Far-end crosstalk modeling based on capacitive and inductive unbalances between pairs in a cable

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.