Detection Theory
Chapter 12 Model Change Detection
Xiang Gao
January 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
H 0 : x  n   A 0  w n 
n  0 , 1,  , N  1
Jump time and DC levels
before and after jump are
known
 A0  w [ n ] n  0 , 1,  , n 0  1
H 1 : x n   
 A0   A  w n  n  n 0 , n 0  1,  , N  1
7
6
5
A=4
4
x[n]
3
A=1
2
1
0
-1
-2
-3
0
10
20
30
40
50
60
Sample, n
70
80
90
100
Example 1: Known DC Level and Jump Time
Neyman-Pearson (NP) test
• Detect the jump and control the amount of false alarm
• Data PDF
p ( x ; A1 , A2 ) 
1
2 
2
N
2

1
exp  
 2
2
 n 0 1
   x n   A1 2 

 n0
• NP detector decides H1
L x  
p  x; H 1 
p x; H 0 
ln L  x  
A

2

p  x ; A1  A0 , A 2  A0   A 
p  x ; A1  A0 , A 2  A0 
N 1
  x n   A  
0
n  n0
N
 n 0  A
2
2
2

N 1
  x n   A 
2
n  n0
2



 
Example 1: Known DC Level and Jump Time
• Test statistic
T x  
N 1
1
N  n0
'




x
n

A



0
n  n0
– Average deviation of data change over assumed jump interval
– Data before jump are irrelavant
• Detection performance
 N 0 ,  2  N  n 0  under Η 0
T x  ~ 
2
 N  n 0  under H 1

N

A
,



PD  Q Q
1
 PFA  
d
2

d 
2

2
Delay time in
detecting a jump
A
2
N
 n0 

N
 n 0  A

2
2
Example 2: Known Variance Jump at Known Time
H 0 : x n   w  n 
n  0 , 1,  , N  1
 w1 n  n  0 , 1,  , n 0  1
H 1 : x n   
 w 2 n  n  n 0 , n 0  1,  , N  1
Energy detector?
4
3
2
1
Variance = 4
0
x[n]
Variance = 1
-1
-2
-3
-4
-5
0
10
20
30
40
50
60
Sample, n
70
80
90
100
Example 2: Known Variance Jump at Known Time
• NP detecor decides H1
Lx  
p  x; H 1 
p  x; H 0 
1
L x  

n0
2 
2
0


exp 
2
1
2
2
0
n 0 1
2
 x  n
n0

1

2  0  
1
n0
2

exp 
 2   2
2

1
2
2
0
n 1
2
1  0 x n 
T  x     

2

2  n0  0
1
  
2 
N 1

n  n0
x n 

n  n0
2
2
0
 


n  n0

2
2
2
0
 
 1
1  N 1 2
1
   x n   2  2
2  n  n 0
  0  0  

n0
x n 
2
2








2
N 1
1
2
 

 x  n 
2
2
 2  0    n  n0

N 1
2
 x  n
n0
N 1
2
N 1
2
0
x n 
N 1
2
0
N  n0
exp

x n  

2
 0 
2
Example 2: Known Variance Jump at Known Time
• Finally, we can get test statistic
T x  
N 1
 x n   
2
'
n  n0
– It is an energy detector
– Same as detecting a Gaussian random signal in WGN (Chapter 5)
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
H 0 : A1  A2
H 1 : A1  A2
• GLRT detector decides H1 if
Aˆ 
1
N
Aˆ 1 
Aˆ 2 
1
n0



N 1
 x n   x
Average over all the data samples
n0
n 0 1
 x n 
Average over data samples before jump
n0
1
N  n0
N 1
 x n 
n  n0

p x ; A 1  Aˆ 1 , A 2  Aˆ 2
LG  x  

ˆ
ˆ
p x ; A 1  A , A2  A
Average over data samples after jump
Example 3: Unknown DC Levels, Known Jump Time
• After some simplification, we decide H1 if
2 ln L G  x  

Aˆ 1  Aˆ 2

2

1
2 1

 


n
N

n
0 
 0
'
• PDF of test statistic
 
 1

1
2

  under H 0

 N  0 ,  
n0
N  n 0  



Aˆ 1  Aˆ 2 ~ 

1
 N  A  A ,  2  1 
  under H 1
2
n

  1
N

n
0
0


 
  12 under H 0
2 ln L G  x  ~  2
  1   under H 1
 
 A1 
 1
2
 
 n0

A2 
2


N  n 0 
1
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
Same as Example 1
p  x ; nˆ 0 , H 1 
L G  x   max
p  x; H 0 
n0
ln L  x ; n 0  
ln L G  x  
A

A

2
2
N 1
 max L  x ; n 0 
n0
  x n   A0  
n  n0
( N  n0 ) A
2
2
2

A

2
N 1
A 



x
n

A




0
2 
n  n0 
N 1
max
N 1
n0
A 



x
n

A




0
2 
n  n0 
A 

T  x   max   x n   A0 

n0
2 
n  n0 
Test statistic is maximized
over all possible values of n0
Final Case: Unknown DC Levels, Unknown Jump Time
• DC levels as well as jump time are unknown
• GLRT decides H1 if
max
n0
Aˆ
1
 Aˆ 2

2
 1

1


 


n
N

n
0
0


2
MLE of DC levels:
Aˆ 1 
Aˆ 2 
1
n0
n 0 1
 x n 
n0
1
N  n0
N 1
 x n 
n  n0
'
Multiple Change Times
Multiple Change Times
Parameter’s value changes more than once in data record
For example: DC levels change multiple times in WGN
9
A=6
8
7
A=4
6
x[n]
5
A=2
4
A=1
3
2
1
0
-1
0
10
20
30
40
50
60
Sample, n
70
80
90
100
Multiple Change Times
• No unknown paramters
– Same as Example 1
• Unknown parameters
– DC levels unknown, change times known
Same as Example 3
– Change times unknown
Computational explosion with the number of change times
Example 5: Unknown DC Levels, Unknown Jump Times
• We have signal embedded in WGN
n  0 , 1,  , n 0  1
 A0

 A1
s n   
 A2
 A3

n  n 0 , n 0  1,  , n1  1
n  n1 , n1  1,  , n 2  1
n  n 2 , n 2  1,  , N  1
• 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 A   A A A A  and n  n n n 
1
Aˆ 
 x n 
n

n
To minimize
T
0
1
2
T
3
0
1
n i 1
2
i
i
J  A, n  
n 0 1
  x n  
n0
A0  
2
n1  1
  x n   A1 
n  n0
n 2 1
2

  x n   A2 
n  n1
N 1
2

  x n  
n  n2
i  1 n  n i 1
A3 
2
Example 5: Unknown DC Levels, Unknwon Jump Times
Dynamic programming
• Not all combinations of n0, n1, n2 need to be evaluated
• Reduce computational complexity
 i n i 1 , n i  1 
n i 1

x n   Aˆ i

2
n  n i 1

   n
J Aˆ , n 
3
i
i 1
, n i  1
Recursion for the minimum
i0
I k L  
k
min
  n
n 0 , n1 ,..., n k 1
n 1  0 , n k  L  1 i  0
i
i 1
, n i  1  min  I k 1 n k 1  1   k n k 1 , L 
n k 1
• Effectively eliminate many possible ”paths”
Problems
•
•
•
•
•
12.1
12.2
12.4
12.6
12.11