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FFT
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FP
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1.
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w
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in
to
f
r
eq
u
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c
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d
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m
ai
n
s
i
g
n
al.
First,
to
d
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o
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p
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in
to
s
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e
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eq
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Fo
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b
asic
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s
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w
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ted
s
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m
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f
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elate
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u
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o
id
s
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c
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m
p
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x
ex
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o
n
e
n
tial
[
1
]
.
T
h
e
Fo
u
r
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tr
an
s
f
o
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m
is
a
w
a
y
to
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a
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n
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to
it
s
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ti
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t
f
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q
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s
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is
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p
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a
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d
i
s
cr
ete
ti
m
e
d
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m
ai
n
s
i
g
n
al
[
2
]
.
DFT
is
a
f
in
ite
d
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atio
n
d
is
cr
ete
f
r
eq
u
en
c
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eq
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ce
w
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ed
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o
f
Fo
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T
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an
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f
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.
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‘
N’
eq
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all
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s
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ac
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ω
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=0
to
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.
(
)
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∑
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)
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w
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etc,
ar
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th
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n
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ati
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6,
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etc.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
E
S
I
SS
N:
2089
-
4864
Op
timiz
a
tio
n
o
f R
eso
u
r
ce
Utiliz
a
tio
n
o
f F
a
s
t F
o
u
r
ier Tr
a
n
s
fo
r
m
(
S
u
b
h
a
s
h
C
h
a
n
d
r
a
Ya
d
a
v
)
187
T
h
e
B
u
tter
f
l
y
d
ia
g
r
a
m
tak
e
s
ad
v
an
ta
g
e
o
f
t
h
is
r
ed
u
n
d
a
n
c
y
an
d
s
y
m
m
etr
y
,
w
h
ic
h
is
p
ar
t
o
f
w
h
a
t
m
ak
e
s
t
h
e
F
FT
p
o
s
s
ib
le.
An
FF
T
is
a
w
a
y
to
co
m
p
u
te
t
h
e
s
a
m
e
r
es
u
lt
m
o
r
e
q
u
ic
k
l
y
co
m
p
u
ti
n
g
a
DFT
of
N
p
o
in
ts
in
th
e
n
ai
v
e
w
a
y
,
u
s
i
n
g
t
h
e
d
ef
i
n
it
io
n
,
ta
k
es
O(
N
2
)
ar
ith
m
etica
l
o
p
er
atio
n
s
,
w
h
ile
a
n
F
FT
ca
n
co
m
p
u
te
t
h
e
s
a
m
e
r
es
u
lt
i
n
o
n
l
y
O(
N
lo
g
N
)
o
p
er
atio
n
s
.
T
h
e
d
if
f
er
en
ce
i
n
s
p
ee
d
ca
n
b
e
s
u
b
s
tan
tial,
e
s
p
ec
iall
y
f
o
r
lo
n
g
d
ata
s
et
s
w
h
er
e
N
m
a
y
b
e
in
t
h
e
th
o
u
s
a
n
d
s
o
r
m
il
lio
n
s
—
in
p
r
ac
tice,
th
e
co
m
p
u
tatio
n
ti
m
e
ca
n
b
e
r
ed
u
ce
d
b
y
s
ev
er
al
o
r
d
er
o
f
m
ag
n
i
tu
d
e
i
n
s
u
c
h
ca
s
es,
a
n
d
th
e
i
m
p
r
o
v
e
m
e
n
t
is
r
o
u
g
h
l
y
p
r
o
p
o
r
tio
n
al
to
N
/
lo
g
(
N
)
.
T
h
is
h
u
g
e
i
m
p
r
o
v
e
m
e
n
t
m
ad
e
m
a
n
y
DFT
-
b
ased
alg
o
r
i
th
m
s
p
r
ac
tical
.
T
h
e
FF
T
is
a
f
ast
alg
o
r
ith
m
to
f
i
n
d
o
u
t
th
e
DFT
o
f
a
s
eq
u
e
n
ce
,
a
d
ir
ec
t
co
m
p
u
tatio
n
o
f
DFT
o
f
a
s
eq
u
en
ce
r
eq
u
ir
ed
(
2
)
co
m
p
le
x
m
u
lt
ip
licatio
n
an
d
(
2
−
)
co
m
p
le
x
ad
d
itio
n
an
d
if
w
e
co
m
p
u
te
D
FT
o
f
a
s
eq
u
e
n
ce
b
y
FF
T
alg
o
r
ith
m
t
h
er
e
r
eq
u
ir
ed
(
2
l
og
2
)
C
o
m
p
lex
M
u
ltip
licat
io
n
a
n
d
(
l
og
2
)
co
m
p
le
x
ad
d
itio
n
.
I
n
t
h
e
ca
s
e
o
f
DFT
th
er
e
ar
e
N
2
co
m
p
le
x
m
u
lt
ip
licatio
n
an
d
N
N
2
c
o
m
p
le
x
ad
d
itio
n
ar
e
r
eq
u
ir
ed
f
o
r
N
p
o
in
t
DFT
.
B
u
t
in
th
e
ca
s
e
o
f
FF
T
th
er
e
ar
e
2
2
N
m
u
ltip
licat
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n
an
d
N
N
2
l
og
ad
d
itio
n
ar
e
r
eq
u
ir
ed
.
Fig
u
r
e
1
.
T
w
id
d
le
f
ac
to
r
as a
r
o
tatin
g
v
ec
to
r
2.
DE
CIM
AT
I
O
N
I
N
T
I
M
E
F
F
T
AL
G
O
RI
T
H
M
Dec
i
m
atio
n
i
n
ti
m
e
F
FT
alg
o
r
ith
m
i
m
p
l
y
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ir
s
t
d
iv
id
i
n
g
t
h
e
in
p
u
t
s
eq
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e
n
ce
i
n
ti
m
e
d
o
m
a
in
[
3
]
a
n
d
th
en
p
r
o
ce
s
s
ed
f
o
r
th
e
F
FT
c
o
m
p
u
tatio
n
.
L
et
b
)
(
n
x
e
a
s
eq
u
en
c
e
w
h
ic
h
w
e
w
a
n
t
to
co
n
v
er
t
i
n
to
f
r
eq
u
e
n
c
y
d
o
m
ai
n
a
n
d
)
(
1
m
f
an
d
)
(
2
m
f
ar
e
th
e
ev
e
n
an
d
o
d
d
p
ar
t
r
esp
ec
tiv
el
y
th
e
n
th
e
DFT
o
f
t
h
e
s
eq
u
e
n
ce
)
(
n
x
is
g
iv
e
n
b
y
E
q
u
a
tio
n
1
as :
1
2
0
)
1
2
(
1
2
0
2
)
1
2
(
)
2
(
)
(
N
m
m
k
N
N
m
km
N
W
m
x
W
m
x
k
X
(
2
)
T
h
e
co
m
b
in
a
tio
n
o
f
eq
u
a
tio
n
(
3
)
,
(
4
)
,
(
5
)
an
d
(
6
)
g
iv
es t
h
e
f
l
o
w
g
r
ap
h
o
f
DI
T
-
FF
T
as sh
o
wn
in
F
ig
u
r
e
-
2
(
a)
.
B
y
t
h
e
p
r
o
p
er
ty
o
f
t
w
id
d
le
f
ac
to
r
W
N
2
=
W
N/2
an
d
t
h
e
d
ef
i
n
itio
n
o
f
DFT
w
e
ca
n
w
r
ite
:
X
(
k
)
=
F
1
(
k
)
+
F
2
(
k
)
(
3
)
Her
e
F
1
(
k
)
is
N
/2
p
o
in
t D
FT
o
f
f
1
(
m
)
an
d
F
2
(
k
)
is
N
/2
p
o
in
t
DFT
o
f
f
2
(
m
).
As N/2
is
t
h
e
p
er
io
d
o
f
th
e
s
eq
u
en
ce
f
1
(
m
)
a
n
d
f
2
(
m
)
s
o
r
ep
la
cin
g
k
b
y
(
k
+
2
)
in
E
q
u
atio
n
3
X
(
k
+
2
)
=
F
1
(
k
)
-
2
(
)
(
4
)
B
y
t
h
e
p
er
io
d
ic
p
r
o
p
er
ty
o
f
D
FT
an
d
t
w
id
d
le
f
ac
to
r
.
Ag
ai
n
d
ec
i
m
ati
n
g
t
h
e
s
eq
u
en
ce
F
1
(
k
)
an
d
F
2
(
k
)
w
e
h
a
v
e
G
11
(
0
)
=
X
(
0
)
+
X
(
2
)
(
5
)
G
11
(
1
)
=
X
(
0
)
−
X
(
2
)
(
6
)
3.
DE
CIM
AT
I
O
N
I
N
F
RE
Q
U
E
NCY
F
F
T
A
L
G
O
RI
T
H
M
Dec
i
m
atio
n
i
n
f
r
eq
u
e
n
c
y
F
FT
alg
o
r
it
h
m
i
n
d
icate
s
f
ir
s
t
co
m
p
u
tatatio
n
o
f
FF
T
an
d
th
e
n
d
i
v
id
in
g
th
e
o
u
tp
u
t
s
eq
u
e
n
ce
in
e
v
en
an
d
o
d
d
p
a
r
t
w
h
ic
h
is
in
f
r
eq
u
en
c
y
d
o
m
ai
n
[
4
]
.
T
h
e
f
lo
w
g
r
ap
h
is
s
h
o
w
n
i
n
Fig
u
r
e
2
.
W
e
can
d
ev
id
e
eq
u
atio
n
(
1
)
in
to
t
w
o
p
ar
ts
as f
o
llo
w
s
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
9
-
4
864
I
J
R
E
S
V
o
l.
6
,
No
.
3
,
N
o
v
e
m
b
er
2
0
1
7
:
1
8
6
–
1
9
0
188
1
2
/
1
2
0
)
(
)
(
)
(
N
N
n
kn
N
N
n
kn
N
W
n
x
W
n
x
k
X
(
7
)
P
u
t n
=
n
+
N/2
,
li
m
it
w
i
ll c
h
a
n
g
e
as
W
h
en
n
=
N/
2
N/2
=
n
+
N/2
th
er
ef
o
r
e
n
=
0
W
h
en
n
=
N
-
1
N
-
1
=
n
+N
/2
th
er
ef
o
r
e
n
=
N
–
1
–
N/
2
=N
/2
-
1
P
u
ttin
g
th
e
s
e
v
a
lu
e
s
in
s
ec
o
n
d
s
u
m
m
atio
n
o
f
ab
o
v
e
eq
.
w
e
g
et
:
1
2
0
)
2
/
(
1
2
0
)
2
(
)
(
)
(
N
n
N
n
k
N
N
n
kn
N
W
N
n
x
W
n
x
k
X
W
h
ich
ca
n
b
e
r
ed
u
ce
d
to
r
N
n
n
x
r
X
2
1
2
0
)
1
(
)
(
[
)
2
(
k
N
n
n
x
k
X
)
1
(
)
(
[
)
(
1
2
0
kn
N
W
N
n
x
)]
2
(
(
8
)
No
w
i
f
w
e
d
ec
i
m
ate
t
h
e
eq
u
at
i
o
n
(
8
)
in
to
ev
en
an
d
o
d
d
p
ar
ts
th
en
w
e
h
a
v
e
r
N
n
n
x
r
X
2
1
2
0
)
1
(
)
(
[
)
2
(
rn
N
W
N
n
x
2
)]
2
(
(
9
)
(
a)
(
b
)
(
c)
Fig
u
r
e
2
.
(
a)
f
lo
w
g
r
ap
h
o
f
DI
T
-
FF
T
(
b
)
f
lo
w
g
r
ap
h
o
f
DI
F
-
F
FT
(
c)
f
lo
w
g
r
ap
h
o
f
G
-
tr
a
n
s
p
o
s
e
FF
T
B
y
p
u
tti
n
g
k
=
2
r
+1
in
eq
.
(
4
.
5
.
1
6
)
w
e
w
il
l
g
et
o
d
d
n
u
m
b
er
o
f
s
eq
u
en
c
e
1
2
1
2
0
)
1
(
)
(
[
)
1
2
(
r
N
n
n
x
r
X
n
r
N
W
N
n
x
)
1
2
(
)]
2
(
(
1
0
)
W
h
er
e
r
is
an
in
te
g
er
w
h
ich
v
a
r
ies f
r
o
m
0
to
N
2
−
1
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
R
E
S
I
SS
N:
2089
-
4864
Op
timiz
a
tio
n
o
f R
eso
u
r
ce
Utiliz
a
tio
n
o
f F
a
s
t F
o
u
r
ier Tr
a
n
s
fo
r
m
(
S
u
b
h
a
s
h
C
h
a
n
d
r
a
Ya
d
a
v
)
189
As
w
e
k
n
o
w
(
-
1
)
2
r
=
1
an
d
(
-
1
)
2
r
+1
=
-
1
P
u
ttin
g
th
e
s
e
v
a
lu
e
s
in
E
q
u
a
ti
o
n
9
an
d
10,
1
2
0
)
(
[
)
2
(
N
n
n
x
r
X
rn
N
W
N
n
x
2
)]
2
(
(
1
1
)
1
2
0
)
(
[
)
1
2
(
N
n
n
x
r
X
n
N
rn
N
W
W
N
n
x
2
)]
2
(
(
1
2
)
Fro
m
E
q
u
atio
n
1
1
an
d
1
2
g
(
n
)
=
(
)
+
(
+
2
)
(
1
3
)
h
(
n
)
=
[
(
)
−
x
(
n
+
N
2
)
]
(
1
4
)
Ag
ai
n
d
ec
i
m
ati
n
g
th
e
s
eq
u
e
n
c
e
g
(
n
)
an
d
h
(
n
)
w
e
h
a
v
e:
X(
k
)
=
g
(
0
)
+
g
(
1
)
(
15
)
X
(
k
)
=
g
(
0
)
–
g
(
1
)
4.
G
-
T
RA
NSPO
S
E
AL
G
O
RI
T
H
M
G
-
T
r
an
s
p
o
s
e
al
g
o
r
ith
m
d
ec
r
ea
s
es
t
h
e
n
u
m
b
er
o
f
t
w
id
d
le
ac
c
ess
es
b
y
a
n
as
y
m
p
to
tic
f
ac
to
r
o
f
lo
g
(
n
)
[
5
]
.
I
d
ea
b
eh
in
d
th
e
G
-
tr
an
s
p
o
s
e
alg
o
r
ith
m
i
s
to
d
ec
r
ea
s
e
th
e
n
u
m
b
er
o
f
t
w
id
d
le
tab
le
ac
ce
s
s
es
b
y
r
estru
ct
u
r
in
g
t
h
e
D
FT
.
T
h
e
G
-
T
r
an
s
p
o
s
e
n
et
w
o
r
k
ap
p
lies
t
h
e
t
w
id
d
le
f
ac
to
r
s
to
ea
c
h
s
u
b
n
et
w
o
r
k
’
s
i
n
p
u
t
s
.
I
n
f
ac
t,
p
er
m
u
t
in
g
th
e
G
-
T
r
an
s
p
o
s
e
n
et
w
o
r
k
’
s
r
o
w
s
in
to
b
it
-
r
ev
er
s
ed
o
r
d
er
w
h
ile
leav
in
g
co
n
n
ec
ti
v
it
y
u
n
c
h
a
n
g
ed
(
t
h
at
i
s
,
r
ed
r
a
w
in
g
th
e
G
-
T
r
an
s
p
o
s
e
n
et
w
o
r
k
s
u
ch
t
h
at
t
h
e
in
p
u
ts
g
i
v
e
n
i
n
t
h
e
o
r
d
er
x
(
0
)
,
x
(
2
)
,
x
(
1
)
,
an
d
x
(
3
)
)
y
ield
s
t
h
e
DI
T
n
et
w
o
r
k
.
T
h
u
s
,
a
G
-
T
r
an
s
p
o
s
e
i
m
p
le
m
en
tatio
n
p
r
o
d
u
ce
s
t
h
e
s
a
m
e
r
esu
lt
s
i
n
f
i
n
ite
p
r
ec
is
io
n
ar
it
h
m
etic
a
s
a
DI
T
im
p
le
m
e
n
tatio
n
b
u
t
p
r
o
c
ess
es
i
n
p
u
t/o
u
tp
u
t
d
ata
lik
e
a
DI
F
i
m
p
le
m
en
ta
tio
n
.
Fi
g
u
r
e
2
(
c)
s
h
o
w
s
th
e
f
lo
w
g
r
ap
h
o
f
G
T
al
g
o
r
ith
m
.
G
T
al
g
o
r
ith
m
g
iv
e
a
n
as
y
m
p
to
tic
r
ed
u
ctio
n
i
n
th
e
n
u
m
b
er
o
f
t
w
id
d
le
f
ac
to
r
lo
ad
s
r
eq
u
ir
ed
f
o
r
f
ir
s
t
s
tag
e
o
f
d
ec
i
m
atio
n
w
h
ic
h
is
r
esp
o
n
s
i
b
le
f
o
r
m
i
n
i
m
izi
n
g
m
e
m
o
r
y
s
ize
as
co
m
p
ar
e
to
C
o
o
le
y
-
T
u
k
e
y
d
ec
i
m
atio
n
-
in
-
t
i
m
e
an
d
d
ec
i
m
atio
n
-
in
-
f
r
eq
u
e
n
c
y
FF
T
.
5.
SI
M
UL
AT
I
O
N
R
E
S
UL
T
S
AND
DIS
CUSS
I
O
N
W
e
in
v
esti
g
ated
G
-
T
r
an
s
p
o
s
e,
C
o
o
ley
-
T
u
k
e
y
Dec
i
m
a
tio
n
-
in
-
T
i
m
e
an
d
Dec
i
m
atio
n
-
in
-
Fre
q
u
en
c
y
FF
T
alg
o
r
ith
m
s
,
W
e
h
a
v
e
tak
en
f
o
llo
w
i
n
g
p
ar
a
m
eter
f
o
r
s
i
m
u
latio
n
o
f
all
al
g
o
r
ith
m
s
R
a
d
ix
=
2
,
N
=
4
(
4
p
o
in
t
FF
T
)
I
n
p
u
t
s
eq
u
e
n
ce
x
(
n
)
=
{2
,
2
,
4
,
0
}
T
w
id
d
le
Facto
r
W
4
0
=
1
an
d
W
4
1
=
-
j
.
T
h
e
r
esu
lt
o
f
s
i
m
u
latio
n
in
MA
T
L
A
B
f
o
r
all
t
h
e
th
r
ee
al
g
o
tith
m
ar
e
s
a
m
e
an
d
g
iv
e
n
as
{8
,
-
2
-
2
j
,
4
,
-
2
+2
j
}
as
s
h
o
w
n
in
Fig
u
r
e
3
,
f
o
r
m
e
m
o
r
y
o
p
ti
m
iza
tio
n
s
i
m
u
la
ti
o
n
o
f
all
T
h
r
ee
al
g
o
r
ith
m
is
s
u
cc
es
s
f
u
ll
y
d
ev
elo
p
ed
u
s
i
n
g
VHDL
o
n
XI
L
I
NX
Sp
ar
tan
-
3
XC
3
S4
0
0
0
l
-
4
f
g
9
0
0
FP
GA
d
ev
elo
p
m
en
t
b
o
ar
d
.
T
h
e
s
i
m
u
la
tio
n
r
esu
lt
f
o
r
all
th
e
th
r
ee
alg
o
r
ith
m
s
ar
e
s
a
m
e
w
h
ich
m
ea
n
s
t
h
e
r
es
u
lt
o
f
FF
T
is
n
o
t
a
f
f
ec
ted
b
u
t
th
e
r
eso
u
r
ce
u
tili
za
tio
n
o
f
all
th
r
ee
al
g
o
r
ith
m
s
is
d
if
f
er
e
n
t.
T
ab
le
1
s
h
o
w
s
th
e
r
e
s
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[1
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[2
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.
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[3
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J.
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.
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[5
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Ke
v
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J.
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Dr
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tran
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.
Evaluation Warning : The document was created with Spire.PDF for Python.