Detection of signals in Noise

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# Detection of signals in Noise - PowerPoint PPT Presentation

Detection of signals in Noise. 指導老師 : 黃文傑 姓名 : 吳政修. outline. Review Bayes’ formula Decision rules Maximum a posteriori probability (MAP) ML detection in AWGN channel. Review Bayes’ formula. Conditional probability: Total probability: Bayes’ formula:.

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## PowerPoint Slideshow about ' Detection of signals in Noise' - anais

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Presentation Transcript

### Detection of signals in Noise

outline
• Review Bayes’ formula
• Decision rules
• Maximum a posteriori probability (MAP)
• ML detection in AWGN channel
Review Bayes’ formula
• Conditional probability:
• Total probability:
• Bayes’ formula:
We apply the Bayes’ formula. First we assume that the observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.
Decision rules
• The optimum detector chooses to minimize

or equivalently, to maximize

• The corresponding probability of being correct is
Maximum a posteriori probability (MAP)
• The probability of the decision be correct, given that observing vector x, is
• The probability of error is as follows
• Thus the optimum decision observes the particular received vector X=x and the output chooses to maximize the .
MAP detection rule
• MAP

if for all

Thus if all transmitted symbols occur equally likely, i.e.

Then the decision is equivalent to the maximum likelihood decision rule.

if for all

In an AWGN channel
• In an AWGN channel
• AWGN ML detection

if for all

The decision is to choose a message point closest to the received signal point, which is intuitively.