Intern
ati
o
n
a
l
Journ
a
l of
Re
con
f
igur
able
and Embe
dded
Sys
t
ems
(I
JRES)
V
o
l.
4, N
o
. 1
,
Mar
c
h
20
15
,
pp
. 6
~
1
2
I
S
SN
: 208
9-4
8
6
4
6
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJRES
An Effi
cient VLS
I
Archit
ectu
r
e f
o
r Nonbinary L
D
PC Decoder
with Ad
aptive Message Control
R. V
a
ra
th
ar
aj
an
Department o
f
ECE, Sri
Lakshmi Ammal Engineering
College, Ch
ennai, Ind
i
a
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Sep 26, 2014
Rev
i
sed
No
v
13
, 20
14
Accepte
d Dec 2, 2014
A new decoder
architecture fo
r
nonbinar
y
low-d
e
nsity
par
i
ty
check (LDPC)
codes is prese
n
ted in
this p
a
per
to
reduce the h
a
rdware operational
complexity
and
power consumption. Ad
aptiv
e m
e
ssage control (
A
MC) is to
achi
e
ve the low
decoding com
p
lexit
y
th
at d
y
na
m
i
call
y
tr
im
s
the m
e
s
s
a
ge
length of b
e
li
ef
inform
ation to r
e
duce
the
am
ount of m
e
m
o
r
y
a
ccesses an
d
arithmetic oper
a
tions. A new horiz
ontal
nonbinar
y
LD
PC decoder
archi
t
ec
ture
is develop
e
d to
i
m
p
le
m
e
nt AMC. Ke
y com
p
o
n
ents in th
e
archi
t
ec
ture
hav
e
be
en d
e
s
i
gned
with
the
consid
eration of
variab
le message
lengths to
lev
e
r
a
ge th
e ben
e
fit
of
the proposed
AMC. Simulation results
demonstrate
that the proposed
nonbinar
y
LDP
C
decod
e
r ar
chitecture
can
significantly
r
e
duce hardware opera
tions an
d power consumption as
compared with existing work with negligible p
e
rf
ormance d
e
grad
ation
.
Keyword:
Ada
p
t
i
v
e m
e
ssage c
ont
rol
Decoding
E
x
t
e
n
d
ed
mi
n
-
s
u
m
No
n bi
nary
l
o
w-
de
nsi
t
y
pari
t
y
-
check (L
DPC
)
code
s
VLSI a
r
chitecture
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
R.
V
a
r
a
th
ar
a
j
an
Pro
f
ess
o
r
&
H
ead
Depa
rt
m
e
nt
of
EC
E,
Sri
L
a
ks
hm
i
A
m
m
a
l En
gi
nee
r
i
n
g C
o
l
l
e
ge, C
h
e
nna
i
,
I
nda
Em
a
il: v
a
rath
u2
1@yahoo
.com
1.
INTRODUCTION
Lo
w-
den
s
i
t
y
pari
t
y
-chec
k
(L
DPC
)
co
des
[
1
]
,
[
2
]
are c
o
nsi
d
e
r
ed
as
o
n
e
of t
h
e m
o
st
po
we
rf
ul
capacity-approaching c
o
des.
LDPC c
o
des
can
b
e
co
nstructed
in
bo
th b
i
nary do
m
a
in
and
Galo
is
field
s
(i.e.,
GF(
2
m
), w
h
e
r
e
m
>1). B
i
nary
L
D
PC
c
ode
s ha
ve
be
en st
u
d
i
e
d e
x
t
e
nsi
v
el
y
an
d
ado
p
t
e
d i
n
m
a
n
y
com
m
uni
cat
i
on p
r
ot
ocol
s
,
su
ch as D
V
B
-
T2
,
W
i
M
a
x, et
c.
I
n
ge
neral
,
a ve
ry
l
o
n
g
co
de l
e
ngt
h i
s
necessa
ry
fo
r
bi
na
ry
LDPC
code
s t
o
a
p
p
r
oac
h
t
h
e c
h
a
nnel
ca
paci
t
y
. Lo
w-
Den
s
i
t
y
Pari
t
y
-C
hec
k
(LD
P
C
)
c
o
d
e
s ha
ve
recently received a lot of attention
b
eca
use of their a
d
m
i
ra
ble perform
a
nce
and ha
ve bee
n
widely consi
d
ere
d
as a pr
om
i
s
i
ng candi
dat
e
er
ro
r-c
or
rect
i
ng c
o
di
n
g
sc
hem
e
f
o
r m
a
ny
real
appl
i
cat
i
o
ns i
n
t
e
l
ecom
m
uni
cati
o
n
s
and m
a
gnet
i
c
s
t
ora
g
e.
No
n
b
i
n
ary
LDPC
c
o
d
e
s co
nst
r
uct
e
d
i
n
Gal
o
i
s
fi
el
d
s
of
fer i
m
pro
v
e
d pe
rf
o
r
m
a
nce at
a
m
oderat
e
code
l
e
ngt
h. I
n
a
d
d
i
t
i
on,
no
n
b
i
n
ar
y
LDPC
c
odes
can
be com
b
i
n
ed
wi
t
h
hi
g
h
or
der m
o
d
u
l
a
t
i
ons
t
o
increase t
h
e
bandwidt
h e
ffic
i
ency. Due t
o
these feat
ures
,
desi
g
n
a
n
d i
m
pl
em
ent
a
t
i
on of
n
o
nbi
nary
LDPC
code
s
have
bec
o
m
e
cri
t
i
cal
for m
a
ny
em
ergi
ng
ap
pl
i
cat
i
ons
suc
h
as
u
n
d
er
wat
e
r ac
o
u
st
i
c
com
m
uni
cat
i
ons.
A
key
chal
l
e
n
g
e i
n
t
h
e a
p
pl
i
cat
i
on
of
n
o
n
b
i
nary
L
D
PC
c
ode
s i
s
t
h
ei
r hi
gh
dec
o
di
n
g
c
o
m
p
l
e
xi
t
y
, as
each sym
bol in the c
ode
word is dec
o
ded
using a l
o
ng m
e
ssage.
A lot
of researc
h
effort aim
s
at reducing the
deco
di
n
g
com
p
l
e
xi
t
y
of n
o
n
b
i
n
a
r
y
LDPC
code
s at
t
h
e
alg
o
rith
m
lev
e
l. To
d
eal with
th
e p
r
ob
lem th
at
co
m
p
u
t
atio
n
a
l
co
m
p
lex
ity in
creases ex
pon
en
tially with
, the extended Mi
n-Sum
(EMS)
was propose
d
in [3
]
whe
r
e
o
n
l
y
t
h
e m
o
st
si
gni
fi
c
a
nt
n
m
ent
r
i
e
s
i
n
a m
e
ssage
were
us
ed
i
n
t
h
e
dec
odi
ng
.
A
dec
odi
ng
t
echni
qu
e
devel
ope
d i
n
[
4
]
con
d
u
ct
ed t
h
e EM
S
with
a redu
ced
co
mp
lex
ity o
f
o(n
m
lo
g
2
n
m
) wi
t
h
m
i
nor
per
f
o
r
m
a
nce
d
e
grad
atio
n. It sh
ou
ld
b
e
n
o
t
ed
th
at these alg
o
r
ith
m
-
lev
e
l tech
n
i
q
u
es do
no
t exp
licitly co
n
s
ider the
com
p
l
e
xi
t
y
i
n
t
h
e i
m
pl
em
ent
a
t
i
on
of
n
o
nbi
na
ry
LD
PC
deco
ders
.
Diffe
re
nt fr
om
these existing
wo
rk
s targetin
g ha
rd
wa
re imp
l
em
en
tatio
n
co
st, th
e fo
cus o
f
th
is p
a
p
e
r
is to
redu
ce the h
a
rdware operatio
n
a
l co
m
p
lex
ity in
nonbi
n
ary L
D
PC
de
code
r a
r
chitect
ures
. T
h
is e
n
a
b
les
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
64
I
J
RES
Vo
l.
4
,
N
o
.
1
,
Mar
c
h 20
15
:
6 – 12
7
effi
ci
ent
deco
d
i
ng s
u
i
t
a
bl
e
fo
r
em
ergi
ng a
ppl
i
cat
i
ons s
u
ch
a
s
u
nde
r
w
at
er a
c
ou
st
i
c
sens
or
net
w
or
ks
[5]
t
h
at
are
un
de
r t
h
e se
ve
re res
o
urce
(e.
g
., e
n
e
r
gy
)
co
nst
r
ai
nt
s
.
It
was reported that
m
e
m
o
ry ac
cesses and a
r
ithm
e
tic
o
p
e
ration
s
are
th
e two
m
a
j
o
r
co
n
t
ribu
to
rs to th
e
op
eratin
g
cost in L
D
PC
decode
rs. As
the am
ount of
me
m
o
ry
accesses and a
r
ithm
e
tic operations is largely determ
ined
by the m
e
ssage length, re
duc
i
ng m
e
ssage lengt
h is
deem
ed as an effective
way for ef
ficien
t d
ecod
i
ng
.
Differen
t
fro
m
th
e
EMS wh
ich
main
tain
s a con
s
tan
t
messag
e
len
g
t
h
fo
r ev
ery sy
m
b
o
l; th
e p
r
op
o
s
ed
AMC ad
ju
sts th
e m
e
ssag
e
leng
th
adap
tiv
ely, wh
ich
can
reduce t
h
e m
e
ssage le
ngt
h at t
h
e
require
d
perform
a
nce.
In t
h
i
s
pa
per
,
a
ho
ri
zo
nt
al
seq
u
ent
i
a
l
VL
SI a
r
ch
itecture for
the nonbi
nary
LDPC
deco
de
r
em
pl
oy
i
n
g
t
h
e AM
C
i
s
devel
o
ped
.
The
desi
g
n
o
f
t
h
e
key
com
pone
nt
s i
n
t
h
i
s
arc
h
i
t
ect
ure, s
u
c
h
as vari
abl
e
n
ode a
n
d
ch
eck
no
d
e
u
p
d
a
te un
its, is op
ti
m
i
zed
b
y
ex
p
l
o
itin
g
th
e
variab
le leng
th
so
rters, wh
ich
can
b
e
d
y
n
a
m
i
cally
con
f
i
g
ure
d
i
n
di
ffe
re
nt
fu
nct
i
onal
uni
t
s
t
o
a
ccom
m
odat
e
vari
abl
e
m
e
ssage l
e
ngt
h
s
. T
h
e
AM
C
i
s
im
pl
em
ent
e
d
by
a l
o
w-c
o
m
p
l
e
xi
t
y
app
r
o
x
i
m
a
ti
on m
e
t
h
o
d
t
o
avoi
d ha
rd
ware
ove
rhe
a
ds an
d per
f
o
rm
ance im
pact
. A
m
a
ppi
n
g
t
a
bl
e base
d ap
pr
oac
h
i
s
pr
op
ose
d
t
o
co
nd
uct
sear
chi
n
g o
p
erat
i
o
ns wi
t
h
l
o
w co
m
p
l
e
xi
ty
. W
e
appl
y
AM
C
t
o
EM
S
t
o
ad
dres
s t
h
e
m
e
m
o
ry
and t
h
r
o
ug
h
put
i
ssu
es cause
d by
t
h
e w
o
r
s
t
case m
e
ssage l
e
ngt
h.
Not
e
t
h
at
AM
C
ca
n
al
so
be em
pl
oy
ed i
n
ot
he
r
dec
odi
ng
up
dat
e
r
u
l
e
s s
u
c
h
as
t
h
e M
i
n-M
a
x al
g
o
ri
t
h
m
.
I
n
a
ddi
t
i
on,
t
h
e pr
o
pose
d
AM
C
can be
gene
ral
l
y
appl
i
e
d t
o
m
o
st
exi
s
t
i
ng
deco
de
r archi
t
ect
u
r
es
(seq
uent
i
a
l
,
p
a
rt
i
a
l
p
a
rallel and
fu
lly p
a
rallel).
2.
R
E
V
I
EW OF
N
O
N
B
INARY
LDPC COD
E
S
A no
nb
in
ary LDPC cod
e
is defin
e
d
b
y
its p
a
rity ch
eck
m
a
trix
(PCM)
H
= [h
ij
],
wh
ich
is an
M × N
sp
arse m
a
trix
with
low
d
e
n
s
ity o
f
n
o
n
z
ero
en
tries. Th
e non
zero
en
tries of
H
t
a
ke
val
u
e
s
fr
om
a Gal
o
i
s
fi
el
d
GF(
2
m
). A leng
th-N vecto
r
x
with
en
tries hav
i
ng
v
a
lu
es fro
m
GF(2
m
) i
s
a code
wo
r
d
i
f
and
onl
y
i
f
Hx
=
0
.
Each entry in t
h
e codeword is
calle
d a sym
b
ol
. A
n
LDPC
c
ode
wi
t
h
PC
M
H
can
b
e
rep
r
esen
ted
b
y
a b
i
partite
gra
p
h cal
l
e
d
T
a
nne
r
gra
p
h ,
whi
c
h c
o
n
s
i
s
t
s
o
f
t
w
o cat
e
g
o
r
i
e
s of
n
o
d
es;
t
h
at
i
s
,
N
vari
a
b
l
e
n
o
d
es
vi
,
1
≤
i
≤
N
and M chec
k
node
s cj ,
1
≤
j
≤
M
.
A va
ri
abl
e
n
ode
vi
i
s
c
o
nnect
e
d
wi
t
h
a
chec
k n
o
d
e c
j
i
f
an
d
onl
y
i
f
hji
i
n
th
e PCM
H
i
s
no
nze
r
o
.
The
dec
odi
ng
pr
ocess
o
f
n
o
n
b
i
n
a
r
y
L
D
P
C
co
des
ope
r
a
t
e
s o
n
t
h
e
m
e
ssages t
h
at
r
e
prese
n
t
t
h
e
p
r
ob
ab
ility d
i
stribu
tio
n
o
f
sy
m
b
o
l
s. A
m
e
ssag
e
is a len
g
t
h-2
m
vector recordi
ng the
2
m
bel
i
e
f i
n
fo
rm
at
ion
of a
sym
bol subj
ect
to channel noi
se, whe
r
e each belief inform
a
tion indicates the probability
of this noisy sym
bol
to
b
e
on
e of
the 2
m
ele
m
ents
in GF(2
m
). Th
e dec
odi
ng
pr
o
cess i
s
essent
i
a
l
l
y
i
t
e
rat
i
v
e
m
e
ssage e
x
c
h
an
ge a
n
d
up
dat
e
bet
w
ee
n t
h
e
chec
k
n
o
d
es a
n
d
va
ri
abl
e
n
odes
i
n
t
h
e
Tan
n
er
g
r
ap
h
r
e
prese
n
t
a
t
i
o
n.
The m
e
ssages
on
t
h
e
Tan
n
er g
r
a
ph
excha
n
ge an
d up
dat
e
i
n
t
w
o
di
rect
i
o
ns. Va
r
i
abl
e
no
de m
e
ssages (
V
NM
s
)
, de
n
o
t
e
d by
q
, pass
fr
om
t
h
e vari
abl
e
no
des t
o
t
h
e check n
o
d
e
s;
and chec
k n
o
d
e m
e
ssages (C
NM
s), de
n
o
t
e
d by
r
, pass f
r
om
t
h
e
ch
eck nod
es t
o
th
e
v
a
riab
le no
d
e
s.
VNMs are in
itialized
with
ch
ann
e
l
messag
e
s,
wh
ich
are th
e inpu
t to
t
h
e
decode
r. T
h
en, VNMs are se
nt to the
check
nodes to
updat
e
the CNMs. T
h
e
ne
w CNMs
are then se
nt back to
t
h
e vari
a
b
l
e
n
ode
s an
d use
d
t
oget
h
e
r
wi
t
h
t
h
e chan
n
e
l
messag
e
s to
up
d
a
te th
e
VNMs. Th
is pro
c
ed
ure is
k
now
n as t
h
e
belief
p
r
op
agatio
n.
There a
r
e m
a
inl
y
t
w
o t
y
pes
of
dec
odi
ng al
go
ri
t
h
m
s
– su
m
-
prod
uct
al
g
o
ri
t
h
m
(SPA
)
and m
i
n-sum
(MS),
of
which the latter one
is a
m
a
the
m
atical appr
oxim
a
tion of SPA where the s
u
m
of product is replaced
b
y
th
e m
a
x
i
mu
m
p
r
o
d
u
c
t term
. Du
e to
its relativ
ely si
mp
le op
eration
,
MS is o
f
ten
ch
o
s
en
for practical
ap
p
lication
s
. Fu
rt
h
e
r redu
ction
in
th
e co
m
p
lex
ity o
f
MS is d
e
sirab
l
e; on
e su
ch
ap
pro
ach is th
e ex
ten
d
e
d
MS
(EM
S
)
alg
o
rith
m
.
3.
M
I
N-
SUM
WITH ADA
PTIV
E
M
E
SSA
G
E C
O
N
T
ROL
In t
h
i
s
sect
i
o
n
,
an ada
p
t
i
v
e m
e
ssage co
nt
r
o
l
m
e
t
hod t
h
at
dy
nam
i
call
y
adju
st
s t
h
e
m
e
ssage l
e
ngt
h o
f
a
sym
bol
du
ri
n
g
t
h
e decodi
n
g
i
t
e
rat
i
on was
devel
o
pe
d. A
t
r
uncat
i
o
n sc
hem
e
i
s
prop
o
s
ed f
o
r t
h
e p
r
op
ose
d
AMC-EMS al
go
rith
m
.
A.
A
d
a
p
tive
Mess
age
C
o
n
t
rol (
A
MC
)
In
t
u
itiv
ely, when
th
e d
i
stributio
n
o
f
b
e
lief in
fo
rm
atio
n
is
m
o
re co
n
c
en
t
r
ated
, a m
e
ssa
g
e
with
a
sm
aller num
ber of e
n
tries m
i
ght be s
u
fficie
n
t to retain
the
sam
e
am
ount of the
be
lief inform
ation. E
x
ploiting
t
h
i
s
fact
, we
p
r
o
p
o
se t
o
a
d
ap
t
i
v
el
y
cont
r
o
l
t
h
e m
e
ssage l
e
ngt
h d
u
ri
ng t
h
e deco
di
n
g
i
t
e
r
a
t
i
on.
Ou
r ap
p
r
oac
h
leads to t
w
o major a
d
vanta
g
e
s
. First, due t
o
the casual
nat
u
re o
f
c
h
a
nnel
n
o
i
s
e, c
h
a
nnel
m
e
ssages f
o
r
d
i
ffere
nt
sym
bol
s m
a
y
have
di
f
f
ere
n
t
st
at
i
s
t
i
c
s. In com
p
ari
s
on
wi
t
h
t
h
e EM
S w
h
i
c
h m
a
i
n
t
a
i
n
s a fi
xed
num
ber o
f
en
tries fo
r all ch
ann
e
l m
e
ssa
g
e
s, th
e
p
r
op
osed
AMC can
lo
wer th
e av
erag
e m
e
ssag
e
l
e
n
g
t
h
wh
ile retain
ing
the sam
e
a
m
o
unt of belief
inform
ation. Second, as
the
decoding proceeds, th
e belief inform
atio
n will
gra
d
ually conc
entrate around
the correct element in th
e case of c
o
nve
r
ge
nce. T
h
us, fe
wer entries are
neede
d
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RES
I
S
SN
:
208
8-8
7
0
8
An
Efficien
t
VLS
I
Arch
itecture fo
r N
o
n
b
i
n
a
ry LDPC Decod
e
r with
Adap
tive …
(
R
. Var
a
t
har
aj
a
n
)
8
i
n
t
h
e m
e
ssage of a sy
m
bol
,
i
.
e., t
h
e m
e
ssage can
be
truncated eve
n
m
o
re. T
h
ese feat
ures are es
senti
a
l for
red
u
ci
n
g
t
h
e c
o
m
put
at
i
onal
c
o
m
p
l
e
xi
t
y
of
d
ecodi
ng
n
o
nbi
nary
L
D
PC
co
des.
The m
a
jor
dec
odi
ng ope
r
ation at a
chec
k node is t
o
refi
ne t
h
e estim
ated
message
of a
sym
bol based
on t
h
e m
e
ssages of
ot
he
r sy
m
bol
s t
h
at
are correlated by t
h
e PCM H.
The el
e
m
entary
check node ope
r
ation
i
n
the Min-Sum
(MS)
decodi
ng
algorithm
[3], [4] can be
expre
ssed as
r=q
1
q
2
(
1
)
Whe
r
e q
1
and q
2
are
t
h
e l
e
ngt
h
2
m
vari
ab
l
e
no
de m
e
ssages, a
n
d t
h
e
basi
c o
p
e
r
at
i
o
n i
s
defi
ne
d a
s
r
α
=
max
(
q1
β
+q2
γ
), whe
r
e r
α
, q1
β
,q2
γ
are th
e en
t
r
ies in
m
e
ssag
e
s
r,
q1
,q
2 corresp
ond
ing
to
α
,
β
,
γ
res
p
ectively.
On
t
h
e o
t
her
h
a
nd
, each
v
a
riab
le nod
e im
p
r
o
v
e
s th
e
fid
e
lity o
f
b
e
lief in
fo
rm
atio
n b
a
sed
on
th
e
receive
d m
e
ssa
ges from
m
u
ltiple chec
k
node
s connecte
d
by
the PCM. T
h
e
ele
m
entary operation at a
va
riable
no
de ca
n
be e
x
press
e
d
as
[3]
,
[4]
q
=
r
1
+
r
2
(
2
)
Wh
ich
su
m
s
u
p
th
e
b
e
lief in
fo
rm
atio
n
asso
ciated
wi
t
h
the
sam
e
finite field ele
m
ent. The
ha
rd
ware
co
m
p
lex
ity in
i
m
p
l
e
m
en
tin
g
(1
) an
d
(2
) is determin
ed
b
y
th
e len
g
t
h
s
of variab
le no
d
e
messag
e
s (VNMs) and
ch
eck
no
d
e
m
e
ssag
e
s
(CNMs). In
t
u
itiv
ely, wh
en
t
h
e d
i
stri
bu
tio
n
o
f
b
e
lief
in
fo
rm
atio
n
is m
o
re co
n
c
en
trated
, a
sh
orter m
e
ssag
e
mig
h
t
b
e
su
fficien
t to
retain
m
o
st o
f
th
e
b
e
l
i
ef in
form
at
io
n.
The ba
sic idea
of AMC is to keep as few e
n
trie
s in a m
e
s
s
age as possi
bl
e with
ou
t in
curring
m
u
ch
i
n
f
o
rm
at
i
on l
o
ss. It
has bee
n
dem
onst
r
at
e
d
t
h
at
m
e
ssage t
r
uncat
i
o
n can be i
m
pl
ement
e
d by
fi
n
d
i
ng t
h
e
min
i
m
a
l n
th
at satisfies
q(
n+
1)
≤
q(
1
)
+
l
n
(1-
ζ
)/
ζ
(
3
)
Whe
r
e
ζ
is the confi
d
ence
factor that determines
the tradeoff
bet
w
een
performance and ope
r
ational
com
p
l
e
xi
t
y
, and
q(
k)
i
n
di
cat
es t
h
e
kt
h e
n
t
r
y
i
n
t
h
e
l
o
g
do
m
a
i
n
rep
r
ese
n
t
a
t
i
on
of t
h
e m
e
ssage
q t
h
at
i
s
so
rt
e
d
in
ord
e
r.
In th
is case, the trun
cation
criteria can
b
e
recast as
q(
n+
1)
≤
ln
(1-
ζ
)/
ζ
(
4
)
whe
r
e t
h
e thres
hol
d is
use
d
t
o
truncate m
e
ssages.
Th
e
op
er
ation
o
f
A
M
C i
n
a
no
nb
in
ar
y LD
PC
dec
ode
r ca
n
be s
u
m
m
ari
zed as f
o
l
l
o
ws.
Initia
liza
t
io
n
• Channel m
e
s
s
age truncation:
ḟ
j
=AMC(f
j
) whe
r
e f
j
is the
receive
d c
h
annel
m
e
ssage.
• V
a
r
i
ab
le
no
de m
e
ssag
e
: q
ij
=
ḟ
j
, where q
ij
i
s
t
h
e
vari
a
b
l
e
n
ode
m
e
ssage fr
om
t
h
e vari
abl
e
n
ode
v
j
to
th
e c
h
eck
no
de c
i
.
Iterations
•
Per
m
ut
at
i
on q
ij
α
q
ij
α
*hij
, where
h
ij
is th
e
n
o
n
z
ero
PCM ele
m
en
t, an
d
th
e m
u
lt
ip
licati
o
n
is con
d
u
c
ted
i
n
GF(
2
m
).
• Check node
update
r
ij
=
j
i
M
k
\
)
(
q
ij
(
5
)
wh
ere M
(
i)\j
is th
e set
of
n
e
ig
hbo
uring
v
a
ri
ab
le nod
es
o
f
th
e
check node c
i
e
x
cluding
the variable node v
j
.
•
Inve
rse
perm
ut
at
i
on r
ij
α
r
ij
α
/hij
, whe
r
e h
ij
i
s
t
h
e no
nze
r
o
PC
M
el
em
ent, an
d t
h
e di
vi
s
i
on i
s
co
n
duct
e
d i
n
GF(
2
m
).
• V
a
r
i
ab
le node up
d
a
te
q
ij
=AMC (
ḟ
j
+
)
\
)
(
kj
r
amc
i
j
N
k
(
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
64
I
J
RES
Vo
l.
4
,
N
o
.
1
,
Mar
c
h 20
15
:
6 – 12
9
wh
ere N(j
)
\i is th
e set o
f
n
e
ig
hbo
ri
n
g
ch
eck
nod
es of th
e v
a
riab
le
n
o
d
e
v
j
e
x
cluding the chec
k
node
c
i
, and
∑
AMC
m
ean
s that th
e AMC is
ap
p
lied to
all th
e in
term
ed
iat
e
resu
lts.
•
Tent
at
i
v
e
dec
odi
ng
c
j
=Max (
ḟ
j
+
)
)
(
kj
r
j
N
k
(
7
)
4.
VLSI
A
R
C
H
I
TECTURE
F
O
R
SEQ
U
EN
TIAL A
M
C-B
A
SED
DE
CO
DER
In th
is section
,
a seq
u
e
n
tial n
onb
in
ary LDPC
dec
ode
r
arc
h
itecture for
the propos
e
d AMC
is
prese
n
t
e
d
.
We
fi
rst
di
scuss t
h
e t
o
p l
e
vel
a
r
chi
t
ect
ure a
n
d
t
h
en
det
a
i
l
t
h
e
desi
g
n
of
sev
e
ral
key
c
o
m
pone
nt
s
su
ch
as th
e v
a
riab
le len
g
t
h
sor
t
er
,
v
a
r
i
ab
le no
d
e
upd
ate un
it (
V
N
U
)
,
and
ch
eck nod
e update u
n
ite (
C
N
U
)
.
Th
e
pr
o
pose
d
AM
C
can be a
ppl
i
e
d t
o
di
ffe
re
nt
no
n
bi
na
ry
LD
PC
dec
odi
ng s
c
hed
u
l
e
s, s
u
c
h
as seq
u
ent
i
a
l
,
part
i
a
l
l
y
p
a
rallel and
fu
lly p
a
rallel architectu
r
es.
A.
Horiz
o
ntal Nonbinar
y
L
D
P
C
Decoder Ar
chitecture
The desi
g
n
of
ho
ri
zo
nt
al
no
n
b
i
n
a
r
y
LDPC
deco
de
r em
pl
oy
i
ng wi
t
h
AM
C
i
s
show
n i
n
fi
g
u
re 1
.
I
n
t
h
i
s
ab
ove a
r
c
h
i
t
ect
ure c
onsi
s
t
s
of
vari
a
b
l
e
no
de
up
dat
e
uni
t
,
c
h
eck
n
o
d
e u
p
d
at
e u
n
i
t
,
chec
k s
u
m
uni
t
,
tentative dec
oding
unit, pe
rmutation,
invers
e pe
rm
utation and RAM. Ran
dom
access m
e
m
o
ry is used t
o
store
t
h
e dat
a
. I
n
t
h
i
s
archi
t
ect
ure
R
A
M
m
e
m
o
ry
i
s
di
vi
ded i
n
t
o
fou
r
di
vi
si
o
n
s.
R
A
M
b
i
s
used
t
o
st
ore t
h
e ch
annel
messag
e
s; RAMa is to
sto
r
e
so
m
e
b
its, wh
i
c
h
are
g
i
v
e
n
to th
e RAMd
t
o
g
e
n
e
rate th
e interm
ed
iate ch
e
c
k
nod
e
m
e
ssages .pe
r
m
u
t
a
t
i
on i
s
used t
o
rear
ra
ng
e t
h
e or
der o
f
i
nput
s
,
w
h
i
c
h
are com
i
ng fr
om
t
h
e vari
abl
e
no
de
up
dat
e
u
n
i
t
.
I
n
verse
pe
rm
ut
ati
on i
s
u
s
ed
t
o
arra
nge
t
h
e
o
r
der
o
f
bi
t
s
,
wh
i
c
h are
f
r
om
t
h
e chec
k
n
ode
up
dat
e
uni
t
.
Te
nt
at
i
v
e
dec
odi
ng i
s
u
s
ed t
o
d
o
t
h
e
i
t
e
rat
i
on
pr
oce
ss. C
h
ec
k s
u
m
u
n
i
t
i
s
use
d
t
o
c
h
eck t
h
e c
h
an
nel
messag
e
s
with
th
e v
a
l
u
es i
n
that u
n
it.
Fi
gu
re
1.
B
l
oc
k
di
ag
ram
of h
o
ri
z
ont
al
no
n
b
i
n
ary
L
D
PC
de
code
r em
pl
oy
i
n
g
wi
t
h
AM
C
At
t
h
e be
gi
n
n
i
ng
of t
h
e
dec
odi
ng
, t
h
e t
r
u
n
cat
ed
ch
ann
e
l
m
e
ssag
e
s are lo
ad
ed
in
to
th
e
m
e
m
o
ry
R
A
M
b
. Te
nt
at
i
v
e dec
odi
ng i
s
con
duct
e
d wi
t
h
t
h
e cha
n
nel
m
e
ssages an
d t
h
e res
u
l
t
s
are s
t
ore
d
i
n
t
h
e m
e
m
o
ry
RAMc. If tenta
tive decoding s
u
ceeds,
decodi
ng te
rm
inates
and R
A
Mc out
puts the final
result. Ot
herwis
e, the
d
ecod
e
r in
itializes th
e i
n
termed
iate ch
eck
no
d
e
m
e
ssag
e
s (ICNMs) i
n
the m
e
m
o
ry RAMd
with th
e ch
ann
e
l
messag
e
s.
After th
e in
itializatio
n
,
t
h
e ch
eck
n
o
d
e
u
p
d
a
te un
it (CNU)
read
s IC
NMs from RAMd
to
perfo
r
m
the check
node update. The
CNMs fr
om
the CNU are inversely perm
uted,
an
d t
h
e
n
pa
ssed t
o
t
h
e va
r
i
abl
e
no
de
u
pdat
e
u
n
i
t
al
o
n
g
wi
t
h
t
h
e ass
o
ci
at
ed
chan
nel
m
e
ssage t
o
per
f
o
r
m
vari
a
b
l
e
n
o
d
e
up
dat
e
.
The
va
ri
abl
e
RAMb
RAMa
perm
I
nvperm
RAMd
CNU
VNU
CSU
R
A
M
c
TDU
I
npu
t
Out
put
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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RES
I
S
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208
8-8
7
0
8
An
Efficien
t
VLS
I
Arch
itecture fo
r N
o
n
b
i
n
a
ry LDPC Decod
e
r with
Adap
tive …
(
R
. Var
a
t
har
aj
a
n
)
10
no
de
up
dat
e
u
n
i
t
(V
NU
) al
s
o
ge
ne
rat
e
s t
h
e
m
e
ssages f
o
r t
h
e t
e
nt
at
i
v
e
deco
di
n
g
uni
t
(TD
U
) t
o
c
o
nd
uct
tentative decoding. If tentative d
ecoding doe
s
not succee
d,
the VNMs fro
m
VNU are then pe
rm
uted and se
nt
back to the C
NU to update
the corresponding IC
NMs, which are the
n
sto
r
ed
in
th
e RAMd
.
After this, th
e
d
ecod
e
r pro
ceed
s
to
an
o
t
h
e
r
variab
le nod
e. Th
is pro
cess con
tin
u
e
s un
til th
e ch
eck
su
m
u
n
it (CSU) d
eci
d
e
s t
o
terminate beca
use
of either
successful dec
o
ding or
reach
i
n
g
the lim
it of iterations.
B.
Vari
able No
d
e
Up
da
te Unit
Fi
gu
re 2.
Im
pl
em
ent
a
t
i
on of
t
h
e VN
U u
n
i
t
The desi
gn o
f
VN
U i
s
sho
w
n
i
n
fi
gure
2. T
h
e fu
nct
i
o
n o
f
VN
U i
s
t
o
com
put
e t
w
o
m
e
ssages,
whe
r
e
the belief i
n
form
ation ass
o
ciated with t
h
e
sam
e
Ga
lo
is field
ele
m
en
t in
th
e t
w
o m
e
ssag
e
s is su
mmed
.
Vari
a
b
l
e
no
de
up
dat
e
u
n
i
t
i
s
use
d
t
o
t
r
u
n
cat
e t
h
e
m
e
ssage l
e
ngt
h
.
I
n
t
h
i
s
uni
t
re
gi
st
er ar
ray
,
m
a
ppi
ng t
a
bl
e,
m
u
l
tip
lex
e
r, sorter, co
m
p
arato
r
are
presen
ted
.
Th
e m
a
p
p
i
ng
tab
l
e is u
tilized
to
su
m
u
p
th
e en
t
r
ies associated
with
th
e sam
e
fin
ite filed
elemen
t. Th
e
fin
a
l resu
lts are
sen
t
to
t
h
e v
a
riab
le leng
th
sorter with th
e asso
ciated
Galo
is field
ele
m
en
ts an
d
AMC is th
en
app
lied
to
t
h
e
outp
u
t
s
o
f
th
e
sorter, starting
from th
e larg
est en
try in
th
e sorter,
wh
en
th
e en
try is smaller th
an
th
e th
resh
o
l
d
,
t
h
e
following e
n
tri
e
s are
discarde
d. T
h
is re
duce
s the
har
d
ware
o
p
er
at
i
ons i
n
t
h
e
V
N
U
.
C.
Check
N
o
de
Upd
a
te
U
n
it
Th
e CNU pro
d
u
ces th
e larg
est ele
m
en
ts a
m
o
n
g
all th
e com
b
in
atio
n
s
from two
in
pu
t messag
e
s. The
pr
o
pose
d
desi
g
n
of C
N
U i
s
sho
w
n i
n
fi
g
u
re
3. Fi
g
u
re sh
o
w
s t
h
at
t
h
e co
m
p
l
e
xi
ty
of C
NU ca
n be re
d
u
ced t
o
additions and insertion ope
r
ations if
the two input m
e
ssages are sorte
d
in
the descending orde
r. He
re
adopt
t
h
i
s
m
e
t
hod i
n
t
h
e C
N
U
desi
g
n
.
The
com
p
l
e
xi
t
y
of
t
h
e C
N
U
depe
n
d
s
o
n
t
w
o
fact
ors
.
Fi
gu
re
3.
Im
pl
em
ent
a
t
i
on
of t
h
e C
NU
u
n
i
t
The fi
rst
fact
o
r
i
s
t
h
e
num
ber o
f
o
u
t
put
s,
whi
c
h det
e
rm
ines
ho
w m
a
ny
i
n
sert
i
o
n o
p
e
rat
i
ons
ar
e
neede
d
.
The se
cond
factor is t
h
e le
ng
th
o
f
t
h
e so
rter,
wh
ich d
e
term
in
es h
o
w m
a
n
y
co
m
p
arison
s an
d
sh
i
f
tin
g
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
089
-48
64
I
J
RES
Vo
l.
4
,
N
o
.
1
,
Mar
c
h 20
15
:
6 – 12
11
ope
rations are
invol
ved i
n
each inse
rti
on
ope
ration. Em
ploying the pr
opose
d
AMC,
the
m
e
ssage lengt
h
decrease
s
thus
becom
e
s s
m
aller during the
iteration.
Th
is redu
ces th
e
n
u
m
b
er o
f
arithmetic o
p
e
ratio
n
s
and
in
sertion
s
. To
ad
dress th
e seco
nd
fact
o
r
, th
e
so
rter is
co
nfigu
r
ed
to
b
e
th
e
sam
e
len
g
t
h
o
f
th
e sh
orter m
e
ssage
th
ereb
y redu
cin
g
th
e co
m
p
lex
ity o
f
each
insertio
n
o
p
e
ra
tio
n. Th
e CNU
also
p
e
rform
s
ad
d
ition
s
i
n
t
h
e Gal
o
is
field.
Howe
ver, additions i
n
the Gal
o
is field are esse
n
tially
b
it-wise
XOR
o
p
e
ration
s
, t
h
u
s
th
e co
m
p
lex
ity is
m
u
ch
lo
wer than
th
e real
num
b
e
r ad
d
ition
s
of
b
e
lief informatio
n
.
D.
V
a
ria
b
le Lengt
h
So
rt
er
Sort
i
ng i
s
an i
m
port
a
nt
o
p
era
t
i
on i
n
t
h
e C
N
U a
nd
V
N
U
.
I
n
t
h
e C
N
U, t
h
e so
rt
er c
h
o
o
se
s n
u
m
b
er o
f
ele
m
en
ts with
th
e larg
est
b
e
lief inform
atio
n
as requ
ir
e
d
for c
h
eck node
update. In the
VNU, truncati
o
n is
carried out and it discards t
h
e
sm
a
ller ele
m
en
ts and t
h
e re
m
a
ining
elem
ents
are s
o
rted
i
n
or
d
e
r
as th
e i
npu
t of
CNU.
The basic sorting
ope
ration is to
insert a new ele
m
ent into a vect
or according to its
m
a
g
n
itude
. The
p
r
op
o
s
ed
v
a
riab
le len
g
t
h
so
rt
er is illu
strate
d
in
figu
re 4, wh
ere on
ly th
e d
a
ta p
a
th
o
f
b
e
lief in
formatio
n
is
sh
own
as it d
e
termin
es th
e sh
ifting
o
p
e
ratio
n. Th
e sorter
consists of
num
b
er of
stages
to acc
omm
odate the
longest m
e
ssa
ge length.
Eac
h
stage c
ontai
ns a com
p
arat
or a
nd a
re
gi
st
er. Eac
h
t
i
m
e
new
bel
i
e
f i
n
f
o
rm
at
i
on
co
m
e
s in
, it is
co
m
p
ared
wit
h
all th
e b
e
lief in
fo
rm
atio
n
in
th
e activ
e stag
es in
p
a
rallel. Th
e con
t
en
t
o
f
t
h
e first
stag
e h
a
v
i
n
g
l
a
rg
er b
e
lief
i
n
fo
rm
atio
n
will
b
e
rep
l
aced b
y
t
h
e
n
e
w
in
pu
t,
wh
ereas its orig
in
al
b
e
lief
in
fo
rm
atio
n
will b
e
sh
ifted
to
th
e righ
t stag
e
an
d
so
o
n
. Sort
er is an
im
p
o
r
tan
t
op
eration
in
ch
eck
nod
e up
d
a
t
e
uni
t
an
d
vari
a
b
l
e
n
ode
u
p
d
a
t
e uni
t
.
S
o
rt
e
r
con
s
i
s
t
s
of
de
code
r,
regi
st
e
r
,
com
p
arat
or
,
m
u
lt
i
p
l
e
xer,
x
o
r
gat
e
.
The
ope
rat
i
on
i
s
paral
l
e
l
pr
oc
ess. Em
pl
oy
i
ng AM
C
,
t
h
e m
e
ssage l
e
n
g
t
h
r
e
duce
s
g
r
ad
ual
l
y
.
Thus
o
n
l
y
t
h
e l
a
st
st
ages are
ena
b
l
e
d.
Thi
s
i
s
c
ont
rol
l
e
d
by
t
h
e l
o
n
g
er
o
n
e
o
f
t
h
e t
w
o m
e
ssages i
n
t
h
e c
h
eck
no
de
up
da
t
e
uni
t
(
C
NU
)
or
v
a
r
i
ab
le
nod
e u
p
d
a
t
e
un
it
(V
NU
).
Fig
u
re
4
.
Variab
le leng
th
sort
er
Th
e m
a
j
o
r op
eratio
n
s
in
th
e so
rt
er a
r
e com
p
arisons and shi
f
tings
. The num
ber of these ope
rations is
m
a
i
n
l
y
det
e
r
m
ined
by
t
h
e l
e
n
g
t
h
of t
h
e m
e
ssage t
o
be
pr
oce
ssed. T
h
e
pr
op
ose
d
AM
C
dy
n
a
m
i
cal
ly
adjust
s t
h
e
messag
e
leng
th du
ri
n
g
th
e iteratio
n
,
t
h
ereb
y
red
u
c
i
n
g th
e
com
p
lex
i
t
y
o
f
sortin
g
.
E.
Searching
Searc
h
i
n
g i
s
a
not
her
m
a
jor
o
p
erat
i
o
n i
n
t
h
e
check
n
o
d
e
up
dat
e
u
n
i
t
(C
N
U
)
an
d
vari
a
b
l
e
n
ode
u
p
d
at
e
u
n
it (VNU).
In th
e VNU, th
e
b
e
lief inform
at
io
n
i
n
on
e m
e
ssage
need
s t
o
s
earch
f
o
r i
t
s
c
o
unt
e
r
pa
rt
i
n
a
n
ot
he
r
m
e
ssage t
o
su
m
s
up t
h
e
bel
i
e
f i
n
fo
rm
at
i
on. The
o
u
t
p
ut
of
th
e so
rter is co
n
s
i
d
ered
v
a
li
d
if and
on
ly if the
cor
r
es
po
n
d
i
n
g
Gal
o
i
s
fi
el
d
el
em
ent
has n
o
t
bee
n
gen
e
r
a
t
e
d p
r
evi
o
usl
y
. A sea
r
chi
n
g
ope
rat
i
o
n
ha
s t
o
be
con
d
u
ct
ed
o
n
t
h
e c
u
r
r
e
n
t
out
put
t
o
c
o
m
p
are i
t
wi
t
h
t
h
e
pre
v
i
o
us
o
u
t
p
ut
s
of
t
h
e s
o
rt
e
r
.T
he
m
a
ppi
ng
in
fo
rm
atio
n
b
e
tween
th
e
Gal
o
is field
elem
e
n
ts and
th
eir i
n
d
e
x
e
s in
a m
e
ssag
e
is m
a
in
tain
ed
b
y
a map
p
i
n
g
t
a
bl
e of
w
o
r
d
s wi
t
h
wo
rd l
e
ngt
h o
f
bi
t
s
. I
n
t
h
e C
N
U
,
n
u
m
b
er of
di
f
f
e
r
ent
el
em
ent
s
need t
o
be ge
nerat
e
d
according to varia
b
le node
m
e
ssages. The output of th
e sorter is considere
d
val
i
d if and onl
y
if the
cor
r
es
po
n
d
i
n
g
Gal
o
i
s
fi
el
d
el
em
ent
has n
o
t
bee
n
gen
e
r
a
t
e
d p
r
evi
o
usl
y
. A sea
r
chi
n
g
ope
rat
i
o
n
ha
s t
o
be
conducted
on t
h
e c
u
rrent
out
put
t
o
com
p
are it with the
prev
i
ous
o
u
t
p
ut
s of
t
h
e s
o
rt
er.
Mapp
ing
tab
l
e is to
records th
e statu
s
of Gal
o
is field
ele
m
en
ts;
it is referre
d
to as statu
s
tab
l
e in
th
e CNU.
Th
e ex
isten
ce
o
f
t
h
e
el
em
ent
needs
t
o
be det
e
rm
i
n
ed;
a
m
a
ppi
n
g
t
a
bl
e wi
t
h
o
n
l
y
num
ber o
f
bi
t
s
can be c
onst
r
uct
e
d. T
h
e
pr
o
pos
ed
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
RES
I
S
SN
:
208
8-8
7
0
8
An
Efficien
t
VLS
I
Arch
itecture fo
r N
o
n
b
i
n
a
ry LDPC Decod
e
r with
Adap
tive …
(
R
. Var
a
t
har
aj
a
n
)
12
map
p
i
ng
-b
ased search
i
n
g
sche
m
e
is n
a
tu
rally lo
w co
m
p
le
x
ity in
co
m
p
a
r
ison
with
d
i
rect search
ing
o
f
the
m
e
ssage, es
pec
i
al
l
y
when t
h
e
m
e
ssage i
s
ver
y
l
o
n
g
.
5.
RESULT AND DIS
C
USSI
ON
Fi
gu
re 5.
H
o
ri
z
ont
al
no
n
b
i
n
a
r
y
LDPC
dec
o
d
e
r
em
pl
oy
i
ng wi
t
h
AM
C
Tab
l
e
1
.
C
o
m
p
arisio
n Resu
lt
Fo
r Variou
s Meth
od
s
Para
m
e
ter
MS
EMS
AMC-EMS
M
e
m
o
ry
Size
3232
32
2143
68
1604
76
Power
1
0.
7
56
m
W
6.
CO
NCL
USI
O
N
A n
e
w
n
onb
inary lo
w
d
e
n
s
ity p
a
rity ch
eck (LDPC)
decoding
arc
h
itecture
is p
r
op
osed to
r
e
d
u
ce
h
a
rdw
a
r
e
op
eratio
n
an
d
p
o
wer
co
n
s
u
m
p
tio
n
.
A
ho
r
i
zo
n
t
al sequ
en
tial VLSI
ar
ch
itectu
r
e for
th
e nonb
in
ar
y
LDPC
dec
o
der
em
pl
oy
s t
h
e AM
C
.
The d
e
s
i
gn o
f
t
h
e key
com
ponent
s i
n
t
h
i
s
archi
t
ect
ure
,
suc
h
as v
a
ri
abl
e
n
o
d
e
and
ch
eck
n
o
d
e
upd
ate un
its, is
op
timized
b
y
ex
p
l
o
itin
g th
e
v
a
riab
le leng
th sorters, wh
ich can
b
e
dy
nam
i
cal
ly
con
f
i
g
ure
d
i
n
di
ffe
rent
fu
nct
i
o
nal
u
n
i
t
s
t
o
ac
com
m
odat
e
va
ri
abl
e
m
e
ssage l
e
ngt
hs.
T
h
e
AM
C
i
s
im
pl
em
ent
e
d by
a l
o
w-c
o
m
p
l
e
xi
t
y
app
r
o
x
i
m
a
ti
on m
e
t
hod t
o
av
oi
d ha
r
d
wa
re o
v
er
hea
d
s an
d pe
rf
or
m
a
nce
im
pact
. Her
e
Ada
p
t
i
v
e
M
e
s
s
age C
o
nt
rol
m
e
t
hod
i
s
us
ed
t
o
r
e
du
ce t
h
e m
e
ssag
e
leng
th
. Th
us th
e
messag
e
l
e
ngt
h
i
s
re
d
u
c
e
d
by
ap
pl
y
i
ng
ada
p
t
i
v
e m
e
ssage c
ont
rol
t
o
t
h
e va
ri
abl
e
n
o
d
e
up
dat
e
uni
t
,
t
h
en
t
h
e t
r
u
n
c
a
t
i
on
i
s
achi
e
ve
d.
Si
m
u
l
a
t
i
on re
sul
t
s an
d sy
nt
hesi
s
re
po
rt
s are
ana
l
y
zed.
REFERE
NC
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