Detection Theory Chapter 12 Model Change Detection

1. Detection TheoryChapter 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

3. 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

4. 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 Delay time in detecting a jump

8. Example 2: Known Variance Jump at Known Time Energy detector? Variance = 1 Variance = 4

9. Example 2: Known Variance Jump at Known Time • NP detecor decides H1

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)

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

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 • 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