Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
9,
No.
5,
October
2019,
pp.
4138
4146
ISSN:
2088-8708,
DOI:
10.11591/ijece.v9i5.pp4138-4146
r
4138
A
computationally
efficient
detector
f
or
MIMO
systems
Samer
Alabed
Colle
ge
of
Engineering
and
T
echnology
,
American
Uni
v
ersity
of
the
Middle
East,
K
uw
ait
Article
Inf
o
Article
history:
Recei
v
ed
Sep
13,
2018
Re
vised
Apr
9,
2019
Accepted
Apr
28,
2019
K
eyw
ords:
MIMO
systems
MIMO
detectors
Zero
forcing
decoder
Maximum
lik
elihood
decoder
Sphere
decoder
Minimum
mean
squared
error
ABSTRA
CT
AIn
this
w
ork,
a
ne
wly
designed
multiple-input
multiple-output
(MIMO)
detector
for
implementation
on
softw
are-defined-radio
platforms
is
proposed
and
its
performance
and
comple
xity
are
studied.
In
particular
,
we
are
interested
in
proposing
and
e
v
aluating
a
MIMO
detector
that
pro
vides
the
optimal
trade-of
f
between
the
decoding
comple
xity
and
bit
error
rate
(BER)
performance
as
compared
to
the
state
of
the
art
detectors.
The
proposed
MIMO
decoding
technique
appears
to
find
the
optimal
compromise
between
competing
interests
encountered
in
the
implementation
of
adv
anced
MIMO
detectors
in
practical
hardw
are
systems
where
it
i)
e
xhibits
deterministic
decoding
comple
xity
,
i.e.,
deterministic
latenc
y
,
ii)
enjo
ys
a
good
comple
xity–performance
trade-of
f,
i.e.,
it
k
eeps
the
comple
xity
considerably
lo
wer
than
that
of
the
maximum
lik
elihood
detec-
tors
with
almost
optimal
performance,
iii)
allo
ws
fully
parameterizable
performance
to
comple
xity
trade-of
f
where
t
he
performance
(or
comple
xity)
of
the
MIMO
detector
can
be
adapti
v
ely
adjusted
without
the
requirement
of
changing
the
implementation,
i
v)
enjo
ys
simple
implementation
and
fully
supports
parallel
processing,
and
v)
allo
ws
simple
and
ef
ficient
e
xtension
to
soft-bit
output
generation
for
support
of
turbo
decod-
ing.
From
the
simulation
results,
the
proposed
MIMO
decoding
techni
que
sho
ws
a
substantially
impro
v
ed
comple
xity–performance
trade-of
f
as
compared
to
the
state
of
the
art
techniques.
Copyright
c
2019
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Samer
Alabed,
Assistant
professor
,
Department
of
Electrical
Engineering,
Colle
ge
of
Engineering
and
T
echnology
,
American
Uni
v
ersity
of
the
Middle
East,
K
uw
ait.
Phone:
+965
2225
1400
Ext.:
1790
Email:
Samer
.Al-Abed@aum.edu.kw
1.
INTR
ODUCTION
In
the
last
decade,
cooperati
v
e
and
multiple-input
multiple-output
(MIMO)
techniques
ha
v
e
been
e
x-
tensi
v
ely
studied
as
their
impro
v
ements
in
performance
do
not
require
additional
po
wer
or
frequenc
y
spec-
trum
[1–13].
In
this
w
ork,
the
performance
of
e
xisting
linear
and
nonlinear
decoders
[2,
14–20]
for
MIMO
systems
is
compared
with
the
ne
wly
proposed
decoder
that
is
particularly
suitable
for
implementation
on
softw
are-defined-radio
architectures.
The
maximum
lik
elihood
(ML)
decoder
is
the
optimal
detector
for
MIMO
systems
[2,
15].
In
this
decoder
,
a
search
o
v
er
all
possible
combination
of
transmitted
symbol
v
ectors
is
per
-
formed.
The
ML
detection
pro
v
es
to
be
optimal,
ho
we
v
er
,
at
the
cost
of
high
comple
xity
which
increases
e
xponentially
with
the
increase
of
the
modulation
size
and
the
number
of
transmit
antennas
[15,
16].
On
the
other
hand,
linear
detectors
such
as
the
zero
forcing
(ZF)
and
minimum
mean
squared
error
(MMSE)
detectors
are
the
simplest
and
widely
used
detectors
with
reasonably
lo
wer
bit
error
rate
(BER)
performances
at
v
ery
lo
w
computational
comple
xity
[2,
4,
17,
18].
Correspondingly
,
the
v
ertical
Bell
laboratories
layered
space-time
(V
-BLAST)
technique
uses
an
iterati
v
e
detector
that
implements
the
concept
of
successi
v
e
interference
can-
cellation
(SIC)
to
find
a
good
trade-of
f
between
comple
xity
and
performance
[2,
18–20].
SIC
decoder
can
be
J
ournal
homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
4139
further
impro
v
ed
by
incorporating
appropriate
ordering
of
the
symbols,
i.e.,
first
decoding
the
symbols
that
e
xhibit
small
estimation
error
before
detecting
the
weak
er
symbols.
In
this
w
ork,
we
are
interested
in
implementing,
de
v
eloping
and
e
v
aluating
a
MIMO
detector
tha
t
pro
vides
the
optimal
trade-of
f
between
the
decoding
comple
xity
and
BER
performance
as
compared
to
the
state
of
the
art
detectors.
Therefore,
we
introduce
a
ne
w
MIMO
decoding
technique
which
i)
enjo
ys
a
good
comple
xity-performance
trade-of
f,
ii)
allo
ws
fully
parameterizable
performance
configuration,
in
the
sense
that,
the
performance
of
the
MIMO
detector
can
be
adapti
v
ely
adjusted
without
the
requirement
of
changing
the
implementation,
iii)
enjo
ys
simple
implementation
and
fully
supports
massi
v
e
parallel
processing,
i
v)
e
xhibits
a
fix
ed
comple
xity
,
i.e.,
unlik
e
the
popular
sphere
decoder
,
the
dec
o
di
ng
comple
xity
is
deterministic
and
does
not
depend
on
the
particular
realizations
of
f
ading
or
noise
en
vironments,
and
v)
allo
ws
natural
e
xtension
to
soft-bit
decoding
required
for
modern
channel
decoders.
2.
SYSTEM
MODEL
Let
us
consider
a
MIMO
system
with
nT
x
transmit
and
nR
x
recei
v
e
antennas
as
illustrated
in
Figure
1
and
assume
frequenc
y
non-selecti
v
e
flat
f
ading
channels.
If
a
signal
v
ector
x
is
sent
from
the
transmit
antenna
array
where
symbol
x
j
emitted
from
the
j
th
transmit
antenna
and
y
i
is
recei
v
ed
by
the
i
th
antenna,
then
the
signal
at
the
recei
v
e
antennas
can
be
e
xpressed
as
y
=
H
x
+
n
(1)
where
y
=
[
y
1
;
y
2
;
;
y
nR
x
]
T
,
x
=
[
x
1
;
x
2
;
;
x
nT
x
]
T
,
n
=
[
n
1
;
n
2
;
;
n
nR
x
]
T
,
and
H
denotes
the
MIMO
channel
matrix
which
describes
the
input-output
relation.
In
this
representation,
n
denotes
the
noise
v
ector
which
is
modeled
as
independent,
zero-mean,
comple
x
Gaussian
random
v
ariables
with
unit
v
ariance.
Figure
1.
System
model
of
MIMO
channel
Let
H
be
a
nR
x
nT
x
matrix
(
nR
x
nT
x
)
with
rank
(
H
)
=
nT
x
.
H
can
be
decomposed
into
the
QR
decomposition,
such
that
H
=
QR
(2)
where
Q
is
an
nR
x
nR
x
unitary
matrix,
R
is
an
(
nR
x
nT
x
)
upper
triangular
matrix,
and
I
nT
x
is
an
(
nT
x
nT
x
)
identity
matrix.
Making
use
of
the
QR
decomposition,
we
can
trans
form
the
channel
model
(1)
into
an
equi
v
alent
triangular
channel,
such
that
~
y
=
Q
H
y
=
Q
H
(
H
x
+
n
)
=
Rx
+
~
n
(3)
where
~
n
=
Q
H
n
.
After
preprocessing
the
recei
v
ed
data,
model
(3)
becomes
in
a
triangularized
form.
3.
THE
PR
OPOSED
RANDOMIZA
TION
B
ASED
MMSE
DECODER
In
this
section,
let
us
introduce
a
ne
w
MIMO
decoding
algorithm
where
this
algorithm
carries
out
the
follo
wing
steps:
3.1.
Step
1:
Pr
epr
ocessing
(Nulling/Channel
equalization)
Nulling,
i.e.
channel
equalization,
is
used
to
remo
v
e
the
channel
ef
fect
from
the
recei
v
ed
signal
v
ector
.
This
process
is
performed
using
MMSE
channel
matrix
in
v
ersion.
The
linear
recei
v
er
W
MMSE
is
computed
to
minimize
the
mean
squared
error
(MSE)
[2]
of
the
recei
v
ed
signal,
gi
v
en
by
f
MMSE
(
x
)
=
E
n
(
W
MMSE
y
x
)
(
W
MMSE
y
x
)
H
o
(4)
A
computationally
ef
ficient
detector
for
...
(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
4140
r
ISSN:
2088-8708
where
E
fg
denotes
statistical
e
xpectation.
Therefore,
the
equalization
matrix
W
MMSE
corresponding
to
the
MMSE
decoder
is
e
xpressed
as
W
MMSE
=
(
H
H
H
+
2
I
)
1
H
H
(5)
where
I
denotes
the
identity
matrix.
According
to
the
principle
of
linear
recei
v
ers,
W
MMSE
gi
v
en
in
(5)
is
multiplied
by
y
gi
v
en
in
(1)
to
reconstruct
the
symbol
v
ector
x
by
remo
ving
the
channel
ef
fect
and
suppressing
noise
enhancement,
such
that
^
x
MMSE
=
W
MMSE
y
.
F
or
con
v
enience
of
representation
and
for
our
deri
v
ations
in
the
follo
wing,
the
MMSE
decoder
can
be
formulated
in
another
form.
Making
use
of
QR
decomposition
e
xplained
in
Sec.
2.,
H
can
be
e
xpressed
as
H
=
H
I
=
Q
1
Q
2
R
=
Q
R
(6)
where
the
partitioning
is
such
that
Q
=
C
2
nR
x
nT
x
,
Q
1
=
C
nR
x
nT
x
1
,
Q
2
=
C
nR
x
nT
x
2
and
R
=
C
nT
x
nT
x
.
Considering
(6),
the
equalization
matrix
W
MMSE
corresponding
to
y
in
(1)
can
be
e
xpressed
as
W
MMSE
=
(
H
H
H
)
1
H
H
=
(
H
H
H
+
2
I
)
1
H
H
=
(
R
H
R
)
1
R
H
Q
H
1
=
R
1
Q
H
1
(7)
where
H
is
gi
v
en
in
(6).
Making
use
of
the
equalization
matrix
(7),
the
soft-decoded
symbol
after
MMSE
decoding
becomes
^
x
MMSE
=
W
MMSE
y
=
x
+
R
1
Q
H
1
n
=
x
+
e
(8)
where
x
denotes
the
true
transmitted
symbol
v
ector
and
the
esti
mation
error
v
ector
e
=
^
x
x
is
Gaussian
distrib
uted
with
zero
mean
and
error
co
v
ariance
matrix
gi
v
en
by
E
ee
H
=
2
(
H
H
H
+
2
I
)
1
=
2
(
R
H
R
)
1
:
(9)
This
step
is
carried
out
only
once
and
before
the
randomization
is
started.
3.2.
Step
2:
Generating
random
v
ector
In
this
step,
the
decoder
generates
a
number
of
instances
of
a
random
v
ector
e
k
,
f
k
=
1
;
:::;
N
o
r
and
g
with
mean
and
v
ariance
equal
to
those
of
the
estimation
error
e
in
(9),
i.e.,
e
k
2
N
0
;
2
(
R
H
R
)
1
where
N
o
r
and
is
the
number
of
generated
random
v
ectors
and
e
k
denotes
the
k
th
generated
random
v
ector
.
From
f
e
k
g
N
o
r
and
k
=1
,
a
corresponding
set
for
random
v
ectors
is
computed
according
to
^
x
k
=
^
x
MMSE
+
e
k
for
k
=
1
;
:
:
:
;
No
rand
(10)
Note
that
generating
more
instances
of
a
random
v
ector
e
k
will
increase
the
probability
to
ha
v
e
one
of
them
as
close
as
possible
to
the
optimal
one.
By
doing
this,
the
o
v
erall
BER
performance
will
impro
v
e.
3.3.
Step
3:
Hard
decoding
In
this
step,
the
decoder
con
v
erts
for
k
=
1
;
:
:
:
;
N
o
r
and
the
soft
decoded
randomized
symbol
v
ector
^
x
k
generating
according
to
(10)
to
hard
decoded
symbol
v
ector
~
^
x
k
by
finding
the
nearest
constellation
point
for
each
soft
decoded
symbol
as
sho
wn
in
Figure
2.
Note
that
this
is
a
symbol
by
symbol
processing
step
performed
using
the
round
operation
with
almost
no
additional
computational
comple
xity
.
3.4.
Step
4:
Selection
In
this
step,
the
decoder
selects
among
the
hard
decoded
symbol
v
ector
~
^
x
k
for
k
=
1
;
:
:
:
;
N
o
r
and
the
v
ector
x
prop
:
that
maximizes
the
ML
metric,
such
as
x
prop
:
=
arg
min
~
^
x
1
;:::;
~
^
x
N
o
r
and
~
y
Rx
2
:
(11)
The
abo
v
e
described
main
procedure
in
Steps
2-3
can
be
ef
ficiently
implemented
using
either
an
iterati
v
e
or
a
parallelized
implementation
as
sho
wn
in
Figure
2.
The
estimate
of
a
symbol
obtained
by
using
MMSE
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
4138
–
4146
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
4141
filter
has
a
bias
or
mean
and
v
ariance.
The
randomization
algorithm
says
that
there
is
a
high
probability
to
get
closer
to
the
actual
symbol
by
searching
randomly
for
a
symbol
set
ha
ving
same
mean
as
our
estimated
symbol
set
and
within
the
limits
of
the
v
ariance
circle.
By
doing
this,
a
better
estimate
of
symbols
can
be
found.
It
is
clear
that
we
can
impro
v
e
the
performance
of
the
decoder
by
increasing
the
number
of
randomiza-
tion
instances,
i.e.,
the
v
alue
of
N
o
r
and
.
This,
ho
we
v
er
,
comes
at
the
e
xpense
of
increased
decoding
com-
ple
xity
.
Therefore,
the
performance
(or
comple
xity)
of
this
decoder
can
be
adapti
v
ely
adjusted
by
changing
the
number
of
randomization
instances
without
the
require
ment
of
changing
the
structure
of
the
implemen-
tation,
i.e.,
t
he
performance
to
comple
xity
trade-of
f
can
be
adjusted
using
system
parameter
N
o
r
and
.
Fur
-
thermore,
the
proposed
algorithm
enjo
ys
simple
implementation
based
on
the
widely
used
MMSE
technique.
The
proposed
algorithm
of
fers
a
fix
ed
decoding
comple
xity
that
does
not
depend
on
the
quality
of
the
recei
v
ed
signal
v
ector
y
.
Figure
2.
Block
diagram
of
the
proposed
decoder
4.
SIMULA
TION
RESUL
TS
In
the
simulations,
let
us
considered
MIMO
systems
with
independent
flat
Rayleigh
f
ading
channels
and
either
four
transmit
and
four
recei
v
e
antennas,
eight
transmit
and
eight
recei
v
e
antennas,
or
20
transmit
and
20
recei
v
e
antennas.
All
MIMO
detectors
using
4-QAM,
16-QAM,
and
64-QAM
constellations
are
compared.
In
the
proposed
decoder
,
the
symbols
can
also
be
dra
wn
from
an
y
M-PSK
constellation.
In
all
illustrated
figures,
i
t
can
be
observ
ed
that
the
ZF
decoder
e
xhibits
the
w
orst
performance,
ho
we
v
er
,
with
linear
decoding
comple
xity
.
On
the
other
hand,
the
curv
es
which
enjo
y
the
optimal
decoding
performance
in
an
y
of
the
figures
represent
the
sphere
decoder
or
ML
decoder
,
ho
we
v
er
,
ML
decoder
suf
fers
from
e
xtremely
high
(e
xponential)
decoding
comple
xity
.
An
y
other
decoder
has
a
performance
and
decoding
comple
xity
between
that
of
the
ML
and
the
ZF
decoder
.
In
the
follo
wing
figures,
ZF
,
MMSE,
SIC,
SD,
ML,
K,
and
R
and
=
L
denote
the
ZF
decoder
,
MMSE
decoder
,
SIC
decoder
,
sphere
decoder
,
ML
decoder
,
K-best
decoder
with
K
nT
x
iterations
and
the
proposed
decoder
using
randomization
technique
wi
th
L
iterations.
From
the
figures,
it
is
observ
ed
that
i)
the
decoders
using
MMSE
outper
form
those
using
ZF
due
to
their
rob
ustness
wi
th
respect
to
noise
enhancement
as
compared
to
the
ZF
decoders,
ii)
the
ML
decoder
enjo
ys
optimal
performance
at
the
cost
of
v
ery
high
decoding
comple
xity
,
iii)
the
proposed
decoder
can
im
pro
v
e
the
comple
xity-performance
trade-of
f
where
it
k
eeps
the
comple
xity
considerably
lo
wer
than
that
of
the
ML
detectors
with
almost
optimal
performance,
and
i
v)
the
sphere
decoder
which
enjo
ys
the
optimal
performance
does
not
ha
v
e
a
fix
ed
comple
xity
and
in
specific
cases
the
comple
xity
may
be
as
lar
ge
as
the
comple
xity
of
the
ML
decoder
,
ho
we
v
er
,
some
sub-optimal
sphere
decoders
enjo
ys
fix
ed
decoding
comple
xity
,
e.g.,
K-best
decoder
with
suboptimal
performance
as
sho
wn
in
Figure
3
[20].
A
computationally
ef
ficient
detector
for
...
(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
4142
r
ISSN:
2088-8708
5.
COMPLEXITY
-PERFORMANCE
TRADE-OFF
From
Figures
3,
4,
5
and
6,
both,
sphere
decoder
and
ML
decoder
are
optimal
and
ha
v
e
e
xactly
the
same
BER
performance
and
the
proposed
decoder
using
only
10
iterations
outperforms
the
suboptimal
decoders,
i.e.,
ZF
,
MMSE,
and
SIC
with
and
without
ordering
using
ZF
or
MMSE.
W
e
emphasize
that,
in
the
proposed
randomization
based
decoder
,
the
nulling
step,
i.e.,
the
matrix
in
v
ersion
step
using
QR-
decomposition,
is
carried
out
only
once
and
bef
o
r
e
the
randomization
is
started
as
discus
sed
in
Sec.
3.,
while
SIC
decoder
performs
the
same
step
e
v
ery
layer
[2].
In
all
in
v
estig
ated
decoders,
i.e.,
ZF
decoder
,
MMSE
decoder
,
ML
decoder
,
sphere
decoder
,
SIC
decoder
,
K-best
decoder
and
the
proposed
randomization
based
decoder
,
the
QR
decomposition
stage
described
in
Sec.
2.
requires
O
((
nT
x
i
)
3
)
operations.
Once
the
QR
decomposition
of
H
is
obtained,
the
remainder
stages
in
the
SIC
decoder
at
the
i
th
layer
,
the
ZF/MMSE
decoder
,
the
ML
decoder
,
K-best
decoder
,
and
the
proposed
decoder
require
O
((
nT
x
i
)
2
)
operations,
O
((
nT
x
)
2
)
operations,
O
((
nT
x
)
2
:
38
)
M
nT
x
operations,
O
((
nT
x
)
2
:
38
)
K
nT
x
+
nT
x
O
((
nT
x
)
2
)
opera-
tions,
and
O
((
nT
x
)
2
:
38
)
N
o
r
and
+
O
((
nT
x
)
3
)
operations,
respecti
v
ely
,
as
sho
wn
in
T
able
1.
Note
that,
in
the
proposed
decoder
,
computing
the
in
v
erse
of
the
channel
matrix,
i.e.,
H
1
=
R
1
Q
H
,
requires
O
((
nT
x
)
3
)
after
obtaining
the
QR
decomposition
of
H
.
From
Figure
3,
it
is
observ
ed
that
the
BER
performance
of
the
proposed
decoder
with
N
o
r
and
=
50
iterations
enjo
ys
the
same
performance
of
K
-best
decoder
with
K
4
=
10
4
iterations.
Clearly
,
the
performance
of
the
propose
d
decoder
outperforms
K
-best
decoder
at
the
same
decoding
comple
xity
where
K
-best
detection
algorithm
suf
fers
from
tw
o
main
problems
which
are
the
e
xpansion
and
the
sorting
operations
.
K
-best
algo-
rithm
e
xpands
each
K
retained
paths
to
its
K
possible
children
at
each
le
v
el.
The
pre
vious
step
requires
sorting
the
children
in
each
layer
before
selecting
the
best
K
paths.
Therefore,
its
decoding
comple
xity
increases
e
x-
ponentially
with
the
increase
of
the
v
alue
K
where
a
high
decoding
comple
xity
is
required
to
enumerate
the
children
nodes
especially
in
the
case
of
lar
ge
number
of
transmit
antennas
and
high
constellation
sizes
as
sho
wn
in
T
able
I,
while
the
comple
xity
of
the
proposed
decoder
increases
linearly
with
the
v
alue
of
N
o
r
and
.
It
can
be
observ
ed
from
Figures
3,
4,
5
and
T
able
2
that
the
proposed
decoder
using
only
200
iterations
achie
v
es
almost
the
same
performance
as
the
optimal
ML
decoder
which
requires
64
4
=
16777216
iterations.
From
T
able
1,
it
is
observ
ed
for
lo
w
constellation
sizes,
that
the
comple
xity
of
the
proposed
decoder
is
similar
to
that
of
the
costly
ML
decoder
.
This
is
also
the
case
if
N
o
r
and
=
M
nT
x
.
Ho
we
v
er
,
the
v
alue
e
xponential
gro
wth
of
M
nT
x
with
the
increas
e
of
the
number
of
transmit
antennas
and
the
constellation
size
is
generally
much
lar
ger
than
the
corresponding
gro
wth
rate
of
N
o
r
and
required
to
achie
v
e
similar
performance.
P
articularly
for
a
lar
ge
constellation
size
and
a
lar
ge
number
of
transmit
antennas
as,
e.g.,
in
the
case
64-QAM
constella-
tions
and
nT
x
=
4
transmit
antennas
as
sho
wn
in
T
able
2,
the
performance
of
the
proposed
decoder
using
only
N
o
r
and
=
200
iterations
enjo
ys
similar
performance
as
the
optimal
ML
decoder
.
Figure
3.
Performance
comparison
of
dif
ferent
decoding
schemes
using
4
4
system
and
16-QAM
constellation
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
4138
–
4146
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
4143
Figure
4.
Performance
comparison
of
dif
ferent
decoding
schemes
using
4
4
system
and
64-QAM
constellation
Figure
5.
Performance
comparison
of
dif
ferent
decoding
schemes
using
8
8
system
and
4-QAM
constellation
Figure
6.
Performance
comparison
of
dif
ferent
decoding
schemes
using
4-QAM
constellation
and
20
20
system
A
computationally
ef
ficient
detector
for
...
(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
4144
r
ISSN:
2088-8708
T
able
1.
Coding
comple
xity
of
the
in
v
estig
ated
MIMO
detectors
ZF
or
MMSE
decoder
SIC
decoder
ML
decoder
K-best
decoder
The
proposed
decoder
Decoding
O
(
nT
x
)
3
O
(
nT
x
)
3
O
(
nT
x
)
2
:
38
M
(
nT
x
)
O
(
nT
x
)
2
:
38
K
nT
x
O
(
nT
x
)
2
:
38
N
o
r
and
+
+
+
+
+
comple
xity
O
(
nT
x
)
2
nT
x
1
P
i
=0
O
(
nT
x
i
)
2
O
(
nT
x
)
3
nT
x
1
P
i
=0
O
(
nT
x
)
2
2
O
(
nT
x
)
3
+
O
(
nT
x
)
3
T
able
2.
Comparison
of
the
relati
v
e
decoding
comple
xity
of
the
proposed
decoder
with
respect
to
the
ML
decoder
Number
of
iterations
Relati
v
e
comple
xity
of
Scenario
N
o
r
and
the
proposed
decoder
w
.r
.t.
the
ML
decoder
(4
4)
MIMO
with
16
-QAM
200
0
:
3052
%
(4
4)
MIMO
with
64
-QAM
200
0
:
0012
%
(8
8)
MIMO
with
4
-QAM
200
0
:
3052
%
(20
20)
MIMO
with
4
-QAM
5000
4
:
5
10
7
%
6.
EFFECT
OF
NOISE
V
ARIANCE
In
this
section,
let
us
compare
the
rob
ustness
of
the
proposed
decoder
with
respect
to
a
mismatch
between
the
true
and
the
noise
v
ariance
at
the
recei
v
er
.
In
the
simulati
ons,
let
us
consider
a
M
IMO
system
with
four
transmit
and
four
recei
v
e
antennas
and
a
true
noise
v
ariance
of
0
-dB.
W
e
assume
that
the
SNR
at
the
r
ecei
v
er
amounts
to
S
N
R
=
P
t
=
2
=
12
dB
in
Figure
7
and
S
N
R
=
P
t
=
2
=
17
dB
in
Figure
8,
and
the
estimated
(presumed)
SNR
at
the
recei
v
er
side
is
v
aried
between
0
-dB
to
20
-dB.
From
Figures
7
and
8,
it
is
observ
ed
that
for
a
number
of
randomization
instances
e
xceeding
N
r
and
=
20
the
performance
of
the
proposed
algorithm
in
terms
of
BER
remains
approximately
constant
as
the
estimated
recei
v
e
SNR
is
v
aried
across
the
entire
range
considered
in
the
simulations.
This
sho
ws
that
the
proposed
algorithm
is
f
airly
rob
ust
with
respect
to
a
mismatch
in
the
noise
v
ariance
or
SNR
estimation.
This
is
due
to
the
idea
of
the
proposed
decoder
which
depends
on
generating
random
v
ectors.
These
random
v
ectors
could
be
f
ar
a
w
ay
from
the
optimal
one,
ho
we
v
er
,
there
is
a
high
probability
that
some
of
them
will
lie
v
ery
close
to
it
e
v
en
if
the
v
ariance
of
the
noise
is
changed.
Figure
7.
Rob
ustness
of
the
proposed
decoder
to
a
mismatch
between
the
true
SNR
v
alue
(
P
t
=
2
=
12
dB
)
and
the
presumed
SNR
v
alue
in
a
4
4
system
using
4-QAM
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
4138
–
4146
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
4145
Figure
8.
Rob
ustness
of
the
proposed
decoder
to
a
mismatch
between
the
true
SNR
v
alue
(
P
t
=
2
=
17
dB
)
and
the
presumed
SNR
v
alue
in
a
4
4
system
using
16-QAM
7.
CONCLUSION
The
proposed
decoder
appears
to
find
the
optimal
compromise
between
competing
interests
encoun-
tered
in
the
implementation
of
adv
anced
MIMO
detectors
in
practical
hardw
are
systems.
The
proposed
de-
tector
e
xhibits
a
number
of
desirable
properties
such
as:
i)
determini
stic
latenc
y
where
the
propos
ed
decoder
e
xhibits
configurable
and
fully
deterministic
decoding
comple
xity
,
which
of
fers
the
benefit
of
a
fix
ed
decoding
comple
xity
,
ii)
full
parameterizable
performance/comple
xity
tradeof
f
where
the
modification
of
the
number
of
randomization
instances
used
in
the
proposed
decoder
allo
ws
to
balance
at
runtime
the
tradeof
f
between
perfor
-
mance
and
computational
comple
xity
,
iii)
simple
implementation
where
the
proposed
algorithm
enjo
ys
simple
implementation
with
a
minimum
requirement
of
control
structures
and
the
proposed
detector
allo
ws
a
high
de
gree
of
parallelization,
i
v)
e
xtension
to
soft-bit
output
where
the
proposed
decoder
can
naturally
be
e
xtended
to
create
soft-bit
outputs
as
required
in
modern
cellular
communication
standards.
REFERENCES
[1]
S.
Alabed,
J.
P
aredes,
and
A.
B.
Gershman,
”A
simple
distrib
uted
space-time
coded
strate
gy
for
tw
o-w
ay
relay
channels,
”
IEEE
T
r
ansactions
on
W
ir
eless
Communications
,
pp.
1260-1265,
v
ol.
11,
no.
4,
April,
2012.
[2]
Ale
x
B.
Gershman
and
N.
Sidiropoulos,
”Space-time
processing
for
MIMO
communications,
”
J
ohn
W
ile
y
and
Sons,
Ltd,
2005.
[3]
S.
Alabed,
”Performance
analysis
of
dif
ferential
beamforming
in
decentralized
netw
orks,
”
International
J
ournal
of
Electrical
and
Computer
Engineering
,
pp.
1692-1700,
v
ol.
8,
no.
3,
June
2018.
[4]
S.
Alabed,
”Computationally
ef
ficient
multi-antenna
techniques
for
multi-user
tw
o-w
ay
wireless
relay
netw
orks,
”
International
J
ournal
of
Electrical
and
Computer
Engineering
,
pp.
1684-1691,
v
ol.
8,
no.
3,
June
2018.
[5]
S.
Alabed,
”Performance
analysis
of
tw
o-w
ay
DF
relay
selection
techniques,
”
Special
Issue
on
ICT
Con
ver
g
ence
in
the
Internet
of
Things
(IoT),
Else
vier
,
pp.
91-95,
vol.
2,
no.
3
,
2016.
DOI:
10.1016/j.icte.2016.08.008.
[6]
S.
Alabed,
M.
Pesa
v
ento,
and
A.
Gershman,
”Distrib
uted
dif
ferential
space-time
coding
techniques
for
tw
o-w
ay
wireless
relay
netw
orks,
”
In
Pr
oceedings
of
the
F
ourth
IEEE
International
W
orkshop
on
Com-
putational
Advances
in
Multi-Sensor
Adaptive
Pr
ocessing
(CAMSAP
11)
,
pp.
221-224,
San
Juan,
Puerto
Rico,
2011.
[7]
S.
Alabed,
M.
Pesa
v
ento,
and
A.
Klein,
”Distrib
uted
dif
ferential
space-time
coding
for
tw
o-w
ay
relay
net-
w
orks
using
analog
netw
ork
coding,
”
In
Pr
oceedings
of
the
1st
Eur
opean
Signal
Pr
ocessing
Conf
er
ence
(EUSIPCO’13)
,
Marrak
ech,
Morocco,
Sep.
9-13,
2013.
[8]
S.
Alabed,
M.
Pesa
v
ento,
and
A.
Klein,
”Relay
selection
based
space-time
coding
for
tw
o-w
ay
wireless
relay
netw
orks
using
digital
netw
ork
coding”,
The
T
enth
International
Symposium
on
W
ir
eless
Communi-
cation
Systems,
TU
Ilmenau,
Ilmenau,
Germany
,
A
ug
.
27-30
,
2013.
[9]
S.
Alabed
and
M.
Pesa
v
ento,
”A
simple
distrib
uted
dif
ferential
transmit
beamforming
technique
for
tw
o-
A
computationally
ef
ficient
detector
for
...
(Samer
Alabed)
Evaluation Warning : The document was created with Spire.PDF for Python.
4146
r
ISSN:
2088-8708
w
ay
wireless
relay
netw
orks,
”
In
the
16th
International
IEEE/ITG
W
orkshop
on
Smart
Antennas
(WSA
2012)
,
pp.
243-247,
Dresden,
German
y,2012
[10]
A.
Schad,
S.
Alabed,
H.
De
genhardt,
and
M.
Pesa
v
ento,
”Bi-directional
dif
ferential
beamforming
for
multi-antenna
relaying,
”
40th
IEEE
International
Confer
ence
on
Acoustics,
Speec
h
and
Signal
Pr
ocess-
ing
,
2015.
[11]
S.
Alabed,
M.
Pesa
v
ento,
and
A.
Klein,
”Non-coherent
distrib
uted
space-time
coding
techniques
for
tw
o-
w
ay
wireless
relay
netw
orks,
”
EURASIP
Special
Issue
on
Sensor
Arr
ay
Pr
ocessing
,
2013.
[12]
X.
W
en,
K.
La
w
,
S.
Alabed,
and
M.
Pes
a
v
ent
o,
”Rank-tw
o
beamforming
for
single-group
multicast-
ing
netw
orks
using
OSTBC,
”
Pr
oc.
of
the
7th
IEEE
Sensor
Arr
ay
and
Multic
hannel
Signal
Pr
ocessing
W
orkshop
(SAM)
,
pp.
65-68,
Jun.
2012.
[13]
D.
T
aleb,
S.
Alabed,
and
M.
Pesa
v
ento,
”Optimal
general-rank
transmit
beamforming
t
echnique
for
multicasting
service
in
modern
wireless
netw
orks
using
STTC,
”
Pr
oceedings
of
the
19th
International
IEEE/ITG
W
orkshop
on
Smart
Antennas
(WSA
2015),
Ilmenau,
German
y
,
March
2015.
[14]
N.
Miyazaki,
S.
Y
oshiza
w
a,
and
Y
.
Miyanag
a,
”Lo
w-po
wer
dynamic
MIMO
detection
for
a
44
MIMO-
OFDM
recei
v
er
,
”
IEICE
T
r
ansactions
on
Fundamentals
of
Electr
onics,
Communications
and
Computer
Sciences,
v
ol.
E97.A,
no.
1,
pp.
306-312,
2014.
[15]
S.
Alabed,
J.
P
aredes,
and
A.
Gershman,
“
A
lo
w
comple
xity
decoder
for
quasi-orthogonal
space-time
block
codes,
”
IEEE
T
r
ansactions
on
W
ir
eless
Communications
,
v
ol.
10,
no.
3,
March
2011.
[16]
M.
Neina
v
aie
and
M.
Derakhtian,
”ML
performance
achie
ving
algorithm
with
the
zero-forcing
comple
xity
at
high
SNR
re
gime,
”
IEEE
T
r
an.
on
W
ir
eless
Comm.,
v
ol.
15,
no.
7,
pp.
4651-4659,
July
2016.
[17]
M.
Ammari
and
P
.
F
ortier
,
”Lo
w
comple
xity
ZF
and
MMSE
detectors
for
the
uplink
MU-MIMO
systems
with
a
time-v
arying
number
of
acti
v
e
users,
”
IEEE
T
r
an.
on
V
ehicular
T
ec
h.,
v
ol.
66,
no.
7,
pp.
6586-6590,
July
2017
.
[18]
S.
Ahmed
and
S.
Kim,
”Ef
ficient
SIC-MMSE
MIMO
detection
with
three
iterati
v
e
loops,
”
International
J
ournal
of
Electr
onics
and
Communications,
v
ol.
72,
pp.
65-71,
Feb
.
2017.
[19]
X.
Zhang,
M.
Zhang,
Q.
Zhao,
et
al.,
”Comparison
of
V
-BLAST/OSIC
algorithm
and
the
QR
decompo-
sition
algorithm,
”
Int.
Conf
.
on
Measur
ement
Information
and
Contr
ol
,
pp.
1158-1162,
October
2013.
[20]
K.
W
ong,
C.
Tsui,
R.
Cheng,
and
W
.
Mo
w
,
”A
VLSI
architecture
of
a
K-best
lattice
decoding
algorithm
for
MIMO
channels,
”
in
Pr
oc.
IEEE
Int.
Symp.
Cir
cuits
Syst.
v
ol.
32,
pp.
273-276,
May
2002.
BIOGRAPHY
OF
A
UTHORS
Samer
Alabed
Samer
Alabed
joined
American
Uni
v
ersity
of
the
Middle
East
as
an
assistant
pro-
fessor
of
electrical
engineering
in
2015.
He
recei
v
ed
his
PhD
de
gree
in
electrical
engineering
and
information
technology
with
great
honor
(”magna
cum
laude”),
from
Darms
tadt
Uni
v
ersity
of
T
ech-
nology
,
Darmstadt,
German
y
and
his
Bachelor
and
Master
de
gree
with
great
honor
.
During
the
last
16
years,
he
has
serv
ed
as
an
assistant
professor
,
(post-doctoral)
researcher
,
teaching
assistant,
and
lecturer
in
se
v
eral
uni
v
ersities
in
German
y
and
Middle
East
and
supervised
tens
of
master
theses
and
se
v
eral
PhD
students.
Dr
.
Alabed
recei
v
ed
se
v
eral
a
w
ards
from
IEE,
IEEE,
D
AAD
...
etc.,
where
the
last
one
w
as
the
best
paper
a
w
ard
from
the
International
IEEE
WSA
in
March,
2015.
Dr
.
Alabed
has
w
ork
ed
as
a
researcher
in
se
v
eral
uni
v
ersities
and
companies
and
w
as
in
vited
to
man
y
conferences
and
w
orkshops
in
Europe,
US,
and
North
Africa.
The
main
idea
of
his
research
is
to
de
v
elop
ad-
v
anced
DSP
algorithms
in
the
area
of
wireless
communication
systems
and
netw
orks
including
(Mas-
si
v
e)
MIMO
systems,
distrib
uted
system
s,
co-operati
v
e
communications,
relay
netw
orks,
space-time
block
and
trellis
coding,
dif
fe
rential
and
blind
multi-antenna
techniques,
MIMO
channel
estima-
tion,
MIMO
decoders,
channel
coding
and
modulation
techniques,
distrib
uted
communication
sys-
tems,
tw
o-w
ay
relayi
ng,
baseband
communications,
multi-carrier
transmission
(OFDM),
modeling
of
wireless
channel
characteristics,
adapti
v
e
beamforming,
sensor
array
processing,
transcei
v
er
de-
sign,
multi-user
and
multi-carrier
wireless
communication
systems,
con
v
e
x
optimization
algorithms
for
signal
processing
communications,
channel
equalizat
ion,
and
other
kinds
of
distortion
and
inter
-
ference
mitig
ation.
Further
information
on
his
homepage:
http://drsameralabed.wixsite.com/samer
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
5,
October
2019
:
4138
–
4146
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