Transmitting Conditions of Cable Tree

Transcription

1 Transmitting Conditions of Cable Tree Vaclav Kreelka and Miloslav Filka FITCE CZ, Mahenova, Brno Deartment of Telecommunications, Purkynova 8, Brno Faculty of Electrical Engineering and Communication Brno University of Technology Abstract. Research and develoment of ectrum Administration for metallic access networks analyzed ossibilities of matrix reresentation for transmission circumstances. ome interesting remarks were gained as a side outut of this work. Keywords: Cable tree, signal sectrum function, noise sectrum function, transmitting function, PWD. Cable tree Key term Cable tree is defined in the last mile toology. Cable tree itself consists of sequence couled cable sections concentrated in site of Main distribution frame in common trunk cable, see Fig.. ote: Cable tree may be equied by otical network units OU, where otical signal is transformed into electrical and vice versa resectively; and which is in OU-MDF received or transmitted into/from airs of tided metallic cable. TP- TP- MDF TP-3 Fig.. Basic architecture of cable tree TP-5 TP-4

2 Analytical Performance of Useful ignals & Disturbing oises Circumstances of mutual influences uon symmetric metallic cable are erformed in Fig. : Downstream Pair k Pair m Ustream Fig. : ignals a Coulings between two Pairs The simlest generalizable case of bidirectional transmission alongside two symmetric airs is erformed here. These airs are terminated in exchange site in v DLAM, in subscriber site in two network terminations. Three transmission characteristics belong to each of four here erformed terminating oints: Transmitted sectrum of signals ) or maybe f) on wanted disturbing noise ) Receiver sensitivity R) as well as inter air crosstalk coulings ear end crosstalk EXT) C ) Far end crosstalk FEXT) C F ) Indexes k and m define mutual osition of airs; indexes D and U discriminate downstream and ustream transmission directions; indexes T and R assign aroriate characteristics to transmission and receive of signal. Transmitting circumstances in site of subscriber termination, in Fig. ointed T are exressed by following equation: k ) T CTk ) k ) a) Equally in site of exchange termination DLAM) is valid: k ) T CTk ) k ) b) where T CT is a vector of cable tree transmitting functions

3 oise circumstances in site of T may be described by following equation: k ) m ) C mk ) + m ) C Dmk ) a) as well as dually in site exchange termination DLAM k ) m ) C Umk ) + m ) C mk ) b) uch circumstances are only excetional in reality; maybe only in last cable section led from subscriber cabinet suosing one-quad subscriber cable and both airs exloited by broadband service. All coulings in real cable tree are multilied and combined many times. We are able to exress these circumstances by matrixes. These matrixes will be of equal order to number of airs in trunk cable of cable tree. This number of airs may reach hundreds as well as exceed thousand in metroolitan networks. Comosition of such matrix exceeds real ossibility of calculation. ote: Variable will be again left out in following text The equation a is able to be erformed by roduct of diagonal matrix: TCT T CT TCT 3a) Equally equation b: TCT T CT T CT 3b) The equation a erforming noise circumstances by matrixes of th order for airs of cable tree trunk cable as a roduct of left diagonal signal matrix multilied by right crosstalk matrix with zero diagonal. Resulting noise matrix is again with zero diagonal in accordance with multilying rule for matrixes.

4 ) ) ) ) + + ) ) U U U U U U 4a) The equation b may be erformed dually as follows: ) ) ) ) U U U U U U + ) ) 4b)

5 ote: Due to the fact of invalidity commutative law for multilying of matrixes, this sequence should be abided. On the other hand change of this obligatory sequence leads to same henomenon as by changing disturbing and disturbed lines, called as crosstalk measurement straight and across or marked and x. 3 The Condition for Information Recoverability The coule of equations and is simlified for next consideration, again missing variable for useful signal in downstream as well as ustream: Ri T CT Ti where i D, U 5) and analogous noise circumstances in site of subscriber exchange: + D 6a) and again dually in site of network termination. U + 6b) The condition for information transmission secured in the simlest configuration as to Fig. in sufficient quality may be erformed for downu)stream as follows: U)k ) R DU)k ) for Э Ω 7) These inequalities may be overwritten for cable tree generally: R D 8a) R U 8b) Induction of both inequalities 8 into coule of equations 6 enables determination for highest assigned signals, there define masks of Power sectral density of transmitted signals: and U R U UR D D for Э Ω 9a)

6 D U R D R U U for Э Ω Inequality system 9 may be altered by other constants roduced by legislative measures of non-technical character. Matrixes of higher order are roosed to be calculated generally; therefore the volume of necessary calculations may exceed realizable measures. List of symbols: D U R D R U T CT - matrix of transmitting functions of FEXT in downstream - matrix of transmitting functions of FEXT in ustream - matrix of transmitting functions of EXT in downstream - matrix of transmitting functions of EXT in ustream - noise sectrum vector in site of receivers in downstream - noise sectrum vector in site of receivers in ustream - receiver sensitivity vector in downstream - receiver sensitivity vector in ustream - signal sectrum vector in site of receivers in downstream - signal sectrum vector in site of receivers in ustream - signal sectrum vector in site of transmitters in downstream - signal sectrum vector in site of transmitters in ustream - transmitting functions vector of cable tree - radian frequency Ω - frequency interval transmitted band) 9b) References [] Křeelka, V.: Doctoral Theses, VUT FEKT ÚTKO, Brno, 5 [] Meninger, M.: Úvod do srávy sektra metalické sítě, Telekomunikace /3, Praha, I 4-59