Re e ences
1
MARTiN-NEIRA,
M.,
and
GOUTOULE,
J.M.:
'MIUS
-
a wo-dimensional
ape u e-syn hesis adiome e o soil-mois u e and ocean salini y
obse a ions',
ESA
Bull.,
1997, (92), pp. 95-104
2
SIVESTRIN,
p.,
BERGER, M.,
ERR,
Y.,
and
FONT,
J.:
'ESA's
Second Ea h
Explo e Oppo uni y Mission: The Soil Mois u e and Ocean Salini y
Mission
-
SMOS',
IEEE
Geosci. Remo e Sens.
Newsl.,
2001,
(1
18),
pp. 11-14
CAMPS, A., BAe,
J.,
CORBELLA,
I.,
and
TORRES, F.:
'The p ocessing o
hexagonally sampled signals wi h s anda d ec angula echniques:
applica ion o
2D
la ge ape u e syn hesis in e e ome ic adiome e s',
IEEE
T ans. Geosci. Remo e Sens.
GRS-35, 1997, pp. 183-190
4
BA~,
J.,
CAMPS,
A.,
TORRES,
F.,
and
CORBELLA,
I.:
'Angula esolu ion
o
wo-dimensional hexagonally sampled in e e ome ic adiome e ',
Radio Sci.,
1998, 33,
(5),
pp. 1459-1473
BA~,
J.:
'Reliabili y analysis in ape u e syn hesis in e e ome ic
adiome e s: applica ion o L-band
MIRAS
ins umen ',
Radio
Sci.,
3
5
VALL-LLOSSERA, M., DUFFO,
N.,
CAMPS, A., CORBELLA,
I.,
TORRES,
E,
and
2001,36,
(I),
pp. 107-1 17
~,
Ku osis
KS
signi icance
le el
(%)
SAR
in e e ome ic phase s a is ics in
wa ele domain
~,
.,
Ku osis KS
Ku osis
KS
signi icance
signi icance
le el
(%)
le el
(%)
C.
Lopez and
X.
Fab egas
Scale
1
1
2.98
Syn he ic ape u e ada
(SAR)
in e e ome y is employed o ob ain
opog aphic in o ma ion. Owing
o
noise, in e e ome ic in o ma ion
has o be il e ed. The wa ele ans o m can be employed
o
il e
he
in e e ome ic phase, main aining he spa ial esolu ion, bu new
signal models ha e o be s udied in his domain o u he p ocessing.
87.93
1
2.98
1
79.83 2.94
1
98.34
In oduc ion:
Syn he ic ape u e ada in e e ome y (InSAR) is an
es ablished echnique o ob ain in o ma ion abou he ea h's su ace
opog aphy. The in e e ome ic phase is calcula ed as he phase
di e ence be ween
wo
complex SAR images om he same a ea,
bu aken om sligh ly di e en posi ions. Owing o he lack o
in e e ome ic cohe ence
(y
I
be ween bo h SAR images, he in e -
e ome ic phase is noisy. In addi ion, he in e e ome ic phase is
only known wi hin he in e al
[-z,
n),
i being necessa y o unw ap
i o eco e unambiguously he heigh in o ma ion. The unw apping
p ocess
is
also
a ec ed
by
phase noise, since
i
induces phase
esidues. Phase il e ing is hus necessa y o educe noise e ec s.
In he las decade, he wa ele ans o m (WT) has shown a big
po en ial o image p ocessing applica ions. In he ield o SAR da a
p ocessing, he use o he WT is eme ging since i allows p ocessing o
SAR image y, keeping he spa ial esolu ion and image de ails.
Since he physics behind SAR da a is comple ely di e en om ha
o
op ical images, any da a p ocessing has o ake his in o accoun .
Thus, i is necessa y o e iew o e en o de ine new noise models
adap ed o his p oblem. In his Le e , we p o ide a s udy o a signal
model o he in e e ome ic phase in he wa ele domain. This model
is alida ed wi h eal in e e ome ic da a.
equency plane.
N,
has a one- o-one ela ion wi h he cohe ence
IyJ
p o iding, hus, he same in o ma ion [3,4].
Fo a cons an in e e ome ic phase and homogeneous noise (i.e.
cons an
Iyl),
he pa ame e
Nc,
as well as he e ms COS(C$;) and sin
(4:)
a e cons an . The e o e, signal andomness is only due o
,"
and
,".
The
disc e e wa ele ans o m (DWT) can be seen as he addi ion o
(weigh ed) andom a iables. By he cen al limi heo em, he weigh ed
sum
o iden ically dis ibu ed andom a iables can be app oxima ed by
a Gaussian dis ibu ion. The e o e,
,"
and
,"
a e app oxima ely
Gaussian dis ibu ed. To es i , a oiding any in e e ence om he
phase
d;,
a cons an slope p oducing
20
pixel inges, co up ed wi h a
noise equi alen o a cohe ence
IyI
=
0.6,
has been simula ed. Table
1
shows a s a is ical es applied o e he eal pa o he in e e ome ic
complex phase in he wa ele domain. As shown, since he use ul signal
is concen a ed in he low equency band (LL), he wa ele bands
(HL,
LH
and
HH)
p esen a ku osis close o 3 and he signi icance
le els o he Kolmogo o -Smimo (KS) es , assuming a Gaussian
dis ibu ion, a e high. These esul s demons a e ha
,"
and
,"
can be
desc ibed by a Gaussian dis ibu ion. The same ag eemen is obse ed
o any o he alue o
Iyl,
and o he imagina y pa
o
he complex
in e e ome ic phase in he wa ele domain. The LL band dese es
special a en ion.
In
his case, as he e is signal con en , he signal
model will be ep esen ed by he eal and imagina y pa s o exp
(j&')
plus a Gaussian noise. The e o e, in his case, he ampli ude
lpl
+jp21
has a Rice dis ibu ion. This esul is equally alid o he es o he
wa ele bands. The ampli ude in he wa ele domain will be Rayleigh
dis ibu ed o
IyI
=
0,
in a pa icula space- equency egion, and Rice
dis ibu ed o
IyJ
>
0
Table
1:
Ku osis and
KS
signi icance le els (Gaussian assump-
ion) o eal pa
o
simula ed complex in e e ome ic
phase amp in wa ele domain
I
Ho izon al band
(HW
I
Ve ical
band
(LH)
I
Diaaonal
band
(LW
I
I I I I I
I
Scale
3
I
3.08
I
82.87
I
2.95
I
85.32
I
3.00
I
82.87
Complex in e e ome icphase signal model:
The opog aphic model
assumed p e iously (i.e. cons an slope) does no ake in o accoun
spa ial de ails, which a e impo an , o ins ance, in u ban a eas. Since
no in o ma ion is a ailable abou he dis ibu ion o he ' ue'
opog aphic phase
4x,
an
a p io i
model o he spa ial de ails is
no a ailable in he spa ial domain. This d awback can be o e come in
he wa ele domain. As men ioned in he p eceding Sec ion, he DWT
can be in e p e ed
as
a weigh ed
sum
o andom a iables. The e o e,
DWT2D{N,
cos(&)} and
DWT2D{Nc
sin(&)} can be supposed o be
de e mined as a i s app oxima ion, by a Gaussian dis ibu ion. Tes s
wi h eal da a (see Tables
2
and
3)
show ha wa ele s a is ics ha e
ku osis highe han
3.
To ake in o accoun his de ia ion om
Gaussian beha iou , a double s ochas ic model is p oposed o he
wa ele coe icien s
x:
Complex in e e ome icphase noise model:
Ea h opog aphy can be
ep esen ed locally by a cons an slope
[I],
hus he in e e ome ic
phase
4,
can be assumed o be a cons an phase amp. In he spa ial
domain he measu ed in e e ome ic phase complies wi h he model
dZ
=
q5x
+
[2], whe e
is a phase noise e m. The eal and imagina y
(5)
pa s o he measu ed phase
4z
coded in he uni ci cle, de ined as he
complex in e e ome ic phase, can be modelled by [3,
41:
whe e
P,(X~~~)
ep esen s he Gaussian dis ibu ion
o
he wa ele
coe icien s and
~,d( ~)
is a gene alised gamma dis ibu ion (GGD)
(1)
(2)
modelling he ahabili6 o he- a iance h ough he phase image.
px(x)
canno be ob ained in a gene al o m. Nume ical in eg a ion o
(9,
see
Fig. 1, indica es ha
px(x)
can be assumed o be a GGD model. To es
he wa ele ans o ms o which a e [3]: he alidi y o his model,
wo
eal in e e ome ic phase images aken
wi h he Ge man senso E-SAR om DLR ha e been employed. The
i s image is an X-band in e e og am o Moun E na (I aly) and he
second one is an L-band in e e og am o he Obe p a enho en es
si e (Ge many), we e man-made s uc u es a e p esen . These
wo
in e e ome ic phases we e il e ed wi h he algo i hm p esen ed in
[3],
which is based
on
main aining he spa ial esolu ion. Tables
2
and 3
p esen he ku osis and he
KS
es signi icance le els, assuming a
GGD model, applied o he eal pa o he il e ed complex in e e o-
/I,
=
DWT,,{COS(~,)]
=
2'Nc
COS(^:)
+
V:
p2
=
DWT2D{sin(&)]
=
2'Nc
sin(+,")
+
y
(3)
(4)
whe e
i
ep esen s he wa ele scale.
,"
and
,"
a e noise e ms
independen
om
he wa ele scale. The phase e m
4:
ep esen s
he in e e ome ic phase in he wa ele domain, which con ains he
same in o ma ion as
q5=
The WT is able o localise
dx
in he space-
ELECTRONICS LETTERS
26 h
Sep embe
2002
Vol.
38
No. 20
1207
me ic phase in he wa ele domain. These esul s show ha he GGD
model is well adap ed o eal da a. Tables
2
and
3
also show ha he
Obe p a enho en image p esen s highe ku osis alues. This esul is
explained by he ac ha his image con ains mo e spa ial de ails (as
buildings o oads) han he Moun E na image, which con ains only
opog aphic in o ma ion.
Ho izon al
band (HL)
Ku osis
KS
signi icance
-1
I
I
I
I
-0.4
-0.2
0
0.2
0.4
a
20.0
Ve ical
band
(LH) Diagonal
band
(HH)
Ku osis
KS
Ku osis
KS
simi icance signi icance
I
I I
I
,
I I I
I
-0.100-0.075 -0.050-0.025
0
0.025 0.050 0.075 0.100
b
Fig.
1
Dis ibu ion compa ison be ween GGD model and nume ical
in eg a ion
o
double s ochas ic model
a
Low ku osis
b
High ku osis
__~
GGD model
~
double s ochas ic model
Ho izon al
band
(HL)
Ku osis
KS
signi icance
le el
(“A)
Ve ical
band
(LH)
Diagonal
band
(LL)?
Ku osis
KS
Ku osis
KS
signi icance
signi icance
le el
(“A)
le el
(“A)
I
l; el(%)
I I
G el(%)
I
I
I; el(%)
Scale
1
I
10.51
I
70.61
I
11.30
I
77.16
I
12.88
I
56.61
Scale
2
I
6.39
I
86.51
I
6.42
I
79.89
I
6.12
I
98.33
Scale
3
I
4.42
I
97.34
I
5.62
I
60.95
I
3.92
I
80.70
Table
3:
Ku osis
and
KS
signi icance le els
(GGD
assump ion)
o eal pa o Obe p a enho en complex in e e o-
me ic phase in wa ele domain.
Scale
1
I
22.50
I
10.53
1
21.44
I
31.14
I
26.95
I
14.76
Scale
2
I
13.82
I
16.24
I
11.51
I
53.28
I
12.68
I
91.26
I
I
I
Scale
3
I
9.25
I
85.52
I
13.09
I
37.32
I
8.56
I
11.75
Conclusions:
We ha e de eloped a s ochas ic model o he complex
in e e ome ic phase in he wa ele domain. In he i s pa o his
Le e , assuming he opog aphy locally as a cons an slope, a
e e ence signal model is p oposed and alida ed. To inc ease i s
lexibili y, a double s ochas ic signal model, which leads o a GGD
model, is es ed o he wa ele coe icien s. In he wa ele domain,
complex phase noise is Gaussian dis ibu ed, whe eas he use ul
signal complies wi h a GGD model. The GGD assump ion o wa ele
coe icien s ag ees wi h obse a ions epo ed by o he esea che s.
Finally, his ep esen a ion can be employed as
a p io i
in o ma ion
o u he p ocessing inside he wa ele domain.
Acknowledgmen s:
This wo k was unded by he Spanish go e nmen
in he CICYT p ojec TIC1999-1050-C03-01 and he Ca alonian
go e nmen in a DURSI ellowship.
0
IEE
2002
Elec onics Le e s Online
No:
20020820
DOI: 10.1049/e1:20020820
C.
Lopez and
X.
Fab egas
(Signal Theo y and Communica ions
Depa men , Elec omagne ics
&
Pho onics Enginee ing G oup,
Uni e si a Poli kcnica de Ca alunya
(UPC),
Building 03
Room
11
8,
Jo di Gi ona
1-3,
E-08034 Ba celona, Spain)
E-mail: [email p o ec ed]
27
July
2002
Re e ences
GATELLI,
E,
GUARNERI, A.M., PARIZZI, E, PASQUALI,
P,
PRATI, C.,
and
ROCCA,
F.:
‘The wa enumbe shi in
SAR
in e e ome y’,
IEEE T ans.
Geosci. Remo e Sens.,
1994,32, (4), pp. 855-865
LEE,
J.s.,
PAPATHANASSIOU, K.P.,
AMSWORTH,
T.L.,
GRUNES,
M.R.,
and
REIGBER,
A,:
‘A
new echnique o noise il e ing o
SAR
in e e ome ic phase images’,
IEEE T ans. Geosci. Remo e Sens.,
LOPEZ-MART~EZ,
C.,
and
F~REGAS-CANOVAS,
EX.:
‘Modeling and
educ ion o SAR in e e ome ic phase noise in he wa ele domain’,
Accep ed o publica ion IEEE T ans. Geosci. Remo e
Sens.,
2001
LOPEZ MART~EZ, C.,
and
FABREGAS CANOVAS,
X.:
‘Resul s on SAR
in e e ome ic phase noise educ ion using wa ele ans o m’.
EUSAR’02, Cologne, Ge many, 2002, Vol. 1
1998,36,
(5),
pp. 1456-1465
Adap i e a ay an enna based on adial
basis unc ion ne wo k as mul iuse
de ec ion o WCDMA
Chang-Jun
Ahn
and Iwao Sasase
An adap i e a ay an enna is p oposed based on he adial basis
unc ion
(RBF)
ne wo k as a mul iuse de ec o o a
WCDMA
sys em. The p oposed sys em calcula es he op imal combining
weigh
coe icien s
using
sample
ma ix
in e sion
wi h
a
common
co ela ion ma ix algo i hm and ob ains he channel esponse ec o
using
he
RBF
ou pu
signal.
In oduc ion:
Wideband code di ision mul iple access (WCDMA)
communica ion sys ems ha e ecen ly a ac ed conside able a en ion
as mobile cellula and IMT-2000 communica ion sys ems due o hei
abili y o supp ess a wide a ie y o in e e ing signals including
na owband in e e ence, mul iple access in e e ence (MAI), and
mul ipa h in e e ence (MPI).
One
o he app oaches o imp o ing
WCDMA sys em pe o mance is he use o spa ial il e ing a
a
base
s a ion wi h an adap i e an enna a ay. The adap i e a ay an enna is
widely accep ed, since i p o ides many p omising ea u es such
as
high capaci y, high spec um e iciency, and mo e deg ees o eedom
o adjus cell co e age cha ac e is ics, leading o mo e e icien
use
o
adio esou ces. To ob ain he op imal combining weigh coe icien s
used in his echnique, se e al adap i e algo i hms such as he sample
ma ix in e sion (SMI) and leas mean squa e (LMS) algo i hm ha e
been p oposed. In a ay an ennas using hese con en ional
algo i hms, weigh p ocessing is indi idually ca ied ou o each
use . The e o e, he base s a ion mus ha e many independen weigh
p ocesso s, which inc ease compu a ional complexi y.
To
sol e his
p oblem, a sample ma ix in e sion wi h
a
common co ela ion ma ix
(CCM-SMI) has been p oposed. This algo i hm has low compu a-
ional complexi y, as weigh con e gence, and good
BER
pe o -
mance a he base s a ion in a mul iuse en i onmen . Howe e , he
CCM-SMI-based adap i e a ay an enna sys em has a poo channel
esponse ec o (as does SMI) due o he ac i e mul iuse . Since he
MF ou pu signal includes MA1 wi h inc easing use s, he sys em
1208
ELECTRONICS LETTERS 26 h Sep embe
2002
Vol.
38
No.
20