Digital Signal Processing Reference
In-Depth Information
By setting derivatives with respect to the unknowns
u
0
¼
(
fH
0
(
S
j
)
g
,
s
0
) to zero and
solving, the updated parameters are found as
X
i
y
i
Pr(
y
,
j
i
¼
S
j
j
u
(
m
)
)
(
mþ
1)
(
S
j
)
¼
X
i
Pr(
y
,
j
i
¼ S
j
ju
(
m
)
)
H
X
i
,
j
(3
:
7)
(
mþ
1)
(
S
j
)]
2
Pr(
y
,
j
i
¼ S
j
ju
(
m
)
)
N
X
i
,
j
Pr(
y
,
j
i
¼ S
j
ju
(
m
)
)
[
y
i
H
(
s
0
2
)
(
mþ
1)
¼
:
Finally, note that if each marginal likelihood evaluation Pr(
y
,
j
i
¼ S
j
ju
(
m
)
) is scaled by
the same factor, the expressions for the updated parameters do not change. Choosing
the scale factor as 1
/
Pr(
y
) then gives
Pr(
y
,
j
i
¼
S
j
j
u
(
m
)
)
Pr(
y
)
¼
Pr(
j
i
¼ S
j
jy
,
u
(
m
)
)
:
This is the state transition probability encompassed by
j
i
, and is available from the
forward-backward algorithm applied to the trellis diagram for the channel. Using
the three-tap channel from Figure 3.8, for example, these state transition probabilities
become
Pr(
j
i
¼ S
0
jy
,
u
)
/ a
i
1
(0)
g
i
(0, 0)
b
i
(0)
Pr(
j
i
¼ S
1
jy
,
u
)
/ a
i
1
(0)
g
i
(0, 1)
b
i
(1)
Pr(
j
i
¼ S
2
jy
,
u
)
/ a
i
1
(1)
g
i
(1, 2)
b
i
(2)
Pr(
j
i
¼ S
3
jy
,
u
)
/ a
i
1
(1)
g
i
(1, 3)
b
i
(3)
Pr(
j
i
¼ S
4
jy
,
u
)
/ a
i
1
(2)
g
i
(2, 0)
b
i
(0)
Pr(
j
i
¼ S
5
jy
,
u
)
/ a
i
1
(2)
g
i
(2, 1)
b
i
(1)
Pr(
j
i
¼ S
6
jy
,
u
)
/ a
i
1
(3)
g
i
(3, 2)
b
i
(2)
Pr(
j
i
¼ S
7
jy
,
u
)
/ a
i
1
(3)
g
i
(3, 3)
b
i
(3)
using the terms
a
,
b
and
g
as per Section 3.5.1.
The modifications to the turbo equalizer are now straightforward. At each iteration,
the means
fH
(
mþ
1)
(
S
j
)
g
and variance
s
0
2
are calculated from the inner decoder using
(3.7) (along with the extrinsic values to be passed to outer decoder, as usual), and these
values are used for the likelihood calculations
g
i
of the inner decoder for the next
iteration [as per (3.4)]. In this way, a blind turbo equalizer is obtained.
B
EXAMPLE 3.10
Obtaining the Channel Coefficients.
In the parameter estimation step we con-
tented ourselves with finding the noise-free output symbols
H
(
S
j
) rather than
the channel coefficients themselves, as this is all that is necessary for the likelihood
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