TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.5, May 2014, pp
. 3460 ~ 34
6
6
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i5.4596
3460
Re
cei
v
ed O
c
t
ober 1, 20
13;
Revi
se
d De
cem
ber 8, 201
3; Acce
pted
De
cem
ber 2
8
,
2013
A Novel Decoding Algorithm for BICM-ID Embedded
Turbo Codes
Jian Wan
g
*, Jianping Li, Chao
shi Cai
Schoo
l of Information En
gi
ne
erin
g, Commu
nicati
on Un
iver
sit
y
of Chi
na, B
e
iji
ng, Ch
ina, 1
000
24
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
w
a
ng
jia
nsu
p
e
r@12
6.com
A
b
st
r
a
ct
Bit-interl
eave
d
code
d mod
u
l
a
tion
iterative
deco
d
in
g (BIC
M-ID) embed
d
ed turb
o cod
e
is w
i
dely
used i
n
w
i
reles
s
communic
a
ti
on bec
aus
e of its excell
ent pe
rforma
nce. T
h
i
s
paper
prop
o
s
es an i
m
prov
e
d
log
a
rith
mic
ma
ximu
m a
post
e
riori (
Log-MAP
)
alg
o
rith
m for
BICM-ID e
m
b
edd
ed tur
b
o
c
odes. It ca
n yi
el
d
excell
ent bit er
ror rate (BER) perfor
m
a
n
ce w
i
th muc
h
low
e
r
comp
lexity. T
he prop
osed a
l
g
o
rith
m expl
oits a
line
a
r i
n
terp
ola
t
ion a
nd
Least
Squar
es a
ppr
oxi
m
ati
on f
unct
i
on to r
epl
ace
the lo
garit
hmic
correctio
n in t
h
e
Jacob
i
an
lo
gar
ithmic functi
on,
w
h
ich avo
i
ds
complic
at
ed
lo
garith
m
l
ook-
u
p tabl
e o
perati
ons i
n
Lo
g-MA
P
alg
o
rith
m. Si
mulati
on r
e
sults
show
that the
nove
l
a
l
gor
ith
m
can
offer a
l
mo
st equ
ival
ent
p
e
rformanc
e to t
h
e
opti
m
a
l
alg
o
rit
h
m w
i
th much
less computa
t
ion. Co
mp
are
d
w
i
th the improve
d
MAX-L
og-MAP al
gori
t
h
m
prop
osed
by
T
a
lako
ub, th
e
pro
pose
d
alg
o
rith
m ca
n r
e
duce
ab
out
3
6
%
of co
mp
utation
a
l c
o
mpl
e
xity,
me
anw
hi
le it
a
c
hiev
es 0.1
db-
0.16d
b
p
e
rfor
mance ga
ins.
In add
ition, it
obt
ains
0.35-
0.4d
b ga
ins th
an M
AX-
Log-MAP a
l
gor
ithm.
Ke
y
w
ords
: BICM-ID, Log-M
AP, interpol
atio
n f
unction, l
eas
t squares, turb
o deco
d
i
n
g
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Bit-interleave
d
co
ded
mo
dulation ite
r
a
t
ive
deco
d
in
g (BICM
-
ID) [1] is an
effective
transmissio
n
schem
e
without b
and
wi
dth expa
nsi
o
n, whi
c
h
co
mbine
s
bit
-
in
terleaved
co
ded
modulatio
n (BICM) [2] wi
th iterative deco
d
ing.
Tu
rbo co
de [3], which wa
s first propo
se
d
by
C.Berrou in 1
993, ma
kes
good u
s
e of the Shann
on
cha
nnel
codi
ng theory a
n
d
is clo
s
e to
the
Shanno
n ca
pacity limit. Naturally, people combin
e BICM-ID
with turbo
code so as
to
simultan
eou
sl
y achieve l
a
rge codin
g
gai
n and
high
b
and
width effi
cien
cy, whi
c
h
lead
s to BICM-
ID embe
dde
d turbo
co
de
s. It is a very promi
s
ing t
e
ch
nolo
g
y co
mpared to th
e cla
s
sical turbo
cod
e
s. As
an
improved
struct
ure, BICM
-ID em
bedd
e
d
turbo
co
de
has b
een
utilized in m
a
n
y
wirel
e
ss com
m
unication systems
to
a
c
hieve
bette
r perfo
rman
ce
[4,
5].
The structu
r
e of
th
e
BICM-ID em
b
edde
d turbo
code is
sho
w
n
in Figure 1.
In the d
e
codi
ng al
gorith
m
s, BCJR alg
o
rithm
[6] i.e.
the maxim
u
m a p
o
ste
r
io
ri (MAP
)
algorith
m
ha
s the be
st pe
rf
orma
nce but
it can
not de
cod
e
until the
decode
r receives the
enti
r
e
bit sequ
en
ce
. This lead
s to large d
e
co
ding d
e
la
y. The Log-MAP [7] algorithm is o
n
e
approximatio
n of MAP alg
o
rithm, and its perfo
rm
a
n
ce is nearby MAP. Howev
e
r, readi
ng d
a
ta
from a big log
a
rithm table i
s
a time con
s
um
ing process for the Log
-MAP. The MAX-Log
-MAP [8
]
is anoth
e
r d
e
s
ira
b
le
candi
date be
cau
s
e
of its simpli
ci
ty, while it lose
s lots of pe
rforma
nce. T
he
improve
d
MAX-Log
-MAP algorithm [
9
] achi
eve
s
the balance betwe
en compl
e
xity and
perfo
rman
ce,
but it involves lots of redu
nda
nt computation
s
without pe
rfo
r
man
c
e g
a
in
s.
Additionally, its pe
rform
a
n
c
e i
s
eq
ual to
the MAX-
Lo
g-MAP un
de
r som
e
ci
rcum
stan
ce
s. So it is
essential
to p
r
opo
se
a
nov
el alg
o
rithm
that
o
w
n
s
sim
p
le compl
e
xity
and su
peri
o
r perfo
rma
n
c
e
for BICM-ID e
m
bedd
ed turbo co
de
s.
In this pap
er, we pro
p
o
s
e a novel d
e
co
ding al
go
rithm ba
sed
on Log
-MAP
and the
improve
d
MAX-Log
-MAP a
l
gorithm. Ou
r algorithm
p
r
opo
se
s a no
vel approxim
ation functio
n
of
logarith
m
ic correctio
n
u
s
i
ng math
ema
t
ical me
th
od.
It is m
o
re
su
peri
o
r th
an
conve
n
tio
nal
decodin
g
alg
o
rithm
s
in terms of perfo
rmance and
complexity.
The rest
of this p
ape
r i
s
orga
nized a
s
follo
ws. In
Section
2, we bri
e
fly intro
duce the
traditional turbo de
codin
g
algorith
m
s. T
he pro
p
o
s
ed
algorithm is presented in
Section 3. We
analyze com
p
lexity and compa
r
e the
perfo
rman
ce
of the propo
sed al
gorith
m
with the other
decodin
g
alg
o
rithm
s
in Section 4. Finall
y
, the conclu
sion is given in
Section 5.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Novel
De
co
ding Algo
rith
m
for BICM-ID Em
bedded
Turb
o Co
de
s (Jia
n Wa
ng
)
3461
Figure 1. The
Structure of t
he BICM-I
D Embedd
ed T
u
rbo
Cod
e
s
2. The Conv
entional Dec
oding Algori
t
hms
At pre
s
ent, th
e cla
s
sical tu
rbo
de
codi
ng
algo
rithms
mainly in
clud
e Log
-MAP a
l
gorithm,
MAX-Log
-MA
P
algorithm, and the imp
r
o
v
ed MAX-Log
-MAP algorith
m
.
The goal of th
e Log-MAP algorithm i
s
to comp
ute log
-
l
i
kelih
ood ratio (LL
R
) [9, 10
]:
]
ln[
]
ln[
)
(
1
),
,
(
)
,
(
)
(
)
(
1
),
,
(
)
,
(
)
(
)
(
1
1
*
*
1
1
*
1
1
*
*
1
1
*
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
u
s
s
s
s
s
s
u
s
s
s
s
s
s
l
e
e
u
L
,
(1)
Whe
r
e
l
u
is the information b
i
ts,
l
S
and
1
l
S
d
enote the stat
e at
l
th and
1
l
th tim
e
instant. T
o
compute E
q
u
a
tion (1),
we
nee
d to
re
cursively
cal
c
ulate forwa
r
d
and
ba
ckward
metrics, den
o
t
ed as
)
(
l
l
s
and
)
(
l
l
s
.
Define the foll
owin
g functio
n
:
)
1
ln(
)
,
max(
)
ln(
)
,
(
max
-
*
y
x
y
x
e
y
x
e
e
y
x
,
(2)
Whe
r
e
)
1
ln(
|
|
y
x
e
is a correctio
n
, whi
c
h ma
ke
s Lo
g-MAP be an
optimal algo
ri
thm.
Acco
rdi
ng to Equation (2), the forwa
r
d a
nd ba
ckwa
rd
metrics can b
e
comp
uted a
s
:
))
(
)
,
(
(
max
)
(
ln
)
(
))
(
)
,
(
(
max
))
(
ln(
)
(
1
*
1
1
*
*
1
*
1
1
*
*
1
1
1
1
l
l
l
l
S
l
l
l
l
l
l
l
l
S
l
l
l
l
S
S
S
S
S
S
S
S
S
S
l
l
l
l
,
(3)
Whe
r
e
1
l
and
l
are
colle
ction
of all state
s
at the mom
e
n
t
1
l
and
l
res
p
ec
tively,
and
is the branch metri
cs.
Finally, we rewrite Equation (1) as
:
)]
(
)
,
(
)
(
[
max
)
(
1
*
1
1
*
*
1
),
(
1
l
l
l
l
l
l
u
S
S
l
S
S
S
S
u
L
l
l
l
)]
(
)
,
(
)
(
[
max
1
*
1
1
*
*
1
),
,
(
1
l
l
l
l
l
l
u
S
S
S
S
S
S
l
l
l
.
(4)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3460 – 34
66
3462
The MAX
-
Lo
g-MAP al
gori
t
hm is obtai
n
ed by
omittin
g
the l
oga
rith
mic
part
of E
quation
(2), so it is a sub
optimal al
gorithm.
The im
prove
d
MAX-L
og-MAP is a
n
al
gorithm
whi
c
h modifie
s
Eq
uation
(2) int
o
Equatio
n
(5) u
s
in
g the MacL
au
rin Serie
s
[9].
)
2
1
2
ln
,
0
max(
)
,
max(
)
,
(
max
*
y
x
y
x
y
x
(5)
3. The Propo
sed Algori
t
h
m
For conveni
e
n
ce, we let
t
y
x
|
|
,
and then, Eq
uation (5
) is e
x
presse
d as:
3862
.
1
2
ln
2
),
,
max(
2
ln
2
,
2
/
|
|
2
ln
)
,
max(
)
,
(
max
*
t
y
x
t
y
x
y
x
y
x
(6)
Whe
n
3862
.
1
t
, after removin
g
the absolute valu
e, we get the followin
g
equ
ation:
2
ln
2
/
)
(
2
/
|
|
2
ln
)
,
max(
y
x
y
x
y
x
(7)
Whe
n
3862
.
1
t
, the improve
d
MA
X-Log
-MAP
algorith
m
is
compl
e
tely e
qual to the
MAX-Log
-MA
P
algorithm, so it is subo
ptimal too.
We ad
d the correctio
n
to solve this pro
b
l
em, then Equation (6
) be
comes:
3862
.
1
),
1
ln(
)
,
max(
3862
.
1
,
2
ln
2
/
)
(
)
,
(
max
|
|
*
t
e
y
x
t
y
x
y
x
y
x
.
(8)
However,
other
problem
s
will follow. Fi
rstly,
lookup tables are requ
ired for a wi
de
range
of operating signal-to
-
n
o
ise
ratios
(SNRs), whi
c
h
in
cre
a
se
s the ha
rd
ware cost. Se
con
d
ly, savin
g
the re
sults
of
)
1
ln(
|
|
y
x
e
in a loo
k
up
table wo
uld
introdu
ce
a
quanti
z
ation
error
cau
s
e
d
by
truncation [9].
Thirdly, rea
d
i
ng d
a
ta from
loga
rithm
ta
bles is
a time
co
nsuming
p
r
ocess. So
it
is
necessa
ry to exploit anot
h
e
r functio
n
in
stead of
)
1
ln(
|
|
y
x
e
. We dedu
ce a
s
follows:
4
/
1
0
2
ln
2
t
e
t
(9)
Let
m
e
t
, then we can get:
4
/
1
0
),
1
ln(
)
1
ln(
|
|
m
m
e
y
x
(10)
Figure 4. The
Compa
r
i
s
on
of the
Approx
imation Straig
ht Lines
0
0.
05
0.
1
0.
15
0.
2
0.
2
5
0
0.
0
5
0.
1
0.
1
5
0.
2
0.
2
5
y=
l
n
(
1
+
m
)
y
=
0.
89
65m
y
=
0.
90
5m
y
=
0.
89
26m
y
=
0.
89
66m
y
=
0.
89
8m
y
=
0.
90
85m
y
=
0.
87
65m
y
=
0.
90
0m
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Novel
De
co
ding Algo
rith
m
for BICM-ID Em
bedded
Turb
o Co
de
s (Jia
n Wa
ng
)
3463
In
(0,
1/4
)
su
ch a small ra
nge,
)
1
ln(
m
will p
r
o
bably b
e
a
ccurately a
p
p
r
o
x
imated by a
linear fun
c
tio
n
. Sele
cting
eight p
o
ints
arou
nd
1/4,
we
ca
n o
b
tai
n
eig
h
t st
rai
ght line
s
by
the
method of interpol
ation fun
c
tion ap
proxi
m
ation
[11]. The line
s
are
shown in the Figure 4.
Observing
ca
refully, we find the optim
al approximatio
n straig
ht line
is:
m
m
8966
.
0
)
1
ln(
.
(11)
By Equation (11) we ca
n g
e
t:
t
t
e
e
8966
.
0
)
1
ln(
.
(12)
More
over, we ca
n find an
optimal line
a
r
lea
s
t sq
uares a
pproxima
t
ion functio
n
repla
c
e
t
e
of equation (12) in (1.386
2, 2.3862)
ra
nge. De
rivation pro
c
e
s
s is as follows.
Let
S
be the
sub
s
p
a
ce of all linea
r functions in
C
[1.3
862,2.38
62]. Although the
function
s
1 and t span
S
, they are not orthogo
nal.
We se
ek a f
unctio
n
of the form
a
t
that is
orthog
onal
to 1.
a
a
t
a
t
8862
.
1
)
(
,
1
3862
.
2
3862
.
1
(13)
Thus a
=
1.8
8
62. Since:
0833
.
0
8862
.
1
t
,
(14)
It follows
that:
)
8862
.
1
(
0833
.
0
1
)
(
1
)
(
2
1
t
t
f
t
f
(15)
Form a
n
orth
onormal ba
si
s for
S
.
Let,
)
5
.
1
5
.
0
(
0833
.
0
1
)
(
)
(
3862
.
2
3862
.
1
3862
.
2
3862
.
1
2
2
3862
.
2
3862
.
1
3862
.
2
3862
.
1
1
1
e
e
dt
e
t
f
c
e
e
dt
e
t
f
c
t
t
.
(16)
The projec
tion:
3862
.
2
3862
.
1
,
4514
.
0
1555
.
0
)
0024
.
6
0072
.
18
(
3217
.
10
9652
.
32
)
(
)
(
3862
.
1
3862
.
2
3862
.
1
3862
.
2
2
2
1
1
t
t
t
e
e
e
e
t
f
c
t
f
c
e
t
(17)
Is the best lin
er lea
s
t squ
a
res ap
proxim
a
t
ion to
t
e
in (1.3862, 2.386
2).
Similarly,
in
t
he ran
ge (2.3
862,
3.3
862
) and (3.3
862, 4.3862
),
t
e
ca
n
be
expresse
d a
s
follows
:
3862
.
4
3862
.
3
,
1032
.
0
0210
.
0
-
3862
.
3
3862
.
2
,
2233
.
0
0572
.
0
-
t
t
t
t
e
t
.
(18)
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3460 – 34
66
3464
A
cco
rdi
ng t
o
p.
Rob
e
rt
s
on’
s st
u
d
y
,
just
sele
cting th
e
t betwee
n
0 a
nd 5
coul
d ob
tain the
ideal ap
proxi
m
ation [12], so we obtai
n the followi
ng e
quation:
3862
.
4
,
0
t
e
t
.
(19)
Combi
ne th
e
expre
s
sion
s (8
)(1
2)(17
)
(18)
and
(19
)
, we
de
du
ce the
app
ro
ximately
expre
ssi
on of
)
,
(
max
*
y
x
as
follows
:
3862
.
4
),
,
max(
3862
.
4
3862
.
1
,
)
,
max(
3862
.
1
,
2
ln
2
/
)
(
)
,
(
max
*
t
y
x
t
b
kt
y
x
t
y
x
y
x
.
(20)
Here, the values of
b
k
,
are
shown in Tabl
e 1.
Table 1. The
Values of k, b
Range of t
k
b
(1.3862,2.
3862)
-0.1394
0.4047
(2.3862,3.
3862)
-0.0515
0.2002
(3.3862,4.
3862)
-0.0188
0.0925
With (20
)
cal
c
ulatin
g
the
)
(
l
l
s
,
)
(
l
l
s
and
the
LLR, we ca
n
a
cco
mpli
sh
better
perfo
rman
ce
with much lo
wer
com
p
lexity than the co
nventional d
e
c
odi
ng algo
rit
h
ms.
4. Simulation Resul
t
s
In the sim
u
lat
i
on, two
syste
m
atic recursi
v
e
convol
utio
nal (RSC) en
cod
e
rs a
r
e e
m
ployed.
We con
s
ide
r
two ci
rcum
sta
n
ce
s which th
e gene
rator p
o
lynomial
s
re
spe
c
tively are
g1 = [7,5] and
g2=[1
1,13]. Simulation
s were re
spe
c
ti
vely
execut
e
d
for code
rate(de
note
R) of
1/2
an
d
3/4
respe
c
tively.
The interle
a
ving length is
set to 1024 an
d the maximum numbe
r of iteration
s
is 6
.
We only consider BPSK, additive
white gaussian
noise
(AWGN)
channel
s in
the transmi
s
sion
environ
ment.
4.1. Perform
a
nce Compa
r
ison
Figure 3. BER Perfo
r
man
c
e Compa
r
i
s
on, N=102
4,
Six Iterations, AWGN Cha
nnel
s, g1=[7,
5
]
0.5
1
1.5
2
2.5
3
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
SN
R
(
d
b
)
BE
R
M
L
M
R
=
1
/
2 N
=
10
24
I
M
LM
R
=
1/
2 N
=
1
024
I
L
M
R
=
1
/
2 N
=
10
24
LM
R
=
1/2 N
=
102
4
LM
R
=
3/4 N
=
102
4
I
M
LM
R
=
3/
4 N
=
1
024
I
L
M
R
=
3
/
4 N
=
10
24
M
L
M
R
=
3
/
4 N
=
10
24
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
A Novel
De
co
ding Algo
rith
m
for BICM-ID Em
bedded
Turb
o Co
de
s (Jia
n Wa
ng
)
3465
Figure 4. BER Perfo
r
man
c
e Compa
r
i
s
on, N=102
4,
Six Iterations, AWGN
Cha
n
nels, g2
= [11,
13]
The p
e
rfo
r
ma
nce
re
sult
s
wi
th different
scheme
s
a
r
e
sh
own
in Fi
gure
3 an
d Fig
u
re
4. We
employ the following
abb
re
viations to
re
place the four algorithm
s.
IMLM: the improved MAX-Log-MAP alg
o
rithm
ILM: the improved Log
-MA
P
algorithm
LM: the Log-MAP algorith
m
MLM: the MAX-Log
-MAP a
l
gorithm
Figure 3 sho
w
s BER pe
rf
orma
nce for the gene
rato
r polynomial
g1=[7,5]. The
curve
s
sho
w
the BE
R for ILM
co
mpared with
LM, MLM an
d IMLM for R=1/2 an
d R=3/4 re
spe
c
tively.
As can
be
seen from Fi
g
u
re
3, ILM, I
M
LM
a
nd L
M
have
obvi
ous pe
rform
a
nce
advanta
ge in
comp
ari
s
o
n
to MLM. ILM can offe
r alm
o
st equival
e
n
t
perform
an
ce com
pared
with LM an
d
it
obtain
s
abo
ut 0.35db pe
rfo
r
man
c
e g
a
in
s than MLM an
d 0.1db gai
ns than IMLM.
Figure 4
presents th
e
sam
e
re
sult
s
whil
e it
s
gen
erat
or p
o
lynomi
a
l is
g2
=[11,13
]. Fro
m
Figure 4, we can
see that ILM achieve
s
about 0.4 db
and 0.16db
gain
s
than MLM and IM
LM
respe
c
tively.
With the increasi
ng of SNR, the
perfo
rmance of the four algo
rit
h
ms
conve
r
g
e
s
grad
ually.
In concl
u
si
o
n
, the propo
sed alg
o
rith
m is sup
e
rio
r
than MLM
and theIMLM
,
and its
perfo
rman
ce i
s
clo
s
e to the
optimal algo
rithm.
4.2. Comple
xit
y
Analy
s
is
We obtain the probabilities of
cal
c
ulati
ng
,
and AP
P L-value
when t lo
cate
s in
different ran
ge by stati
s
tics to
co
mp
ute the
complexity. Selec
t
ing 5 S
NRs
to mak
e
large
numbe
rs
of statistics, and then,
we
re
sp
ectively calculate the
av
e
r
age
in e
a
ch
rang
e a
nd
write
as
5
4
3
2
1
,
,
,
,
p
p
p
p
p
8285
.
0
0365
.
0
0394
.
0
0390
.
0
0562
.
0
5
4
3
2
1
p
p
p
p
p
(33)
Usi
ng the
s
e d
a
ta, we ca
n compute the computation
calling the formula (3
2) on
ce.
The num
ber
of addition is:
3430
.
0
)
1
(
2
5
p
(
3
4
)
The num
ber
of multiplication is:
1715
.
0
)
1
(
1
5
p
(
3
5
)
0.
5
1
1.
5
2
2.
5
3
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
S
N
R
(
db)
BE
R
M
L
M
R
=
1/
2 N
=
1024
LM
R
=
1/
2 N
=
1024
LM
R
=
3/
4 N
=
1024
I
M
LM
R
=
3/
4 N
=
1024
I
L
M
R
=
3/
4 N
=
1024
M
L
M
R
=
3/
4 N
=
1024
LM
R
=
1/
2 N
=
1024
I
L
M
R
=
1/
2 N
=
1024
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3460 – 34
66
3466
The num
ber
of compa
r
i
s
o
n
is:
1365
.
2
1
1
-
1
-
2
4
3
4
2
5
1
）
（
）
（
）
（
i
i
i
i
p
p
p
p
(36)
Throu
gh co
mplex cal
c
ula
t
ing, we obtai
n the Table 2
by referring t
o
[10, 12], wh
ere
M
is
the con
s
train
t
length of encod
er.
Table 2. The
Compl
e
xity of Turbo
De
cod
i
ng Algorithm
s
Algorithms
LM
MLM
IMLM
ILM
Comparisons
2
2
5
M
2
2
5
M
4
2
10
M
273
.
4
2
6825
.
10
M
Additions
9
2
15
M
11
2
10
M
15
2
20
M
6002
.
2
2
143
.
11
M
Multiplicat
ions
8
8
6
2
4
M
657
.
7
2
686
.
0
M
Look-ups
2
2
5
M
0
0
0
From T
able
2
we
can
see t
hat ILM avoid
s
t
he lo
garith
m
cal
c
ulatio
n
for LM. Addit
i
onally,
the compl
e
xity of ILM is reduced ab
out 36% com
pared with IMLM
.
5. Conclu
sion
We p
r
op
ose
an imp
r
oved
Log-MAP alg
o
rithm for BI
CM-I
D emb
e
dded tu
rbo
code in thi
s
pape
r. The n
o
vel method
utilize
s
a ne
w functio
n
to
expand the
corre
c
tion fu
nction of the
Log-
MAP algorith
m
by the method of interp
o
l
ation f
unctio
n
and Le
ast
Squares
a
pproximation. As are
sho
w
n in the
compl
e
xity analysis a
nd
simulation resu
lts, the prop
o
s
ed al
go
rithm
can a
c
compli
sh
almost eq
uivalent perfo
rmance to the optimal
al
gorithm
with
a much lo
wer
com
p
lexi
ty.
In
addition, it o
ffers a
bout
0.1-0.16
db p
e
rfor
m
a
n
c
e
gain
s
than t
he imp
r
oved
MAX-Log
-M
AP
algorith
m
with only
74%
computation
s
.
To
sum
up, t
he n
o
vel al
go
rithm h
a
s obv
ious adva
n
ta
ges
in co
ntra
st to
the traditio
nal
algo
rithms a
nd
it can b
e
easily im
plem
ented in
BICM-ID e
m
be
dd
ed
turbo code
s.
Referen
ces
[1]
X
Li, A Chindapol, JA Ritce
. Bit-interleav
ed coded modulation
w
i
th it
erative decoding and 8PSK
sign
ali
ng.
IEEE Trans. on Comm
un.
200
2; (50): 125
0-1
257
.
[2]
E Zehavi. 8-PS
K trellis cod
e
s for a Ra
yl
eig
h
chan
nel.
IEEE Trans.Commun.
1992; (40): 8
7
3
-88
4
.
[3]
Ji-Hoo
n Kim. In-Che
olPark,
Expr
essBri
efs. Bit-Level e
x
t
r
insic inf
o
rmat
i
on e
x
c
h
a
nge
method for
dou
ble-
bi
nar
y turbo co
des.
IEEE Transactions.
2009; (5
6): 81–
85.
[4]
Shah C. P T
s
i
m
eni
dis CC, Sharif BS, Neas
ham JA
. Lo
w
-
Compl
e
xit
y
iter
ative receiv
er structure for
time-var
yin
g
fr
equ
enc
y-se
lect
ive sh
all
o
w
un
der
w
a
t
e
r aco
u
s
tic chan
nels
usin
g BICM-ID
:
desig
n a
nd
exper
imenta
l
results.
IEEE Oceanic Engineering
. 201
1; (36): 406-4
21.
[5]
Li
ya
o W
ang,
Jian
pin
g
L
i
.
A new
iterative
sched
uli
ng fo
r BICM-ID embed
de
d turbo
codes
, IEEE
Internatio
na
l C
onfere
n
ce. 20
1
2
; 39: 953-
956.
[6]
L Bahl, J Cock
e, J Jelin
ek, J Raviv, F
Raviv
.
Optimal deco
d
in
g of lin
ear c
odes for mi
nim
i
zing s
y
m
b
o
l
error rate.
IEEE Trans. Inform. Theory.
19
7
4
; (20): 284-2
8
7
.
[7]
Xi
ao
yi
Li, Z
h
ao
F
ang. Res
earc
h
on
impr
ove
d
turbo-co
des
d
e
cod
i
ng
al
gorit
hm.
Co
mp
uter
. 200
9; (2
5):
230-
232.
[8]
Rob
e
rtson P,
Hoe
her P. Optimal a
nd s
ub-o
p
tima
l ma
ximu
m a poster
i
ori
alg
o
rithm suitable for tur
b
o
dccod
i
ng.
Eur.
T
r
ansactions. T
e
leco
mmun.
199
7; (8): 119-
125.
[9]
S T
a
lakoub, L
Sabeti, B S
h
ahrrav
a
, M Ahmad
i
. An im
prove
d
Ma
x-L
og-MAP a
l
gor
i
t
hm for turb
o
deco
d
in
g a
nd t
u
rbo
equ
aliz
ati
on.
IEEE Trans
. on Instrument
ation
and
mea
s
ure
m
e
n
t.
200
7; (56): 10
58-
106
3.
[10]
S Lin, Da
nie
l
J Costell
o
Jr. 200
4. Error C
ontrol C
o
d
i
ng.
Jian Ya
n, Yua
n
zhi H
e
, Yah
a
n
Pan. Ch
in
a
Machi
ne Press
.
2007.
[11]
Istvan Hall
er, Sergi
u
Ned
e
vs
chi. Desig
n
of in
terp
olati
on fu
nctions for sub
p
i
x
el-
a
ccur
a
c
y
stereovis
io
n
s
y
stems.
IEEE Transactions on image processing.
20
12; (2
1): 889- 89
8.
[12]
Rob
e
rtson P,
Vill
ebru
n
E,
Hoe
her P.
A co
mp
aris
on of
opti
m
al a
nd s
ub-
Optima
l MAP
deco
d
in
ga
lgor
ithms o
per
ating
in the Lo
g do
main.
IEEE International Confer
ence on 1822.
1995: 1009-
101
3.
Evaluation Warning : The document was created with Spire.PDF for Python.