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Detection of signals in NoisePowerPoint Presentation

Detection of signals in Noise

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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 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) 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 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 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 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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