TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.5, May 2014, pp
. 3467 ~ 34
7
5
DOI: http://dx.doi.org/10.11591/telkomni
ka.v12i5.2547
3467
Re
cei
v
ed Ma
r 3, 2013; Re
vised Decem
ber 8, 201
3; Acce
pted De
cem
ber 2
9
, 2013
Two Novel Decoding Algorithms for Turbo Codes
Based on Taylor Series in 3GPP LTE System
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
T
h
is pa
per
pro
poses tw
o n
o
v
e
l
meth
ods t
o
simply
Log
arith
m
ic M
a
xi
mum
a post
e
rior
i (Lo
g
-MAP)
alg
o
rith
m for turbo co
des
in
the T
h
ird Gen
e
ratio
n
Partner
ship Pro
j
ect L
o
ng T
e
r
m
Evol
u
t
ion (3GPP LT
E).
F
i
rstly, w
e
exploit a n
e
w
functi
on to re
plac
e t
he l
ogar
ith
m
ic t
e
rm
in th
e Jac
obi
an l
o
g
a
rith
mic fu
nctio
n
ba
sed
on T
a
ylor seri
e
s
, w
h
ich has the best appr
oxi
m
ate
d
accur
a
c
y
comp
ared w
i
th the existin
g
meth
ods. W
i
th this
meth
od, w
e
g
e
t algor
ith
m
I. Secon
d
ly, to further
si
mp
lify the co
mpl
e
xity, w
e
propose a
new
piece-w
i
s
e
lad
der fu
nctio
n
to
ap
proxi
m
ate the
l
ogar
ithmic ter
m
acc
o
rdi
ng to
a
l
go
rithm I. In this
w
a
y, w
e
obt
a
i
n
algorithm
II. Sim
u
lation results show t
hat th
e perform
ance of the algorithm
I is m
o
st close to the optim
a
l
alg
o
rith
m. Alg
o
r
ithm II ow
ns the co
mp
lexity
w
h
ich is
si
mil
a
r to the MAX-Log-MAP al
gorit
hm,
me
anw
hil
e
it
can offer 0.37-
0.4db p
e
rfor
ma
nce ga
ins tha
n
MAX-Log-MAP
algor
ith
m
.
Ke
y
w
ords
: 3GPP LT
E, turbo codes, Lo
g-MA
P, T
a
ylor serie
s
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
The Thi
r
d G
eneration Pa
rtnershi
p
Pro
j
ect Lo
ng T
e
rm Evolutio
n (3GPP L
T
E) [1],
sup
porte
d by
the most tel
e
com
m
uni
cat
i
on ope
rato
rs
from the wh
ole wo
rld, is
investigate
d
in
orde
r to ensu
r
e the com
p
e
t
itiv
eness of Universal Mobile Tele
com
m
unication
s System (UM
T
S)
for the next
10 years a
n
d
beyond. LT
E is a high
-d
ata-rate, low-latency an
d
packet-optimi
z
ed
radio
a
c
cess techn
o
logy [
2
]. Turbo
co
de [3], ca
pab
le of a
c
hievi
ng cl
ose-to
-S
hann
on
cap
a
c
ity
and ame
nabl
e to hard
w
a
r
e-efficient i
m
pleme
n
tatio
n
, has be
en
adopted by
many wirel
e
ss
comm
uni
cati
on stan
dards,
includi
ng HS
DPA [4] and LTE [5].
The 3GPP
worki
ng g
r
ou
p
adopt
s the 1/
3 cod
e
rate t
u
rbo
co
de
s to obtain the
high data
rate i
n
con
s
id
eration
of th
ei
r p
o
we
rful
error
co
rrecting
cap
ability [1]. In additio
n
, L
T
E empl
oys t
he
turbo
cod
e
with a ne
w
conte
n
tion-f
r
e
e
intern
al int
e
rleave
r
ba
sed on q
uad
ratic pe
rmutat
ion
polynomial
(QPP), whi
c
h
requi
re
s smal
l para
m
eter
storage, p
r
ovi
des th
e excel
l
ent perfo
rma
n
ce
[6, 7]. The e
n
c
odi
ng
and
d
e
co
ding
structure of
3GPP
LTE turb
o
co
des is sim
p
ly
sho
w
n
in Fi
g
u
re
1 [4], where
k
x
and
k
L
rep
r
e
s
ent the
syst
ematic
bits
and the
Lo
g
-
likeli
hoo
d ra
tio (LL
R
),
r
e
spec
tively.
The
symbol-by-symb
ol L
og-MAP al
go
rithm is
opti
m
al for ite
r
at
ive decodin
g
in white
Gau
ssi
an n
o
i
s
e [8,
9]. Ho
wever,
re
adin
g
data f
r
om
a
big tabl
e i
s
a
time con
s
um
ing p
r
o
c
e
s
s a
nd
logarith
m
is
not ea
sy to i
m
pleme
n
t in
hard
w
a
r
e. Its sub
-
o
p
timal
variants, th
e
Max-Lo
g-MA
P
algorith
m
[10
], redu
ce
s th
e co
mputatio
nal compl
e
xity greatly. It is repo
rted th
at the Max-L
og-
MAP has a perform
a
nce degradation about 0.4dB
[1
0, 11], which
will bring
almost 10%
capacit
y
loss in the
system [11]. To
improve the
perfo
rman
ce
of Max-Lo
g-MAP algorith
m
, many effort
s
have bee
n de
voted in litera
t
ures in
clu
d
in
g [12-16].
In this pap
er,
we p
r
opo
se
two novel d
e
c
odi
ng alg
o
rit
h
ms fo
r 3GP
P
LTE turbo
cod
e
s,
whi
c
h can yield good BE
R perfo
rma
n
c
e with lo
we
r complexity. The propo
se
d algorith
m
s
can
offer the b
e
st
app
roximate
d pe
rform
a
n
c
e to Lo
g-MA
P and it n
e
e
d
not
comp
ute the lo
garith
m
ic
term.
This pap
er
is orga
nized as follows.
In
Se
ction
2, there
is a
n
introdu
ct
ion of exi
s
tin
g
turb
o
decodin
g
alg
o
rithm. Secti
on 3 describe
s
the pro
p
o
s
ed algo
rithms. Simulation result
s are
sh
own
in Section 4
and we p
r
e
s
ent de
sig
n
architectu
re i
n
Section 5.
The con
c
lu
sion i
s
given
in
Section 6.
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: 3467 – 34
75
3468
Figure 1. The
Encodin
g
an
d De
codi
ng Structu
r
e of 3G
PP LTE Turb
o Cod
e
s
2. Existing Turbo De
codi
ng Algorith
m
s
We b
r
iefly re
view serval
classic tu
rbo
dec
odin
g
alg
o
rithm
s
. Deta
iled explan
ations
are
given in [13].
2.1. The Opti
mal Algorith
m
Log-MAP alg
o
rithm is th
e optimal alg
o
ri
thm for turb
o
code. T
he g
oal of the Lo
g-MAP
algorith
m
is to comp
ute lo
g-likeliho
od ratio (LL
R
) [13
]
:
]
ln[
]
ln[
)
(
1
),
,
(
)
,
(
)
(
)
(
1
),
,
(
)
,
(
)
(
)
(
1
1
*
*
1
1
*
1
1
*
*
1
1
*
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
u
s
s
s
s
s
s
u
s
s
s
s
s
s
k
e
e
u
L
.
(1)
Whe
r
e
k
u
is the information
bits,
k
s
and
1
k
s
den
ote
the state at
k
th and
1
k
th time
instant. To
compute th
e
Equation
(1
), we
nee
d
to recursively
calcul
ate
forward and ba
ckward
metrics, den
o
t
ed as
)
(
k
k
s
and
)
(
k
k
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 co
rre
ction, which make
s Lo
g-MAP optimal.
Acco
rdi
ng to the Equation
(2),
the forward and ba
ckward met
r
ics can be comput
ed as:
))
(
)
,
(
(
max
)
(
ln
)
(
))
(
)
,
(
(
max
))
(
ln(
)
(
1
*
1
1
*
*
1
*
1
1
*
*
1
1
1
1
k
k
k
k
S
k
k
k
k
k
k
k
k
S
k
k
k
k
S
S
S
S
S
S
S
S
S
S
k
k
k
k
.
(3)
Whe
r
e
1
k
and
k
are
colle
ction
of all state
s
at
the mome
nt
1
k
and
k
re
spe
c
tiv
e
ly
,
and
is the bra
n
ch met
r
ics.
Finally, we rewrite Equation (1) as
:
)]
(
)
,
(
)
(
max[
)
(
1
*
1
1
*
k
k
k
k
k
k
k
S
S
S
S
u
L
)]
(
)
,
(
)
(
max[
1
*
1
1
*
k
k
k
k
k
k
S
S
S
S
.
(4)
2.2. The Suboptimal Algo
rithms
The subopti
m
al algo
rithm
s
main
conta
i
n Max-Lo
g-MAP [10], lin
ear L
og-MAP
[12], the
improve
d
Ma
x-Log
-MAP [
13], non
-line
a
r
Lo
g-MAP
[1
4] and
the
co
nstant
Log
-M
AP [15, 16] a
nd
they are obtai
ned by the followin
g
expre
s
sion
s to repl
a
c
e
)
1
ln(
|
|
y
x
e
in turn.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Two Novel Deco
ding Algo
rithm
s
for Turbo Co
de
s Based o
n
Tayl
or Series in
… (Jian
Wan
g
)
3469
0
)
1
ln(
y
x
e
(5)
)
4
1
2
ln
,
0
max(
)
1
ln(
y
x
e
y
x
(6)
)
2
1
2
ln
,
0
max(
)
1
ln(
y
x
e
y
x
(7)
y
x
y
x
e
2
2
ln
)
1
ln(
(8)
2
,
0
2
,
375
.
0
)
1
ln(
y
x
y
x
e
y
x
(9)
5
.
1
,
0
5
.
1
,
5
.
0
)
1
ln(
y
x
y
x
e
y
x
(10)
3. The Propo
sed Algori
t
h
m
s
As is known t
o
all,
)
1
ln(
|
|
y
x
e
in Log-MAP brings l
o
ts of
unde
sirable problem
s. Firstly,
saving
the
re
sults of
)
1
ln(
|
|
y
x
e
in a lo
oku
p
tabl
e
would i
n
volve
a qu
antizatio
n erro
r
cau
s
e
d
by
truncation
of the inp
u
t of th
e loo
k
up
tabl
e. Seco
ndly, lookup tabl
es
are
req
u
ired f
o
r a
wid
e
ran
ge
of
ope
rating
sign
al-to
-
noi
se
ratio
s
(SNRs), whi
c
h i
n
creases the h
a
r
dware
co
st [
13]. La
st but
not
the only one, readi
ng data
from loga
rith
m tables i
s
a time con
s
umi
ng pro
c
e
s
s. So we find a
ne
w
function to re
place it. Derivation pro
c
e
ss is as follo
ws.
Let:
t
t
t
f
1
1
ln
)
(
.
(11)
Comp
uting its derivative, we can
g
e
t the followin
g
expression
s:
2
'
1
2
)
(
t
t
f
(12)
Acco
rdi
ng to the Taylor
seri
es:
1
1
,
...
1
-
1
1
3
2
t
t
t
t
t
t
n
,
(13)
Equation (12) is modified in
to Equation (14).
1
1
),
...
1
(
2
)
(
2
4
2
'
t
t
t
t
t
f
n
(14)
We obtai
n the followin
g
expre
ssi
on thro
ugh co
mputin
g the integral
of Equation (14).
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3467 – 34
75
3470
1
1
...)
5
1
3
1
(
2
)
(
5
3
t
t
t
t
t
f
，
(15)
Omitting the high-order te
rms, we rewrit
e Equation (1
5) as Eq
uatio
n (16
)
1
1
,
2
)
(
t
t
t
f
.
(16)
Let:
t
t
e
m
1
1
ln
)
1
ln(
,
(17)
Then:
m
e
t
5
.
0
5
.
0
.
(18)
Combi
n
ing E
quation (16
)
with Equation
(18), we get the followi
ng a
pproxim
ation.
m
m
e
e
5
.
0
1
)
1
ln(
(19)
Inspired by o
b
se
rving the
curv
e of the
exact co
rrecti
on term, we
prop
ose the followin
g
c
o
rrec
tion func
tion.
m
m
e
e
5
.
0
025
.
1
)
1
ln(
(20)
To simply the com
putat
ional co
mple
xity
, we further p
r
opo
se
a ladder functio
n
approximatio
n.
m
m
e
e
5
.
0
025
.
1
)
1
ln(
(21)
Whe
r
e
m
is the large
s
t intege
r that is small
e
r or eq
ual to
m
. As shown i
n
Figure 2, these two
corre
c
tion terms are more accurate than
equat
ion
s
(5
), (6), (7), (8),
(9) a
nd (1
0).
Figure 2. The
Compa
r
i
s
on
of the Approx
imations
0
1
2
3
4
5
0
0.
1
0.
2
0.
3
0.
4
0.
5
0.
6
0.
7
Lo
g-
M
A
P
I
m
p
r
ove
d
M
a
x-
Log
-
M
A
P
li
nea
r
Lo
g-
M
A
P
no
n-
lin
ea
r
Lo
g-
M
A
P
Eq
u
.
(2
0
)
0
.
375
-
c
ons
t
a
nt
Log
-
M
A
P
0
.
5-
c
o
n
s
t
ant
Lo
g-
M
A
P
Eq
u
.
(2
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Two Novel Deco
ding Algo
rithm
s
for Turbo Co
de
s Based o
n
Tayl
or Series in
… (Jian
Wan
g
)
3471
Acco
rdi
ng to
P. Robe
rtson’
s
study, just
sele
ctin
g the t
betwe
en 0
an
d 5
coul
d o
b
tain th
e
ideal ap
proxi
m
ation [10], so we can get:
5
,
0
5
,
5
.
0
025
.
1
)
1
ln(
m
m
e
e
m
m
(22)
5
,
0
5
,
5
.
0
025
.
1
)
1
ln(
m
m
e
e
m
m
.
(23)
Thro
ugh
Equ
a
tion
(22
)
a
n
d
(23) to
co
mpute
th
e lo
garithm
in E
q
uation
(2
), we get t
w
o
novel alg
o
rith
ms, an
d
we
donate
a
s
al
gorithm
I an
d algo
rithm II
re
spe
c
tively. In practi
ce,
we
coul
d ch
oo
se
different alg
o
rithm
s
acco
rding
to ou
r requireme
nt. For the sy
ste
m
s whi
c
h n
e
e
d
higher reliabil
i
ty such as satellite comm
unication
s, we can em
ploy
algorithm I decode. For the
system
whi
c
h
need hi
ghe
r
validity syste
m
su
ch a
s
p
o
w
er li
ne
com
m
unication
s, we
could
ch
o
o
se
algorith
m
II.
4. Simulation Resul
t
s
4.1. Perform
a
nce Compa
r
ison
Figure 3 sho
w
the simula
ted performa
n
ce un
de
r AWG
N
ch
ann
el for the propo
se
d
algorith
m
s
a
nd othe
rs, in
cludi
ng Lo
g-MAP [9], Max-Log
-MAP [10], linear
L
og-MAP [12],
the
improve
d
Ma
x-Log
-MAP [13], non-li
nea
r Log
-MAP [1
4] and the
co
nstant L
og-M
AP [15, 16]. The
bit erro
r rate (BER) perfo
rm
ance is simul
a
ted in
a rate
-1/3, 8-states
turbo code
d system with the
gene
rato
r [7,
5]. The frame
si
ze i
s
N=10
24 a
nd th
e m
a
ximum n
u
m
ber
of iteratio
ns fo
r d
e
codi
ng
was
s
e
t to 6.
Figure 4 show the perform
ance for algorithm I, algori
thm II, Log-MAP and
Max-Log-
MAP algo
rith
m. Figu
re
4
has simil
a
r si
mulated
env
ironment
with f
i
gure
5,
while
its frame
si
ze is
N=512.
As sh
own in
Figure 3, al
gorithm I off
e
rs
almo
st th
e sam
e
pe
rf
orma
nce a
s
Log-MAP
algorith
m
. Th
e extra codi
ng gai
n is
a
bout 0.
4d
b compa
r
ed to t
he Max-Log
-MAP algorith
m
.
0.15db to th
e linear L
og-MAP, 0.12db to the impr
o
v
ed Max-Lo
g
-
MAP, 0.1db
to the const
ant
Log-MAP, 0.08db to the non-linear L
og-MAP. It is also slightly
superior to the algorithm II.
As ca
n be
se
en in Fig
u
re
4, algorithm I
and alg
o
rith
m II can offer similar
pe
rfo
r
man
c
e,
whi
c
h is
alm
o
st equ
al to the Log
-MAP
algorith
m
. an
d they have nearly 0.3
7
d
B
gain over
Max-
Log-MAP alg
o
rithm.
Figure 3. BER Perfo
r
man
c
e
s
for Algorit
hm I, Algorithm II and the Existing Turb
o De
codi
ng
Algorithm
s
0.
5
0.
6
0.
7
0.
8
0.
9
1
1.
1
1.
2
10
-5
10
-4
10
-3
10
-2
10
-1
S
N
R
(
db)
BE
R
Log
-
M
A
P
A
l
gor
i
t
h
m
I
no
n-
l
i
ne
a
r
L
o
g
-
M
A
P
A
l
gor
i
t
h
m
I
I
0.
375-
co
n
s
t
a
n
t
L
og-
M
A
P
0.
5-
con
s
t
a
n
t
-
L
o
g
-
M
A
P
i
m
pr
ov
ed M
a
x
-
Log
-
M
A
P
l
i
n
ear
Log
-
M
A
P
M
a
x
-
Log
-
M
A
P
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3467 – 34
75
3472
Figure 4. BER Performances for Algorit
hm I, Algorithm II, Log-MAP and Max-Log-MAP
4.2. Comple
xit
y
Analy
s
is
In order to
calculate the
complexity of al
gor
ithm
II, we gain the
probability of
cal
c
ulating
)
(
l
l
s
,
)
(
l
l
s
and LL
R wh
en t locate
s in different ra
nge by
statistics. We
sele
ct SNR=1d
b to
make
statisti
cs, then we g
e
t
the Table 1.
Table 1. The
probability of cal
c
ulating
)
(
l
l
s
,
)
(
l
l
s
and LL
R, SNR=1d
b
(0,1)
(1,2)
(2,3)
(3,4)
(4,5)
(5,
)
)
(
l
l
s
0.0540
0.0532
0.0527
0.0547
0.0542
0.7312
)
(
l
l
s
0.0565
0.0555
0.0511
0.0543
0.0601
0.7226
LLR
0.0708
0.0550
0.0601
0.0610
0.0610
0.6921
Ac
c
o
rding to Table I, we obtain
6
5
4
3
2
1
,
,
,
,
,
p
p
p
p
p
p
.
7153
.
0
0584
.
0
0567
.
0
0546
.
0
0545
.
0
0604
.
0
6
5
4
3
2
1
p
p
p
p
p
p
(24)
Usi
ng the
s
e
data, we
can
cal
c
ulate
the
com
putation
invokin
g
the
Equation
(2
3) o
n
ce.
The num
ber
of addition is:
2847
.
0
2
7118
.
0
)
1
(
)
1
2
5
.
2
(
6
M
M
p
(
2
5
)
The num
ber
of compa
r
i
s
o
n
is:
371
.
2
1
1
-
1
2
5
4
3
2
3
1
6
i
i
i
i
i
i
p
p
p
p
）
（
）
（
(26)
0.
5
0.
6
0.
7
0.
8
0.
9
1
1.
1
1.
2
1.
3
10
-5
10
-4
10
-3
10
-2
10
-1
S
NR(
d
b
)
BER
Log
-
M
A
P
A
l
gor
i
t
h
m
I
I
M
a
x
-
Log
-
M
A
P
A
l
gor
i
t
h
m
I
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Two Novel Deco
ding Algo
rithm
s
for Turbo Co
de
s Based o
n
Tayl
or Series in
… (Jian
Wan
g
)
3473
At las
t, we obtain the Table 2 by referring to [10].
Table 2. The
Compl
e
xity of Turbo
De
cod
i
ng Algorithm
s
Algorithm Comparisons
Additi
ons Multiplicat
ions
Look-ups
Log-MAP
2
2
5
M
9
2
15
M
8
2
2
5
M
Max
-
L
og-MAP
2
2
5
M
11
2
10
M
8 0
Algorithm II
742
.
4
2
855
.
11
M
7153
.
0
2
7118
.
11
M
8 0
As we all
know, comp
uting the “com
pari
s
on
” is
a
l
most no tim
e
con
s
u
m
ing
.for the
comp
uter.
From a
bove ta
b
l
e we
can
see
that the
co
m
p
lexity of algo
rithm II i
s
al
m
o
st e
qual
to t
he
Max-Lo
g-MA
P algorithm.
5. Design ar
chitec
ture
We
de
scribe
the d
e
tail d
e
sig
n
a
r
chite
c
ture
of al
go
rithm II in thi
s
se
ction. Th
e blo
ck
diagram in
Fi
gure
5
sho
w
s the no
de m
e
tric
cal
c
ulatio
n units [12]. In this fig
u
re,
j
s
refers
to the
state j at time k, while
'
j
s
and
'
'
j
s
refer to tho
s
e perviou
s
states at
time k-1, whi
c
h e
n
ter state at
time k. The
delay elem
en
ts sh
own by
“D” in th
i
s
fig
u
re a
r
e
empl
oyed in o
r
de
r to provide t
he
node m
e
tri
c
value
s
at time
k-1 [1
3]. In our meth
od, we com
p
a
r
e
y
x
with 5 usi
n
g
co
mparer1
first,
than we comp
are
y
x
with intege
r 2, 4
,
3, 1 in turn
s by the meth
od of di
choto
m
ization.
With this met
hod, we
can
further imp
r
o
v
e the compu
t
ational efficie
n
cy.
The detaile
d architectu
re
of each blo
ck is sho
w
n in Figure 6 and
)
(
s
k
is determin
e
d
usin
g the
sa
me structu
r
e
in ba
ckwa
rd
recursio
n.
Th
is figure p
r
ov
es that the i
m
pleme
n
tatio
n
of
algorith
m
II is much
simpl
e
r than Lo
g-M
AP that r
equi
res
multiple l
ookup table
s
for a wid
e
ra
nge
of SNRs. In this way, algorithm
II reduce
s
the implem
entation cost.
Figure 5. Nod
e
Metric
Cal
c
ulati
on Unit for n Differe
nt States
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 5, May 2014: 3467 – 34
75
3474
Figure 6. Det
a
iled Archite
c
ture of Algorit
hm II
6. Conclusio
n
In this pape
r,
we propo
se
two novel de
codi
ng alg
o
rit
h
ms for tu
rbo
code in 3
G
P
P
LTE
system, which result in almost equival
ent perfo
rma
n
ce to the op
timum algorit
hm and avoid
hig
h
compl
e
xity. F
i
rstly, we expl
oit a new a
p
p
r
oxim
ated
co
rrectio
n
term
s for the Log
-MAP algorith
m
,
and it
can
offer the
be
st a
pproxim
ated
accuracy i
n
contra
st to the
existing
algo
rithms. T
hen,
a
novel wi
se-pi
e
ce la
dde
r fu
nction i
s
pro
posed to re
pl
ace the lo
ga
rithmic term.
The sim
u
latio
n
s
sho
w
that the
novel deco
d
i
ng schem
es
are supe
ri
o
r
to Max-Lo
g-M
AP algorithm
in perfo
rman
ce
with
slightly i
n
crea
sed
co
mplexity. In addition,
th
e
prop
osed
alg
o
rithm
s
a
r
e
very flexible,
we
coul
d ch
oo
se
different algo
rithms a
c
cord
ing to differen
t
systems.
Referen
ces
[1]
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ang,
J Liu.
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oved R
a
te Matchin
g
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ith
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ird Internatio
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Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Two Novel Deco
ding Algo
rithm
s
for Turbo Co
de
s Based o
n
Tayl
or Series in
… (Jian
Wan
g
)
3475
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