Signal Model Mismatch and Maximum-Likelihood Mean .... D. Richmond-3 MIT Lincoln Laboratory Tuesday...

MIT Lincoln LaboratoryC. D. Richmond-1Tuesday 6th June

ASAP 2006

Signal Model Mismatch and Maximum-Likelihood

Mean-Squared Error Performance

Christ D. Richmond

The Adaptive Sensor Array Processing Workshop MIT Lincoln Laboratory

Session 2A: Adaptive Detection and Estimation 3:50pm, Tuesday 6th June 2006

*This work was sponsored by NAVSEA PEO-IWSS 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.


ASAP 2006

Outline

• Introduction

– Problem description

– Goals of analysis

• Mean-squared error prediction

• Numerical examples

• Summary


ASAP 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

( )data, maxargˆ θθθ

MLML t=

Scan Angle

θ θT

Noise FreeAmbiguityFunction

Cramér-RaoBound

θ θT ˆ θ ML

High SNRSmallErrors

Driven by LocalMainlobe ErrorsDriven by LocalMainlobe Errors


ASAP 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

Array Element Positions

Array ElementPositions

No Mismatch Signal Mismatch

TrueAssumed

θT θT SidelobeTarget

SidelobeTarget

Mismatch affects threshold and asymptotic region leading to atypical performance curves

Mismatch affects threshold and asymptotic region leading to atypical performance curves

Cramér-Rao

Cramér-RaoBound

No Information

Threshold

Asymptotic

CRB

SNR (dB)SNRTH

Mea

n Sq

uare

d Er

ror (

dB)

MismatchSNRTH


ASAP 2006

Goals of Analysis

• Problem:– Maximum-Likelihood (ML) mean-squared error (MSE) performance

in presence of arbitrary deterministic signal mismatch unknown Non-adaptive and adaptive case unknown

• Goal:– Use Method of Interval Errors (MIE) to predict MSE

ClairvoyantMatched

Filter

Adaptive Matched

Filter

( ){ }2ˆ TMLE θθ −( ){ }2ˆ TMLE θθ −

Mean Squared Error:

( )( ) ( )θθ

θθ

θ vRv

xRv1

21

ˆ

ˆ maxargˆ

−

−

=H

H

ML

Known Noise Covariance Unknown Noise Covariance

( )( ) ( )θθ

θθ

θ vRvxRv

1

21

maxargˆ−

−

= H

H

ML

x = Sd + n x = Sd + n

v θ( )≠ d v θ( )≠ d

Test Cell:

for all θ allowed


ASAP 2006

Outline

• Introduction


– Method of interval errors (MIE)


• Summary


ASAP 2006

Nonlinear Estimation and Ambiguity Functions

MLAmbiguity Function

• ML involves nonlinear search on finite interval

• PDF aggregates density around maxima of ambiguity function (AF)

• Divide interval into K sub-intervals

– “No Interval Error” (NIE) Local Errors

– “Intervals of Error” (IE) Global Errors

• Use intervals to approximate MSE– Estimation approximately a K-ary

hypothesis testing problem

( )( ) ( )θθ

θθ

θ vRvxRv

1

21

maxargˆ−

−

= H

H

ML

θθ

( ) ( )( ) ( )θθ

θθvRv

vRv1

2

11

−

−

H

H

θ1θ1θ2θ2 θ3θ3 θK−1 θK−1θ4θ4 θK θK IE IE IE IE IE IE

NIE

(dB)

Histogram of ML Estimatesθθ

≡ψAF θ( )


ASAP 2006

Nonlinear Estimation and Ambiguity Functions

MLAmbiguity Function

• ML involves nonlinear search on finite interval

• PDF aggregates density around maxima of ambiguity function (AF)

• Divide interval into K sub-intervals

– “No Interval Error” (NIE) Local Errors

– “Intervals of Error” (IE) Global Errors

• Use intervals to approximate MSE– Estimation approximately a K-ary

hypothesis testing problem

( )( ) ( )θθ

θθ

θ vRvxRv

1

21

maxargˆ−

−

= H

H

ML

θθ

( ) ( )( ) ( )θθ

θθvRv

vRv1

2

11

−

−

H

H

θ1θ1θ2θ2 θ3θ3 θK−1 θK−1θ4θ4 θK θK IE IE IE IE IE IE

NIE

Approx. PDF of ˆ θ ML

(dB)

Histogram of ML Estimatesθθ

≡ψAF θ( )


ASAP 2006

Method of ML MSE Prediction:Based on Interval Errors

• MSE given by ( ){ } ( ) ( )∫ −≡− ωωθωθθ θ dpE ˆ2

1

2

1ˆ

≈ ω −θ1( )2 × dω∫NIEIE IE

“Global Errors”

( ) ( ) ( ) ( ) ( )21

211

2

211

2

1ˆˆ1ˆ θθθθθθσθθθθθθ −⋅=+⋅⎥

⎦

⎤⎢⎣

⎡ =−≅⎭⎬⎫

⎩⎨⎧ − ∑∑

==k

K

kkMLML

K

kkMLML ppE

“Local Errors”

Cramér-Rao Bound not useful when there’s signal mismatch

σ ML2 θ1( )= ?

and asymptotic MSE:

( ) ?ˆ1 == θθθ kMLp

• Challenge is calculation of error probabilities


ASAP 2006

Error Probabilities and No Mismatch:A Union Bound (UB) Approximation

θ1

True SignalDirection

GlobalMaximum

θ θk

Array Element Positions

θ1 ψAF θ( )(dB)

ML Ambiguity Fnc No Mismatch

*Exact formulae derived in Richmond, IEEE T-IT, Vol. 52, No. 5, May 2006

No Mismatch

• UB based approximation of interval error probabilities*:

( ) ( ) ( ) ( )[ ]1111 Prˆ1ˆ θθθθθθθθθ MLkMLkMLkML ttpp >≈≠−==

Dominant pairwise error probability relative to true signal location


ASAP 2006

Error Probabilities and Signal Mismatch:A Union Bound (UB) Approximation

ψAF θ( )(dB)

ML Ambiguity Fnc with Mismatch

θθ1


θG

GlobalMaximum

BiasArray ElementPositionsTrue

Assumed

θk

θ1

*Exact formulae derived in Richmond, IEEE T-IT, Vol. 52, No. 5, May 2006

Signal Mismatch

( ) ( ) ( ) ( )[ ]111 Prˆ1ˆ θθθθθθθθθ GMLkMLkMLkML ttpp >≈≠−==

• UB based approximation of interval error probabilities*:

Dominant pairwise error probability relative to global maximum in general


ASAP 2006

ψAF θ( )(dB)

ML Ambiguity Fnc with Mismatch

θθ1


θG

GlobalMaximum

Bias

Asymptotic ML MSE with Signal Mismatch

( ) ( ){ }2

112 ˆ θθθσ −= MLML E

( ){ }[ ] ( ){ }22

1ˆ ˆ GMLML EE θθθθ −+−=

Bias Variance

• Asymptotic MSE (jitter) given by

Goal is to determine the necessary expectations

( ){ } { } ( )11ˆ ˆ θθθθθθ −+−=− GGMLML EEwhere bias can be written as


ASAP 2006

ML Asymptotic MSE:A Taylor Series Based Approach

( ) ( ) ( )[ ]{ +Δ−≈− GTGH

TGGML Kf

K θθθ

θθ wRw0R

Re /1,,,

/2ˆ

• Following Taylor Series approach it can be shown that

MSE obtained from Mean and Variance of GML θθ −ˆ

Known R

Unknown R

*Richmond, IEEE T-SP, Vol. 44, No. 7, July 1996

,ˆ HT xxR =

( ) ( )[ ] ( )[ ]twRtwtR ++≡ θθχχθ TH

Tf ˆ,,ˆ,• Quadratic function defined

{ } , 2 HT

H SE ddRRxx +=≡

( ) ( )( ) ( )

, 1

1

θθθθ

vRvvRw

−

−

≡H

where

χ and t are random quantities* modeling errors in w(θ) due to unknown R

( ) ( )( ) ( )

( )[ ]twvRv

vRw +=≡−

−

θχθθ

θθ1

1

ˆ

ˆˆ

H

H

LXXR 1ˆ ≡

( )[ ] ( ) ( )[ ]( )}KGTGH

TGH /1 Re Re −+ χθθθ wRwtRw


ASAP 2006

Outline

• Introduction



– known noise covariance

– unknown noise covariance

• Summary


ASAP 2006

Broadside Planewave Signal in White Noise: No Mismatch, 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


ASAP 2006

Broadside Planewave Signal in White Noise: Perturbed ULA

• N=18 element ULA positions perturbed by 3-D Gaussian noise– Zero mean with stand. dev. 0.1λ– Array perturbation from single realization of Gaussian noise process

• CRB no longer useful, while MIE accurate above and below threshold• Mismatch can increase threshold SNR and introduce asymptotic bias

RM

SE in

Bea

mw

idth

s(d

B)


ThresholdSNR



( )233 , RMSn N σI0z +

λσ 1.0=RMS

Dis

tanc

e (in

uni

ts o

f λ)

ULAElement

Positions


ASAP 2006

• N=18 element uniform linear array (ULA), (λ/2.25) element spacing– Sidelobe target 2dB above noise @ 75degs

• SINR loss and threshold region results from competing targets• Asymptotic MSE ∝ 1/SNR due to signal living on ML manifold


RM

SE in

Bea

mw

idth

s(d

B)

ThresholdSNRFrom 4000

Monte CarloSimulations

Signal in White Noise: Unknown 2dB Sidelobe Target


Dis

tanc

e (in

uni

ts o

f λ)ULA

ElementPositions

zn


ASAP 2006

Signal in White Noise: Perturbed ULA + 2dB Sidelobe Target (SLT)

• N=18 element ULA positions perturbed by 3-D Gaussian noise– Zero mean with stand. dev. 0.1λ– Array perturbation from single realization of Gaussian noise process– Sidelobe target 2dB above noise @ 75degs

• CRB no longer useful, while MIE accurate above and below threshold• Mismatch affects both threshold region and asymptotic performance

RM

SE in

Bea

mw

idth

s(d

B)


ThresholdSNRFrom 4000

Monte CarloSimulations


( )233 , RMSn N σI0z +

λσ 1.0=RMS

Dis

tanc

e (in

uni

ts o

f λ)

ULAElement

Positions


ASAP 2006

Outline

• Introduction



– known noise covariance

– unknown noise covariance

• Summary


ASAP 2006

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)


ThresholdSNR


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


ASAP 2006

Signal in White Noise: Perturbed ULA + 2dB SLT, 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 – Sidelobe target 2dB above noise @ 75degs– Estimated colored noise covariance from L = 3N samples

• Note @ ~15dB SNR adaptivity loss limits beam split ratio to 20:1 as opposed to 63:1 when R is known

RM

SE in

Bea

mw

idth

s(d

B)



AdaptivityLoss

ThresholdSNR

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


ASAP 2006

Outline

• Introduction



• Summary


ASAP 2006

Summary

• Developed theory to predict impact of signal mismatch on ML estimation accuracy

– arbitrary deterministic signal mismatch allowed

– above and below threshold SNR performance considered

– finite sample effects due to noise covariance estimation included

• Generalizes previous work on ML threshold region performance and provides tools useful for system design and analysis

• Circumvents need for time consuming Monte Carlo simulations

• Asymptotic ML MSE easily extended to stochastic signal mismatch

Signal Model Mismatch and Maximum-Likelihood Mean .... D. Richmond-3 MIT Lincoln Laboratory Tuesday...

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