mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
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166 ADVANCED TOPICS<br />
Notice we used the Markov property to drop Y*~ from the conditioning.<br />
We can interpret (7.48) as a recursive way of computing alpha terms at time t<br />
using alpha terms at time t — 1. To see how to get started, we substitute t = 1 into<br />
(7.48) and realize that Yj" is an empty set, giving<br />
:(<br />
Ql(m) = Σ<br />
Pl i S " = m'bi(m',m). (7.49)<br />
m'=()<br />
To keep the same form as (7.48), (7.49) implies<br />
a n {m) â Pr{5n = m'}, (7.50)<br />
the initial state probabilities. If b\ is preceded by a set of known symbols, then the<br />
initial state probability for the state m! associated with those symbols would be 1<br />
and the remaining initial state probabilities would be zero. If b\ is preceded by a<br />
set of unknown symbols, then the initial state probabilities would all be equal to<br />
1/4.<br />
Thus, (7.48) and (7.50) define a forward recursion (start with t = 1 and increase<br />
t forward in time). Similarly, one can derive a backward recursion for computing<br />
the beta values (see homework problems). Specifically,<br />
:(<br />
ßt(m) = £A +1 (mb(m,m) (7.51)<br />
m=()<br />
ß T (m) = Pr{5 T = rh\S T -i = m). (7.52)<br />
If b T and 6 T _! are known symbols, then S T is known and only one of the conditional<br />
final state probabilities ß T (m) will be nonzero. If these last two symbols are<br />
unknown, then all conditional final state probabilities (for valid state transitions)<br />
will be equal to 1/4.<br />
It is common to implement (7.37) in the log domain, so that multiplies becomes<br />
additions. This is sometimes referred to as log-MAP. Notice that the log domain<br />
cannot be used throughout, due to the summations in alpha and beta recursions.<br />
Also notice that such conversions require knowing iVo or the SNR, so that exponentials<br />
are formed properly. However, if we approximate these summations with<br />
the largest term in the summation (the maximum term), then log domain calculations<br />
can be used throughout (and we do not need to know N (t ). This is sometimes<br />
referred to as the log-MAX approximation. Observe that in this case, the forward<br />
recursion becomes the Viterbi algorithm, in which the log of the alphas are the<br />
path metrics and the log of the gammas are the branch metrics. Unlike MLSD, the<br />
Viterbi algorithm must be run twice, once forward and once backward.<br />
Once the joint state transition/data probabilities have been computed, symbol<br />
likelihoods can be determined by summing the transition probabilities that correspond<br />
to a particular symbol value.<br />
7.3.2 Soft information<br />
With M-ary modulation, bit-level soft information can be extracted from the symbol<br />
metrics by summing the symbol metrics corresponding to a certain bit value.