14 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
FAR-END CROSSTALK MODELING BASED ON CAPACITIVE AND
INDUCTIVE UNBALANCES BETWEEN PAIRS IN A CABLE
Pa el LAFATA.1
1 Depa men o Telecommunica ion Enginee ing, Facul y o Elec ical Enginee ing, Czech Technical Uni e si y in
P ague, Technická 2, P ague, 166 27, Czech Republic
la a pa @ el.c u .cz
Abs ac . This a icle deals wi h new ways o a -end
c oss alk (FEXT) modeling in mul i-pai and mul i-quad
me allic cables. Cu en s anda d modeling me hods
p o ide only ough es ima ions o FEXT cha ac e is ics
based on a e age alues o c oss alk o he whole cable.
Howe e , o p ac ical implemen a ion o ec o disc e e
mul i- one modula ion (VDMT) is necessa y o p edic and
simula e FEXT cha ac e is ics wi h su icien accu acy
and simula e FEXT ans e unc ions indi idually o
each combina ion o symme ical pai s in a cable. This
a icle con ains a heo e ical analysis and desc ip ion o
he p oblem and sugges s a new me hod o modeling o
FEXT c oss alk using capaci i e and induc i e unbalances
be ween pai s in a cable. This p oposed model o e s mo e
accu a e and ealis ic esul s o c oss alk. Theo e ical
simula ions and esul s a e also compa ed wi h he
measu ed cha ac e is ics o speci ic me allic cable.
Keywo ds
C oss alk, FEXT, T ansmission line, VDMT, xDSL.
1. In oduc ion
The c oss alk can be gene ally desc ibed as a
nega i e phenomenon, when a pa o he signal
ansmi ing a he dis u bing line pene a es h ough
induc i e and capaci i e couplings o he pa allel dis u bed
pai [1]. The in luence o nea -end c oss alk (NEXT) can
be well limi ed by sepa a ing ansmission di ec ions by
using di e en equency bands, bu he educ ion o a -
end c oss alk (FEXT) is no so easy and he e o e FEXT is
a dominan sou ce o dis u bance in cu en xDSL lines
and i signi ican ly educes he maximum ansmission
speed achie ed by hese sys ems [2]. The s anda d model
o FEXT c oss alk is based only on a e age alues o
c oss alk ac oss all pai s and hei combina ions o he
whole cable. I uses only one c oss alk pa ame e gi en o
he whole cable so i is ob ious ha such model canno be
e y accu a e and ha i p o ides only app oxima e and
no e y ealis ic esul s, as p esen ed in [3], allowing only
limi ed es ima ions o i s impac on he esul ing sys em
esponse. One o he mos p omising solu ions o he
elimina ion o FEXT is Vec o ed DMT modula ion
(VDMT) o VDSL2 connec ions. Howe e , his me hod
equi es e y accu a e p edic ion o c oss alk beha io and
ealis ic modeling o FEXT o all combina ions o pai s in
a cable and indi idually o each ansmission channel [4].
This pape p esen s a new inno a i e me hod o
FEXT modeling, which is based on simula ions and
calcula ions o capaci i e and induc i e unbalances
be ween pai s in a cable and using cascade ma ices o a
ansmission line. The i s pa is ocused on de i a ion o
a gene al desc ip ion o he cu en si ua ion and c oss alk
cu en s o a pai o symme ical pai s loca ed wi hin a
coppe cable. This de i a ion will also be compa ed wi h
he o mulas o he s anda d model o FEXT c oss alk, as
p esen ed in [5]. The nex pa deals wi h he
implemen a ion o a new me hod o modeling FEXT
c oss alk. This model will espec he in e nal s uc u e o
a cable and will conside he alue o he a iable
capaci i e and induc i e coupling be ween pai s along he
leng h o a cable. The esul s o simula ions will be also
compa ed wi h s anda d FEXT model as well as wi h
measu ed esul s o he cable wi h TCEPKPFLE
speci ica ion.
2. Gene al exp ession o a -end
c oss alk cu en s in a cable
The elemen a y uni o a s anda d elecommunica ion
cable is gene ally wo insula ed wi es wis ed uni o mly o
o m a balanced pai . By wis ing ou insula ed wi es
oge he uni o mly a s a -quad is o med. Se e al quads
a e ypically wis ed oge he o o m a subg oup o pai s
(o quads), hese subg oups can be u he wis ed and
ga he ed acco ding o a cable’s in e nal s uc u e and hey
can be also co e ed wi h sc eening, shee ing o aping o
o m g ounded shielding and o sepa a e each subg oup o
pai s. In e s ices be ween pai s, quads and subg oups a e
usually illed wi h a gel o ai [6]. Du ing he p ocess o
cable’s manu ac u ing, se e al pa ame e s ha e o be
INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 15
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
measu ed and checked, and mus mee speci ied
ole ances. Based on hese ole ances, pai s, quads and
subg oups in a cable demons a e owa ds hemsel es
small i egula i ies and unbalances. These unbalances a e
caused mainly by i egula i ies o conduc o s and
dielec ic, de ec s o dimensions and posi ional di e ences
o conduc o s, wi es, pai s and quads. The second pa o
hese unbalances comes om cable’s imp op ia e placing,
ex e nal o in e nal de o ma ions and some andom
in luences. Capaci i e and induc i e unbalances and
couplings a e he main sou ce o c oss alk be ween hem.
These capaci i e and induc i e couplings in a quad o ou
wi es o m an unbalanced b idge [7]. Using he s a -
polygon ans o ma ion i is possible o exp ess esul ing
capaci i e unbalance Cub and induc i e unbalance Mub.
The calcula ion o hese unbalances is based on he
geome ical s uc u e o he quad and o he pa ame e s,
such as pe mi i i y and pe meabili y o he ma e ials. In
his case, he Cub unbalance is calcula ed using ou
indi idual capaci i e unbalances be ween single
conduc o s, he e o e i is equal o 4·
D
C(13,14,23,24) o hese
unbalances [15]. The impac o induc i e coupling can be
modelled by an addi ional capaci ance unbalance and bo h
capaci i e and induc i e pa s can be included in he
summa y capaci i e unbalance C´ [8].
The a -end c oss alk is caused by dis u bing
cu en s, which pene a e om he dis u bing pai o he
pa allel dis u bed pai , hanks o he capaci i e and
induc i e unbalances be ween hem. This si ua ion is
desc ibed in he nex schema ic.
Fig. 1. The schema ic si ua ion o pa allel dis u bing and dis u bed
pai s
We can assume he si ua ion wi h wo pa allel pai s
in a cable, whe e he nea -end o he dis u bing pai
con ains he sou ce o signal u1 wi h o al cu en i1. The
pai is co ec ly e mina ed on i s a -end by he
cha ac e is ic impedance o his pai ZC1. The dis u bed
pai is p ope ly e mina ed on i s bo h ends by i s
cha ac e is ic impedance ZC2. The p opaga ion cons an o
dis u bing pai is
g
1, while he p opaga ion cons an o
dis u bed pai is
g
2. The leng h o bo h pai s is l. The
in ini e elemen
D
x con ains a o al capaci i e unbalance
Cub
D
x h ough which he capaci i e c oss alk cu en iC
p opaga es om he dis u bing pai in o he dis u bed pai .
This elemen also con ains he induc i e unbalance Mub
D
x,
which causes he o igina ion o induc i e c oss alk ol age
uM in he dis u bed pai . The sum o bo h c oss alk
dis u bances is he o al c oss alk cu en ix, which
p opaga es along he dis u bed pai o i s nea -end as a
cu en iN whe e i causes he nea -end c oss alk, NEXT
and ano he pa p opaga es also o he a -end as a cu en
iF whe e i causes he a -end c oss alk, FEXT.
The c oss alk cu en iCx, which comes om he
capaci i e unbalance Cub
D
x, can be exp essed [9]:
2
12C
ub
Cx
Cx Z
xCj
u
i
+
D
=
w
.(1)
The e m wi h ZC2 in he denomina o can be
neglec ed and he exp ession simpli ied:
CxubCx uxCji ×D=
w
.(2)
The ol age p esen ed in he capaci i e unbalance in
he elemen
D
x is gi en:
x
CCx eiZu 1
11
g
-
××= .(3)
and he e o e he equa ion (1) can be exp essed:
x
CubCx eiZxCji 1
11
g
w
-
×××D= .(4)
This cu en is di ided; one pa p opaga es o he
nea -end, while he second one o he a -end o he
dis u bed pai , as i is p esen ed in [9], [15]. The
di e ence be ween he FEXT cu en and he NEXT
cu en is gi en by he posi i e/nega i e di ec ion o he
induc i e unbalance cu en - iMF. The cu en , which is
caused by capaci i e unbalance and appea s a he a -end
-iCF, can be he e o e calcula ed:
)1(
11 21
2
1xx
CubCF eeiZxCji --- ××××D=
gg
w
. (5)
I is also possible o exp ess he c oss alk ol age
uMx, which comes om he induc i e unbalance Mub
D
x [9]:
x
ubMxubMx eixMjixMju 1
1
g
ww
-
××D×=×D= . (6)
The e o e he c oss alk cu en coming om he
induc i e unbalance and appea ing a he a -end - iMF can
be calcula ed:
)1(
1
22
21
22
xx
C
ub
C
Mx
MF eei
Z
xM
j
Z
u
i--- ×××
D
-=-=
gg
w
. (7)
Based on he p e ious equa ions (5) and (7) i is
possible o de i e he summa y a -end c oss alk cu en
om bo h unbalances o igina ing in he elemen
D
x:
.
2
1
2
1
)1(
121 ÷
÷
ø
ö
ç
ç
è
æD
-D××××=
=+=
---
C
ub
ubC
xx
MFCFF
Z
xM
xCZeeij
iii
gg
w
(8)
16 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
2.1 S anda d simple FEXT model
To ob ain he s anda d FEXT model, i is necessa y
o modi y he equa ion (8) and o conside some
simpli ying assump ions, as desc ibed in [5]. Capaci i e
Cub and induc i e Mub unbalances in a eal me allic cable
a e gene ally a ying along he cable, so hey can be
exp essed as a unc ion o hei posi ion x. Bu in case o
he simpli ied s anda d FEXT model i is possible o
assume bo h unbalances cons an and equal o hei mean
alues o he whole leng h o a cable l, so hey a e
cons an and independen on hei posi ions x. Thanks o
his assump ion, i is possible o conside he elemen
D
x
as in ini ely sho and o exp ess i by using di e en ial
e m dx. Ano he simpli ica ion conside s he ansmission
pa ame e s o bo h pai s wi hin he same cable o be
iden ical (
g
,ZC).
Acco ding o hese simpli ica ions, he equa ion (8)
can be modi ied:
÷
÷
ø
ö
ç
ç
è
æ-×××××= ---
2
)1(
12
1
C
ub
ub
xx
CF Z
M
CeeiZji
gg
w
. (9)
The o mula (9) can be u he simpli ied:
´
1
2
1
´
2
1Ceuj
Z
M
Ceuji l
Calancesumma yunb
C
ub
ub
l
Fx ×××=
÷
÷
ø
ö
ç
ç
è
æ-×××= -
-
-
gg ww
4434421
. (10)
The FEXT c oss alk powe ans e unc ion is
de ined [14]:
()
(
)
()
P
P
H
N
FEXT
FEXT
1
2=.(11)
In which PFEXT( ) ep esen s he powe unc ion o
a -end c oss alk and P1( ) he inpu powe unc ion a he
nea -end o a dis u bing pai . The FEXT powe ans e
unc ion can be ob ained by an in eg a ion o c oss alk
con ibu ions (10) o he leng h l [8]:
() ()
ò-
-××××=×=
l
x
CFCFEXT dxeC uZiZ P
0
22´2
1
22
g
w
.(12)
Assuming elec ically long symme ical pai s [8]
and (12) i is possible o exp ess FEXT powe ans e
unc ion (11) as:
() () ()
()
.
2
1
2
2´2
1
2
2
1
2
2
C
C
C
FC
FEXT
Z
u
HlC uZ
Z
u
iZ
H ×××××
=
×
=
w
() ()
2
2´2
22 HlCZ H CFEXT ××××=
w
. (13)
()()
.
2
2
2 Hl K H FEXTFEXT ×××=
Whe e KFEXT is a c oss alk pa ame e (a cons an o
he selec ed combina ion o pai s), which ep esen s he
summa y a e o capaci i e and induc i e couplings
be ween speci ic pai s. |H( )|2 is he powe ans e
unc ion o a pai , is he equency and l ep esen s he
leng h o bo h pai s. Following he p e ious modi ica ions,
i is ob ious, ha [8]:
2´2
24CZK CFEXT ××=
p
.(14)
The e o e KFEXT c oss alk pa ame e is exp essed
h ough he in eg a ion o capaci i e and induc i e
unbalances in (12). The equa ion (13) ep esen s he
s anda d simple FEXT model, which is p esen ed in [5].
3. FEXT model based on cascade
ma ices and capaci i e unbalances
The p e iously de i ed s anda d FEXT model uses
se e al simpli ica ions and assump ions. The mos
nega i e condi ion is he conside a ion o cons an
capaci i e and induc i e unbalances and hei
independence on he posi ion x. Howe e , o accu a e and
ealis ic FEXT modeling, i is necessa y o assume a ying
unbalances along a cable. Ne e heless, analy ical
exp ession o hese unc ions Cub(x),Mub(x), could be
ma hema ically qui e di icul . The alues o hese
unc ions a e p obably a ying pseudo- andomly in he
in e al gi en by manu ac u ing ole ances and o he
in luences in a cable. I is possible o assume ha he
cha ac e o hese unc ions would ha e p obably he
beha io o a no mal dis ibu ion wi h he de ia ion gi en
by hese ole ances and impe ec ions o a cable. F om his
eason, i is no possible o use he ope a ion o in eg a ion
o he c oss alk con ibu ions.
The main idea o his p oposed FEXT model is
di iding he whole cable in o se e al ansmission sub-
sec ions wi h ansmission lines, c oss alk coupling and
he b idge aps om he unused ends o bo h symme ical
pai s. Each sec ion is desc ibed by i s cascade ma ix and
he inal c oss alk cu en is calcula ed by hei
mul iplica ion. Fi s , se e al assump ions a e necessa y.
The model does no include he impac o a c oss alk
h ough he hi d lines (ci cui s) in a cable, o an indi ec
e ec o he c oss alk o igina ing om e lec ions om he
ends o he unused lines. To al c oss alk coupling is
summa ily exp essed by i s induc i e and capaci i e
componen s, bu he induc i e pa is app oxima ed by he
capaci i e unbalance. This assump ion is based on
p e ious heo e ical conside a ions [10], acco ding o
which he impac o induc i e coupling can be modeled by
an addi ional capaci ance unbalance and hese wo pa s
a e included in he summa y capaci i e unbalance C´ [11].
The las simpli ica ion o he model conce ns he ques ion
o simula ion and de e mina ion o he capaci i e
INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 17
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
unbalance. I could be e y complica ed o exp ess i s
alues ma hema ically. Mo eo e , hese alues a e usually
pseudo- andom and a e in luenced by many in e nal
and/o ex e nal e ec s. Tha is why a simple me hod by
gene a ing pseudo- andom alues using o mulas o
no mal dis ibu ion and he p ope s a is ical alues is used
in he model. These assump ions will be u he e i ied by
compa ing he esul s o simula ions wi h he eal
cha ac e is ics o c oss alk measu ed o a cable wi h
TCEPKPFLE speci ica ion.
Based on he p e ious assump ions i is possible o
p o ide a schema ic model o he whole si ua ion, Fig. 2.
S anda d models o c oss alk be ween wo pai s a e
usually based on he desc ip ion o 4-po ne wo k, o wo
coupled 2-po ne wo ks, bu o he basic c oss alk
modeling, he simple 2-po model is su icien .
Fig. 2. The cascade elemen s o p oposed FEXT model
The signal gene a o wi h ou pu ol age u0 and
in e nal impedance Zg is loca ed a he inpu o dis u bing
pai . The inpu impedance o he whole sys em Z1 p o ides
he o al cu en i1 and ol age u1. The summa y capaci i e
coupling C´, which is ep esen ed by he impedance Zub, is
si ua ed in he posi ion x om he beginning o a cable and
l-x om he a -end o a cable, while l is he leng h o a
cable. This unbalance is si ua ed in se ies wi h he
gene a o om he pe spec i e o FEXT c oss alk. The
i s b idge ap, which consis s o he unused pa o he
dis u bing pai and has leng h l-x, is connec ed o he
unbalance in pa allel. Also he unused sec ion o he
dis u bed pai , which o ms he second b idge ap o he
leng h l, is connec ed in pa allel. The es o he dis u bed
pai wi h leng h l-x is connec ed in se ies om he
pe spec i e o FEXT c oss alk. The a -end o he
dis u bed pai is e mina ed by he load impedance ZZ. The
p opaga ion cons an o dis u bing pai is
g
1 and dis u bed
pai
g
2. The ends o bo h b idge aps a e opened, bu he
model could be u he modi ied by e mina ing he aps by
impedances ZC1 and ZC2.
Now, i is possible o exp ess he cascade ma ices o
he si ua ion desc ibed in he Fig. 2 using p e ious
o mulas.
The cascade ma ix o he ansmission sec ion o
dis u bing pai wi h he leng h x:
(
)
(
)
(
)
(
)
(
)
()( )
() ()( )
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
×
×
×××
=x
Z
x
x Zx
P
C
C
1
1
1
111
1cosh
sinh
sinhcosh
g
g
gg
. (15)
The cascade ma ix o he i s b idge ap, which
consis s o he unused sec ion o dis u bing pai wi h he
leng h l-x:
() ( )( )( )
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
-××
=1
co h
101
11
1
xl Z
O
C
g
.(16)
The cascade ma ix o he coupling impedance Zub:
÷
÷
ø
ö
ç
ç
è
æ
=10
1ub
Z
V.(17)
In which he impedance Zub acco ding o he p e ious
assump ions can be calcula ed:
´
1
Cj
Zub
w
=.(18)
The cascade ma ix o he second b idge ap, which
ep esen s he unused nea -end o he dis u bed pai wi h
he leng h x:
() ()( )
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
××
=1
co h
101
22
2
x Z
O
C
g
.(19)
And inally, he cascade ma ix o he es
ansmission pa o he dis u bed pai , which is
e mina ed by he impedance ZZ a i s a -end:
(
)
(
)
(
)
(
)
(
)
(
)
(
)
( )( )( )
() ( )( )( )
÷
÷
÷
ø
ö
ç
ç
ç
è
æ
-×
-×
-××-×
=xl
Z
xl
xl Zxl
P
C
C
2
2
2
222
2cosh
sinh
sinhcosh
g
g
gg
.(20)
The esul ing cascade ma ix W can be exp essed by
he mul iplica ion o p e ious cascade ma ices o all
sec ions:
.
.
2221
1211
2211
÷
÷
ø
ö
ç
ç
è
æ
=
××××=
ww
ww
W
POVOPW
(21)
The p ima y pa ame e s can be calcula ed using
B i ish Telecom model and by using app op ia e o mulas,
i is possible o ob ain he cha ac e is ic impedances ZC1,
ZC2 and he p opaga ion cons an s
g
1 and
g
2. Acco ding o
(14), i is possible o calcula e summa y capaci i e
unbalance om he c oss alk pa ame e KFEXT. Based on
he p e ious conclusions abou he in luence o in e nal
s uc u e o a cable on esul ing FEXT c oss alk [3], he
KFEXT pa ame e can be calcula ed o h ee main
ca ego ies - he pai s wi hin he same subg oup, pai s om
18 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
wo su ounding subg oups and pai s om wo dis an
subg oups. The alue o capaci i e unbalance C´ o each
ca ego y can be he e o e calcula ed using measu ed KFEXT
pa ame e and equa ion (22), which is based on (14):
2
2
´
4
p
×
=
C
FEXT
Z
K
C,
[
]
W,;/ kmkmF .(22)
The a -end c oss alk cu en , which comes om one
unbalance si ua ed in he posi ion x, can be calcula ed:
()
()
() ()
() () () ()(
() () () () ()
)
....
...
.
2221
12110
2221
1211
1
1
w Z w Z Z
w w Z Z
Z Z
u
i
i
u
ww
ww
i
u
gZg
ZZ
Zg
Fx
Fx
Fx
×+××
++××
+
=
÷
÷
ø
ö
ç
ç
è
æ
×
÷
÷
ø
ö
ç
ç
è
æ
=
÷
÷
ø
ö
ç
ç
è
æ
(23)
To calcula e FEXT a enua ion, i is necessa y o
summa ize all con ibu ions o c oss alk cu en s o he
whole leng h l:
(
)
()
(
)
()
å
=
l
FxF
u
i
u
i
00
.(24)
The e o e, he FEXT a enua ion can be exp essed:
()
(
)
()
P
P
A
FEXT
N
FEXT
1
log10×=
[
]
WWdB ,; . (25)
Wi h e e ence o (11), he equa ion (25) can be
modi ied by exp essing he powe unc ions wi h ol ages
and p ope impedances:
() ()
()
(
)
()
()
()
Z
u
Z
u
P
P
A
C
F
C
FEXT
N
FEXT
2
2
1
2
0
1log10log10 ×=×= . (26)
Assuming he same cha ac e is ic impedances Zc o
bo h pai s in a cable (Zc1=Zc2=Zc), i is possible o simpli y
(26) and o exp ess he summa y FEXT ol age uF( ) wi h
(24):
()
(
)
()
()
()
()
()
() ()
() () ()
()
.log20
.log20log10
0
0
0
0
2
2
1
2
0
å
××
×=
×=×=
l
Fx
z
FEXT
F
C
F
C
FEXT
u
i
u Z
u
A
u
u
Z
u
Z
u
A
(27)
Which inally leads in o he exp ession o FEXT
a enua ion:
() () ()
()
å
×
×=
l
Fx
z
FEXT
u
i
Z
A
0
1
log20
[
]
dB . (28)
3.1 Resul s o p oposed me hod o FEXT
modeling
The esul s ob ained by p esen ed me hod o FEXT
c oss alk modeling a e p esen ed o me allic cable wi h
he speci ica ion TCEPKPFLE 75x4x0.4 and leng h l =
400 m. The i s s ep equi es di iding he cable in o
se e al sub-sec ions wi h di e en c oss alk couplings. Fo
ha eason, he whole cable was di ided in o sec ions o 1
m each, which means 399 capaci i e unbalances (400-1)
o he whole cable wi h he o leng h 400 m. The c oss alk
cu en s om all sec ions a e hen summa ized. Acco ding
o (22), i is possible o calcula e summa y capaci i e
unbalance om he c oss alk pa ame e KFEXT. Based on
he p e ious conclusions abou he in luence o in e nal
s uc u e o a cable on esul ing FEXT c oss alk, he KFEXT
pa ame e can be calcula ed o h ee main ca ego ies - he
pai s wi hin he same subg oup, pai s om su ounding
subg oups and pai s om dis an subg oups. The alue o
capaci i e unbalance C´ o each ca ego y can be he e o e
calcula ed using measu ed KFEXT pa ame e and equa ion
(22). The KFEXT pa ame e is usually de i ed o a cable
wi h he leng h o 1000 m ha ’s why i is necessa y o
p o ide ecalcula ion o he si ua ion o capaci i e
unbalance o sec ions - 1 m in his case, he o mula
comes om he exp ession o FEXT [8], [12]. The
equa ion (22) could be hence modi ied o ge he capaci i e
unbalance o he e e ence leng h o 1m:
[ ] [ ]
.
10004
1000
4
1000
/
/
2
2
2
2
´
´
××
=
=
×
==
p
p
C
FEXT
C
FEXT
Z
K
Z
K
kmFC
kmFC (29)
Tab. 1. The calcula ion o capaci i e unbalances
The ecalcula ion K
FEXT
C´ [F/√m]
Pai s wi hin he same subg oup 9.9462.10-17 5.0194.10-13
Pai s om su ounding
subg oups
1.292.10-17 1.8090.10-13
Pai s om dis an subg oups 3.2040.10-18 9.0087.10-14
As i was desc ibed be o e, he beha io o capaci i e
unbalance is a ying along he cable in he in e al o
alues wi h pseudo- andom cha ac e is ic, which can be
p edic ed using he o mulas o no mal dis ibu ion.
The e o e, he alues o capaci i e unbalance C´ in he
Tab. 1 we e subsequen ly used as a s anda d de ia ion o

INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011 19
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
gene a ing he cha ac e o capaci i e unbalance C´(x)
wi h he ze o mean alue. The alues o pa ame e KFEXT
we e ob ained om measu ed cha ac e is ics o
TCEPKPFLE cable and by using s a is ical p ocessing.
Based on p e ious equa ions o p oposed ad anced FEXT
model (21), (23), (24) and (28) oge he wi h he pseudo-
andomly gene a ed C´(x) cha ac e is ic acco ding o he
alues in Tab. 1, se e al examples o esul s we e
ob ained. These esul s we e compa ed wi h he measu ed
cha ac e is ic o a cable and also wi h he s anda d FEXT
model exp essed by (13). The compa isons o di e en
in e nal ca ego ies a e p esen ed in he ollowing g aphs
Fig. 3, 4 and 5. All esul s a e gi en o he equency
band om 12.9375 kHz o app ox. 5.89 MHz.
Fig. 3. The compa ison o p oposed FEXT model, s anda d
FEXT model and measu ed esul s o pai s wi hin he same
subg oup
Fig. 4. The compa ison o p oposed FEXT model, s anda d FEXT
model and measu ed esul s o pai s om wo su ounding subg oups
Fig. 5. The compa ison o p oposed FEXT model, s anda d FEXT
model and measu ed esul s o pai s om wo dis an subg oups
4. Conclusion
The equa ions om (15) o (29) we e implemen ed
in o he simula ion p og am in MATLAB en i onmen .
Based on he p e ious conclusions and measu ed esul s
p esen ed in chap e 2 o speci ic me allic cable o
TCEPKPFLE ype, he s a is ical pa ame e s o
gene a ing he alues C´ capaci i e unbalance o each
cons uc ional ca ego y. These alues we e subsequen ly
used in he cascade ma ices and equa ions o p oposed
FEXT model o ob ain inal esul s o simula ions.
P e ious cha ac e is ics in he Fig. 3, 4 and 5 gi e an
example o p esen ed me hod o FEXT modeling, s anda d
FEXT model and measu ed esul s o he equency band
o app ox. 6 MHz. I is ob ious ha unlike he s anda d
FEXT model (p esen ed in he g aphs as a ed line), he
p oposed modeling me hod p o ides mo e accu a e and
ealis ic esul s. The s anda d model comes om only
a e age alues o he whole cable, he inno a i e me hod
based on he a ying unc ion C´(x) o bo h unbalances
oge he wi h he in luence o in e nal s uc u e o he
cable p o ides inal esul s e y close o he cha ac e is ics
in eal applica ions. The p oposed model b ings mo e
accu a e esul s and eaches ealis ic beha io o he
ansmission and c oss alk cha ac e is ics in a cable. The
accu acy o he model could be u he imp o ed by mo e
complex me hod o capaci i e unbalance C´(x) simula ion
and calcula ion as well as by espec ing o he in luences in
mul i-pai and mul i-quad me allic cables. The p oposed
model could also se e o he simula ions and calcula ions
o FEXT c oss alk and o p epa e ealis ic esul s o
implemen ing VDMT modula ion in o VDSL2 digi al
lines. The esul s o p esen ed model we e used in [16],
[17] o es ima e and calcula e he ansmission capaci y o
VDSL2 lines wi h VDMT modula ion o FEXT
cancella ion.
Acknowledgmen s
This wo k was suppo ed by he G an Agency o he
Czech Technical Uni e si y in P ague, g an No. SGS
10/275/OHK3/3T/13.
Re e ences
[1] ŠIMÁK, B., VODRÁŽKA, J., SVOBODA, J. Digi ální účas nické
přípojky xDSL, 1. díl: Me ody přenosu, popis přípojek HDSL,
SHDSL, ADSL, VDSL. Sdělo ací echnika, P aha 2005. ISBN 80-86-
64507-X.
[2] VODRÁŽKA, J., ŠIMÁK, B. Digi ální účas nické přípojky xDSL, 2.
díl: Vlas nos i přenoso ého p os ředí a jejich měření. Sdělo ací
echnika, P aha 2007. ISBN 80-86645-16-9.
[3] LAFATA, P., VODRÁŽKA, J. Simula ions and S a is ical
E alua ions o FEXT C oss alk in xDSL Sys ems Using Me allic
Cable Cons uc ional A angemen . TSP - 31s In e na ional
Con e ence Telecommunica ions and Signal P ocessing [CD-ROM].
20 INFORMATION AND COMMUNICATION TECHNOLOGIES AND SERVICES, VOL. 9, NO. 1,MARCH 2011
©2011 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING ISSN 1804-3119
Budapes : Asszisz encia Sze ező K ., 2008, ISBN 978-963-06-5487-
6.
[4] GINIS, G., CIOFFI, J., M. Vec o ed T ansmission o Digi al
Subsc ibe Line Sys em. IEEE JSAC [online], [ci . 2011-01-27].
A ailable om:
<h p://isl.s an o d.edu/~cio i/dsm/ ec o pap/ ec o .ps>.
[5] CHEN, W., Y. DSL: Simula ion Techniques and S anda ds
De elopmen o Digi al Subsc ibe Line Sys em. Macmillan
Technology Se ies, Indianopolis, USA, 1998. ISBN 1-57870-017-5.
[6] HUGHES, H. Telecommunica ions Cables: Design, Manu ac u e
and Ins alla ion. John Wiley&Sons L d., Chiches e , England, June
1997. ISBN 0-471-97410-2.
[7] KVASIL, J. Teo ie sdělo acích edení. Přednášky. Naklada els í
ČVUT, P aha 1980.
[8] RAUSCHMAYER, D., J. ADSL/VDSL P inciples: A P ac ical and
P ecise S udy o Asymme ic Digi al Subsc ibe Lines and Ve y High
Speed Digi al Subsc ibe Lines. Macmillan Technical Publishing,
Indianapolis, USA, No embe 1998. ISBN 1-57870-015-9.
[9] SCHLITTER, M. Telekomunikační edení. Přednášky. Naklada els í
ČVUT, 2. ydání, P aha 1986. Číslo publikace 5615.
[10] KAISER, K., L. T ansmission Lines, Ma ching, and C oss alk.
Taylo &F ancis CRC, Boca Ra on, USA, 2006. ISBN 0-84-936362-4.
[11] STARR, T., SORBARA, M., CIOFFI, J., M., SILVERMAN, P. DSL
Ad ances. Uppe Saddle Ri e , USA: P en ice Hall, 2002. ISBN 0-13-
093810-6.
[12] STARR, T., CIOFFI, J., M., SILVERMAN, P., J. Unde s anding
Digi al Subsc ibe Line Technology. P en ice Hall PTR, Uppe
Saddle Ri e , USA, Janua y 1999. ISBN 0-13-780545-4.
[13] LAFATA, P., VODRÁŽKA, J. Modeling o T ansmission Func ions
and C oss alk in Me allic Cables o Implemen a ion o MIMO
Concep . Radioenginee ing. 2009, ol. 18, no. 4, p. 491-496. ISSN
1210-2512.
[14] CIOFFI, J., GOLDEN, P., DEDIEU, H., JACOBSEN, K.
Fundamen als o DSL Technology. Aeu bach Publica ions, 2005.
ISBN 978-0-8493-1913-6.
[15] ONODA, S., INOUE, K., AITA, K., NAKADA, T. C oss alk Con ol
o High Speed LAN Connec o s. IEICE T an. on Elec onics, ol.
E90-C, no. 7, July 2007.
[16] LAFATA, P., JAREŠ, P. Use o C oss alk Ad anced Model o
Modeling o VDMT Modula ion Bene i s. In TSP 2010 - 33 d
In e na ional Con e ence on Telecommunica ions and Signal
P ocessing [CD-ROM]. Budapes : Asszisz encia Sze ező K ., 2010,
p. 386-390. ISBN 978-963-88981-0-4.
[17] LAFATA, P., JAREŠ, P. Analysis o Simula ion Me hods o Fa -end
C oss alk Cancella ion. Radioenginee ing. 2011, ol. 20, no. 1. ISSN
1210-2512.
Abou Au ho s ...
Pa el LAFATA was bo n in České
Budějo ice, Czech Republic in
1982. He ecei ed his Mas e (Ing.)
deg ee in 2007 and is expec ed o
ecei e Ph.D. in 2011 a FEE,
Czech Technical Uni e si y (CTU)
in P ague, specializing in
Telecommunica ion Enginee ing.
Cu en ly he is an assis an
p o esso and junio esea ch
assis an a he Depa men o Telecommunica ion
Enginee ing o he CTU in P ague. He is a membe o he
T ansmission Media and Sys ems scien i ic g oup a he
Depa men . His esea ch ac i i ies a e ocused mainly on
p oblems o dis u bance and c oss alk in me allic cables
o digi al subsc ibe lines and op ical access ne wo ks.