TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5448 ~ 54
5
7
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.581
4
5448
Re
cei
v
ed
Jan
uary 18, 201
4
;
Revi
sed Fe
brua
ry 26, 20
14; Accepted
March 22, 20
14
QC LDPC Code
s for MIMO and Cooperative Networks
using Two Way Normalized Min-Sum Decoding
Waheed Ullah*
1
,
Yang Fengfan
2
, Abid
Yah
y
a
3
1,2
Colleg
e
of Electronics a
nd Informatio
n
Eng
i
ne
erin
g
Nanj
in
g Univ
er
sit
y
of Aero
na
u
t
ics and Astron
autics Na
nji
ng
210
01
6, Chin
a
2
Universiti Ma
l
a
y
s
ia Per
lis (U
niMAP), Mala
ysia
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: uet_
w
a
h
e
e
d
@
yah
oo.com
A
b
st
r
a
ct
T
h
is pap
er is based o
n
the magn
itud
e overe
s
timati
on corre
ction of
the
var
i
abl
e mess
ag
e by
us
i
n
g
tw
o nor
mal
i
z
e
d
factors i
n
eac
h iter
ation
for
LDPC
min-s
u
m
dec
odi
ng
al
gorith
m
. T
h
e v
a
ria
b
le
messa
ge
i
s
mo
difi
ed w
i
th a nor
ma
li
z
e
d
factor w
hen the
r
e is a sig
n
ch
ang
e an
d w
i
th anoth
e
r nor
ma
li
z
e
d factor w
h
e
n
there is no si
g
n
chan
ge d
u
rin
g
any tw
o con
s
ecutive it
er
ati
ons. T
h
is pa
pe
r incorp
orates
QC LDPC cod
e
s
usin
g this new
deco
d
in
g al
gor
ithm for flat fad
i
ng
mult
i
p
l
e
inp
u
t multi
p
l
e
out
put (MIMO) ch
ann
el an
d sin
g
l
e
relay co
oper
ati
v
e commun
i
ca
tion netw
o
rks for impr
ovi
ng
the bit error p
e
rform
anc
e. MIMO flat
fadin
g
chan
nel
is us
e
d
w
i
th
z
e
ro for
c
ing (Z
F
)
spati
a
l d
e
co
d
i
n
g
for
nois
e
su
ppres
sion. T
he
perfo
rma
n
ce
is gre
a
t
ly
enh
anc
ed
by u
s
ing th
e n
e
w
mi
n-su
m
alg
o
ri
thm for
me
d
i
u
m
and
short
le
ngth C
o
o
perati
v
e co
mmu
n
icat
i
o
n
netw
o
rk and M
I
MO LDPC cod
e
s.
Ke
y
w
ords
:
co
ded MIMO, QC LDPC, mi
n-su
m LDP
C
, chan
nel co
din
g
, coo
perativ
e co
mmunic
a
tion
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. Introduction
With the a
d
vent of wirele
ss commu
nica
tion, efforts h
a
ve alway
s
b
een ma
de to
transmit
maximum data with maximum reliability. To
achiev
e the maximum data rate, MIMO wirel
e
ss
system
s hav
e gained p
o
p
u
larity as its theoreti
c
-ca
p
a
city increa
se linearly wit
h
increa
se in
th
e
numbe
r of a
n
t
ennae [1
-3].
The e
rro
r p
e
rforman
c
e
of
MIMO syste
m
ca
n be
gre
a
tly improved
by
error
co
rrecti
on code
s. Co
operative co
mmuni
cation
[4
] is on
e of
the faste
s
t g
r
owi
ng a
r
e
a
s of
resea
r
ch, an
d it is li
kely to be a
key e
nablin
g techn
o
logy for
efficient
spe
c
tru
m
use in futu
re.
The
key id
ea
in u
s
er-coo
pe
ration i
s
th
at
of re
sou
r
ce-sharin
g a
m
ong
multiple
nod
es i
n
a
net
wo
rk.
Low
De
nsity
Parity Ch
e
c
k (L
DPC)
code
s a
r
e
o
n
e
of the m
o
st po
we
rful
error
co
rrecti
o
n
techni
que
s, fi
rst
pro
p
o
s
ed
by Galla
ger [
5
] an
d
we
re
reinvente
d
by
Ma
ckay & Neal [6, 7]
. L
D
PC
have ta
ken
consi
derable
a
ttention rece
ntly due
to
it
s p
o
werful
error corre
c
ting
ca
pabilitie
s
and
their nea
r Sh
anno
n limit perform
an
ce [6, 8] with b
e
lief prop
aga
tion (BP) decoding alg
o
rith
m.
The Belief Propagation (B
P) or the Sum-Product
decodi
ng algorit
h
m (SPA)
performs well
but
at
the co
st of high hardware,
long pro
c
e
s
sing ti
me an
d has de
pen
den
cy on the noise varia
n
c
e.
The L
D
PC
decodin
g
alg
o
rithm
whi
c
h
offers mu
ch
lowe
r h
a
rdware
com
p
lexity at the cost
of
perfo
rman
ce
degradatio
n is the min-su
m algorit
hm
(MSA)[9, 10] It is indepe
ndent of noi
se
varian
ce a
s
well. Efforts h
a
ve been m
a
de to ac
hieve
optimum tra
deoff betwe
e
n
com
p
lexity and
bit error pe
rforma
nce (BE
R
)
of L
D
PC
decode
rs.
Se
veral a
pproa
che
s
attempted to
kee
p
t
h
e
performanc
e
c
l
os
e to SPA with less
hardware
c
o
mplexity
for prac
tic
a
l applications
[11-14]
Different m
e
thods
are u
s
ed to
bri
n
g
the si
m
p
lified form
of
the algo
rith
m clo
s
e i
n
perfo
rman
ce
to the o
r
igin
al
BP or sum p
r
odu
ct al
go
rithm. The
mo
st popul
ar
app
roa
c
he
s
are t
he
norm
a
lized
min-sum
(No
r
mali
zed MS
A) and th
e o
ffs
et min-su
m (Offset M
SA) algo
rith
ms. To
redu
ce the
magnitud
e
of overestimati
on, the
che
c
k messa
ge i
s
modified d
u
ring the iteration
process which bri
n
gs th
ese min-sum al
gorithm
s
close in
per
formance to the st
andard SPA
and
make
s them
suitabl
e for p
r
acti
cal ap
plications
a
nd h
a
rd
wa
re impl
ementation [1
5-17]. Mod
e
rate
length mi
n-su
m LDP
C
d
e
coding
algo
rith
m [18, 19
], d
ue to its
re
du
ced
co
mplexi
ty, has gai
ne
d
popul
arity in the wid
e
are
a
of wirele
ss co
mmuni
cation.
Figure 1 bel
ow sho
w
s th
e typical flow for LDP
C
e
n
co
ded MIM
O
tran
smitte
d data an
d L
D
PC
decode
d data
output after correctio
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
QC L
D
PC Co
des for MIM
O
and Co
ope
ra
tive Net
w
orks using T
w
o
Way…
(Wahee
d
Ullah
)
5449
Figure 1. MIMO-L
DPC in
Typical Com
m
unication S
y
stem
In this pap
er,
the ne
w min-sum d
e
codin
g
algo
rithm[1
9
] has b
e
en
e
m
bedd
ed
with MIMO.
Zero
forcin
g
(ZF)
sp
atial d
e
co
ding
is u
s
ed fo
r
sup
p
re
ssi
ng th
e
noi
s
e i
n
flat fa
di
ng a
n
d
additi
ve
white G
aussi
an (A
WG) MI
MO ch
ann
els. The pro
p
o
s
ed algo
rithm
offers b
e
tter
BER perfo
r
m
ance
and can be e
a
sily implem
e
n
ted in hardware.
The
pro
p
o
s
e
d
L
D
PC met
hod [1
9] with
MIMO
co
rre
c
ts
overestim
a
ted m
a
gnitu
de of th
e
variable
me
ssag
e d
u
rin
g
two
con
s
e
c
ut
ive iteration
s
.
Wh
en th
e
si
gns of the
prese
n
t an
d th
e
previou
s
me
ssage
are th
e same th
en
the p
r
e
s
ent
me
ssa
ge i
s
scale
d
a
n
d
upd
a
ted
wit
h
a
norm
a
lized fa
ctor, ho
weve
r when the
sig
n
s a
r
e differe
nt then the two me
ssage
s are add
ed
and
scaled
with a norm
a
lized fa
ctor, which i
s
di
fferent from the firs
t s
c
aling fac
t
or.
2. Introduction
To LDPC
2.1.
Repr
esen
ta
tion of LDP
C
Code
s
LDPC
code
s are a
type
of linea
r
blo
c
k
cod
e
s an
d
are
rep
r
e
s
e
n
ted by
pa
rity che
c
k
matrix. LDCP
code
ca
n be
denoted in
g
eneral a
s
(
N,
d
v
, d
c
); where
N
is the le
n
g
th of the co
de
equal to the n
u
mbe
r
of colu
mns in the pa
rity check ma
trix,
d
c
is the
numbe
r of on
es in a colum
n
of the matrix;
d
v
is the n
u
m
ber
of one
s in a ro
w of t
he matrix. L
D
PC
cod
e
s
can be
reg
u
lar or
irre
gula
r
. If the num
be
r of
one
s in
ea
ch
ro
w a
nd
co
l
u
mn of
a p
a
ri
ty che
c
k matrix are th
e
sa
me
then it is a
re
gular
co
de a
nd othe
rwi
s
e
it is calle
d irregula
r
code.
Followi
ng i
s
an exampl
e o
f
an
irre
gula
r
LDP
C
co
de.
12
7
1
2
3
11
1
0
1
0
0
01
1
1
01
0
10
1
1
0
0
1
T
xx
x
c
c
c
H
Hx
0
(
1
)
The co
de is v
a
lid only if H. cod
e
T
=
0.
The spa
r
se parity
che
ck matrix
is be
st
r
epresente
d
by a bi
part
ite
grap
hs know as
Tanne
r g
r
ap
h
s
[20]. Each
row of the p
a
ri
ty check
matrix repre
s
e
n
ts
the variabl
e
n
ode an
d ea
ch
colum
n
repre
s
ent
s the
ch
eck no
de.
Th
e 1s in e
a
ch
row or col
u
m
n
re
pre
s
e
n
ts
the co
nne
ctivity
betwe
en vari
able a
nd che
ck
nod
es. Th
e set of bit n
o
des
co
nne
cti
ng to che
ck
n
ode
m
is
d
eno
te
d
by
N
(
m
)=
{
n|h
mn
=1} and the
set of che
ck
node
s
con
n
e
c
ting to bit n
o
d
e
n
is
b
y
M
(
m
)={
m|h
mn
=1}
.
A
typical Tann
e
r
gra
ph is
sho
w
n in the Fig
u
re 2.
Thi
s
graph is for
(3, 7) irregul
ar L
D
PC code.
Figure 2. Tan
ner G
r
ap
h Re
pre
s
ent
atio
n of Parity Che
ck M
a
trix
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5448 – 54
57
5450
In algebraic f
o
rm, it can be
demon
strate
d as:
2.2.
T
w
o
Way
Normalized Min
-
sum Algori
t
hm (MSA)
Let
C
= {
c
1
, c
2
…..c
n
} be
a
tran
smitted
cod
e
ove
r
a
n
additive
whit
e Ga
ussia
n
(AWG
N)
cha
nnel.
Y=
C+
n
; where
n
is
an AWGN.
No
w L
D
PC
min-sum d
e
coding[1
9
] ca
n be
stat
ed
in the follo
wing
step
s f
o
r a
parity
che
c
k mat
r
ix
H
mn
; where
m
is the numbe
r of rows an
d
n
is the num
ber of column
s.
Step
1
: Initializ
a
tion: Set
R
n
=
Y
as initial
log
li
kelihood
ratio
(L
LR
) a
nd fo
r ea
ch
(
m, n
)
∈
{(
m , n
)|
h
mn
=1}
V
0
mn
=R
n
(1)
Set
i
=
0
to
I
ma
x
,
I
ma
x
is the maximum nu
mber of iterati
o
n
Step
2
: Ho
rizontal pro
c
e
ss: check no
de
update:
For
m
=
0
to
M
-1, update
C
i
mn
for each
n
N
(
m
)
''
'
'
'
'
11
()
()
()
.
m
i
n
|
|
ii
i
mn
mn
m
n
nN
m
nN
m
nn
nn
Cs
i
g
n
V
V
(2)
Step
3
: vertical pro
c
e
ss: bi
t node upd
ate
For
n
=
0
to
N
-1
, update
~
()
ii
Rn
n
m
n
mM
n
RC
(3)
No
w
upd
ating
V
i
mn
for
each
m
∈
M
(
n
):
,
it
m
p
i
i
mn
n
m
n
VR
C
(4)
The
sig
n
s of
the
pre
s
e
n
t me
ssage
,
it
m
p
mn
V
and th
e p
r
ev
ious me
ssag
e
1
i
mn
V
are
then
comp
ared.
If s
i
gn (
,
it
m
p
mn
V
)= =si
gn(
1
i
mn
V
)
Then up
date
the pre
s
ent m
e
ssag
e as:
,
1
()
ii
t
m
p
mn
mn
Vs
f
V
(5)
E
l
se if
sign (
,
it
m
p
mn
V
)
≠
sig
n
(
1
i
mn
V
)
Then up
date
the pre
s
ent m
e
ssag
e as:
,1
2
()
ii
t
m
p
i
mn
mn
mn
Vs
f
V
V
(6)
The scali
n
g
factors
sf
1
and
sf
2
are
cho
s
en
su
ch that they can b
e
con
v
eniently
impleme
n
ted
in hard
w
a
r
e
and at the
same time
provide g
ood
approximation to the error
perfo
rman
ce.
Now if the sign
s are different
then t
he ch
ang
e in magnitud
e
is large a
n
d is
modified
with
a
small
e
r fa
ctor to redu
ce the
ov
erest
i
mation effe
ct
. The
scalin
g
facto
r
s u
s
ed
for
the s
i
mulations
in this
paper are
sf
1
=0.5
and
sf
2
=
0
.25.
11
2
3
5
22
3
4
6
31
3
4
7
:0
:0
:0
cx
x
x
x
cx
x
x
x
cx
x
x
x
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
QC L
D
PC Co
des for MIM
O
and Co
ope
ra
tive Net
w
orks using T
w
o
Way…
(Wahee
d Ullah
)
5451
No
w Equatio
n (5) a
nd Equ
a
tion(6
)
can b
e
re-written a
s
:
,
0.
5
(
)
ii
t
m
p
mn
m
n
VV
(5a)
,1
0.
25
(
)
ii
t
m
p
i
m
n
mn
mn
VV
V
(6a)
Equation
(5a
)
an
d Equ
a
tion (6a
)
give
s g
ood
pe
rforma
nce a
c
h
i
evement
whi
l
e the
co
st f
o
r
hard
w
a
r
e i
s
very low. Thi
s
bri
n
g
s
further imp
r
ove
m
ent to the MSA in both lower
and u
ppe
r
regio
n
of SNR by usin
g two scaling fa
ctors.
The su
mmati
on in Equatio
n (6a
)
with the pr
eviou
s
m
e
ssag
e doe
s not affect sig
n
ificantl
y
the de
codi
ng
perfo
rman
ce.
Thu
s
the
su
mmation
with
the p
r
eviou
s
messag
e
can
be m
odified
as
sho
w
n b
e
low
in Equation (6b) which re
d
u
ce
s t
he time
latency and
still offer better re
sult
s.
1
0.
2
5
(
)
ii
mn
mn
VV
(
6
b
)
Step 4
: Ha
rd
De
cisi
on:
0,
f
o
r
R
0
ˆ
1,
f
o
r
R
0
n
n
n
c
(
7
)
Step 5
: Stop con
d
ition: If the parity ch
e
ck e
quatio
n is satisfied.
12
ˆ
ˆˆ
.
(
.
......
.....
)
0
T
n
Hc
c
c
(8)
O
r
ma
xi
m
u
m i
t
e
r
a
t
i
o
n (
I
ma
x
) is re
ached
then terminat
e the
de
codi
ng o
r
oth
e
rwi
s
e
i
=
i
+1
and go b
a
ck to step 2.
3.
MIMO Communication
Net
w
o
r
k
The multipl
e
-i
nput an
d mul
t
iple-outp
u
t (MIMO) i
s
the
use
of multi
p
le ante
nna
s at both
the tran
smitte
r and
re
ceive
r
to imp
r
ove
comm
uni
cati
on pe
rform
a
n
c
e. It is o
ne
of seve
ral forms
of sma
r
t a
n
tenna te
ch
nology. MIM
O
tech
nolo
g
y has
attra
c
ted attentio
n in wi
rel
e
ss
comm
uni
cati
ons, be
ca
use it offers si
gnifica
nt
increases in
dat
a throu
ghp
ut and lin
k ra
nge
without a
dditi
onal b
and
wi
dth or t
r
an
smit power
. B
e
ca
use of th
ese
prope
rtie
s, MIMO i
s
a
n
important part of mode
rn wirel
e
ss
communi
ca
tion standards
such
as IEEE 802.11n (Wifi),
WiMAX etc
.
Con
s
id
er a
fl
at
fading
MIMO system model with
N
t
tran
smit an
d
N
r
re
ceive
antenn
as
.The re
ceived
signal ve
ctor at each in
sta
n
t of time is given by:
R=
H
x
+n
(9)
Whe
r
e
R
i
s
ithe re
ceived
signal , H is th
e cha
nnel m
a
trix (
N
t
xN
r
)
and
n
is the
additive
white
Gau
s
si
an n
o
ise
(AWG
N).
MIMO
commu
nication
system
for th
e Eq
ua
tion (9) is sh
ow i
n
Figure (3
).
Whe
r
e
r
is
1
r
N
received
signal
vector,
H
is a
rt
NN
channel
re
spon
se matrix,
x
is a
1
t
N
tran
smitted signal
vector an
d
n
is the additive white
Gaussia
n
noise
(AWG
N).Typical L
D
PC cod
ed MIMO co
mmuni
cation
system i
s
sh
o
w
in Figu
re 1.
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Figure 3. MIMO Com
m
un
ication Syste
m
Con
s
id
er th
a
t
we
have
a
tran
smi
ssi
on
se
que
nce, for exa
m
ple.
12
,
n
x
xx
. In normal
transmissio
n, we
will be
sendin
g
1
x
in the firs
t time s
l
ot,
2
x
in the seco
nd time sl
ot and
so o
n
.
Ho
wever, a
s
we n
o
w h
a
ve
2 tran
smit a
n
tenna
e,
we
may gro
up th
e symbol
s int
o
gro
u
p
s
of two.
In the firs
t time
s
l
ot, send
1
x
and
2
x
from
the first an
d
se
con
d
a
n
ten
na. In
se
co
n
d
time
slot,
sen
d
3
x
and
4
x
fro
m
the first a
n
d
second
ant
enna,
se
nd
5
x
and
3
x
in the thi
r
d time
slot a
nd
s
o
on. Notice that as we a
r
e
groupi
ng two symbol
s an
d sen
d
ing th
em in one time slot, we
need
only
/2
n
time slo
t
s to
compl
e
te the tran
smi
ssi
on
– data
rate is dou
ble
d
! This form
s the
simple
explanation
o
f
a prob
able
MIMO tran
smissi
on
sche
me with 2 t
r
a
n
smit ante
n
n
a
s a
nd 2
re
ceive
antenn
as. T
he two tran
smitted sym
bols inte
rf
ered with e
a
ch other
call
ed inter
ch
a
nnel
interferen
ce (ICI).
The
ch
annel
is
flat
fading
– In
simple te
rm
s, it mea
n
s th
at the m
u
ltip
ath
cha
nnel
ha
s
only on
e tap.
So, the
conv
olution
ope
rat
i
on
red
u
ce
s to a
sim
p
le
m
u
ltiplication
a
nd
the chan
nel e
x
perien
c
e by each tran
smit
antenna
is in
depe
ndent from the cha
n
n
e
l experie
nce
d
by other tran
smit anten
na
s. For th
e
it
h
tran
smit ante
nna to
j
th
re
ceive a
n
tenn
a, each
transmitted
symbol get
s m
u
ltiplied by
a
rand
omly varying co
mplex
numb
e
r
j
i
h
.As the chan
nel
unde
r co
nsi
d
eration i
s
a Rayleigh chan
nel given by:
2
22
ex
p
[
(
]
fo
r
0
()
2
0
o
t
h
e
rw
i
s
e
rr
r
pr
(10)
Whe
r
e
r
is th
e envelope a
m
plitude of the re
ceived signal, and
2
2
is the pre-d
e
te
ction mea
n
power of the
multipath si
gnal. The re
al and imagi
nary pa
rts of
j
i
h
are Gau
s
sia
n
distrib
u
ted
having m
ean
0
ji
h
an
d va
ria
n
ce
2
1/
2
ji
h
. The
ch
annel
expe
ri
enced
betwe
en e
a
ch
transmit to the receive ant
enna i
s
inde
p
ende
nt and randomly vary
ing in time.
On the
re
cei
v
e antenn
a, the noi
se
n
has the
Ga
ussian
proba
bility density fu
nctio
n
with:
2
2
()
2
2
1
()
e
x
p
2
n
pn
(
1
0
a
)
Whe
r
e
0
and
2
0
2
N
.
The
ch
ann
el
j
i
h
is kn
own at the receiver.
4.
Zero Forcing (ZF) Equalizer
For a
2x2 MI
MO commu
ni
cation
syste
m
, in the first time sl
ot, the receive
d
si
gn
al on th
e
first re
ceive a
n
tenna i
s
,
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046
QC L
D
PC Co
des for MIM
O
and Co
ope
ra
tive Net
w
orks using T
w
o
Way…
(Wahee
d Ullah
)
5453
1
1
1
1
1
12
2
1
11
12
1
1
[,
]
x
yh
x
h
x
n
h
h
n
x
(
1
1
)
The re
ceive
d
sign
al on the
se
con
d
re
cei
v
e antenna i
s
:
1
2
2
1
1
22
2
2
21
22
2
2
[,
]
x
yh
x
h
x
n
h
h
n
x
(
1
2
)
Whe
r
e,
12
,
yy
, are the re
cei
v
ed symbol o
n
the first and
second a
n
te
nna re
sp
ectiv
e
ly,
11
h
is the ch
ann
e
l
from 1
st
tran
smit antenn
a to 1
st
receive
antenn
a,
12
h
is the ch
ann
e
l
from 2
nd
transmit antenn
a to 1
st
receive
antenn
a,
21
h
is the ch
ann
e
l
from 1
st
tran
smit antenn
a to 2
nd
receive
antenna,
22
h
is the ch
ann
e
l
from 2
nd
transmit antenn
a to 2
nd
receive
antenna,
12
,
x
x
are the tra
n
smitted symbo
l
s and
12
,
nn
is the noi
se o
n
1
st
and 2
nd
receive a
n
ten
nas.
We a
s
sume
that the re
ce
iver kn
ows
11
h
,
12
h
,
21
h
and
22
h
.
The receiv
e
r
al
so
kno
w
s
12
&
yy
. T
h
e
u
n
k
n
ow
ns
ar
e
12
&
x
x
. So we
have
two
equ
ation
s
a
nd t
w
o
u
n
kn
owns.
Fo
r
conve
n
ien
c
e,
the above eq
uation can be
r
epresented i
n
matrix notation as follo
ws:
11
1
1
2
1
1
22
1
2
2
2
2
y
hh
x
n
y
hh
x
n
(
1
3
)
Equivalently,
rH
x
n
(
1
4
)
To solve for
x
, we
kno
w
th
a
t
we ne
ed to
find a matrix
W
whi
c
h sat
i
sf
ie
s
WH
I
. The
Zero F
o
rcing
(ZF) lin
ear d
e
t
ector for m
e
eting this con
s
traint i
s
given by,
1
()
HH
WH
H
H
(
1
5
)
This mat
r
ix is also kno
w
n a
s
the pseud
o inverse for a
gene
ral
m x
n
matrix.
5.
QC LDPC Coded 2x2 MIMO using Tw
o
Wa
y
Nor
m
alized MSA
The pe
rform
a
nce of the MI
MO syste
m
can
be en
han
ced by u
s
ing
state of the a
r
t error
corre
c
tion te
chniqu
e like Q
uasi
cycli
c
(Q
C) L
D
PC
cod
e
s a
s
sh
own in Figure 4.
Figure 4. 2x2 LDPC
Cod
e
d
MIMO
System with ZF Equalizer
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57
5454
The system
has
be
en si
mulated
fo
r
a Q
C
LD
PC pa
r
i
ty c
h
e
ck ma
tr
ix
261
522
H
su
ch
that code
length
=
52
2, and e
a
ch su
b-matrix
si
ze
is 87. F
o
llo
wing Q
C
pa
rity che
c
k matrix
(H) is de
sig
ned
su
ch that E2
1= E32
=
0 a
n
d
all others a
r
e compo
s
e
d
of circularly
shifted id
e
n
tity sub- mat
r
ices,
each of size 87.
H = [E11 E12
E13 E14 E15 E16
E21 E22 E23 E24 E25 E26
E31 E32 E33 E34 E35 E36]
And the cha
n
nel matrix for
flat fading MIMO is: C = [ h11 h1
2 ; h21 h22 ]
Figure 6. Performa
nce Co
mpari
s
o
n
of LDPC
Code
d MIMO and Si
mple MIMO
with ZF
The g
r
aph i
n
Figu
re 6
sho
w
s impro
v
ed perfo
rm
ance for L
D
PC co
ded
MIMO in
comp
ari
s
o
n
to simpl
e
MI
MO with Z
F
decodin
g
. Th
e LDP
C
de
coding
algo
rithm used a
r
e
the
stand
ard SP
A and the new improve
d
MSA [19]
me
ntioned in se
ction 2.2 whi
c
h sh
ows bet
ter
perfo
rman
ce
for medium a
nd sho
r
t lengt
h code
s an
d are suitable f
o
r the MIMO cha
nnel
s to split
data into sma
ll packets a
n
d
transmit ind
e
pend
ently.
6.
Cooperative Commun
i
cation Using
Joint Lay
e
red Decoding
We
con
s
id
er a one
relay
coo
perative comm
uni
cati
on sy
stems [
21, 22] a
s
shown in
Figure 7.
The
typical
dista
n
c
e fo
r th
e
cod
ed
cop
per
ative sy
stem
are
mentione
d in
[23]. He
re the
distan
ce
s bet
wee
n
so
urce,
relay and
de
stination a
r
e
su
ch that the
distan
ce b
e
twee
n so
urce
to
destin
a
tion is
norm
a
lie
zed t
o
1 as sho
w
n
in equation b
e
low.
10
2
11
1
,
,
a
n
d
(1
)
dd
Figure 7. Single Rel
a
y Co
oper
ative Co
mmuni
cation
System
0
2
4
6
8
10
12
14
16
18
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
S
NR(
d
B
)
BE
R
2*
2 M
I
M
O
-
L
D
P
C
S
i
m
u
l
a
t
i
on
L
D
PC
(
N
M
SA)
M
I
M
O
L
D
PC
(
SPA
)
S
i
mp
l
e
MI
M
O
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
QC L
D
PC Co
des for MIM
O
and Co
ope
ra
tive Net
w
orks using T
w
o
Way…
(Wahee
d Ullah
)
5455
Whe
r
e
is the
path lo
ss a
n
d
is u
s
ually taken in the
ran
ge 2
~
3.The
g
r
aph i
n
the fi
gure
s
9 & 1
0
are
sim
u
lated
for i
deal
a
n
d
non
-id
eal
co
operativ
e
co
mmuni
cation.
The
di
stan
ce
between
R-D in
a non
-ide
al coope
ration i
s
su
ch that it receiv
e
s
4d
b
more
power.
The di
stan
ce
betwe
en R-D is
su
ch that it receive
s
po
wer 1d
b mo
re
than S-D
for
both ideal
an
d non
-ide
al situation. A pa
rity
che
c
k mat
r
ix
f
o
r t
he
so
ur
c
e
en
cod
e
r
is
12
&
H
H
is
sele
cted
with 2
50
ro
ws a
nd 5
00
co
lumns
s
u
c
h
that:
12
250
500
250
500
250
500
250
500
&
SR
HH
I
H
H
I
(
1
6
)
Whe
r
e
12
H
H
and
2
H
is the row perm
u
tation matix obtained from
1
H
and both are
regul
ar m
a
tri
c
e
s
. The
nu
mber
of one
s in
ro
w an
d
colu
mns are
equal i
n
bot
h the matri
c
es
4 a
n
d
8
vc
dd
.
I
is the ident
itity matrix ,
R
H
is the i
r
reg
u
lar sy
stema
t
ic pai
rty che
ck
matrix at the
relay
encod
er ,
S
H
is the i
r
regula
r
syste
m
atic p
a
ir
ty
che
c
k m
a
tri
x
at theso
u
rce
encode
r. The
final matrix
at destinatio
n
D is
D
H
and is
given by:
1
250
250
250
250
2
2
50
2
5
0
2
5
0
25
0
0
0
D
HI
H
HI
(
1
7
)
Figure 8. Typical Way for Showi
ng the Di
stan
ce
s between Sou
r
ces(S), Relay(R)
and
De
stination (D)
A decod
e /re
-
en
cod
e
/forward st
rategy
has b
een ad
opted at the relay ch
ann
el
. At th
e
relay a me
ssage is d
e
o
c
oded a
nd the
n
re-en
c
ode
the parity onl
y and then transmit the p
a
i
rty
bits to the d
e
s
tination
wh
e
r
e it is
com
b
i
ned
with
the
messag
e fro
m
the so
urce
s such that th
e
r
p
is se
nt by relay R:
s
r
code
s
p
p
(
1
8
)
This code is decod
ed by
parity che
c
k matrix
in Equation (17)
by LDPC no
rmali
z
ed
layered
min
-
sum
de
codi
n
g
metho
d
.
The
cha
nnel
s a
r
e
sim
u
lated for the
rayleig
h
fa
ding
coeffici
ents such
that:
yh
x
n
(19)
Whe
r
e
h
is the rayleigh fa
ding chan
nel
coefficie
n
t,
n
is the additiv
e
white gu
assian
noise,
x
is
th
e in
fo
r
m
a
t
ion
b
i
ts
. T
h
e
s
i
mu
la
tio
n
re
su
lts
sh
ow
s
the
c
o
mpa
r
is
on
fo
r
th
e
c
o
ded
coo
peration
with ideal, n
on-id
eal an
d
non-coo
p
e
r
ative commu
nicatio
n
(dei
rect
sou
r
ce to
destin
a
tion) u
nder the
sam
e
chan
nel
co
ndition
s.
Thi
s
sam
e
cod
e
h
a
s
bee
n si
mu
lated u
nde
r t
he
same channel conditions f
o
r la
ye
red
mi
n sum d
e
codi
ng alg
o
rithm
whi
c
h h
a
s fa
st convergen
ce
and better
re
sults a
s
this i
s
free of noi
se varian
ce. So prio
r ch
ann
el informatio
n
is not requi
re
d to
initialize the i
n
formatio
n bits. The graph
in Figur
e 10 shows the pe
rforma
nce com
pari
s
on fo
r this
pratical type of LDPC de
coder.
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57
5456
Figure 9. BER Perfo
r
man
c
e for O
ne Relay-coop
erat
ion
and Non
-
coo
peration Systems
for Cod
e
Length
=
7
50, rate (at S and
R)=2/3, iterat
ions=1
0
Figure 10. BER Perfo
r
man
c
e for O
ne Relay-c
oop
erat
ion
and Non
-
coo
peration Systems
for
Cod
e
Len
gth
=
75
0, rate (at
S and R)=2/
3
, iterations=10
7.
Conclusion
In this p
ape
r, two
way no
rmalize
d
min
-
sum
and j
o
int
layere
d L
D
P
C
de
co
ding
a
l
gorithm
has
bee
n sim
u
lated for
MIMO and
coop
erative comm
uni
cation respectively a
s
it is fre
e
of noi
se
varian
ce
and
give si
gnifica
ntly low
co
mp
lexity. For MI
MO
comm
uni
cation,
an i
m
proved
two
wa
y
norm
a
lized
min-sum de
coding h
a
s b
een u
s
ed to
sho
w
the p
e
rform
a
n
c
e
comp
ari
s
o
n
with
stand
ard L
D
PC sum
-
pro
duct algo
rith
m. T
he bit
error graph
clea
rly sho
w
s an improved
perfo
rman
ce
for the
ne
w
schem
e. A joi
n
t layered
min
-
sum
LDP
C
d
e
co
ding
app
roach i
s
u
s
e
d
fo
r
coo
perative communi
catio
n
for a
c
hievi
ng fast
d
e
co
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Ne
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