Digital Signal Processing Reference
In-Depth Information
1
Moreover, since
is a positive scaling factor and does not influence the
2
p
(
x
(
n
))
sign of
f
(
x
(
n
))
, (7.22) can be rewritten as
p
x
s
n
d
=+
1
−
p
x
s
n
d
=−
1
f
1
(
x
(
n
))
=
(
n
)
|
−
(
n
)
|
−
(7.23)
Using (7.20) and (7.21) in (7.23), we have
p
x
c
j
−
1
N
s
1
N
s
f
(
x
(
n
))
=
(
n
)
|
p
(
x
(
n
)
|
c
i
)
C
d
C
d
c
j
∈
c
i
∈
2πσ
η
−
K
/
2
exp
x
c
j
2
1
N
s
−
(
n
)
−
=
2σ
η
C
d
c
j
∈
2πσ
η
−
K
/2
exp
2
1
N
s
−
x
(
n
)
−
c
i
−
2σ
η
C
d
c
i
∈
⎧
⎨
⎫
exp
x
c
j
exp
⎬
2
2
−
(
n
)
−
−
c
i
x
(
n
)
−
=
−
⎩
2σ
η
2σ
η
⎭
C
d
C
d
c
j
∈
c
i
∈
2πσ
η
−
K
/2
1
N
s
×
(7.24)
N
s
2πσ
η
−
K/
2
in (7.24) is always positive and, hence, can be
suppressed without altering the decision function sign, which can be written
as [233]
1
The term
exp
exp
−
x
c
j
2
2
(
n
)
−
−
x
(
n
)
−
c
i
f
(
x
(
n
))
=
−
2σ
η
2σ
η
C
d
C
d
c
j
∈
c
i
∈
w
j
exp
x
c
j
2
N
s
−
(
n
)
−
=
(7.25)
2σ
η
j
=
1
C
d
. Equation 7.25 is the Bayesian
equalizer decision function. As can be noted, this decision function is non-
linear and completely defined in terms of the channel states and noise
statistics.
C
d
and
w
i
=−
where
w
i
=+
1if
c
j
∈
1if
c
j
∈
