Kevin Buckley  2007
2
Since we know that the sequence
{
v
n
}
forms a sufficient statistic for MLSE of the
sequence
{
I
n
}
, we can formulate this MLSE problem, at time
n
, in terms of the joint
PDF of
v
n
= [
v
1
,
v
2
,
· · ·
,
v
n
]
T
conditioned on
I
n
= [
I
1
, I
2
,
· · ·
, I
n
]
T
:
p
(
v
n
/I
n
) =
1
(2
πσ
2
n
)
n
e

∑
n
k
=1

v
k

f
H
I
k,L

2
/
2
σ
2
n
.
(3)
Concerning notation, here we represent current time (i.e. the most recent symbol time
that we want to optimuze up to) as
n
, and
k
represents all symbol time up to
n
. We
then consider incrementing to to the next current time
n
+ 1.
The MLSE problem, at symbol time
n
, is
max
I
n
p
(
v
n
/I
n
)
.
(4)
Taking the negative natural log and eliminating constant terms that do not effect the
relative costs for different
I
n
’s, we have the equivalent problem
1
min
I
n
Λ
n
(
I
n
) =
CM
(
I
n
) =

PM
(
I
n
) =
n
summationdisplay
k
=1

v
k

f
H
I
k,L

2
,
(5)
where Λ
n
(
I
n
) is the
cost
of sequence
I
n
.
1
In Chapters 5 & 10 of the Course Text, where MLSE and the Viterbi algorithm are discussed, several
notations are used to represent the measure or metric to be optimized to. When
PM
is used, it is maximized,
and typically refers to a probability metric. When
CM
is used, it is minimized, and is sometimes referred
to as a corelation of crosscorrelation metric.
CM
is often equivalent to the Euclidean distance
D
. Here, I
start using the notation Λ and refer to it as a cost to be mimimized. As used, it is close to if not the same
as
CM
or
D
of the Course Text. I choose Λ so as to get away from the variety of Course Text notations,
and because I’ve used this notation in other places to represent cost.