Adaptive Array Detection, Estimation and Beamformingweb.mit.edu/sea06/agenda/talks/Richmond.pdf · Adaptive Array Detection, Estimation and Beamforming ... Workshop on Stochastic

MIT Lincoln LaboratoryC. D. Richmond-1Monday 10th July

SEA@MIT 2006

Adaptive Array Detection,Estimation and Beamforming

Christ D. Richmond

Workshop on Stochastic Eigen-Analysis and its Applications

3:30pm, Monday, July 10th 2006

*This work was sponsored by Defense Advanced Research Projects Agency under Air Force contract FA8721-05-C-0002. Opinions, interpretations, conclusions, and recommendations are those of the author and are not necessarily endorsed by the United States Government.


SEA@MIT 2006

Outline

• Introduction

– Radar/Sonar problem

• Detection algorithms

• Estimation algorithms

• Open problems

• Summary


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Airborne Surveillance Radars

• RAdio Detection And Ranging = RADAR• Goals / Mission:

– Long range surveillance– Airborne Moving Target Indication (AMTI)– Ground Moving Target Indication (GMTI)– Synthetic Aperture Radar (SAR) Imaging

• RAdio Detection And Ranging = RADAR• Goals / Mission:

– Long range surveillance– Airborne Moving Target Indication (AMTI)– Ground Moving Target Indication (GMTI)– Synthetic Aperture Radar (SAR) Imaging


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Transmit Power Pattern

Airborne Surveillance Radars: Signals and Interference

Target

v

θAzimuth

sTX t( )= Re ˜ p t( )⋅ e j 2πfc t{ }

TX/RXWaveform

sRX t( )= Re α˜ p t − τ( )⋅ e j 2π ( fc + fd )t{ }

Time Delay(Range)

Doppler(Velocity)

GroundClutter

HostileJamming Interferer


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Two-dimensional filtering required to cancel ground clutter

-0.5

0

0.5 -0.5

0

0.50

10

20

30

40

50

-1

1

50

0

Primary snapshot (target range gate)

NOISEJAMMERCLUTTER

Radar Data Modeland Optimum Linear Filter

*Brennan and Reed, IEEE T-AES 1973First to propose this for Radar Sig. Proc

GroundClutter

xT = Sv θT , fT( )+ nE xT{ }= Sv θT , fT( )cov xT( )= E nnH{ }= R

-1

1

50

0

Space-Time Adaptive Processing(STAP)

Space-Time Adaptive Processing(STAP)

ClutterNull Jammer

Null

Filter Response of w ∝R−1v

Sin (Azimuth) Doppler (Hz)

Pow

er (d

B)

TARGET


SEA@MIT 2006

Outline

• Introduction



• Open problems

• Summary


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Adaptive Detection Problem

•Two Unknowns R & S

H0 : xT = nH1 : xT = SvT + n

Test Cell:

cov(xT) = R

Assumptions:

• All Data Complex Gaussian

• Training Samples

– Homogeneous with Test Cell → cov(xT) = cov(xi)

• Perfect Look ( v = vT )

Analogy to 1-DCFAR StatisticAnalogy to 1-DCFAR Statistic

2

Cells#

2 ˆ|| Nt ση ⋅<≥

⋅= ∑cov(xi) = R[ ]LxxxX ||| 21=

•Use Noise Only Training Set


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Summary of Adaptive Detection Algorithms

•Adaptive Matched Filter(AMF) Robey, et. al. IEEE T-AES 1992Reed & Chen 1992, Reed et. al. 1974

•Generalized Likelihood RatioTest (GLRT) Kelly IEEE T-AES 1986, Khatri 1979

•Adaptive Cosine Estimator (ACE) Conte et. al. IEEE T-AES 1995,Scharf et. al. Asilomar 1996

•Adaptive Sidelobe Blanker(ASB) Kreithen, Baranoski, 1996Richmond Asilomar 1997

tAMF =

|vH ˆ R −1xT |2

vH ˆ R −1v

tGLRT =tAMF

1L

+ xTH ˆ R −1xT

f (tAMF , tACE )

Each Algorithm is a function of the

Sample Covariance

tACE =

tAMF

xTH ˆ R −1xT

ˆ R = 1

Lx1x1

H + x2x2H + + xLxL

H( )

More


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10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

Adaptive Detection Performance:An Example

•Random matrix theory predicts performance loss due tocovariance estimation

N=10, L=2N,PFA=1e-6

Output Array SINR (dB)

PD

GLRTASB, Fixed Thr.ACEAMFMax ASB PD

PD vs SINR

Optimal MF

R unknownR unknown

R knownR known

Loss Due toCovarianceEstimation

Loss Due toCovarianceEstimation


SEA@MIT 2006

Outline

• Introduction



• Open problems

• Summary


SEA@MIT 2006

No Information

Threshold

Asymptotic

Mea

n Sq

uare

d Er

ror (

dB)

SNR (dB)SNRTH

Cram�r-Rao Bound

Mean-Squared Error Performance:No Mismatch vs Mismatch

θT

No MismatchArray Element

Positions

θ θT ˆ θ ML

Low SNR

LargeErrors

Driven by GlobalAmbiguity/Sidelobe

Errors

Driven by GlobalAmbiguity/Sidelobe

Errors

θ θT ˆ θ ML

High SNRSmallErrors

Driven by LocalMainlobe ErrorsDriven by LocalMainlobe Errors

ˆ θ ML = argmax

θ tML θ,data( )

Scan Angle

θ θT

Noise FreeAmbiguityFunction


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Mean-Squared Error Performance:No Mismatch vs Mismatch

Array Element Positions

Array ElementPositions

No Information

Threshold

Asymptotic

CRB

SNR (dB)SNRTH

Mea

n Sq

uare

d Er

ror (

dB)

MismatchSNRTH

No Mismatch Signal Mismatch

TrueAssumed

θT θT

No Information

Threshold

Asymptotic

Mea

n Sq

uare

d Er

ror (

dB)

SNR (dB)SNRTH

Cram�r-Rao Bound

SidelobeTarget

SidelobeTarget

Mismatch affects threshold and asymptotic region leading to atypical performance curves

Mismatch affects threshold and asymptotic region leading to atypical performance curves


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Maximum-Likelihood Signal Parameter Estimation

• Complex Gaussian data model: All snapshots N x 1– Arbitrary N x N Colored Covariance– Deterministic Signal (“Conditional”)

• Colored noise only training samples available

π −N R −1 exp − x − Sv θ( )[ ]H R−1 x − Sv θ( )[ ]{ }

θML = argmax tMF θ( ) tMF θ( )=

vH θ( )R−1x2

vH θ( )R−1v θ( )

Data Model:

ML Estimator*:

S unknown

ClairvoyantMatched Filter

π −N (L +1) R −(L +1) exp − x − Sv θ( )[ ]H R−1 x − Sv θ( )[ ]− tr R−1XXH( ){ }

θML = argmax tAMF θ( ) tAMF θ( )=

vH θ( )ˆ R −1x2

vH θ( )ˆ R −1v θ( ) ˆ R ≡ 1

LXXH

Data Model:

ML Estimator*:

R unknown

S unknown

AdaptiveMatched Filter

*See Swindlehurst & Stoica Proc. IEEE 1998

Test Cell Training Data


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Approximating MSE Performance:Based on Interval Errors

• MSE given by E ˆ θ −θ1( )2⎧ ⎨ ⎩

⎫ ⎬ ⎭ ≡ ω −θ1( )2 p ˆ θ

ω( )dω∫

≈ ω −θ1( )2 × dω∫NIEIE IE

“Global Errors”

E ˆ θ ML −θ1( )2

θ1⎧ ⎨ ⎩

⎫ ⎬ ⎭

≅ 1− p ˆ θ ML = θk θ1( )k= 2

K

∑⎡

⎣ ⎢

⎤

⎦ ⎥ ⋅σ ML

2 θ1( )+ p ˆ θ ML = θk θ1( )k= 2

K

∑ ⋅ θk −θ1( )2

“Local Errors”

and asymptotic MSE:

σ ML2 θ1( )= ?

Both are functions of the estimated covarianceBoth are functions of the estimated covariance

p ˆ θ ML = θk θ1( )= ?

• Challenge is calculation of error probabilities


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Broadside Planewave Signal in White Noise: No Mismatch, R known, ULA

• N=18 element uniform linear array (ULA), (λ/2.25) element spacing– 3dB Beamwidth ≈ 7.2 degs, search space [60 120] degs– 0dB white noise, True Signal @ 90 degs (broadside)

• Asymptotic ML MSE agrees with CRB above threshold SNR• MIE MSE predictions very accurate above and below threshold

Element Level SNR (dB)

RM

SE in

Bea

mw

idth

s (d

B)

From 4000Monte CarloSimulations

ThresholdSNR

Var. UniformCRBAsympt. MSEMSE PredictionMonte Carlo

Dis

tanc

e (in

uni

ts o

f λ)ULA

ElementPositions

zn


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Signal in White Noise: Perturbed ULA,R unknown, L = 3N

• N=18 element ULA positions perturbed by 3-D Gaussian noise– Zero mean with stand. dev. 0.1λ; use single realization – Estimated colored noise covariance from L = 3N samples

• Note @ ~15dB SNR adaptivity loss limits beam split ratio to 16:1 as opposed to 22:1 when R is known

RM

SE in

Bea

mw

idth

s (d

B)

Element Level SNR (dB)

ThresholdSNR

From 4000Monte CarloSimulations

AdaptivityLoss

MC Known RMC Unknown RAsympt. MSEMSE PredictionCRB

( )233 , RMSn N σI0z +

λσ 1.0=RMS

Dis

tanc

e (in

uni

ts o

f λ)

ULAElement

Positions


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The Capon-MVDR Algorithm

R = E x l( )xH l( ){ } l =1,2,…,L

• Capon proposed filterbank approach to spectral estimation that designs linear filters optimally:

Given N x 1 vector snapshots with covariance

for x l

choose filter weights w according to

( )

minw HRw wHv θ( )=1

Minimum Variance Distortionless Response

w θ( )=

R−1v

such that

θ( )vH θ( )R−1v θ( )

E wH θ( )x l( )2{ }=

1vH θ( )R−1v θ( )

Solution well-known: Average Output Power of Optimal Filter:

PCapon θ( )≡

1vH θ( )ˆ R −1v θ( )

Capon’s Spectrum:

ˆ R = 1

Lx l( )xH l( )

l=1

L

∑where is sample covariance matrix

• Parameter estimate given by location of maximum power

θ θ θTˆ θ θT

AmbiguityFunction

Scan Angle

Estimation Error

Scan Angle

ˆ θ

1vH θ( )ˆ R −1v θ( )

1vH θ( )ˆ R −1v θ( )

1vH θ( )R−1v θ( )

1vH θ( )R−1v θ( )

Capon1969

θT


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Diagonally Loaded Capon Algorithm

ˆ R α = α ⋅ I +

1L

x l( )xH l( )l=1

L

∑ PCapon

I θ,α( )=1

vH θ( )ˆ R α−1v θ( )

PCaponI θ,α( )=

1vH θ( )ˆ R α

−1v θ( )

• In practice it is common to diagonally load the sample covariance:

• Diagonal loading mitigates undesired finite sample effects*

– Slow convergence of small/noise eigenvalues (DL compresses)– High sidelobes (DL provides sidelobe [white noise gain] control)– Excessive loading can degrade performance

*Cox, IEEE T-SP 1987, Carlson, IEEE T-AES 1988

• Diagonal loading is necessary to invert matrix in snapshot deficient aacase, i.e. L ≤ N

• Eigenvectors of sample covariance remain unaffected by diagonal aaloading

• Featherstone et al. showed diagonally loaded Capon to be a robust aadirection finding algorithm

*“Robustify”Processing


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Single Signal Broadside to Arrayin Spatially White Noise, L = 0.5N

Output Array SNR (dB) Output Array SNR (dB)

L = 0.5Nα = -10dB

L = 0.5Nα = +10dB

RM

SE in

Bea

mw

idth

s (d

B)

RM

SE in

Bea

mw

idth

s (d

B)

• N=18 element uniform linear array (ULA), (λ/2.25) element spacing– 3dB Beamwidth ≈ 7.2 degs– 0dB white noise, True Signal @ 90 degs (broadside)– 4000 Monte Carlo simulations

• VB MSE prediction not applicable for L < N

Threshold SNRs

MSE Prediction

Monte CarloCRB


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Mismatch Example:Perturbed Array Positions

8dB Error in VB Prediction of15:1 Beamsplit Ratio SNR

Output Array SNR (dB)

RM

SE in

Bea

mw

idth

s (d

B)

L = 1.5Nα = +10dB

MSE PredictionMonte Carlo

VB MSE Prediction zn zn + en

en ~ N 3 0,I3σ RMS2( ),

AssumedNominal

Array Position

ActualPerturbed

Array Position

Based on Single Realization of Gaussian Perturbation:

• N=18 element ULA with perturbed positions but assumed straight• VB MSE prediction can lead to large errors in required SNRs• DL Capon is more robust DF approach: 18:1 vs 28:1 @ 40dB ASNR

10dB Error for 17:1

σ RMS = 0.04λ


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Outline

• Introduction



• Open problems

• Summary


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What About Robust Detection ?

•Adaptive Matched Filter(AMF) Robey, et. al. IEEE T-AES 1992Reed & Chen 1992, Reed et. al. 1974

•Generalized Likelihood RatioTest (GLRT) Kelly IEEE T-AES 1986, Khatri 1979

•Adaptive Cosine Estimator (ACE) Conte et. al. IEEE T-AES 1995,Scharf et. al. Asilomar 1996

•Adaptive Sidelobe Blanker(ASB) Kreithen, Baranoski, 1996Richmond Asilomar 1997

tAMF α( )=

|vH ˆ R α−1xT |2

vH ˆ R α−1v

tGLRT α( ) =

tAMF

1 + xTH ˆ R α

−1xT

f t AMF α( ), t ACE α( )[ ]

Each Algorithm is a function of the

Sample Covariance

tACE α( ) =

tAMF

xTH ˆ R α

−1xT

ˆ R α = α ⋅ I +

1L

x l( )xH l( )l=1

L

∑


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Source True Location

False Peaks

• Inflated Cortical Surface • LCMV Cost Function

- Based on 74 Channel Dual Sensor Magnes II Biomagnetometer- SNR = -23 dB

• Dipolar source located in the center of the Somatosensory Region

(dB)

x (meters)

y (meters)

z (m

eter

s)PLCMV θ( )

Magneto-encephalography (MEG)


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Cost Function:Output of LCMV spatial filter as signal location hypothesis

is varied when using true R

Cost Function:Output of LCMV spatial filter as signal location hypothesis

is varied when using true R

Large Errors DueTo False Peaks of

Cost Function

Large Errors DueTo False Peaks of

Cost Function

Residual ErrorDue to Jitter About

True Source Location

Residual ErrorDue to Jitter About

True Source Location

Composite Localization Accuracy vs Signal-to-Noise Ratio (SNR)

No Information:No Signal

Threshold:

Weak Signal

Asymptotic:

Strong Signal

Loca

lizat

ion

MSE

(dB

)

SNR (dB)THRESHOLDSNR

∝ -log(SNR)

Cost FncHeight

High

Low

Pr tr V1

H ˆ R −1V1( )−1⎡ ⎣ ⎢

⎤ ⎦ ⎥ > tr V2

H ˆ R −1V2( )−1⎡ ⎣ ⎢

⎤ ⎦ ⎥

⎧ ⎨ ⎩

⎫ ⎬ ⎭

= ?OutstandingProblem


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Outline

• Introduction



• Open problems

• Summary


SEA@MIT 2006

Summary

• Random matrix theory provides insight into the performance of adaptive arrays systems

– Finite random matrix theory has been most common approach

• Infinite random matrix theory quickly gaining momentum as tool for analyses and design of robust signal processing algorithms


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Distributions of 1-D DetectorsHomogeneous Case

˜ δ β = β⋅ | S | ⋅ v HR-1v

˜ t GLRT =d

F1,K −1 ( ˜ δ β )

tAMF =d

F1,K −1 ( ˜ δ β ) /β

˜ t ACE =d

F1,K −1 ( ˜ δ β ) /(1 − β )

Distributions of Adaptive Detectors

PD ASB = Pr( t ACE > η ace , t AMF > η amf )

)1/(~GLRTGLRTGLRT ttt −≡ )1/(~

ACEACEACE ttt −≡ 2+−= NLK

•Recall that PD of ASB is

•Define the following

•It can be shown that

where

Richmond Asilomar 1997Richmond IEEE SP 2000

Requires knowledge of Dependence!

Found in thisSummary!


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The AMF Detector

=TestRatio

Likelihood

XXH = ˆ R →R vRv

xRv1

21

ˆ

ˆ

−

−

=H

TH

AMFt

Known as the Adaptive Matched Filter (AMF) detector

Form the optimal Neyman-Pearson test statistic, that is, the LRT.

Assume complex Gaussian statistics

gH0= π −N R −1 exp −xT

HR−1xT[ ]

gH1

= π −N RT−1 exp − xT − vS( )H R−1 xT − vS( )[ ]

H0 :

H1 :

Simply replace true data covariance with an estimate

MatchedFilter

Return

maxS

gH1

gH0

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

=vH ˆ R −1xT

2

vH ˆ R −1v≡ tMF


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The Generalized LRT (GLRT)

M = vS | 0[ ]

tGLRT =max

S,RgH1

maxR

gH0

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1L +1

=1+ xT

H ˆ R −1xT

1+ xTH ˆ R −1xT −

vH ˆ R −1xT

2

vH ˆ R −1v

gH0= π −N (L +1) R −(L +1) exp −trR−1X0X0

H[ ]

g H 1

= π − N ( L +1) R − ( L +1) exp − trR −1 X 0 − M( ) X 0 − M( )H[ ]H0 :

H1 :

Form the LRT based on the totality of data:

Assume homogeneous complex gaussian statistics Test Cell | Interference Training Set[ ]= xT | X[ ]≡ X0

Maximize likelihood functions over all unknown parameters:

where

Known as Kelly’s / Khatri’s GLRT Return


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The Adaptive Cosine Estimator (ACE)

ψv

Target array response

Mea

sure

d da

ta v

ecto

r x • The ACE statistic provides a measure of correlation between the test data vector xT and the assumed target array response v

• Inner product space defined wrt inverse of data covariance

– in whitened space

tACE =

vH ˆ R −1xT

2

xTH ˆ R −1xT( )vH ˆ R −1v( )

= cosψ 2

Return


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Simplest: The AMF Detector

• AMF Computationally Attractive: Linear Filter

• AMF is an Adaptive Beamformer

– Measures Power in Assumed Target Direction

– Interference Suppression Based on Covariance Estimate

• Inhomogeneities Frustrate Interference Suppression

– Covariance Estimate Uncharacteristic of Data

– Results in high False Alarm Rates

Practical Issues:


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Classical Sidelobe Blanking

Directional Channel

Omni-directional Channel

θ

θ

≥<

Threshold

≥<

Threshold

Comparator

OutputGate

Input

θTime

Typical Comparator Input

Channel Magnitude Response

Azimuth

Power in TargetDirection

Total Power from All Directions

StrongSignal

Ch 1Ch 2Ch 1

Ch 2

StrongSignal


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2-D ASB Detection Algorithm

Fails AMF & ACE

PassesACE

Passes AMF

Region of DeclaredDetections

tACE

tAMF

ηace

ηamf

1

0“S

idel

obe

Bla

nkin

g”

Step 1: Beamforming

Step 2 : “Sidelobe Blanking”

tAMF > η amf

tAMF > ηace ⋅ xTH ˆ R -1xT



Total Power FromAll Directions

2-D ASB Detector

“Directional Beamformer”

Return


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The Complex Wishart Random Matrix

L ˆ R ≡ XXH = xkxk

H

k=1

L

∑

dˆ R ( )= d ˆ R 11d ˆ R 22 d ˆ R NN ⋅ d Re ˆ R 12( )d Im ˆ R 12( )⋅ d Re ˆ R 13( )d Im ˆ R 13( )

dRe ˆ R N−1,N( )d Im ˆ R N−1,N( )

X ~ CN 0,IL ⊗ R( )

NL ≥

If the training data is complex Gaussian s.t.

then the sample covariance matrix is the Maximum-Likelihoodestimator of the covariance parameter R:

If then its PDF exists and is given by

L ˆ R

L−NR −L / ˜ Γ N (L)⎡

⎣ ⎢ ⎤ ⎦ ⎥ exp −tr R−1 ˆ R L( )[ ] 0 < ˆ R where

and the differential volume element is given by

Documents

Adaptive Array Detection, Estimation and Beamformingweb.mit.edu/sea06/agenda/talks/Richmond.pdf · Adaptive Array Detection, Estimation and Beamforming ... Workshop on Stochastic