Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
2,
April
2020,
pp.
2139
2150
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i2.pp2139-2150
r
2139
Efficient
po
wer
allocation
method
f
or
non
orthogonal
multiple
access
5G
systems
Maan
A.
S.
Al-Ad
wany
Department
of
Computer
and
Information
Engineering,
Nine
v
ah
Uni
v
ersity
,
Iraq
Article
Inf
o
Article
history:
Recei
v
ed
Apr
10,
2019
Re
vised
Oct
19,
2019
Accepted
Oct
27,
2019
K
eyw
ords:
IDMA
NOMA
Po
wer
allocation
ABSTRA
CT
One
of
the
hot
research
topics
for
the
upcoming
5G
(fifth-generation)
wireless
communication
netw
orks
is
the
non
orthogonal
multiple
access
(NOMA)
systems,
where
it
ha
v
e
attracted
both
industrial
and
academic
fields
to
impro
v
e
the
e
xisting
spectral
ef
ficienc
y
.
In
f
act,
the
multiuser
detection
process
for
NOMA
systems
is
lar
gely
af
fected
by
the
po
wer
distrib
ution
of
the
recei
v
ed
signals.
In
this
paper
,
a
ne
w
method
has
been
proposed
to
control
the
transmit
po
wer
among
acti
v
e
users
in
one
of
the
promising
NOMA
systems;
the
interlea
v
e
di
vision
multiple
access
(IDMA)
which
has
been
adopted
here
for
consideration.
Unlik
e
con
v
entional
methods,
where
tedious
mathematical
computations
are
required;
a
simple
and
direct
method
has
been
deri
v
ed.
The
proposed
method
has
been
applied
to
IDMA
system
with
dif
ferent
FEC
codes.
The
obtained
results
sho
w
that
the
proposed
method
outperforms
the
con
v
entional
one
as
compared
to
optimal
results.
Copyright
c
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Maan
Ahmed
Shehathah
Al-Adw
an
y
,
Assistant
Professor
at
Department
of
Computer
and
Information
Engineering,
Colle
ge
of
Electronics
Engineering,
Nine
v
ah
Uni
v
ersity
,
Mosul,
Iraq.
Email:
maanaladw
an
y@yahoo.com
1.
INTR
ODUCTION
Researchers
ha
v
e
considered
mul
tiuser
detection
techniques
for
spread
spectrum
wireless
comm
un
i
-
cations
systems
and,
recently
,
for
non
orthogonal
multiple
access
(NOMA)
systems
.
In
f
act,
the
multiuser
detection
process
is
lar
gely
af
fected
by
the
po
wer
of
the
recei
v
ed
signals
[1-3].
Thus,
managing
these
po
wers
is
crucial
for
interference
suppression
and
the
stability
of
the
NOMA
systems
[4,
5].
In
addition,
one
of
the
major
design
concerns
is
the
po
wer
consumption
reduction
for
mobile
de
vices.
Thus,
minimization
of
total
po
wer
recei
v
ed
by
uplink
r
ecei
v
er
is
required
while
maintaining
the
SINR
(signal
to
interference
plus
noise
ratio)
on
certain
desired
le
v
el
at
final
iteration
of
multiuser
detector
.
That
is
so
called
the
’po
wer
allocation
problem’.
Recently
,
researchers
spare
no
ef
fort
to
solv
e
the
po
wer
allocation
problem
in
NOMA
system
[1,2,6-9],
while
early
researches
on
po
wer
allocation
problem
were
equally
important
that
can
be
found
else
where
[10-15].
Although
the
methods
that
ha
v
e
been
proposed
in
the
literatureare
ha
v
e
in
v
olv
ed
dif
ficul
t
computations;
the
y
did
not
lead
to
e
xact
solutions
as
one
may
e
xpect.
This,
in
turn,
encouraged
me
to
deri
v
e
a
simple
method
to
solv
e
po
wer
allocation
problem
in
IDMA
as
one
of
the
promising
NOMA
systems
for
5G.
The
contrib
ution
of
this
paper
can
be
summerized
with
the
follo
wing
points:
(a)
This
paper
presents
a
method
which
could
be
an
ef
ficient
alternati
v
e
to
the
e
xisting
po
wer
allocation
methods
for
IDMA
system.
(b)
The
proposed
method
dra
ws
its
strength
from
being
simple
and
direct.
It
depends
only
on
the
forw
ard
error
correcting
(FEC)
rate
to
solv
e
po
wer
allocation
problem
in
IDMA
system;
whereas
other
methods
usually
use
linear
programing
or
comple
x
analytical
approaches.
J
ournal
homepage:
http://ijece
.iaescor
e
.com/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
2140
r
ISSN:
2088-8708
This
paper
is
or
g
anized
as
follo
ws:
after
the
introduction,
the
IDMA
uplink
system
is
briefly
described
in
section
2
in
order
for
a
reader
to
focus
only
on
the
po
wer
allocation
problem
which
is
presented
in
section
3.
Whereas
a
modelling
of
the
proposed
method
is
considered
in
section
4.
Numerical
results
are
then
discussed
in
section
5.
Finally
,
the
conclusions
are
dra
wn
in
section
6.
2.
IDMA
UPLINK
SYSTEM
Interlea
v
e
Di
vision
Multiple
Access
(IDMA)
is
a
promising
non
orthogonal
multiple
access
system
[16-19],
so
it
has
been
adopted
for
consideration
in
this
research.
It
can
be
considered
as
a
special
case
of
spread
spectrum
systems.
In
IDMA
system
the
users
are
identified
by
their
unique
int
erlea
v
ers
[20-27],
while
in
spread
spectrum
systems
such
as
CDMA
the
users
are
identified
by
their
unique
codes.
The
mechanism
of
the
IDMA
uplink
system
can
be
clarified
according
to
the
schematic
diagram
sho
wn
in
Figure
1.
M
u
lt
ip
le
A
cces
s
Cha
nne
l
FE
C
e
nc
o
de
r
Co
nv
o
l
ut
i
o
na
l
E
n
c
od
e
r
Rep
e
ti
t
i
on
C
od
e
I
n
t
e
r
l
e
a
ve
r
1
h
1
U
se
r
1
F
E
C e
nc
o
de
r
Co
nv
o
l
ut
i
o
na
l
E
n
c
od
e
r
R
e
p
e
t
i
t
i
on
C
od
e
I
n
t
e
r
l
e
a
ve
r
2
h
2
U
se
r
2
F
E
C e
nc
o
de
r
Co
nv
o
l
ut
i
o
na
l
Enc
o
de
r
R
e
p
e
t
i
t
i
on
C
od
e
I
n
t
e
r
l
e
a
ve
r
K
h
K
U
se
r
K
E
l
e
m
e
n
t
a
r
y
S
ig
n
a
l
E
st
i
m
a
t
o
r
(E
S
E
)
D
e
c
o
d
e
r
De
-
i
n
t
e
r
l
e
a
ve
r
2
U
se
r
2
I
n
t
e
r
l
e
a
ve
r
2
D
e
c
o
d
e
r
De
-
i
n
t
e
r
l
e
a
ve
r
K
U
se
r
K
I
n
t
e
r
l
e
a
ve
r
K
D
e
c
o
d
e
r
De
-
in
t
e
r
l
e
a
ve
r
1
U
se
r
1
I
n
t
e
r
l
e
a
ve
r
1
Figure
1.
The
structure
of
the
IDMA
system
At
the
transmit
side,
the
information
bits
of
each
user
are
encoded
using
certain
FEC
(forw
ard-error
-
correction)
codes
that
are
assumed
the
same
for
all
users.
Each
FEC
code
is
achie
v
ed
by
serial
concatena-
tion
of
con
v
olut
ional
code
of
rate
R
c
along
with
a
repetition
code
of
rate
R
p
resulting
a
total
FEC
code
rate
of
R
t
=
R
c
R
p
.
Consequently
,
the
system
can
be
considered
to
ha
v
e
a
total
spreading
f
actor
S
F
=
1
=R
t
.
Figure
2.
illustrates
some
well-kno
wn
FEC
codes
that
ha
v
e
been
used
in
this
paper
.
After
that,
the
encoded
bits
(also
called
chips)
are
or
g
anized
into
frames
each
has
length
of
1024;
to
be
prepared
for
the
interlea
ving
process
where
ea
ch
user
is
assigned
a
unique
interlea
v
er
.
In
IDMA,
the
k
e
y
principle
is
that
the
user
specific
in-
terlea
v
ers
should
ha
v
e
minimum
cross
correlation
and
must
be
generated
randomly
and
independently
.
In
f
act,
these
user
specific
interlea
v
ers
mak
e
a
dispersion
in
the
encoded
sequences
which
produces
an
approximately
uncorrelated
adjacent
chips;
this
in
turn
f
acili
tates
the
recei
v
er
detection
process.
The
interlea
v
ers
at
recei
v
er
and
transmitter
sides
should
be
stored
some
where
at
t
he
base
station
by
utilizing
a
memory
prepared
for
this
purpose.
Throughout
the
duration
of
the
initial
link
setting-up
stage,
the
base
stat
ion
and
the
mobile
stations
must
e
xchange
messages
to
tell
each
other
about
the
emplo
yed
interlea
v
ers.
In
this
paper
,
in
order
to
focus
on
the
po
wer
allocation
problem,
it
is
assumed
that
the
emplo
yed
interlea
v
ers
are
optimum.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2139
–
2150
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2141
C
o
nv
o
l
ut
i
o
na
l
C
o
de
R
ep
eti
ti
o
n
C
o
d
e R
a
te
(
R
p
)
T
o
ta
l
C
o
d
e
R
a
te
(
R
t
=R
c
R
p
)
To
t
al
S
pr
e
a
di
ng
F
a
ct
o
r
S
F
=
1/R
t
k
m
ax
C
o
de
g
en
er
a
to
r
(o
ct
a
l
)
C
o
ns
t
r
a
i
nt
l
en
g
th
F
e
e
dba
ck
C
o
d
e R
a
te
(
R
c
)
F
E
C1
(
237
345)
8
8
(
237)
8
1/
2
1/
8
1/
16
16
12
F
E
C2
(
27
31)
8
5
(
27)
8
1/
2
1/
8
1/
16
16
13
F
E
C3
(5
3
)
8
3
(5
)
8
1/
2
1/
8
1/
16
16
18
F
E
C4
(
23
35)
8
5
(
23)
8
1/
2
1/
4
1/
8
8
7
Figure
2.
Some
FEC
codes
that
were
used
in
this
research
The
interlea
v
er
output
is
then
scaled
by
coef
ficient
for
each
user
h
k
;
which
is
assumed
to
constitute
the
combined
ef
fect
of
the
channel
loss
and
po
wer
control
f
actor
.
In
other
w
ords,
the
distrib
ution
of
h
2
k
can
be
fitted
via
po
wer
control
process.
The
signals
of
all
acti
v
e
users
are
assumed
to
be
superimposed
into
a
wireless
flat
f
ading
channel.
At
the
recei
v
e
side,
the
recei
v
er
is
composed
of
elementary
signal
estimator
(ESE)
along
with
a
bank
of
K
decoders
(DECs)
corresponding
to
K
users.
The
DECs
are
a
posteriori
probability
(APP)
decoders.
A
related
point
to
consider
,
is
that
the
operation
of
the
recei
v
er
is
mainly
based
on
multiple
access
channel
and
the
FEC
code
constraints.
The
outcomes
of
the
ESE
and
DECs
are
combi
ned,
via
an
iterati
v
e
process,
to
retrie
v
e
the
transmitted
bits.
It
is
suitable
here
to
mention
that
we
ha
v
e
designed
and
achie
v
ed
the
IDMA
uplink
system
simulator
using
MA
TLAB,
b
ut
it
has
not
been
included
in
this
paper
to
a
v
oid
redundanc
y
.
3.
PR
OBLEM
FORMULA
TION
As
mentioned
before,
the
performance
of
IDMA
system
can
be
enhanced
through
suitable
po
wer
allocation.
The
proper
po
wer
distrib
ution
among
users
will
highly
increase
the
chance
for
correct
signal
detection
at
the
recei
v
er
.
Strong
signals
are
firstly
detected.
Consequently
,
their
interference
ef
fect
on
weak
signals
can
be
correctly
remo
v
ed
which
in
turn
increase
the
chance
of
weak
signal
detection.
The
main
research
question
here
is:
ho
w
to
allocate
the
po
wer
le
v
els
among
users?.
In
general,
the
po
wer
allocation
problem
can
be
solv
ed
by
di
viding
the
total
number
of
multiple
access
users
into
groups,
where
for
each
a
certain
po
wer
le
v
el
is
assigned.
As
a
consequence,
the
former
research
question
triggers
the
follo
wing
three
sub-questions:
(a)
What
is
the
proper
size
for
each
group?
(b)
What
is
the
proper
po
wer
le
v
el
for
each
group
to
achie
v
e
feasible
tar
get
BER?
(c)
If
1
and
2
are
satisfied,
has
the
total
transmitted
po
wer
(
P
t
)
from
all
acti
v
e
users
been
minimized
as
a
result?
Early
researches
tried
to
solv
e
this
issue;
for
instance
Sc
hle
gel
et
al.
[28]
pro
v
ed
that
the
minimum
total
po
wer
can
be
achie
v
ed
when
the
groups
ha
v
e
an
equal
number
of
users;
so
one
can
adopt
this
assumption
as
a
foundation
to
meet
other
sub-questions.
Therefore,
the
objecti
v
e
of
this
research
becomes
ho
w
to
find
the
proper
po
wer
le
v
el
for
each
group
so
that
the
minimum
total
po
wer
condition
is
satisfied
with
feasible
BER
(typically
10
6
).
4.
MODELLING
OF
PR
OPOSED
METHOD
Unlik
e
the
e
xisting
researches,
this
paper
ha
v
e
re
v
ealed
a
ne
w
feature
for
FEC
codes
which
can
be
utilized
to
solv
e
the
po
wer
allocation
problem.
The
feature
tells
us
that
all
FEC
codes
can
support
limitted
number
of
users
(users
with
same
po
wer
le
v
el).
Ho
we
v
er
,
e
xceeding
this
limit
leads
to
de
gradation
in
system
performance.
Figure
3
sho
ws
the
simulation
results
of
the
BER
with
number
of
equal
po
wer
users
for
dif-
ferent
FEC
codes.
One
can
observ
e
that
for
each
FEC
code,
a
tar
get
BER
of
10
6
can
be
achie
v
ed
as
long
as
the
number
of
users
do
not
e
xceed
certain
t
h
r
eshold
say
k
max
.
Abo
v
e
this
threshold,
the
BER
jumps
to
infeasible
v
alues.
In
this
w
ork,
such
a
feature
has
been
e
xploited
to
solv
e
the
po
wer
allocation
problem.
It
is
ob
vious,
from
the
former
figure,
that
k
max
for
the
codes
FEC1,
FEC2,
and
FEC3,
are
12,
13,
and
18,
respecti
v
ely
.
Ef
ficient
power
allocation
method
for
non
ortho
gonal...
(Maan
A.
S.
Al-Adwany)
Evaluation Warning : The document was created with Spire.PDF for Python.
2142
r
ISSN:
2088-8708
Figure
2
demonstrates
some
FEC
codes
with
their
corresponding
k
max
s.
T
o
clarify
our
proposed
method,
it
is
suitable
to
discuss
the
follo
wing
e
xample.
Assume
an
IDMA
(NOMA)
system
with
BPSK
mapping
of
S
N
R
=
10
dB
and
with
the
follo
wing
s
pecifications:
the
total
number
of
users
(
K
)
is
32;
a
FEC
code
consisting
of
con
v
olutional
code
of
rate
1
=
2
serially
concatenated
with
a
repetition
code
of
length
8
(i.e
R
p
=
1
=
8
).
Thus
the
total
FEC
rate
is
R
t
=
1
=
16
.
In
order
to
distrib
ute
the
po
wer
among
acti
v
e
users;
first,
one
should
di
vide
the
total
number
of
users
K
into
groups.
W
e
start
with
the
case
of
32
groups,
each
group
consists
of
one
user
.
No
w
,
assume
that
the
po
wer
set
f
p
1
p
2
:::
p
32
g
represents
a
solution
to
the
po
wer
allocation
problem.
The
po
wer
set
should
satisfy
tw
o
conditions:
a
minimum
total
po
wer;
and
con
v
er
gence
to
feasible
BER.
By
assuming
that
the
up-
link
users
ha
v
e
the
same
transmission
rate,
one
can
pro
v
e
that
the
po
wer
set
should
follo
w
a
constant
ratio
according
to
[29];
hence:
p
2
=p
1
=
p
3
=p
2
::::
=
p
K
=p
K
1
=
where
represents
po
wer
ratio.
Assume
p
1
represents
the
minimum
transmit
po
wer
allo
wed
for
each
user
(
p
min
),
hence
the
po
wer
set
can
be
re
written
as:
f
p
min
p
min
2
p
min
:::::
31
p
min
g
or
p
min
f
1
2
:::::
31
g
The
po
wer
ratio
(
)
in
this
case
is
equal
to
.
It
is
w
orth
mentioning
that
p
min
can
be
determined
according
to
the
wireless
system
requirements.
Thus,
solution
of
the
po
wer
allocation
problem
is
mainly
achie
v
ed
by
finding
.
Since
total
po
wer
should
be
minimized,
hence
the
po
wer
allocation
problem
can
be
solv
ed
by
finding
the
minimum
(i.e.
min
),
or
for
the
sak
e
of
generality
min
.
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
32
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
N
B
E
R
F
E
C
1
F
E
C
2
F
E
C
3
k
m
a
x
1
k
m
a
x
3
k
m
a
x
2
S
F
=
1
6
Figure
3.
BER
vs.
number
of
equal
po
wer
users
4.1.
Finding
minimum
po
wer
ratio
(
min
)
Going
forw
ard
to
find
min
,
assume
that
g
represents
the
number
of
users
per
group
(or
the
group
size).
Thus,
for
the
same
t
otal
number
of
users
K
=
32
,
in
or
d
e
r
to
increase
the
group
size
from
1
to
2
one
should
raise
the
po
wer
ratio
from
to
2
as
sho
wn
in
Figure
4;
starting
from
rightmost
column
.
This
is
logically
acceptable
if
one
thinks
of
it
as
to
ensure
that
each
group
wi
ll
ha
v
e
enough
po
wer
for
con
v
er
ging
to
the
tar
get
BER.
By
the
same
w
ay
,
in
order
to
increase
g
to
4,
the
po
wer
ratio
should
be
increased
to
4
,
and
so
on.
Figure
4
sho
ws
the
possible
group
sizes
along
with
the
corresponding
po
wer
rat
ios.
In
general,
one
can
observ
e
that
for
certain
group
size
(
g
);
the
corres
ponding
po
wer
ratio
should
be
g
.
Figure
5
illustrates
the
ef
fect
of
group
size
on
the
po
wer
ratio.
It
is
ob
vious
that
as
the
group
size
increases,
the
required
po
wer
ratio
also
increases.
While
for
certain
FEC
code,
as
the
group
size
approaches
k
max
,
the
required
po
wer
ratio
f
airly
increases.
Thus,
in
order
to
a
v
oid
unw
anted
increas
e
in
total
po
wer
,
we
recommend
that
the
maximum
group
size
should
not
approach
k
max
for
gi
v
en
FEC
code.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2139
–
2150
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2143
To
t
a
l
num
ber
o
f
u
s
e
r
s
=
3
2
2
g
r
o
u
p
s
4
g
r
o
u
p
s
8
g
r
o
u
p
s
16 g
r
o
u
p
s
32 g
r
o
u
p
s
g=
16
u
s
e
r
s
/
g
r
o
u
p
g=
8
u
s
e
r
s
/
gr
o
u
p
g=
4
u
s
e
r
s
/
gr
o
u
p
g=
2
u
s
e
r
s
/
gr
o
u
p
g=
1
u
s
e
r
s
/
gr
o
u
p
P
o
w
er
Ra
t
i
o
(Ψ
)
=β
16
P
o
w
er
Ra
t
i
o
(Ψ
)
=β
8
P
o
w
er
Ra
t
i
o
(Ψ
)
=β
4
P
o
w
er
Ra
t
i
o
(Ψ
)
=β
2
P
o
w
er
Ra
t
i
o
(Ψ
)
=β
β
0
β
0
β
0
β
0
β
0
β
1
β
2
β
2
β
3
β
4
β
4
β
4
β
5
β
6
β
6
β
7
β
8
β
8
β
8
β
8
β
9
β
10
β
10
β
11
β
12
β
12
β
12
β
13
β
14
β
14
β
15
β
16
β
16
β
16
β
16
β
16
β
17
β
18
β
18
β
19
β
20
β
20
β
20
β
21
β
22
β
22
β
23
β
24
β
24
β
24
β
24
β
25
β
26
β
26
β
27
β
28
β
28
β
28
β
29
β
30
β
30
β
31
P
t
5
= g
P
m
i
n
(β
0
+
β
16
)
P
t
4
= g
P
m
i
n
(β
0
+
β
8
+β
16
+β
24
)
P
t
3
= g
P
m
i
n
(β
0
+
β
4
+β
8
+
...
.+
β
28
)
P
t
2
= g
P
m
i
n
(β
0
+
β
2
+β
4
+
...
.+
β
30
)
P
t
1
= g
P
m
i
n
(β
0
+
β
1
+β
2
+
...
.+
β
31
)
Figure
4.
Estimated
po
wer
ratios
for
dif
ferent
group
sizes
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
5
10
15
20
25
30
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
P
o
w
e
r
R
a
t
i
o
(
Ψ
)
F
E
C
1
F
E
C
3
k
m
a
x
1
k
m
a
x
3
Figure
5.
Po
wer
ratio
vs.
group
size
for
dif
ferent
FEC
codes
Ef
ficient
power
allocation
method
for
non
ortho
gonal...
(Maan
A.
S.
Al-Adwany)
Evaluation Warning : The document was created with Spire.PDF for Python.
2144
r
ISSN:
2088-8708
The
important
note
here
is
that
for
certain
FEC
code
(say
FEC3),
the
sum
of
the
users’
normalized
po
wers
should
not
e
xceed
k
max
as
s
een
by
the
recei
v
er;
otherwise
the
BER
jumps
to
infeasible
v
alues
as
sho
wn
in
Figure
3.
So;
P
T
otal
P
max
k
max
(1)
No
w
,
assume
that
the
number
of
groups
is
decreased
from
32
do
wn
to
2.
Thus,
each
of
the
resulting
tw
o
groups
w
ould
ha
v
e
the
maximum
number
of
users
g
max
(where
g
max
=
16
in
our
e
xample).
Hence,
one
can
deri
v
e;
(
g
max
P
min
+
g
max
P
max
)
P
max
k
max
(2)
where
in
this
case
P
max
=
g
max
P
min
,
thus;
g
max
g
max
+
g
max
k
max
(3)
which
leads
to;
(
g
max
k
max
g
max
)
1
g
max
(4)
Since
total
po
wer
should
be
minimized,
which
means
minimum
po
wer
ratio
among
groups,
hence
one
can
write;
min
=
(
g
max
k
max
g
max
)
1
g
max
(5)
where
g
max
<
k
max
.
Kno
wing
that
g
max
and
k
max
are
i
n
t
e
gers,
and
g
max
should
ha
v
e
maximum
v
alue;
thus
g
max
=
k
max
1
.
Substituting
for
g
max
;
min
=
(
k
max
1)
(
1
k
max
1
)
(6)
It
is
ob
vious
that
one
can
find
a
solution
for
po
wer
allocation
problem
for
an
y
FEC
code
by
kno
wing
its
k
max
,
which
is
the
o
bj
ecti
v
e
of
this
research.
Ho
we
v
er
,
by
kno
wing
that
e
v
ery
FEC
code
has
its
o
wn
k
max
,
which
may
be
some
what
dif
ficult
to
find;
hence
it
is
preferable
if
one
can
mak
e
further
simplification
to
equation
(6).
4.2.
Simplification
of
equation
(6)
In
order
to
understand
(6),
it
is
preferable
to
study
its
characteristics.
Figure
6
sho
ws
the
v
ariation
of
the
minimum
po
wer
ratio
min
v
ersus
range
of
k
max
inte
ger
v
alues.
One
can
observ
e
that
for
k
max
>
4
(which
is
the
case
in
pract
ical
FEC
codes),
increasing
k
max
al
w
ays
results
in
decreasing
the
po
wer
ratio
min
.
This
will
in
turn
reduce
t
he
total
po
wer
.
Although
lar
ger
v
alues
of
min
ensure
iterati
v
e
recei
v
er
con
v
er
gence
to
the
tar
get
BER
with
fe
wer
iterations,
b
ut
it
will
be
at
the
e
xpense
of
unw
anted
i
ncrease
in
total
po
wer
.
Thus,
a
reasonable
trade-of
f
should
be
achie
v
ed.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2139
–
2150
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2145
0
4
8
12
16
20
24
28
32
36
40
44
48
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
k
m
a
x
M
i
n
i
m
u
m
P
o
w
e
r
R
a
t
i
o
(
β
m
i
n
)
p
o
w
e
r
r
a
t
i
o
=
β
∗
m
i
n
k
m
a
x
=
S
F
Figure
6.
min
vs.
k
max
In
order
to
mak
e
a
proper
trade-of
f,
Figure
3
should
be
studied
carefully
.
By
looking
from
the
line
N
=
16
=
S
F
,
one
can
note
that
strong
FEC
codes
such
as
FEC1
and
FEC2
ha
v
e
k
max
<
S
F
.
Whereas
weak
FEC
codes
(such
as
FEC3)
ha
v
e
k
max
>
S
F
.
Thus,
it
may
be
suitable
to
consider
S
F
as
the
a
v
erage
v
alue
for
k
max
for
a
range
of
FEC
codes;
from
the
strongest
one
to
weak
est.
According
to
the
pre
vious
discussion,
an
acceptable
trade-of
f
can
be
achie
v
ed
by
replacing
k
max
in
(6)
by
S
F
.
This
replacement
leads
to
simplify
(6)
to;
min
(
S
F
1)
(
1
S
F
1
)
(7)
Thus,
in
order
to
find
the
po
wer
ratio
min
we
only
need
to
kno
w
the
total
spreading
f
actor
(
S
F
=
1
=R
c
R
p
)
rather
than
kno
wing
k
max
for
each
code.
So,
generally
speaking,
to
find
the
m
inimum
po
wer
ratio
min
for
certain
group
size
(
g
),
the
rule
can
be
re
written
in
the
follo
wing
final
general
form;
min
j
g
=
(
min
)
g
(
S
F
1)
(
g
S
F
1
)
(8)
It
is
w
orth
mentioning
that
the
rule
in
equation
(8)
is
deri
v
ed
under
assumption
that
the
users
who
under
go
the
po
wer
control
process
should
utilize
the
same
total
FEC
rate.
4.3.
Pr
oposed
po
wer
allocation
algorithm
Figure
7
sho
ws
the
proposed
po
wer
allocation
algorithm.
It
starts
by
entering
total
number
of
users,
total
FEC
rate,
and
the
minimum
transmit
po
wer
allo
wed
for
each
user
.
Then
an
initial
v
alue
for
group
size
(
g
)
is
assumed
to
be
1.
After
that,
the
algorithm
runs
to
find
the
group
size
that
satisfies
the
minimum
total
po
wer;
then
allocates
the
po
wer
among
users
accordingly
.
5.
RESUL
TS
AND
DISCUSSION
T
o
v
erify
the
ability
of
our
proposed
method
to
solv
e
the
po
wer
allocation
problem,
the
method
has
been
applied
to
dif
ferent
FEC
codes.
F
or
the
sak
e
of
simplicity
,
FEC3
is
discuss
ed.
Figure
8
sho
ws
tw
o
methods
of
finding
the
po
wer
ratio;
our
proposed
method
(green
color)
and
one
of
the
con
v
entional
methods
for
NOMA
(bro
wn
color)
[1,2,4,6-10,28]
compared
to
the
corresponding
optim
um
method
(blue
color).
Compared
with
the
optimal
results,
the
proposed
method
seems
in
good
agreement.
Ho
we
v
er
,
saying
”in
good
agreement”
can
not
be
considered
as
a
measure.
So,
one
should
put
a
measure
to
describe
the
de
gree
of
agreement
precisely
.
Thus,
in
this
paper
a
term
coincidence
(
C
)
has
been
used
to
represent
this
measure
of
agreement;
more
generally
the
percentage
coincidence
which
can
be
e
xpressed
as
C
=
(1
)
100%
,
where;
=
estimated
optimum
optimum
(9)
C
=
1
estimated
optimum
optimum
100%
(10)
Ef
ficient
power
allocation
method
for
non
ortho
gonal...
(Maan
A.
S.
Al-Adwany)
Evaluation Warning : The document was created with Spire.PDF for Python.
2146
r
ISSN:
2088-8708
St
a
r
t
S
e
t C
o
u
n
te
r
=
1
.
D
is
tr
ib
u
te
U
s
e
r
s
i
n
to
G
r
o
u
p
s:
G
r
o
u
p
S
i
ze
=
2
(
−
1
)
.
Ca
l
cu
l
at
e t
h
e M
i
n
i
m
u
m
P
o
w
e
r
R
a
tio
:
=
(
−
1
)
1
−
1
.
C
al
cu
l
at
e t
h
e P
o
w
er
R
a
tio
(
)
A
c
c
or
di
ng
to
th
e
G
r
o
u
p
S
iz
e
:
(
)
=
.
=
×
∗
(
−
1
)
×
=
1
C
al
cu
l
at
e T
o
t
al
P
o
w
er
:
i
s P
t
mi
n
i
mu
m
?
2
>
2
?
=
+
1
F
o
llo
w
t
he
St
e
p
s
to
A
llo
c
a
te
t
he
P
o
w
e
r
A
m
o
ng U
se
r
s
:
1
-
D
is
tr
ib
u
te
U
s
e
r
s
i
n
to
G
r
ou
ps
.
2
-
F
ol
l
o
w
T
a
bl
e
2 f
or
S
u
ita
b
le
P
o
we
r
A
llo
c
a
tio
n
.
YE
S
NO
YE
S
NO
E
nt
e
r
K
:
T
ot
a
l
N
um
be
r
of
U
s
e
r
s
.
T
o
ta
l F
E
C
R
a
te
:
=
×
.
P
mi
n
:
M
i
n
i
mu
m T
r
a
n
s
mi
t
P
o
we
r
Al
l
o
we
d
F
o
r U
s
e
r.
C
a
lc
u
la
te
t
h
e
S
p
r
ead
i
n
g
F
act
o
r
:
=
1
.
Figure
7.
Proposed
po
wer
allocation
algorithm
2
4
6
8
10
12
14
16
0
2
4
6
8
10
12
14
16
18
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
P
o
w
e
r
R
a
t
i
o
(
Ψ
)
Ψ
c
a
l
c
u
l
a
t
e
d
u
s
i
n
g
N
O
M
A
p
r
o
p
o
s
e
d
m
e
t
h
o
d
Ψ
c
a
l
c
u
l
a
t
e
d
u
s
i
n
g
N
O
M
A
c
o
n
v
e
n
t
i
o
n
a
l
m
e
t
h
o
d
s
Ψ
O
p
t
i
m
u
m
Figure
8.
Comparison
among
dif
ferent
methods
of
finding
the
po
wer
ratio
for
FEC3
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2139
–
2150
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2147
Figure
9
illustrates
the
de
gree
of
coincidence
of
the
proposed
method
as
compared
to
the
con
v
enti
onal
one.
It
is
clear
that
for
the
con
v
entional
method,
the
coincidence
(
C
)
is
e
xcellent
for
g
=
4
or
fe
wer;
whereas
it
drops
do
wn
for
g
lar
ger
than
4.
Ho
we
v
er
,
our
proposed
method
sho
ws
better
performance
especially
for
g
lar
ger
than
4.
Accordingly
,
one
may
decide
to
distrib
ute
users
into
sm
all
groups
(lo
w
v
alues
of
g
)
for
the
sak
e
of
maximum
coincidence.
Ho
we
v
er
,
in
the
po
wer
allocation
problem
some
other
parameters
must
be
tak
en
into
account.
One
of
these,
is
the
total
po
wer
consumed
by
act
i
v
e
users
(
P
t
).
It
is
kno
wn
that
the
purpose
behind
the
po
wer
allocation
is
to
distrib
ute
po
wer
among
users
to
f
acilitate
the
multi-user
detection
process,
with
minimum
total
po
wer
consumption.
Thus,
one
should
choose
minimum
P
t
.
Figure
10
illustrates
that
our
proposed
met
hod
can
achie
v
e
a
minimum
total
po
wer
of
25
dB
at
g
=
16
,
whereas
the
con
v
entional
method
achie
v
es
27
dB
at
g
=
1
.
2
4
6
8
10
12
14
16
0
10
20
30
40
50
60
70
80
90
100
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
C
o
i
n
c
i
d
e
n
c
e
%
N
O
M
A
C
o
n
v
e
n
t
i
o
n
a
l
M
e
t
h
o
d
N
O
M
A
P
r
o
p
o
s
e
d
M
e
t
h
o
d
Figure
9.
Percentage
coincidence
vs.
g
for
FEC3
2
4
6
8
10
12
14
16
20
25
30
35
40
45
50
55
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
R
e
l
a
t
i
v
e
T
o
t
a
l
P
o
w
e
r
[
d
B
]
N
O
M
A
C
o
n
v
e
n
t
i
o
n
a
l
M
e
t
h
o
d
N
O
M
A
P
r
o
p
o
s
e
d
M
e
t
h
o
d
Figure
10.
Relati
v
e
total
po
wer
vs.
g
for
FEC3
It
is
clear
that
our
proposed
method
reduces
the
t
otal
po
wer
by
2
dB
at
g
=
16
.
According
to
Figure
8,
g
=
16
corresponds
to
significant
increase
in
the
po
wer
ratio
(green
color)
as
compared
to
the
optimum
one.
F
ortunately
,
this
increase
will
f
acilitate
the
detection
process
by
reducing
the
number
of
itera-
tions
needed
to
reach
the
tar
get
BER
as
sho
wn
in
Figure
11.
1
5
10
15
10
15
20
25
30
35
40
45
50
55
60
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
N
u
m
b
e
r
o
f
I
t
e
r
a
t
i
o
n
s
N
O
M
A
O
p
t
i
m
a
l
M
e
t
h
o
d
N
O
M
A
P
r
o
p
o
s
e
d
M
e
t
h
o
d
Figure
11.
Recei
v
er
iterations
vs.
group
size.
A
comparison
between
the
proposed
method
and
the
simulation
results
for
FEC3
It
is
important
here
to
mention
that
the
‘optimum
method’
denotes
to
the
method
that
achie
v
es
the
minimum
total
po
wer
re
g
ardless
the
number
of
recei
v
er
iterations
required
to
reach
tar
get
BER.
No
w
,
ref
erring
to
Figure
11,
our
proposed
method
sho
ws
considerable
reduction
in
the
number
of
iterations
o
v
er
the
possible
v
alues
of
g
.
This
can
be
interpreted
as
follo
ws:
from
the
iterati
v
e
recei
v
er
point
of
vie
w
,
the
increase
in
the
po
wer
ratio
increases
the
recei
v
er
ability
to
discriminate
among
users
groups.
This
in
turn
Ef
ficient
power
allocation
method
for
non
ortho
gonal...
(Maan
A.
S.
Al-Adwany)
Evaluation Warning : The document was created with Spire.PDF for Python.
2148
r
ISSN:
2088-8708
enhances
the
recei
v
er
ability
to
con
v
er
ge
to
the
tar
get
BER
(
10
6
)
with
fe
wer
number
of
iterations.
In
other
w
ords,
it
means
reducing
the
time
for
recei
v
er
to
achie
v
e
its
assigned
task.
Figure
12
illustrates
the
percentage
reduction
in
the
time
required
by
the
recei
v
er
to
con
v
er
ge
to
the
tar
get
BER
which
is
called
here
’T
ask
T
ime
Reduction’.
It
is
clear
that
choosing
g
=
16
can
reduce
task
time
by
47%.
Moreo
v
er
,
to
further
v
erify
the
v
ali
d
i
ty
of
our
proposed
method;
we
ha
v
e
applied
it
to
dif
ferent
FEC
codes.
Figure
13
and
Figure
14
sho
w
the
relati
v
e
total
po
wer
graphs
for
FEC2
and
FEC1,
respecti
v
ely
.
It
is
clear
that
our
proposed
method
outperforms
the
con
v
entional
one
by
increasing
the
FEC
code
constraint
length.
In
other
w
ords,
the
proposed
method
performs
better
as
the
FEC
code
becomming
stronger
.
1
5
10
15
45
50
55
60
65
70
75
80
85
90
95
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
T
a
s
k
T
i
m
e
R
e
d
u
c
t
i
o
n
%
Figure
12.
The
percentage
T
ask
T
ime
reduction
vs.
g
for
FEC3
2
4
6
8
10
12
20
25
30
35
40
45
50
55
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
R
e
l
a
t
i
v
e
T
o
t
a
l
P
o
w
e
r
[
d
B
]
N
O
M
A
C
o
n
v
e
n
t
i
o
n
a
l
M
e
t
h
o
d
N
O
M
A
P
r
o
p
o
s
e
d
M
e
t
h
o
d
Figure
13.
Relati
v
e
total
po
wer
vs.
g
for
FEC2
1
2
3
4
5
6
7
8
9
10
20
25
30
35
40
45
50
55
60
N
u
m
b
e
r
o
f
u
s
e
r
s
p
e
r
g
r
o
u
p
(
g
)
R
e
l
a
t
i
v
e
T
o
t
a
l
P
o
w
e
r
[
d
B
]
N
O
M
A
C
o
n
v
e
n
t
i
o
n
a
l
M
e
t
h
o
d
N
O
M
A
P
r
o
p
o
s
e
d
M
e
t
h
o
d
Figure
14.
Relati
v
e
total
po
wer
vs.
g
for
FEC1
6.
CONCLUSIONS
In
this
paper
,
a
ne
w
method
has
been
deri
v
ed
to
simplify
the
solution
of
the
po
wer
allocation
problem
in
one
of
the
promising
NOMA
systems;
that
is
the
IDMA
syst
em.
Compared
with
the
con
v
entional
methods,
the
proposed
method
is
pro
v
ed
to
be
ef
ficient
and
easy
to
apply
because
it
needs
only
total
system
FEC
rate
to
find
a
solution
to
the
po
wer
allocation
problem.
Finally
,
the
obtained
numerical
results
sho
w
that
this
simple
method
is
more
feasible
than
its
counterparts
and
could
be
applied
right
a
w
ay
.
REFERENCES
[1]
J.
Zhu,
et
al.,
”On
optimal
po
wer
allocation
for
do
wnlink
non-orthogonal
multiple
access
systems,
”
IEEE
Journal
on
Sel.
Areas
in
Comm.
,
v
ol.
35,
no.
12,
pp.
2744–2757,
2017.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
2,
April
2020
:
2139
–
2150
Evaluation Warning : The document was created with Spire.PDF for Python.