1. Detection Theory Chapter 12 Model Change Detection
Xiang Gao
January 18, 2011

2. 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
Summarize the chapters we studied before

3. Outline Basic problem Extension to basic problem Multiple change times
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
Dynamic programming for parameters estimation to reduce the computation
Problems

4. (No Unknown Parameters)
Basic Problem
(No Unknown Parameters)

5. Example 1: Known DC Level and Jump Time
Jump time and DC levels before and after jump are known
A = 4
A = 1

6. 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

7. 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
Tradeoffs in parameter change detection
Delay time in detecting a jump

8. Example 2: Known Variance Jump at Known Time
Energy detector?
Variance = 1
Variance = 4
Detect power change in noise, ex. SNR, guess energy detector

9. Example 2: Known Variance Jump at Known Time
NP detecor decides H1
Detection performance, UMP

10. 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)
Chapter 5, UMP

11. Extensions to Basic Problem
(Unknown Parameters Present)

12. 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 if
Average over all the data samples
Average over data samples before jump
Average over data samples after jump

13. Example 3: Unknown DC Levels, Known Jump Time
After some simplification, we decide H1 if
PDF of test statistic
Explain lamda, best performance occurs when n0 is at mid-point of data record, different from example 1

14. 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
Test statistic is maximized over all possible values of n0

15. 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:

16. Multiple Change Times

17. Multiple Change Times
Parameter’s value changes more than once in data record
For example: DC levels change multiple times in WGN
A = 6
A = 4
A = 2
A = 1

18. Multiple Change Times No unknown paramters Unknown parameters
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

19. 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

20. 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”
Recursion for the minimum

21. Problems
12.1
12.2
12.4
12.6
12.11