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o
o
ley
-
T
u
k
e
y
al
g
o
r
ith
m
.
T
h
e
h
ar
d
w
ar
e
i
m
p
le
m
e
n
tatio
n
o
f
m
i
x
ed
r
ad
ix
F
FT
s
w
ith
co
r
es
o
f
r
ad
ix
5
an
d
r
ad
ix
3
as
w
ell
as
t
h
e
s
tan
d
ar
d
r
ad
ix
2
co
r
e
h
as
b
ee
n
p
r
esen
ted
in
[
3
]
.
T
h
e
m
i
x
ed
r
ad
ix
FF
T
is
m
o
r
e
co
s
tl
y
th
a
n
th
e
r
ad
ix
2
i
m
p
le
m
en
ta
tio
n
.
A
m
i
x
ed
r
ad
ix
FF
T
o
f
1
2
0
0
p
o
in
ts
n
ee
d
3
6
r
ea
l
m
u
l
tip
lier
s
in
t
h
e
i
m
p
le
m
en
ta
tio
n
o
f
p
ip
elin
ed
FF
T
w
h
er
ea
s
a
2
0
4
8
r
ad
ix
2
FF
T
n
ee
d
s
3
0
r
ea
l
m
u
ltip
lier
s
.
A
r
ad
ix
-
3
FF
T
h
as b
e
en
d
escr
ib
ed
in
[
4
]
w
h
ic
h
th
e
ele
m
en
t
th
r
ee
-
p
o
in
t
DFT
’
s
n
ee
d
s
n
o
m
u
ltip
licat
i
o
n
s
.
T
h
is
r
esu
lt
s
in
a
r
ed
u
ctio
n
in
t
h
e
n
u
m
b
er
o
f
m
u
ltip
licatio
n
s
b
u
t
a
co
n
c
u
r
r
en
t
i
n
cr
ea
s
e
in
t
h
e
n
u
m
b
er
o
f
ad
d
itio
n
s
.
T
h
e
al
g
o
r
ith
m
w
i
ll
s
h
o
w
a
n
ad
v
an
tag
e
o
f
p
r
o
ce
s
s
o
r
s
w
h
ic
h
r
eq
u
ir
e
m
o
r
e
tim
e
f
o
r
m
u
ltip
licatio
n
th
a
n
ad
d
itio
n
.
A
n
o
v
el
F
FT
alg
o
r
ith
m
h
as
b
e
en
d
ev
elo
p
ed
in
[
5
]
to
g
eth
er
w
it
h
t
h
e
d
esig
n
o
f
p
ip
eli
n
ed
ar
ch
itect
u
r
e.
T
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
h
a
s
b
ee
n
u
s
ed
to
r
ed
u
ce
t
h
e
n
u
m
b
e
r
o
f
co
m
p
le
x
m
u
lt
ip
lier
s
i
n
ad
d
itio
n
to
th
e
s
ize
o
f
t
w
id
d
le
f
ac
to
r
R
OM
s
.
I
t
is
p
r
o
v
ed
to
b
e
ap
p
r
o
p
r
iate
f
o
r
lar
g
e
s
ize
o
f
FF
T
VL
SI
i
m
p
le
m
en
tatio
n
.
T
h
ese
FF
T
ar
ch
itect
u
r
es a
r
e
d
esi
g
n
ed
f
o
r
OFDM
ap
p
licatio
n
s
i
n
[
6
]
-
[
7
]
.
A
n
o
v
el
ar
ch
itectu
r
e
f
o
r
e
f
f
ic
i
en
t
m
et
h
o
d
o
f
Fa
s
t
Fo
u
r
ier
T
r
an
s
f
o
r
m
(
FF
T
)
p
r
o
ce
s
s
o
r
[
8
]
to
g
ath
er
th
e
n
e
ce
s
s
itie
s
o
f
h
i
g
h
s
p
ee
d
w
ir
el
ess
co
m
m
u
n
icatio
n
s
y
s
te
m
s
ta
n
d
ar
d
s
.
T
h
is
p
ap
er
d
ev
elo
p
s
a
n
o
p
ti
m
al
co
n
s
ta
n
t
m
u
lt
ip
licatio
n
ar
ith
m
etic
d
esig
n
to
m
u
ltip
l
y
a
f
i
x
ed
p
o
in
t in
p
u
t b
y
m
ea
n
s
o
f
o
n
e
o
f
th
e
n
u
m
er
o
u
s
cu
r
r
en
t t
w
id
d
le
f
ac
to
r
co
n
s
tan
ts
.
3.
RADIX
-
3
F
F
T
AL
G
O
R
I
T
H
M
R
ad
ix
-
3
FF
T
alg
o
r
ith
m
is
u
s
ed
to
co
m
p
u
te
Di
s
cr
ete
Fo
u
r
ier
T
r
an
s
f
o
r
m
(
DFT
)
.
I
t
tak
es
les
s
m
u
ltip
licatio
n
t
h
an
t
h
e
n
o
r
m
a
l
o
n
e.
R
ad
i
x
-
3
F
FT
alg
o
r
ith
m
is
m
ain
l
y
b
ased
o
n
d
iv
id
e
a
n
d
co
n
q
u
er
m
et
h
o
d
.
I
t
d
ec
o
m
p
o
s
es
an
N
-
p
o
in
t
DF
T
in
to
s
eq
u
en
tiall
y
s
m
a
ller
DF
T
s
.
A
s
s
o
o
n
as
th
e
n
u
m
b
er
o
f
d
ata
p
o
in
ts
is
p
o
w
er
o
f
3
(
i.e
.
,
3
n
)
.
T
h
e
r
ad
ix
-
3
al
g
o
r
ith
m
f
o
r
r
ea
lizatio
n
o
f
DFT
o
f
le
n
g
th
N=
3
n
(
n
=1
,
2
,
3
,
…)
.
T
h
e
DFT
o
f
le
n
g
th
N
ca
n
b
e
r
ea
lized
f
r
o
m
th
r
ee
DFT
s
eq
u
en
ce
s
,
ea
c
h
o
f
le
n
g
th
N/3
.
I
f
t
h
e
i
n
p
u
t
s
i
g
n
a
l
h
as
len
g
t
h
N,
d
ir
ec
t
ca
lcu
latio
n
o
f
D
FT
n
ee
d
s
O(
N2
)
co
m
p
le
x
m
u
lt
ip
licatio
n
s
.
R
ad
ix
-
3
alg
o
r
it
h
m
w
h
ich
i
s
u
tili
ze
d
to
r
ed
u
ce
th
e
m
u
ltip
licatio
n
s
.
T
h
e
p
r
o
ce
s
s
i
n
g
ti
m
e
a
n
d
h
ar
d
w
ar
e
co
m
p
lex
it
y
f
o
r
i
m
p
le
m
e
n
ti
n
g
r
ad
ix
-
3
DFT
alg
o
r
ith
m
ca
n
b
e
r
ed
u
ce
d
.
T
h
e
DFT
o
f
N
p
o
in
ts
is
g
iv
e
n
b
y
e
N
kn
j
N
n
n
x
k
Z
2
1
0
,
f
o
r
k
=0
,
1
,
2
,
…,
N
-
1
(
1
)
W
h
er
e
j
=√
-
1
Z
(
k
)
=
T
r
an
s
f
o
r
m
ed
Data
Su
b
s
ti
tu
te
W
e
N
kn
j
kn
N
2
in
(
1
)
W
kn
N
N
n
n
x
k
Z
1
0
(
2
)
W
kn
N
is
k
n
o
w
n
as t
w
id
d
le
f
ac
to
r
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
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J
E
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&
C
o
m
p
Sci
I
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N:
2502
-
4752
A
N
o
ve
l A
r
ch
itectu
r
e
o
f R
a
d
ix
-
3
S
in
g
lep
a
th
Dela
y
F
ee
d
b
a
ck
(
R
3
S
DF
)
F
F
T…
(
K
.
P
eriya
r
s
elva
m
)
39
T
h
e
o
u
tp
u
t
co
m
p
o
n
e
n
ts
ar
e
Z
(
0
)
,
Z
(
1
)
,
Z
(
2
)
,
….
.
,
Z
(
N
-
1
)
ar
e
ar
r
an
g
ed
in
t
h
r
ee
g
r
o
u
p
s
n
a
m
el
y
Z
(
3
r
)
,
Z
(
3
r
+1
)
an
d
Z
(
N
-
3r
-
1
)
,
w
h
er
e
r
=0
,
1
,
2
,
…,
N/3
-
1.
T
h
e
f
o
llo
w
i
n
g
e
x
p
r
ess
io
n
s
ca
n
b
e
d
er
iv
ed
f
r
o
m
eq
u
at
io
n
s
(
2
)
,
1
3
0
3
2
3
3
N
n
W
kn
N
N
n
N
n
x
n
x
r
Z
(
3
)
W
h
er
e
k
=3
r
an
d
r
=0
,
1
,
2
,
…,
N/3
-
1.
1
3
0
3
2
3
/
2
3
3
/
1
3
N
n
W
kn
N
N
n
W
N
k
N
N
n
x
W
N
k
N
n
x
r
Z
(
4
)
W
h
er
e
k
=3
r
+1
an
d
r
=0
,
1
,
2
,
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N/3
-
1.
1
3
0
3
/
2
3
/
3
2
3
1
3
N
n
kn
N
N
k
N
N
k
N
W
W
W
N
n
N
n
x
n
x
r
N
Z
(
5
)
W
h
er
e
k
=
(
N
-
3r
-
1
)
an
d
r
=0
,
1
,
2
…
N/3
-
1.
Fig
u
r
e
1
.
Flo
w
Gr
ap
h
o
f
R
ad
i
x
-
3
F
FT
4.
P
RO
P
O
SE
D
P
I
P
E
L
I
NE
D
S
T
RUC
T
UR
E
O
F
RADIX
-
3
SDF
F
F
T
USI
NG
M
CSL
A
I
n
th
is
p
ap
er
,
a
n
e
w
al
g
o
r
ith
m
o
f
p
ip
elin
ed
s
tr
u
ctu
r
e
b
ased
R
ad
ix
-
3
SDF
FF
T
h
as
b
ee
n
d
esig
n
ed
f
o
r
i
m
p
r
o
v
i
n
g
t
h
e
s
p
ee
d
.
R
ad
ix
-
3
FF
T
,
w
h
ich
i
s
u
s
ed
to
r
ed
u
ce
th
e
n
u
m
b
er
o
f
m
u
ltip
licatio
n
.
Sin
g
le
p
at
h
Dela
y
Feed
b
ac
k
FF
T
is
a
p
i
p
elin
ed
b
ased
f
r
eq
u
en
c
y
tr
a
n
s
f
o
r
m
atio
n
tech
n
iq
u
e.
I
n
SD
F
FF
T
,
th
e
in
p
u
t
s
ar
e
g
iv
e
n
in
s
er
ial
m
an
n
er
.
T
h
e
SD
F
F
FT
p
r
o
v
id
es
h
ig
h
s
p
ee
d
o
p
er
atio
n
.
T
h
is
FF
T
s
tr
u
ct
u
r
e
co
n
s
u
m
es
m
o
r
e
d
ela
y
a
n
d
p
o
w
er
co
n
s
u
m
p
tio
n
[
9
]
d
u
e
to
u
tili
zin
g
o
r
s
to
r
in
g
b
u
l
k
o
f
u
n
w
a
n
ted
in
ter
m
ed
iate
p
r
o
ce
s
s
i
n
g
s
ig
n
als.
SD
F
FF
T
s
tr
u
ctu
r
es
h
a
v
e
th
e
m
o
s
t
p
r
o
f
icien
t
m
e
m
o
r
y
u
til
izatio
n
f
o
r
p
ip
elin
ed
FF
T
p
r
o
ce
s
s
o
r
s
.
F
ig
u
r
e
2
s
h
o
w
s
th
a
t
th
e
ar
c
h
i
tectu
r
e
o
f
R
ad
ix
-
3
p
ip
elin
ed
SDF
F
FT
.
T
h
is
ar
ch
itect
u
r
e
c
o
n
s
is
ts
o
f
P
r
o
ce
s
s
in
g
E
le
m
e
n
t
(
P
E
)
,
De
la
y
an
d
T
w
id
d
le
Facto
r
v
alu
es.
A
d
d
itio
n
an
d
s
u
b
tr
ac
tio
n
o
p
er
atio
n
h
as
b
ee
n
p
er
f
o
r
m
ed
i
n
t
h
e
p
r
o
ce
s
s
in
g
e
le
m
e
n
t.
I
n
itia
ll
y
,
t
h
e
i
n
p
u
t
d
at
a
o
f
r
ea
l
an
d
i
m
a
g
i
n
ar
y
v
al
u
e
s
ar
e
g
iv
e
n
to
t
h
e
f
ir
s
t
s
tag
e.
T
h
en
t
h
e
i
n
p
u
t
v
alu
es a
r
e
d
elay
ed
b
y
1
.
T
h
e
d
elay
ed
v
alu
es a
r
e
g
iv
e
n
to
th
e
p
r
o
ce
s
s
i
n
g
ele
m
e
n
t.
I
n
t
h
e
p
r
o
ce
s
s
i
n
g
ele
m
en
t,
th
e
ad
d
itio
n
an
d
s
u
b
tr
ac
tio
n
o
p
er
atio
n
s
ar
e
d
o
n
e.
Af
ter
t
h
at
t
h
e
v
alu
e
s
ar
e
m
u
ltip
lied
b
y
t
w
id
d
le
f
ac
to
r
v
al
u
es.
Fi
n
all
y
,
t
h
e
f
ir
s
t
s
tag
e
o
f
o
u
tp
u
t
v
al
u
es
is
co
n
tr
o
lled
b
y
u
s
in
g
m
u
ltip
lex
o
r
.
T
h
e
f
ir
s
t
s
ta
g
e
o
u
tp
u
t
i
s
f
ed
b
ac
k
to
t
h
e
i
n
p
u
t
o
f
s
ec
o
n
d
s
tag
e.
Si
m
ilar
l
y
,
th
e
f
i
r
s
t
s
ta
g
e
o
p
er
atio
n
h
as
b
ee
n
d
o
n
e
in
th
e
s
ec
o
n
d
s
tag
e
an
d
th
ir
d
s
ta
g
e.
T
h
e
m
ai
n
d
is
ad
v
an
ta
g
e
o
f
s
in
g
le
p
ath
d
ela
y
f
ee
d
b
ac
k
FF
T
is
lar
g
e
p
o
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a
l
Co
n
f
e
re
n
c
e
o
n
S
ig
n
a
l
P
r
o
c
e
ss
in
g
.
2
0
0
8
;
4
1
4
-
4
1
7
.
[6
]
S
a
lh
A
,
A
u
d
a
h
L
,
S
h
a
h
NS,
H
a
m
z
a
h
S
A
.
M
a
x
i
m
izin
g
En
e
rg
y
Eff
icie
n
c
y
f
o
r
Co
n
su
m
p
ti
o
n
Ci
rc
u
it
P
o
w
e
r
in
Do
w
n
li
n
k
M
a
ss
iv
e
M
IM
O
W
ire
les
s
Ne
t
w
o
rk
s
.
In
ter
n
a
ti
o
n
a
l
J
o
u
rn
a
l
o
f
El
e
c
trica
l
a
n
d
Co
m
p
u
t
e
r
En
g
i
n
e
e
rin
g
(
IJ
ECE
).
2
0
1
7
;
7
(
6
).
[7
]
Ra
u
t
S
N,
a
n
d
Ja
ln
e
k
a
r
R
M
.
P
e
rf
o
r
m
a
n
c
e
En
h
a
n
c
e
m
e
n
t
in
S
U
a
n
d
M
U
M
IM
O
-
OFDM
T
e
c
h
n
iq
u
e
f
o
r
W
irele
ss
Co
m
m
u
n
ica
ti
o
n
.
A
Rev
iew.
In
ter
n
a
ti
o
n
a
l
J
o
u
r
n
a
l
o
f
El
e
c
trica
l
a
n
d
Co
mp
u
ter
E
n
g
in
e
e
rin
g
.
2
0
1
7
;
7
(5
)
:
2
4
5
9
.
[8
]
A
l
g
n
a
b
i
Y S
,
A
ld
a
a
m
e
e
F
A
,
T
e
y
m
o
u
rz
a
d
e
h
R,
Oth
m
a
n
M
,
a
n
d
Isla
m
M
S
.
No
v
e
l
a
rc
h
it
e
c
tu
re
o
f
p
i
p
e
li
n
e
Ra
d
ix
-
22
S
DF
FF
T
b
a
se
d
o
n
d
i
g
it
-
slici
n
g
tec
h
n
iq
u
e
.
IEE
E
In
tern
a
ti
o
n
a
l
Co
n
f
e
re
n
c
e
o
n
S
e
m
ico
n
d
u
c
to
r
El
e
c
tro
n
ics
(ICS
E)
.
2
0
1
2
;
4
7
0
-
4
7
4
.
[9
]
S
u
m
a
n
S
,
S
h
a
rm
a
K
G
,
G
h
o
sh
P
K.
2
5
0
M
H
z
M
u
lt
i
p
h
a
se
De
lay
L
o
c
k
e
d
L
o
o
p
f
o
r
L
o
w
P
o
w
e
r
A
p
p
li
c
a
ti
o
n
s
.
In
ter
n
a
t
io
n
a
l
J
o
u
rn
a
l
o
f
E
lec
trica
l
a
n
d
C
o
mp
u
ter
En
g
in
e
e
rin
g
(
IJ
ECE
).
2
0
1
7
;
7
(6
).
[1
0
]
Ja
y
a
k
u
m
a
r
D,
a
n
d
L
o
g
a
sh
a
n
m
u
g
a
m
n
E.
De
sig
n
a
n
d
Im
p
le
m
e
n
tatio
n
o
f
P
i
p
e
li
n
e
d
Ra
d
ix
-
2
S
DF
-
S
DC
F
F
T
Us
in
g
M
o
d
if
ied
Ca
rry
S
e
lec
t
A
d
d
e
r.
IIOAB
.
2
0
1
6
;
7
:1
3
7
-
1
4
3
.
[1
1
]
Na
ik
M
V
K.
De
sig
n
o
f
c
a
rr
y
se
lec
t
a
d
d
e
r
f
o
r
lo
w
-
p
o
we
r
a
n
d
h
i
g
h
s
p
e
e
d
VL
S
I
a
p
p
l
ica
ti
o
n
s
.
I
EE
E
In
tern
a
ti
o
n
a
l
Co
n
f
e
re
n
c
e
o
n
El
e
c
tri
c
a
l,
Co
m
p
u
ter an
d
C
o
m
m
u
n
ica
ti
o
n
T
e
c
h
n
o
l
o
g
ies
(ICECCT
).
2
0
1
5
;
1
-
4.
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