Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
Vol
.
5
,
No
. 3,
J
une
2
0
1
5
,
pp
. 54
8~
56
1
I
S
SN
: 208
8-8
7
0
8
5
48
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Blind Signal Processing Algo
rith
ms bas
e
d on Recursive
Gradien
t
Estim
a
tion
Nam
y
on
g
Ki
m*,
Mi
n
g
oo
K
a
n
g
*
*
* Division of
Electron
i
c, Informa
tion
and
Commnication
Engineering,
Kangwon
National Univer
sity
, Korea
** Division of
I
n
formation and
Telecommnicati
on Engin
eering
,
Hanshin Univers
i
ty
, Korea
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Ja
n 12, 2015
Rev
i
sed
Ap
r
19
, 20
15
Accepte
d
May 4, 2015
Blind algorithm
s
based on
the
Euclid
ean
distan
ce (
E
D) between the outpu
t
distribution fun
c
tion and a set of Di
rac delta functions have a heav
y
computation
a
l burden of
O
(
NM
)
due to so
me do
uble summation operation
s
for the sample size
N
and
M
s
y
mbol points. In this paper, a r
e
cursive
approach
to the
estimation of th
e ED
and its gradient is proposed to reduce
the com
putat
ion
a
l com
p
lexi
t
y
fo
r effici
ent im
ple
m
entation of the
algorithm
.
The ED of th
e algorithm is comprised of
information potentials (I
Ps), and the
IP
s
at the next iterat
i
on can be c
a
lcu
l
at
ed recurs
i
v
el
y
b
a
s
e
d on the curren
t
l
y
obtain
e
d IPs. Ut
iliz
ing th
e r
ecur
s
ivel
y
estim
ated
IPs, the
next
st
ep grad
ien
t
for the weight u
pdate of th
e a
l
g
o
rithm
can be
es
tim
ated r
ecurs
iv
el
y with th
e
pres
ent gradi
e
nt.
W
ith this
recurs
ive approa
ch, th
e com
putation
a
l com
p
lex
i
t
y
of gradien
t
calculation h
a
s only
O
(
N
). The sim
u
lation
results s
how that th
e
proposed gradient estimation
me
thod holds significantly
redu
ced
com
putation
a
l
c
o
m
p
lexit
y
k
eep
ing th
e
s
a
m
e
perform
ance
as
the
block
processing meth
od.
Keyword:
B
l
i
nd si
g
n
al
pr
ocessi
n
g
Co
m
p
u
t
atio
n
a
l co
m
p
lex
ity
Dirac d
e
lta
functio
n
Euclidea
n distance
Recursi
v
e gra
d
ient
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Nam
y
ong Ki
m
,
Di
vi
si
o
n
of
El
e
c
t
r
o
n
i
c
, I
n
fo
rm
at
i
on a
n
d
C
o
m
m
uni
cat
i
on En
gi
nee
r
i
n
g,
Kangwon
Natio
n
a
l Un
iv
ersity,
3
4
6
Jun
g
ang
R
o
ad, Sam
c
h
e
ok city, K
a
ngw
on, 245
-71
1
,
K
o
rea.
Em
ail: nam
y
ong@ka
ngwon.a
c
.kr
1.
INTRODUCTION
The i
n
f
o
rm
ation
-
t
h
e
o
ret
i
c
l
e
arni
ng
(
I
TL
)
m
e
t
hod
ha
s
be
en
de
vel
o
ped
base
d
on
a
co
m
b
i
n
at
i
on
o
f
p
r
ob
ab
ility d
e
n
s
ity fun
c
tion
s
(PDFs) and
a pro
c
edu
r
e to
co
m
p
u
t
e in
formatio
n
po
ten
t
i
a
l (IP) t
h
at is
d
e
fi
n
e
d
base
d
on t
h
e
c
once
p
t
that
dat
a
sam
p
les inte
ract
with each
othe
r a
s
a
pair of physical
pa
rticles in a
pot
e
ntial
field [1].
T
h
e
construction
of P
D
Fs is
done by the
ke
rne
l
density estim
ation
m
e
thod e
m
ploying Ga
ussia
n
k
e
rn
el [2
].
The IT
L
m
e
t
hod
has y
i
el
ded
m
a
ny
perf
orm
a
nce cri
t
e
ri
a s
u
ch as
K
u
l
l
b
ac
k-Lei
b
l
e
r (
K
L
)
di
ver
g
e
n
ce
to
esti
m
a
te
m
u
tu
al in
fo
rm
atio
n
and
Eu
clid
ian
d
i
stan
ce
(E
D)
bet
w
ee
n t
w
o P
D
Fs t
o
m
e
asure t
h
eir similarity
[3]
,
[
4
]
.
T
h
r
o
u
gh m
i
nim
i
zi
ng t
h
e KL
di
ve
r
g
ence
o
r
t
h
e
ED, al
go
ri
t
h
m
s
fo
r s
upe
rvi
s
e
d
t
r
ai
ni
ng
ha
v
e
been
devel
ope
d
f
o
r
cl
assi
fi
cat
i
on i
n
bi
om
edi
cal
p
r
o
b
l
e
m
s
[4]
,
[
5
]
.
In
bl
i
n
d si
g
n
al
pr
ocessi
ng
wh
ere t
r
ai
ni
n
g
dat
a
set
s
are n
o
t
avai
l
a
bl
e t
h
e P
D
F
of
desi
re
d
si
gnal
nee
d
s
be
ge
nerat
e
d
wi
t
h
o
u
t
k
n
o
w
i
n
g
t
h
e
exact
d
e
si
red
si
g
n
al
.
For
c
o
m
m
uni
cat
i
on a
p
pl
i
cat
ions
whe
r
e t
h
e
sym
bol
poi
nts to
be tra
n
sm
itted are identica
lly and inde
pe
nde
ntly dist
ributed, the
desire
d PDF c
a
n be
re
placed
with
a
set
of Di
rac
-
de
l
t
a
funct
i
o
ns [
6
]
.
The bl
i
n
d al
go
ri
t
h
m
devel
o
ped
base
d o
n
t
h
e ED cri
t
e
ri
o
n
bet
w
ee
n t
h
e
PDF
of
out
put sam
p
les and
Dirac-delta func
tions
in place of
the desire
d PD
F ha
s shown superi
or lea
r
ni
ng
per
f
o
r
m
a
nce i
n
va
ri
o
u
s e
nvi
ro
nm
ent
s
.
Howev
e
r, in
the ED esti
m
a
tio
n
of th
e b
lind
alg
o
rith
m
,
there is a co
m
putational problem
that hinders
th
e efficien
t im
p
l
e
m
en
tatio
n
o
f
th
e algo
rit
h
m
.
Th
e
m
e
th
o
d
requ
ires
h
e
av
y co
m
p
u
t
atio
n
a
l co
m
p
lex
ity d
u
e
to
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
54
8 – 5
6
1
54
9
double summ
a
tion ope
r
ations at each
iteration tim
e. In our initia
l work [7], a rec
u
rsi
v
e
approac
h
to t
h
e ED
est
i
m
a
ti
on
has
been
i
n
t
r
o
duce
d
as
an
ef
fi
ci
en
t
m
e
t
hod
re
duc
i
ng t
h
e c
o
m
put
at
i
onal
c
o
m
p
l
e
xi
t
y
.
The E
D
est
i
m
at
i
on ca
n be
use
d
t
o
c
h
ec
k
ho
w
wel
l
t
h
e
out
pu
t
PDF m
a
t
c
hes t
h
e de
si
red
P
D
F.
That
i
s
,
th
e esti
m
a
ted
ED
can
b
e
u
s
ed
as an
in
d
i
cat
o
r
of
th
e conver
g
en
ce state o
f
th
e e
m
p
l
o
y
ed alg
o
r
ith
m
.
Th
er
efo
r
e
t
h
e rec
u
rsi
v
e
e
s
t
i
m
a
ti
on
of
E
D
i
n
[
7
]
d
o
es
not
pl
ay
t
h
e
role o
f
weigh
t
adj
u
stm
e
n
t
in
th
e b
lind
al
g
o
rithm
.
Fo
r
t
h
at
pu
r
pose
,
g
r
adi
e
nt
est
i
m
a
ti
on esse
nt
i
a
l
f
o
r t
h
e wei
ght
up
dat
e
p
r
ocess
of t
h
e bl
i
n
d al
go
ri
t
h
m
i
s
needed t
o
b
e
d
ealt
with.
After
d
e
ri
v
a
tiv
es
o
f
th
e ED are tak
e
n, the re
l
a
t
i
o
n
s
hi
p
bet
w
ee
n t
h
e c
u
r
r
ent
a
n
d t
h
e
ne
xt
t
i
m
e
com
pone
nt
s p
r
od
uce
d
fr
om
the de
ri
vat
i
o
n i
s
i
nvest
i
g
at
e
d
t
o
fi
g
u
re
out
whet
her
gra
d
i
e
nt
est
i
m
a
ti
on c
a
n be
carried
ou
t recu
rsi
v
ely with
red
u
c
ed
co
m
p
u
t
atio
n
a
l co
m
p
lex
ity. And
th
en
it is ex
p
e
rim
e
n
t
ed
to
prov
e that th
e
pr
o
pose
d
m
e
t
hod
pr
o
duc
es t
h
e sam
e
equal
i
zat
i
on pe
rf
o
r
m
a
nce wi
t
h
t
h
e si
gni
fi
cant
l
y
red
u
ced c
o
m
put
at
i
o
n
s
com
p
ared t
o
t
h
e bl
oc
k
p
r
oces
si
ng
m
e
t
hod i
n
t
r
o
duce
d
i
n
[
6
]
.
Thi
s
pape
r
i
s
o
r
ga
ni
zed
as fol
l
ows.
Sect
i
o
n 2 prese
n
t
s
t
h
e defi
ni
t
i
on of
E
u
cl
i
d
ean
di
st
a
n
ce
bet
w
ee
n
the PDF of the
equalizer
output sam
p
les an
d a set o
f
Dirac
d
e
lta fun
c
tion
s
. Th
e esti
m
a
tio
n
of th
e
ED is
shown
to be able to be carried
out re
cursi
v
ely in Section 3,
and
the recu
rsiv
e calcu
latio
n
o
f
th
e
g
r
ad
ien
t
of th
e ED is
p
r
op
o
s
ed
for
th
e w
e
i
g
h
t
update in
Sectio
n
4
.
Section
5
rep
o
r
t
s sim
u
lati
o
n
r
e
su
lts an
d d
i
scu
ssi
o
n
s. Fin
a
lly
,
concl
udi
ng
re
m
a
rks a
r
e
pres
ent
e
d i
n
Sect
i
o
n6
.
2.
ED BETWEEN THE E
Q
UALIZ
E
R OUTPUT AND
A SET OF DIRAC
DELTA FUNCTIONS
The si
m
i
l
a
ri
t
y
m
easure bet
w
een
t
w
o
di
st
ri
but
i
o
n
f
u
n
c
t
i
o
ns o
f
t
h
e
desi
red
sy
m
bol
s
a
n
d
e
qual
i
zer
out
put
sam
p
l
e
s can be est
i
m
at
ed t
h
r
o
ug
h the ED calculation. The
ED
be
tween
D
f
the PDF of t
h
e desi
re
d
sym
bol
an
d
Y
f
t
h
e PD
F
of t
h
e e
qual
i
zer
o
u
t
p
ut
i
s
de
fi
ne
d as
d
f
f
f
f
ED
Y
D
Y
D
2
)]
(
)
(
[
)
,
(
d
f
f
d
f
d
f
Y
D
Y
D
)
(
)
(
2
)
(
)
(
2
2
(1)
The
o
u
t
p
ut
PD
F can
be
co
nst
r
uct
e
d
by
t
h
e
ker
n
el
densi
t
y
est
i
m
a
ti
on m
e
tho
d
fo
r
o
u
t
p
ut
sam
p
l
e
s at
ti
m
e
k
1
1
,...,
,
N
k
k
k
y
y
y
with
th
e
sam
p
le size
N
[2
].
k
N
k
i
i
Y
y
y
N
y
f
1
2
2
]
2
)
(
exp[
2
1
1
)
(
(2
)
The
PD
F
o
f
t
h
e
desi
re
d
sy
m
bol
s can
be
exp
r
esse
d
as Dirac
delta functions when
t
h
e
M
sy
m
bol
poi
nt
s
M
s
s
s
,...,
,
2
1
are equa
lly likely [6].
...
)
(
)
(
[
1
)
(
2
1
s
d
s
d
M
d
f
D
)]
(
...
)
(
M
m
s
d
s
d
(3
)
Sub
s
titu
tin
g (2) an
d (3
) in
to (1
), each
term
of (1)
b
eco
m
e
s
d
f
D
)
(
2
M
1
(4
)
d
f
Y
)
(
2
k
N
k
i
k
N
k
j
i
j
y
y
N
11
2
2
2
]
4
)
(
exp[
2
1
1
(5
)
d
f
f
Y
D
)
(
)
(
M
m
m
Y
s
f
M
1
)
(
1
M
m
k
N
k
i
i
m
y
s
N
M
11
2
2
]
2
)
(
exp[
2
1
1
1
(6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Blin
d
S
i
gna
l
Pro
cessi
n
g
Algorith
ms ba
sed
on
Recu
rsive Gra
d
i
en
t
Estima
tio
n
(Nam
yo
ng
Kim
)
55
0
For convenie
nce, th
e e
q
uat
i
o
n
(
5
)
an
d
(
6
) a
r
e
defi
ne
d as
k
A
and
k
B
, respectively,
t
h
en
k
Y
D
f
f
ED
)
,
(
at
t
i
m
e
k bec
o
m
e
s
k
k
k
Y
D
B
A
M
f
f
ED
1
)
,
(
(7
)
It is noticeable
that
k
A
in (5
) an
d
k
B
in
(6) are calcu
lated
th
rou
gh
do
ubl
e s
u
m
m
a
t
i
on o
p
erat
i
o
n
s
so
t
h
at
t
h
i
s
bl
oc
k pr
ocessi
ng
m
e
tho
d
dem
a
nds h
eavy
com
put
at
i
onal
b
u
r
d
e
n
.
One of the basic ideas of t
h
e
ITL m
e
th
od
is th
at d
a
ta sa
m
p
les can
b
e
con
s
id
ered as p
h
y
sical
p
a
r
ticles in
physics. Th
at is, th
e sam
p
le v
a
lues
i
y
and
j
y
in
(5
) are con
s
id
ered as in
fo
rm
atio
n
p
a
rticles p
l
aced
at th
e lo
catio
ns o
f
i
y
an
j
y
.
The
Gaus
si
an ker
n
el
]
4
)
(
exp[
2
1
2
2
i
j
y
y
pr
od
uces a
n
ex
p
one
nt
i
a
l
decay with the
distance
bet
w
een the
pa
rticles. T
h
is leads
us to consi
d
er
the Ga
ussia
n
kernel as
a
pote
n
tial
fi
el
d i
n
d
u
ci
n
g
i
n
t
e
ract
i
o
n a
m
ong t
h
e
i
n
f
o
rm
ati
on
part
i
c
l
e
s. Th
en
k
N
k
j
i
j
y
y
1
2
2
]
4
)
(
exp[
2
1
is
co
rresp
ond
ing to
th
e su
m o
f
in
teraction
s
on
th
e i th
p
a
rticle, an
d
k
N
k
i
k
N
k
j
i
j
y
y
N
11
2
2
2
]
4
)
(
exp[
2
1
1
in
(5) is th
e su
m
o
f
all p
a
irs o
f
in
teracti
o
ns. Th
is ov
erall
p
o
t
en
tial en
ergy is referred
t
o
as in
form
at
io
n
p
o
t
en
tial [1
].
Accord
ing
to th
e co
ncep
t
o
f
i
n
fo
rm
atio
n
p
o
t
en
tial,
the
ED
(7) ca
n
be
re
garde
d
as
a c
o
m
b
ination of
in
fo
rm
atio
n
poten
tials;
M
1
cau
sed
b
y
th
e p
a
i
r
s of tran
sm
itte
d
sym
b
o
l
po
ints,
k
A
by
t
h
e
pai
r
s
of
o
u
t
p
ut
sam
p
les, and
k
B
by
t
h
e
pai
r
s
of
sym
bol
p
o
i
n
t
s
and
o
u
t
p
ut
sa
m
p
l
e
s.
W
h
e
n
a ne
w
out
put
s
a
m
p
l
e
1
k
y
com
e
s
in
to
th
e co
m
b
in
ed po
ten
tial field
at ti
m
e
k
+
1
,
1
k
A
and
1
k
B
of
1
)
,
(
k
Y
D
f
f
ED
are r
e
newe
d by
c
o
m
put
i
ng
i
n
t
e
ract
i
o
n
s
a
m
ong al
l
corre
spo
n
d
i
n
g sam
p
l
e
pai
r
s.
We ca
n o
b
se
rve t
h
at
t
h
e ol
d
o
u
t
p
ut
sam
p
l
e
1
N
k
y
leaves
th
e field
wh
ile th
e
n
e
w sam
p
le
1
k
y
co
m
e
s in
to
it. Th
is i
n
d
i
cat
es th
at t
h
e i
n
teractio
n
s
b
e
tween
1
N
k
y
and
th
e o
t
h
e
r samp
les are d
i
scard
e
d
wh
ile th
e n
e
w in
teraction
s
b
e
tween
1
k
y
and t
h
e ot
he
rs ar
e adde
d t
o
t
h
e
com
b
ined pote
ntial field. It is
noticea
ble that each ne
w inform
ation pote
ntial at time k+1,
1
k
A
and
1
k
B
m
i
ght
be cal
cul
a
t
e
d just
by
a
ddi
ng
new i
n
t
e
ract
i
ons
rel
a
t
e
d wi
t
h
1
k
y
to
th
e
cu
rren
t informatio
n
po
ten
tials
k
A
and
k
B
, an
d s
u
b
t
ract
i
ng
ol
d
i
n
t
e
ract
i
ons
rel
a
t
e
d
wi
t
h
1
N
k
y
fr
om
the c
u
r
r
ent
i
n
f
o
rm
ati
on
pot
e
n
t
i
a
l
s
.
Thi
s
p
o
i
n
t
s
o
u
t
t
h
at
a si
m
p
l
e
r m
e
t
hod
fo
r
)
,
(
Y
D
f
f
ED
cal
cul
a
t
i
on ca
n
be
po
ssi
bl
e t
h
a
n
t
h
e bl
oc
k p
r
oce
ssi
ng
m
e
t
hod
o
f
car
r
y
i
ng
out
d
o
u
b
l
e
sum
m
ati
on
o
p
erat
i
o
ns
f
o
r
(
5
)
an
d
(6
) at
ea
ch i
t
e
rat
i
o
n t
i
m
e.
3.
RECURSIVE ED
ESTIMATION
During
th
e time in
terv
al
N
k
1
, only k output sa
m
p
les are availa
ble at time k. The
r
efore two
cases will b
e
dealt with
, th
at i
s
, th
e
I
k
A
and
I
k
B
are
for th
e in
itial state
N
k
1
, and
S
k
A
and
S
k
B
are for the
steady state
N
k
, a
s
f
o
llows
.
k
i
k
j
i
j
I
k
y
y
k
A
11
2
2
2
]
4
)
(
exp[
2
1
1
(8
)
k
i
i
N
j
i
m
I
k
y
s
kN
B
1
2
2
]
2
)
(
exp[
2
1
2
(9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
54
8 – 5
6
1
55
1
k
N
k
i
k
N
k
j
i
j
S
k
y
y
N
A
11
2
2
2
]
4
)
(
exp[
2
1
1
(1
0)
M
m
k
N
k
i
i
m
S
k
y
s
N
M
B
11
2
2
]
2
)
(
exp[
2
1
1
1
(1
1)
In
th
e i
n
itial
state
N
k
1
, th
e term
s
o
f
1
)
,
(
k
Y
D
f
f
ED
,
I
k
A
1
and
I
k
B
1
can be estimated
recursively as
1
1
1
1
2
2
2
1
]
4
)
(
exp[
2
1
)
1
(
1
k
i
k
j
i
j
I
k
y
y
k
A
k
i
k
j
i
j
y
y
k
k
k
1
1
1
2
2
2
2
2
]
4
)
(
exp[
2
1
)
1
(
1
1
2
2
1
]
4
)
(
exp[
2
1
k
j
k
j
y
y
k
i
k
j
i
j
y
y
k
k
k
11
2
2
2
2
2
]
4
)
(
exp[
2
1
)
1
(
k
i
i
k
y
y
1
2
2
1
]
4
)
(
exp[
2
1
1
1
2
2
1
2
2
2
]
4
)
(
exp[
2
1
)
1
(
k
j
k
j
y
y
k
k
k
k
i
i
k
I
k
y
y
k
A
k
k
1
2
2
1
2
2
2
]
4
)
(
exp[
2
1
)
1
(
1
)
1
(
k
j
k
j
y
y
k
1
2
2
1
2
]
4
)
(
exp[
2
1
)
1
(
1
]
4
)
(
exp[
2
1
2
2
1
1
k
k
y
y
(1
2)
k
i
i
k
I
k
I
k
y
y
k
A
k
k
A
1
2
2
1
2
2
2
1
]
4
)
(
exp[
2
1
)
1
(
2
)
1
(
2
1
)
1
(
1
2
k
(1
3)
wh
ere t
h
e eq
u
a
lity
]
4
)
(
exp[
]
4
)
(
exp[
2
2
1
2
2
1
k
i
i
k
y
y
y
y
is
u
tilized
in
(12
)
fo
r (13).
Si
m
ilarly,
1
1
2
2
1
]
2
)
(
exp[
2
1
)
1
(
2
k
i
i
M
m
i
m
I
k
y
s
M
k
B
k
i
i
M
m
i
m
y
s
kM
k
k
1
2
2
]
2
)
(
exp[
2
1
)
1
(
2
M
m
k
m
y
s
M
k
1
2
2
1
]
2
)
(
exp[
2
1
)
1
(
2
M
m
k
m
I
k
y
s
M
k
B
k
k
1
2
2
1
]
2
)
(
exp[
2
1
)
1
(
2
)
1
(
(1
4)
The eq
uat
i
o
ns
(1
3) a
nd
(1
4
)
sho
w
t
h
at
1
)
,
(
k
Y
D
f
f
ED
in
th
e in
itial stat
e can
b
e
calculated
with
1
k
y
and
the c
u
rre
nt
ED
term
s
I
k
A
and
I
k
B
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Blin
d
S
i
gna
l
Pro
cessi
n
g
Algorith
ms ba
sed
on
Recu
rsive Gra
d
i
en
t
Estima
tio
n
(Nam
yo
ng
Kim
)
55
2
In the steady state, each com
pone
nt
S
k
A
1
and
S
k
B
1
of
1
)
,
(
k
Y
D
f
f
ED
can be di
vi
de
d i
n
t
o
t
h
e
1
k
y
related
term
s, th
e
1
N
k
y
related
term
s an
d
th
e
remain
in
g.
1
2
1
2
2
2
2
1
]
4
)
(
exp[
2
1
1
k
N
k
i
k
N
k
j
i
j
S
k
y
y
N
A
k
N
k
i
k
N
k
j
i
j
y
y
N
11
2
2
2
]
4
)
(
exp[
2
1
1
]
4
)
(
exp[
2
1
2
2
1
k
i
y
y
]
4
)
(
exp[
2
1
2
2
1
N
k
i
y
y
k
N
k
j
j
k
y
y
N
1
2
2
1
2
]
4
)
(
exp[
2
1
1
]
4
)
(
exp[
2
1
2
2
1
1
k
k
y
y
]
4
)
(
exp[
2
1
2
2
1
1
N
k
k
y
y
k
N
k
j
j
N
k
y
y
N
1
2
2
1
2
]
4
)
(
exp[
2
1
1
]
4
)
(
exp[
2
1
2
2
1
1
k
N
k
y
y
]
4
)
(
exp[
2
1
2
2
1
1
N
k
N
k
y
y
(1
5)
Utilizin
g
th
e eq
u
ity
]
4
)
(
exp[
]
4
)
(
exp[
2
2
1
1
2
2
1
1
k
N
k
N
k
k
y
y
y
y
in
(
15)
,
S
k
A
1
can
b
e
rewritten
as
k
N
k
j
k
i
S
k
S
k
y
y
G
N
A
A
1
1
2
2
1
)
(
2
k
N
k
j
N
k
i
y
y
N
1
2
2
1
2
]
4
)
(
exp[
2
1
2
2
1
2
]
4
)
(
exp[
2
1
2
2
2
2
1
1
2
N
y
y
N
N
k
k
(1
6)
Si
m
ilarly,
1
21
2
2
1
]
2
)
(
exp[
2
1
2
k
N
k
i
M
m
i
m
S
k
y
s
NM
B
k
N
k
i
M
m
i
m
y
s
NM
11
2
2
]
2
)
(
exp[
2
1
2
M
m
k
m
y
s
NM
1
2
2
1
]
2
)
(
exp[
2
1
2
M
m
N
k
m
y
s
NM
1
2
2
1
]
2
)
(
exp[
2
1
2
M
m
k
m
S
k
y
s
NM
B
1
2
2
1
]
2
)
(
exp[
2
1
2
]
2
)
(
exp[
2
1
2
2
1
N
k
m
y
s
(1
7)
The e
quat
i
o
ns
(1
6) a
n
d (
1
7)
pr
o
v
es t
h
at
1
)
,
(
k
Y
D
f
f
ED
in the steady state can
be calculated wi
t
h
in
teractio
ns with
1
k
y
and
1
N
k
y
, ba
sed
o
n
t
h
e
cu
rre
nt
i
n
fo
rm
ati
on
pot
e
n
t
i
a
l
s
S
k
A
and
S
k
B
[7]
.
In s
u
m
m
ary
,
the eq
uat
i
o
ns (
1
3
)
,
(1
4)
, (
1
6)
and
(1
7
)
are a
recu
rsi
v
e
vers
i
on
of t
h
e
bl
oc
k-
pr
ocess
e
d
calculation of
1
)
,
(
k
Y
D
f
f
ED
. Th
at is, informatio
n
po
ten
tials in
th
e Eu
cl
id
ean
d
i
stan
ce can
b
e
estim
a
t
ed
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
54
8 – 5
6
1
55
3
recu
rsively
.
Fu
rtherm
ore,
the
com
put
at
i
onal
com
p
l
e
xi
t
y
of
i
t
s
recu
rsi
v
e
versi
o
n
has
o
n
l
y
)
(
N
O
wh
ich
is
com
p
ared a
p
pa
rently with
)
(
NM
O
of
t
h
e bl
oc
k-
pr
oce
ssed ED.
4.
BLIND
ALG
O
RITH
MS
B
A
SED
O
N
RE
CU
RSI
V
E
G
R
A
D
IE
NT O
F
ED
The output
k
y
at tim
e
k
of linear
equalizer st
ruc
t
ures s
u
ch as
a tappe
d delay
line (TDL
) with
L
weigh
t
s, th
e inp
u
t
v
ector
k
X
T
L
k
k
k
x
x
x
]
,...,
,
[
1
1
and
k
W
T
k
L
k
k
w
w
w
]
,...,
,
[
,
1
,
1
,
0
can be
e
x
pres
sed
as
k
T
k
k
y
X
W
. B
l
i
nd al
go
ri
t
h
m
s
searchi
n
g f
o
r i
t
s
opt
i
m
u
m
wei
ght
vect
or ar
e deri
ve
d f
r
o
m
t
h
e
m
i
nim
i
zat
i
on pr
ocess
o
f
per
f
o
r
m
a
nce
cri
t
e
ria, for
whic
h the E
u
clidea
n
distance
)
,
(
Y
D
f
f
ED
in (7
)
is
em
pl
oy
ed i
n
t
h
i
s
pa
per.
T
h
at
i
s
, t
h
e
m
i
nim
i
zat
i
on
of
k
k
B
A
M
1
with
resp
ect to th
e equ
a
lizer
weigh
t
s can
be ca
rried out
recursi
v
ely by
using the
gra
d
ie
nt of
t
h
e criteri
a and the
steepest desce
n
t m
e
thod a
s
W
W
W
)
(
1
k
k
k
k
B
A
(1
8)
whe
r
e
M
1
i
s
o
m
i
t
t
e
d beca
use i
t
’
s not
a fu
nct
i
o
n of wei
ght
an
d
is the step-size for c
o
nverge
nc
e
cont
rol
.
Wh
en
th
e grad
ien
t
at ti
me
k
W
W
W
k
k
k
k
B
A
B
A
)
(
i
s
cal
cul
a
t
e
d by
t
h
e bl
oc
k-
pr
ocessi
n
g
m
e
t
hod
di
rect
l
y
on
(
5
)
an
d
(
6
), i
t
i
s
obt
ai
ned
as
k
N
k
i
k
N
k
j
i
j
k
y
y
N
A
,
11
2
2
)
(
2
1
W
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
(1
9)
k
N
k
i
M
m
i
m
k
y
s
NM
B
,
11
2
)
(
2
W
i
i
m
y
s
X
]
2
)
(
exp[
2
1
2
2
(2
0)
The weight update equatio
n (18
)
w
ith gr
adients (19) and (2
0) ha
s b
een introduced in the wor
k
[6]. In this section, a r
ecursive version
of t
h
e algorithm is pro
p
o
s
ed
b
y
using a similar appr
oach that was
used in the rec
u
rsive ED esti
mation intr
od
uced in the previous section.
It can
be
noticed that t
h
e gra
d
ient
s
(1
9)
an
d (
2
0) a
r
e
desc
ri
be
d f
o
r
the st
eady state so t
h
at they are
corres
ponding to
W
S
k
A
and
W
S
k
B
, resp
ectiv
ely. Th
e g
r
ad
ien
t
s in
th
e i
n
itial state
W
I
k
A
and
W
I
k
B
are written
as
k
i
k
j
i
j
I
k
y
y
k
A
11
2
2
)
(
2
1
W
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
(2
1)
k
i
M
m
i
m
I
k
y
s
kM
B
11
2
)
(
2
W
i
i
m
y
s
X
]
2
)
(
exp[
2
1
2
2
(2
2)
The
n
1
1
1
1
2
2
1
)
(
)
1
(
2
1
k
i
k
j
i
j
I
k
y
y
k
A
W
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
(2
3)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Blin
d
S
i
gna
l
Pro
cessi
n
g
Algorith
ms ba
sed
on
Recu
rsive Gra
d
i
en
t
Estima
tio
n
(Nam
yo
ng
Kim
)
55
4
1
11
2
1
)
(
)
1
(
2
k
i
M
m
i
m
I
k
y
s
M
k
B
W
i
i
m
y
s
X
]
2
)
(
exp[
2
1
2
2
(2
4)
Using
th
e similar ap
pro
a
ch
as in
(1
2), th
e g
r
ad
ien
t
s
W
I
k
A
1
and
W
I
k
B
1
can be
di
vi
d
e
d i
n
t
o
t
h
e
term
s related
with
1
k
y
an
d th
e
remain
in
g
.
k
i
k
j
i
j
I
k
y
y
k
k
k
A
1
1
1
2
2
2
2
1
)
(
2
)
1
(
W
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
1
1
1
2
2
2
2
)
(
2
)
1
(
k
j
k
j
y
y
k
k
k
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
k
k
j
y
y
X
X
k
i
k
j
i
j
y
y
k
k
k
11
2
2
2
2
)
(
2
)
1
(
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
k
i
i
k
y
y
k
k
k
1
1
2
2
2
2
)
(
2
)
1
(
)
(
]
4
)
(
exp[
2
1
1
2
2
1
k
i
i
k
y
y
X
X
k
j
k
j
y
y
k
k
k
1
1
2
2
2
2
)
(
2
)
1
(
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
k
k
j
y
y
X
X
)
(
2
)
1
(
1
1
2
2
2
2
k
k
y
y
k
k
k
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
k
k
k
k
y
y
X
X
W
I
k
A
k
k
2
2
)
1
(
k
i
i
k
y
y
k
1
1
2
2
)
(
)
1
(
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
k
i
i
k
y
y
X
X
(2
5)
The e
q
uat
i
o
n
(
2
5
)
i
ndi
cat
es t
h
at
t
h
e
ne
xt
gr
adi
e
nt
ca
n
di
vi
ded
i
n
t
o
t
h
e
t
e
rm
of t
h
e
cu
rr
ent
g
r
a
d
i
e
n
t
and
t
h
e
ot
he
r t
e
rm
of
1
k
y
.
On
t
h
e ot
he
r h
a
nd
,
W
I
k
B
1
k
i
M
m
i
m
y
s
M
k
11
2
)
(
)
1
(
2
i
i
m
y
s
X
]
2
)
(
exp[
2
1
2
2
1
2
2
1
1
1
]
2
)
(
exp[
2
1
)
(
k
k
m
M
m
k
m
y
s
y
s
X
W
I
k
B
Mk
M
k
2
)
1
(
2
2
2
1
2
2
1
1
1
]
2
)
(
exp[
2
1
)
(
k
k
m
M
m
k
m
y
s
y
s
X
M
m
k
m
I
k
y
s
M
k
B
k
k
1
1
2
)
(
)
1
(
2
1
W
1
2
2
1
]
2
)
(
exp[
2
1
k
k
m
y
s
X
(2
6)
Th
rou
g
h
(2
5) an
d (26
)
, th
e i
n
itial state g
r
ad
ien
t
s
W
I
k
A
1
and
W
I
k
B
1
can
be obtained rec
u
rsively.
Sim
ilarly, the steady
state
gradients
W
S
k
A
1
and
W
S
k
B
1
can be inve
stigated whet
her they can be
sep
a
rated
in
t
o
t
h
e term
s related
with
1
k
y
an
d th
e
o
n
e
s related wi
th
1
N
k
y
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
54
8 – 5
6
1
55
5
Fr
o
m
(
19)
,
1
,
2
1
2
2
2
1
)
(
2
1
k
N
k
i
k
N
k
j
i
j
S
k
y
y
N
A
W
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
k
N
k
i
k
N
k
j
i
j
y
y
N
,
1
1
2
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
1
2
1
2
2
)
(
2
1
k
N
k
j
k
j
y
y
N
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
k
k
j
y
y
X
X
1
2
1
2
2
)
(
2
1
k
N
k
j
N
k
j
y
y
N
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
N
k
N
k
j
y
y
X
X
k
N
k
i
k
N
k
j
i
j
y
y
N
,
11
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
2
2
j
i
i
j
y
y
X
X
)
(
]
4
)
(
exp[
2
1
)
(
1
2
2
1
1
k
i
i
k
i
k
y
y
y
y
X
X
)
(
]
4
)
(
exp[
2
1
)
(
1
2
2
1
1
N
k
i
i
N
k
i
N
k
y
y
y
y
X
X
k
N
k
j
k
j
y
y
N
1
1
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
k
k
j
y
y
X
X
)
(
2
1
1
1
2
2
k
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
k
k
k
k
y
y
X
X
)
(
2
1
1
1
2
2
k
N
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
N
k
k
k
N
k
y
y
X
X
k
N
k
j
N
k
j
y
y
N
1
1
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
N
k
N
k
j
y
y
X
X
)
(
2
1
1
1
2
2
N
k
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
k
N
k
N
k
k
y
y
X
X
)
(
2
1
1
1
2
2
N
k
N
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
N
k
N
k
N
k
N
k
y
y
X
X
W
S
k
A
k
N
k
j
i
k
y
y
N
1
1
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
k
i
i
k
y
y
X
X
k
N
k
j
i
N
k
y
y
N
1
1
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
N
k
i
i
N
k
y
y
X
X
k
N
k
j
k
j
y
y
N
1
1
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
k
k
j
y
y
X
X
)
(
2
1
1
1
2
2
k
N
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
N
k
k
k
N
k
y
y
X
X
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Blin
d
S
i
gna
l
Pro
cessi
n
g
Algorith
ms ba
sed
on
Recu
rsive Gra
d
i
en
t
Estima
tio
n
(Nam
yo
ng
Kim
)
55
6
k
N
k
j
N
k
j
y
y
N
1
1
2
2
)
(
2
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
j
N
k
N
k
j
y
y
X
X
)
(
2
1
1
1
2
2
N
k
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
k
N
k
N
k
k
y
y
X
X
W
S
k
A
k
N
k
j
i
k
y
y
N
1
1
2
2
)
(
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
k
i
i
k
y
y
X
X
k
N
k
j
i
N
k
y
y
N
1
1
2
2
)
(
1
)
(
]
4
)
(
exp[
2
1
1
2
2
1
N
k
i
i
N
k
y
y
X
X
)
(
1
1
1
2
2
k
N
k
y
y
N
)
(
]
4
)
(
exp[
2
1
1
1
2
2
1
1
N
k
k
k
N
k
y
y
X
X
(2
7)
From
(2
7)
,
we
can
o
b
ser
v
e t
h
at
t
h
e
ne
xt
g
r
adi
e
nt
o
f
i
n
fo
rm
ati
on
pot
e
n
t
i
a
l
W
S
k
A
1
can be
ac
quired
recursively from the curre
nt gradie
nt
W
S
k
A
and
ot
her t
e
rm
s rel
a
t
e
d wi
t
h
1
k
y
and
1
N
k
y
. S
i
m
i
larly,
W
S
k
B
1
is analyzed t
o
s
ee if it can be
obtain
ed
rec
u
rsively
as descri
bed below.
1
21
2
1
)
(
2
k
N
k
i
M
m
i
m
S
k
y
s
NM
B
W
i
i
m
y
s
X
]
2
)
(
exp[
2
1
2
2
k
N
k
i
M
m
i
m
y
s
NM
11
2
)
(
2
i
i
m
y
s
X
]
2
)
(
exp[
2
1
2
2
1
2
2
1
1
1
]
2
)
(
exp[
2
1
)
(
k
k
m
M
m
k
m
y
s
y
s
X
1
2
2
1
1
1
]
2
)
(
exp[
2
1
)
(
N
k
N
k
m
M
m
N
k
m
y
s
y
s
X
W
S
k
B
M
m
k
k
m
k
m
y
s
y
s
NM
1
1
2
2
1
1
2
]
2
)
(
exp[
2
1
)
(
2
X
1
2
2
1
1
]
2
)
(
exp[
2
1
)
(
N
k
N
k
m
N
k
m
y
s
y
s
X
(2
8)
I
n
su
mm
ar
y, t
h
e weigh
t
up
date in
(1
8)
can b
e
car
r
i
ed ou
t
w
ith
th
e gr
ad
i
e
n
t
W
)
,
(
Y
D
f
f
ED
th
at is
recu
rsi
v
el
y
obt
ai
ned
t
h
r
o
ug
h (2
5)
, (2
6)
, (2
7)
an
d (2
8)
,
as
o
p
p
o
se
d
t
o
t
h
e
bl
oc
k-
pr
ocesse
d gra
d
i
e
nt
s
(
1
9
)
a
n
d
(2
0)
. Al
s
o
t
h
e
recu
rsi
v
e e
qua
t
i
ons sh
o
w
t
h
a
t
t
h
e recur
s
i
v
e
versi
on
of t
h
e
gra
d
i
e
nt
has s
i
gni
fi
ca
nt
l
y
reduc
e
d
co
m
p
u
t
atio
n
a
l co
m
p
lex
ity
o
f
)
(
N
O
whi
l
e
t
h
e
bl
oc
k-
pr
ocesse
d
E
D
has
)
(
NM
O
.
In t
h
e following section, it wi
ll be investigated
if the
proposed E
D
estim
a
tion m
e
thod in whic
h the
in
fo
rm
atio
n
p
o
ten
tials as co
mp
on
en
ts of th
e
ED are cal
cu
lated
recu
rsi
v
ely yield
s
th
e same esti
m
a
t
i
o
n
resu
lts
as the bloc
k-processe
d ED est
i
m
a
tion
m
e
thod does.
And then it will be shown
that the
gradient of the E
D
for
t
h
e wei
g
ht
u
p
d
at
e has e
q
ual
resul
t
s
f
o
r t
h
e t
w
o cases
o
f
t
h
e
bl
oc
k-
pr
ocessi
n
g
m
e
t
hod a
n
d t
h
e
re
cursi
v
e
calculation m
e
thod.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
IJEC
E V
o
l
.
5, No
. 3,
J
u
ne 2
0
1
5
:
54
8 – 5
6
1
55
7
5.
R
E
SU
LTS AN
D ANA
LY
SIS
In
th
is section
,
si
m
u
latio
n
resu
lts in
th
e same en
v
i
ron
m
e
n
t o
f
b
lin
d
equ
a
lizatio
n
u
s
i
n
g
th
e MED2
alg
o
rith
m
as
i
n
[6
] are illu
strated
fo
r th
e t
w
o
ap
pro
a
ch
es o
f
th
e b
l
o
c
k-p
r
o
cessi
n
g
an
d recu
rsi
v
e m
e
t
h
od
s to
est
i
m
a
t
e
t
h
e ED
bei
n
g m
i
nim
i
zed an
d t
h
e
gr
adi
e
nt
of
ED
c
o
m
i
ng t
o
ze
ro
.
Th
e tran
sm
itte
d
sym
b
o
l
po
ints are
}
1
,
3
{
so th
at
th
e
p
r
ob
ab
ility d
e
nsity fun
c
tio
n
(3)
b
ecomes
)]
3
(
)
1
(
)
1
(
)
3
(
[
4
1
)
(
d
d
d
d
d
f
D
. The
cha
n
nel
im
pul
se res
p
ons
e
i
h
is ch
o
s
en
to
be
]}
3
.
3
/
)
2
(
2
cos[
1
{
2
/
1
i
h
i
,
3
,
2
,
1
i
[8
].
Th
e data-
b
lo
ck
size
20
N
, the c
o
nvergence
param
e
ter
005
.
0
, and t
h
e
kerne
l
size is set to
be
6
.
0
.
In Fi
gure 1, t
h
e trace of Euc
lidean
distance for the t
w
o m
e
thods
i
s
depicted
where the bloc
k-
p
r
o
cessin
g
m
e
t
h
od
is
b
y
(4
),
(5
)
and
(6)
,
an
d th
e r
e
cu
rsiv
e
calcu
latio
n
is
b
y
(1
3)
,
(1
4)
,
(1
6)
an
d (1
7)
.
A
s
the
M
E
D2 al
g
o
ri
t
h
m
desi
gne
d t
o
m
i
nim
i
ze t
h
e
ED co
nv
er
ges,
the distance decreases ra
pidl
y with iterations. T
h
e
t
w
o m
e
t
hods
y
i
el
d t
h
e sam
e
di
st
ance est
i
m
a
ti
on p
e
r
f
o
r
m
a
nce sh
owi
n
g n
o
di
ffe
re
n
ce bet
w
ee
n t
h
e t
w
o
m
e
thods
, but we can
observe a slight difference i
n
Figur
e 2 that is focuse
d in th
e i
n
itial part of i
t
eration
)
1
(
N
k
. That is, the two m
e
thods produce diffe
re
nt esti
m
a
tion results but getting close
r
to each othe
r
u
n
til th
e iteratio
n
n
u
m
b
e
r
2
0
, and
i
n
the stead
y state
)
(
N
k
they yield exactly the sa
me estimation
perform
a
nce. We see that the di
fferen
ce
du
ri
n
g
th
e tim
e
in
terv
al
N
k
1
i
s
du
e t
o
t
h
e num
ber of
out
put
sam
p
les b
e
in
g
av
ailab
l
e at time k
.
F
o
r f
a
ir
co
mp
a
r
is
on
,
w
e
n
e
ed to
prese
n
t c
o
nve
rgence
curves
for all
)
11
(
L
wei
ght
g
r
adi
e
nt
s, but
onl
y
t
h
e
gra
d
i
e
nt
of
t
h
e ce
nt
e
r
t
a
p
wei
g
ht
i
s
prese
n
t
e
d
j
u
st
for t
h
e limited
roo
m
o
f
t
h
is
pap
e
r. Sim
ilar resu
lts
are ob
ser
v
ed i
n
Fi
g
u
re 2 t
h
at
sho
w
s t
h
e t
r
ac
e of cent
e
r t
a
p
gra
d
i
e
nt
cal
cul
a
t
e
d by
t
h
e t
w
o m
e
t
hods;
t
h
e
bl
ock
-
p
r
o
cessi
n
g
m
e
th
od
b
y
(1
9)
an
d (2
0)
,
and th
e r
e
cur
s
iv
e calcu
latio
n
by (
2
5
)
, (26
)
, (2
7)
an
d (2
8)
.
A
s
th
e
gra
d
i
e
nt
c
o
nve
rges
by
t
h
e M
E
D
2
al
g
o
r
i
t
h
m
,
fl
uct
u
at
i
o
n
s
o
f
t
h
e
gra
d
i
e
nt
decrease
a
n
d
t
h
e
g
r
adi
e
n
t
val
u
e
conce
n
t
r
at
es a
t
zero a
f
t
e
r t
h
e i
t
e
rat
i
on
nu
m
b
er 40
00
, f
r
o
m
whi
c
h t
h
e
ED c
o
n
v
e
r
ge
s al
so as i
n
F
i
gu
re
1
.
Tho
ugh
a slight d
i
fference is
o
b
s
erv
e
d in
t
h
e in
itial p
a
rt of i
t
eratio
n
)
20
1
(
k
as show
n in
Fi
g
u
r
e
4, th
e two
m
e
t
hods
gi
ve t
h
e sam
e
gra
d
i
e
nt
est
i
m
at
i
on r
e
sul
t
s
i
n
t
h
e st
e
a
dy
st
at
e
)
20
(
k
as ob
serve
d
i
n
E
D
e
s
t
i
m
a
ti
on.
0
2
00
0
400
0
6
0
0
0
8
0
0
0
1
0
000
0.
1
0.
2
0.
3
0.
4
Euc
lid
ea
n d
i
sta
n
c
e
Iter
a
ti
o
n
s (
num
b
e
r
of
s
a
m
p
l
e
s
)
B
l
o
c
k
-
pr
oc
e
s
sing
me
t
h
o
d
Re
c
u
r
s
ive c
a
lc
ul
a
t
io
n
Fi
gu
re
1.
Eucl
i
d
ean
di
st
a
n
ce
f
o
r
t
h
e
bl
oc
k-
processing m
e
thod and t
h
e
recursive
one
Evaluation Warning : The document was created with Spire.PDF for Python.