Detection Theory

Chapter 12 Model Change Detection

Xiang GaoJanuary 18, 2011

Examples of Model Change Detection

• So far, we have studied detection of a signal in noise• Model change detection

– Detection of system parameters change in time or space• In this chapter we study detection of

– DC level change– Noise variance change

• Examples in wireless communication– Synchronization– Detection of user presence

Outline

• Basic problem– Known DC level jump at known time– Known variance jump at known time– NP approach

• Extension to basic problem– Unknown DC levels and known jump time– Known DC levels and unknown jump time– GLRT approach

• Multiple change times• GLRT approach• Dynamic programming for parameters estimation to reduce the

computation• Problems

Basic Problem(No Unknown Parameters)

Example 1: Known DC Level and Jump Time

0 10 20 30 40 50 60 70 80 90 100-3

-2

-1

0

1

2

3

4

5

6

7

Sample, n

x[n]

1,,1,1,,1,0][

:

1,,1,0:

000

001

00

NnnnnwAAnnnwA

nxH

NnnwAnxH

A = 1

A = 4

Jump time and DC levels before and after jump are known

Example 1: Known DC Level and Jump Time

Neyman-Pearson (NP) test• Detect the jump and control the amount of false alarm• Data PDF

• NP detector decides H1

1

0

12

22

1222

21

0

021exp

2

1),;(n

n

N

nnN AnxAnxAAxp

0201

0201

0

1

,;,;

;;

AAAAxpAAAAAxp

HxpHxpxL

1

2

20

020

2ln

N

nn

AnNAnxAxL

Example 1: Known DC Level and Jump Time

• Test statistic

– Average deviation of data change over assumed jump interval– Data before jump are irrelavant

• Detection performance

'1

00 0

1

N

nn

AnxnN

xT

102

002

under,under,0

~HnNANΗnNN

xT

21 dPQQP FAD

2

20

02

22

AnN

nNAd

Delay time in detecting a jump

Example 2: Known Variance Jump at Known Time

0 10 20 30 40 50 60 70 80 90 100-5

-4

-3

-2

-1

0

1

2

3

4

Sample, n

x[n]

1,,1,1,,1,0

:

1,,1,0:

002

011

0

Nnnnnwnnnw

nxH

NnnwnxH

Variance = 1 Variance = 4

Energy detector?

Example 2: Known Variance Jump at Known Time

• NP detecor decides H1

0

1

;;HxpHxpxL

1

021exp

22

1

1

21exp

22

11

021exp

22

1

2200

20

0

222

00220

0 2200

20

N

nnxn

N

nnnxnN

n

nnxn

xL

220

20

12

1

220

21

20

2

1

020

21

220

21

020

2

1121

21

21

0

00

0

0

N

nn

N

nn

N

nn

N

n

N

nn

n

n

nx

nxnx

nxnxnxxT

Example 2: Known Variance Jump at Known Time

• Finally, we can get test statistic

– It is an energy detector– Same as detecting a Gaussian random signal in WGN (Chapter 5)

'1

2

0

N

nn

nxxT

Extensions to Basic Problem(Unknown Parameters Present)

Example 3: Unknown DC Levels, Known Jump Time

• Assume n0 is known but DC levels before the jump A1 and after the jump A2 are unknown

• GLRT detector decides H1 if211

210

::

AAHAAH

AAAAxpAAAAxpxLG ˆ,ˆ;

ˆ,ˆ;

21

2211

xnxN

AN

n

1

0

1ˆ

1

001

01ˆn

n

nxn

A

1

02

0

1ˆN

nn

nxnN

A

Average over all the data samples

Average over data samples before jump

Average over data samples after jump

Example 3: Unknown DC Levels, Known Jump Time

• After some simplification, we decide H1 if

• PDF of test statistic

'11

ˆˆln2

00

2

2

21

nNn

AAxLG

100

221

000

2

21

under11,

under11,0~ˆˆ

HnNn

AAN

HnNn

NAA

121

021

underunder

~ln2H

HxLG

00

2

221

11nNn

AA

Example 4: Known DC Levels, Unknown Jump Time

• Now the case is: A0 and ΔA are known, but n0 is unknown• This is classical synchronization problem• GLRT detector decides H1 if

0

0

10 ;max;,ˆ;max

00

nxLHxpHnxpxL

nnG

1

022

20

1

02000

22)(;ln

N

nn

N

nn

AAnxAAnNAnxAnxL

1

020

0 2maxln

N

nnnGAAnxAxL

1

00

0 2max

N

nnn

AAnxxT Test statistic is maximized over all possible values of n0

Same as Example 1

Final Case: Unknown DC Levels, Unknown Jump Time

• DC levels as well as jump time are unknown• GLRT decides H1 if

MLE of DC levels:

'11

ˆˆmax

00

2

2

21

0

nNn

AAn

1

02

1

001

0

0

1ˆ

1ˆ

N

nn

n

n

nxnN

A

nxn

A

Multiple Change Times

Multiple Change Times

0 10 20 30 40 50 60 70 80 90 100-1

0

1

2

3

4

5

6

7

8

9

Sample, n

x[n]

Parameter’s value changes more than once in data recordFor example: DC levels change multiple times in WGN

A = 1

A = 4

A = 2

A = 6

Multiple Change Times

• No unknown paramters– Same as Example 1

• Unknown parameters– DC levels unknown, change times known

Same as Example 3

– Change times unknownComputational explosion with the number of change times

Example 5: Unknown DC Levels, Unknown Jump Times

• We have signal embedded in WGN

• GLRT can be used if we can determine the MLE of change times• Focus on estimation of DC levels and change times• Joint MLE of

To minimize

1,,1,1,,1,1,,1,

1,,1,0

223

2112

1001

00

NnnnAnnnnAnnnnA

nnA

ns

1

23

12

2

12

1

1

0

20

2

2

1

1

0

0

,N

nn

n

nn

n

nn

n

n

AnxAnxAnxAnxnAJ

TT nnnnAAAAA 2103210 and

1

1 1

1ˆi

i

n

nniii nx

nnA

Example 5: Unknown DC Levels, Unknwon Jump Times

Dynamic programming• Not all combinations of n0, n1, n2 need to be evaluated• Reduce computational complexity

• Effectively eliminate many possible ”paths”

1 2

11

ˆ1,i

i

n

nniiii Anxnn

3

01 1,,ˆ

iiii nnnAJ

LnnInnLI kkkkn

k

iiii

Lnnnnnk

kkk

,1min1,min 1110

11,0

,...,, 11

110

Recursion for the minimum

Problems

• 12.1• 12.2• 12.4• 12.6• 12.11