TELK
OMNIKA
Indonesian
Journal
of
Electrical
Engineering
V
ol.
12,
No
.
5,
Ma
y
2014,
pp
.
3955
3961
DOI:
http://dx.doi.org/10.11591/telk
omnika.v12.i5.5375
3955
A
Stud
y
on
P
eak-to-A
vera
g
e
P
o
wer
Ratio
in
D
WT
-OFDM
Systems
Filber
t
H.
J
uw
ono
*
,
Rand
y
S.
Putra
,
and
Dadang
Guna
wan
Depar
tment
of
Electr
ical
Engineer
ing,
Univ
ersity
of
Indonesia
Kampus
Bar
u
UI
Depok,
16424
*
Corresponding
author
,
e-mail:
filber
t@ieee
.org
Abstract
Or
thogonal
frequency
division
m
ult
iple
xing
(OFDM)
systems
suff
er
from
large
peak-to-a
v
er
age
po
w
er
r
atio
(P
APR).
In
this
paper
w
e
study
the
discrete
w
a
v
elet
tr
ansf
or
m
(D
WT)-based
OFDM
systems
.
In
par
ticular
,
w
e
discuss
the
eff
ect
of
the
decomposition
le
v
el
of
each
w
a
v
elet
f
amily
in
the
D
WT
-based
OFDM
regarding
the
P
APR.
The
sim
ulation
results
sho
w
that,
in
gener
al,
there
is
a
decomposit
ion
le
v
el
that
minimiz
e
the
P
APR
in
e
v
er
y
w
a
v
elet
f
amily
.
In
addition,
w
e
also
analyz
e
the
eff
ect
of
clipping
nonlinear
ities
,
i.e
.
con
v
entional
clipping
and
deep
clipping,
as
P
A
PR
reduct
ion
method
in
D
WT
-OFDM
systems
.
The
re-
sults
sho
w
that
the
clipping
nonline
ar
ities
giv
e
a
noticeab
le
P
APR
reduction.
Ho
w
e
v
er
,
as
D
WT
-OFDM
itself
has
lo
w
er
P
APR
compared
to
the
con
v
entional
discrete
F
our
ier
tr
ansf
or
m
(DFT)-based
OFDM,
the
clipping
nonlinear
ity
subsystem
ma
y
not
be
essential
as
it
deg
r
ades
the
system
perf
or
mance
.
K
e
yw
or
ds:
OFDM,
P
APR,
D
WT
-OFDM
Cop
yright
c
2014
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
v
ed.
1.
Intr
oduction
Or
thogonal
frequency
division
m
ultiple
xing
(OFDM)
is
a
popular
modulation
technique
f
or
broadband
ser
vices
in
wireless
comm
unicati
ons
,
such
as
D
VB-T
[1]
and
wireline
comm
uni-
cations
,
such
as
optical
comm
unications
[2]
and
po
w
er
line
c
o
mm
unications
[3].
OFDM
divides
the
total
bandwidth
into
some
par
allel
narro
wband
subcarr
iers
so
that
the
symbol
dur
ation,
T
s
,
is
smaller
than
the
m
ultipath
dela
y
.
As
a
result,
it
o
v
ercomes
intersymbol
interf
erence
(ISI)
prob
lem
in
m
ultipath
f
ading
en
vironment.
Ho
w
e
v
er
,
OFDM
also
has
tw
o
main
dr
a
wbac
ks
,
i.e
.
the
sensitivity
to
frequency
offset
and
large
peak-to-a
v
er
age
po
w
er
r
atio
(P
APR).
F
requency
offset
deals
with
loss
of
the
or
thogonality
betw
een
subcarr
iers
[4].
Meanwhile
,
large
P
APR
causes
inefficiency
in
po
w
er
amplifier
.
In
OFDM-
based
comm
unication
systems
,
P
APR
reduction
is
needed
to
perf
or
m
po
w
er
sa
vings
[5].
Some
P
APR
reduction
methods
ha
v
e
been
proposed.
Gener
ally
,
the
reduction
methods
can
be
divided
into
three
categor
ies:
distor
tion
method,
such
as
clipping
and
filter
ing;
distor-
tionless
or
probabilistic
method,
such
as
selectiv
e
mapping;
and
coding
method
such
as
Gola
y
complementar
y
sequences
[6].
An
o
v
er
vie
w
of
some
P
APR
reduction
methods
can
be
f
ound
in
[7].
All
the
P
APR
reduction
methods
descr
ibed
in
[7]
deal
wit
h
discrete
F
our
ier
tr
ansf
or
m
(DFT)-based
OFDM.
Another
v
ar
iant
of
OFDM,
called
discrete
w
a
v
elet
tr
ansf
or
m
(D
WT)-based
OFDM,
w
as
studied
in
[8–10].
In
par
ticular
,
D
WT
-OFDM
w
as
b
asically
intended
to
deal
with
the
narro
wband
interf
erence
as
w
ell
as
intercarr
ier
intersymbol
(ICI)
[8].
In
contr
ast
to
DFT
-OFDM,
no
cyclic
prefix
(CP)
is
needed
in
D
WT
-OFDM
so
that
impro
ving
t
he
spectr
al
efficiency
[9,
10].
Moreo
v
er
,
the
use
of
D
WT
in
place
of
DFT
can
also
reduce
P
APR
[8].
In
[8],
D
WT
-OFDM
systems
with
three
w
a
v
elet
functions
w
ere
compared
in
ter
ms
of
P
APR
distr
ib
ution.
The
w
a
v
elet
functions
used
w
ere
daubechies
1
(Haar),
symlets
,
and
coiflets
.
The
sim
ulation
results
sho
w
ed
that
the
Haar
w
a
v
elet
yielded
the
minim
um
P
APR.
Ho
w
e
v
er
,
the
ef-
f
ect
of
decomposition
le
v
el
f
or
each
w
a
v
elet
f
amily
regarding
the
P
APR
distr
ib
ution
has
not
been
discussed
i
n
[8].
In
this
paper
,
w
e
will
sim
ulate
the
distr
ib
ution
of
the
P
APR
f
or
e
v
er
y
decompo-
sition
le
v
el
to
obtain
the
best
decomposition
le
v
el
f
or
each
w
a
v
elet
f
amily
,
i.e
.
the
decomposition
Receiv
ed
No
v
ember
16,
2013;
Re
vised
December
19,
2013;
Accepted
J
an
uar
y
9,
2014
Evaluation Warning : The document was created with Spire.PDF for Python.
3956
ISSN:
2302-4046
QAM
/P
S
K
M
od
ul
atio
n
IDWT
Ze
r
o
pa
dd
in
g
IDFT
CP
A
dd
itio
n
Cl
ip
pi
ng
DF
T-
OFD
M
system
DWT-
OFD
M
system
op
t
i
on
al
b
lo
ck
Figure
1.
DFT
-
and
D
WT
-OFDM
System
le
v
el
that
yields
the
lo
w
est
P
APR.
In
addition,
clipping
nonlinear
ity
subsystem
ma
y
be
added
in
the
D
WT
-OFDM
systems
to
obtain
more
P
APR
reduction.
W
e
will
sho
w
the
sim
ulation
results
f
or
P
APR
distr
ib
ution
of
a
D
WT
-OFDM
system
using
tw
o
clipping
nonlinear
ity
functions
,
i.e
.
con
v
en-
tional
clipping
and
deep
clipping
and
compare
them
with
D
WT
-OFDM
and
DFT
-OFDM
systems
.
The
rest
of
this
paper
is
organiz
ed
as
f
ollo
ws
.
Section
II
discusses
the
DFT
-based
and
D
WT
-based
OFDM
system
models
.
Section
III
compares
and
analyz
es
the
sim
ulation
results
.
The
conclusions
are
giv
en
in
Section
IV
.
2.
System
Model
2.1.
DFT
-
and
D
WT
-OFDM
The
k
-th
unmodulated
par
allel
subcarr
ier
signal
in
OFDM
systems
is
giv
en
b
y
[12]
~
g
k
(
t
)
=
(
e
j
2
k
f
t
;
if
8
t
2
[0
;
T
s
]
;
0
;
if
8
t
62
[0
;
T
s
]
:
(1)
T
o
o
v
ercome
interb
loc
k
intersymbol
(IBI),
a
guard
inter
v
al
in
f
or
m
of
CP
is
appended
in
the
front
of
each
OFDM
b
loc
k
so
that
the
subcarr
ier
signal
becomes
g
k
(
t
)
=
(
e
j
2
k
f
t
;
if
8
t
2
[
T
g
;
T
s
]
;
0
;
if
8
t
62
[
T
g
;
T
s
]
:
(2)
where
T
g
is
the
CP
length.
Theref
ore
,
the
analog
OFDM
signal
can
be
e
xpressed
as
x
(
t
)
=
1
p
N
N
1
X
k
=0
X
k
e
j
2
k
t
T
s
(3)
where
N
is
the
n
umber
of
subcarr
iers
,
X
k
is
the
QAM/PSK
modulated
signal,
and
j
=
p
1
.
F
rom
(3)
it
is
ob
vious
th
at
w
e
can
implement
the
in
v
erse
discrete
F
our
ier
tr
ansf
or
m
(IDFT),
so
named
DFT
-OFDM,
to
the
mod
ulated
input
signal
X
k
to
obtain
the
OFDM
signal.
The
discrete
DFT
-OFDM
can
be
obtained
b
y
sampling
the
analog
OFDM
signal
at
time
x
[
n
]
=
x
(
nT
s
=
N
)
.
Another
w
a
y
to
f
or
m
OFDM
signal
is
to
replace
t
he
IDFT
b
y
in
v
erse
discrete
w
a
v
elet
tr
ansf
or
m
(ID
WT)
[8–11].
DFT
-
and
D
WT
-OFDM
systems
are
sho
wn
in
Fig.
1.
CP
is
not
required
in
D
WT
-OFDM
system
as
mentioned
bef
ore
.
A
clipping
subsystem
can
be
optionally
added.
W
e
will
discuss
the
discrete
w
a
v
elet
tr
ansf
or
m
in
the
ne
xt
subsection.
TELK
OMNIKA
V
ol.
12,
No
.
5,
Ma
y
2014
:
3955
3961
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
3957
ap
pr
o
xim
a
t
i
on
co
eff
i
cie
nts
de
t
a
il
s
co
eff
i
cie
nts
Figure
2.
D
WT
decomposition
ap
pr
o
xim
a
t
i
on
co
eff
i
cie
nts
de
t
a
il
s
co
eff
i
cie
nts
Figure
3.
D
WT
reconstr
uction
2.2.
Discrete
W
a
velet
T
ransf
orm
As
the
discrete
tr
ansf
or
ms
are
basically
the
sampled
v
ersion
of
the
contin
uous
tr
ansf
or
ms
,
w
e
will
first
present
the
continous
w
a
v
elet
tr
ansf
or
m
(CWT).
CWT
can
be
e
xpressed
as
[13]
W
(
a;
b
)
=
Z
1
1
f
(
t
)
1
p
j
a
j
(
t
b
a
)
dt
(4)
where
(
t
)
is
mother
w
a
v
elet,
f
(
t
)
is
the
input
signal,
and
*
denotes
comple
x
conjugate
oper
ator
.
The
in
v
erse
of
the
w
a
v
elet
tr
ansf
or
m
can
be
e
xpressed
as
f
(
t
)
=
1
C
Z
1
1
Z
1
1
1
j
a
j
2
W
(
a;
b
)
1
p
j
a
j
(
t
b
a
)
dadb
(5)
where
C
=
Z
1
1
j
j
2
j
!
j
d!
and
(
!
)
=
Z
1
1
(
t
)
e
j
!
t
dt
The
mother
w
a
v
elet
m
ust
satisfy
the
three
proper
ties:
1.
The
total
area
under
the
(
t
)
is
z
ero
.
This
implies
that
the
function
m
ust
oscillate
abo
v
e
and
belo
w
the
x
-axis
.
Z
1
1
(
t
)
dt
=
0
(6a)
2.
The
total
area
of
j
(
t
)
j
2
is
finite
that
implies
the
energy
of
the
w
a
v
elet
is
finite
.
Z
1
1
j
(
t
)
j
2
dt
<
1
(6b)
3.
The
admissibility
condition
which
means
C
is
required
to
be
positiv
e
and
finite
.
As
sho
wn
in
(4),
the
CWT
in
v
olv
es
time
shif
ting
and
scaling
f
actor
.
In
D
WT
,
those
oper-
ations
are
implemented
b
y
using
the
lo
wpass
and
highpass
filters
which
are
denoted
b
y
g
[
n
]
and
h
[
n
]
,
respectiv
ely
.
The
decomposition
and
reconstr
uction
(in
v
erse)
filters
of
D
WT
are
sho
wn
in
Fig.
2
and
Fig.
3,
respectiv
ely
.
In
Fig.
2,
input
signal
samples
are
con
v
olv
ed
with
the
lo
wpass
and
highpass
decompo-
sition
filter
coefficients
and
then
the
y
are
do
wnsampled
b
y
a
f
actor
of
tw
o
.
As
a
result,
w
e
ha
v
e
appro
ximation
and
deta
ils
coefficients
.
T
o
do
the
reconstr
uction
process
,
as
sho
wn
in
Fig.
3,
an
upsampling
process
b
y
f
actor
of
tw
o
is
applied
and
then
f
ollo
w
ed
b
y
a
con
v
o
lution
process
with
highpass
and
lo
wpass
reconstr
uction
filters
coefficients
.
A
Study
on
P
eak-to-A
v
er
age
P
o
w
er
Ratio
in
D
WT
-OFDM
Systems
(Filber
t
H.
J
uw
ono)
Evaluation Warning : The document was created with Spire.PDF for Python.
3958
ISSN:
2302-4046
T
ab
le
1.
W
a
v
elet
f
amilies
used
in
this
paper
W
a
velet
F
amilies
W
a
velet
Function
with
Or
der
s
Daubechies
(Db)
Db1,
Db3,
Db5,
Db7,
Db9,
Db11
Symlets
(sym)
sym1,
sym3,
sym5,
sym7,
sym9
Coiflets
(coif)
coif1,
coif2,
coif3,
coif4,
coif5
BiorSplines
(bior)
and
Re
v
erseBior
(rbio)
bior5.5,
bior6.8,
rbio3.7,
rbio3.9,
rbio4.4,
rbio5.5,
rbio6.8
2.3.
W
a
velet
F
amilies
As
discussed
in
the
pre
vious
subsection,
mother
w
a
v
elet
can
be
an
y
function
as
long
as
it
satisfies
the
three
proper
ties
.
In
gener
al,
there
are
t
w
o
categor
ies
of
w
a
v
elet
f
amilies:
or
thogonal
and
bior
thogonal
[13–15].
The
or
thogona
lity
of
the
w
a
v
elet
f
amily
deals
with
the
filter
coefficients
.
The
or
thogonal
w
a
v
elet
f
amily
includes
Daubechies
(Db),
Symlet
(sym),
and
Coiflet
(coif)
while
the
bior
thogonal
w
a
v
elets
are
BiorSplines
(bior)
and
Re
v
erseBior
(rbio).
Or
thogonal
w
a
v
elet
is
char
acter
iz
ed
b
y
a
par
ameter
N
which
is
the
filter
order
while
bior
thogonal
w
a
v
elet
ma
y
ha
v
e
diff
erent
order
f
or
the
decomposition
and
reconstr
uction
filters
,
i.
e
.
N
d
and
N
r
,
respectiv
ely
[16].
The
w
a
v
elet
f
amilies
used
in
this
paper
are
summar
iz
ed
in
T
ab
le
.1.
2.4.
Clipping
Nonlinearity
In
this
paper
w
e
use
tw
o
clipping
nonlinear
ity
functions
,
which
are
con
v
entional
clipping
and
deep
clipping
[17].
The
con
v
entional
clipping
f
or
m
ula
is
giv
en
b
y
y
[
n
]
=
(
x
[
n
]
;
if
j
x
[
n
]
j
T
;
T
e
j
'
[
n
]
;
if
j
x
[
n
]
j
>
T
:
(7)
where
T
is
clipping
threshold
and
'
[
n
]
=
arg
x
[
n
]
.
The
f
or
m
ula
f
or
deep
clipping
is
y
[
n
]
=
8
>
<
>
:
x
[
n
]
;
if
j
x
[
n
]
j
T
;
T
p
(
j
x
[
n
]
j
T
)
e
j
'
[
n
]
;
if
T
<
j
x
[
n
]
j
T
;
0
;
if
j
x
[
n
]
j
>
T
:
(8)
where
p
is
depth
f
actor
and
=
(
p
+
1)
=p
.
The
clipping
threshold
is
char
act
er
iz
ed
b
y
a
par
ameter
called
clipping
r
atio
which
is
defined
as
C
R
=
T
(9)
where
is
the
r
ms
of
the
OFDM
signal.
3.
Results
and
Anal
ysis
In
this
sim
ulation,
w
e
use
16-QAM
modulation,
64
subcarr
iers
,
and
f
our
times
o
v
ersam-
pling.
T
o
analyz
e
P
APR
distr
ib
ution,
a
statistical
par
ameter
called
complementar
y
cum
ulativ
e
distr
ib
ution
function
(CCDF)
is
usually
used.
CCDF
giv
es
a
probability
that
P
APR
e
xceeds
cer
tain
v
alue
.
The
CCDF
results
f
or
D
WT
-OFDM
systems
using
or
thogonal
w
a
v
elet
f
amilies
,
i.e
.
Db
,
sym,
coif
,
compared
with
DFT
-OFDM
are
sho
wn
in
Fig.
4
-
Fig.
6,
respectiv
ely
.
W
e
can
obser
v
e
from
Fig.
4
that
D
WT
-OFDM
using
Db1
has
the
lo
w
est
P
APR.
The
diff
erence
is
about
7.5
dB
at
probability
of
10
3
compared
with
the
con
v
entional
DFT
-OFDM.
Note
that
the
le
v
els
of
P
APR
f
or
odd-order
Daubechies
w
a
v
elet
f
amily
,
from
Db3
to
Db11,
do
not
linear
ly
depend
on
the
filter
order
.
In
Fig.
5,
at
probability
of
10
3
the
lo
w
est
P
APR
distr
ib
ution
in
D
WT
-
OFDM
is
achie
v
ed
b
y
using
sym1,
which
is
about
7.5
dB
lo
w
er
than
DFT
-OFDM,
and
it
is
the
same
as
D
WT
-OFDM
using
Db1
system.
Meanwhile
,
the
P
APR
distr
ib
utions
of
D
WT
-OFDM
using
sym3,
sym5,
sym7,
and
sym9
are
near
ly
the
same
,
which
is
about
4.5
dB
lo
w
er
than
DFT
-OFDM
TELK
OMNIKA
V
ol.
12,
No
.
5,
Ma
y
2014
:
3955
3961
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
3959
0
2
4
6
8
10
12
10
−4
10
−3
10
−2
10
−1
10
0
PAPR [dB]
Pr{PAPR >= x}
db1
db3
db5
db7
db9
db11
DFT−OFDM
Figure
4.
CCDFs
of
P
APR
f
or
Daubechies
D
WT
-
and
DFT
-OFDM
0
2
4
6
8
10
12
10
−4
10
−3
10
−2
10
−1
10
0
PAPR [dB]
Pr{PAPR >= x}
sym1
sym3
sym5
sym7
sym9
DFT−OFDM
Figure
5.
CCDFs
of
P
APR
f
or
symlet
D
WT
-
and
DFT
-OFDM
0
2
4
6
8
10
12
10
−4
10
−3
10
−2
10
−1
10
0
PAPR [dB]
Pr{PAPR >= x}
coif1
coif2
coif3
coif4
coif5
DFT−OFDM
Figure
6.
CCDFs
of
P
APR
f
or
coiflet
D
WT
-
and
DFT
-OFDM
0
2
4
6
8
10
12
10
−4
10
−3
10
−2
10
−1
10
0
PAPR [dB]
Pr{PAPR >= x}
rbio3.7
rbio3.9
rbio4.4
rbio5.5
rbio6.8
bior5.5
bior6.8
Figure
7.
CCDFs
of
P
APR
f
or
bior
thogonal
D
WT
-
and
DFT
-OFDM
A
Study
on
P
eak-to-A
v
er
age
P
o
w
er
Ratio
in
D
WT
-OFDM
Systems
(Filber
t
H.
J
uw
ono)
Evaluation Warning : The document was created with Spire.PDF for Python.
3960
ISSN:
2302-4046
0
2
4
6
8
10
12
10
−4
10
−3
10
−2
10
−1
10
0
PAPR [dB]
Pr{PAPR >= x}
DWT−OFDM Conventional Clipping
DWT−OFDM Deep Clipping
DWT−OFDM
DFT−OFDM
Figure
8.
CCDFs
of
P
APR
f
or
D
WT
-
and
DFT
-OFDM
with
clipping
nonlinear
ity
at
probabiity
of
10
3
.
As
sho
wn
in
Fig.
6,
the
P
APR
distr
ib
utions
of
D
WT
-OFDM
using
coif1-coif5
ha
v
e
only
little
diff
erence
.
Ho
w
e
v
er
,
w
e
can
obser
v
e
that
D
WT
-OFDM
using
coif3
has
the
lo
w
est
P
APR
compared
with
DFT
-OFDM
which
is
about
5
dB
diff
erence
at
probability
of
10
3
.
Fig.
7
sho
ws
the
CCDF
f
or
bior
thogonal
w
a
v
elet
f
amilies
compared
with
DFT
-OFDM.
It
is
ob
vious
that
all
the
results
using
bior
thogonal
w
a
v
elet
functions
yield
almost
the
same
P
APR
distr
ib
ution.
At
probab
ility
of
10
3
,
rbio3.7
yields
about
5
dB
reduction
compared
with
DFT
-OFDM.
In
Fig.
8,
w
e
analyz
e
the
eff
ect
of
clipping
nonlinear
ity
functions
in
D
WT
-OFDM
and
also
compared
it
with
DFT
-OFDM.
W
e
use
con
v
entional
clipping
and
deep
clipping
with
p
=
0
:
6
.
The
clipping
r
atio
,
C
R
,
is
set
to
be
1.4
f
or
both
clipping
functions
.
W
e
use
rbio3.7
w
a
v
elet
function
f
or
this
sim
ulation.
W
e
notice
that
the
clipping
nonlinear
ity
giv
es
additional
P
APR
reduction,
about
2
dB
,
compared
with
the
D
WT
-OFDM
system.
Additionally
,
con
v
entional
clipping
and
deep
clipping
ha
v
e
near
ly
similar
results
.
As
the
D
WT
-OFDM
systems
ha
v
e
smaller
P
APR
t
han
DFT
-OFDM
systems
,
it
is
not
necessar
y
to
perf
or
m
an
additional
P
APR
reduction
technique
b
y
using
clipping
nonlinear
ity
because
it
deg
r
ades
the
syst
em
perf
or
mance
.
Theref
ore
,
D
WT
-OFDM
systems
off
er
an
adv
antage
compared
with
the
DFT
-OFDM
systems
regarding
the
P
APR
distr
ib
ution.
4.
Conc
lusions
W
e
ha
v
e
sim
ulated
the
D
WT
-OFDM
using
or
thogonal
and
bior
thogonal
w
a
v
elet
f
amilies
.
The
sim
ulation
results
sho
w
that
D
WT
-OFDM
reduces
the
P
APR
compared
with
con
v
entional
DFT
-
OFDM.
F
or
each
w
a
v
elet
f
amily
,
the
e
ff
e
ct
of
decomposition
le
v
el
(or
filter
order)
is
also
compared.
F
or
or
thogonal
w
a
v
elet
f
amily
,
the
Db1,
sym1,
and
coif3
yield
the
largest
P
APR
reduction
while
f
or
bior
thogonal
w
a
v
elet,
the
rbio3.7
yields
the
largest
P
APR
reduction.
Finally
,
as
D
WT
-OFDM
reduces
the
P
APR
significantly
,
the
clipping
nonlinear
ity
subsystem
is
not
desir
ab
le
since
the
system
perf
or
mance
is
not
e
xpected
to
deg
r
ade
at
the
receiv
er
.
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erences
[1]
E.
Costa
and
S
.
Pupolin,
”
M
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System
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the
Presence
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f
a
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r
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2002.
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OMNIKA
V
ol.
12,
No
.
5,
Ma
y
2014
:
3955
3961
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
2302-4046
3961
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ma
w
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2005.
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”
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eak-to-A
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er
age
P
o
w
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WT
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t
H.
J
uw
ono)
Evaluation Warning : The document was created with Spire.PDF for Python.