorithms
DOA Estimation Results - Distributed Run 2
0 20 40 60 80 100 120
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0
50
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Time (seconds)
DO
A (
degr
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DOA Estimates using geometric Capon for Run 2 with the failed node (node 2) removed.
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General Source Error Models
Outline
1 IntroductionOutlinePrevious WorkResearch Objectives
2 Wideband DOA EstimationSignal ModelReview of Wideband DOA Estimation Algorithms
3 General Source Error ModelsNon-Ideal Source ModelsGeneral Source Error Coherence
4 Robust Wideband DOA Estimation Methods5 Conclusions and Future Work
Conclusions and SummarySuggestions for Future Work
(5 min)
(10 min)
(10 min)
(10 min)(5 min)
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General Source Error Models
The Need for a General Error Model
The reviewed models have been reduced to an understanding of thetype of error coherence the source has and how this affects thecovariance matrix.
This is important because
• It is useful to have a general model for different errors which youcan tailor to the type error present in a particular scenario
• The basis for developing a particular robust algorithm is expresslydependent on the coherence of the error or mismatch assumedpresent in the data
• The error coherence distinguishes the rank of the source in thecovariance matrix, and therefore determines how to develop analgorithm to match to this source.
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General Source Error Models
Error Models in terms of Source Coherence
Coherence types
• Complete spatial coherence. This response models sources which arespatially coherent and temporally persistent within the observationperiod.
• Sensor position error, phase/gain or other array miscalibration• Near-field effects and local scattering multipath effects
• Complete spatial incoherence. For this case there is no spatialcoherence between source samples during the observation period.
• Uncorrelated reflections of a source off tropospheric scatterers(radiowave source)
• Partial spatial incoherence. The spatial coherence of the source in thisscenario persists infrequently throughout the observation period.
• Multipath for non-local scattering• When the array geometry is flexible and alters within the
observation period
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General Source Error Models Non-Ideal Source Models
Array and Calibration Error Model
• This type of error includes: Sensor position error, arraymiscalibration, quantization errors, gain/phase errors, other errorswithout structure.
• The array signal model for this type of error can be expressed asthe nominal response, A(fj , θ), plus an error matrix, A(fj , θ). Theperturbed sample covariance can be written as
Rxx(fj) = (A(fj , θ)+ A(fj , θ))Rs(fj)(A(fj , θ) + A(fj , θ))H + Rn(fj ). (7)
• The structure of the array manifold has changed, but theperturbed signal covariance is still rank-one.
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General Source Error Models Non-Ideal Source Models
Multipath Model
This multipath model considers local scattering. The array response is thesum of multiple coherent planewaves arriving at nearby angles. Thus themultipath spatial response for the i th source is
vi(fj , φi) =
Ni∑
k=1
αik (fj)a(fj , φi + φik ). (8)
The local scattering enables a first order derivative approximation andresults in the following spatial covariance
Rxx(fj ) = [A(fj ,φ) + D(fj ,φ)]Γ(fj)Rs(fj)ΓH(fj )[A(fj ,φ) + D(fj ,φ)] + Rn(fj). (9)
where Γ(fj ) is a diagonal fading matrix and D(fj ,φ) is the first derivative of thearray manifold (by vector). As is evident, this is still a rank-one matrix. Thiswould be similar for near-field effects.
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General Source Error Models Non-Ideal Source Models
Model for Coherent and Incoherent Sources
Consider the noise-free covariance matrix for the i th source (uncorrelatedfrom other sources)
Ri(fj ,ψ) =
∫
φ∈Φ
∫
φ′∈Φ
a(fj , φ)pi (fj , φ, φ′;ψi)aH(fj , φ′)dφdφ′ (10)
where, for the i th source,
• ψi spreading parameter vector
• pi(fj , φ, φ′;ψi) = E [si(fj , φ;ψ i)s∗i (fj , φ′;ψi)] is the angular auto-correlation
The coherent source signal density can be written as
si(fj , φ;ψi) = γi(fj )g(fj , φ;ψi) (11)
which results in the angular auto correlation function (ACF)
p(fj , φ, φ′;ψ) = η(fj , )g(fj , φ;ψ)g∗(fj , φ′;ψ). (12)
The incoherent source angular ACF is
p(fj , φ, φ′;ψ) = p(fj , φ;ψ)δ(φ − φ′). (13)
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General Source Error Models General Source Error Coherence
Conclusions on Source Error Coherence
Cohrence types
• Complete spatial coherence. This response models sources which arespatially coherent that persist temporally within the observation period.
• Rank one source covariance of unknown structure. Thiscorresponds to the type of error coherence in the data sets of thisstudy.
• Complete spatial incoherence. For this case there is no coherencebetween source samples during the observation period.
• Full rank source covariance.
• Partial spatial incoherence. The coherence in this scenario persistsinfrequently throughout the observation period.
• Multi-rank source covariance, but typically not full rank.
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Robust Wideband DOA Estimation Methods
Outline
1 IntroductionOutlinePrevious WorkResearch Objectives
2 Wideband DOA EstimationSignal ModelReview of Wideband DOA Estimation Algorithms
3 General Source Error ModelsNon-Ideal Source ModelsGeneral Source Error Coherence
4 Robust Wideband DOA Estimation Methods5 Conclusions and Future Work
Conclusions and SummarySuggestions for Future Work
(5 min)
(10 min)
(10 min)
(10 min)(5 min)
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Robust Wideband DOA Estimation Methods
Review of Wideband Robust Capon Algorithm
• The robust Capon is designed to be robust against rank-one errors inthe steering vector, i.e. sensor position error and near-field effects.
• It belongs to the class of diagonal loading approaches for robustestimation and is introduced by Li and Stoica[21].
• It operates based on an ellipsoidal uncertainty constraint for the steeringvector and is formulated as
mina
aH(fj , θ)R−1xx (fj)a(fj , θ) s.t. ‖ a(fj , θ) − a(fj , θ) ‖2≤ ǫ. (14)
The optimal beamformer weight vector is
w(fj , θ) = a(fj , θ) − U(fj )(I + λ(fj )Σ(fj ))−1UH(fj)a(fj , θ). (15)
The wideband robust Capon power spectrum becomes
PGRobust(θ) =
∏Jj=1
1
aH(fj ,θ)U(fj)Σ(fj )[λ−2(fj )+2λ−1(fj )Σ(fj )+Σ2(fj )]−1UH(fj )a(fj ,θ)
.
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Robust Wideband DOA Estimation Methods
Review of Wideband Beamspace Capon
• The beamspace method is a preprocessing method that performsspatial filtering by focusing on a region of interest[6].
• Initially was implemented to counteract the effects of low SNRsources and wind noise.
• The L × M beamspace matrix Bbs is a matrix in CL which projects
the input from the element space to the beamspace, in CM , where
L ≥ M.• A general form of a non-orthogonalized beamspace matrix is
Bno(fj , θ) =[
b(fj , φ−P + θ) · · ·b(fj , φ0 + θ) · · ·b(fj , φP + θ)]
, (16)
where for an arbitrary array geometry and c the speed of sound
b(fj , φp) =
ej2πfj/c(α0 cos(φp)+β0 sin(φp))
ej2πfj/c(α1 cos(φp)+β1 sin(φp))
...ej2πfj/c(αL−1cos(φp)+βL−1sin(φp))
. (17)
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Robust Wideband DOA Estimation Methods
Review of Wideband Beamspace Capon
• To ensure orthogonality of the beamspace matrix, we perform
Bbs = Bno[BHnoBno]−
12 (18)
Clearly, this whitening yields orthogonality in BHbsBbs = IM .
• The beamspace Capon method results in the weight vector
wbs(fj , θ) =R−1
vv (fj , θ)abs(fj , θ)
aHbs(fj , θ)R
−1vv (fj , θ)abs(fj , θ)
, (19)
where Rvv(fj , θ) = BHbs(fj , θ)Rxx(fj)Bbs(fj , θ) and
abs(fj , θ) = BHbs(fj , θ)a(fj , θ) are the transformed sample covariance and
steering vector, respectively.
• The wideband geometric mean beamspace Capon power spectrumoutput then becomes
PGbs(θ) =
J∏
j=1
1
aHbs(fj , θ)R
−1vv (fj , θ)abs(fj , θ)
.
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Robust Wideband DOA Estimation Methods
DOA Estimation Results - Baseline Run 1
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Beamspace Geometric Capon Standard Geometric Capon
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
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150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
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Time (seconds)
DO
A (
degr
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Robust Wideband DOA Estimation Methods
DOA Estimation Results - Baseline Run 1
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200 250 300 350 400
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Robust Geometric Capon Standard Geometric Capon
140 160 180 200 220 240 260 280 300 320 340
0
20
40
60
80
100
120
140
Time (seconds)
DO
A (
degr
ees)
140 160 180 200 220 240 260 280 300 320 340
0
20
40
60
80
100
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Time (seconds)
DO
A (
degr
ees)
Robust Geometric Capon detail Standard Geometric Capon detail(Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 2007 30 / 73
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Robust Wideband DOA Estimation Methods
DOA Estimation Results - Baseline Run 4
0 50 100 150 200
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200
−150
−100
−50
0
50
100
150
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DO
A (
degr
ees)
Beamspace Geometric Capon Standard Geometric Capon
0 50 100 150 200
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200
−150
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−50
0
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A (
degr
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Robust Wideband DOA Estimation Methods
Error Statistics for Run 4
−5 0 5 10 150
0.1
0.2
0.3
0.4
0.5
Error (degrees)
No.
Err
ors
in N
orm
aliz
ed B
ins
Distribution of Error
Geometric CaponGeometric MUSICWSFBeamspace Geo. Capon
Geo. Capon Bmspc Capon Geo. MUSIC WSF
µe 2.8619◦ 2.8426◦ ∗2.5633◦ 2.6842◦
σ2e 5.0469 3.3621 6.8435 ∗ 3.0665
DOA error statistics for algorithms with mean µe and variance σ2e .(Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 2007 32 / 73
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Robust Wideband DOA Estimation Methods
DOA Estimation Results - Baseline Run 4
0 50 100 150 200
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200
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−50
0
50
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150
Time (seconds)
DO
A (
degr
ees)
Robust Geometric Capon Standard Geometric Capon
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Robust Wideband DOA Estimation Methods
Error Statistics for Run 4
−15 −10 −5 0 5 10 15 20 250
0.05
0.1
0.15
0.2
0.25
0.3
Error (degrees)
No.
Err
ors
in N
orm
aliz
ed B
ins
Distribution of Error
Geometric CaponRobust Capon
Geo. Capon Robust Capon
µe ∗ 5.22◦ 5.64◦
σ2e 122.93 ∗ 106.17
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Robust Wideband DOA Estimation Methods
DOA Estimation Results - Distributed Run 2
0 20 40 60 80 100 120
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0
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150
Time (seconds)
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A (
degr
ees)
0 20 40 60 80 100 120
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−50
0
50
100
150
Time (seconds)
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A (
degr
ees)
Beamspace Geometric Capon Standard Geometric Capon
0 20 40 60 80 100 120
−150
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−50
0
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150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120
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A (
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Conclusions and Future Work
Outline
1 IntroductionOutlinePrevious WorkResearch Objectives
2 Wideband DOA EstimationSignal ModelReview of Wideband DOA Estimation Algorithms
3 General Source Error ModelsNon-Ideal Source ModelsGeneral Source Error Coherence
4 Robust Wideband DOA Estimation Methods5 Conclusions and Future Work
Conclusions and SummarySuggestions for Future Work
(5 min)
(10 min)
(10 min)
(10 min)(5 min)
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Conclusions and Future Work Conclusions and Summary
Conclusions and Summary
• The tested benchmark algorithms provided good results however,with some drawbacks and deficiencies.
• The error coherence source model enabled the appropriate choiceof rank-one robust DOA estimation algorithms.
• The wideband robust Capon provided robust DOA estimates in thepresence of sensor location uncertainties and near-field effects.
• Both the wideband beamspace Capon and the robust Caponprovided good DOA estimation performance even when there wasdata loss or corruption.
• Additionally, the beamspace Capon method provided betterperformance in wind noise and in locating far range sources.
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Conclusions and Future Work Suggestions for Future Work
Suggestions for Future Work
• Algorithm for detecting level of coherence and performing DOAestimation with the appropriate matching to whichever errorcoherence is present.
• Expand the application of these algorithms in this research toother wideband data.
• Explore better ways of finding algorithm parameters, namely, thenumber of beams and estimated error in the beamspace androbust Capon methods, respectively.
• Using DOA estimation algorithms for providing input to data fusionand tracking methods.
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Conclusions and Future Work Suggestions for Future Work
Questions and Thanks
Questions?
• My most sincere an deep thanks to Dr. Azimi for his ideas, editingmy writing, and encouragement through this long process.
• Gratitude is also due Dr.’s Scharf and Breidt for their willingness tobe members in my committee.
• Also to the members of the signal and image processinglaboratory: Bryan, Gordon, Jered, Amanda, Jaime, Makoto, Neil,Derek, Mike and Tim; you have made grad school an interestingand great learning experience.
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Appendix Bibliography
References I
T. Pham and B. M. Sadler, “Wideband array processing algorithmsfor acoustic tracking of ground vehicles,” tech. rep., Army ResearchLaboratories, Adelphi, MD, 1997.
M. R. Azimi-Sadjadi, A. Pezeshki, L. Scharf, and M. Hohil,“Wideband DOA estimation algorithms for multiple target detectionand tracking using unattended acoustic sensors,” in Proc. ofSPIE’04 Defense and Security Symposium - Unattended GroundSensors VI, vol. 5417, pp. 1–11, Apr. 2004.
M. R. Azimi-Sadjadi, Y. Jiang, and G. Wichern, “Properties ofrandomly distributed sparse acoustic sensors for ground vehicletracking and localization,” in Proc. of SPIE’06 Defense andSecurity Symposium, vol. 6201, Apr. 2006.
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Appendix Bibliography
References II
T. R. Damarla, J. Chang, and A. Rotolo, “Tracking a convoy ofmultiple targets using acoustic sensor data,” in Proc. of SPIE’03Defense and Security Symposium - Acquisition, Tracking, andPointing XVII, vol. 5082, pp. 37–42, Aug. 2003.
M. E. Hohil, J. R. Heberley, J. Chang, and A. Rotolo, “Vehiclecounting and classification algorithms for unattended groundsensors,” in Proc. of SPIE’03 Defense and Security Symposium -Unattended Ground Sensor Technologies and Applications V,pp. 99–110, Sept. 2003.
H. L. V. Trees, Optimum Array Processing.Wiley Interscience, 2002.
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Appendix Bibliography
References III
R. O. Schmidt, “Multiple emitter location and signal parameterestimation,” IEEE Trans. on Antennas and Propagation, vol. 34,pp. 276 – 280, Mar. 1986.
M. Viberg and B. Ottersten, “Sensor array processing based onsubspace fitting,” IEEE Trans. on Signal Processing, vol. 39,pp. 1110–1121, May 1991.
J. Krolik and D. Swingler, “Multiple broad-band source locationusing steered covariance matrices,” IEEE Trans. on Aoustics,Speech, and Signal Processing, vol. 37, pp. 1481–1494, Oct.1989.
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Appendix Bibliography
References IV
J. Krolik, Focused wideband Array Processing for Spatial SpectralEstimation, ch. 6.in “Advances in Spectrum Analysis and Array Processing, Vol. II”,Prentice-Hall, 1991.
T. Pham and M. Fong, “Real-time implementation of MUSIC forwideband acoustic detection and tracking,” in Proc. of SPIEAeroSense’97 - Automatic Target Recognition VII, vol. 3069,pp. 250–256, Apr. 1997.
H. Wang and M. Kaveh, “Coherent signal-subspace processing forthe detection and estimation of angles of arrival of multiplewide-band sources,” IEEE Trans. on Acoustics, Speech, andSignal Processing, vol. 33, pp. 823 – 831, Aug. 1985.
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Appendix Bibliography
References V
E. D. D. Claudio and R. Parisi, “WAVES: Weighted average ofsignal subspaces for robust wideband direction finding,” IEEETrans. on Signal Processing, vol. 49, pp. 2179–2191, Oct. 2001.
A. L. Swindlehurst and M. Viberg, “Bayesian approaches for robustarray signal processing.” research supported by NSF grantMIP-9408154, 1991.
D. Asztély, “Spatial models for narrowband signal estimation withantenna arrays,” tech. lic. thesis, Royal Institute of Technology,Stockholm, Sweden, Nov. 1997.
D. Asztély, B. Ottersten, and A. L. Swindlehurst, “Generalised arraymanifold model for wireless communication channels with localscattering,” IEE Proc.-Radar, Sonar Navig., vol. 145, pp. 51–57,Feb. 1998.
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Appendix Bibliography
References VI
S. Valaee, B. Champagne, and P. Kabal, “Parametric localization ofdistributed sources,” IEEE Trans. on Signal Processing, vol. 43,pp. 2144 – 2153, Sept. 1995.
A. Pezeshki, B. D. V. Veen, L. L. Scharf, H. Cox, and M. Lundberg,“Eigenvalue beamforming using a multi-rank MVDR beamformerand subspace selection,” to appear in IEEE Trans. on SignalProcessing, submitted Sept. 2006.
L. L. Scharf, A. Pezeshki, and M. Lundberg, “Multi-rank adaptivebeamforming,” in Proc. IEEE 13th Workshop Statistical SignalProcessing, (Bordeaux, France), July 2005.
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Appendix Bibliography
References VII
Y. Meng, P. Stoica, and K. M. Wong, “Estimation of the directionsof arrival of spatially dispersed signals in array processing,” inProceedings of IEE Conf. on Radar, Sonar, and Navig., vol. 143,Feb. 1996.
J. Li, P. Stoica, and Z. Wang, “On robust capon beamforming anddiagonal loading,” IEEE Trans. on Signal Processing, vol. 51,pp. 1702–1715, July 2003.
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Appendix Back Up Slides
Other Wideband Algorithm Power Spectrums
The wideband arithmetic mean Capon power spectrum is
PA(θ) =J
∑
j=1
p(fj , θ) =J
∑
j=1
1
aH(fj , θ)R−1xx (fj )a(fj , θ)
(20)
The wideband harmonic mean Capon power spectrum is
PH(θ) =1
∑Jj=1 wH(fj , θ)Rxx(fj)w(fj , θ)
(21)
The WSF power spectrum is
PWSF (θ) =1
tr{∑J
j=1 P⊥A1
(fj , θ)Us(fj)W(fj )WH(fj )UHs (fj)P⊥
A1(fj , θ)}
, (22)
where PA1(fj , θ) = a(fj , θ)a†(fj , θ) is the rank-one projection matrix.
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Appendix Back Up Slides
Derivation of Arithmetic Mean Wideband Capon I
The wideband arithmetic Capon problem can be cast as
minw(fj ,θ) PA(θ) =∑J
j=1
∑Kk=1 wH(fj , θ)x(fj , k)xH(fj , k)w(fj , θ)
=∑J
j=1 wH(fj , θ)Rxx(fj )w(fj , θ)(23)
under the constraints
wH(fj , θ)a(fj , θ) = 1, ∀j ∈ [1, J ] (24)
It is assumed that w(fj , θ) is independent of k within the observation periodT0. This leads to a constrained minimization problem
minw(fj ,θ)
J∑
j=1
wH(fj , θ)Rxx(fj)w(fj , θ) +
J∑
j=1
λ(fj )(wH(fj , θ)a(fj , θ) − 1)
(25)
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Appendix Back Up Slides
Derivation of Arithmetic Mean Wideband Capon II
where λ(fj )’s are frequency dependent Lagrange multipliers. Thisminimization problem leads the optimal beamformer, but here thisoptimization produces J narrowband rank-one Capon beamformers forw(fj , θ)’s,
w(fj , θ) =R−1
xx (fj)a(fj , θ)
aH(fj , θ)R−1xx (fj )a(fj , θ)
, ∀j ∈ [1, J ] (26)
and the wideband Capon spectrum,
PA(θ) =J
∑
j=1
p(fj , θ) =J
∑
j=1
1
aH(fj , θ)R−1xx (fj )a(fj , θ)
(27)
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Appendix Back Up Slides
MUltiple SIgnal Classification (MUSIC)
The MUSIC algorithm is a type of subspace-based algorithm which uses thedecomposition of the orthogonal (or unitary for this complex case) signal andnoise subspaces some eigendecomposition
Rxx(fj ) = Us(fj)Σs(fj)UHs (fj ) + Un(fj)Σn(fj )UH
n (fj ). (28)
Thus, the squared Euclidean distance between the steering vector a(fj , θ)and the noise subspace,
∂2 = aH(fj , θ)Un(fj )UHn (fj)a(fj , θ), (29)
will be minimum in the direction of a source. The incoherent widebandgeometric mean MUSIC algorithm is formulated by the inverse of thisdistance as
PMUSICG(θ) =
J∏
j=1
pMUSIC(fj , θ) =J
∏
j=1
1aH(fj , θ)Un(fj )UH
n (fj)a(fj , θ).
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Appendix Back Up Slides
Weighted Subspace Fitting (WSF)
Narrowband subspace fitting attempts to fit the data to a search arraymanifold as the minimization problem
minθ
||x(fj , k) − A(fj ,θ)s(fj , k)||2F . (30)
the signal estimate iss(fj , k) = A†(fj ,θ)x(fj , k), (31)
The error in this LS estimation of the signal estimate is given by
e(fj , k) = x(fj , k) − A(fj ,θ)s(fj , k) = P⊥A (fj ,θ)x(fj , k) (32)
where
P⊥A (fj ,θ) = I − PA(fj ,θ) = I − A(fj ,θ)[AH(fj ,θ)A(fj ,θ)]−1AH(fj ,θ) (33)
is the orthogonal projection complement operator onto thesubspace spanned by the columns of the array response matrix A(fj ,θ).
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Appendix Back Up Slides
Weighted Subspace Fitting (WSF)
The algorithm is a multi-dimensional search through θ for the minimumof the squared error between this estimate and the actual signal,
θ = arg minθ
tr{J
∑
j=1
P⊥A (fj , θ)Rxx(fj)P⊥
A (fj , θ)}. (34)
The decomposition of Rxx(fj) allows it to be written approx. as
Rxx(fj) ≈ Us(fj)Σs(fj)UHs (fj). (35)
The algorithm generalizes this decomposition as
Rxx(fj) ≈ Us(fj)W(fj )WH(fj)UHs (fj), (36)
where the weighting matrix, W(fj) = (Σs(fj ) − σ2nI)Σ_s−1/2(fj ), is
commonly used. The weighted subspace replaces Rxx(fj) and themulti-dimensional search becomes
θ = arg minθ
tr{J
∑
j=1
P⊥A (fj ,θ)Us(fj )W(fj)WH(fj )UH
s (fj)P⊥A (fj ,θ)}.
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Appendix Back Up Slides
STeered Covariance Matrix (STCM) Method
• The desired effect of the STCM algorithm is to generate a singlecoherent signal subspace by focusing to a reference frequencythose subspaces at other frequencies.
• Focusing matrices T(fj , θ), j = 1, 2, ..., J, exist so that
T(fj , θ)A(fj , θ) = A(f0, θ), s.t . T(fj , θ)TH(fj , θ) = I. (37)
• The steered or focused spatial covariance matrix may therefore bedefined as
R(θ) =J
∑
j=1
R(fj , θ) =J
∑
j=1
T(fj , θ)Rxx(fj)TH(fj , θ) (38)
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Appendix Back Up Slides
STeered Covariance Matrix (STCM) Method
• Using the array signal model, we can rewrite R(θ) as
R(θ) =J
∑
j=1
A(f0, θ)Rs(fj)AH(f0, θ) + Rnθ. (39)
Rnθis the focused noise covariance.
• The STCM focusing method can be applied in tandem with anyother narrowband DOA estimation algorithm, as its results in asingle covariance matrix.
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Appendix Back Up Slides
Array Geometries
North
−25 −20 −15 −10 −5 0 5 10 15 20−30
−25
−20
−15
−10
−5
0
5
10
0
12
3
4
5
6
78
9
10 11
12
13
14
East−West (meters)
Nor
th−
Sou
th (
met
ers)
Baseline array Distributed array
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Appendix Back Up Slides
Examples of Vehicle Movement and Acoustic Data:Baseline Array
−1500 −1000 −500 0 500 1000 1500 2000 2500 3000
−2500
−2000
−1500
−1000
−500
0
500
1000
node 1
node 2
node 3
start:1start:2
start:3
start:4 start:5
start:6
end:1
end:2end:3
end:4
end:5
end:6
East−West (m)
Nor
th−
Sou
th (
m)
N
Movement path baseline Run 1 Spectrogram of mic 0, baseline Run 1
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Appendix Back Up Slides
Examples of Vehicle Movement and Acoustic Data:Distributed Array
East−West (m)
Nor
th−
Sou
th (
m)
−365 −245 −120 0 120
245
180
120
60
0
−60
−120
−180
−245
NArray Site
Data collection site Spectrogram of mic 0, distributed Run 2
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Appendix Back Up Slides
Bad Time-series Data
0 20 40 60 80 1000
500
1000
1500
2000
2500
3000
3500
4000
4500
Time (sec)
Am
plitu
de
0 20 40 60 80 1002300
2320
2340
2360
2380
2400
2420
2440
Time (sec)
Am
plitu
de
Working node Bad mic amplifier node
0 20 40 60 80 100 120 140 160 180500
1000
1500
2000
2500
3000
3500
4000
Time (sec)
Am
plitu
de
14 14.5 15 15.5 16 16.5 17 17.5 18
1900
2000
2100
2200
2300
2400
2500
2600
Time (sec)
Am
plitu
de
Missing data node Detail of missing data node(Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 2007 58 / 73
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Appendix Back Up Slides
Bearing Response Analysis - Baseline Array
150 160 170 180 190 200 210−0.5
0
0.5
1
Arithmetic Capon Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
150 160 170 180 190 200 210−0.5
0
0.5
1
Geometric Capon Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
150 160 170 180 190 200 210−0.5
0
0.5
1
Harmonic Capon Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
Arithmetic Capon Geometric Capon Harmonic Capon
150 160 170 180 190 200 210−0.5
0
0.5
1
STCM Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
150 160 170 180 190 200 210−0.5
0
0.5
1
Geometric MUSIC Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
150 160 170 180 190 200 210−0.5
0
0.5
1
WSF Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
STCM Geometric MUSIC WSF5-element circular array with two sources at separations of 20◦, 23◦, and 26◦.
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Appendix Back Up Slides
New Alg. Bearing Response Analysis - Baseline Array
150 160 170 180 190 200 210−0.5
0
0.5
1
Beamspace Capon Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
150 160 170 180 190 200 210−0.5
0
0.5
1
Geometric Capon Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
Beamspace Geometric Capon Standard Geometric Capon
150 160 170 180 190 200 210−0.5
0
0.5
1
Robust Capon Bearing Response, 50−250 Hz
BR for sources at 170o and 190o
BR for sources at 169o and 192o
BR for sources at 167o and 193o
Robust Geometric Capon5-element circular array with two sources at separations of 20◦, 23◦, and 26◦.
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Appendix Back Up Slides
Bearing Response Analysis - Random Array
175 176 177 178 179 180 181 182 183 184 185−0.5
0
0.5
1
Arithmetic Capon Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
175 176 177 178 179 180 181 182 183 184 185−0.5
0
0.5
1
Geometric Capon Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
175 176 177 178 179 180 181 182 183 184 185−0.5
0
0.5
1
Harmonic Capon Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
Arithmetic Capon Geometric Capon Harmonic Capon
175 176 177 178 179 180 181 182 183 184 185−0.5
0
0.5
1
STCM Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
175 176 177 178 179 180 181 182 183 184 185−0.5
0
0.5
1
Geometric MUSIC Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
175 176 177 178 179 180 181 182 183 184 185−0.5
0
0.5
1
WSF Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
STCM Geometric MUSIC WSF15-element randomly distributed array with two sources at separations of 1◦, 3◦, and 4◦.
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Appendix Back Up Slides
New Alg. Bearing Response Analysis - Random Array
170 172 174 176 178 180 182 184 186 188 190−2000
−1800
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
Beamspace Capon Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
170 172 174 176 178 180 182 184 186 188 190−2000
−1800
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
Geometric Capon Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
Beamspace Geometric Capon Standard Geometric Capon
170 172 174 176 178 180 182 184 186 188 190−2000
−1800
−1600
−1400
−1200
−1000
−800
−600
−400
−200
0
Robust Capon Bearing Response, 50−250 Hz
BR for sources at 180o and 181o
BR for sources at 179o and 182o
BR for sources at 178o and 183o
Robust Geometric Capon15-element randomly distributed array with two sources at separations of 1◦, 3◦, and 4◦.
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Appendix Back Up Slides
Spatially Coherent Signals
Coherent sources includes slow fading multipath, near-field effects, and eventhe arrays errors discussed previously. In this case the spatial dependentsignal density is given by
si(fj , φ;ψi) = γi(fj )g(fj , φ;ψi) (40)
which results in the angular auto correlation
p(fj , φ, φ′;ψ) = η(fj , )g(fj , φ;ψ)g∗(fj , φ′;ψ) (41)
with η(fj) = E{γ(fj)γ∗(fj)}. The integral is separable as
Ri(fj ,ψi) =
∫
φ∈Φ
∫
φ′∈Φ
a(fj , φ)pi (fj , φ, φ′;ψi)aH(fj , φ
′)dφdφ′
= ηi
∫
φ∈Φ
a(fj , φ)g(fj , φ;ψi)dφ
∫
φ′∈Φ
g∗(fj , φ′;ψi)aH(fj , φ′)dφ′.
and demonstrates that the coherent signal is rank-one.
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Appendix Back Up Slides
Illustration of Coherent Signal
Array
φ0
γ0 g(φ
0; ψ
i)
g(φ; ψi)
δ
φ
Illustration of distributed source with spatial coherence.
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Appendix Back Up Slides
Spatially Incoherent Sources
An incoherent signal exists when the signal rays arriving from differentdirections can be assumed uncorrelated. The angular auto-correlation iswritten as
p(fj , φ, φ′;ψ) = p(fj , φ;ψ)δ(φ − φ′) (42)
The noise-free array correlation matrix for this signal is
Ri(fj ,ψ) =
∫
φ∈Φ
a(fj , φ)pi (fj , φ;ψ i)aH(fj , φ)dφ. (43)
This incoherent source covariance Ri(fj ,ψi) is always full rank. Using anapproximation to the full rank representation is practical for many distributions(and spreading widths).
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Appendix Back Up Slides
Partially Incoherent Sources
As a simple example consider the uniform distribution on [− ε2 , ε
2 ] whichresults in the covariance
Rs(ψi) =
∫
φ∈Φpi(φ;ψi)a(φ)aH(φ)dφ
=1ε
∫ ε/2
−ε/2a(φ)aH(φ)dφ
≈ σ2sUsΛUH
s , (44)
where rank(Rs) ≈ε
2π L = p. The matrix Us is the L × p basis for the pdimensional subspace 〈Us〉 and Λ ≈ I, i.e. signal power is even acrossall distributed components.
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Appendix Back Up Slides
Partially Incoherent Sources
So the generation of a input sample, x, for an partially incoherentnoise-free source looks like
x = Usbs
E [bbH ] = Λs
E [ss∗] = σ2s
Rs = E [xxH ] = σ2sUsΛsUH
s , (45)
where every sample within the observation period is formed from arandom linear combination of the p signal subspace basis vectors.The coherent source is a similar formulation, except that the randomcombining vector b is fixed for the observation period.
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Appendix Back Up Slides
DOA Estimation Results - Baseline Run 3
0 50 100 150 200 250 300 350
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200 250 300 350
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Beamspace Geometric Capon Standard Geometric Capon
0 50 100 150 200 250 300 350
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 50 100 150 200 250 300 350
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Geometric MUSIC WSFDOA estimation is the presence of high wind noise.
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Appendix Back Up Slides
DOA Estimation Results - Distributed Run 3
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Arithmetic Capon Geometric Capon Harmonic Capon
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
STCM Geometric MUSIC WSFDOA estimation for a single extremely near-field source.
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Appendix Back Up Slides
DOA Estimation Results - Distributed Run 3
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Beamspace Geometric Capon Standard Geometric Capon
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
0 20 40 60 80 100 120 140
−150
−100
−50
0
50
100
150
Time (seconds)
DO
A (
degr
ees)
Robust Geometric Capon WSFDOA estimation for a single extremely near-field source.
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Appendix Back Up Slides
DOA Estimation Results - Distributed Run 3
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DOA Estimates using geometric robust Capon for Run 3 with estimated error of 10 (instead of 0.7).
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Appendix Back Up Slides
Additional DOA Estimation Results - Distributed Run 4
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Beamspace Geometric Capon Standard Geometric Capon
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Robust Geometric Capon WSFDOA Estimates on two source distributed array run 4.
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Appendix Back Up Slides
Additional DOA Estimation Results - Distributed Run 5
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Robust Geometric Capon WSFDOA Estimates for case with two sensor nodes that have missing data.
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