28 Convolutional encoder tree diagram representation.
of error. The likelihood functions are given as P(r|xk) where r received sequence of information and
xkone of the possible transmitted sequences of information. The goal of the ML decoder is to choose
a particular transmitted sequence that maximizes the likelihood function. This is accomplished by
exhaustively comparing or searching all the possible code words that could have been transmitted.
Hence for each code word sequence a likelihood value is associated with it. Invoking assumptions of
a memoryless channel and additive white Gaussian noise (AWGN) allows the decoder to accumulate
likelihood values for each path. The likelihood function can be interpreted as a measure of similarity
between all the trellis paths entering each state at time tk and the received signal at time tk.
As one can expect the complexity of such a brute force application can quickly grow. One such
simplification is to discard paths that are ???unlikely,??? this type of decoder is still optimal. When two
paths enter the same state, the most likely path is chosen, and this path is called the surviving path.
Each state selects its surviving path. In 1969, Omura showed that the VA is an ML decoding technique
and thus optimal [14, 15].
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