Detection of signals in Noise

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:

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.

Optimum receiver • For a correlator receiver, we consider that 1.equally likely source symbols 2.AWGN channel Received signal Performed by a correlator receiver

procedure • Set if for all

Detection of signals in Noise