Very Computer

1. Discrete-Time Markov Processes and Forney's Paper on the Viterbi Algorithm

Hi Group,

I have a few very basic questions/misunderstandings about the "discrete-time
Markov process" model Forney uses in his 1973 landmark paper
"The Viterbi Algorithm" (Proceedings of the IEEE, vol 61, no 3,
March, 1973).

On p.269, Forney arrives at a fundamental expression for P(z | x),
where x is a finite input state vector running from time 0 to
time K and z is a finite observation state vector running over
the same interval, the observation being made over a channel with
memoryless noise. His expression involves the transition
states xi[k] (Greek letter xi) where xi[k] = (x[k+1],x[k]).

Here's where I run into a real basic conceptual problem. The
probability we seek to evaluate is P(z | x), i.e., "the probability
of z GIVEN x", does not have ANYTHING to do with the transition
states. That is, we are GIVEN x - it matters not one whit how
probable x is. Now since we are given x it seems that the only
thing needed to evaluate P(z | x) are the *CHANNEL* transition
probabilities P(z[k] | x[k]), NOT the source transition probabilies
P(x[k+1] | x[k]).

In other words, when Forney gives the relation

  P(z | x) = sum_{k=0}^{K-1} P(z[k] | xi[k])

I don't see that xi[k] has anything to do with it. We are GIVEN
x - the likelihood of x is 1!!!!
--
%% Randy Yates
%%  DSP Engineer
%%  Ericsson / Research Triangle Park, NC, USA

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