Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
9,
No.
3,
June
2019,
pp.
1732
1741
ISSN:
2088-8708,
DOI:
10.11591/ijece.v9i3.pp1732-1741
r
1732
Blind
separation
of
complex-v
alued
satellite-AIS
data
f
or
marine
sur
v
eillance:
a
spatial
quadratic
time-fr
equency
domain
appr
oach
Omar
Cherrak
1
,
Hicham
Ghennioui
2
,
Nad
`
ege
Thirion
Mor
eau
3
,
El
Hossein
Abarkan
4
1
LREA,
Institut
sup
´
erieur
du
G
´
enie
Appliqu
´
e,
Maroc
1,2,4
LSSC,
Uni
v
ersit
´
e
Sidi
Mohamed
Ben
Abdellah,
F
acult
´
e
des
Sciences
et
T
echniques,
Maroc
3
Aix-Marseille
Uni
v
ersit
´
e,
CNRS,
France
3
Uni
v
ersit
´
e
de
T
oulon,
France
Article
Inf
o
Article
history:
Recei
v
ed
Jun
30,
2018
Re
vised
No
v
21,
2018
Accepted
Des
15,
2018
K
eyw
ords:
Blind
source
separation
Joint
zero-(block)
diagonalization
Marine
surv
eillance
Matrix
decompositions
Satellite-automatic
identification
system
Spatial
generalized
mean
ambiguity
function
Spatial
time-frequenc
y
based
approach
ABSTRA
CT
In
this
paper
,
the
problem
of
the
blind
separation
of
c
omple
x-v
alued
Satellite-AIS
data
for
marine
surv
eillance
is
addressed.
Due
to
the
specific
properties
of
the
sources
un-
der
consideration:
the
y
are
c
yclo-stationary
signals
with
tw
o
close
c
yclic
frequencies,
we
opt
for
spatial
quadratic
time-frequenc
y
domain
methods.
The
use
of
an
additional
di
v
ersity
,
the
time
delay
,
is
aimed
at
making
it
possible
to
undo
the
mixing
of
signals
at
the
multi-sensor
recei
v
er
.
The
suggested
method
in
v
olv
es
three
main
stages.
First,
the
spatial
generalized
mean
Ambiguity
function
of
the
observ
ations
acros
s
the
array
is
constructed.
Second,
in
the
Ambiguity
plane,
Delay-Doppler
re
gions
of
high
mag-
nitude
are
determined
and
Delay-Doppler
points
of
peak
y
v
alues
are
selected.
Third,
the
mixing
matrix
is
estimated
from
these
Delay-Doppler
re
gions
using
our
proposed
non-unitary
joint
zero-(block)
diagonalization
algorithms
as
to
perform
separation.
Copyright
c
2019
Institute
of
Advanced
Engineering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Omar
Cherrak,
LREA,
Institut
sup
´
erieur
du
G
´
enie
Appliqu
´
e,
279
Bd
Bir
Anzarane,
Casablanca,
Maroc.
omar
.cherrak@ig
a.ac.ma
1.
INTR
ODUCTION
This
paper
concerns
the
spatial
automatic
identification
system
(S-AIS)
dedicated
to
marine
surv
eil-
lance
by
satellite.
It
co
v
ers
a
lar
ger
area
than
the
t
errestrial
automatic
identification
system
[1],
[2].
The
idea
of
switching
to
satellite
monitoring
w
as
introduced
because
of
the
f
ast
and
dynamic
de
v
elopment
of
the
ma-
rine
traf
fic
[3–5].
It
w
as
an
emer
genc
y
to
adopt
a
method
that
operates
a
global
monitoring
with
reliability
,
ef
ficienc
y
and
rob
ustness.
Ho
we
v
er
,
this
generalization
to
space
in
v
olv
es
se
v
eral
phenomena.
Among
these
phenomena,
we
found:
(a)
The
speed
of
the
satellite
mo
v
ement
generates
the
Doppler
ef
fect
which
creates
frequenc
y
of
fsets
at
the
S-AIS
signals
[6],
(b)
The
propag
ation
delay
of
the
signals
and
their
attenuation
due
to
the
satellite
altitude
[7],
(c)
When
a
wide
area
is
co
v
ered
by
the
satellite,
it
certainly
includes
se
v
eral
traditional
AIS
cells.
In
f
act,
the
ti
me
propag
ation
of
signals
transmitted
from
v
essels
to
the
satellite
v
ary
according
to
the
position
J
ournal
homepage:
http://iaescor
e
.com/journals/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1733
of
the
ships
and
the
maximum
co
v
erage
area
of
t
he
satellite
antenna.
Due
to
these
tw
o
problems,
it
mainly
af
fects
the
or
g
anizational
mechanism
of
S-AIS
signals.
It
results
a
collision
data,
as
illustrated
in
the
Figure
1,
issued
by
v
essels
located
in
dif
ferent
AIS
cells
b
ut
recei
v
ed
at
the
antenna
of
the
same
satellite
[8],
[9].
F
or
this
reason,
we
present
ne
w
approaches
to
address
this
problem
where
the
Doppler
ef
fect
and
the
propag
ation
delay
are
also
tak
en
into
consideration.
Figure
1.
Collision
problem:
The
AIS
signals
from
tw
o
dif
ferent
SO-TDMA
cells
recei
v
ed
to
the
satellite
antenna
at
the
same
time.
In
f
act,
to
solv
e
the
collision
problem,
fe
w
w
orks
ha
v
e
focused
on
blind
s
eparation
of
sources
(
BSS
)
methods
[10],
[11].
In
[11],
Zhou
et
al.
present
a
multi-user
recei
v
er
equipped
with
an
array
of
antennas
embedded
in
Lo
w
Orbit
Eart
h
(LEO)
satellite.
The
principle
of
this
recei
v
er
is
to
e
xploit
spatial
multiple
xing
in
a
non-stationary
asynchronous
conte
xt.
Indeed,
the
authors
consider
the
equation
belo
w:
X
=
HG
(
S
)
+
N
;
(1)
where
is
the
Schur
-Hadamard
operator
,
X
=
[
x
1
;
x
2
;
:
:
:
;
x
P
N
]
2
C
M
P
N
,
x
n
=
x
(
nT
s
)
,
1
n
P
N
,
is
the
observ
ation
matrix,
H
=
[
h
1
;
:
:
:
;
h
d
]
2
C
M
d
is
the
matrix
of
antenna
response,
G
=
diag
f
g
1
;
g
2
;
:
:
:
;
g
d
g
2
R
d
d
contains
the
po
wer
of
the
sources,
S
=
[
s
H
1
;
s
H
2
;
:
:
:
;
s
H
d
]
H
2
C
d
P
N
is
the
matrix
of
sources
and
=
0
B
B
B
@
1
'
1
1
:
:
:
'
P
N
1
1
1
'
1
2
:
:
:
'
P
N
1
2
.
.
.
.
.
.
.
.
.
.
.
.
1
'
1
d
:
:
:
'
P
N
1
d
1
C
C
C
A
,
where
'
k
=
e
j
2
f
k
T
s
contains
the
Doppler
frequencies
of
the
sources.
The
principle
of
this
method
is
based
on
joint
diagonalization
(JD)
of
matrices
in
order
to
reconstruct
the
S-AIS
sourc
es
from
separation
matrix
estimation
[12].
Ho
we
v
er
,
because
of
the
v
ery
specific
properties
of
the
S-AIS
signals
(comple
x
and
c
yclo-stationary
with
tw
o
close
c
yclic
frequencies),
we
opt
for
spatial
quadratic
time-frequenc
y
domain
meth-
ods.
Our
aim
is
reshaping
the
collision
problem
into
BSS
problem
more
simpler
than
(1).
W
e
will
sho
w
ho
w
another
type
of
decomposition
matrix
named
joint
zero-diagonalization
(JZD)
of
matrices
set
resulting
from
spatial
quadratic
time-frequenc
y
distrib
utions
allo
ws
the
restitution
of
S-AIS
sources.
2.
TRANSMISSION
SCHEME
2.1.
AIS
Frame
The
AIS
frame
is
a
length
of
256
bits
and
occupies
one
minute.
It
is
di
vided
into
2250
time
s
lots
where
one
slot
equals
26.67
ms
[13].
Its
structure
as
illustrated
in
Figure
2
contains
a
trai
ning
sequence
(TS)
consisting
zero
and
on
e
which
tak
es
24
bits.
The
start
flag
(SF)
and
the
end
flag
(EF)
for
information
tak
es
8
bits.
A
Frame
Check
Sequence
(FCS)
(or
16
bits
Cyclic
Redundanc
y
Code
(CRC))
is
added
to
the
data
information
(168
bits)
in
which
a
zero
is
inserted
after
e
v
ery
fi
v
e
cont
inuous
one.
The
binary
sequence
f
a
k
g
0
k
K
of
the
AIS
frame
tak
es
the
v
alues
f
1
;
+1
g
since
the
NRZI
encoding
is
used.
Moreo
v
er
,
the
modulation
specified
Blind
separ
ation
of
comple
x-valued
satellite-AIS
data
...
(Omar
Cherr
ak)
Evaluation Warning : The document was created with Spire.PDF for Python.
1734
r
ISSN:
2088-8708
by
S-AIS
standard
is
Gaussian
Minimum
Shift
K
e
ying
(GMSK)
[14].
The
encoded
message
is
modulated
and
transmitted
at
9600
bps
on
161.975
MHz
and
162.025
MHz
frequencies
carrier
.
Figure
2.
AIS
Frame.
2.2.
GMSK
modulation
The
resulting
sequence
after
the
bit
st
uf
fing
and
NRZI
coding
procedure
is
modulated
with
GMSK
which
is
a
frequenc
y-shift
k
e
ying
modulation
producing
constant-en
v
elope
and
continuous-phase.
Hence,
the
signal
can
be
written
as
s
g
(
t
)
=
P
+
1
k
=0
a
k
g
(
t
k
T
s
)
,
where
a
k
are
the
transmitted
symboles,
T
s
is
the
symbol
period
and
g
(
t
)
=
q
2
log
2
B
exp
2
log
2
(
B
t
)
2
represents
the
shaping
Gaussian
filter
where
B
is
the
band-
width
of
the
Gaussian
filter
.
The
GMSK
m
odulation
is
described
by
the
bandwidth-time
(BT)
product
where
S-AIS
uses
BT
=
0
:
4
and
T
s
=
1
9600
s
).
Making
the
signal
on
one
of
the
frequencies
carrier
f
c
,
produces
a
signal
of
spectral
characteristic
which
is
adapted
to
the
band-pass
channel
transmission.
The
GMSK
signal
is,
thus,
e
xpressed
as
:
s
(
t
)
=
<f
e
j
(2
f
c
t
+
(
t
))
g
=
I
(
t
)
cos(2
f
c
t
)
Q
(
t
)
sin(2
f
c
t
)
;
where
<fg
is
the
real
part
of
a
comple
x
number
,
(
t
)
=
2
h
P
+
1
k
=0
a
k
g
(
t
k
T
s
)
is
the
instantaneous
phase
of
s
g
(
t
)
where,
in
the
AIS
system,
the
modulation
inde
x
is
theoretically
equal
to
h
=
0
:
5
[15],
I
(
t
)
(resp.
Q
(
t
)
)
modulates
the
frequenc
y
carrier
in
phase
(resp.
in
phase
quadrature).
All
steps
of
the
GMSK
modulati
on
can
be
presented
in
the
Figure
3.
sin
(
t
)
Gaussian
filter
Inte
grator
cos
(
t
)
a
k
s
g
(
t
)
(
t
)
sin(2
f
c
t
)
cos(2
f
c
t
)
I
(
t
)
Q
(
t
)
s
(
t
)
Figure
3.
GMSK
modulator
scheme.
3.
PR
OBLEM
ST
A
TEMENT
:
COLLISION
&
BSS
IN
INST
ANT
ANEOUS
CONTEXT
3.1.
Mathematical
model
of
collision
pr
oblem
The
collision
problem
can
be
simply
e
xpressed
as
follo
ws:
x
(
t
)
=
J
X
j
=1
h
j
s
j
(
t
j
)
e
i
2
f
j
t
+
n
(
t
)
;
(2)
where
x
(
t
)
is
the
recei
v
ed
signal
by
the
satellite,
s
j
is
the
transmitted
signal
by
the
j
th
v
essel,
h
j
,
j
and
f
j
are
respecti
v
ely
the
coef
ficients
of
the
channel,
the
delay
and
the
Doppler
shift
corresponding
to
the
j
th
v
essel
with
J
is
the
number
of
v
essels
and
n
(
t
)
is
an
additi
v
e
stationary
white
Gaussian
noise,
mutually
uncorrelated,
independent
from
the
s
j
,
with
the
v
ariance
2
n
.
3.2.
Reshape
the
collision
pr
oblem
into
BSS
pr
oblem
W
e
sho
w
,
here,
that
(2)
can
be
written
in
BSS
nomenclature
in
which
the
delay
and
the
Doppler
shift
caused
by
the
satellite
speed
are
considered.
Ho
we
v
er
,
before
an
y
reformulation,
we
notice
that
the
mixing
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
3,
June
2019
:
1732
–
1741
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1735
matrix
considered
for
S-AIS
application
is
an
inst
antaneous
mixture
due
to
the
absence
of
obstacles
in
the
ocean.
Thus,
we
set
J
=
n
,
the
collision
problem
can
be
easily
modeled
in
a
BSS
problem
as
follo
ws:
x
(
t
)
=
Hs
(
t
)
+
n
(
t
)
;
(3)
where
H
is
a
(
m
n
)
mixing
matrix,
s
(
t
)
=
[
s
1
(
t
)
;
s
2
(
t
)
;
:
:
:
;
s
n
(
t
)]
T
is
a
(
n
1)
sources
v
ector
with
s
j
(
t
)
=
s
j
(
t
j
)
exp
f
i
2
f
j
t
g
;
8
j
=
1
;
:
:
:
;
n
and
x
(
t
)
=
[
x
1
(
t
)
;
x
2
(
t
)
;
:
:
:
;
x
m
(
t
)]
T
;
n
(
t
)
=
[
n
1
(
t
)
;
n
2
(
t
)
;
:
:
:
;
n
m
(
t
)]
T
are
respecti
v
ely
the
(
m
1)
observ
ations
and
noises
v
ectors.
The
superscript
(
:
)
T
denotes
the
transpose
operator
.
Our
de
v
elopments
are
based
on
the
follo
wing
assumptions:
Assumption
A
:
The
noises
n
j
(
t
)
for
all
j
=
1
;
:
:
:
;
m
are
stationary
,
white,
zero-mean,
mutually
uncorrelated
random
signals
and
independent
from
the
sources
with
v
ariance
2
n
.
Assumption
B
:
F
or
each
s
j
of
the
n
sources,
there
Delay-Doppler
points
of
only
one
source
is
present
in
the
Ambiguity
plane.
Assumption
C
:
The
number
of
sensors
m
and
the
number
of
sources
n
are
both
kno
wn
and
m
n
to
deal
with
an
o
v
er
-determined
model
(the
under
-determined
case
is
outside
of
the
scope
in
this
paper).
4.
PRINCIPLE
OF
THE
PR
OPOSED
METHODS
B
ASED
ON
THE
SP
A
TIAL
GENERALIZED
MEAN
AMBIGUITY
FUNCTION
W
e
sho
w
,
here,
ho
w
the
algorithms
proposed
in
[16],
[17]
adresses
the
problem
of
the
separation
of
instantaneous
mixtures
of
S-AIS
data.
The
principle
of
the
proposed
methods
are
based
on
three
m
ain
steps:
first,
the
SGMAF
of
the
observ
ations
across
the
array
is
constructed.
Second,
in
the
Ambiguity
plane,
Delay-
Doppler
re
gions
of
high
magnitude
are
determined
and
Delay-Doppler
points
of
peak
y
v
alues
are
selected.
Third,
the
mixing
matrix
is
estimated
from
these
Delay-Doppler
re
gions
so
as
to
perform
separation
and
to
undo
the
mixing
of
signals
at
the
multi-sensor
recei
v
er
.
4.1.
The
Spatial
Generalized
Mean
Ambiguity
Function
W
ith
re
g
ard
to
BSS
,
it
has
been
sho
wn
that
spatial
time-frequenc
y
distrib
utions
are
an
ef
fecti
v
e
tool
when
signature
of
the
sources
dif
fer
in
certain
points
of
the
time-frequenc
y
plan
[18].
Ho
we
v
er
,
in
the
c
yclo-
stationary
sources
case,
the
Delay-Doppler
frequenc
y
domain
seems
to
be
a
more
natural
field
for
the
re-
estimation
of
sources
than
the
time-frequenc
y
domain.
As
mentioned
in
[19],
the
approaches
based
on
infor
-
mation
deri
v
ed
from
spatial
Ambiguity
function
(
SAF
)
or
on
SGMAF
should
be
used.
In
f
act,
for
an
y
v
ectorial
comple
x
signal
z
(
t
)
,
the
SGMAF
is
e
xpressed
as
[20–22]:
A
z
(
;
)
=
Z
1
1
r
z
(
t;
)
e
j
2
t
dt
=
E
fh
z
;
s
;
z
ig
;
(4)
where
(
s
;
z
)
is
the
operator
of
elementary
Delay-Doppler
translations
of
z
defined
by
(
s
;
z
)(
t
)
=
z
(
t
)
e
j
2
(
t
)
and
r
z
(
t;
)
=
R
z
t
+
2
;
t
2
=
E
z
t
+
2
z
H
t
2
,
where
R
z
(
t;
)
stands
for
the
correlat
ion
matrix
of
z
(
t
)
,
E
f
:
g
stands
for
the
mathematical
e
xpectation
operator
and
superscript
(
:
)
H
denotes
the
conjug
ate
transpose
operator
.
A
z
(
;
)
characterizes
the
a
v
erage
correlation
of
all
pairs
se
pa-
rated
by
in
time
and
by
in
frequenc
y
[21],
[22].
Notice
that
the
diagonal
terms
of
the
matrix
A
z
(
;
)
are
called
auto-terms,
while
the
other
ones
are
called
cross-terms.
4.2.
Selection
of
peak
y
Delay-Doppler
points
Under
the
linear
data
model
in
(3),
the
SGMAF
of
observ
ations
across
the
array
at
a
gi
v
en
Delay-
Doppler
point
is
a
(
m
m
)
matrix
admits
the
follo
wing
decomposition:
A
x
(
;
)
=
H
A
s
(
;
)
H
H
+
A
n
(
;
)
;
=
H
A
s
(
;
)
H
H
+
R
n
(
)
;
(5)
where
A
s
(
;
)
represents
the
(
n
n
)
SGMAF
of
sources
defined
similarly
to
A
z
(
;
)
in
(4)
and
R
n
(
)
=
2
n
(
)
I
m
with
(
)
=
R
1
1
e
j
2
t
dt
and
I
m
is
the
m
m
identity
matrix.
It
is
kno
wn
that
the
matrix
A
s
(
;
)
for
an
y
and
has
no
special
structure.
Ho
we
v
er
,
there
are
some
Delay-Doppler
points
where
this
matrix
has
a
specific
algebraic
structure
:
(a)
Diagonal,
for
points
where
each
of
them
corresponds
to
a
single
auto-source
term
for
all
source
signals,
Blind
separ
ation
of
comple
x-valued
satellite-AIS
data
...
(Omar
Cherr
ak)
Evaluation Warning : The document was created with Spire.PDF for Python.
1736
r
ISSN:
2088-8708
(b)
Zero-diagonal
for
points
where
each
of
them
correspond
t
o
all
tw
o
by
tw
o
cross-source
term
(this
struc-
ture
is
e
xploited
because
the
signature
of
the
sources
dif
fer
in
cert
ain
points
of
the
Delay-Doppler
plan
on
the
zero-diagonal
part
(as
sho
wn
in
section
5.).
Our
aim
is
to
tak
e
adv
antage
of
these
properties
of
the
A
x
(
t;
)
in
(5)
since
the
element
of
this
is
no
more
(zero)
diagonal
matrices
due
to
the
mixing
ef
fect
in
order
to
estimate
the
separation
matrix
B
(the
pseudo-in
v
erse
of
matrix
H
)
and
restore
the
unkno
wn
sources.
4.3.
Construction
of
M
(set
of
Delay-Doppler
matrices
of
the
obser
v
ations
acr
oss
the
array
at
the
cho-
sen
Delay-Doppler
points)
W
e
use
the
detector
suggested
in
[23]
(denoted
C
Ins
)
for
the
instantaneous
mixture
consi
dered
without
pre-whitening
of
the
observ
ations.
The
idea
is
to
find
“useful”
Delay-Doppler
points
which
consists
in
k
eeping
Delay-Doppler
points
with
a
suf
ficient
ener
gy
,
then
using
the
rank-one
property
to
detect
single
cross-source
terms
(we
don’
t
mak
e
an
y
assumptions
on
the
kno
wledge
of
c
yclic
frequencies)
in
the
follo
wing
w
ay:
8
>
<
>
:
k
A
x
(
t;
)
k
>
1
;
max
A
x
(
t;
)
k
A
x
(
t;
)
k
1
>
2
;
(6)
where
1
,
2
are
(suf
ficiently)
small
positif
v
alues
and
max
f
:
g
is
the
lar
gest
eigen
v
alue
of
a
matrix.
4.4.
Non-unitary
joint
zer
o-(block)
diagonalization
algorithms
(
NU
JZ
(
B
)
D
)
The
matrices
belonging
to
the
set
M
(whose
size
is
denoted
by
N
m
(
N
m
2
N
))
all
admit
a
particular
structure
since
the
y
can
be
decomposed
into
H
A
s
(
;
)
H
H
with
A
s
(
;
)
a
zero-diagonal
matrix
with
only
one
non
null
term
on
its
zero-diagonal.
One
possible
w
ay
to
reco
v
er
the
mixing
matrix
B
is
to
directly
joint
ze
ro-
diagonalize
the
matrix
set
M
.
It
has
to
be
noticed
that
the
reco
v
ered
sources
(after
multiplying
the
observ
ations
v
ector
by
the
estimated
matrix
B
)
are
obtained
up
to
a
permutation
(among
the
classical
indetermination
of
the
BSS
).
Hence,
tw
o
BSS
methods
can
be
deri
v
ed.
The
first
called
JZD
CG
DD
algorithm
based
on
conjug
ate
gradient
approach
[16].
The
second
JZD
LM
DD
algorithm
based
on
Le
v
enbre
g-Marquardt
scheme
[17].
T
o
tackle
that
problem,
we
propose
here,
to
consider
the
follo
wing
cost
function
[16],
[17],
C
Z
B
D
(
B
)
=
P
N
m
i
=1
k
Bdiag
(
n
)
f
BM
i
B
H
gk
2
F
;
where
the
matrix
operator
Bdiag
(
n
)
f
:
g
is
defined
as
follo
ws:
Bdiag
(
n
)
f
M
g
=
0
B
B
@
M
11
0
12
:
:
:
0
1
r
0
21
M
22
.
.
.
0
2
r
.
.
.
.
.
.
.
.
.
.
.
.
0
r
1
0
r
2
:
:
:
M
r
r
1
C
C
A
;
where
M
is
a
N
N
(
N
=
n
(
L
+
L
0
)
where
L
is
the
order
of
the
FIR
filter
and
L
0
is
the
number
of
delays
considered
when
the
con
v
olutif
mixture
is
considered)
square
matrix
whose
block
components
M
ij
for
all
i;
j
=
1
;
:
:
:
;
r
are
n
i
n
j
matrices
(and
n
1
+
:
:
:
+
n
r
=
N
)
denoting
by
n
=
(
n
1
;
n
2
;
:
:
:
;
n
r
)
.
Note
that
when
L
=
0
,
L
0
=
1
we
find
the
instantaneous
model
since
A
x
are
no
more
matrices
b
ut
scalars.
Thus,
it
leads
to
the
minimization
of
the
follo
wing
cost
function:
C
Z
D
(
B
)
=
N
m
X
i
=1
k
Diag
f
BM
i
B
H
gk
2
F
;
(7)
where
M
i
=
A
x
i
is
the
i
th
of
the
N
m
matrices
belonging
to
M
.
W
e
suggest
to
use
conjug
ate
gradient
and
Le
v
enber
g-Marquardt
algorithms
[16],
[17]
to
minimize
the
cost
function
gi
v
en
by
Equation
.(7)
in
order
to
estimate
the
matrix
B
2
C
n
m
.
It
means
that
B
is
re-estimated
at
each
iteration
m
(denoted
B
(
m
)
or
b
(
m
)
when
the
v
ector
b
(
m
)
=
vec
B
(
m
)
is
considered).
The
matrix
B
(or
the
v
ector
b
)
is
updated
according
to
the
follo
wing
adaptation
rule
for
all
m
=
1
;
2
;
:
:
:
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
3,
June
2019
:
1732
–
1741
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1737
Conjugate
gradient
appr
oach
b
(
m
+1)
=
b
(
m
)
(
m
)
d
(
m
)
B
;
d
(
m
+1)
B
=
g
(
m
+1)
+
(
m
)
d
(
m
)
B
;
(8)
where
is
a
positi
v
e
small
f
actor
called
the
step-size,
d
B
is
the
direction
of
search,
is
an
e
xact
line
search
and
g
=
vec
(
r
a
C
Z
D
(
B
))
is
the
v
ectorization
of
the
comple
x
gradient
matrix
G
=
r
a
C
Z
D
(
B
)
=
2
P
N
m
i
=1
[
Diag
f
BM
i
B
H
g
BM
H
i
+
Diag
f
BM
i
B
H
g
H
BM
i
i
(see
the
proof
pro
vided
in
[16]
ho
w
the
optimal
step-size
opt
,
r
a
C
Z
D
(
B
)
and
are
calculated
at
each
iteration).
Le
v
enber
g-Mar
quardt
appr
oach
b
(
m
)
=
b
(
m
1)
h
H
(
m
1)
e
+
I
m
2
i
1
g
(
m
1)
;
(9)
where
[
:
]
1
denotes
the
in
v
erse
of
a
matrix,
is
positi
v
e
a
small
damping
f
actor
,
I
m
2
is
the
m
2
m
2
identity
matrix,
H
e
=
H
e
B
;
B
=
A
00
+
A
T
11
2
H
e
B
;
B
=
A
01
+
A
T
01
2
H
e
B
;
B
=
A
10
+
A
T
10
2
H
e
B
;
B
=
H
e
B
;
B
T
!
is
the
Hessian
matrix
of
C
Z
D
(
B
)
com-
posed
of
four
comple
x
matrices
with:
A
00
=
M
T
i
B
T
I
N
T
T
Bof
f
(
B
M
i
I
N
)
+
M
i
B
T
I
N
T
T
Bof
f
B
M
T
i
I
N
+
M
i
OBdiag
(
n
)
f
BM
i
B
H
g
+
M
T
i
OBdiag
(
n
)
f
BM
H
i
B
H
g
=
A
11
;
(10)
A
10
=
K
T
N
;M
I
N
M
i
B
H
T
T
Bof
f
(
B
M
i
I
N
)
+
K
T
N
;M
I
N
M
H
i
B
H
T
T
Bof
f
B
M
T
i
I
N
=
A
01
;
(11)
where
the
operator
denotes
the
Kroneck
er
product,
K
N
;M
is
a
square
commutation
ma
trix
of
size
N
M
N
M
and
T
Bof
f
=
I
N
2
T
Diag
,
is
the
N
2
N
2
“transformation”
matrix,
with
T
Diag
=
diag
f
vec
(
BDiag
f
1
N
g
)
g
,
1
N
is
the
N
N
matrix
whose
components
are
all
ones,
diag
f
a
g
is
a
square
diagonal
matrix
whose
diagonal
elements
are
the
elements
of
the
v
ector
a
,
I
N
2
=
Diag
f
1
N
2
g
is
the
N
2
N
2
identity
matrix,
and
Diag
f
A
g
is
the
square
diagonal
matrix
with
the
same
diagonal
elements
as
A
.
4.5.
Summary
of
the
pr
oposed
methods
The
proposed
methods
JZD
CG
DD
and
JZD
LM
DD
combine
the
NU
JZD
algorithms
which
are
JZD
CG
and
JZD
LM
together
with
the
detector
C
Ins
.
Its
principles
are
summarized
belo
w:
Data:
Consider
the
N
m
matrices
of
set
M
:
f
A
x
1
;
A
x
2
;
:
:
:
;
A
x
N
m
g
,
stopping
criterion
,
step-size
(for
conjug
ate
gradient),
max.
number
of
iterations
M
max
Result:
Estimation
of
joint
zero
diagonalizer
B
initialize:
B
(0)
;
(0)
;
m
=
0
;
D
(0)
(for
conjug
ate
gradient);
Conjugate
gradient
r
epeat
if
m
mo
d
M
0
=
0
then
restart
else
Calculate
(
m
)
opt
Compute
g
(
m
)
Compute
B
(
m
+1)
Compute
(
m
)
PR
Compute
d
(
m
+1)
B
m
=
m
+
1
;
end
until
((
k
B
(
m
+1)
B
(
m
)
k
2
F
)
ou
(
m
M
max
))
;
Le
v
enber
g-Mar
quardt
r
epeat
Calculate
g
(
m
)
Calculate
the
diagonal
of
H
e
Calculate
b
(
m
+1)
Calculate
the
error
e
(
m
)
=
1
N
m
C
Z
D
(
B
(
m
+1)
)
m
=
m
+
1
;
if
e
(
m
)
e
(
m
1)
then
(
m
)
=
(
m
1)
10
,
e
(
m
)
=
e
(
m
1)
else
(
m
)
=
10
(
m
1)
end
until
((
k
B
(
m
+1)
B
(
m
)
k
2
F
)
ou
(
m
M
max
))
;
5.
COMPUTER
SIMULA
TIONS
Computer
simulations
are
performed
to
illustrate
the
good
beha
vior
of
the
suggested
methods
and
to
compare
them
with
the
same
kind
of
e
xisting
approach
denoted
by
JZD
Chab
riel
DD
proposed
in
[24]
with
the
Blind
separ
ation
of
comple
x-valued
satellite-AIS
data
...
(Omar
Cherr
ak)
Evaluation Warning : The document was created with Spire.PDF for Python.
1738
r
ISSN:
2088-8708
Delay-Doppler
point
C
Ins
detector
.
W
e
consider
m
=
3
mixtures
of
n
=
2
frames
of
256
bits
correspond
to
tw
o
v
essels
with
dif
ferent
characteristics.
The
frames
are
generated
according
to
the
S-AIS
recommendation
as
mentioned
in
the
Figure
2
(see
also
[11],
[10]).
These
frames
are
encoded
with
NRZI
and
modulated
in
GMSK
with
a
bandwidth-bit-time
product
parameter
BT
=
0
:
4
.
The
transmission
bit
rate
is
=
9600
bps
and
the
order
g
aussian
filter
is
OF
=
21
.
The
frequenc
y
carrier
of
the
first
source
(resp.
the
second
source)
is
161.975
MHz
(resp.
162.025
MHz),
taking
into
account
a
delay
of
10
ms
and
a
Doppler
shift
of
4
kHz
(resp.
a
delay
of
0
ms
and
the
Doppler
shift
of
0
Hz).
These
sources
correspond
to
1400
time
samples
which
are
mix
ed
according
to
a
mixture
matrix
H
whose
components
stands
for:
H
=
0
@
1
:
1974
1
:
3646
0
:
8623
1
:
6107
0
:
1568
0
:
9674
1
A
:
(12)
The
real
part
and
the
imaginary
part
of
their
SGMAF
is
gi
v
en
on
the
left
and
on
the
right
of
t
he
Figure
4
respecti
v
ely
.
Then,
the
SGMAF
of
the
observ
ations
x
is
then
calculated
by
(5)
and
finally
the
100
resulting
SGMAF
are
a
v
eraged.
W
e
ha
v
e
chosen
1
=
0
:
07
and
2
=
0
:
08
for
the
detector
C
Ins
in
order
to
construct
the
set
M
to
be
joint
zero-diagonalized.
The
signal-to-noise
ratio
SNR
is
defined
by
SNR
=
10
log
(
1
2
N
)
of
mean
0
and
v
ariance
2
n
.
The
selected
Delay-Doppler
points
using
the
proposed
detector
are
represented
in
the
Figure
5
for
SNR
=
10
dB
and
100
dB.
The SGMAF of the source 1
Cyclic frequency (Doppler) [Hz]
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF between the source 1 and 2
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF between the source 2 and 1
Time in samples
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF of the source 2
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF of the source 1
Cyclic frequency (Doppler) [Hz]
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF between the source 1 and 2
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF between the source 2 and 1
Time in samples
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
The SGMAF of the source 2
100
200
300
400
500
600
700
−6.481
−4.8608
−3.2405
−1.6203
0
1.6203
3.2405
4.8608
x 10
8
Figure
4.
Left
:
The
SGMAF
real
part
of
the
S-AIS
sources.
Right
:
The
SGMAF
imaginary
part
of
the
S-AIS
sources.
T
o
measure
the
quality
of
the
estimation,
the
ensuing
error
inde
x
is
used
[25]
:
I
(
T
)
=
1
n
(
n
1)
2
4
n
X
i
=1
0
@
n
X
j
=1
k
T
i;j
k
2
F
max
`
k
T
i;`
k
2
F
1
1
A
+
n
X
j
=1
0
@
n
X
i
=1
k
T
i;j
k
2
F
max
`
k
T
`;j
k
2
F
1
1
A
3
5
;
(13)
where
(
T
)
i;j
for
all
i;
j
2
1
;
:
:
:
;
n
is
the
(
i;
j
)
-th
element
of
T
=
^
BH
.
The
separation
is
perfect
when
the
error
inde
x
I
(
)
is
close
to
0
in
a
linear
scale
(
1
in
a
log
arithmic
scale).
All
the
displayed
results
ha
v
e
been
a
v
eraged
o
v
er
30
Monte-Carlo
trials.
W
e
plot,
in
the
Figure
6,
the
e
v
olution
of
the
error
inde
x
v
ersus
the
SNR
in
order
to
emphasize
the
influence
of
this
in
the
quality
of
the
estimation.
All
algorithms
are
initialized
using
the
same
initialization
suggested
in
[24].
Int
J
Elec
&
Comp
Eng,
V
ol.
9,
No.
3,
June
2019
:
1732
–
1741
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1739
First,
we
can
deduce
from
the
Figure
4
t
hat
the
di
v
ersity
in
the
Delay-Doppler
re
gions
is
obtained
on
the
zero-diagonal
part
which
supports
the
use
of
zero
diagonalization
algorithms.
Then,
our
analysis
are
e
xamined
on
the
Figure
6
according
to
noiseless
and
noisy
conte
xts
.
F
or
the
noiseless
conte
xt
(when
SNR
=100
dB),
the
JZD
CG
DD
and
JZD
LM
DD
reach
approximately
-64
dB
and
-60
comparing
with
JZD
Chab
ri
e
l
DD
method
which
reaches
'
-20
dB.
From
this
comparison,
we
ha
v
e
check
ed
the
v
alidity
of
the
good
beha
vior
of
JZD
CG
DD
and
JZD
LM
DD
compared
to
the
JZD
Chab
riel
DD
approach.
Moreo
v
er
,
we
observ
e
that
the
JZD
LM
DD
based
on
the
computation
of
e
xact
Hessian
matrices
is
more
ef
ficient
than
the
JZD
CG
DD
approach.
Ev
en
in
a
dif
ficult
(noisy)
conte
xt
(for
e
xample
SNR
=15
dB),
we
note
that
the
best
results
are
generally
obtained
using
the
JZD
LM
DD
(-36
dB)
then
JZD
CG
DD
(-33
dB)
especially
the
JZD
LM
DD
algorithm
based
on
the
computation
of
e
xact
Hessian
matrices.
It
may
be
concluded
that
the
approaches
e
xploiting
the
Delay-Doppler
di
v
ersity
of
S-AIS
signals
seem
rather
promising.
Due
to
its
rob
ustness
to
the
noise,
it
seems
to
be
able
to
solv
e
the
problem
of
BSS
(i.e
the
collision
problem)
in
a
marine
surv
eillance
conte
xt.
400
500
600
700
800
900
1000
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time in samples
Reduced Frequency
400
500
600
700
800
900
1000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time in samples
Reduced frequency
Figure
5.
Delay-Doppler
points
selected
with
the
detector
C
Ins
.
left
:
SNR
=100
dB.
right
:
SNR
=10
dB.
15
20
30
40
60
80
100
SNR [dB]
-70
-60
-50
-40
-30
-20
-10
Error Index I(T) [dB]
JZD
LM
DD
JZD
CG
DD
ZDC
Chabriel
DD
Figure
6.
Comparison
of
the
dif
ferent
methods:
e
v
olution
of
the
error
inde
x
I
(
T
)
in
dB
v
ersus
SNR
.
6.
CONCLUSION
In
this
paper
,
we
ha
v
e
sho
wn
that
the
blind
source
separation
based
on
SGMAF
can
be
performed.
W
e
ha
v
e
considered
comple
x-v
alued
S-AIS
data
for
marine
surv
eillance
which
can
be
recei
v
ed
at
the
same
time-
slot
in
where
the
collision
of
these
data
is
caused.
In
addition,
it
is
presented
that
the
collision
problem
can
be
reshaped
into
BSS
problem.
Moreo
v
er
,
it
is
sho
wn
that
proposed
BSS
methods
are
established
thanks
to
an
automatic
single
cross-term
selection
procedure
combined
with
tw
o
NU
JZD
algorithms
denoted
Conjug
ate
Gradient
and
Le
v
enber
g-Marquardt
which
are
based
on
the
minimization
of
a
least-mean-square
criterion.
Finally
,
we
deduced
that
the
JZD
LM
DD
and
JZD
CG
DD
of
fer
the
best
perform
ances
e
v
en
in
noisy
conte
xts.
As
perspecti
v
e,
a
question
needing
analysis
is
t
o
study
more
realistic
and
comple
x
cases
in
which
the
number
of
S-AIS
messages
recei
v
ed
at
the
antenna
embedded
in
the
satellite
w
ould
be
much
higher
and
mixing
models
could
also
be
considered.
Blind
separ
ation
of
comple
x-valued
satellite-AIS
data
...
(Omar
Cherr
ak)
Evaluation Warning : The document was created with Spire.PDF for Python.
1740
r
ISSN:
2088-8708
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ol.
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3,
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2019
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1732
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Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
1741
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BIOGRAPHIES
OF
A
UTHORS
Omar
Cherrak
w
as
born
in
Fez
Morocco.
He
recei
v
ed
the
Bachelor
de
gree
in
2009
in
the
field
of
Electronics
T
elecommunications
and
Computer
Sciences
and
the
Master
de
gree
of
Microelec-
tronic,
T
elecommunications
and
Computer
Industry
Systems
in
2011,
both
from
the
Uni
v
ersit
´
e
Sidi
Mohamed
Ben
Abdella
h
(USMB
A),
F
acult
´
e
des
Sci
ences
et
T
echniques
(FST),
Fez
Morocco.
He
obtained
his
Ph.D.
de
gree
on
March
2016
in
the
are
a
of
”Signal,
T
elecommunications,
Image
and
Radar”
from
Uni
v
ersit
´
e
de
T
oulon
and
this
thesis
w
as
carried
out
in
cotutelle
with
USMB
A.
His
main
research
interests
are
blind
source
separation,
telecommunications,
joint
matrix
decomposi-
tions,
maritime
surv
eillance
s
ystem,
time-frequenc
y
representation,
smart
grid
and
DoA
estimation.
Hicham
Ghennioui
obtained
the
Ma
ˆ
ıtrise
de
gree
in
T
elecommunications
from
the
F
aculty
of
Sci-
ences
and
T
echnologies
(FST),
Fez,
Morocco,
in
2002.
He
got
the
D.E.S.A.
de
gree
in
Computer
and
T
elecommunications
from
the
F
aculty
of
Sciences,
Rabat,
Morocco,
in
2004
and
the
Ph.D
de
gree
in
engineering
sciences
in
2008,
from
Mohamed
V
–
Agdal
Uni
v
ersi
ty
,
Morocco,
and
the
Uni
v
ersity
of
T
oulon,
France,
respecti
v
ely
.
From
May
2008
to
December
2009,
he
w
as
a
Research
&
De
v
el-
opment
Engineer
at
Amesys
Bull,
and
form
January
2010
to
May
2011,
he
w
as
a
Signal/Image
Researcher
at
Moroccan
foundation
for
Adv
anced
Science,
Inno
v
ation
and
Research
(MASCIR),
Rabat,
Morocco.
Since
2011,
he
is
an
Assistant
Professor
at
the
Electrical
Engineering
Department
of
the
F
aculty
of
Sciences
and
T
echniques,
Sidi
Mohamed
Ben
Abdellah
Uni
v
ersity
,
Fez,
Morocco.
His
main
research
interests
are
signal/image
processing
including
blind
sources
separation,
decon-
v
olution,
deblurring,
time-frequenc
y
representations
and
cogniti
v
e
radio.
Nad
`
ege
Thirion-Mor
eau
w
as
born
in
Montb
´
eliard
France.
S
he
recei
v
ed
the
DEA
de
gree
in
1992
and
the
Ph.D.
de
gree
in
1995,
both
in
the
field
of
signal
processing
and
from
the
Ecole
Nationale
Sup
´
erieure
de
Ph
ysique
(ENSPG)
Institut
National
Polytechnique
de
Grenoble
(IN
PG),
France.
From
1996
to
1998,
she
w
as
an
Assistant
Professor
at
the
Ecole
Sup
´
erieure
des
Proc
´
ed
´
es
Elec-
troniques
et
Optiques
(ESPEO),
Orl
´
eans,
France.
Since
1998,
she
has
been
with
the
Institut
des
Sciences
de
l’Ing
´
enieur
de
T
oulon
et
du
V
ar
(ISITV),
La
V
alette,
France,
as
an
Assistant
Professor
,
in
the
Department
of
T
elecommunications.
Her
main
research
interests
are
in
deterministic
and
statistical
signal
processing
including
array
processing,
blind
sources
separation/equalization,
high-
order
statistics,
nonstationary
signals,
time-frequenc
y
representations,
and
decision/
classificat
ion.
El
Hossain
Abarkan
w
as
born
in
1953,
Nador
,
Morocco.
He
graduated
the
bachelor
de
gree
in
ph
ysics
from
Mohammed
V
Uni
v
ersity
,
Rabat,
Morocco,
in
1978.
He
obtained
the
DEA,
the
Doc-
torate
and
the
Ph.D
de
grees
from
Uni
v
ersity
of
Languedoc,
Montpellier
,
France,
in
1979,
1981
and
1987
respecti
v
ely
.
He
got
the
Ph.D
de
gree
from
the
Uni
v
ersity
of
Sidi
Mohamed
Ben
Abdellah,
Fez,
Morocco,
in
1992.
Cur
rently
,
he
is
a
Professor
in
Sidi
Mohamed
Ben
Abdellah
Uni
v
ersity
.
From
1996
to
2014,
he
w
as
the
founder
and
the
director
of
the
Laboratory
of
Si
gnals,
Systems
and
Com-
ponents,
Sidi
Mohamed
Ben
Abdel
lah
Uni
v
ersity
.
His
main
r
esearch
interests
are
in
electronics,
Modeling,
characterization
and
CAD
in
inte
grated
circuits.
Blind
separ
ation
of
comple
x-valued
satellite-AIS
data
...
(Omar
Cherr
ak)
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