mohatta2015.pdf
signal processing from power amplifier operation control point of view
signal processing from power amplifier operation control point of view
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
142 MAXIMUM LIKELIHOOD SEQUENCE DETECTION<br />
6.4.1 Block form<br />
Consider the situation in which a vector of received samples can be modeled as<br />
r \= Hs + n, (6.64)<br />
where n is zero-mean with covariance C n . ïf H is triangular, we can use a treesearch<br />
approach. In the more general case, the MLSD solution can be found using<br />
a brute-force, block approach, i.e., using<br />
s = arg max (r — Hs m ) C n (r — Hs„<br />
s 7 „es N "<br />
by trying each possible value for s one at a time.<br />
(6.65)<br />
6.4.2 Sphere decoding<br />
The idea with sphere decoding is to reduce the search space in (6.65) to only those<br />
sequences that produce a predicted received vector that falls within a sphere of<br />
radius p from the actual received vector. As long as p is large enough to include at<br />
least one predicted value, the MLSD solution will be found.<br />
Sphere decoding has been explored extensively for the MIMO scenario. Consider<br />
the case of 3 x 3 MIMO after the channel has been triangularized, so that we have<br />
the model<br />
x\<br />
X2<br />
x-Λ<br />
h<br />
/in hi2 h r .i<br />
0 hl2 h-23<br />
0 0 hm<br />
«1<br />
S2<br />
S.'j<br />
+<br />
The sphere decoder looks for all symbol sets such that<br />
n\<br />
«2<br />
m<br />
(6.66)<br />
P = |xi -/insi -/ii 2 s 2 -/ii:tS:)| 2 -|-|a;2-'i22S2-'i23S3| 2 + |a ; :)-/i.').'iS:i| 2 < P 2· (6.67)<br />
With a conventional tree search, we would start with x 3 and consider all possible<br />
values of s.j. Then we would introduce X2 and S2 and so on. No pruning would<br />
occur.<br />
With sphere decoding, we can impose the radius constraint at each stage. This<br />
is because if the final metric must be within the radius, then so must the partially<br />
accumulated metrics. Thus, at the end of stage 1, we can check that<br />
Rt /i:«s :j | 2 < P 1 - (6.68)<br />
Any value of s.) for which this is not true can be eliminated. After the second stage,<br />
we check<br />
\Χ-Λ - hxiSii] 2 + \x2 - h,22S2 - /i2.)S.'i| 2 < p 1 (6.69)<br />
and discard any paths for which this is not true.<br />
If we choose p too large, we won't get to discard anything and have to perform<br />
a full tree search. If we choose p too small, we risk discarding everything at the<br />
end and missing the MLSD solution. Even if we choose a reasonable value for p,<br />
we may not get much pruning, particularly in the early stages. Like the basic tree<br />
search method, we can use the M- and T-algorithms to obtain approximate forms.