Post on 23-Feb-2016
description
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