mohatta2015.pdf

142 MAXIMUM LIKELIHOOD SEQUENCE DETECTION
6.4.1 Block form
Consider the situation in which a vector of received samples can be modeled as
r \= Hs + n, (6.64)
where n is zero-mean with covariance C n . ïf H is triangular, we can use a treesearch
approach. In the more general case, the MLSD solution can be found using
a brute-force, block approach, i.e., using
s = arg max (r — Hs m ) C n (r — Hs„
s 7 „es N "
by trying each possible value for s one at a time.
(6.65)
6.4.2 Sphere decoding
The idea with sphere decoding is to reduce the search space in (6.65) to only those
sequences that produce a predicted received vector that falls within a sphere of
radius p from the actual received vector. As long as p is large enough to include at
least one predicted value, the MLSD solution will be found.
Sphere decoding has been explored extensively for the MIMO scenario. Consider
the case of 3 x 3 MIMO after the channel has been triangularized, so that we have
the model
x\
X2
x-Λ
h
/in hi2 h r .i
0 hl2 h-23
0 0 hm
«1
S2
S.'j
+
The sphere decoder looks for all symbol sets such that
n\
«2
m
(6.66)
P = |xi -/insi -/ii 2 s 2 -/ii:tS:)| 2 -|-|a;2-'i22S2-'i23S3| 2 + |a ; :)-/i.').'iS:i| 2 < P 2· (6.67)
With a conventional tree search, we would start with x 3 and consider all possible
values of s.j. Then we would introduce X2 and S2 and so on. No pruning would
occur.
With sphere decoding, we can impose the radius constraint at each stage. This
is because if the final metric must be within the radius, then so must the partially
accumulated metrics. Thus, at the end of stage 1, we can check that
Rt /i:«s :j | 2 < P 1 - (6.68)
Any value of s.) for which this is not true can be eliminated. After the second stage,
we check
\Χ-Λ - hxiSii] 2 + \x2 - h,22S2 - /i2.)S.'i| 2 < p 1 (6.69)
and discard any paths for which this is not true.
If we choose p too large, we won't get to discard anything and have to perform
a full tree search. If we choose p too small, we risk discarding everything at the
end and missing the MLSD solution. Even if we choose a reasonable value for p,
we may not get much pruning, particularly in the early stages. Like the basic tree
search method, we can use the M- and T-algorithms to obtain approximate forms.