[RP]Lecture Note X - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) X-1 C1002900 RP Lecture Handout X: Detection and Hypothesis Testing Reading: Chapter 8.4 Detection Problem: We want to guess which of finitely many possible causes produced an observed effect. You have a fever (observed effect): do you think you have the flu or a cold or the malaria? A receiver gets a particular waveform; did the transmitter send the bit 0 or the bit 1 ? A natural framework for setting up detection or decision problems is in terms of a hy- pothesis test. In this framework, each of the possible scenarios corresponds to a hypo- thesis. M hypotheses:   ,, , M HH H 01 1 H 0 is often referred to as the “null” hypothesis, particularly in asymmetric problems where it have special significance. Random and Nonrandom Hypothesis Test: 1) In many case, the hypothesis can be viewed as a discrete random variable. Then, we can associate a priori probabilities   mm PH H  . (X.1) The model for the observed data under each hypothesis takes:  for , , , m H fH m M  1 Y y . (X.2) 2) In other case, it is more appropriate to view the hypothesis not as a random varia- ble, but as a deterministic but unknown quantity. In these situations, a priori proba- bilities are not associated with hypotheses.
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) X-2 The probability density model for the observation is parameterized by the hy- pothesis:  for , , , m fH m M  01 1 Y y . (X.3) 10.1. Binary Random Hypothesis Testing: A Bayesian Approach Two pieces of information 1) A priori probabilities:    . PH H P HH P  00 11 0 1 (X.4) Our state of knowledge about the hypothesis before any observed data are available. 2) Measurement model: : :. H H Hf H H Y Y y y (X.5) Likelihood function The solution to a hypothesis test is specified in terms of a decision rule. Optimum Decision Rules: Likelihood Ratio Test (LRT) Decision rule:   ˆ :, H A function ˆ H uniquely maps every possible N -dimensional observation y to one of the two hypotheses. Partitioning the observation space   y into two disjoint decision regions.
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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) X-3 0 1 H 0 possible observation space N H 1  ˆ H y Goal: Design ˆ H in such a way that the best possible performance is achievable.
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[RP]Lecture Note X - Kyung Hee University Department of...

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