Indonesian J
ournal of Ele
c
trical Engin
eering and
Computer Sci
e
nce
Vol. 2, No. 3,
Jun
e
201
6, pp. 636 ~ 64
6
DOI: 10.115
9
1
/ijeecs.v2.i3.pp63
6-6
4
6
636
Re
cei
v
ed Fe
brua
ry 25, 20
16; Re
vised
Ap
ril 29, 201
6; Acce
pted
May 10, 20
16
Reducing Computational Complexity and Enhancing
Performance of IKSD Algorithm for Unc
oded MIMO
Systems
Mohammed Qasim
Sultta
n
Dep
a
rtment of Electron
ic and
Information En
gin
eeri
ng,
Hu
a
z
hon
g Univ
ersi
t
y
of Scienc
e a
nd T
e
chnol
og
y,
W
uhan 4
3
0
074
, China
Univers
i
t
y
of
T
e
chn
o
lo
g
y
, Ba
ghd
ad, Iraq
*Corres
p
o
ndi
n
g
author, e-ma
i
l
: mohamma
da
lsulta
n@
yma
il.
com, I20132
20
29@
hust.ed
u
.cn
A
b
st
r
a
ct
The main c
hallenge in MIMO system
s is how to
design the
MIMO detection
algorithm
s
with lowest
computati
o
n
a
l
compl
e
xity an
d hi
gh
perfor
m
a
n
ce th
at
c
apa
ble
of acc
u
rately
detecti
ng the
trans
mitte
d
sign
als. In last
valu
abl
e rese
a
r
ch results, it h
ad
b
een
prov
e
d
the Max
i
mu
m L
i
kel
i
h
ood
D
e
tection (M
LD)
as
the o
p
ti
mu
m o
ne, b
u
t this
a
l
g
o
rith
m
has
an
expo
ne
ntial
co
mp
lexity
esp
e
c
i
ally
w
i
th i
n
cre
a
sin
g
of
a
nu
mbe
r
of trans
mit
ant
enn
as a
n
d
con
s
tellati
on s
i
z
e
mak
i
n
g
it
an i
m
practica
l
for imple
m
entat
io
n.
How
e
ver, ther
e
ar
e
altern
ative
al
g
o
rith
ms s
u
ch
a
s
the K-
best s
pher
e
det
ectio
n
(KSD)
an
d I
m
pr
ove
d
K-b
e
s
t spher
e
dete
c
tio
n
(IKSD) w
h
ich can ach
i
eve
a
close to Max
i
mu
m Li
ke
li
ho
od (ML) perfor
m
a
n
ce a
nd l
e
ss comp
utatio
nal
compl
e
xity. In this pa
per, w
e
have pr
op
ose
d
an e
n
h
anci
n
g IKSD alg
o
rit
h
m
by ad
di
ng
the co
mb
ini
ng
of
colu
mn n
o
r
m
o
r
derin
g (c
han
n
e
l or
deri
ng) w
i
t
h
Man
hattan
metric
to e
n
h
a
n
ce th
e p
e
rfor
ma
nce
an
d re
duce
the co
mp
utati
ona
l co
mp
lexit
y
. T
he simula
tion res
u
lt
s show
us that the ch
ann
el or
deri
ng a
ppro
a
c
h
enh
anc
es the
perfor
m
a
n
ce
and re
duc
es
the co
mpl
e
xi
ty, and Man
h
a
ttan metrica
l
onec
an re
duc
e the
compl
e
xity. T
herefore, the c
o
mbi
ned ch
an
nel or
deri
ng a
ppro
a
ch w
i
th Manh
attan me
tric enha
nces
the
perfor
m
a
n
ce
a
nd
muc
h
re
duc
es the c
o
mpl
e
x
i
ty more th
an
if w
e
used
the c
han
nel
ord
e
ri
n
g
ap
pro
a
ch
alo
n
e
.
So our pr
op
os
ed al
gor
ith
m
c
an b
e
cons
ide
r
ed a fe
asib
le
compl
e
xity red
u
ction sc
he
me
and su
itab
le f
o
r
practical implem
entation.
Ke
y
w
ords
: MIMO, KSD, ML
D, chann
el or
d
e
rin
g
, and Ma
n
hattan
metric
Copy
right
©
2016 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
In
r
e
c
e
n
t
year
s
,
two
ma
in
r
e
se
ar
ch
ac
tivi
ties have
d
o
minated
the
de
sign
of po
wer an
d
band
width ef
ficientwi
rele
ss commu
nication sy
stem
s:
First, multiple input
s, multiple outp
u
ts
(MIMO) [1] that emb
ody
the mea
n
ing
of co
m
m
uni
cation th
ro
ug
h multiple
a
n
tenna
s. MI
MO
techni
que
pe
rmits
simulta
neou
s tran
smit of multip
l
e
symb
ols from multipl
e
t
r
an
smit a
n
te
nna
s.
This results i
n
a linear in
crease in the chann
el ca
p
a
city commen
s
urate to the n
u
mbe
r
of tran
smit
antenn
as
wh
en there are
a suitable
n
u
mbe
r
of
receive antennas
[2]. Sec
o
nd, the Iterativ
e
detectio
n
, it is a
pra
c
tical
method to i
m
prov
e th
e
symbol
-erro
r
-rate (SE
R
)
p
e
rform
a
n
c
e f
o
r
comm
uni
cati
on syste
m
s.
So the study of co
m
b
ining the It
erative dete
c
tion an
d M
I
MO
techni
que
s to
appro
a
ching
from the ca
pa
city of MIMO cha
nnel
s [3].
MIMO dete
c
tion is a
challen
g
ing
and im
portant topic for
re
se
arch
ers a
n
d
comm
uni
cat
i
on
sy
st
em
d
e
sig
ner
s,
m
a
ssiv
e
r
e
se
ar
c
h
effort
s
were do
ne i
n
the
last ye
ars
gi
ving
the birth to
a variety of
detectio
n
techniqu
es
th
at differ in
strategy ada
pte
d
, com
putati
onal
compl
e
xity, a
nd perfo
rma
n
c
e. In orde
r to solve
the detectio
n
pro
b
lem in MIMO system
s, the
resea
r
chers
have been fo
cu
sed on
sub
optimal dete
c
tion techniq
u
e
s whi
c
h a
r
e
efficient in terms
of both p
e
rfo
r
mance a
nd
computation
a
l
compl
e
xity
, and po
we
rful i
n
term
s of e
r
ror p
e
rfo
r
man
c
e
and are practi
cal for impl
e
m
entation pu
rposes [4].
A novel
and
e
fficient MIMO
dete
c
tion
alg
o
ri
thm
fo
r any
wi
rele
ss com
m
unication sy
stem
s
must in
clu
d
e
so
me im
port
ant featu
r
es
su
ch
as
lo
w-compl
e
xity, near-optimal
p
e
rform
a
n
c
e
a
nd
robu
st sche
me. The ML
D [2] can p
r
ese
n
t outsta
nding p
e
rfo
r
mance; but, it suffers fro
m
high
comp
utationa
l co
mplexity in practi
cal i
m
plementat
io
n
esp
e
ci
ally wh
en in
crea
sing
the n
u
mbe
r
of
transmit ante
nna
s to a
c
hi
eve a g
ood t
r
an
smi
ssi
on
cap
a
city in
M
I
MO syste
m
s. Different
ne
ar-
optimal MIM
O
dete
c
tion t
e
ch
niqu
es
ha
ve been
prop
ose
d
in p
r
evi
ous lite
r
atu
r
e
s
some
of th
em
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
2, No. 3, Jun
e
2016 : 636
– 646
637
based o
n
zero
-forcing
(ZF) [3], minimum m
e
a
n
-s
qua
re
d-e
r
r
o
r (M
MSE)
[3], suc
c
e
s
siv
e
interferen
ce
can
c
ell
a
tion (SIC) [5], parallel inte
rfe
r
e
n
ce
ca
ncellation (PIC) [6] and o
r
de
red
SIC
(OSIC) [7].
Unfortu
nately
,
all of them cann
ot achi
eve the perf
o
rma
n
ce of an MLD. Sp
here
detectio
n
/decod
er (S
D) [8-12] was in
vesti
gated to
achieve the
ML perform
ance by usi
n
g
reliabl
e
radi
u
s
. Th
e id
ea
o
f
SD
wa
s i
n
trodu
ced
in [
1
3] and
it h
a
s
been
furth
e
rmore
de
bate
d
in
variou
s
re
sea
r
ch
es [14,1
5
]. The
K-b
e
st
sphere
de
cod
e
r (KSD)
[12]
for MIMO de
tection app
ea
rs
in the area of
detection techniqu
es be
ca
use of
its fixed throug
hput
and pa
rallel i
m
pleme
n
tatio
n
.
In the other side, the use
of the depth-f
i
rst tr
ee
sea
r
ch in convent
ional SD givi
ng non
-con
stant
throug
hput, which limits the
detection effi
cien
cy. So
instead of u
s
in
g a depth-fi
rst to traverse the
tree, the
KSD exe
c
ute
s
a
breadth
-
first se
arch
an
d
kee
p
s only K
-
be
st n
ode
s i
n
ea
ch
laye
r. In
KSD algo
rith
m to achieve
clo
s
e-
ML p
e
r
forma
n
ce [1
2], the KSD e
s
pe
cially n
e
e
d
s fo
r very la
rge
values
of K, whi
c
h in
turn
lead
s to a
hi
gher
complex
i
ty than that who i
n
the
convention
a
l
SD.
Non
e
thele
s
s, due to adva
n
tage
s of the
KSD algo
rith
m, some va
ri
ants h
a
ve be
en propo
se
d to
improve its p
e
rform
a
n
c
e a
nd/or r
edu
cin
g
its com
p
lexity [16-19].
The computat
ional complex
i
ty of an MIMO
detectio
n
a
l
gorithm
s de
p
end
s on the n
u
mbe
r
of spatially multiplexed
dat
a stream
s (n
umbe
r
of transmi
t
antenn
as) a
nd the sym
bol
con
s
tellatio
n
size, but freq
uently on the
instanta
neo
u
s
MIMO chan
nel re
alizatio
n and the
sig
nal-
to-noi
se
ratio
(SNR) [2
0]. The comp
utational
com
p
lex
i
ty of tree sea
r
ch
alg
o
rithm
s
i
s
dete
r
min
e
d
by two no
rm
s: Firstly, the numb
e
r
of node
s
that have
to
be examined an
d
Secondly, the
operational
cost pe
r nod
e. In SD, the numbe
r of
visited nod
es d
e
pend
s on th
e
choi
ce of init
ial
sph
e
re
radi
us and on th
e d
e
crea
sing
of the ra
diu
s
co
n
s
traint
s du
e to a ra
dius
up
date [21]. Th
e
compl
e
xity of K-best SD algorithm
s depe
nd
s criti
c
ally on the
preprocessi
ng stage
(Q
R
decompo
sitio
n
), the o
r
d
e
ri
ng (ba
c
k-sub
s
titution) i
n
which th
e
com
pone
nts of i
n
formatio
n si
gn
als
are con
s
ide
r
e
d
, and the initial choi
ce
of the radi
us of the sp
here.
In this
wo
rk,
we p
r
o
p
o
s
e e
nhan
cin
g
IKSD al
g
o
rithm, t
h
is
can
be
achieved by
divided th
e
K-be
st SD al
gorithm
wo
rk into two
par
t
s
. The fi
rst p
a
rt is
kn
own
as the
“
p
r
ep
roce
ss pa
r
”
t
, the
prep
ro
ce
ss can be
achiev
ed by ex
e
c
uti
on the
colu
m
n
no
rm o
r
de
ri
ng [22] (ch
a
n
nel o
r
de
ring
)
for
cha
nnel m
a
trix due to that the comp
utation co
mp
lexity is so sen
s
itive to the order
of the
colum
n
s
of the ch
ann
el
matrix. The seco
nd pa
rt is kno
w
n a
s
th
e “
search part”
, it is comp
uted
the ML
soluti
on of tran
smit
ted vecto
r
fro
m
the re
ce
ive
d
vector, in
th
is pa
rt we
pro
pose u
s
ing th
e
Manhattan n
o
r
m to cal
c
ulat
e the ML solu
tion in orde
r to redu
ce the
compl
e
xity of
this part.
The rest of t
h
is p
ape
r is
orga
nized a
s
fo
llows: Section 2 prese
n
ts the m
o
d
e
l of the
MIMO syste
m
and K-B
e
st SD algo
rith
m. The en
ha
nc
e
d
IKSD al
gorithm i
s
p
r
ese
n
t in Secti
on 3.
The
colu
mn
norm
orde
rin
g
(cha
nnel
o
r
de
ring
app
roach)
prepro
c
e
ssi
ng i
s
d
e
scrib
ed i
n
Sub-
se
ction
3.1.
Manhattan
m
e
tric propo
se
to u
s
e
in
search
pa
rt in
Sub-se
ction
3.2. Simul
a
tion
results p
r
e
s
e
n
ted in Sectio
n 4. Finally, in
Section 5,
we present the con
c
lu
sio
n
s.
2. Sy
stem Model and K-Bes
t
SD Alg
o
rithm
We con
s
id
er an
un
co
ded M-QAM
4
×
4
MIMO
syst
e
m
having
tr
an
s
m
it an
d
re
ceiv
e
antenn
as wh
ere
. Under th
e assumption
of a
flat-fading cha
nnel, the re
ceived
vector
can b
e
expre
s
sed a
s
̅
(1)
whe
r
e
̅
̅
,
̅
,
…
..,
̅
denote
s
1
transmitted
vector, and the entrie
s
of
̅
are sele
cted
from a compl
e
x constellati
on,
,
,…..,
denotes
1
complex-val
ued re
ceived
vec
t
or
,
deno
tes the
com
p
lex-value
d
chann
el matrix
with elem
ent
s are a
s
sum
ed to
be inde
pen
d
ent and id
ent
ically dist
ribut
ed (i.i.d.)
co
mplex Gau
ssian varia
b
le
s with ze
ro m
ean
and unit vari
ance, and
is the com
p
lex-valued of ad
ditive white
Gau
ssi
an noi
se (A
WG
N)
with
zero mea
n
an
d
var
i
anc
e
.
For si
mplifying the system
, the comple
x-valued
rece
ived vector is transfo
rme
d
into an
equivalent re
al-value
d received vector by repr
e
s
e
n
ting the real part and the imagina
ry part of
as
(2)
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Red
u
ci
ng Co
m
putational Com
p
lexit
y
a
nd Enhan
cin
g
Perform
ance
of IKSD …
(M.Qas
im Sulttan)
638
whe
r
e the di
mensi
on is d
ouble
d
su
ch t
hat
m =
2
,
n =
2
,
i.
e.,
̅
̅
(3)
whe
r
e
and
denote the rea
l
and imagi
n
a
ry part
s
of [·] respe
c
tively,
̅
̅
denote
s
1
real-val
ued t
r
an
smitted ve
ctor,
denote
s
1
real
-value
d
receive
d
vector
,
denote
s
1
real-valu
ed n
o
ise ve
ctor,
and
deno
tes
real-value
d cha
nnel mat
r
ix [14],
[23].
Assu
me that the receiver
has a p
e
rfe
c
t chan
ne
l
kno
w
le
dge, so the MLD p
r
oblem can be
formulated a
s
a
r
g
m
i
n
∈
∥
∥
(4)
whe
r
e
,
D
is real
-valued
si
gnal con
s
tella
tion set, for 1
6
-QAM
, D
=
[-3,-1,1,3]
.
In SD algorit
hms the sea
r
ch in
clu
d
e
s
only the lattice point
s (
Hs
) insi
de the hypersp
h
e
re
cente
r
ed at the re
ceived v
e
ctor(
y
) with
radiu
s
(
d
) inst
ead of com
p
rehen
sive sea
r
ch fo
r all lattice
points a
s
in
MLD, and
ca
n be written a
s
a
r
g
m
i
n
∈
∥
∥
(5)
By decomp
o
sed the ch
ann
el matrix
usi
ng the stan
da
rd Q
R
de
com
positio
n, we can get
(6)
whe
r
e
,
,
⋅
den
otes He
rmitia
n matrix transpo
sition,
R
is an
upper
triangul
ar ma
trix, and
is
an
orthogo
nal
matrix.Utilizing the
triangular nature of R,
the
left-hand
side
of (6) ca
n be
rewritten as
∥
,
∥
(7)
From (7) we can see
th
e
d
e
tection
p
r
o
b
l
e
m
as
a tree
that has its
ro
ot just above
the m-
th layer and leaves on the 1
st
layer, and each survive
d
candi
date o
f
i-th
layer is
defined as
,
……
,
. The Euclid
ean di
stan
ce
in (7)
can
be co
mputed
iteratively by defining
with the parti
al Euclide
an
distan
ce
s (P
EDs)[24].
∥
∥
,
,
…
.
.
,
1
(8)
The initialization
0
, and the distan
ce in
crements a
r
e
∥
∥
∥
,
∥
(9)
The PED,
, depend on the symbol vecto
r
(
s
) throug
h the partial sy
mbol vector
, t
h
e
SD pro
b
lem
has b
een ch
ange
d into a weighted tre
e
-sea
rch pro
b
lem. The SD algo
rithm with
depth-fi
rst tre
e
search
suffers from
no
n
-
co
ns
ta
nt through
put an
d
non-efficien
cy decoding
[25].
T
o
o
v
er
co
me th
e
s
e
pr
ob
lems
, th
e K-
bes
t SD a
l
go
r
i
th
m is
us
ed
,
w
i
th
ap
p
l
yin
g
th
e
br
ea
th
-
f
ir
s
t
tree
sea
r
ch
strategy. T
h
e K-be
st al
g
o
rithm
simp
li
fies the
com
p
lexity of SD alg
o
rithm
b
y
sho
r
tenin
g
th
e path
s
in ea
ch d
e
tectio
n layer from th
e m-th laye
r
to the 1
st
layer an
d only t
he
smalle
st K node
s are
ke
pt in each la
yer (except the 1
st
layer),
which
will be extended i
n
to
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639
node
s in th
e next layer. We can de
scrib
e
the K-b
e
st algo
rithm
throug
h the followin
g
st
ep
s:
a)
Apply QR de
comp
ositio
n according to
H = Q
R
.
b)
Comp
ute the PEDs
acco
rd
ing to Eq. (6)
to (9) (path e
x
tension
)
.
c)
Prune the p
a
ths that gr
eate
r
than the ra
d
i
us ba
se
d on
the PEDs of step (b).
d)
Do sorting to
K nodes a
nd
cho
o
se the smallest on
e.
e)
Do path u
pda
te by updatin
g
by
1
, If
1
, go back to step
(b).
f) If
1
, go to step (b) a
nd (c), th
en sel
e
ct the
path with mini
mum PED as a deci
s
ion.
If K is not large eno
ugh, the K-be
st SD algorit
hm n
o
t
able to guarantee the sa
me perfo
rma
n
ce
as SD an
d M
L
algorith
m
. Thus, the
r
e i
s
a pre
s
sing
need to find
efficient K-be
st algorith
m
s
tha
t
can d
o
the be
st trade
-off betwee
n
perfo
rmance and
complexity.
3. The Enha
ncing IKSD
Algorithm
As
n
o
ted earlier, we ca
n enha
nce
the
perfo
rm
an
ce
of the IKSD
algorith
m
by
addin
g
cha
nnel
orde
ring
app
roa
c
h to the
prep
rocessin
g p
a
rt, and al
so
a
dding
the M
a
nhattan m
e
tri
c
to
the sea
r
ch p
a
rt. Whe
r
e th
e cha
nnel o
r
derin
g app
ro
ach i
s
workin
g to improve
SER perfo
rm
ance
and re
du
cing
the compl
e
xity, the Manhattan metric i
s
workin
g to re
duce the com
p
lexity more. As
it will be cla
r
i
f
ied in the ne
xt two sub
-
se
ction
s
. The e
nhan
cin
g
IKSD algo
rithm i
s
de
scrib
ed i
n
Algorithm
-I.
Algorithm
-I:
The enha
nci
ng IKSD algo
rithm
Input:
,
H
,
K
,
A
,
∆
, d
Outpu
t
:
Initialization
0
(the bran
ch m
e
tric) and
is the root no
de
(level
);
Π
;
,
_
;
;
0
; and start fro
m
level
w
h
ile
i
≥
1
do
ℓ
1
;
for
j=1
to
length (
)
do
,
,
,
∀
∈
;
ℓ
ℓ
1;
end
sort all the
co
mpone
nts of
in an asce
ndi
ng ord
e
r ;
if
length
Then
Keep all the candid
a
tes in t
r
ee;
else
Only k
e
ep the element
s
whose c
o
s
t
indexes
satis
f
y
∆
in tree ;
end
R
eplac
e the
←
;
1
;
end
Return
←
the 1
st
element in the tree
3.1. The Cha
nnel Orderin
g
Appro
ach
(Prepro
ces
s Part)
The computa
t
ion com
p
lexi
ty of K-best SD is
q
u
ite sensitive to th
e ord
e
r of the
colum
n
s
of the ch
ann
el matrix, whi
c
h rely on b
o
t
h the ch
ann
el matrix an
d
the re
ceived
sign
al. So, the
rand
om dete
c
tion o
r
de
r is not the be
st detection
order, pa
rticul
a
r
ly for low S
NR o
r
hig
h
o
r
de
r
modulatio
n. Usually, re
-a
rra
ngin
g
the
colum
n
s
of
the matrix a
p
p
rop
r
iately i
s
to get a go
od
detectio
n
pro
c
e
ss (low co
mplexity).
The Q
R
d
e
compo
s
ition p
e
rform
a
n
c
e
can be i
m
pro
v
ed if the ch
annel m
a
trix
is p
r
e-
pro
c
e
s
sed b
e
fore Q
R
de
comp
ositio
n. So, we
sug
g
e
st usin
g the
preprocess
of column no
rm
orde
rin
g
(cha
nnel orde
ring
appro
a
ch) [
22] before
Q
R
de
comp
osi
t
ion. The col
u
mns
of cha
nnel
matrix can b
e
reorde
red in
accordan
ce
with the
no
rm
of each
colu
mn, so the si
gnal
s with hig
her
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Red
u
ci
ng Co
m
putational Com
p
lexit
y
a
nd Enhan
cin
g
Perform
ance
of IKSD …
(M.Qas
im Sulttan)
640
sign
al-to
-
noi
se ratio
(SNR) a
r
e
dete
c
te
d first. Thi
s
can
be
a
c
hi
eved by
mul
t
iplying chan
nel
matrix H by a permutation matrix
Π
, i.e.,
ΠH
Q
R
.
The column
norm o
r
d
e
rin
g
method in
clud
es
the f
o
llowin
g
procedure, arran
g
ing th
e
colum
n
s
of the ch
ann
el
matrix
,
…
..,
,
in accordan
ce
wit
h
their E
u
cli
dean
distan
ce
∥
∥
in a
n
asce
nding
manne
r. The
arrangi
ng of
is processe
d
by the perm
u
tation
so that
∥
∥∥
∥
˂
(10
)
Then, the arrangin
g
ch
ann
el matrix is re
pre
s
ente
d
as
Π
(11
)
whe
r
e
Π
is
pe
rmutation m
a
trix such as
Π
,
,……,
, where
is the
colum
n
vecto
r
of whi
c
h ent
ries a
r
e o
ne
i-th
positio
n onl
y and are zero in every other po
sition
s.
One adva
n
ta
ge of ch
annel
orde
ring of t
he gen
erato
r
matrix
H
is th
at the colum
n
norm o
r
de
rin
g
doe
s not di
st
ort or
distu
r
b
the bou
nda
ries of
the fin
i
te lattice ca
n be ea
sily
determi
ned a
n
d
exploited. Thi
s
can
be u
n
d
e
rsto
od
as th
e col
u
mn
arrangin
g
of
H
simply lead
s t
o
a re-arran
gi
ng
of the compo
nents of tra
n
smit signal vectors.
3.2. The Man
h
attan Me
tri
c
(Searc
h
Part)
In this sectio
n, we prop
o
s
e to use M
anhattan met
r
ic (M
M) to cal
c
ulate ML
solution
instea
d of u
s
i
ng the Eu
clid
ean met
r
ic
(E
M), in o
r
de
r to re
duce the
compl
e
xity in sea
r
ch pa
rt. The
purp
o
se of u
s
ing M
M
or E
M
is to calcul
ate t
he weigh
t
s of ea
ch
ca
ndidate
node
[26]. In EM, the
brute
-
force M
L
D can be
co
nverted into a
full tr
ee stru
cture se
arch b
y
using EM such a
s
a
r
g
m
i
n
∈
||
||
a
r
g
min
∈
,
(12
)
From (12) th
e MIMO-ML
D
sea
r
che
s
a candi
date
that minimizes the sq
uared EM
betwe
en
an
d
that i
s
re
ferre
d to a
s
, and
we
ca
n se
e that t
he op
eration
s
perfo
rmed
d
epen
d on
summation a
nd multipli
ca
tion due to
squa
re te
rm.The ha
rd
ware
impleme
n
tation is i
n
fea
s
ibl
e
due to
a log
i
c re
so
urce li
mitation of th
e target
device be
cau
s
e th
ere
are
4
= 1,048,
576 re
al mult
iplicatio
ns (fo
r
16-QA
M) a
r
e requi
re
d to comp
ute all the EM.
Acco
rdi
ng to
(12
)
thi
s
type
of dete
c
tion
a
l
gorithm
is
practically impo
ssi
ble to
impl
ement in
MIM
O
system
s that
utilize
high
orde
r m
odul
ation such
a
s
(16-QAM,
64-QA
M). So
we
ado
pted
a
pra
c
tical
metric li
ke M
M
to
avoid th
e u
s
e of a
r
ithmeti
c
multipli
catio
n
s, the
MM i
s
comp
uted
by
addin
g
ab
sol
u
te values of
and
, as in (13).
a
r
g
m
i
n
∈
|
|
a
r
g
m
i
n
∈
,
(13
)
As sho
w
n in (13), the ope
rations pe
rf
ormed dep
end
only on sum
m
ation and di
dn’t have
a squ
a
re te
rmand the
r
efo
r
e it does n
o
t need for a
r
ith
m
etic multipli
cation
sa
s in (12).
4. Simulation Resul
t
s
In this sectio
n, we di
scu
s
s an
d compa
r
e the
perfo
rmance (th
e
symbol erro
r rate, SER)
and co
mputat
ional co
mple
xity (the number of node
s
visited) for b
o
th traditional
KSD and IKSD
algorith
m
s,
with ca
se
s
of n
o
or
de
ring,
orderin
g, an
d
combine
o
r
de
ring
with MM.
To ma
ke
a fa
ir
comp
ari
s
o
n
for all ca
se
s, suppo
se the in
itial radiu
s
for all cases i
s
the sam
e
.
Firstly we di
scu
ss the effe
ct of column
norm o
r
de
rin
g
and MM on
SER performance in
both alg
o
rith
ms tra
d
itional
KSD and IK
SD. An
un
co
ded 4x4 MIM
O
syste
m
wit
h
16-QAM an
d 64-
QAM are
sim
u
lated ove
r
a
flat Rayleigh f
ading
cha
nne
l. From figure
(1) a
nd figu
re (2
), can
sh
ow
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641
the same S
E
R pe
rforma
nce
in the
case
of no
o
r
derin
g fo
r 16
-QAM a
nd
6
4
-QAM j
u
st t
he
perfo
rman
ce
of the IKSD need
s for
(K=2),
while th
e tradition
al KSD need
s f
o
r (K
=16
)
, an
d we
can
see imp
r
oved the SER perfo
rma
n
ce whe
n
usi
n
g
orde
ring a
n
d
combin
e ord
e
ring
with M
M
.
From figu
re
(1)
we ha
ve obse
r
ved
that for 16
-QAM an
d
at an SER=10
-2
, the
perfo
rman
ce
gain
abo
ut 2.
5dB
whe
n
u
s
i
ng
the
ch
ann
el o
r
de
ring
a
nd
com
b
ine
chann
el o
r
de
ri
ng
with MM com
pare
d
to usin
g of no orderi
ng. From figu
re (2
) can
co
nclu
de that the performan
ce
enha
nces by
3.9dB at an SER=1
0
-1
.
Secon
d
ly, we
discu
ss the
effect of colu
mn no
rm ord
e
ring
and M
M
on comput
ational
compl
e
xity in both al
gorith
m
s traditional
KSD an
d IKSD. Figu
re
(3) a
nd figu
re
(4) sho
w
s t
h
e
comp
ari
s
o
n
of
the com
p
l
e
xity
(visited node
s)
in
tra
d
itional KS
D
and IKSD al
gorithm
s
with
no
orde
rin
g
,
ord
e
ring, and
co
mbine ord
e
ri
ng with
MM.
And it al
so
sh
ows that the
compl
e
xity of the
IKSD algorith
m
with combi
ned o
r
de
ring
and MM a
r
e l
o
we
r than th
at of IKSD with orde
ring
a
n
d
with no orde
ring, and the
complexity of the I
KSD algorithm
wi
th orde
ring i
s
less than t
h
e
compl
e
xity of the IKSD alg
o
rithm with n
o
orde
ri
ng. In
other wo
rd
s, for example (ca
s
e of Figu
re
3), while th
e IKSD algorith
m
with ord
e
ri
ng visit on a
v
erage 4
4
no
des to obtai
n
a performan
ce
comp
arable t
o
IKSD algo
ri
thm with n
o
o
r
de
ring,
a
nd t
he IKSD alg
o
r
ithm with
co
mbined
orderi
ng
and MM
i
s
a
b
le
to a
c
hiev
e the same
perfo
rman
ce
visiting on averag
e 30 no
des while IKSD
algorith
m
wit
h
no orde
rin
g
visits 61 n
ode
s. Fr
om f
i
gure
(3
) we
can
com
pare
the com
p
lexity
betwe
en th
e
curve
s
of n
o
orde
rin
g
ca
se an
d
com
b
i
ne o
r
d
e
rin
g
with MM
of IKSD al
gorith
m
at a
minimum (SNR=0
dB) and maximum
(S
NR=2
5
dB)
di
fference bet
ween the
s
e t
w
o cu
rve
s
. In no
orde
rin
g
IKSD al
gorith
m
searche
s
abo
u
t
(61
an
d 1
8
1
)
n
ode
s, a
nd
in combin
e o
r
derin
g a
nd
M
M
need
s (30 a
n
d
42) nod
es v
i
sited respe
c
tively. So
the propo
se
d IKSD with
co
mbi
ne orde
ring
a
nd
MM nee
ds 5
1
% to 77% fe
wer complexi
ties than
IKSD with
no
ord
e
ring. Al
so,
can comp
are
the
comp
utation
s
betwee
n
no
orde
rin
g
and
orde
rin
g
of
IKSD algo
rith
m, at a minimum (S
NR=0 dB)
and the
maxi
mum (S
NR=25 dB)
difference bet
wee
n
these two
curve
s
, the
n
o
orderi
ng IK
SD
algorith
m
se
a
r
ch
es
abo
ut (61 and
181
)
node
s, and
i
n
orde
ring
nee
ds (44 a
nd 4
8
) no
de
s visit
e
d
respe
c
tively. So the IKSD algorith
m
wit
h
ord
e
rin
g
ne
eds 2
8
% to 73% fewe
r complexities t
han
IKSD with n
o
ord
e
rin
g
. From figure
(4) we
ca
n d
o
the
same
cal
c
ulatio
n a
s
i
n
figure (3),
for
comp
are bet
wee
n
two
cu
rves of n
o
ord
e
ring
and
co
mbine
ord
e
ri
ng with
MM
at SNR=0
d
B
and
SNR=25, it’
s
nee
d (117
9
and
93
23)
node
s
and
(87an
d
8
7
) resp
ectively. So
the com
b
ine
orde
rin
g
with
MM ne
ed
s 9
2
% to 9
9
% fe
wer
compl
e
xities tha
n
n
o
orde
rin
g
, an
d
the
com
pari
s
on
betwe
en n
o
orde
rin
g
a
nd orde
ring
need
s (11
79 an
d 932
3) an
d (599
and 1
123
)
node
s
respe
c
tively.
So the orde
ri
ng nee
ds 4
9
%
to 88% fewer co
mplexiti
es than n
o
orderin
g.
In figure (3) a
nd figure (4)
can see that the co
m
p
lexity of traditional KSD algorith
m
in the
ca
se
s of no o
r
de
ring a
nd o
r
de
ring i
s
the
same an
d it's different co
mpared with I
KSD algorith
m
,
this i
s
d
ue to
the
way of
a
c
count th
e vi
sited
nod
es
i
n
ea
ch
algo
ri
thm, to cla
r
ify that, the visit
e
d
node
s in the
traditional KS
D algo
rithm a
r
e calcul
ated
from all child
node
s that e
x
tend in every
layer and al
so the value of K. But in
IKSD algorith
m
the visited node
s cal
c
ulat
ed from all child
node
s that
e
x
tend in eve
r
y layer a
nd
restri
cted
wi
th
the value
of
K and fixed t
h
re
shol
d [27].
As
depi
cted
in
figure
s
(1
)an
d
(
2), we ca
n note
that
a
s
it has
hap
pe
ned in
othe
r
works [28], [
29],
whe
n
usi
ng MM the perfo
rman
ce
suffer from a sli
ght
degradatio
n, but in our p
r
o
posed work, with
usin
g of
cha
n
nel o
r
de
ring
approa
ch th
e
SER p
e
rf
o
r
m
ance d
o
e
s
no
t suffer any d
egra
dation
u
n
til
with usi
ng M
M
.
In figure (5
)
note that the visited node
s of
IKSD with 64-QAM is
much la
rg
er t
han the
visited no
de
s in 16
-QAM, t
h
is i
s
d
ue to
the differe
nt in co
nstell
atio
n si
ze
betwe
en 16
-QAM
a
n
d
64-QA
M, an
d al
so th
e v
i
sited
nod
es is
directly
prop
ortio
nal t
o
in
cre
a
si
ng
the
size of
the
con
s
tellatio
n
.The 16
-QAM
and 64
-QAM
modulatio
n schem
es
achi
eve different
perfo
rming i
n
the
pre
s
en
ce
of
noise. In pa
rt
icula
r
, 64
-QA
M
(hi
ghe
r o
r
der) mo
dulati
on
scheme
is able to
a
c
hi
eve
highe
r data rates but it's n
o
t robu
st in the pres
e
n
ce
of noise. 16
-QAM (Lo
w
e
r
orde
r) m
odul
ation
scheme give
fewer d
a
ta ra
tes but it is more ro
bu
st in the pre
s
en
ce
of noise. So, figure (6
) sh
ow
us the va
riati
on in the
perf
o
rma
n
ce of the I
KSD alg
o
r
ithm bet
wee
n
two type
s o
f
modulation
16-
QAM and
64-QAM, we
can
see th
at the perfo
rman
ce
of 16-QAM
b
e
tter than the
perfo
rman
ce
of
64-QA
M.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Red
u
ci
ng Co
m
putational Com
p
lexit
y
a
nd Enhan
cin
g
Perform
ance
of IKSD …
(M.Qas
im Sulttan)
642
Figure 1. Performa
nce of KSD and IKSD with No
orde
ring, orderi
n
g
,
and combi
n
e orde
rin
g
wit
h
MM for un
cod
ed 4x4 MIMO
16-QAM
syst
em
Figure 2. Performa
nce of KSD and IKSD with No
orde
ring, orderi
n
g
,
and combi
n
e orde
rin
g
wit
h
MM for un
cod
ed 4x4 MIMO
64-QAM
syst
em
0
5
10
15
20
25
10
-4
10
-3
10
-2
10
-1
10
0
SN
R
SER
16
-
Q
A
M
4x
4M
I
M
O
IKSD K
=2 NoOrder
IKSD K
=2 Ordering
IKSD K
=2 Ordering
+Manhattan
KSD K=
16 Ordering
KSD K=
16 NoOrder
0
5
10
15
20
25
10
-3
10
-2
10
-1
10
0
SN
R
SE
R
64-Q
A
M
4x
4M
I
M
O
I
K
S
D
K
=
2 N
o
O
r
de
r
I
K
S
D
K
=
2
O
r
der
i
n
g
I
K
S
D
K
=
2
O
r
der
i
n
g
+
M
anh
at
t
a
n
KS
D
K=1
6
O
r
d
e
r
i
n
g
KS
D
K=1
6
N
o
O
r
d
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
2, No. 3, Jun
e
2016 : 636
– 646
643
Figure 3. Co
mplexity of KSD and IKSD with No
orde
ring, orderi
n
g
,
and combi
n
e orde
rin
g
wit
h
MM for un
cod
ed 4x4 MIMO
16-QAM
syst
em
Figure 4. Co
mplexity of KSD and IKSD with No
orde
ring, orderi
n
g
,
and combi
n
e orde
rin
g
wit
h
MM for un
cod
ed 4x4 MIMO
64-QAM
syst
em
0
5
10
15
20
25
10
1
10
2
10
3
SN
R
A
v
erag
e nu
m
ber o
f
n
ode
s
v
i
s
i
t
e
d
16-Q
A
M
4
x
4M
I
M
O
I
KSD
K=
2
NoO
rde
r
I
KSD
K=
2
Ord
eri
ng
I
KSD
K=
2
Ord
eri
ng
+Ma
nha
tt
an
K
SD
K=1
6
Ord
eri
ng
K
SD
K=1
6
NoO
rde
r
0
5
10
15
20
25
10
1
10
2
10
3
10
4
SN
R
A
v
er
a
ge n
u
m
ber
of
n
ode
s
v
i
s
i
t
e
d
64
-
Q
A
M
4x
4M
I
M
O
I
K
SD
K=
2
N
o
O
r
d
e
r
I
K
S
D
K
=
2 O
r
de
r
i
ng
I
K
S
D
K
=
2 O
r
de
r
i
ng
+
M
an
ha
ttan
KSD
K=
1
6
O
r
d
e
r
i
n
g
KSD
K=
1
6
N
o
O
r
d
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4
752
Red
u
ci
ng Co
m
putational Com
p
lexit
y
a
nd Enhan
cin
g
Perform
ance
of IKSD …
(M.Qas
im Sulttan)
644
Figure 5. Co
mpare the co
mplexity of IKSD wi
th No o
r
de
ring, orde
ring, and com
b
ine orde
ring
with MM for u
n
co
ded 4x4
MIMO 16-QAM and 64
-QA
M
system
Figure 6. Co
mpare the SER perfo
rma
n
ce of IKSD
with No orde
ring
, orderi
ng, an
d combi
ne
orde
rin
g
with
MM for un
cod
ed 4x4 MIMO
16-QAM a
n
d
64-QAM
syst
em
5. Conclusio
n
s
In this pape
r, we pro
p
o
s
e an enha
nci
n
g
IKSD by adding the com
b
i
n
ing of colu
m
n
norm
orde
rin
g
an
d
Manh
attan
metric to e
n
han
ce th
e p
e
rform
a
n
c
e
a
nd redu
ce
th
e computatio
nal
compl
e
xity, in order to
make
this
kind of
al
gori
t
hms m
o
re
suitabl
e in t
e
rm of
FPG
A
0
5
10
15
20
25
10
1
10
2
10
3
10
4
SN
R
A
v
er
ag
e n
u
m
b
e
r
of
n
o
d
e
s
v
i
s
i
t
e
d
IKSD 16
QAM NoO
rder
IKSD 16
QAM Ord
ering
IKSD 16
QAM Ord
ering+M
anhatta
n
IKSD 64
QAM NoO
rder
IKSD 64
QAM Ord
ering
IKSD 64
QAM Ord
ering+M
anhatta
n
0
5
10
15
20
25
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SN
R
SE
R
IKSD 16QAM NoOr
der
IKSD 16QAM Orde
ring
IKSD 16QAM Orde
ring+Manhattan
IKSD 64QAM NoOr
der
IKSD 64QAM Orde
ring
IKSD 64QAM Orde
ring+Manhattan
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 25
02-4
752
IJEECS
Vol.
2, No. 3, Jun
e
2016 : 636
– 646
645
impleme
n
tation.Fro
m
the
simulatio
n
re
sults that a
p
p
ear i
n
thi
s
wo
rk,
we
ca
n
se
e that the
col
u
mn
norm
ord
e
rin
g
method i
s
simple to i
m
pleme
n
t bu
t very effective in term o
f
enhan
cing
the
perfo
rman
ce
and redu
ce t
he complexit
y
. And we
can al
so see
the effect of
combi
n
ing t
he
colum
n
n
o
rm
ord
e
rin
g
an
d
Manh
attan
metric i
n
term of enh
an
ci
ng the
perfo
rmance a
nd
more
redu
cin
g
of complexity than the colum
n
norm o
r
de
rin
g
method alo
ne.
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