mohatta2015.pdf

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minimum sequence metric and subtract it from all sequence metrics before forming
symbol metrics.
Revisiting the Alice and Bob example, we would take the sequence metrics in
Table 6.1, identify 40 as the minimum value, and then subtract it from all sequence
metrics, giving the normalized sequence metrics in Table 7.4.
Table 7.4
Example of normalized sequence metrics
Index Hypothesis Metric
~WWW~
~ +1 +1 +1 Ö
2 +1 +1 -1 640
3 +1 -1 +1 428
4 +1 -1 -1 348
5 -1 +1 +1 396
6 -1 +1 -1 1036
7 -1 -1 +1 104
8 -1 -1 -1 24
It turns out for a lot of operating scenarios, MLSD and MAPSD provide similar
performance in terms of sequence and symbol error rate. Only at very low SNR
does each provide an advantage in terms of the error rate it minimizes. However, as
we will see in the next section, MAPSD is helpful in understanding soft information
generation.
7.2.2 Soft information
To explore soft information further, we start with the log-likelihood ratio, a common
way of representing soft bit information. Then, an example of an encoder and
decoder is introduced. Finally, the notion of joint demodulation and decoding is
considered.
7.2.2.1 The log-likelihood ratio Notice that we had two soft values when using
MAPSD symbol metrics but only one soft value when using other forms of equalization.
This is because the other equalization approaches were producing a soft
value proportional to the log of the ratio of the two bit likelihoods. We call this
ratio the log-likelihood ratio (LLR).
LLR = log U{ Sm = -iwJ
(7·8)
= log(Pr{ Sm = +l|r})-log(Pr{ Sm = -l|r}). (7.9)
Here is a summary of the advantages of using LLRs.
1. We only need to find something proportional to the likelihood, as any scaling
factors divide out when taking the ratio.
2. We only need to store one soft value per bit, rather than two.
3. The sign of the LLR gives the detected symbol value.