Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
V
ol.
4,
No
.
1,
October
2016,
pp
.
104
115
DOI:
10.11591/ijeecs
.v4.i1.pp104-115
104
Optimal
P
o
wer
and
Modulation
Adaptation
P
olicies
with
Receiver
Diver
sity
o
ver
Ra
yleigh
F
ading
Channel
Muhammad
Imran
T
ariq
*
,
Razv
an
Beuran
,
and
Y
oic
hi
Shinoda
J
apan
Adv
anced
Institute
of
Science
and
T
echnology
1-1
Asahidai,
Nomi
city
,
Ishika
w
a,
J
apan
*Corresponding
author
,
e-mail:
imr
an@jaist.ac.jp
Abstract
Efficient
bandwidth
utilization
is
par
amount
in
wireless
comm
unication
systems
,
par
ticular
y
in
f
ading
en
vironments
,
since
f
ading
is
one
of
the
major
constr
aints
that
impair
comm
unication
in
wireless
systems
.
The
bandwidth
efficiency
of
a
wireless
comm
unication
system
can
be
enhanced
sign
ificantly
b
y
emplo
ying
po
w
er
and
modulation
adaptation
policies
with
div
ersity
combining
gain.
In
this
w
or
k,
first
w
e
e
xamine
an
analytically-der
iv
ed
solution
f
or
Maxim
um
Combining
Ratio
(MRC)
div
ersity
t
echnique
f
or
the
capacity
per
unit
bandwidth.
Then,
w
e
design
an
adaptiv
e
tr
ansmission
system
to
utiliz
e
the
div
ersity
combining
gain
while
retaining
the
target
BER
b
y
adapting
po
w
er
and
constellation
siz
e
using
contin
uous
po
w
er
,
channel
in
v
ersion
with
fix
ed
r
ate
and
contin
uous
po
w
er
and
disrecte-r
ate
.
By
consider
ing
the
eff
ect
of
div
ersity
combining
gain,
the
designed
system
yields
a
reasonab
le
spectr
al
efficiency
with
respect
to
target
BER
that
g
ro
ws
as
the
n
umber
of
div
ersity
le
v
els
increase
.
Fur
ther
more
,
the
presented
results
sho
w
contin
uous
po
w
er
and
discrete-r
ate
adaptation
policy
reduces
probability
of
outage
unlik
e
its
achie
v
ed
spectr
al
efficiency
is
close
to
other
selected
policies
,
which
r
a
tifies
the
optimiz
ed
s
witching
thresholds
and
mak
es
it
best
candidate
f
or
imperf
ect
channel
conditions
.
K
e
yw
or
ds:
Adaptiv
e
tr
ansmission,
div
ersity
combining,
bit
error
r
ate
,
s
witching
threshold,
Ra
yleigh
f
ading
Cop
yright
c
2016
Institute
of
Ad
v
anced
Engineering
and
Science
1.
Intr
oduction
TWith
the
adv
ent
of
high
data
tr
ansmission,
quadr
uple-pla
y
applications
in
par
ticular
and
high
quality
contents
r
ich
streams
in
wireless
comm
unication,
in
gener
al
are
highly
demanded
and
will
contin
ue
in
the
future
.
The
demand
f
or
those
applications
e
xpeditiously
ha
v
e
been
b
ur-
geoning.
Consequently
,
the
limited
wireless
r
esources
are
increasingly
demanded.
Theref
ore
,
the
capacity
is
a
major
f
actor
dur
ing
designing
wireless
comm
unication
systems
,
since
wireless
links
are
g
reatly
impaired
due
to
f
ading.
Thus
,
the
current
f
ading
mitigation
schemes
are
utiliz
ed
in
wireless
comm
unication
systems
such
as
Digital
Video
Broadcasting-Satellite
V
ersion
2
(D
VB-S2)
[1],
WiMAX
[2]
and
L
TE
[3].
Div
ersity
combining
and
adaptiv
e
modulation
are
capacity
impro
ving
techniques
that
can
be
used
to
o
v
ercome
f
ading
in
wireless
comm
unication
systems
.
Figure
1.
sho
ws
such
adaptiv
e
systems
acclimating
to
f
ading
b
y
utilizing
div
ersity
combining
gain,
optimal
po
w
er
adaptation
and
diff
erent
component
channel
codes
ha
ving
v
ar
ious
constellations
siz
es
.
Increasing
t
he
system
capacity
and
compensating
f
ading
in
wireless
comm
unication
sys-
tems
,
is
a
thoroughly
studied
topic
and
in
v
estigated
in
the
recent
liter
ature
[4,
5,
6,
7,
8,
9,
10,
11].
A
d
a
p
t
iv
e
M
o
d
u
l
a
t
io
n
a
n
d
C
o
d
in
g
P
o
w
e
r
O
p
t
im
iz
a
t
io
n
R
a
y
le
ig
h
Fa
d
in
g
C
h
a
n
n
e
l
∑
De
m
o
d
u
la
t
i
o
n
C
h
a
n
n
e
l
E
s
t
im
a
t
o
r
A
d
a
p
t
iv
e
T
r
a
n
s
m
is
s
io
n
M
o
d
u
le
I
n
p
u
t
Da
t
a
O
u
t
p
u
t
Da
t
a
E
r
r
o
r
fr
ee
F
ee
d
b
a
c
k
p
a
t
h
Figure
1.
Ov
er
vie
w
of
adaptiv
e
tr
anmission
system
[4].
Receiv
ed
J
une
12,
2016;
Re
vised
September
10,
2016;
Accepted
September
23,
2016
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
105
In
[12,
13],
Goldsmith
and
V
ar
aiy
a
presented
some
of
the
adaptiv
e
tr
ansmission
policies
,
namely:
optimal
tr
ansmit
po
w
er
with
r
ate
adaptation
-
constant
po
w
er
with
optimal
r
ate
adaptation
-
chan-
nel
in
v
ersion
with
fix
ed
r
ate
.
The
idea
of
adapting
optimal
po
w
er
combining
with
modulation
and
coding
r
ate
w
as
to
deter
mine
the
maxim
um
a
v
er
age
spectr
al
efficiency
(MASE)
o
v
er
flat
f
ad-
ing
channels
ha
ving
perf
ect
channel
state
inf
or
mation
(CSI)
with
respect
to
both
the
tr
ansmitter
and
receiv
er
.
A
pr
actical
approach
in
deter
mining
the
spectr
al
efficiency
is
adapted
b
y
[14],
f
or
discrete-r
ate
m
ultile
v
el
quadr
ature
amplitude
modulation
(MQAM)
f
or
f
ading
channels
.
While
in
[8]
using
trellis
coded
modulation
ha
v
e
o
v
ercome
the
obser
v
ed
gap
betw
een
the
achie
v
ab
le
spectr
al
efficiency
and
MASE.
Fur
thmore
,
Hole
in
[9]
e
xtended
[14,
8]
to
de
v
elope
a
gener
al
techniqu
e
to
e
v
aluate
a
v
er
age
spectr
al
efficiency
of
the
coded
modulation
f
or
2
L
-D
trellis
codes
o
v
er
Nakagami
m
ultipath
f
ading
(NMF).
One
of
the
f
ading
compen
sation
techniques
as
w
ell
as
boosting
link
perf
or
mance
is
the
div
ersity
combining
technique
which
accum
ulates
signals
receiv
ed
from
se
v
er
al
paths
.
In
[4],
the
authors
use
div
ersity
combining
techniques
to
design
a
system
f
or
maxim
um
spectr
al
efficiency
.
The
y
ha
v
e
e
v
olv
ed
closed-f
or
m
e
xpressions
f
or
the
gener
al
theor
y
of
adaptiv
e
tr
ansmission
from
[13]
into
three
adaptiv
e
tr
ansmission
and
div
ersity
combining
techniques
f
or
Ra
yleigh
f
ading
chan-
nels
.
Although,
div
ersity
pro
vides
large
capacity
gain
f
or
each
unit
bandwidth
o
v
er
all
the
tech-
niques
,
ho
w
e
v
er
in
[15,
4]
it
has
been
pro
v
ed,
there
m
ust
be
a
tr
adeoff
betw
een
the
conflicting
targets;
comple
xity
and
capacity
of
the
adaptation
methods
.
The
e
v
aluation
of
v
ar
ious
po
w
er
and
r
ate
adaptation
policies
w
ere
also
considered
in
[6],
where
closed-f
or
m
solutions
w
ere
der
iv
ed
f
or
maxim
um-combining
r
atio
(MRC)
of
the
gener
aliz
ed
Rician
f
ading
channel.
It
is
repor
ted
that
tr
uncated
channel
in
v
ersion
adaptation
policy
ha
s
perf
or
mance
adv
antage
o
v
er
the
diff
erent
sin-
gle
antenna
reception
policies
.
Moreo
v
er
,
channel
in
v
ersion
with
fix
ed
r
ate
policy
is
the
pref
err
ab
le
MRC
policy
and
equal
gain
combining
(EGC)
div
ersity
techniques
[16].
F
ading
mitigation/compensation
methods
with
div
ersity
combining
enab
le
the
adaptiv
e
tr
ansimssion
policies
to
achie
v
e
capcity
close
to
shanon
limits
in
flat
f
ading
channel
[15].
Shanon
capacity
is
considered
idealistic
limit
f
or
comm
unication
systems
,
and
also
a
de-f
acto
ref
erence
to
compare
adaptiv
e
schemes
with
respect
to
spectr
al
efficiency
.
Unlik
e
pr
ior
mentioned
w
or
k
[4]-
[16],
design
adaptiv
e
t
r
ansmission
system
emplo
ying
f
ading
compensation
methods
(e
.g
po
w
er
and
r
ate
adaptation)
that
adequately
consider
the
target
BER
in
conjuction
with
MRC
div
ersity
.
The
design
also
optimiz
ed
adaptiv
e
po
w
er
and
const
ellation
siz
e
f
or
contin
uous
po
w
er
and
r
ate
adaptation
while
handling
tr
uncated
channel
in
v
ersion
with
fix
ed
r
ate
and
contin
uous
po
w
er
and
discrete
r
ate
adaptation.
Hence
,
the
pr
imar
y
pur
pose
of
this
adaptiv
e
tr
ansmission
system
is
to
balance
the
link
b
udget
in
real-time
through
adaptiv
e
v
ar
iation
of
the
v
ar
ious
impor
tant
par
ameters
such
as
symbol
r
ate
,
tr
ansmit
po
w
er
,
constellation
siz
e
,
coding
r
ate
,
or
an
y
combination
of
these
par
ameters
[14,
4].
Moreo
v
er
,
a
suitab
le
model
of
the
div
ersity
combining
adaptiv
e
process
w
as
also
outlined
to
e
xtend
the
obtained
results
to
diff
erent
adaptiv
e
tr
ansmission
policies
with
respect
to
those
considered.
The
proposed
mechanism
tak
es
the
influence
of
channel
in
ter
ms
of
atten
uation
caused
b
y
m
ultipath
f
ading
into
account,
and
also
e
xamines
the
diminshing
eff
ect
of
BER
in
conjunction
with
MRC
div
ersity
on
selected
adaptiv
e
tr
ansmission
schemes
.
These
consider
a
tions
f
or
m
ulate
adaptiv
e
constellation
siz
e
f
or
tr
ansmitting
optimal
symbols
to
impro
v
e
the
spectr
al
efficiency
when
consider
ing
a
cer
tain
f
ading
distr
ib
ution.
The
main
contr
ib
utions
of
this
paper
are
as
f
ollo
ws:
The
target
BER
and
div
ersity
co
mbining
gain
at
Ra
yleigh
f
ading
channel
are
assumed
while
der
iving
closed-f
or
m
e
xpressions
using
constr
ained
spectr
al
efficiency
maximization
of
adaptiv
e
tr
ansmission
policies;
The
constellation
s
witching
thresholds
and
its
ada
ptiv
e
po
w
er
f
or
discrete-r
ate
po
licy
are
optimiz
ed
to
maintain
the
target
BER
in
conjunction
with
div
ersity
combining.
A
ser
ies
of
n
umer
ical
results
that
v
alidate
the
oper
ations
of
proposed
system,
sho
w
the
introduction
of
po
w
er
adaptation
with
respect
to
div
ersity
combining
and
target
BER
in
the
selected
policies
ha
v
e
good
ag
reement
in
ter
ms
of
spectr
al
efficiency
increase
and
reducing
the
possibility
of
no
tr
ansmission,
e
v
en
when
the
n
umber
constellations
siz
e
are
finite
.
Optimal
P
o
w
er
and
Modulation
Adaptation
P
olicies
with
Receiv
er
...
(Muhammad
Imr
an
T
ar
iq)
Evaluation Warning : The document was created with Spire.PDF for Python.
106
ISSN:
2502-4752
The
rest
of
this
paper
is
str
uctured
as
f
ollo
ws
.
In
section
2,
the
system
model
is
pre-
sented
b
y
f
ocusing
the
prob
lem
that
is
in
v
estigated.
Section
3
discusses
and
der
iv
es
the
adaptiv
e
tr
ansmission
policies
with
respect
to
the
target
BER
and
MRC
div
ersity
.
In
section
4,
obtained
n
umer
ical
results
are
presented
with
detail
disucssion.
Finally
,
in
section
5
the
conclusions
are
dr
a
wn.
2.
System
Model
and
Pr
ob
lem
Form
ulation
2.1.
System
Model
In
this
section,
a
single-link
wireless
comm
unication
model
will
be
considered
as
sho
wn
in
Figure
1,
similar
to
the
one
descr
ibed
in
[4,
5].
In
a
giv
en
wireless
comm
unication
model,
the
discrete-time
channel
oscillates
slo
w
er
deg
ree
than
the
data
r
ate
.
In
this
analysis
,
it
is
assumed
to
be
slo
w
v
ar
ying
and
frequency-flate
Ra
yleigh
f
ading
channel.
By
the
probity
of
these
assumption
the
distr
ib
ution
of
the
receiv
ed
Signal-to-Noise
Ratio
(SNR)
is
represented
b
y
an
e
xponential
distr
ib
ution
[17].
f
(
)
=
1
e
=
;
(1)
Where
is
the
a
v
er
age
receiv
ed
SNR
and
represents
the
instantaneous
receiv
ed
SNR.
In
this
analysis
,
MRC
combining
technique
is
considered
where
the
amplitudes
and
phases
of
the
receiv
ed
signals
are
assumed
kno
wn
b
y
the
receiv
er
perf
ectly
.
By
vir
tue
of
this
assumption
w
e
can
ter
m
it
perf
ect
combining
.
It
is
pro
v
en
in
[17,
15,
4],
that
MRC
yields
max-
im
um
spectr
al
efficiency
impro
v
ement
relativ
e
to
all
combining
techniques
as
a
result
of
perf
ect
combining
.
It
requires
the
receiv
ed
signals
to
be
w
eighted
b
y
their
o
wn
SNR
(w
eighting
par
am-
eter)
independent
from
each
div
ersity
br
anch
to
compose
the
output
d
ecision
v
ar
iab
le
.
F
or
a
Ra
yleigh
f
ading
channel,
the
output
of
a
linear
ly
combined
MRC
combiner
with
L
-br
ach
is
giv
en
in
[17]
as
a
distr
ib
ution
of
the
instantaneous
receiv
ed
SNR.
f
mr
c
(
)
=
1
(
L
1)!
L
1
L
e
=
;
(2)
W
e
simply
consider
throughout
in
our
study
,
as
in
[14,
13],
the
channel
state
is
perf
ectly
tr
ac
k
ed
b
y
the
tr
ansmitter
via
error-free
f
eedbac
k
channel.
Accordingly
,
the
proposed
model
coordinattes
with
po
w
er
adaptation
scheme
P
(
)
to
adapt
tr
ansmit
po
w
er
.
Let
N
denotes
the
quantization
le
v
els
of
a
v
ailab
le
constellations
M
n
that
represent
the
instantaneous
receiv
ed
SNR
.
The
r
ange
is
par
titioned
into
N
+1
nono
v
er
lapping
successiv
e
f
ading
regions
,
with
boundr
y
points
denoted
b
y
the
s
witching
thresholds
f
n
T
g
N
+1
n
=0
.
Specifically
,
constellation
n
ha
ving
spectr
al
efficiency
S
E
,
is
chosen
when
2
n
T
;
n
+1
T
and
within
this
region,
the
tr
ansmission
r
ate
does
not
change;
y
et
the
adaptiv
e
policy
might
change
the
tr
ansmit
po
w
er
in
order
to
compensate
f
or
the
f
ading.
T
o
o
v
ercome
strong
channel
f
adings
,
data
w
ould
be
b
uff
ered
f
or
output
SNR
of
f
ading
regions
0
T
<
1
T
.
F
or
con
vience
,
w
e
suppose
0
T
=
0
and
n
T
=
1
.
2.2.
Pr
ob
lem
Form
ulation
This
section
sho
ws
capacity
of
flat
f
adding
channel
ha
ving
a
perf
ect
channel
state
inf
or-
mation
(CSI)
with
respect
to
the
tr
ansmitter
side
and
the
receiv
er
as
w
ell.
Which
is
widely
repre-
sented
as
C
(
)
=
B
log
2
(1
+
(
P
(
)
=
P
)
)
data
bits/s
.
This
capacity
is
considered
as
a
y
ardstic
k
to
e
v
aluate
adaptiv
e
tr
ansmission
schemes
regarding
its
spectr
al
efficiency
[14].
By
taking
this
eff
ec-
tiv
e
div
ersity
combining
based
approach,
there
e
xists
constellation
siz
e
that
can
achie
v
e
spectr
al
efficiencies
that
can
achie
v
e
C
(
)
bits/s
,
while
maintaining
cer
tain
target
BER.
The
presence
of
such
constellation/modulation
is
assured
according
to
the
w
ell
kno
wn
Shannon’
s
theorem
(chan-
nel
coding).
Our
aim
of
this
paper
is
to
optimiz
e
a
set
of
capacities
with
tr
ansmission
modulation
r
ates
and
s
witching
threshold
to
guar
antee
the
desired
BER
and
po
w
er
adaptation
shcemes
.
This
should
maximiz
e
the
corresponding
spectr
al
efficiency
in
the
giv
en
f
ading
distr
ib
ution.
The
outage
probability
P
out
of
adaptiv
e
policy
with
target
BER
can
only
appear
when
the
set
of
channel
states
is
belo
w
the
optimal
cutoff
SNR
0
,
which
nor
mally
appears
within
the
IJEECS
V
ol.
4,
No
.
1,
October
2016
:
104
115
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
107
first
inter
v
al
of
f
ading
regions
.
Only
at
that
time
data
is
b
uff
ered
f
or
the
corresponding
per
iod.
The
associated
coding
r
ate
of
other
f
ading
regions
accommodate
fluctuations
.
Thus
,
f
or
adaptiv
e
systems
the
resulting
spectr
al
efficiency
can
be
e
xpressed
as
S
E
=
N
X
n
=1
M
n
P
n
(3)
Where
M
n
=
log
2
(
M
n
)
represents
the
con
stellation
siz
e
or
modulation
code
,
and
P
n
is
defined
as
the
proability
of
selecting
code
n
:
P
n
=
Z
n
+1
T
n
T
f
mr
c
(
)
d
(4)
3.
Adaptive
T
ransmission
P
olicies
with
MRC
diver
sity
The
center
of
attention
in
this
paper
is
on
the
p
erf
or
mance
char
acter
izatoin
of
an
adaptiv
e
system
with
MRC
div
ersity
.
The
f
or
mer
studies
emphasiz
e
on
maximizing
spectr
al
efficiency
of
adaptiv
e
tr
ansmission
under
constant
po
w
er
constr
aints
while
maintaining
target
BER
[18]
and
dif-
f
erent
po
w
er
le
v
els
[19]
unlik
e
those
studies
,
this
research
is
interested
in
e
v
aluating
the
adaptiv
e
system
with
MRC
div
ersity
while
sustaining
a
cer
tain
BER,
in
selected
adaptation
policies
.
Ac-
cordingly
,
it
is
first
argued
in
[20,
21]that
the
appro
ximate
BER
of
a
rectangular
MQAM
uncoded
modulation
in
A
WGN
can
be
e
xpressed
b
y
the
f
or
m
P
b
a:exp
b
(
M
1)
(5)
Where
a
and
b
are
positiv
e
fix
ed
constants
used
to
appro
ximate
the
bounds
of
the
e
x-
pression
and
M
is
the
siz
e
of
a
constellation.
The
desired
bound
within
1
dB
it
is
achie
v
e
d
with
a
=
0.2
and
b
=
1.5
f
or
M
4
and
receiv
ed
SNR
0
30
dB
[12].
Though,
e
v
aluations
of
P
b
e
.g
lose
or
tighter
bounds
descr
ibed
in
liter
ature
clear
ly
argue
that
using
cur
v
e
fitting
techniques
w
e
optimiz
e
the
v
alue
of
a
and
b
such
that
the
e
xpression
yields
good
accur
acy
e
v
en
f
or
lo
w
.
The
error
probability
f
or
MRC
div
ersity
combining
with
L
-br
anches
,
in
independent
and
identically
distr
ib
uted
(i.i.d)
Ra
yleigh-f
ading
channels
can
be
der
iv
ed
b
y
substituting
(2)
the
pdf
of
into
(5).
That
w
a
y
f
or
the
A
WGN
channel
with
uncoded
modulation,
the
error
r
ates
w
ould
be
a
v
er
aged
out.
It
is
possib
le
to
obtain
closed-f
or
m
e
xpression
b
y
simplifying
as
in
the
f
ollo
wing
P
mr
c
b
=
Z
1
0
P
b
:f
mr
c
(
)
d
a:exp
b
(
M
1)
+
1
L
(6)
Accordingly
,
w
e
use
this
e
xpression
when
needed,
since
it
is
easy
to
in
v
er
t.
Hence
,
f
or
a
giv
en
target
P
b
,
adaptiv
e
tr
ansmission
policy
and
L
-br
aches
or
div
ersity
combining
le
v
els
,
the
required
M
constellation
le
v
el
can
be
deter
mined
n
umer
ically
.
By
selecting
the
adaptiv
e
tr
ansmission
policies
,
no
w
w
e
are
ready
to
deter
mine
the
r
ate
and
po
w
er
which
fluctuate
according
to
time-v
ar
iations
of
the
channel.
The
tr
ansmitter
compen-
sates/reciprocates
to
channel
fluctations
b
y
adjusting
the
constellation
siz
e
M
(
)
and
the
tr
ans-
mission
po
w
er
P
(
)
relativ
e
to
div
ersity
combining
and
B
E
R
T
.
At
the
receiv
er
,
the
receiv
ed
SNR
becomes
P
(
)
P
,
thus
the
appro
ximation
of
instantaneous
P
b
can
be
appro
ximated
b
y
(6)
f
or
each
v
alue
of
as:
P
mr
c
b
(
)
a:exp
b
(
M
1)
P
(
)
P
L
(7)
Optimal
P
o
w
er
and
Modulation
Adaptation
P
olicies
with
Receiv
er
...
(Muhammad
Imr
an
T
ar
iq)
Evaluation Warning : The document was created with Spire.PDF for Python.
108
ISSN:
2502-4752
By
re-arr
anging
the
abo
v
e
in
ter
ms
of
M
,
w
e
obtain
an
e
xpression
f
or
maxim
um
constellation
siz
e
as
a
function
of
target
BER
B
E
R
T
,
div
ersity
le
v
el
L
and
instantneous
receiv
ed
SNR
:
M
(
)
=
1
+
K
n
P
(
)
P
(8)
Where
K
n
,
b:L
p
a:B
E
R
T
represents
MRC
div
ersity
combining
and
a
=
5.0
and
b
=
1.5,
to
po
w
er
diminishing
par
ameter
in
v
ersely
propotion
to
B
E
R
T
.
3.1.
Contin
uous
po
wer
and
rate
adaptation
polic
y
In
the
f
ollo
wing,
w
e
maximizing
spect
r
al
efficiency
of
the
MQAM
scheme
f
or
a
specified
a
v
er
age
tr
ansmission
po
w
er
with
t
he
target
BER
B
E
R
T
in
the
first
adaptation
policy
.
The
spectr
al
efficiency
of
a
f
ading
channel
and
receiv
ed
SNR
distr
ib
ution
f
mr
c
(
)
with
respect
to
div
ersity
combining,
is
maximiz
ed
b
y
maximizing
(8)
as
f
ollo
ws
[12]:
E
[log
2
M
(
)]
=
Z
log
2
1
+
K
n
P
(
)
P
f
mr
c
(
)
d
(9)
Introducing
a
maxim
um
spectr
al
efficiency
scheme
(9)
,
which
subject
to
a
v
er
age
po
w
er
constr
aint
P
(
)
,
then
constr
ict
the
a
v
er
age
po
w
er
(using
an
equal)
r
ather
than
just
boundnig
(using
inequality)
@
J
@
P
(
)
=
0
Lag
r
angian
op
timization
solution.
Optimal
po
w
er
adaptation
f
or
MQAM
mentioned
in
[12]
can
be
e
xpressed
as
f
ollo
ws:
P
(
)
P
=
(
1
0
1
K
n
0
=K
n
0
<
0
=K
n
(10)
Where
0
=K
n
is
the
optimal
le
v
el
SNR
at
which
at
the
cutoff
f
ade
.
Be
y
ond
that
le
v
el
tr
ansmission
w
ould
be
def
erred.
This
optimal
cutoff
and
MRC
combining
0
f
ollo
w:
Z
+
1
0
1
0
1
K
n
f
mr
c
(
)
d
=
1
(11)
F
or
the
abo
v
e
equation
of
optimal
cutoff
SNR
while
maintaining
BER
perf
or
mance
,
a
closed-f
or
m
e
xpression
adopts
the
n
umer
ical
root
finding
techniques
and
substitutes
(2)
in
(11)
to
find
0
:
(
L;
0
)
0
1
K
n
(
L
1
;
0
)
=
(
L
1)!
(12)
Where
(
;
x
)
=
R
1
x
t
1
e
t
dt
is
a
complementar
y
incomplete
gamma
function
[15],[22].
Let
x
=
0
and
define:
f
mr
c
K
n
(
x
)
=
(
L;
x
)
x
1
K
n
(
L
1
;
x
)
(
L
1)!
(13)
Note
that
@
f
mr
c
K
n
(
x
)
@
0
<
0
8
x
>
0
and
L
2
.
By
vir
tue
of
lim
x
!
0
+
f
mr
c
K
n
(
x
)
=
+
1
and
lim
x
!
+
1
f
mr
c
K
n
(
x
)
=
(
L
1)!
,
hence
it
is
assumed
that
there
is
a
unique
positiv
e
0
such
that
f
mr
c
K
n
(
0
)
=
0
and
restr
icted
b
y
(12).
Gn
u
Scientific
Libr
ar
y
(GSL)
optimization
routines
[23]
are
used
to
obtain
n
umer
ical
results
.
The
results
sho
w
0
with
MRC
div
ersity
and
BER
perf
or
mance
alw
a
ys
lies
in
the
inter
v
al
[0
;
1]
.
Based
on
the
result
of
(13),
w
e
can
de
vise
the
closed-f
or
m
e
xpression
of
(9)
b
y
substitut-
ing
(2)
in
(9),
f
or
the
channel
capacity
h
C
i
CPRA
as:
h
C
i
mrc
CPRA
=
J
L
K
n
K
n
L
log
2
(
e
)(
L
1)!
(14)
IJEECS
V
ol.
4,
No
.
1,
October
2016
:
104
115
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
109
Where
K
n
=
0
=K
n
is
a
cutoff
SNR
threshold.
Be
y
ond
that
thershold
def
err
ing
of
the
data
tr
ansmission
occurs
.
B
represents
to
channel
bandwidth
(in
Her
tz).
Letting
=
K
n
>
0
,
using
the
e
v
aluation
of
J
L
(
)
which
is
giv
en
in
Appendix
A
[22],
and
rearr
anging
the
(14).
F
or
MRC
div
ersity
,
w
e
obtain
the
SE
[bits/s/Hz]
and
BER
perf
or
mance
b
y
using
contin
uous
po
w
er
and
r
ate
adaptation
policy
as:
h
C
i
mrc
CPRA
B
=
log
2
(
e
)
E
1(
)
+
L
1
X
k
=1
P
k
(
)
k
!
(15)
Where
P
k
(
)
is
P
oisson
distr
ib
ution
defined
as
P
k
(
)
=
e
k
1
P
j
=0
j
j
!
and
E
1(
)
=
R
1
1
e
t
t
dt
is
an
e
xponential
integ
r
al
function
of
first-order
[22].
T
o
achie
v
e
the
capacity
(15),
instantaneous
receiv
ed
SNR
should
not
f
all
belo
w
K
n
.
Since
,
no
data
tr
ansmission
occurs
when
<
K
n
,
the
policy
is
vulner
ab
le
to
outage
.
Outage
probability
under
B
E
R
T
perf
or
mance
f
or
the
MRC
div
ersity
case
is
e
xpressed
belo
w:
f
out
mr
c
=
Z
K
n
0
f
mr
c
(
)
d
=
1
Z
1
K
n
f
mr
c
(
)
d
(16)
Substituting
(2)
in
(16),
f
or
the
probability
of
outage
,
w
e
obtain
the
f
ollo
wing
closed
f
or
m
e
xpres-
sion:
f
out
mr
c
=
1
P
L
(
)
(17)
3.2.
T
runcated
c
hannel
in
ver
sion
polic
y
with
fix
ed
rate
Another
adaptation
policy
where
the
tr
ansmitter
in
v
er
ts
the
channel
higher
than
the
f
ade
depth
0
to
maintain
the
SNR
receiv
ed.
This
po
w
er
adaptation
policy
is
ter
med
T
r
uncated
Channel
In
v
ersion
with
Fix
ed
Rate
(TCIFR)
that
is
emplo
y
ed
in
inner-loop
po
w
er
control
mechanism
[24],
which
adapts
the
tr
ansmiting
po
w
er
to
achie
v
e
a
desired
SNR
at
the
receiv
er
.
Then,
adaptiv
e
sys-
tem
tr
ansmits
fix
ed-r
ate
MQAM
constellation
that
maintains
the
B
E
R
T
with
div
ersity
combining.
Hence
,
introducing
P
(
)
=
P
=
=
in
(8)
and
m
ultiplying
b
y
the
probability
that
>
0
,
w
e
obtain
a
ne
w
e
xpression
f
or
channel
capacity
e
xpressed
b
y:
h
C
i
mrc
tcifr
=
B
log
2
0
@
1
+
K
n
1
R
1
K
n
f
mr
c
(
)
d
1
A
(1
f
out
mr
c
)
(18)
Where
f
out
mr
c
is
calculated
as
in
(16)
to
maximiz
e
(18).
In
order
to
find
the
closed-f
or
m
e
xpres-
sion,
w
e
substitute
(2)
into
(16)
and
rearr
ange
to
obtain
the
SE
[bits/s/Hz]
under
TCIFR
policy
and
B
E
R
T
perf
or
mance
f
or
MRC
div
ersity
as:
h
C
i
mrc
tcifr
B
=
log
2
1
+
(
M
1)!
(
M
1
;
0
=
)
(
M
;
0
=
)
(
M
1)!
(19)
Using
proper
ties
of
(
;
)
of
complementar
y
incomplete
gamma
function
[15][22],
(19)
fur
ther
can
be
simpliefied
as
f
ollo
ws
h
C
i
mrc
tcifr
B
=
log
2
1
+
(
M
1)
P
M
1
(
0
=
)
P
M
(
0
=
)
(20)
3.3.
Contin
uous
po
wer
and
discrete-rate
adaptation
polic
y
W
e
no
w
proceed
to
e
xtend
the
design
of
adaptiv
e
system
f
or
contin
uous
po
w
er
and
discrete-r
ate
adaptatin
policy
,
b
y
restr
icting
adaptiv
e
MQAM
to
N
f
ading
regions
,
whose
con-
stellations
siz
e
M
n
=
2
2(
n
1)
and
whose
BER
can
be
appro
ximated
b
y
(7)
f
or
n
=
2
;
:::;
N
1
Optimal
P
o
w
er
and
Modulation
Adaptation
P
olicies
with
Receiv
er
...
(Muhammad
Imr
an
T
ar
iq)
Evaluation Warning : The document was created with Spire.PDF for Python.
110
ISSN:
2502-4752
under
div
ersity
combining.
Recall
that
instantaneous
SNR
is
par
tationed
into
N
f
ading
regions
[12],
[9],
the
modulation
siz
e
M
n
is
used
whene
v
er
2
[
n
T
;
n
+1
T
)
.
Thus
,
the
adaptiv
e
system
is
lo
w
er
bounded
b
y
n
T
to
maintain
the
B
E
R
T
with
div
ersity
combining,
after
tr
ansmission
po
w
er
adaptation.
F
ollo
wing
(8),
this
policy
also
requires
the
boundr
ies
of
s
witching
threshold
be
deter
mined.
Theref
ore
,
b
y
substituting
(10)
into
(8)
yields
the
constellation
siz
e
f
or
a
giv
en
as:
M
(
)
=
(21)
Where
is
a
optimiz
ed
par
ameter
f
ound
b
y
n
umer
ical
methods
to
optimiz
e
the
s
witching
thresh-
olds
and
maximiz
e
spectr
al
efficiency
.
F
or
mer
,
the
f
ading
region
n
of
s
witching
threshold
and
as-
sociated
constellation
siz
e
M
n
are
deter
mined,
then
w
e
obtain
the
po
w
er
adaptation
policy
based
on
(8)
to
maintain
the
fix
ed
B
E
R
T
and
satifisfies
the
a
v
er
age
po
w
er
constr
aint
R
P
(
)
f
mr
c
(
)
as
f
ollo
ws:
P
n
(
)
P
=
(
(
M
n
1)
K
n
M
n
<
M
n
+1
0
M
n
=
0
(22)
Where
P
n
(
)
represents
post
adaptation
po
w
er
,
and
when
<
no
tr
ansmission
ocurrs
and
data
is
b
uff
ered.
The
maximiz
ed
spectr
al
efficiency
SE
of
CPD
A
policy
f
or
a
giv
en
div
ersity
le
v
el
and
B
E
R
T
with
distr
ib
ution
f
mr
c
(
)
,
is
defined
as
the
sum
of
spectr
al
efficiencies
a
s
so
ciated
with
each
of
the
f
ading
regions:
h
C
i
mrc
CPD
A
B
=
N
1
X
n
=1
log
2
(
M
n
)
f
mr
c
(
M
n
<
M
n
+1
)
(23)
Since
M
n
is
a
function
of
,
b
y
maximizing
(23)
with
respect
to
,
w
e
arr
iv
e
at
the
f
ollo
w-
ing
optimization
prob
lem
where
(23)
is
maximiz
ed
subject
to
po
w
er
constr
aint
m
ust
be
satisfied:
N
1
X
n
=1
Z
M
n
+1
M
n
P
n
(
)
f
mr
c
(
)
d
=
1
(24)
If
w
e
assume
equ
ation
(2)
Ra
yleigh
f
ading
channel
with
div
ersity
combining
the
closed-f
or
m
e
x-
pression
of
(24)
can
be
wr
itten
as
f
ollo
ws:
f
mr
c
(
)
=
N
1
X
n
=1
M
n
1
K
n
(
L
1)!
(
L
1
;
M
n
+1
)
(
L
1
;
M
n
)
(25)
With
the
increase
of
div
ersity
g
ain,
relativ
e
to
others
polices
,
the
discrete-r
ate
adaptiv
e
policy
will
ha
v
e
reasonab
le
ag
reement
in
ter
m
spectr
al
efficiency
b
ut
significantly
has
less
p
robability
of
outage
as
w
e
sho
w
in
results
section.
4.
Numerical
Results
and
Discussion
In
this
section,
w
e
obtain
the
n
umer
ically
e
v
aluated
results
f
or
contin
uous
po
w
er
and
r
ate
adaptation
(CPRA),
tr
uncated
channel
in
v
ersion
with
fix
ed-r
ate
adaptation
(TCIFR)
and
contin
u-
ous
po
w
er
and
discrete-r
ate
adaptation
(CPD
A)
at
target
B
E
R
T
.
F
or
the
f
ollo
wing,
a
Ra
yleigh
f
ading
distr
ib
ution
and
MRC
div
ersity
combining
technique
ha
v
e
been
assumed.
Using
descr
ibed
techniques
in
[15]
and
[4],
w
e
ha
v
e
designed
an
ada
ptiv
e
MQAM
system
which
tak
es
the
target
B
E
R
T
into
account
and
utiliz
es
the
div
ersity
combining
gain,
to
maximiz
e
IJEECS
V
ol.
4,
No
.
1,
October
2016
:
104
115
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IJEECS
ISSN:
2502-4752
111
0
2
4
6
8
10
5
10
15
20
25
Spectral Efficiency [bits/s/Hz]
Average SNR [dB]
BER=10
-2
BER=10
-3
BER=10
-4
BER=10
-5
Figure
2.
Spectr
al
efficiency
of
contin
uous
po
w
er
and
r
ate
adaptation
f
or
v
ar
ious
target
B
E
R
T
with
div
ersity
le
v
el
L=4
,
v
ersus
a
v
er
age
receiv
ed
SNR
.
0
2
4
6
8
10
5
10
15
20
25
Spectral Efficiency [bits/s/Hz]
Average SNR [dB]
L = 2
L = 4
L = 6
CPRA
TCIRA
CPDRA
Figure
3.
Spectr
al
efficiency
of
v
ar
ious
adaptiv
e
policies
with
v
ar
ious
div
ersity
le
v
els
and
B
E
R
T
=
10
4
,
v
ersus
a
v
er
age
receiv
ed
SNR
.
the
spectr
al
efficiency
.
As
can
be
s
e
en
in
Figure
2,
the
gap
betw
een
the
spectr
al
efficiencies
is
induced
b
y
a
eff
ectiv
e
po
w
er
loss
par
ameter
K
n
,
since
it
is
a
function
of
B
E
R
T
and
n
umber
of
L
div
ersity
le
v
els
.
Moreo
v
er
,
w
e
also
obser
v
e
that
K
n
diminshes
the
perf
or
mance
within
4
dB
with
constant
L
as
B
E
R
T
adv
ances
.
Hence
,
w
e
fix
the
target
B
E
R
T
in
the
f
ollo
wing
,
and
e
v
aluate
its
eff
ect
on
spectr
al
efficiency
and
po
w
er
adaptation
f
or
proposed
ada
ptiv
e
system
with
div
ersity
combining
techniques
.
Figure
3
presents
the
resulting
channel
capacity
f
or
each
bandwidth
unit
or
spectr
al
efficiency
acquired
using
closed-f
or
m
e
xpression
(15),
(19)
and
(23)
.
The
capacity
per
unit
band-
width
increases
with
th
e
n
umber
of
div
ersity
le
v
els
.
W
e
noted
that
the
spectr
al
efficiency
cur
v
e
f
or
CPRA
policy
has
increasing
trend
at
lo
w
rece
iv
ed
SNR
(0
dB
to
10
dB)
in
the
beginning,
af-
terw
ards
g
r
adually
comes
close
to
other
policies
k
eeping
slightly
larger
spectr
al
efficiency(when
SNR
is
g
reater
than
10
dB).
The
cause
of
this
eff
ect
is
the
CPRA
policy
of
le
v
er
aging
the
w
ater-
filling
nature
at
lo
w
er
receiv
ed
SNR
and
it
only
holds
this
instict
f
or
div
ersity
le
v
el(L
=
1
or
L
=
2).
In
Figure
3,
the
discrete-r
ate
policy
has
similair
beha
viour
e
v
en
in
div
ersity
combining
[12],
b
y
restr
icting
it
to
the
f
ading
regions
and
it
yields
lo
w
spectr
al
efficiency
.
That
resulting
efficence
y
is
1
dB
compared
to
that
produced
b
y
CPRA.
Ho
w
e
v
er
,
the
spectr
al
efficiency
cur
v
es
are
v
er
y
close
to
each
other
in
almost
all
div
ersity
le
v
els
.
The
optimal
s
witching
thresholds
f
n
T
g
N
n
=1
f
or
CPD
A
policy
which
is
suitab
le
f
or
adaptiv
e
system,
are
sho
wn
in
Figure
4,
at
targer
t
B
E
R
T
=
10
4
and
f
or
0
dB
<
<
25
dB
under
v
ar
ious
div
ersity
le
v
els
.
Obtain
the
thresholds
using
e
xpression
(21)
with
4
f
ading
regions
and
optimiz
ed
po
w
er
adaptation
(22),
and
f
ound
through
n
umer
ical
methods
,
where
the
threshold
n
T
equals
Optimal
P
o
w
er
and
Modulation
Adaptation
P
olicies
with
Receiv
er
...
(Muhammad
Imr
an
T
ar
iq)
Evaluation Warning : The document was created with Spire.PDF for Python.
112
ISSN:
2502-4752
-5
0
5
10
15
20
25
5
10
15
20
25
Switching thresholds
{γ
n
}
N
n=1
[dB]
Average SNR [dB]
L=2
L=3
L=4
L=5
L=6
Figure
4.
Switching
thresholds
f
n
T
g
N
n
=1
under
contin
uous
po
w
er
adap
tation
policy
with
target
B
E
R
T
=
10
4
as
a
functin
of
v
ar
ious
div
ersity
le
v
els
.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
5
10
15
20
25
P(
γ
)/
–
P
Average SNR [dB]
L=2
L=3
L=4
L=5
L=6
Figure
5.
P
o
w
er
adaptation
scheme
P
(
)
=
P
with
v
ar
ious
div
ersity
le
v
els
with
respect
to
a
v
er
age
receiv
ed
SNR,
plotted
f
or
an
a
v
er
age
SNR
=
15
dB
.
the
minim
um
target
SNR
to
obtain
the
desired
B
E
R
T
.
As
,
it
is
ob
vious
in
Figure
4
n
T
in
each
div
ersity
le
v
el
e
xploits
the
div
ersity
gain
after
3
dB
and
it
monotonically
widens
with
the
increment
of
div
ersity
le
v
el.
By
taking
the
le
v
er
age
of
div
ersity
gain,
the
adaptiv
e
system
s
witches
the
modulation
code
when
M
n
+1
>
M
n
and
maximiz
es
the
spectr
al
efficiency
b
y
the
probability
R
n
+1
T
n
T
f
mr
c
(
)
d
that
the
instantenous
receiv
ed
SNR
f
alls
in
region
n
.
F
or
each
of
interest,
designed
the
constellation
siz
e
abo
v
e
f
or
spectr
al
efficiency
.
The
optimiz
ed
policy
f
or
po
w
er
adaptation
is
teh
u
s
e
d
and
presented
in
(10).
Figure
5,
highlights
the
contin
uous
po
w
er
adaptation
policy
used
in
designing
the
adaptiv
e
system,
f
or
=
15
dB
.
W
e
noticed
the
r
ange
of
the
optimiz
ed
tr
ansmission
po
w
er
P
(
)
is
0
to
1.4
P
.
F
ollo
wing
the
w
ater-fillin
g
nature
,
it
can
be
seen
that
most
f
o
the
po
w
er
is
allocated
to
the
best
SNR
channels
regardless
div
ersity
gain.
In
addition,
it
is
interesting
to
point
out
that
with
the
upsurge
of
div
ersity
gains
the
optimiz
ed
po
w
er
becomes
unif
or
m
especially
when
15
dB
.
It
is
per
tinent
to
mention
that
due
to
lo
w
div
ersity
combing
gain,
35
%
more
tr
ansmit
po
w
er
is
allo
cated
when
div
ersity
le
v
el
is
L
=
2.
In
this
ana
ylsis
,
it
is
also
noticed
that
b
y
vir
tue
of
v
ar
ious
div
ersity
gain,
P
(
)
does
not
tak
e
r
igorous
peak
v
alues
.
Figure
6
depicts
the
nor
maliz
ed
spectr
al
efficiency
with
div
ersity
le
v
el
L
of
MRC
div
ersity
relation
to
cutoff
SNR
0
.
The
optimal
cutoff
SNR
(the
channel
is
not
used
belo
w
that)
is
n
umer-
ically
f
ound
using
(20)
f
or
a
cer
tain
v
alue
of
0
to
maximiz
e
spectr
al
efficiency
.
It
w
as
f
ound
that
spectr
al
efficiency
amelior
ates
b
y
increasing
the
div
ersity
le
v
els
.
Fur
ther
more
,
it
can
be
obser
v
ed
the
g
r
adual
f
all
of
the
cur
v
es
in
Figure
6
as
a
result
of
higher
probability
of
outage
.
As
,
it
w
as
stated
in
[12],
[4],
tr
uncated
channel
in
v
ersion
policy
yields
more
spectr
al
ef-
IJEECS
V
ol.
4,
No
.
1,
October
2016
:
104
115
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
113
0
1
2
3
4
5
5
10
15
20
25
Normalized Spectral Efficiency [bits/s/Hz]
Cut-off SNR [dB]
L=2
L=3
L=4
L=5
Figure
6.
Nor
maliz
ed
spectr
al
efficiency
of
TCIF
A
policy
with
v
ar
ious
div
ersity
le
v
els
and
=
12
dB
with
cutoff
SNR
0
10
-4
10
-3
10
-2
10
-1
10
0
5
10
15
20
25
Probability of outage [P
out
]
Average SNR [dB]
L=2
BER=10
-4
CPRA
TCIRA
CPDRA
Figure
7.
Probability
of
outage
of
selected
adaptatio
policies
with
div
ersity
le
v
el
L=2
and
B
E
R
T
=
10
4
,
as
a
function
of
a
v
er
age
receiv
ed
SNR
ficiency
sacr
ificing
the
probability
of
outage
,
as
e
xplained
in
Figure
7.
The
co
rresponding
prob-
ability
of
outage
P
out
f
or
CPRA,
TCIF
A
and
CPD
A
policies
are
calculated
according
to
(17)
with
respect
to
target
B
E
R
T
in
Figure
7.
The
giv
en
adaptiv
e
policies
then
are
rectified
to
achie
v
e
the
maxim
um
spectr
al
efficiency
under
the
giv
en
constr
aint
of
probability
of
outage
.
F
or
mer
ly
,
w
e
e
xplained
TCIF
A
policy
has
higher
probability
of
outage
compared
with
CPRA
and
CPD
A
poli-
cies
,
because
it
in
v
er
ts
the
f
ading
channel
and
compensates
the
f
ading
to
maximiz
e
the
spectr
al
efficiency
.
5.
Conc
lusion
In
this
ar
ticle
,
po
w
er
and
r
ate
adaptation
policies
ha
v
e
been
de
vised
to
maximiz
e
the
spectr
al
efficiency
of
MQAM
system
in
Ra
yleigh
f
ading
channel
with
MRC
div
ersity
combining
technique
.
In
par
ticular
,
closed-f
or
m
e
xpressions
of
selected
adaptiv
e
policies
are
obtained,
in
order
to
maintain
target
BER
a
nd
maximiz
e
spectr
al
efficiencies
in
conjunction
with
div
ersity
com-
bining.
The
results
w
ere
compared
f
or
a
contin
uous
po
w
er
and
r
ata
adaptation
policy
under
the
constr
aint
of
BER
and
its
eff
ect
on
spectr
al
efficiency
under
div
ersity
gain.
W
e
concluded
that
the
perf
or
mance
of
the
proposed
adaptiv
e
system
significantly
influenced
b
y
K
n
v
alue
.
On
the
other
hand,
w
e
obser
v
ed
that
the
diff
erence
in
each
step
of
BER
alone
e
xtends/reduces
the
system
spectr
al
efficiency
alsmost
4
dB
.
Additionally
,
w
e
optimiz
ed
the
s
witching
threshold
f
or
disrete-r
ate
adaptation
policy
and
po
w
er
adaptation
policies
through
n
umer
ical
techniques
.
It
is
interesting
to
note
that
the
contin
uous
po
w
er
and
r
ate
adaptation
policy
has
only
a
little
better
spectr
al
efficiency
compared
to
discrete-r
ate
and
tr
uncated
channel
in
v
ersion
policy
,
despite
the
Optimal
P
o
w
er
and
Modulation
Adaptation
P
olicies
with
Receiv
er
...
(Muhammad
Imr
an
T
ar
iq)
Evaluation Warning : The document was created with Spire.PDF for Python.