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 Summarize the chapters we studied before

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

(No Unknown Parameters) Basic Problem (No Unknown Parameters)

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

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

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

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

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

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

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 if 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 Explain lamda, best performance occurs when n0 is at mid-point of data record, different from example 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 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 MLE of DC levels:

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 A = 6 A = 4 A = 2 A = 1

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

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

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

Problems 12.1 12.2 12.4 12.6 12.11