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u
en
tl
y
,
m
o
r
e
e
f
f
o
r
ts
ar
e
n
ee
d
ed
in
th
e
ar
ea
o
f
e
n
er
g
y
ef
f
icie
n
t
w
ir
ele
s
s
co
m
m
u
n
icatio
n
s
to
li
m
it th
e
e
n
v
ir
o
n
m
e
n
tal
i
m
p
ac
t o
f
t
h
e
m
o
b
ile
b
r
o
a
d
b
an
d
co
m
m
u
n
icati
o
n
s
.
Ne
w
d
is
tr
ib
u
ted
co
n
ce
p
ts
a
r
e
e
m
er
g
i
n
g
s
u
c
h
a
s
co
g
n
i
tiv
e
r
ad
io
[
7
]
an
d
d
ev
ice
to
d
ev
ice
co
m
m
u
n
icatio
n
s
(
D2
D)
[
8
]
,
t
h
an
k
s
to
s
m
ar
ter
eq
u
ip
m
e
n
t
(
UE
)
.
B
esid
es,
s
m
all
ce
ll
ar
e
m
as
s
i
v
el
y
d
ep
lo
y
ed
an
d
t
h
e
ce
ll
s
ize
is
co
n
ti
n
u
o
u
s
l
y
r
ed
u
ce
d
w
h
ic
h
i
m
p
l
y
les
s
UE
s
p
er
b
ase
s
tat
io
n
(
B
S)
e
s
p
ec
iall
y
in
d
en
s
e
u
r
b
an
ar
ea
.
T
h
e
f
ea
s
ib
ilit
y
o
f
p
o
w
er
co
n
tr
o
l
w
a
s
s
t
u
d
ies
i
n
[
9
]
is
f
o
cu
s
ed
o
n
a
ce
n
tr
alize
d
f
e
m
to
ce
ll
n
et
w
o
r
k
w
it
h
a
m
u
l
ti
-
ch
a
n
n
el
u
s
er
.
T
h
er
ef
o
r
e,
r
e
d
u
cin
g
t
h
e
p
r
o
ce
s
s
i
n
g
lo
ad
at
t
h
e
B
S
lev
el
i
s
co
n
v
e
n
ien
t
an
d
p
r
ac
tical
f
o
r
in
cr
ea
s
i
n
g
s
y
s
te
m
s
ca
lab
ilit
y
a
n
d
r
ed
u
cin
g
co
m
p
lex
it
y
,
a
n
d
d
is
tr
ib
u
ted
d
ec
is
io
n
s
s
h
o
u
ld
b
e
m
a
d
e
at
th
e
ter
m
in
al
lev
el.
P
o
w
er
co
n
tr
o
l
is
o
f
p
ar
a
m
o
u
n
t
i
m
p
o
r
ta
n
ce
in
o
r
d
er
to
m
iti
g
ate
i
n
ter
f
er
e
n
ce
in
t
h
e
s
a
m
e
ch
a
n
n
e
l
an
d
th
e
n
ea
r
-
f
ar
e
f
f
ec
t.
T
h
e
d
is
tr
ib
u
te
d
en
er
g
y
ef
f
icie
n
t
p
o
w
er
co
n
tr
o
l
d
esig
n
g
i
v
e
t
h
e
w
ir
ele
s
s
u
s
e
r
s
th
e
p
o
s
s
ib
ilit
y
to
b
en
ef
it
f
r
o
m
a
n
i
m
p
r
o
v
ed
q
u
alit
y
o
f
ex
p
er
ien
ce
b
y
r
is
i
n
g
th
eir
b
atter
y
li
f
e
f
o
r
th
e
s
a
m
e
v
o
l
u
m
e
o
f
d
ata
co
n
s
u
m
ed
w
h
ile
i
m
p
r
o
v
in
g
ac
ce
s
s
to
co
m
m
o
n
r
eso
u
r
ce
a
n
d
r
ed
u
cin
g
co
m
p
lex
it
y
m
a
n
ag
ed
b
y
t
h
e
B
S.
T
h
e
D2
D
(
d
ev
ice
to
d
ev
ice
)
th
at
w
a
s
in
tr
o
d
u
ce
d
f
r
o
m
v
er
s
io
n
1
3
o
f
3
GP
P
[
1
0
]
is
a
r
e
m
ar
k
ab
le
tech
n
o
lo
g
y
w
h
ic
h
ai
m
s
to
d
eliv
er
,
a
d
ir
ec
t
co
n
n
ec
tio
n
b
et
w
e
en
n
ea
r
b
y
u
s
er
s
,
u
n
d
er
l
y
i
n
g
th
e
ce
llu
lar
n
et
w
o
r
k
.
T
h
u
s
,
D2
D
i
m
p
r
o
v
es
t
h
e
E
E
o
f
th
e
co
m
m
u
n
ica
tio
n
.
T
h
is
a
r
ticle
d
is
cu
s
s
e
s
en
er
g
y
-
e
f
f
icie
n
t
p
o
w
er
co
n
tr
o
l
i
n
th
e
u
p
li
n
k
o
f
a
w
ir
ele
s
s
n
et
w
o
r
k
t
h
at
o
f
f
er
s
co
n
v
e
n
tio
n
a
l
ce
l
lu
lar
a
n
d
D2
D
c
o
m
m
u
n
icat
io
n
s
at
th
e
s
a
m
e
ti
m
e.
T
h
e
D2
D
c
o
m
m
u
n
icatio
n
m
o
d
e
allo
w
s
n
ea
r
b
y
u
s
er
s
to
ex
ch
a
n
g
e
d
ata
b
y
p
as
s
i
n
g
th
e
o
p
er
ato
r
s
’
ce
llu
lar
in
f
r
astru
ct
u
r
e,
w
h
ic
h
in
cr
ea
s
e
s
th
r
o
u
g
h
p
u
t,
E
E
an
d
r
ed
u
ce
s
d
elay
an
d
p
av
es
t
h
e
w
a
y
to
n
e
w
u
s
a
g
e
s
s
u
c
h
as
d
is
aster
r
elief
[
1
1
]
an
d
v
eh
ic
le
-
to
-
v
e
h
icle
co
m
m
u
n
icatio
n
s
(
V2
V)
[
1
2
]
.
D
2
D
h
as
b
etter
r
an
g
es
a
n
d
b
etter
th
r
o
u
g
h
p
u
t th
a
n
co
m
p
eti
to
r
tech
n
o
lo
g
ies s
u
c
h
as W
if
i
d
ir
ec
t
[
1
3
]
.
T
h
e
tech
n
o
lo
g
ica
l
tech
n
iq
u
es
f
o
r
r
eso
u
r
ce
allo
ca
tio
n
,
i
n
ter
f
er
en
ce
h
a
n
d
lin
g
,
m
o
b
ili
t
y
m
an
ag
e
m
e
n
t
an
d
o
th
er
s
y
s
te
m
-
lev
e
l
co
m
p
o
n
en
t
p
er
m
it
p
r
o
f
icie
n
t
D2
D
f
u
n
ctio
n
n
in
g
[
1
4
]
.
I
n
th
e
v
ast
m
aj
o
r
tit
y
o
f
th
e
r
elate
d
liter
atu
r
e,
D2
D
co
m
m
u
n
icatio
n
r
eu
s
e
s
s
p
ec
tr
al
r
eso
u
r
ce
o
f
ce
llu
lar
tr
an
s
m
i
s
s
i
o
n
s
,
w
h
ic
h
lead
s
to
m
o
r
e
s
p
ec
tr
u
m
e
f
f
icien
c
y
.
T
h
e
r
eso
u
r
ce
s
h
ar
i
n
g
b
et
w
ee
n
D
2
D
co
m
m
u
n
icatio
n
a
n
d
ce
ll
u
l
ar
o
n
e
p
o
s
es
a
m
aj
o
r
ch
alle
n
g
e
i
n
ter
m
o
f
in
ter
f
er
en
ce
m
an
a
g
e
m
en
t,
th
e
p
o
w
er
co
n
tr
o
l is p
o
w
er
f
u
l to
o
l to
s
o
lv
e
th
is
p
r
o
b
le
m
atic.
I
n
th
e
p
r
esen
t
p
ap
er
,
w
e
s
tu
d
y
th
e
u
p
lin
k
o
f
a
m
u
lti
-
ca
r
r
ier
s
y
s
te
m
w
it
h
m
i
n
i
m
u
m
Qo
S
r
e
q
u
ir
e
m
e
n
t
co
m
p
o
s
ed
o
f
b
o
th
ce
l
lu
lar
an
d
D2
D
t
ier
s
.
OF
DM
t
ec
h
n
o
lo
g
y
h
a
s
s
e
v
er
al
m
er
its
:
lo
w
co
m
p
le
x
it
y
,
co
m
p
atib
il
it
y
w
it
h
MI
MO
.
I
t
t
h
u
s
s
tr
o
n
g
l
y
m
o
tiv
a
tes
5
G
N
R
(
n
e
w
r
ad
io
)
s
till
ch
o
o
s
i
n
g
OF
DM
as
t
h
e
b
asis
o
f
n
e
w
w
av
e
f
o
r
m
d
esi
g
n
[
1
5
]
.
C
o
n
s
eq
u
e
n
tl
y
,
w
e
b
eliev
e
t
h
at
OFDM
A
tech
n
o
lo
g
y
w
i
ll
r
e
m
ain
a
f
o
u
n
d
atio
n
t
o
5
G
s
y
s
te
m
s
.
T
h
e
f
r
a
m
e
w
o
r
k
o
f
t
h
e
n
o
n
-
co
o
p
er
ativ
e
g
a
m
es
is
ap
p
lied
to
in
cr
ea
s
e
t
h
e
E
E
.
T
h
r
ee
m
e
th
o
d
s
w
er
e
co
m
p
ar
ed
f
o
r
n
et
w
o
r
k
E
E
m
a
x
i
m
izatio
n
:
a)
Din
k
elb
ac
h
Me
th
o
d
b
)
Fo
r
m
E
x
p
r
ess
io
n
c)
I
n
v
er
s
e
W
ater
-
f
illi
n
g
.
As
f
ar
a
s
w
e
k
n
o
w
t
h
e
co
m
p
a
r
is
o
n
b
et
w
ee
n
th
e
s
e
al
g
o
r
ith
m
s
ex
i
s
ts
in
t
h
e
r
elate
d
p
ap
er
in
t
h
e
liter
at
u
r
e
b
u
t
h
as
n
ev
er
b
ee
n
e
v
alu
a
ted
an
d
ass
es
s
ed
.
T
h
e
m
ai
n
co
n
tr
ib
u
tio
n
o
f
t
h
is
ar
ticle
is
th
er
e
f
o
r
e
th
e
i
m
p
le
m
en
tatio
n
a
n
d
th
e
ev
a
lu
at
io
n
o
f
th
e
clo
s
ed
ex
p
r
ess
io
n
-
f
o
r
m
alg
o
r
it
h
m
a
n
d
th
e
u
s
e
o
f
W
r
ig
h
t
O
m
e
g
a
i
n
s
tead
o
f
L
a
m
b
er
t
-
W
f
u
n
ctio
n
.
W
h
er
ea
s
t
h
e
clo
s
ed
ex
p
r
ess
io
n
o
f
p
o
w
er
w
as
d
es
cr
ib
ed
in
th
eo
r
y
i
n
[
1
6
]
,
it
w
a
s
n
o
t
i
m
p
le
m
en
ted
.
Gr
ap
h
ical
co
m
p
ar
is
o
n
o
f
th
e
E
E
,
SE,
p
o
w
er
an
d
ti
m
e
e
x
ec
u
tio
n
w
as
d
o
n
e
f
o
r
Din
k
elb
ac
h
an
d
C
lo
s
ed
f
o
r
m
al
g
o
r
it
h
m
s
f
o
r
ce
llu
lar
u
s
er
s
a
n
d
also
f
o
r
D2
D
u
s
er
s
.
A
b
asel
in
e
al
g
o
r
ith
m
w
as
u
s
ed
:
t
h
e
I
n
v
er
s
e
w
ater
f
ill
in
g
u
s
i
n
g
e
x
ac
t
m
i
n
i
m
u
m
r
ate
r
eq
u
ir
e
m
en
ts
.
2.
RE
L
AT
E
D
WO
RK
S
I
n
th
e
w
o
r
k
o
f
[
1
7
]
,
th
e
E
E
m
etr
ic
f
r
eq
u
en
tl
y
e
n
co
u
n
ter
ed
in
later
w
o
r
k
,
is
d
ef
i
n
ed
as
th
e
d
ata
s
u
cc
e
s
s
f
u
ll
y
tr
an
s
m
i
tted
(
b
it)
o
v
er
th
e
e
n
er
g
y
co
n
s
u
m
ed
(
J
o
u
le)
,
also
a
u
th
o
r
s
ap
p
lied
n
o
n
-
co
o
p
er
ativ
e
g
a
m
e
s
th
eo
r
y
to
in
cr
ea
s
e
E
E
i
n
th
e
c
ase
o
f
d
ata
tr
a
n
s
m
is
s
io
n
.
T
h
is
w
o
r
k
w
a
s
la
ter
ex
ten
d
ed
b
y
[
1
8
]
u
s
in
g
a
li
n
ea
r
p
r
icin
g
f
u
n
ctio
n
co
r
r
esp
o
n
d
in
g
to
th
e
p
o
w
er
th
at
i
s
tr
an
s
m
i
tted
.
T
h
e
au
th
o
r
s
o
f
[
1
9
]
ex
ten
d
ed
later
th
e
w
o
r
k
o
f
[
1
7
]
to
m
u
lt
i
-
ca
r
r
ier
C
DM
A
s
y
s
te
m
s
.
T
h
e
au
t
h
o
r
s
o
f
[
2
0
]
s
tu
d
ied
th
e
Ga
m
e
s
o
f
p
o
w
er
allo
ca
ti
o
n
f
o
r
MI
MO
(
m
u
l
tip
le
i
n
p
u
t
m
u
lt
ip
le
o
u
tp
u
t
)
.
T
h
e
au
th
o
r
s
o
f
[
2
1
]
u
s
ed
th
e
p
o
ten
tial
g
a
m
e
s
to
tack
le
r
eso
u
r
ce
allo
ca
tio
n
f
o
r
a
m
u
lt
icell
s
y
s
te
m
u
s
in
g
OF
DM
A
(
o
r
th
o
g
o
n
a
l
f
r
eq
u
en
cy
m
u
lt
i
p
l
e
ac
c
e
s
s
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e
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T
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a
u
th
o
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o
f
[
2
2
]
p
r
o
p
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d
a
c
r
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s
-
l
ay
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r
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et
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ib
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ted
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g
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f
ic
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o
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r
in
n
et
w
o
r
k
s
ex
p
er
ien
c
in
g
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n
t
er
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er
en
ce
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
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lec
&
C
o
m
p
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n
g
,
Vo
l.
10
,
No
.
4
,
A
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g
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s
t 2
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0
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4
1
1
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4135
4120
T
h
e
au
th
o
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s
o
f
[
5
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s
tu
d
y
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ter
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li
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ited
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v
ir
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ativ
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m
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t
h
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tr
ad
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o
f
f
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p
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icie
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d
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.
T
h
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a
u
t
h
o
r
s
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f
[
2
3
]
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p
ly
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t
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alize
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F
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ased
HetNe
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h
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co
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m
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w
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t
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ied
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th
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f
[
2
4
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w
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er
e
th
e
m
a
x
i
m
izatio
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o
f
E
E
i
s
d
o
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e
w
h
ile
r
esp
ec
ti
n
g
m
in
i
m
u
m
Qo
S
r
eq
u
ir
e
m
e
n
ts
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T
h
e
y
d
e
f
i
n
e
a
s
o
l
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tio
n
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ased
o
n
Deb
r
eu
E
q
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il
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m
t
h
at
g
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er
alize
s
Nas
h
eq
u
ilib
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iu
m
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w
h
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h
lead
s
to
a
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ater
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il
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lik
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est
r
esp
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s
e.
T
h
e
tr
ad
e
-
o
f
f
b
et
w
ee
n
s
p
ec
tr
al
ef
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icie
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c
y
(
SE)
an
d
e
n
er
g
y
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f
f
icie
n
c
y
(
E
E
)
is
i
n
v
e
s
ti
g
ated
f
o
r
d
ev
ice
-
to
-
d
ev
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(
D2
D)
co
m
m
u
n
ica
tio
n
s
r
eu
s
i
n
g
u
p
li
n
k
c
h
a
n
n
el
o
f
ce
l
lu
lar
n
et
w
o
r
k
s
i
n
[
2
5
]
.
T
h
e
r
est
o
f
t
h
e
ar
t
icle
i
s
s
tr
u
ct
u
r
ed
as
f
o
llo
w
i
n
g
.
T
h
e
n
o
tatio
n
s
ad
o
p
ted
ar
e
d
ef
i
n
ed
i
n
t
h
e
s
ec
tio
n
3
.
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h
e
s
y
s
te
m
m
o
d
el
is
d
is
cu
s
s
ed
in
s
ec
tio
n
4
.
I
n
th
e
s
ec
tio
n
5
th
e
p
o
w
er
co
n
tr
o
l
g
a
m
e
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d
escr
ib
ed
,
th
en
th
e
Nas
h
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ilib
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iu
m
s
o
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o
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tain
ed
u
s
i
n
g
f
r
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tio
n
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l
p
r
o
g
r
am
m
i
n
g
t
h
eo
r
y
in
s
ec
t
i
o
n
6
.
T
h
e
p
r
ac
tica
l
asp
ec
t
an
d
th
e
d
esi
g
n
o
f
th
e
alg
o
r
ith
m
ar
e
g
i
v
e
n
i
n
Sectio
n
7
.
T
h
e
s
i
m
u
l
at
io
n
s
r
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u
lts
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e
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m
m
e
n
ted
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n
|
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e
s
ec
tio
n
8
.
W
co
n
clu
d
e
f
i
n
a
ll
y
i
n
s
ec
tio
n
9
.
3.
NO
T
AT
I
O
NS
T
h
e
n
o
tatio
n
in
t
h
e
s
eq
u
e
l o
f
th
e
p
r
esen
t o
f
th
e
p
r
esen
t p
ap
er
ar
e
as f
o
llo
w
s
:
B
o
ld
letter
s
ar
e
u
s
ed
f
o
r
m
a
tr
ices
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d
v
ec
to
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s
L
a
m
b
er
t
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f
u
n
ctio
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s
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en
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te
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(
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+
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m
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s
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,
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it
h
:
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|
is
u
s
ed
f
o
r
th
e
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r
d
in
alit
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f
a
s
et
4.
SYST
E
M
M
O
DE
L
T
h
e
s
y
s
te
m
w
e
ad
o
p
t
is
a
s
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g
le
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ll
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er
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ed
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y
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n
e
B
S
(
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ase
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tatio
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),
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llu
lar
UE
s
an
d
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tr
an
s
m
itter
/r
ec
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v
er
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air
s
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h
e
UE
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o
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th
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y
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m
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g
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th
e
f
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n
g
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e
t:
=
{
1
,
2
,
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,
}
T
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et
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f
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D
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air
s
in
th
e
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y
s
te
m
i
s
d
en
o
te
d
as:
=
{
1
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,
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}
Sp
ec
tr
al
r
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r
ce
allo
ca
tio
n
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s
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ased
o
n
t
h
e
O
FDM
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tech
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n
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s
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b
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ca
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ier
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a
l
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ch
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lar
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o
r
e,
n
o
i
n
ter
f
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ce
ex
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s
ts
i
n
t
h
e
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llu
lar
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ier
.
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y
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n
tr
ar
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v
e
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y
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s
m
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tter
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o
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to
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s
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e
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tio
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to
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er
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r
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ed
,
b
u
t
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ath
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tr
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n
s
m
it
r
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n
d
o
m
l
y
t
h
eir
p
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s
o
v
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th
e
s
u
b
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r
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ier
s
.
T
h
e
T
ab
le
1
g
i
v
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t
h
e
m
ai
n
n
o
tatio
n
s
ad
o
p
ted
.
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ab
le
1.
No
tatio
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s
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a
t
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M
e
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B
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
E
n
erg
y
efficien
t p
o
w
er c
o
n
tr
o
l fo
r
d
ev
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to
d
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o
mmu
n
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ca
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n
in
5
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…
(
Mo
h
a
med
A
min
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h
a
r
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r
)
4121
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h
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s
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th
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d
s
ig
n
al
is
wr
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f
o
r
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E
k
a
s:
y
k
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h
k
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k
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x
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i
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i
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n
k
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)
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ith
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k
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d
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ite
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k
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th
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s
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n
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n
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(
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it
h
n
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is
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h
a
v
i
n
g
t
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a
m
e
m
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d
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n
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as
n
k
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h
e
ch
an
n
e
ls
ar
e
ass
u
m
ed
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m
p
lex
Ga
u
s
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ia
n
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ea
l
p
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m
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g
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ar
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ian
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.
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r
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ch
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air
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en
o
ted
(
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v
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all
s
u
b
ca
r
r
ier
s
)
:
=
[
,
,
,
,
…
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(
3
)
a
n
d
th
e
v
ec
to
r
p
o
w
er
o
f
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D
p
air
s
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clu
d
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g
t
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e
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air
is
:
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=
[
,
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,
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,
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,
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,
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(
4
)
T
h
e
tr
an
s
m
i
tti
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g
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ec
to
r
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r
r
esp
o
n
d
in
g
to
all
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air
s
:
=
[
,
−
]
(
5
)
T
h
e
to
tal
p
o
w
er
o
f
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D
p
air
ac
co
u
n
ti
n
g
th
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cir
c
u
it
r
elate
d
co
n
s
u
m
p
tio
n
:
(
)
=
∑
,
+
=
(
6
)
a
n
d
th
e
p
o
w
er
v
ec
to
r
o
f
UE
s
e
x
ce
p
t
UE
k
i
s
:
−
=
[
p
1
c
,
…
,
−
1
,
+
1
,
…
,
]
(
7
)
T
h
e
g
lo
b
al
p
o
w
er
o
f
ce
ll
u
lar
u
s
er
s
is
d
en
o
ted
:
=
[
p
k
c
,
−
]
(
8
)
T
h
e
to
tal
p
o
w
er
o
f
UE
k
ac
co
u
n
ti
n
g
t
h
e
cir
cu
it
r
elate
d
co
n
s
u
m
p
tio
n
:
(
)
=
+
(
9
)
T
h
e
SIN
R
(
Sig
n
al
to
I
n
ter
f
er
e
n
ce
p
lu
s
No
i
s
e
R
atio
)
o
f
UE
k
at
th
e
n
th
s
u
b
-
ca
r
r
ier
:
=
|
ℎ
|
2
∑
|
,
|
2
,
+
2
=
1
(
1
0
)
W
e
d
en
o
te
th
e
SIN
R
o
f
D2
D
p
air
at
th
e
n
th
s
u
b
-
ca
r
r
ier
:
,
=
|
|
2
,
∑
|
,
|
2
,
+
|
ℎ
|
2
+
2
=
1
≠
(
1
1
)
L
et
u
s
d
ef
i
n
e
t
h
e
ef
f
ec
tiv
e
c
h
a
n
n
el
g
ai
n
o
f
t
h
e
ce
ll
u
lar
u
s
er
UE
k
as
f
o
llo
w
i
n
g
:
=
|
ℎ
|
2
∑
|
,
|
2
,
+
2
=
1
(
12
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
10
,
No
.
4
,
A
u
g
u
s
t 2
0
2
0
:
4
1
1
8
-
4135
4122
T
h
e
ef
f
ec
tiv
e
c
h
an
n
el
g
ai
n
o
f
D2
D
p
air
k
is
d
ef
in
ed
as
:
,
=
|
|
2
∑
|
,
|
2
,
+
|
ℎ
|
2
+
2
=
1
≠
(
1
3
)
T
h
e
SE
(
Sp
ec
tr
al
E
f
f
icien
c
y
)
u
tili
t
y
f
u
n
ctio
n
o
f
UE
k
is
:
(
,
−
,
)
=
l
og
2
(
1
+
Γ
)
(
14
)
W
ith
Γ
th
e
SIN
R
g
ap
w
.
r
.
t B
E
R
as d
ef
in
ed
i
n
[
2
6
]
:
Γ
=
−
ln
(
5
B
ER
)
1
.
5
T
h
e
SE
(
Sp
ec
tr
al
E
f
f
icien
c
y
)
u
tili
t
y
f
u
n
ctio
n
o
f
th
e
k
th
D2
D
p
air
is
:
(
,
−
,
)
=
∑
2
(
1
+
,
)
=
1
(
1
5
)
T
h
e
E
E
(
E
n
er
g
y
E
f
f
icie
n
c
y
)
u
t
ilit
y
f
u
n
ct
io
n
o
f
th
e
u
p
li
n
k
co
m
m
u
n
icatio
n
o
f
UE
k
is
:
(
,
−
,
)
=
(
,
−
,
)
(
)
(
1
6
)
T
h
e
E
E
(
E
n
er
g
y
E
f
f
icie
n
c
y
)
u
t
ilit
y
f
u
n
ct
io
n
o
f
th
e
k
th
D2
D
p
air
:
(
,
−
,
)
=
(
,
−
,
)
(
)
(
17
)
E
ac
h
UE
w
il
l
in
cr
ea
s
e
it
s
en
er
g
y
e
f
f
icie
n
c
y
a
n
d
at
th
e
s
a
m
e
ti
m
e
r
esp
ec
t
th
e
m
in
i
m
u
m
r
ate
r
eq
u
ir
e
m
en
ts
.
T
h
e
UE
k
w
i
ll
tr
y
to
m
a
x
i
m
ize
it
s
g
lo
b
al
en
er
g
y
e
f
f
icie
n
c
y
w
h
ile
r
esp
e
ctin
g
m
i
n
i
m
u
m
r
ate
co
n
s
tr
ain
t.
C
o
n
s
q
u
e
n
tl
y
,
t
h
e
f
o
llo
w
i
n
g
p
r
o
b
le
m
h
a
s
to
b
e
s
o
lv
ed
f
o
r
UE
k
:
M
a
ximi
ze
(
,
−
,
)
s
.t
.(
C
1A
)
p
k
c
∈
[
0
,
p
m
ax
c
]
(
18
)
(
C
2
A
)
(
,
−
,
)
≥
Si
m
i
lar
l
y
,
t
h
e
f
o
llo
w
i
n
g
p
r
o
b
l
e
m
h
a
s
to
b
e
s
o
lv
ed
f
o
r
k
th
D2
D
p
air
:
M
a
ximi
ze
(
,
−
,
)
s
.t
.(
C
1B
)
∈
[
0
,
]
(
19
)
(
C
2
B
)
(
,
−
,
)
≥
Hen
ce
,
t
h
er
e
ar
e
(
+
)
o
p
ti
m
izati
o
n
p
r
o
b
lem
s
to
b
e
j
o
in
tl
y
s
o
lv
ed
in
th
e
s
y
s
te
m
.
T
h
e
m
u
lti
v
ar
iate
o
p
tim
izatio
n
ca
n
n
o
t
b
e
ap
p
lied
;
as
th
er
e
is
n
o
ce
n
tr
al
e
n
tit
y
th
a
t
is
co
n
tr
o
llin
g
all
v
a
r
ia
b
les
o
f
th
e
s
y
s
te
m
,
in
s
tead
ea
c
h
h
as p
ar
tial c
o
n
tr
o
l o
n
h
is
o
w
n
v
ar
iab
les,
w
h
ic
h
l
ea
d
s
to
th
e
ap
p
licatio
n
o
f
Ga
m
e
T
h
eo
r
y
.
5.
G
AM
E
T
H
E
O
RY
F
O
RM
UL
AT
I
O
N
T
h
e
n
o
n
-
co
o
p
er
ativ
e
g
a
m
e
t
h
eo
r
y
ap
p
ea
r
s
as
th
e
n
at
u
r
al
to
o
l
to
tack
le
to
p
r
o
b
lem
as
w
e
h
av
e
in
d
ep
en
d
en
t,
r
atio
n
al
p
la
y
er
s
th
at
co
n
tr
o
l
th
eir
v
ar
iab
les.
T
h
e
f
o
llo
w
i
n
g
tr
ip
let
d
ef
i
n
es
t
h
e
n
o
n
-
co
o
p
er
ativ
e
g
a
m
e
f
o
r
m
al
l
y
a
s
:
=
{
,
{
}
∈
{
1
,
.
.
,
}
,
{
}
∈
{
1
,
.
.
,
}
}
w
h
er
e
:
P
lay
er
s
:
=
{
,
}
A
ctio
n
s
:
=
[
0
,
]
∈
[
0
,
]
∈
Utilit
ie
s
: T
h
e
EE
u
tili
t
y
f
u
n
cti
o
n
(
b
it/J
/Hz)
.
Fo
r
e
ac
h
∈
{
1
,
.
.
,
}
th
e
ac
tio
n
s
p
ac
e
co
r
r
esp
o
n
d
in
g
to
o
th
er
p
la
y
er
s
:
−
=
∏
=
1
≠
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
E
n
erg
y
efficien
t p
o
w
er c
o
n
tr
o
l fo
r
d
ev
ice
to
d
ev
ice
c
o
mmu
n
i
ca
tio
n
in
5
G
…
(
Mo
h
a
med
A
min
e
C
h
a
r
a
r
)
4123
I
n
o
r
d
er
to
g
et
a
s
o
lu
tio
n
,
w
e
h
av
e
to
f
in
d
th
e
eq
u
il
ib
r
iu
m
o
f
th
e
g
a
m
e.
I
n
d
ee
d
,
th
e
im
p
o
r
tan
ce
o
f
th
e
eq
u
ilib
r
iu
m
i
s
g
a
m
e
is
i
m
p
o
r
ta
n
t
a
s
ag
e
n
t
s
w
il
l
b
e
co
m
e
p
r
ed
ictab
le
an
d
s
tab
le.
T
h
e
n
et
w
o
r
k
o
p
er
ates
ef
f
ec
tiv
e
l
y
w
h
e
n
r
ea
ch
i
n
g
eq
u
ilib
r
iu
m
[
2
7
]
.
T
h
e
p
o
w
er
s
tr
ateg
y
o
f
ea
ch
u
s
er
m
u
s
t
b
e
f
ea
s
ib
le
b
y
r
esp
ec
ti
n
g
m
i
n
i
m
u
m
Qo
S
(
q
u
alit
y
o
f
s
er
v
ice
)
co
n
d
itio
n
s
.
T
h
is
lead
s
u
s
to
u
s
e
a
d
if
f
er
e
n
t
Ga
m
e
th
eo
r
y
s
o
lu
tio
n
th
at
allo
w
s
ea
ch
p
la
y
er
to
ch
o
o
s
e
its
s
tr
ateg
y
k
n
o
w
i
n
g
th
e
s
tr
ate
g
y
o
f
o
th
er
p
la
y
er
s
w
h
ile
r
esp
ec
tin
g
f
ea
s
ib
ili
t
y
co
n
d
itio
n
s
:
t
h
e
G
N
E
(
g
en
er
alize
d
Na
s
h
eq
u
ilib
r
i
u
m
)
.
B
y
s
u
p
p
o
s
i
n
g
th
at
ea
ch
p
l
a
y
er
’
s
s
tr
ate
g
y
m
a
y
d
ep
en
d
o
n
t
h
e
o
t
h
er
p
la
y
er
s
’
,
th
e
GNE
g
o
es
b
e
y
o
n
d
th
e
co
m
m
o
n
NE
(
Nas
h
eq
u
ilib
r
i
u
m
)
[
2
8
]
.
No
t
o
n
l
y
u
tili
t
y
f
u
n
ct
io
n
s
ar
e
d
ep
en
d
e
n
t
b
u
t
also
t
h
e
p
la
y
er
s
’
ac
tio
n
s
ets
ar
e
co
u
p
le,
w
h
ic
h
is
d
if
f
e
r
en
t
f
r
o
m
clas
s
ical
n
o
n
-
co
o
p
er
ativ
e
g
a
m
es
[
2
9
]
.
W
e
w
i
ll
d
en
o
te
b
y
(
−
)
th
e
s
et
o
f
u
s
er
UE
k
act
io
n
s
r
esp
e
ctin
g
th
e
m
i
n
i
m
u
m
r
ate
an
d
th
e
m
a
x
i
m
u
m
p
o
w
er
r
eq
u
ir
e
m
en
ts
w
i
th
r
esp
ec
t to
o
th
er
p
la
y
er
s
’
s
tr
ate
g
y
:
(
−
)
=
{
∈
:
(
,
−
)
re
s
p
e
ct
s
(C
1
)
&
C
2
)
}
Def
i
n
itio
n
.
Gen
er
alize
d
Nas
h
eq
u
ilib
r
iu
m
T
h
e
m
atr
i
x
o
f
p
o
w
er
∗
=
(
∗
,
−
∗
)
is
a
G
en
er
alize
d
Nash
eq
u
ilib
r
i
u
m
i
f
:
−
∗
∈
(
−
)
an
d
:
∀
p
k
∈
k
G
(
−
k
∗
)
:
E
E
k
(
∗
)
≥
E
E
k
(
,
∗
)
(
2
0
)
W
h
ich
m
ea
n
s
t
h
at
n
o
p
la
y
er
h
as
th
e
i
n
ter
est
to
d
ev
iate
u
n
ilater
all
y
f
r
o
m
a
G
NE
to
an
o
th
er
f
ea
s
ib
le
p
o
in
t
(
r
esp
ec
tin
g
t
h
e
co
n
s
tr
ain
t
s
)
.
6.
F
RACTI
O
N
AL
P
RO
G
RAM
M
I
NG
F
O
RM
UL
AT
I
O
N
A
ND
G
AM
E
SO
L
U
T
I
O
N
T
h
e
E
E
o
b
j
ec
tiv
e
f
u
n
ctio
n
i
n
(
1
8
)
an
d
(
1
9
)
ar
e
f
r
ac
tio
n
al
an
d
n
o
n
-
co
n
v
ex
.
T
h
e
m
o
s
t
s
u
itab
le
m
at
h
e
m
a
tical
f
r
a
m
e
w
o
r
k
to
t
ac
k
le
th
e
o
p
ti
m
izatio
n
o
f
s
u
c
h
f
u
n
c
tio
n
s
is
th
e
f
r
ac
tio
n
al
p
r
o
g
r
a
m
m
in
g
,
w
h
ich
p
r
o
v
id
e
p
o
ly
n
o
m
ial
co
m
p
le
x
it
y
alg
o
r
it
h
m
s
w
h
en
t
h
e
n
u
m
er
a
to
r
is
co
n
ca
v
e
an
d
d
en
o
m
in
a
to
r
is
co
n
v
e
x
[
2
9
]
.
First,
let
u
s
g
i
v
e
a
f
o
r
m
al
d
ef
i
n
itio
n
o
f
Fra
ctio
n
al
p
r
o
g
r
a
m
m
in
g
:
Def
i
n
itio
n
.
Fra
ctio
n
al
p
r
o
g
r
am
m
i
ng
L
et
∈
ℝ
co
n
v
e
x
s
u
b
s
et
an
d
let
→
ℝ
th
e
f
o
llo
w
i
n
g
f
u
n
ct
io
n
s
:
:
→
(
)
a
n
d
g
:
→
(
)
(
2
1
)
A
f
r
ac
tio
n
al
p
r
o
g
r
a
m
i
s
t
h
e
o
p
ti
m
izat
io
n
p
r
o
b
lem
:
M
a
ximi
ze
x
∈
C
f
(
x
)
g
(
x
)
(
2
2
)
T
h
e
f
r
ac
tio
n
al
p
r
o
g
r
a
m
m
in
g
allo
w
s
u
s
to
s
o
lv
e
eq
u
i
v
ale
n
t
f
o
r
m
o
f
t
h
e
p
r
o
b
le
m
th
at
is
less
co
m
p
le
x
t
h
a
n
th
e
o
r
ig
i
n
al
o
n
e:
P
r
o
p
o
s
itio
n
1
.
T
h
e
v
ec
to
r
∗
∈
s
o
l
v
es
t
h
e
(
2
2
)
if
an
d
o
n
ly
i
f
:
(
(
∗
)
−
∗
(
∗
)
)
=
0
,
∗
th
e
ze
r
o
o
f
th
e
f
u
n
ctio
n
:
(
)
=
∈
(
)
−
(
)
(
2
3
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Fo
r
a
g
i
v
e
n
v
al
u
e,
th
e
UE
k
w
il
l
s
o
lv
e
t
h
e
f
o
llo
w
in
g
t
r
an
s
f
o
r
m
ed
f
r
ac
tio
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p
r
o
g
r
am
m
in
g
p
r
o
b
le
m
in
s
tead
o
f
(
1
8
)
:
M
a
ximi
ze
p
k
c
S
E
k
c
(
p
k
c
,
−
,
)
−
λ
k
c
P
tot
(
p
k
c
)
s
.t.
(
C
1A
)
&
(
C
2A
)
(
24
)
Fo
r
th
e
o
p
ti
m
al
v
a
lu
e,
,
∗
w
e
g
et:
S
E
k
c
(
p
k
c
,
−
,
)
−
λ
k
c
,
∗
P
tot
(
p
k
c
)
=
0
(
2
5
)
T
h
is
m
ea
n
s
th
at
t
h
e
b
est
r
esp
o
n
s
e
p
o
w
er
,
∗
g
iv
e
n
o
th
er
t
r
an
s
m
itter
s
(
−
,
)
v
er
if
ie
s
th
e
f
o
llo
w
i
n
g
ex
p
r
ess
io
n
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
10
,
No
.
4
,
A
u
g
u
s
t 2
0
2
0
:
4
1
1
8
-
4135
4124
,
∗
=
(
,
−
,
)
(
)
(
2
6
)
Si
m
i
lar
l
y
,
f
o
r
a
g
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e
n
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al
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e
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h
e
eq
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iv
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n
t
p
r
o
b
lem
o
f
o
p
tim
izatio
n
th
at
th
e
k
th
D2
D
p
air
w
il
l
s
o
lv
e
in
s
tead
o
f
(
1
9
)
:
M
a
ximi
ze
(
,
−
,
)
−
(
)
s
.t.
(
C
1B
)
&
(
C
2B
)
(
2
7
)
Fo
r
th
e
o
p
ti
m
al
v
a
lu
e,
λ
k
d
,
∗
w
e
g
et:
(
,
−
,
)
−
,
∗
(
)
=
(
28
)
T
h
is
m
ea
n
s
t
h
at
t
h
e
b
est
r
e
s
p
o
n
s
e
p
o
w
er
g
i
v
en
o
t
h
er
tr
an
s
m
itter
s
(
−
,
)
v
er
i
f
ies
t
h
e
f
o
ll
o
w
i
n
g
ex
p
r
ess
io
n
:
λ
k
d
,
∗
=
S
E
k
d
(
,
−
,
)
(
)
(
29
)
P
r
ac
tically
,
th
e
s
o
l
u
tio
n
o
f
(
2
4
)
an
d
(
2
7
)
is
g
o
t
w
h
en
t
h
e
p
la
y
er
s
ar
e
g
i
v
in
g
th
eir
b
est
r
esp
o
n
s
e
to
ea
c
h
o
th
er
.
A
s
f
o
llo
w
s
,
t
h
e
k
th
u
s
er
b
est
r
esp
o
n
s
e
f
u
n
c
tio
n
w
h
ic
h
is
t
h
e
b
est
ac
tio
n
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ad
o
p
t
w
h
e
n
th
e
s
tr
ate
g
y
o
f
th
e
o
t
h
er
p
la
y
er
s
is
−
∈
−
(
)
:
(
−
)
=
∈
(
−
)
(
,
−
)
(
3
0
)
T
h
e
n
ec
ess
ar
y
co
n
d
it
io
n
s
ar
e
m
o
r
e
ea
s
il
y
o
b
tai
n
ed
in
t
h
e
ca
s
e
o
f
a
s
i
n
g
le
ca
r
r
ier
.
P
r
o
p
o
s
itio
n
2
.
I
n
th
e
s
i
n
g
le
c
ar
r
ier
ca
s
e,
th
e
u
n
iq
u
en
e
s
s
an
d
ex
is
ten
ce
o
f
Gen
er
alize
d
N
ash
eq
u
ilib
r
i
u
m
ar
e
g
u
ar
a
n
teed
,
as th
e
B
R
f
u
n
ct
io
n
s
o
f
t
h
e
u
s
er
k
ar
e
s
ta
n
d
ar
d
an
d
v
er
if
y
t
h
e
f
o
llo
w
i
n
g
p
r
o
p
er
ties
:
C
o
n
ca
v
it
y
:
(
−
)
is
co
n
ca
v
e
s
tr
ictl
y
in
−
P
o
s
itiv
it
y
:
(
−
)
>
0
Mo
n
o
to
n
icit
y
: i
f
−
>
−
th
e
n
(
−
)
>
(
−
)
Scalab
ilit
y
: Fo
r
all
>
1
,
(
−
)
>
(
−
)
P
r
o
o
f
.
T
h
e
p
r
o
o
f
is
g
i
v
en
i
n
[
3
0
]
I
n
th
e
ca
s
e
o
f
m
u
l
ti
-
ca
r
r
ier
,
t
h
e
eq
u
il
ib
r
iu
m
ca
n
b
e
ch
ar
ac
t
r
ized
b
y
g
i
v
i
n
g
th
e
e
x
p
r
ess
io
n
o
f
p
o
w
er
f
o
r
ea
ch
s
u
b
-
ca
r
r
ier
an
d
f
o
r
ea
ch
u
s
er
a
s
in
t
h
e
f
o
llo
w
i
n
g
p
r
o
p
o
s
itio
n
.
P
r
o
p
o
s
itio
n
3
.
T
h
e
b
est p
o
w
er
o
f
th
e
UE
k
i
s
g
i
v
en
b
y
th
e
f
o
ll
o
w
i
n
g
w
h
e
n
th
e
g
a
m
e
ad
m
it
s
a
GNE
:
p
k
c
,
∗
=
min
(
p
m
ax
c
,
(
1
λ
k
c
,
∗
−
1
ν
k
,
n
c
)
+
)
(
31
)
T
h
e
o
p
tim
al
v
alu
e
,
∗
:
,
∗
=
min
(
,
,
)
(
32
)
=
(
l
o
g
(
ex
p
(
−
1
)
)
)
(
33
)
W
ith:
=
l
og
(
)
(
34
)
=
(
−
1
,
)
(
35
)
is
ca
lcu
lated
w
h
en
t
h
e
r
ate
r
eq
u
ir
e
m
e
n
ts
ar
e
ac
ti
v
e:
,
=
,
2
−
(
36
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
E
n
erg
y
efficien
t p
o
w
er c
o
n
tr
o
l fo
r
d
ev
ice
to
d
ev
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c
o
mmu
n
i
ca
tio
n
in
5
G
…
(
Mo
h
a
med
A
min
e
C
h
a
r
a
r
)
4125
P
r
o
p
o
s
itio
n
4
.
T
h
e
f
o
llo
w
in
g
f
o
r
m
u
la
g
iv
e
s
th
e
o
p
ti
m
al
p
o
w
er
o
f
th
e
D2
D
k
f
o
r
ea
ch
s
u
b
-
ca
r
r
ier
n
:
∀
∈
:
,
,
∗
=
min
(
,
(
1
,
∗
−
1
,
)
+
)
(
37
)
The
opt
i
ma
l
va
l
ue
,
∗
:
,
∗
=
min
(
,
,
)
(
38
)
=
(
l
o
g
(
ex
p
(
−
1
)
)
)
(
3
9
)
is
th
e
s
et
o
f
ac
ti
v
e
s
u
b
ca
r
r
ier
s
(
i.e
.
w
ith
s
tr
ictl
y
p
o
s
itiv
e
p
o
wer
)
o
f
D2
D
k
w
h
e
n
:
,
∗
=
∗
=
{
∈
|
(
1
−
1
,
)
>
0
}
(
40
)
w
it
h
:
=
l
o
g
(
∏
∈
∗
,
)
|
∗
|
(
41
)
=
(
−
∑
1
,
∈
∗
)
|
∗
|
(
42
)
W
h
en
t
h
e
n
o
r
m
a
lized
r
ate
co
n
s
tr
ain
t
s
ar
e
ac
t
iv
e,
,
h
as th
e
f
o
ll
o
w
i
n
g
f
o
r
m
u
latio
n
,
=
(
2
−
|
m
ax
|
m
in
∏
,
∈
m
ax
)
1
m
ax
(
43
)
w
ith:
m
ax
=
{
n
∈
K
c
|
(
1
λ
k
d
,
m
ax
−
1
ν
k
,
n
d
)
>
0
}
(
44
)
P
r
o
o
f
.
T
h
e
p
r
o
o
f
is
g
i
v
en
i
n
[
3
1
]
ap
p
lied
to
o
u
r
m
o
d
el.
7.
AL
G
O
RI
T
H
M
S DES
I
G
N
A
ND
ANAL
YSI
S
7
.
1
.
P
r
a
ct
ica
l desig
n
Fro
m
p
r
ac
tical
p
o
in
t
o
f
v
ie
w
,
ea
ch
p
la
y
er
g
iv
e
it
s
b
est
r
esp
o
n
s
e
to
s
o
lv
e
E
E
m
a
x
i
m
izatio
n
p
r
o
b
lem
k
n
o
w
i
n
g
o
th
er
p
la
y
er
s
’
s
tr
ate
g
y
.
W
e
co
n
s
id
er
a
s
c
h
e
m
e
t
h
at
is
as
y
n
c
h
r
o
n
o
u
s
as
t
h
e
u
s
e
r
ch
o
o
s
es
its
p
o
w
er
s
tr
ateg
y
at
ti
m
e
b
ased
o
n
th
e
o
th
er
p
la
y
er
s
’
s
tr
ateg
y
o
f
−
1
.
I
n
d
ee
d
,
an
i
n
s
ta
n
ta
n
eo
u
s
f
ee
b
ac
k
is
is
v
er
y
d
if
f
ic
u
lt
in
p
r
ac
tice.
W
e
co
n
s
id
er
th
at
ea
ch
p
la
y
er
k
n
o
w
s
h
is
i
n
d
iv
id
u
al
C
SI
(
ch
a
n
n
el
s
tate
in
f
o
r
m
a
tio
n
)
w
h
it
h
i
n
th
e
SIN
R
o
n
ea
ch
s
u
b
ca
r
r
ier
.
A
s
i
n
tial
v
alu
e,
t
h
e
ce
llu
lar
u
s
er
s
c
h
o
o
s
e
an
d
D2
D
u
s
er
s
ch
o
o
s
e
.
I
n
ea
ch
iter
atio
n
,
th
e
u
s
er
UE
k
est
i
m
ate
s
th
e
ef
f
ec
ti
v
e
ch
an
n
el
g
ai
n
b
ased
o
n
th
e
p
r
ev
io
u
s
v
al
u
e
an
d
b
ased
o
n
th
e
p
r
ev
io
u
s
SIN
R
:
̂
(
)
=
(
−
1
)
(
−
1
)
(
45
)
T
h
e
u
s
er
UE
k
e
s
ti
m
ates t
h
e
E
E
as f
o
llo
w
in
g
a
n
d
tr
ies to
in
c
r
ea
s
e
it
w
.
r
.
t to
co
n
d
itio
n
s
(
C
1
A
)
an
d
(
C
2
A
)
:
̂
(
)
=
̂
(
)
,
(
)
+
(
46
)
w
it
h
:
̂
(
)
=
l
og
2
(
1
+
̂
(
)
(
)
)
(
47
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
10
,
No
.
4
,
A
u
g
u
s
t 2
0
2
0
:
4
1
1
8
-
4135
4126
As in
p
r
o
p
o
s
itio
n
3
:
,
∗
(
+
1
)
=
min
(
,
(
1
,
∗
(
+
1
)
−
1
̂
(
+
1
)
)
+
)
(
4
8
)
Si
m
i
lar
l
y
,
th
e
u
s
er
D2
D
k
esti
m
ates
t
h
e
E
E
a
s
f
o
llo
w
i
n
g
a
n
d
tr
ies
to
i
n
cr
ea
s
e
i
t
w
.
r
.
t
t
o
co
n
d
itio
n
s
(
C
1
A
)
an
d
(
C
2
A
)
:
,
̂
(
)
=
,
(
−
1
)
,
(
−
1
)
(
49
)
T
h
e
u
s
er
D2
D
k
es
ti
m
ates t
h
e
E
E
as f
o
llo
w
in
g
an
d
tr
ies to
i
n
cr
ea
s
e
it
w
.
r
.
t to
co
n
d
itio
n
s
(
C
1
B
)
an
d
(
C
2
B
):
̂
(
)
=
̂
(
)
∑
,
(
)
+
=
1
(
50
)
w
it
h
:
̂
(
)
=
∑
l
og
2
(
1
+
,
̂
(
)
,
(
)
=
1
(
51
)
As in
p
r
o
p
o
s
itio
n
4
:
∀
∈
:
,
,
∗
(
)
=
(
,
(
1
,
∗
(
)
−
1
,
̂
(
)
)
+
)
(5
2
)
T
h
e
ch
allen
g
e
h
er
e
is
to
f
in
d
,
∗
(
)
an
d
,
∗
(
)
,
f
o
r
th
at
w
e
u
s
e
t
h
r
ee
m
et
h
o
d
s
:
T
h
e
f
ir
s
t
m
et
h
o
d
th
at
w
a
s
u
s
e
d
b
y
[
2
4
]
r
elies o
n
Din
k
elb
ac
h
alg
o
r
ith
m
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