mohatta2015.pdf

THE IDEA 153
Traditionally, we still call it MAPSD, even though strictly speaking its ML symbol
detection. In general, ML detection can be viewed as a special case of MAP
detection in which a symbol or sequence values are assumed equi-likely.
7.1.2 Soft information
So far, we have concentrated on recovering the binary symbols being transmitted.
In reality, we are interested on underlying information being sent. Often a code is
used that relates the information to the transmitted symbols. Consider the Alice
and Bob example in Table 1.1, in which there are three possible messages. Notice
that two binary symbols are used to represent the message. Because two binary
symbols can represent up to four messages, there is one pattern that is not used
(s\ = —1, S2 = +1). This fact can be used when decoding the message.
In the Alice and Bob example, when we perform MAPSD, the detected values are
Si = —1, S2 — +1, an invalid combination. That means there is an error somewhere
in the detected sequence. But where? To answer this question, we need to know
more than just the detected symbol values. We need to know how confident we are
of each symbol value.
Soft information generation is about assigning a confidence level or likelihood to
each symbol value. Ideally it is a function of the likelihood that a symbol equals
a certain value, given the received signal (Pr{s 2 = +1|τ~})- But wait a minute. In
the previous subsection, we used symbol metrics related to symbol likelihoods in
MAPSD. In fact, they were proportional to symbol likelihoods. Thus, if we are
using MAPSD, then we can use the symbol metrics as the soft information!
7.1.2.1 Using soft information But what do we do with the soft information?
Consider the Alice and Bob example in Table 1.1. Suppose we use MAPSD and
form the symbol metrics given in Table 7.1. Notice that the symbol metrics for the
two symbol values do not add to one. That is because we used metrics proportional
to a posteriori likelihoods. Though not necessary, we could normalize each pair by
its sum to get likelihoods that add to one.
To decode the message using the soft information, we form message metrics
by multiplying the symbol metrics corresponding to a particular message. For
example, to form a metric for message 1, we would multiply the symbol metrics
L(s 1 = +1) = 0.96975 and L(s 2 = -1) = 0.90783, giving 0.8804. We would need to
form metrics for all possible messages, as shown in Table 7.2. The decoded message
is the one with the largest metric, in this case message 3, which is correct.
Table 7.1
Examplo of MAPSD symbol metrics
Symbol Metric Value
L(s 1 = +1) 0.96975
L(si = -1) 1.4529
LÍS2 = +1) 1.5149
L(s 2 = -1) 0.90783