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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

(Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 2007 1 / 73

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

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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



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]



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]


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


Wideband DOA Estimation

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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)


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.



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


. (6)

incoherent averaging across frequency using the geometric mean is

PG(θ) =J

∏

j=1

pCapon(fj , θ) =J

∏

j=1

1


.



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



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



DOA Estimation Results - Baseline Run 1

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Arithmetic Capon Geometric Capon Harmonic Capon

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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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STCM Geometric MUSIC WSF



Error Statistics for Run 4

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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



DOA Estimation Results - Distributed Run 2

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DOA Estimates using geometric Capon for Run 2 with the failed node (node 2) removed.


General Source Error Models

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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.



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


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.



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.



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)


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.


Robust Wideband DOA Estimation Methods

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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)

.



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)



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 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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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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Geometric CaponRobust Capon

Geo. Capon Robust Capon

µe ∗ 5.22◦ 5.64◦

σ2e 122.93 ∗ 106.17




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Conclusions and Future Work

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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.


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.


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.


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.



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.



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.



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.



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.



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.



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.


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.



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)



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)


, ∀j ∈ [1, J ] (26)

and the wideband Capon spectrum,

PA(θ) =J

∑

j=1

p(fj , θ) =J

∑

j=1

1


(27)



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).



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).



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)}.



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)



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.



Array Geometries

North

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Examples of Vehicle Movement and Acoustic Data:Baseline Array

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Movement path baseline Run 1 Spectrogram of mic 0, baseline Run 1



Examples of Vehicle Movement and Acoustic Data:Distributed Array

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Missing data node Detail of missing data node(Colorado State University) Wideband DOA Estimation for UAS Thesis Defense, Sep 2007 58 / 73


Bearing Response Analysis - Baseline Array

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Arithmetic Capon Bearing Response, 50−250 Hz

BR for sources at 170o and 190o



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Geometric Capon Bearing Response, 50−250 Hz




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Harmonic Capon Bearing Response, 50−250 Hz





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STCM Bearing Response, 50−250 Hz




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Geometric MUSIC Bearing Response, 50−250 Hz




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WSF Bearing Response, 50−250 Hz




STCM Geometric MUSIC WSF5-element circular array with two sources at separations of 20◦, 23◦, and 26◦.



New Alg. Bearing Response Analysis - Baseline Array

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Beamspace Capon Bearing Response, 50−250 Hz




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Robust Capon Bearing Response, 50−250 Hz




Robust Geometric Capon5-element circular array with two sources at separations of 20◦, 23◦, and 26◦.



Bearing Response Analysis - Random Array

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Arithmetic Capon Bearing Response, 50−250 Hz




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Harmonic Capon Bearing Response, 50−250 Hz





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STCM Bearing Response, 50−250 Hz




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Geometric MUSIC Bearing Response, 50−250 Hz




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WSF Bearing Response, 50−250 Hz




STCM Geometric MUSIC WSF15-element randomly distributed array with two sources at separations of 1◦, 3◦, and 4◦.



New Alg. Bearing Response Analysis - Random Array

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Robust Capon Bearing Response, 50−250 Hz




Robust Geometric Capon15-element randomly distributed array with two sources at separations of 1◦, 3◦, and 4◦.



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.



Illustration of Coherent Signal

Array

φ0

γ0 g(φ

0; ψ

i)

g(φ; ψi)

δ

φ

Illustration of distributed source with spatial coherence.



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).



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.



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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STCM Geometric MUSIC WSFDOA estimation for a single extremely near-field source.




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Robust Geometric Capon WSFDOA Estimates on two source distributed array run 4.



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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