mohatta2015.pdf

154 ADVANCED TOPICS
Table 7.2
Example of message metrics formed from MAPSD metrics
Message Symbol Sequence Message Metric
Î +1 -1 0.880
2 -1 -1 1.319
3 +1 +1 1.469
Observe what happened. With MAPSD, we detected si = —1 and ¿2 = +1,
which is not a valid message. When we used soft information, we only considered
valid messages and determined the message to be «i = +1, S2 = +1> which is
correct. Thus, we corrected the error that MAPSD made in s\. The soft information
allowed us to indirectly find and correct the detection error.
7.1.2.2 Soft information from other equalizers What if we aren't using MAPSD?
Then how do we get soft information? For MF, DFE and LE, it turns out the
decision variable can be interpreted as a scaled estimate of the log of the a posteriori
likelihood that the symbol is +1. The soft value for the symbol being a —1 is simply
the negative of the decision variable. Because these are log values, decoding involves
adding soft values rather than multiplying. With this approach, we don't have to
worry about what the scaling factor is, as it won't change the result as long as the
scaling is positive.
As an example, suppose the decision variables are the values given in Table 4.1
for MMSE LE. Then the message metric for message 1 would be (—0.18914) +
(—0.26488) = —0.454. The message metrics for all possible messages are given in
Table 7.3. Observe that despite errors in individual symbols, the correct message
is decoded.
Table 7.3
Example of message metrics formed from MMSE LE metrics
Message Symbol Sequence Message Metric
Ï +1 -1 -0.454
2 -1 -1 -0.0757
3 +1 +1 0.0757
What about soft information for MLSD? As the sequence metrics are related
to log-likelihoods of sequences, one approach for obtaining soft information for a
symbol, say S2, is to take the difference of two sequence metrics. The first sequence
metric is the detected sequence metric. The second sequence metric is obtained by
setting all the symbols to their detected values, except for symbol S2, which is set
to the opposite of the detected value. For example, consider the sequence metrics
given in Table 6.1. The detected sequence is s = +1, sj = —1, and S2 = +1 which has sequence metric 468. The
soft value magnitude would be |468 — 40| or 428. The soft value sign would be the