Chap 4 - Detection-Classification

8/3/2019 Chap 4 - Detection-Classification

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Advanced signal processing

Dr. Mohamad KAHLILIslamic University of Lebanon


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Outline

Random variables Histogram, Mean, Variances, Moments, Correlation,

types, multiple random variables

Random functions Correlation, stationarity, spectral density estimation

methods

Signal modeling: AR, MA, ARMA, Detection and classification in signals

Advanced applications on signal processing: Time frequency and wavelet


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Chapter 4: detection andclassification in random signals

Detection Definition

Statistical tests for detection Likelihood ratio

Example of detection when change in mean

Example of detection when change in variances Multidimensional detection


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Detection: definition

Hi i: Hj j:

H

H

i i

j j

:

:

Hypotheses :

Known or unknown

estimated


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Gaussian distributionsNormal distributions

2

2

21

2

1

)(

mx

exfx

2)()( XVmTE

)1;0();( NmXmNXsi


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Loi du Chi 2 (Khi-two of Pearson)

)1;0(,...,, 21 NZZZSi k

2)(

1

2k

k

i

iZ

chi2 with k degree of freedom

Chi2 distributions

15 dof

10 dof

0

1

2/12/

2/

2

)(

)(

)2/(21

dxexk

exk

xk

xk

kk

E[chi2]=kVariance of Chi2=2k


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Fisher Test

Student distribution

)()1;0(2

kt

k

ZNZSik

Student with k degree of freedom

Fisher-Sndcor Distribution

);(2

2

lkF

l

k

l

k

Fisher with k and l degree of freedom

Example: Detection in signals

F(6,7)

F(6,10)


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Detection: definition

Hi i: Hj j:

H

H

i i

j j

:

:

Hypotheses :

Known or unknown

estimated


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Parameters definition

False alarm Detect H1, H0 is correct

Detection Detect H1, H1 is correct

Miss detection Detect H0, H1 is correct


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Likelihood ratio

Detection in signals

)0/(

)1/()(

HxP

HxPx

0)1(

)0(

)0/(

)1/(

1)1(

)0(

)0/(

)1/(

)0()0/()1()1/(

)/0()/1(1

)1(

)()/1()1/(

)0(

)()/0()0/(

)(

)()/()/(

HxHp

Hp

Hxp

Hxp

HxHp

Hp

Hxp

Hxp

HpHxpHpHxp

xHpxHpHx

HP

xPxHPHxp

HP

xPxHPHxp

BP

APABPBAP


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Variation in mean

Detection in meanH0: z(t) = 0 + b(t) = b(t)H1: z(t) = m + b(t)

1)()()()(

1

01

0

1

0

D

D

D

D

zsoitHPHPz

2

2

2

2

2

2

2

)2(exp

2exp

2

)(exp

)(

zmm

z

mz

z

2'0.21)(

1

0

1

0

2

1

0

mzodmmzz

D

D

D

D

D

D


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Detection in variance

Detection in variance1)(

1

0

D

D

z

01

11

)(

HZ

P

HZP

zn

N

n

nN

n

2

0

2

00

2

1

2

11

2

exp

.2

1

2exp

.2

1

nn

nn

z

H

ZP

z

HZ

P

ZZz t

N

..2

exp)(0

21

2

02

12

1

0

0

1

20

21

2

1

2

0

1

0

1

0

ln2

..1)( NZZz

D

D

t

D

D

1

01

2D

D

N

n

nzS


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Parameters

False Alarm probability

Detection probability

dPP Hfa )(

0

0

dPP Hd )(

0

1


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Parameters


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Neymen pearson method

Fix the probability of false alarm

Estimate the threshold

)0/(

)1/()(

HzP

HzPz

dH

zPHDP

001

)(/


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Detection: multidimensional case

Multidimensional case


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Distribution de Fisher-Snedecor

= 0,05


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DISTRIBUTION DU KHI-DEUX


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DISTRIBUTION DU KHI-DEUX (suite)


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LOI NORMALE CENTR ERDUITE

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Chap 4 - Detection-Classification