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h
o
w
e
v
er
it
r
eq
u
ir
e
s
m
o
r
e
ti
m
e.
Fig
u
r
e
1
.
P
r
o
d
u
ct
co
d
e
p
r
in
cip
le
Fig
u
r
e
2
.
Stru
ct
u
r
e
o
f
th
e
t
u
r
b
o
d
ec
o
d
er
3.
T
H
E
S
I
SO
DE
CO
DE
R
3
.
1
.
T
hres
ho
ld De
co
din
g
A
lg
o
rit
h
m
T
h
e
SISO
(
So
f
t
I
n
-
So
f
t
O
u
t)
d
ec
o
d
er
is
th
e
b
ase
ele
m
e
n
t
o
f
th
e
tu
r
b
o
d
ec
o
d
in
g
.
W
e
u
s
ed
SISO
d
ec
o
d
er
f
o
r
w
h
ich
t
h
e
t
h
r
esh
o
l
d
d
ec
o
d
in
g
al
g
o
r
ith
m
[
3
]
,
[
4
]
,
[
1
3
]
is
g
iv
en
b
elo
w
:
Fo
r
e
ac
h
j =
n
t
o
1
a.
C
alcu
la
te
th
e
te
r
m
s
B
i
an
d
ω
i
w
ith
iє
{
1
,
.
.
,
M
}
b.
C
alcu
la
te
B
0
an
d
w
0
c.
C
alcu
la
te
th
e
ex
t
r
in
s
ic
in
f
o
r
m
atio
n
M
i
i
i
j
B
y
W
1
)
2
1
(
)
(
d.
C
alcu
la
te
j
j
j
Wy
y
Es
y
LLR
N
0
4
)
(
e.
I
f
(
L
L
R
(
y
j
)
>
0
)
H
ar
d
De
cis
io
n
=
1
else
Ha
r
d
D
ec
is
io
n
=
0
W
h
e
r
e:
a.
n
: Co
d
e
l
en
g
th
;
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
J
E
C
E
Vo
l.
7
,
No
.
4
,
A
u
g
u
s
t
2
0
1
7
:
1
8
2
4
–
1
8
3
2
182
6
b.
M:
n
u
m
b
er
o
f
o
r
th
o
g
o
n
al
E
q
u
atio
n
s
;
c.
B
i:
th
e
o
r
th
o
g
o
n
al
E
q
u
atio
n
o
n
th
e
i
th
b
it,
a
f
ter
r
e
m
o
v
e
o
f
i
th
b
it;
d.
ω
i: p
r
o
p
o
r
tio
n
al
to
th
e
r
eliab
ilit
y
o
f
i
th
p
ar
it
y
E
q
u
atio
n
,
e.
W
(
y
j
)
: e
x
tr
in
s
ic
in
f
o
r
m
at
io
n
r
ep
r
esen
tin
g
t
h
e
esti
m
ated
o
r
th
o
g
o
n
al
E
q
u
atio
n
s
o
n
t
h
e
s
y
m
b
o
l y
j;
f.
L
L
R
(
y
i
)
th
e
d
ec
is
io
n
f
u
n
ctio
n
o
n
th
e
s
y
m
b
o
l
y
j;
An
d
:
ni
k
k
i
k
ik
ni
k
k
i
k
ik
L
L
i
,
1
,
1
)
2
t
a
n
h
(
1
)
2
t
a
n
h
(
1
ln
(
1
)
3
.
2
.
Str
uct
ura
l D
ia
g
ra
m
o
f
SI
SO
Dec
o
der
W
e
h
av
e
d
esi
g
n
ed
,
i
n
VHD
L
,
a
SISO
d
ec
o
d
er
,
w
h
ich
c
an
b
e
u
s
ed
i
n
an
iter
ati
v
e
p
r
o
ce
s
s
.
T
h
is
iter
ativ
e
p
r
o
ce
s
s
w
e
u
s
e
f
o
llo
w
s
th
e
m
o
d
el
p
r
o
p
o
s
ed
b
y
P
y
n
d
iah
[
3
]
; See
Fig
u
r
e
3
.
T
h
e
s
o
f
t
in
p
u
t a
n
d
r
esp
ec
ti
v
el
y
t
h
e
s
o
f
t
o
u
tp
u
t o
f
t
h
e
q
th
s
tep
(
h
al
f
-
iter
atio
n
)
o
f
th
e
iter
ati
v
e
d
ec
o
d
in
g
ar
e
g
iv
e
n
b
y
:
)
(
)
(
)
1
(
q
W
q
R
q
R
)
(
)
(
4
)
(
0
q
W
q
R
N
E
q
LLR
s
W
h
er
e
R
r
ep
r
esen
t
s
t
h
e
li
n
es
o
r
co
lu
m
n
s
o
f
t
h
e
r
ec
eiv
ed
an
d
q
u
an
ti
f
ied
w
o
r
d
,
W
(
q
)
th
e
ex
tr
i
n
s
ic
in
f
o
r
m
atio
n
ca
lcu
la
ted
b
y
t
h
e
p
r
ev
io
u
s
d
ec
o
d
er
an
d
α
is
a
co
ef
f
icie
n
t
w
h
ic
h
v
ar
ies
w
i
t
h
ea
ch
iter
atio
n
.
I
n
ad
d
itio
n
to
its
th
r
es
h
o
ld
o
u
tp
u
t
,
th
e
d
ec
o
d
er
h
as a
h
ar
d
d
ec
is
io
n
o
u
tp
u
t t
h
at
w
e
u
s
ed
f
o
r
its
v
alid
atio
n
.
Fig
u
r
e
3
.
SISO
Dec
o
d
er
(
elem
en
tar
y
ce
ll o
f
tu
r
b
o
d
ec
o
d
in
g
)
3
.
3
F
P
G
A
I
m
ple
m
e
nta
t
io
n o
f
t
he
SI
SO
Dec
o
der
T
o
r
ed
u
ce
th
e
co
m
p
le
x
it
y
o
f
t
h
e
alg
o
r
it
h
m
,
w
e
u
s
ed
,
to
s
i
m
p
lify
t
h
e
ex
p
r
es
s
io
n
(
1
)
,
th
e
E
q
u
atio
n
(
2
)
b
elo
w
p
r
o
p
o
s
ed
b
y
[
1
2
]
an
d
ap
p
lied
to
th
e
m
aj
o
r
it
y
lo
g
ic
co
d
es b
y
[
1
3
]
,
[
1
1
]
,
[
2
1
]
.
T
h
e
s
i
m
p
li
f
ied
ex
p
r
ess
io
n
o
f
o
u
r
d
ec
o
d
er
b
ec
o
m
es
:
(
2
)
T
h
e
ar
ch
itectu
r
e
o
f
t
h
e
p
r
o
p
o
s
ed
SISO
d
ec
o
d
er
w
as
d
escr
ib
e
d
in
VHD
L
an
d
i
m
p
le
m
en
ted
o
n
FP
GA
u
s
i
n
g
Qu
ar
t
u
s
I
I
to
o
l
f
r
o
m
Al
ter
a.
W
e
u
s
ed
T
h
e
E
P
C
4
C
E
1
1
5
F2
9
C
7
FP
GA
t
y
p
e,
co
n
tai
n
in
g
ab
o
u
t
1
1
5
,
0
0
0
n
k
ik
ik
ni
k
L
s
i
g
n
L
i
1
1
)
(
*
m
i
n
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
E
C
E
I
SS
N:
2
0
8
8
-
8708
V
HDL
Desig
n
a
n
d
F
P
GA
I
mp
leme
n
ta
tio
n
o
f a
Hig
h
Da
ta
R
a
te
Tu
r
b
o
Dec
o
d
er …
.
(
A
.
B
o
u
d
a
o
u
d
)
1827
L
E
s
.
W
e
s
tu
d
ied
t
h
e
co
m
p
le
x
it
y
e
v
o
lu
tio
n
o
f
o
u
r
d
ec
o
d
er
s
f
o
r
d
if
f
e
r
e
n
t
q
u
a
n
tizat
io
n
b
it
n
u
m
b
er
u
s
i
n
g
C
(
7
,
3
)
C
(
7
,
3
)
,
C
(
2
1
,
1
1
)
C
(
2
1
,
1
1
)
an
d
C
(
7
3
,
4
5
)
C
(
7
3
,
4
5
)
p
r
o
d
u
ct
co
d
es.
T
h
e
r
esu
lts
o
b
tain
ed
,
af
ter
th
e
Qu
ar
tu
s
I
I
s
y
n
th
e
s
is
,
ar
e
s
u
m
m
ar
ized
in
T
ab
le
1
.
Fig
u
r
e
4
s
h
o
w
s
a
f
u
n
ctio
n
al
s
i
m
u
latio
n
ex
a
m
p
l
e
o
f
o
u
r
SISO
d
ec
o
d
er
f
o
r
DSC
(
7
,
3
)
co
d
e
o
n
w
h
ich
i
t
is
r
ep
o
r
ted
th
at
th
e
o
u
tp
u
ts
ar
e
u
p
d
ated
at
ea
ch
r
is
in
g
ed
g
e
o
f
t
h
e
clo
ck
,
h
e
n
ce
t
h
e
laten
c
y
i
s
o
n
e
clo
ck
c
y
cle.
Fig
u
r
e
4
.
Si
m
u
latio
n
ex
a
m
p
le
o
f
o
u
r
SISO
d
ec
o
d
er
f
o
r
th
e
DSC
(
7
,
3
)
co
d
e
,
w
ith
5
b
its
o
f
q
u
an
t
i
z
atio
n
.
T
ab
le
1
.
T
h
e
Ob
tain
ed
Data
f
o
r
Dif
f
er
e
n
t SI
SO D
ec
o
d
er
s
4.
T
H
E
I
N
T
E
RL
E
AV
E
R
4
.
1
.
T
he
I
nte
rlea
v
er
P
rinciple
An
in
ter
lea
v
er
is
a
s
y
s
te
m
t
h
a
t
r
ec
eiv
es
a
s
eq
u
en
ce
o
f
s
y
m
b
o
ls
in
its
in
p
u
t
an
d
p
r
o
v
id
es
a
n
o
th
er
o
f
th
e
s
a
m
e
alp
h
ab
et
to
t
h
e
o
u
tp
u
t
i
n
a
co
m
p
lete
l
y
d
if
f
er
e
n
t
o
r
d
er
.
I
n
o
u
r
ca
s
e
o
f
p
r
o
d
u
ct
co
d
es,
th
e
o
p
er
atio
n
o
f
th
e
i
n
ter
lea
v
er
is
s
u
m
m
ar
ized
in
t
w
o
s
tag
e
s
:
r
ec
eip
t
o
f
al
l
s
y
m
b
o
ls
o
f
t
h
e
p
r
o
d
u
ct
co
d
e
m
atr
i
x
n
2
*
n
1
,
an
d
s
w
ap
p
in
g
r
o
w
-
co
lu
m
n
o
f
t
h
is
m
atr
i
x
.
4
.
2
.
F
P
G
A
I
m
ple
m
e
nta
t
io
n o
f
t
h
e
I
nte
rlea
v
er
T
h
e
u
s
ed
in
ter
leav
er
s
ar
e
d
escr
ib
ed
in
VHDL
an
d
i
m
p
le
m
en
ted
o
n
th
e
s
a
m
e
FP
G
A
cir
cu
it
u
s
in
g
Qu
ar
tu
s
I
I
to
o
l
f
o
r
C
(
7
,
3
)
C
(
7
,
3
)
,
C
(
2
1
,
1
1
)
C
(
2
1
,
1
1
)
an
d
C
(
7
3
,
4
5
)
C
(
7
3
,
4
5
)
p
r
o
d
u
ct
co
d
es.
T
h
e
F
ig
u
r
e
5
s
h
o
w
s
t
h
e
e
x
ter
n
al
s
tr
u
ct
u
r
e
o
f
th
e
in
ter
lea
v
er
d
esi
g
n
f
o
r
th
e
C
(
7
,
3
)
C
(
7
,
3
)
p
r
o
d
u
ct
co
d
e.
S
I
S
O
D
e
c
o
d
e
r
s
C
h
a
r
a
c
t
e
r
i
st
i
c
s
Q
u
a
n
t
i
z
a
t
i
o
n
b
i
t
n
u
mb
e
r
4
B
i
t
s
5
B
i
t
s
6
B
i
t
s
7
B
i
t
s
D
S
C
(
7
,
3
)
C
o
mp
l
e
x
i
t
y
(
L
E
)
7
1
4
9
9
4
1
1
4
1
1
3
0
2
F
r
e
q
u
e
n
c
y
max
.
(
M
H
z
)
4
7
.
6
2
4
3
.
7
1
4
1
.
5
1
4
1
.
4
6
t
h
r
o
u
g
h
p
u
t
m
a
x
.
(
G
b
/
s)
0
.
3
3
0
.
3
1
0
.
2
9
0
.
2
9
L
a
t
e
n
c
y
(
n
s)
2
1
.
0
0
2
2
.
8
8
2
4
.
0
9
2
4
.
1
2
D
S
C
(
2
1
,
1
1
)
C
o
mp
l
e
x
i
t
y
(
L
E
)
5
3
3
4
6
4
9
6
7
7
4
9
8
8
6
3
F
r
e
q
u
e
n
c
y
max
.
(
M
H
z
)
2
4
.
9
9
2
4
.
3
0
2
3
.
5
8
2
3
.
34
t
h
r
o
u
g
h
p
u
t
m
a
x
.
(
G
b
/
s)
0
.
5
2
0
.
5
1
0
.
5
0
0
.
4
9
L
a
t
e
n
c
y
(
n
s)
4
0
.
0
2
4
1
.
1
5
4
2
.
4
1
4
2
.
8
4
D
S
C
(
7
3
,
4
5
)
C
o
mp
l
e
x
i
t
y
(
L
E
)
5
2
5
1
6
6
3
1
5
5
7
8
8
3
3
9
0
5
1
3
F
r
e
q
u
e
n
c
y
max
.
(
M
H
z
)
2
0
.
0
2
1
8
.
5
6
1
6
.
8
6
1
6
.
1
2
t
h
r
o
u
g
h
p
u
t
m
a
x
.
(
G
b
/
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I
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N
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2
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8
8
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it
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y
.
RE
F
E
R
E
NC
E
S
[1
]
C
Be
rro
u
,
A
.
G
la
v
ieu
x
,
P
.
T
h
it
i
m
a
jsh
ima
,
“
Ne
a
r
S
h
a
n
n
o
n
L
imit
Erro
r
Co
rr
e
c
ti
n
g
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d
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n
g
a
n
d
D
e
c
o
d
in
g
:
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u
r
b
o
Co
d
e
s”
,
IE
EE
I
n
tern
a
ti
o
n
a
l
C
o
n
f
e
re
n
c
e
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n
Co
m
m
u
n
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ti
o
n
ICC9
3
,
v
o
l.
2
/3
,
M
a
y
1
9
9
3
.
[2
]
R.
P
y
n
d
iah
,
A
.
G
l
a
v
ieu
x
,
A
.
P
i
c
a
rt
,
S
.
Ja
c
q
,
“
Ne
a
r
Op
ti
m
u
m
D
e
c
o
d
in
g
o
f
P
r
o
d
u
c
t
C
o
d
e
s”
,
Glo
b
e
c
o
m
'
9
4
,
S
a
n
F
ra
n
sisc
o
,
1
9
9
4
.
[3
]
M.
Be
lk
a
s
m
i,
M
.
L
a
h
m
e
r,
F.
Ay
o
u
b
,
“
Iter
a
ti
v
e
T
h
re
sh
o
ld
De
c
o
d
i
n
g
o
f
Pro
d
u
c
t
C
o
d
e
s
Co
n
stru
c
te
d
fro
m
M
a
j
o
rity
L
o
g
ic De
c
o
d
a
b
le Co
d
e
s”
,
ICCT
A
c
o
n
f
,
Da
m
a
sc
u
s,S
y
ria
,
p
p
.
2
3
7
6
-
2
3
8
1
,
A
p
ril
2
0
0
6
.
[4
]
J.
L
.
M
a
ss
e
y
,
“
T
h
re
sh
o
ld
De
c
o
d
i
n
g
”
,
M
.
I.
T
.
P
re
ss
,
Ca
m
b
rid
g
e
,
M
a
ss
a
c
h
u
se
t
ts,
1
9
6
3
.
[5
]
E.
J
.
.
W
e
ld
o
n
Jr,
“
Dif
f
e
re
n
c
e
-
S
e
t
C
y
c
li
c
Co
d
e
s”
,
Bell
S
y
ste
m T
e
c
h
n
ica
l
J
o
u
r
n
a
l
,
V
o
l
.
4
5
,
p
p
.
1
0
4
5
_
1
0
5
5
,
se
p
1
9
6
6
.
[6
]
R.
L
.
G
ra
h
a
m
,
F.
J.
M
a
c
W
il
li
a
m
s
,
“
On
th
e
Nu
m
b
e
r
o
f
In
f
o
rm
a
ti
o
n
S
y
m
b
o
ls
in
Diffe
re
n
c
e
-
S
e
t
C
y
c
li
c
Co
d
e
s”
,
Bell
S
y
ste
m T
e
c
h
n
ica
l
J
o
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rn
a
l
,
v
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l.
4
5
,
p
p
.
1
0
5
7
-
1
0
7
0
,
S
e
p
1
9
6
6
.
[7
]
J.
S
in
g
e
r,
“
A
T
h
e
o
re
m
in
F
in
it
e
P
r
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jec
ti
v
e
G
e
o
m
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tr
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n
d
S
o
m
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A
p
p
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c
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ti
o
n
s
to
Nu
m
b
e
rs
T
h
e
o
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”
,
AM
S
T
ra
n
s
.
,
v
o
l.
4
3
,
p
p
.
3
7
7
-
3
8
5
,
1
9
3
8
.
[8
]
F
.
J.
M
a
c
W
il
li
a
m
s,
“
A
T
a
b
le
o
f
P
rim
it
iv
e
Bin
a
ry
Id
e
m
p
o
ten
ts
o
f
o
d
d
L
e
n
g
th
n
,
7
≤n
≤
5
1
1
”
,
IEE
E
T
ra
n
s.
I
n
f
o
rm
.
T
h
e
o
ry
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v
o
l.
IT
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2
5
,
p
p
.
1
1
8
–
1
2
3
,
Ja
n
.
1
9
7
9
.
[9
]
F
.
J.
M
a
c
W
il
li
a
m
s,
J.
S
e
e
r
y
,
“
Th
e
w
e
i
g
h
t
Distrib
u
ti
o
n
s
o
f
so
m
e
m
in
i
m
a
l
C
y
c
li
c
C
o
d
e
s,”
IEE
E
T
ra
n
s.
I
n
fo
rm
.
T
h
e
o
ry
,
v
o
l.
IT
-
2
7
,
p
p
.
7
9
6
–
8
0
6
,
No
v
.
1
9
8
1
[1
0
]
C.
Clarc
k
,
B.
Ca
i
n
,
“
Err
o
r
-
Co
rre
c
ti
o
n
Co
d
in
g
f
o
r
Dig
it
a
l
Co
m
m
u
n
i
c
a
ti
o
n
s,”
P
len
u
m
P
re
ss
,
1
9
8
3
.
[1
1
]
M
.
El
h
a
ro
u
ss
i,
M
.
Be
lk
a
sm
i,
“
V
HD
L
De
sig
n
a
n
d
F
P
G
A
I
m
p
l
e
m
e
n
tatio
n
o
f
a
F
u
l
ly
P
a
ra
ll
e
l
A
rc
h
it
e
c
tu
re
f
o
r
Itera
ti
v
e
De
c
o
d
e
r
o
f
M
a
jo
rit
y
L
o
g
ic
Co
d
e
s
f
o
r
Hig
h
Da
ta
Ra
te
A
p
p
li
c
a
ti
o
n
s”
,
J
o
u
rn
a
l
o
f
W
ire
le
ss
Ne
two
rk
in
g
a
n
d
Co
mm
u
n
ica
ti
o
n
s (
J
W
NC)
,
2
(4
),
p
p
.
3
5
-
4
2
,
2
0
1
2
[1
2
]
J.
Ha
g
e
n
a
u
e
r,
E.
O
ff
e
r,
L
.
P
a
p
k
e
,
“
Itera
ti
v
e
D
e
c
o
d
in
g
o
f
Bin
a
r
y
Blo
c
k
a
n
d
Co
n
v
o
lu
ti
o
n
a
l
C
o
d
e
s”
,
IEE
E
T
ra
n
s.
In
fo
rm
.
T
h
e
o
ry
,
M
a
rs 1
9
9
6
,
V
o
l
.
4
2
,
p
p
.
4
2
9
-
4
4
6
.
[1
3
]
A
.
Bo
u
d
a
o
u
d
,
E.
A
b
d
e
lm
o
u
n
im
,
A.
Ba
ra
z
z
o
u
k
,
J.
Zb
it
o
u
,
M
.
Be
lk
a
sm
i
,
“
F
P
G
A
I
m
p
le
m
e
n
tatio
n
o
f
H
IHO
a
n
d
S
IHO
De
c
o
d
e
rs f
o
r
DSC
Co
d
e
s”
,
IEE
E
e
x
p
lo
re
r
,
p
p
.
1
4
6
1
–
1
4
6
4
,
A
p
ril
2
0
1
4
[1
4
]
L.
D.
Ru
d
o
lp
h
,
“
G
e
o
m
e
tri
c
Co
n
f
ig
u
ra
ti
o
n
a
n
d
M
a
j
o
rit
y
L
o
g
ic
De
c
o
d
a
b
le
Co
d
e
s”
,
P
h
D
t
h
e
sis,
Un
iv
e
rsity
o
f
Ok
lah
o
m
a
,
No
r
m
a
n
,
1
9
6
4
.
[1
5
]
G
.
P
re
s
c
h
e
r,
T
.
G
e
m
m
e
k
e
,
T
.
G
.
No
ll
,
“
A
p
a
ra
me
triza
b
le L
o
w
p
o
w
e
r Hig
h
-
th
ro
u
g
h
p
u
t
T
u
rb
o
-
d
e
c
o
d
e
r,”
in
IEE
E
In
t
.
Co
n
f
.
A
c
o
u
stics
,
S
p
e
e
c
h
,
a
n
d
S
ig
n
a
l
P
ro
c
e
ss
in
g
(ICA
S
S
P
),
v
o
l
u
m
e
5
,
p
a
g
e
s 2
5
-
2
8
,
M
a
r.
2
0
0
5
.
[1
6
]
B.
Bo
u
g
a
rd
,
A
.
G
iu
li
e
tt
i,
V
.
De
ru
d
d
e
r,
J.
W
il
lem
,
S
.
Du
p
o
n
t,
F
.
Ca
tt
h
o
o
r,
L
.
H
o
ll
e
v
o
e
t,
L
.
V
.
d
e
r
P
e
rre
,
H.
D.
M
a
n
,
R.
L
a
u
w
e
r
e
in
s,
“
A
sc
a
l
a
b
le
8
.
7
n
j/
b
i
t
7
5
.
6
M
b
/s
p
a
r
a
ll
e
l
c
o
n
c
a
te
n
a
ted
c
o
n
v
o
lu
t
io
n
a
l
(
tu
rb
o
-
)c
o
d
e
c
,
”
in
IEE
E
In
t
.
S
o
li
d
-
S
tate
Circu
i
t
Co
n
f
.
,
p
p
.
1
5
2
–
4
8
4
,
F
e
b
.
2
0
0
3
.
[1
7
]
Ra
h
u
l
S
h
re
sth
a
,
Ro
y
P
P
a
il
y
,
“
De
sig
n
a
n
d
Im
p
lem
e
n
ta
ti
o
n
o
f
a
Hig
h
S
p
e
e
d
M
AP
De
c
o
d
e
r
Arc
h
it
e
c
tu
re
fo
r
T
u
r
b
o
De
c
o
d
in
g
”
,
2
6
th
IEE
E
In
tern
a
t
io
n
a
l
Co
n
f
e
re
n
c
e
o
n
V
L
S
I
De
sig
n
a
n
d
th
e
1
2
t
h
IEE
E
In
tern
a
ti
o
n
a
l
Co
n
f
e
re
n
c
e
o
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.
Evaluation Warning : The document was created with Spire.PDF for Python.