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n
t
h
e
p
ap
er
is
o
r
g
an
ized
a
s
f
o
llo
w
s
.
I
n
s
ec
tio
n
I
I
,
th
e
p
ap
er
d
escr
ib
es
ab
o
u
t
th
e
r
ad
ix
-
2
al
g
o
r
ith
m
.
Sec
tio
n
I
I
I
d
is
cu
s
s
ed
ab
o
u
t
th
e
r
ad
ix
-
4
alg
o
r
ith
m
.
Secti
o
n
I
V
e
x
p
lain
s
ab
o
u
t
t
h
e
r
ad
i
x
-
2
i
alg
o
r
it
h
m
s
(
in
c
lu
d
i
n
g
r
ad
ix
-
2
^2
an
d
r
a
d
ix
-
2
^3
)
.
Sec
tio
n
V
clea
r
l
y
g
i
v
es
t
h
e
id
ea
r
eg
ar
d
in
g
t
h
e
p
r
o
p
o
s
ed
w
o
r
k
.
Sectio
n
VI
r
ep
r
esen
ts
th
e
s
y
n
t
h
esi
s
r
esu
lt
o
f
d
if
f
er
en
t
r
ad
ices a
n
d
s
ec
tio
n
VI
I
in
cl
u
d
es c
o
n
clu
s
io
n
.
2.
RADIX
-
2
AL
G
O
RI
T
H
M
R
ad
ix
-
2
F
FT
alg
o
r
ith
m
s
i
m
p
l
e
r
ad
ix
i
n
u
s
ed
in
FF
T
.
T
h
e
o
r
ig
in
a
l
i
n
p
u
t
v
ec
to
r
,
x
(
n
)
i
s
d
i
v
id
ed
in
to
t
w
o
N/2
le
n
g
t
h
v
ec
to
r
s
i.e
.
ev
e
n
an
d
o
d
d
in
p
u
t te
r
m
s
(
x
e(
n
)
,
x
o
(
n
)
)
[
6
,
7
].
T
h
e
eq
u
atio
n
is
d
ef
in
ed
as
,
X(
K)
=
∑
(
1
)
X
o
d
d
(
n
)
=
X
(
2
n
)
X
ev
en
(
n
)
=
X(
2
n
+1
)
n
=
0
,
1
N/2
-
1
T
h
e
r
ad
ix
-
2
DI
T
FF
T
is
r
ew
r
it
ten
b
y
d
er
iv
i
n
g
th
e
eq
u
a
tio
n
∑
∑
(
2
)
T
h
e
ab
o
v
e
eq
u
atio
n
d
iv
id
es r
a
d
ix
-
2
i
n
ev
e
n
i
n
d
ex
i
n
p
u
t
s
an
d
th
e
o
d
d
in
d
ex
in
p
u
ts
a
n
d
th
e
n
co
m
b
i
n
es
th
e
t
w
o
r
esu
lts
to
p
r
o
d
u
ce
th
e
en
tire
DFT
s
eq
u
en
ce
[
8
,
9
]
.
Fro
m
t
h
e
f
ig
u
r
e,
it
is
o
b
s
er
v
ed
th
at
t
h
e
s
ec
o
n
d
in
p
u
t
g
e
ts
m
u
lt
ip
lied
w
it
h
th
e
t
w
id
d
le
f
ac
to
r
an
d
ad
d
ed
w
i
t
h
th
e
f
ir
s
t
in
p
u
t
to
g
et
th
e
f
ir
s
t
o
u
tp
u
t.
Si
m
i
lar
l
y
,
th
e
s
ec
o
n
d
o
u
tp
u
t
i
s
o
b
tain
ed
b
y
s
u
b
tr
ac
ti
n
g
th
e
m
u
ltip
lied
ter
m
w
it
h
t
h
e
f
ir
s
t
i
n
p
u
t.
Fi
g
u
r
e
1
s
h
o
w
s
s
i
g
n
al
f
lo
w
g
r
ap
h
o
f
1
6
P
o
in
t r
ad
ix
-
2
DI
T
-
FFT
.
Fig
u
r
e
1
.
Sig
n
al
f
lo
w
g
r
ap
h
o
f
1
6
p
o
in
t r
ad
ix
-
2
DI
T
-
FFT
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
R
ec
o
n
f
i
g
u
r
ab
le
&
E
m
b
ed
d
ed
Sy
s
t
I
SS
N:
2089
-
4864
Lo
w
p
o
w
er a
n
d
h
ig
h
p
erfo
r
ma
n
ce
F
F
T w
ith
d
iffer
en
t ra
d
ices
(
Md
.
Za
kir Hu
s
s
a
in
)
101
3.
R
ADIX
-
4
F
F
T
AL
G
O
R
I
T
H
M
T
h
e
r
ad
ix
-
4
FF
T
alg
o
r
ith
m
co
n
s
is
ts
o
f
t
w
o
s
tag
e
s
w
h
en
co
m
p
ar
ed
to
r
ad
ix
-
2
FF
T
.
I
n
r
ad
ix
-
4
alg
o
r
ith
m
th
e
p
o
in
t
s
ize
in
cr
e
ases
a
s
t
h
e
m
u
ltip
le
o
f
f
o
u
r
.
Hen
ce
,
t
h
e
r
ad
i
x
-
4
FF
T
r
eq
u
i
r
es
f
e
w
er
s
ta
g
es
a
n
d
b
u
tter
f
lie
s
th
a
n
th
e
r
ad
ix
-
2
alg
o
r
ith
m
.
T
h
e
m
a
in
id
ea
o
f
r
ad
ix
-
4
DI
T
FF
T
is
to
d
iv
id
e
th
e
o
r
ig
in
al
i
n
p
u
t
s
eq
u
en
ce
in
to
f
o
u
r
s
m
aller
s
u
b
s
eq
u
en
ce
s
[
1
0
].
Fig
u
r
e
2
s
h
o
w
s
s
i
g
n
a
l
f
lo
w
g
r
ap
h
o
f
r
ad
ix
-
4
FF
T
f
o
r
N=
4
.
Fig
u
r
e
3
s
h
o
w
s
s
ig
n
al
f
lo
w
g
r
ap
h
o
f
1
6
p
o
in
t r
ad
ix
-
4
DI
F
-
F
FT
.
X(
K)
=∑_
(
N=
0
)
^(
N
-
1
)
▒
〖
x
(
n
)
W
_
N^n
k
〗
,
w
h
er
e
k
=
0
,
1
,
…
.
N
-
1
Fig
u
r
e
2
.
Sig
n
al
f
lo
w
g
r
ap
h
o
f
r
ad
ix
-
4
FF
T
f
o
r
N=
4
T
h
e
r
ad
ix
-
4
alg
o
r
ith
m
eq
u
atio
n
is
d
er
iv
ed
b
y
r
e
w
r
iti
n
g
(
1
)
∑
∑
∑
∑
Fig
u
r
e
3
.
Sig
n
al
f
lo
w
g
r
ap
h
o
f
1
6
p
o
in
t
r
ad
ix
-
4
DI
F
-
FFT
0
0
0
0
0
1
2
3
0
2
4
6
0
3
6
9
S
T
A
G
E
1
S
T
A
G
E
2
x
[
0
]
x
[
1
]
x
[
2
]
x
[
3
]
x[
4
]
x
[
5
]
x
[
6
]
x
[
7
]
x
[
8
]
x
[
9
]
x
[
1
0
]
x
[
1
1
]
x
[
1
2
]
x[
1
3
]
x
[
1
4
]
x
[
1
5
]
X
[
0
]
X
[
4
]
X
[
8
]
X
[
1
2
]
X
[
1
]
X
[
5
]
X
[
9
]
X
[
1
3
]
X
[
2
]
X
[
6
]
X
[
1
0
]
X
[
1
4
]
X
[
3
]
X
[
7
]
X
[
1
1
]
X
[
1
5
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4864
I
n
t J
R
ec
o
n
f
i
g
u
r
ab
le
&
E
m
b
ed
d
ed
Sy
s
t
Vo
l.
8
,
No
.
2
,
J
u
ly
2
0
1
9
:
9
9
–
106
102
4.
RADIX
-
F
F
T
AL
G
O
R
I
T
H
M
4
.
1
.
Ra
dix
-
2
^2
a
lg
o
rit
h
m
R
adi
x
-
2^2appr
oa
ch
pr
op
os
ed
by
H
e
and
T
or
k
el
son.
B
y
usi
ng
l
i
nea
r
m
appi
ng
t
ec
hni
ques
,
t
he
t
w
o
bu
t
t
e
r
f
l
y
uni
t
s
a
r
e
com
put
ed
t
o
o
ne
b
ut
t
er
f
l
y
uni
t
i
n
r
ad
i
x
-
22
[
11
]
.
F
or
N
=16,
r
a
di
x
-
22
i
s
com
put
ed
i
n
t
w
o
s
t
ag
es
but
w
i
t
h
d
i
f
f
er
ent
t
w
i
dd
l
e
f
ac
t
o
r
swhe
n
com
par
ed
t
o
r
adi
x
-
2
al
g
or
i
t
hm
.
Fig
u
r
e
4
s
h
o
w
s
s
i
g
n
al
f
lo
w
g
r
ap
h
o
f
1
6
P
o
in
t
r
a
d
ix
-
2
^2
DI
F
-
F
FT
.
Fig
u
r
e
5
s
h
o
w
s
s
ig
n
al
f
lo
w
g
r
ap
h
o
f
1
6
P
o
in
t
r
ad
ix
-
2
^3
DI
T
-
FFT
.
x
[
0
]
x
[
1
]
x
[
2
]
x
[
3
]
x
[
4
]
x
[
5
]
x
[
6
]
x
[
7
]
x
[
8
]
x
[
9
]
x
[
1
0
]
x
[
1
1
]
x
[
1
2
]
x
[
1
3
]
x
[
1
4
]
x
[
1
5
]
0
0
0
0
0
0
0
0
0
0
0
0
4
4
4
4
0
0
0
0
0
2
4
6
0
1
2
3
0
3
6
9
0
0
0
4
0
0
0
4
0
0
0
4
0
0
0
4
X
[
0
]
X
[
8
]
X
[
4
]
X
[
1
2
]
X
[
2
]
X
[
1
0
]
X
[
6
]
X
[
1
4
]
X
[
1
]
X
[
9
]
X
[
5
]
X
[
1
3
]
X
[
3
]
X
[
1
1
]
X
[
7
]
S
T
A
G
E
1
S
T
A
G
E
2
S
T
A
G
E
3
S
T
A
G
E
4
X
[
1
5
]
Fig
u
r
e
4
.
Sig
n
al
f
lo
w
g
r
ap
h
o
f
1
6
p
o
in
t r
ad
ix
-
2
^2
DI
F
-
FFT
T
h
e
r
ad
ix
-
2
2
is
d
er
iv
ed
b
y
w
r
i
tin
g
th
e
eq
u
a
tio
n
∑
∑
∑
∑
[
]
W
h
er
e
[
(
)
]
(
)
(
)
4
.
2
.
RADIX
-
2
^3
AL
G
O
RI
T
H
M
Fro
m
th
e
f
i
g
u
r
e,
it i
n
d
icate
s
t
h
at
in
r
ad
ix
-
2
^3
alg
o
r
ith
m
t
h
e
t
w
id
d
le
f
ac
to
r
ex
is
t
s
in
t
h
ir
d
s
t
ag
e
[
1
2
]
.
T
h
e
eq
u
atio
n
o
f
r
ad
ix
-
2
^3
al
g
o
r
ith
m
ca
n
b
e
d
er
iv
ed
as f
o
llo
w
s
∑
W
h
er
e
(
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
R
ec
o
n
f
i
g
u
r
ab
le
&
E
m
b
ed
d
ed
Sy
s
t
I
SS
N:
2089
-
4864
Lo
w
p
o
w
er a
n
d
h
ig
h
p
erfo
r
ma
n
ce
F
F
T w
ith
d
iffer
en
t ra
d
ices
(
Md
.
Za
kir Hu
s
s
a
in
)
103
4
16
x
[
0
]
X
[
4
]
X
[
2
]
X
[
6
]
X
[
1
]
X
[
5
]
X
[
3
]
X
[
7
]
x
[
1
]
x
[
9
]
x
[
5
]
x
[
1
3
]
x
[
3
]
x
[
1
1
]
x
[
7
]
x
[
1
5
]
X
[
0
]
X
[
1
]
X
[
2
]
X
[
3
]
X
[
4
]
X
[
5
]
X
[
6
]
X
[
7
]
X
[
8
]
X
[
9
]
X
[
1
0
]
X
[
1
1
]
X
[
1
2
]
X
[
1
3
]
X
[
1
4
]
X
[
1
5
]
0
0
0
0
0
0
0
0
0
0
0
0
4
1
4
1
0
0
0
0
0
0
0
0
0
0
0
8
12
20
24
28
0
0
0
4
4
4
4
3
6
9
12
15
18
24
S
T
A
G
E
1
S
T
A
G
E
2
S
T
A
G
E
3
S
T
A
G
E
4
Fig
u
r
e
5
.
Sig
n
al
f
lo
w
g
r
ap
h
o
f
1
6
p
o
in
t r
ad
ix
-
2
^3
DI
T
-
FFT
5.
P
RO
P
O
SE
D
WO
RK
I
n
th
i
s
p
ap
er
,
th
e
p
r
o
p
o
s
ed
w
o
r
k
d
e
m
o
n
s
tr
ates
u
s
i
n
g
t
h
e
d
if
f
er
en
t
co
m
b
i
n
atio
n
s
o
f
t
h
e
co
m
p
u
tatio
n
a
l
ele
m
e
n
ts
a
n
d
i
m
p
le
m
e
n
tat
io
n
o
f
t
h
o
s
e
co
m
p
u
tatio
n
a
l
ele
m
en
ts
(
m
u
ltip
lier
s
a
n
d
ad
d
er
s
)
in
d
i
f
f
er
e
n
t
r
ad
ices
s
u
c
h
as
r
ad
ix
-
2
,
r
ad
ix
-
4
,
r
ad
ix
-
2
2
an
d
r
ad
ix
-
2
^3
FF
T
ar
ch
itect
u
r
e
s
f
o
r
1
6
-
p
o
in
t.
T
h
e
o
v
er
all
i
m
p
ac
t
an
d
p
er
f
o
r
m
a
n
ce
i
s
co
n
s
id
er
ed
in
d
if
f
er
e
n
t
r
ad
ices
u
s
in
g
d
i
f
f
er
en
t
co
m
p
u
tatio
n
al
ele
m
e
n
ts
.
T
h
e
w
o
r
k
f
o
c
u
s
es
o
n
th
e
F
FT
ar
ch
itectu
r
e
a
n
d
th
e
co
m
p
u
tatio
n
s
to
b
e
d
o
n
e
i
n
e
ac
h
b
u
tter
f
l
y
u
n
it
in
th
e
r
ad
i
x
u
s
i
n
g
t
h
e
d
if
f
er
en
t
c
o
m
b
i
n
atio
n
s
o
f
m
u
ltip
lier
s
.
Fro
m
th
e
ab
o
v
e
s
ig
n
al
f
lo
w
g
r
ap
h
s
,
it
i
s
ill
u
s
tr
ated
t
h
at
t
o
co
m
p
u
te
t
w
id
d
le
f
ac
to
r
w
ith
t
h
e
i
n
p
u
t,
t
h
e
m
u
lt
ip
licatio
n
is
n
ec
es
s
ar
y
.
So
e
f
f
i
cien
t
m
u
ltip
lier
s
h
o
u
ld
b
e
co
n
s
id
er
ed
to
h
av
e
an
ef
f
icien
t r
ad
ix
in
FF
T
p
r
o
ce
s
s
o
r
s
.
Firs
tl
y
,
d
if
f
er
en
t
co
m
p
u
tat
io
n
al
ele
m
e
n
ts
(
m
u
ltip
lier
s
a
n
d
ad
d
er
s
)
h
av
e
b
ee
n
s
t
u
d
ied
an
d
u
tili
ze
d
i
n
d
if
f
er
e
n
t
r
ad
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to
ac
q
u
ir
e
t
h
e
ef
f
icie
n
t
F
FT
ar
ch
itectu
r
e.
I
n
t
h
is
p
ap
er
,
th
e
d
if
f
er
en
t
m
u
l
tip
lier
s
u
s
ed
s
u
ch
as
B
o
o
t
h
m
u
ltip
lier
,
Mo
d
if
ied
B
o
o
th
[
1
3
]
,
C
an
o
n
ical
s
i
g
n
ed
Dig
it
(
C
SD)
[
6
]
,
m
u
ltip
lier
s
to
co
m
p
u
te
t
w
id
d
le
f
ac
to
r
in
b
u
t
ter
f
l
y
u
n
i
t.
T
h
e
ad
d
er
u
s
ed
is
ca
r
r
y
s
a
v
e
ad
d
er
as
it
is
f
a
s
ter
an
d
m
o
r
e
e
f
f
icie
n
t
w
h
e
n
co
m
p
ar
ed
to
ca
r
r
y
-
lo
o
k
a
h
ea
d
ad
d
er
.
5
.
1
.
T
w
idd
le
F
a
ct
o
r
M
ultiplica
t
io
n
T
w
id
d
le
f
ac
to
r
m
u
l
tip
licatio
n
p
lay
s
s
i
g
n
i
f
ica
n
t
r
o
le
in
s
o
l
v
i
n
g
t
h
e
b
u
t
ter
f
l
y
u
n
it
in
ea
ch
s
tag
e
o
f
t
h
e
d
if
f
er
e
n
t
r
ad
ices.
W
h
ile
m
u
lti
p
ly
in
g
t
h
e
t
w
id
d
le
f
ac
to
r
w
i
t
h
t
h
e
v
al
u
e,
an
ef
f
icie
n
t
m
u
lti
p
lier
is
u
s
ed
i
n
t
h
e
r
ad
ix
.
Dif
f
er
en
t t
w
id
d
le
f
ac
to
r
s
u
s
ed
i
n
1
6
-
p
o
in
t r
ad
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r
e
r
ep
r
esen
ted
as
:
Fro
m
th
e
ab
o
v
e
f
i
g
u
r
e,
it
is
o
b
s
er
v
ed
th
at
h
o
w
t
h
e
m
u
ltip
li
ca
tio
n
an
d
ad
d
itio
n
p
r
o
ce
s
s
o
cc
u
r
in
t
h
e
b
u
tter
f
l
y
u
n
it
s
in
t
h
e
r
ad
ix
.
W
h
er
ev
er
th
e
ad
d
itio
n
r
eq
u
ir
e
s
ad
d
er
is
u
s
ed
in
th
at
p
lace
an
d
f
o
r
m
u
ltip
licat
io
n
d
if
f
er
e
n
t
ab
o
v
e
m
e
n
tio
n
m
u
lt
ip
lier
s
ca
n
u
s
ed
.
T
w
id
d
le
f
ac
t
o
r
v
alu
es
ar
e
r
ep
r
esen
ted
as
0
.
7
0
7
,
0
.
9
2
3
.
0
.
3
8
2
,
th
ese
v
al
u
es
ar
e
co
n
v
er
ted
i
n
to
b
in
ar
y
f
o
r
m
a
n
d
th
e
n
r
ep
r
ese
n
te
d
i
n
Q
-
f
o
r
m
at
(
f
i
x
ed
p
o
in
t
r
ep
r
esen
tatio
n
)
[
1
4
]
.
T
h
e
t
w
id
d
le
f
ac
to
r
v
al
u
es
r
ep
r
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ted
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Q
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f
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r
m
at
ar
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s
h
o
w
n
b
elo
w
i
n
t
h
e
T
ab
le
1
.
Fig
u
r
e
6
s
h
o
w
s
d
iag
r
a
m
o
f
b
u
tter
f
l
y
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n
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t
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
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2
0
8
9
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I
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9
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–
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104
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u
r
e
6
.
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it
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1
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e
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to
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.
2
.
F
ix
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po
int
M
ultiplica
t
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Fix
ed
p
o
in
t
m
u
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licatio
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h
as
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et
h
o
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w
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s
i
g
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r
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ig
n
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n
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m
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s
.
Fix
ed
p
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in
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m
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r
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tatio
n
t
h
e
n
u
m
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er
o
f
d
i
g
its
o
r
b
its
ar
e
f
ix
ed
eit
h
er
b
ef
o
r
e
o
r
af
ter
th
e
r
ad
ix
p
o
in
t
i.e
.
b
in
ar
y
p
o
in
t.
Fi
g
u
r
e
7
s
h
o
w
s
f
o
r
m
at
o
f
f
i
x
ed
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p
o
i
n
t n
u
m
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er
.
S
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gn
b
i
t
M
-
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r
bi
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s
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f
r
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t
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l
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i
t
s
B
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r
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N
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b
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Fig
u
r
e
7
.
Fo
r
m
a
t o
f
f
ix
ed
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p
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t n
u
m
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er
6.
RE
SU
L
T
S
Th
is
p
ap
er
d
elib
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ates
th
e
i
m
p
le
m
en
ta
tio
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o
f
1
6
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p
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s
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m
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t
s
i
n
th
e
b
u
tter
f
l
y
u
n
it
o
f
FF
T
ar
ch
itect
u
r
e.
T
h
e
an
al
y
s
i
s
a
n
d
co
m
p
ar
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o
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ad
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b
et
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n
d
if
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er
en
t
r
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o
f
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.
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h
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f
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ct
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Xili
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1
4
.
7
to
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s
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Sp
ar
tan
6
f
a
m
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.
Fro
m
th
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s
2
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s
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ated
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t
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Ved
ic
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o
r
1
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its
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ar
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s
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ter
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co
m
p
ar
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to
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r
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h
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er
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m
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ab
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ab
le
s
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d
5
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it is
o
b
s
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v
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at
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o
m
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ar
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lt o
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b)
R
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in
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r
eq
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en
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ar
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to
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ad
ix
-
2.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
R
ec
o
n
f
i
g
u
r
ab
le
&
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m
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I
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2089
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4864
Lo
w
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105
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f
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n
t
m
u
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lier
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Ty
p
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.
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fa
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2
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.
c
irc
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