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

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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:


Detection of signals in noise

  • 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

Decision rules

  • 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)

  • 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

  • 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

  • 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

  • For a correlator receiver, we consider that

    1.equally likely source symbols

    2.AWGN channel

    Received signal

    Performed by a correlator receiver


Detection of signals in noise

  • procedure

  • Set if for all


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