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

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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

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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

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Basic Problem (No Unknown Parameters)

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

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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

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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

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

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

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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)

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Extensions to Basic Problem (Unknown Parameters Present)

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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

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Example 3: Unknown DC Levels, Known Jump Time After some simplification, we decide H1 if PDF of test statistic

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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 Test statistic is maximized over all possible values of n0 Same as Example 1

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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:

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Multiple Change Times

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

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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

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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

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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

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Problems 12.1 12.2 12.4 12.6 12.11

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