turbo1 - Turbo Coding and MAP Decoding 1 Intuitive Guide to...

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Turbo Coding and MAP Decoding www.complextoreal.com 1 Intuitive Guide to Principles of Communications www.complextoreal.com Turbo Coding and MAP decoding - Part 1 Baye’s Theorem of conditional probabilities Let’s first state the theorem of conditional probability, also called the Baye s theorem. ) ( ) ( ) ( A given B P A P B and A P Which we can write in more formal terminology as ( , ) ( ) ( | ) P P A B A B P A A where P ( B | A ) is referred to as the probability of event B given that A has already occurred. If event A always occurs with event B, then we can write the following expression for the absolute probability of event A. ( ) ( , ) B P A P A B B If events A and B are independent from each other then (A) degenerates to ) ( ) , ( A P B A P B P A P B A P ( ) ( ) , ( ) C The relationship C is very important and we will use it heavily in the explanation of Turbo decoding in this chapter.
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Turbo Coding and MAP Decoding www.complextoreal.com 2 If there are three independent events A, B and C, then the Baye’s rule becomes ) , | ( ) | ( ) | , ( C A B P C A P C B A P D A-priori and a-posteriori probabilities Here is Bayes' theorem again. Pr ( , ) ( ) ( ) ( ) ( ) a priori obability of a posteriori probability of both A and B probability of event B event A a priori a posteriori probability of probability of event A event B P A B P AB P B P B A P A   E The probability of event A conditioned on event B, is given by the probability of A given B times the probability of event a. The probability of A, or P(A) is the base probability of even A and is called the a-priori probability. The term P(A,B) the conditional probability is called the a-posteriori probability or APP . One is independent probability, the other depends on some event occurring. We will be using the acronym APP a lot, so make sure you remember that is the same a-posteriori probability. In other words, the APP of an event is a function of an another event also occurring at the same time. We can write (E) as ( ) ( ) ( , ) ( ) () APP P B A P A P A B P AB PB  F This says that we can determine the APP of an event by taking the conditional probability of that event divided by it’s a -priori probability. What these mean is best explained by the following two quotes. In epistemological terms “ A priori” and “a posteriori” refer primarily to how, or on what basis, a proposition might be known. In general terms, a proposition is knowable a priori if it is knowable independently of experience, while a proposition knowable a posteriori is knowable on the basis of experience. The distinction between a priori and a posteriori knowledge thus broadly corresponds to the distinction between empirical and non- empirical knowledge.” [2] “But how do we decide when we have gathered enough data to justify modifying our prediction of the probabilities? That is one of the essential problems of decision theory.
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turbo1 - Turbo Coding and MAP Decoding 1 Intuitive Guide to...

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