mohatta2015.pdf

THE MATH 161
still take advantage of the entire received signal and the fact that other, discrete
symbols are being sent.
The implication of the second difference is that MAPSD will allow us to introduce
prior information about the symbol likelihoods. Mathematically, MAPSD gives
s(m) — arg max Pr{s(m) = q(m)\r(t) Vi}, (7-16)
q(m,)eS
where we use Pr{·} to denote likelihood, which can be either a discrete probability
(sum to one) or a PDF value (integrates to one).
Thus, we find the hypothetical symbol value q(m) that is most likely, given the
received signal. Applying Bayes' rule, we can rewrite this as
= arg max P r {r(«)Vtk(m) = g(m)}l>rMm) = g(m)}
V ' g(m)es Pr{r(í)Ví}
v ;
Each of the three terms on the r.h.s. is discussed separately.
It helps to start with the second term in the numerator. This term is the a priori
or prior information regarding the symbol. If we knew that the possible symbol
values are not equi-likely, we would introduce that information here.
The first term in the numerator looks like the MLSD criterion, except the likelihood
of the received signal is conditioned on a single symbol, rather than a sequence.
Thus, this first term is really an ML symbol detection metric. We will see later that
this term can also include prior information, prior information about other symbols
besides s(m).
The term in the denominator is the likelihood of the received signal. As it is an
unconditional likelihood, it will be the same for each value of q(m). Thus, we can
ignore this term when performing the maximization operation.
We can rewrite the first term in the numerator in a way that relates it to sequence
detection. Let s m denote the subsequence of s that excludes s(m), and let q m
denote a hypothetical value for s m . Also, let S^'~ x denote the set of all possible
subsequences for s m . Then, we can write
Pr{r(i) Vt|« = q(m)} = £ Pr{r(i) Vt|«(m) = (?(m), s m = q m }
xPr{s m = q m }. (7.18)
Using this result and dropping the denominator term gives the MAP symbol metric
P{s = q(m)\r(t) Vi} = £ Pr{r(t) Vt|e(m) = q(m),s m = q} m
xPr{s(m) = 9(m)}Pr}s m = q m }. (7.19)
Observe that the metric consists of the sum of a product of three terms. The first
term is the MLSD metric for a particular sequence. However, it is only evaluated for
sequences for which s(m) = q(m). The second and third terms give prior sequence
likelihood information. Here is where prior information about other symbols can
be introduced.