mohatta2015.pdf

THE MATH 125
We would then identify the best candidate using
jb = arg min C m (io,j). (6.38)
We would then store the path metric for state ¿o at time m,
and update the path history
P m (.ia) = C m (ia,jb), ( 6·39 )
Qm(io) = [qm(io),Qm-l(jb)\- (6.40)
This process would be repeated for each possible state i at time m. When finished,
m would be incremented and overall process would be repeated.
The last iteration would be at symbol period m^. After updating the path
metrics and path histories, we would then determine which state has the best path
metric:
ib = arg min P m (i)- (6-41)
i=0 ... Ns(m)
The detected values would then be determined from Q m {ib)·
We can shorten the path history at iteration m by only storing symbol values
from m — (L — 2) and earlier, as the current state determines symbol values for
times m through m — (L — 2) and the previous state determines the symbol value
for time m — (L — 1).
6.3.2 SISO TDM scenario
Now let's consider MLSD for the SISO TDM scenario. The MLSD solution is
the hypothetical sequence of symbols that maximizes the likelihood of the received
signal, given the transmitted sequence equals the hypothesized sequence. Unlike
the LE and DFE formulations, we will not assume that the received signal has been
pre-processed by a filter matched to the pulse shape. However, we will find that
the result can be expressed in terms of such pre-processing.
We basically follow the derivation given in [Ung74]. Let S p denote the set of
possible sequences. For a block of N s symbols with M-ary modulation, there are
M N ' elements in the set. Using q to denote an hypothesized sequence, the MLSD
solution is
{s} = arg max Pr{r(í) Vt|s = q}. (6.42)
qSSp
Next we note that for a particular value of t = to, using model equations (1.27) and
(1.23) gives
ΛΓ,-1
Pr{r(i 0 )|s = q} = Pr{n(t„) = r(t 0 ) - y/Ë~ s ]Γ h(t 0 - mT)q{m)}
m=0
1 rvn Í -Irfo) - ν^ΓΣ^Γο 1 Ht - mT)q{m)\*
~ nN eXP 0 \ N 0
Since the noise is assumed white, the likelihood of multiple received values is simply
the product of the individual likelihoods. Working in the log domain, we end up