mohatta2015.pdf

166 ADVANCED TOPICS
Notice we used the Markov property to drop Y*~ from the conditioning.
We can interpret (7.48) as a recursive way of computing alpha terms at time t
using alpha terms at time t — 1. To see how to get started, we substitute t = 1 into
(7.48) and realize that Yj" is an empty set, giving
:(
Ql(m) = Σ
Pl i S " = m'bi(m',m). (7.49)
m'=()
To keep the same form as (7.48), (7.49) implies
a n {m) â Pr{5n = m'}, (7.50)
the initial state probabilities. If b\ is preceded by a set of known symbols, then the
initial state probability for the state m! associated with those symbols would be 1
and the remaining initial state probabilities would be zero. If b\ is preceded by a
set of unknown symbols, then the initial state probabilities would all be equal to
1/4.
Thus, (7.48) and (7.50) define a forward recursion (start with t = 1 and increase
t forward in time). Similarly, one can derive a backward recursion for computing
the beta values (see homework problems). Specifically,
:(
ßt(m) = £A +1 (mb(m,m) (7.51)
m=()
ß T (m) = Pr{5 T = rh\S T -i = m). (7.52)
If b T and 6 T _! are known symbols, then S T is known and only one of the conditional
final state probabilities ß T (m) will be nonzero. If these last two symbols are
unknown, then all conditional final state probabilities (for valid state transitions)
will be equal to 1/4.
It is common to implement (7.37) in the log domain, so that multiplies becomes
additions. This is sometimes referred to as log-MAP. Notice that the log domain
cannot be used throughout, due to the summations in alpha and beta recursions.
Also notice that such conversions require knowing iVo or the SNR, so that exponentials
are formed properly. However, if we approximate these summations with
the largest term in the summation (the maximum term), then log domain calculations
can be used throughout (and we do not need to know N (t ). This is sometimes
referred to as the log-MAX approximation. Observe that in this case, the forward
recursion becomes the Viterbi algorithm, in which the log of the alphas are the
path metrics and the log of the gammas are the branch metrics. Unlike MLSD, the
Viterbi algorithm must be run twice, once forward and once backward.
Once the joint state transition/data probabilities have been computed, symbol
likelihoods can be determined by summing the transition probabilities that correspond
to a particular symbol value.
7.3.2 Soft information
With M-ary modulation, bit-level soft information can be extracted from the symbol
metrics by summing the symbol metrics corresponding to a certain bit value.