Sequential decoding: Difference between revisions

Revision as of 20:09, 6 July 2010

Revision as of 10:03, 17 July 2010

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For a [[binary symmetric channel]] (with error probability <math>p</math>) the Fano metric can be derived via [[Bayes theorem]]. We are interested in following the most likely path <math>P_i</math> given an explored state of the tree <math>X</math> and a received sequence <math>{\mathbf r}</math>. Using the language of [[probability]] and [[Bayes theorem]] we want to choose the maximum over <math>i</math> of:

:<math>\Pr(P_i|X,{\mathbf r}) \propto \Pr({\mathbf r}|P_i,X)\Pr(P_i|X)</math>

<math>\Pr(P_i|X,{\mathbf r}) \propto \Pr({\mathbf r}|P_i,X)\Pr(P_i|X)</math>

We now introduce the following notation:

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Therefore:

:<math>

<math>

\begin{align}

\Pr(P_i|X,{\mathbf r}) &\propto p^{d_i} (1-p)^{n_ib-d_i} 2^{-(N-n_i)b} 2^{-n_iRb} \\

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We can equivalently maximise the log of this probability, ie

:<math>

<math>

\begin{align}

&d_i \log_2 p + (n_ib-d_i) \log_2 (1-p) +n_ib-n_iRb