Detection of signals in noise
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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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Detection of signals in noise

Detection of signals in Noise

指導老師:黃文傑

姓名:吳政修


Outline
outline

  • Review Bayes’ formula

  • Decision rules

  • Maximum a posteriori probability (MAP)

  • ML detection in AWGN channel


Review bayes formula
Review Bayes’ formula

  • Conditional probability:

  • Total probability:

  • Bayes’ formula:



Decision rules
Decision rules observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.

  • The optimum detector chooses to minimize

    or equivalently, to maximize

  • The corresponding probability of being correct is


Maximum a posteriori probability map
Maximum a posteriori probability (MAP) observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.

  • 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 detection rule observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.

  • 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 observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.

  • 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.


Optimum receiver
Optimum receiver observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.

  • For a correlator receiver, we consider that

    1.equally likely source symbols

    2.AWGN channel

    Received signal

    Performed by a correlator receiver


  • procedure observation vector x can take on a finite number of values, then given x, the probability that the symbol was transmitted.

  • Set if for all


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