Sometimes students get confused while studying receivers design in Digital communication system. I have just tried to explain things in a concise and simple method.
Lets consider a receiver model as:
so received signal is the additive sum of transmitted symbols and white noise added by the channel.
suppose we received a symbol r and the transmitted symbol was s.
Probability of correct decision given symbol r is received can be written as P(s/r), mathematically
where , p(r) , probability density function of received vector which is always positive.
Thus , The optimal receiver will be one which maximizes
.
The receiver knows the set of symbols which are being used by the transmitter to send data.To decide on s , it is designed to select s among all possible values of s, such that the conditional probability
is maximum. Mathematically find 
such that,
here , we can drop p(r) as the value of p(r) is always positive. Thus the decision rule equation becomes
From the above rule equation we design system for MAP(Maximum aposterior probability) RULE and also ML(Maximum Likelihood) RULE.
Lets consider a receiver model as:
so received signal is the additive sum of transmitted symbols and white noise added by the channel.
suppose we received a symbol r and the transmitted symbol was s.
Probability of correct decision given symbol r is received can be written as P(s/r), mathematically
where , p(r) , probability density function of received vector which is always positive.
Thus , The optimal receiver will be one which maximizes
The receiver knows the set of symbols which are being used by the transmitter to send data.To decide on s , it is designed to select s among all possible values of s, such that the conditional probability
here , we can drop p(r) as the value of p(r) is always positive. Thus the decision rule equation becomes
From the above rule equation we design system for MAP(Maximum aposterior probability) RULE and also ML(Maximum Likelihood) RULE.
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