Crosstalk Identification and Modified Crosstalk Model for xDSL-Channels

ALFRED VOGLGSANG, RUDI KNORR, STEFAN LINDENMEIER* Fraunhofer Institute for Communication Systems ESK Hansastrasse 32, 80686 Munich GERMANY *DaimlerChrysler Research and Technology, Wilhelm-Runge Str.11, 89081 Ulm, Germany [email protected], [email protected] http://www.esk.fhg.de [email protected] http://www. DaimlerChrysler.com

Abstract: - The performance of xDSL-systems is determined by the existing crosstalk interference. xDSL- systems are optimised to the existing crosstalk interferences in a cable bundle. The crosstalk interference leads to unwanted restrictions in the number of xDSL-systems, which are able to operate parallel in a cable bundle. Realistic crosstalk models are necessary for xDSL-system tests and for the design of future telephone networks. With the modified crosstalk model introduced in this paper, the crosstalk coupling functions between the twisted pairs of a cable bundle can be estimated with low effort in the frequency domain. The crosstalk coupling functions are calculated from the measured spectra of the transmitted signals along the interfering wire pairs and from the spectra of the crosstalk interference on the interfered twisted pair. The modified crosstalk model can be implemented in a flexible impairment generator, which reproduces realistic crosstalk interferences for xDSL-system tests in the laboratory. Furthermore, the modified crosstalk model can be deployed in a tool for design and maintenance of telephone networks.

Key-Words: - xDSL, crosstalk model, crosstalk identification, multiple linear regression

1 Introduction opposite cable-end. Fig. 1 shows the principle of Under the heading of xDSL, the technologies ISDN, crosstalk between two twisted pairs of a multi-pair ADSL (asymmetric digital subscriber lines), HDSL cable. (high-bit-rate digital subscriber line), SDSL (single- NEXT-source FEXT-source pair digital subscriber line), SHDSL (single-pair high- (xDSL-system) Cable binder (xDSL-system) speed digital subscriber line) and VDSL (very-high- pair 1 bit-rate digital subscriber line) are summarised. NEXT (f) 12 FEXT12 (f) xDSL-techniques use the existing telephone network pair 2 for data-transfer. The telephone cables consist of twisted and unshielded wire pairs, which have been disturbed xDSL-system xDSL-system conceived for the transfer of analogous voice signals in a frequency-band from 300 Hz to 3.4 kHz Fig. 1 Crosstalk in multi-pair telephone originally. In contrast to the analogous telephone cable signals, the xDSL-techniques also use frequencies over 3.4 kHz. At the receiver, NEXT interferences are more The transmit signals on the twisted pairs generate powerful than FEXT interferences [3]. Thus, the electromagnetic fields, which surround the twisted following investigation is focusing on the investigation pairs. As a consequence of the existing capacitive and of NEXT. inductive coupling between the twisted pairs, the The crosstalk interferences increase together with electromagnetic fields induce currents as well as the frequency [1]. Since xDSL-systems (an xDSL- voltages on adjacent wire pairs. This effect is called modem on the customer side or an xDSL-line card of crosstalk. The twisting of the wire pairs reduces the a Digital Subscriber Line Access Multiplexer crosstalk interferences in a cable bundle. Crosstalk is (DSLAM) in the central office) use transmit differentiated into near-end crosstalk (NEXT) at the frequencies up to several MHz, the performance of an cable-start and far-end crosstalk (FEXT) at the xDSL-system – e.g. the reach or the data thruput – is dominated by the existing crosstalk influences. 2 Standard Crosstalk Model The design of xDSL-systems is optimised with The capability of xDSL-systems is tested also in the regard to the present telephone loops and the existing presence of crosstalk interferences, with the aid of crosstalk interference. Thereby the performance of an loop simulators and impairment generators. For the xDSL-system is tested with standardised tests. In standardised system tests, the present impairment these tests the crosstalk interferences are simulated generators reproduce crosstalk interferences in with crosstalk models of the xDSL-standards of ETSI, accordance with the 1% worst case power-sum model ANSI or ITU-T [1] - [4]. The problem is that of the xDSL-standards of ETSI, ANSI, ITU-T [2] - crosstalk interferences on real telephone loops can [4]. easily be several dB lower than the interferences The 1% worst case power-sum model is based on simulated with the standardised models [5]. the crosstalk model of J. H. W. Unger [8]. The overall With the modified near-end crosstalk model crosstalk PSD on a twisted pair is the sum of the signal (NEXT model) proposed in this paper, the NEXT PSDs on the adjacent pairs multiplied by the interferences measurable on real telephone loops associated NEXT coupling functions [1], [9]. The could be realistically estimated and simulated. This ANSI T1E1.4 NEXT power-summed coupling modified NEXT model can be deployed in a flexible function of the 1% worst case power-sum model is [1], impairment generator. The flexible impairment [2]: generator reproduces the crosstalk interferences measurable on real cable bundles for system tests in 2 0.6 − H ( f ) = K ⋅ f 1.5 = ()N ⋅10 13 ⋅ f 1.5 . (1) the laboratory. The modified NEXT model can also NEXT NEXT 49 be used in a tool for network planning and the maintenance of xDSL-loops. The NEXT interferences Equation (1) represents the NEXT power-summed appearing in real cable bundles can be evaluated with coupling function for a 50-pair cable. While N is the this tool. Thus, cable bundles can be covered with number of crosstalk interferers, K NEXT is denoted as xDSL-systems so that the mutual crosstalk the coupling constant. The number of interfering interferences of the transmission systems in the cables xDSL-systems is taken into account by the size N are reduced. The modified crosstalk model, which ()≤ . works in the frequency domain, is of relatively low N 50 complexity for an efficient implementation in soft- The parameters of the 1% worst case power-sum and hardware. model were empirically fixed after measurements, so For the derivation of the model-parameters, the that 99% of the examined telephone cables show interfering transmit signals on the adjacent twisted crosstalk interferences that are either smaller than or pairs and the crosstalk signal on the interfered pair equal to the 1% worst case model simulated crosstalk resulting from it are measured and stored. In practice, interferences [6], [7]. the power spectra densities (PSDs) could be measured If the crosstalk interferences measurable in real with a corresponding measurement module, which cable bundles are smaller than the simulated could be implemented into a DSLAM. The near-end interferences, xDSL-systems are tested in this case in crosstalk coupling functions (NEXT coupling a too pessimistic crosstalk environment. However, functions) between the interfering and the interfered manufacturers of xDSL-systems are interested in the wire pairs are calculated with the stored measuring performance of an xDSL-system under realistic data. The NEXT coupling functions represent the interferences. Furthermore, the providers of xDSL- power transfer function between the twisted pairs at services must be able to appraise the crosstalk the same cable end, in this context. interferences appearing on the cable beforehand, in This paper is organised as follows. Chapter 2 order to optimally cover the cable bundles with reviews the present NEXT crosstalk model of the xDSL-systems. If the estimation of the potential xDSL-standards. The modified crosstalk model is crosstalk interferences occurs with the 1% worst case described in chapter 3. Chapter 4 shows the NEXT power-sum model, a too pessimistic crosstalk coupling functions simulated with the modified environment is conceivably assumed. This leads to an NEXT crosstalk model. Chapter 5 contains the inefficient utilisation of the cable. summary of the paper. 3 Modified Crosstalk Model The crosstalk interferences appearing in a cable bundle depend on the cable construction, the wire-diameters, the isolation and the twisted-length. However, tolerances of the cable manufacturing may lead to different crosstalk behaviour within the same cable In the modified crosstalk model each NEXT coupling type. The crosstalk coupling functions of real cable is individually taken into account by a NEXT transfer 2 bundles can be individually determined with the function H ( f ) and each of the interfering transmit modified crosstalk model described in the following i ()∈[]K chapters. signals by Si ( f ) i 1, , N . The squared magnitudes of the NEXT coupling 2 functions H ( f ) are approximated here by the 3.1 Non Linear Regression Model i Crosstalk is regarded as a stochastic and unpredictable potential functions noise-process. For the description of the statistical 2 qualities of noise, the autocorrelation function or the ≈ ⋅ xi (3) Hi ( f ) ki f power spectral density is used [1], [7], [10]. The representation of the PSD takes place in the frequency domain. The modified NEXT model is based on Fig. 2. with the individual coupling constants ki and

To provide a clear description, only one interfered individual exponents xi . To the examination of the twisted pair was represented in Fig. 2. simplifying assumption of equation (3), the NEXT coupling functions of a telephone cable DIN VDE 0815 J-2Y(ST) 50 x 2 x 0.6 mm STIII were measured S1 ( f ) 2 H1 ( f ) in a frequency band from 9 kHz to 30 MHz. The measuring of the NEXT coupling functions is equivalent with measuring the s-parameters [9]. The S ( f ) 2 2 measurements have been accomplished with a vector H2 ( f ) SNEXT ( f ) analyser at 401 linear distributed points in the + frequency domain. The twisted pairs were terminated S3 ( f ) 2 H3 ( f ) with the characterised impedance to avoid signal reflections [6], [9]. The squared magnitudes of the measured NEXT M transfer functions were approximated by the potential functions defined in equation (3). To the calculation of SN ( f ) 2 the parameters ki and xi the substitutions H N ( f ) = u log10 ( f ) (4) Fig. 2 Block diagram of the modified = ( 2 ) (5) crosstalk model vi log10 Hi ( f ) = (6) ci log10 (ki )

SNEXT ( f ) NEXT PSD on the interfered pair S ( f ) Transmit PSD on the interfering pair i are introduced. With the substitutions (4) - (6), the i potency-function from equation (3) results in a straight 2 Hi ( f ) Squared magnitude of the crosstalk line coupling function from the interfering = ⋅ + pair i to the interfered pair vi (u) xi u ci (7) N Number of crosstalk interferers on the logarithmic scaled frequency axis. The With N crosstalk interferers, the NEXT power-sum parameters xi , ci and the resulting value of the SNEXT ( f ) is the linear sum of the transmit PSDs coupling constants ki are derived from the measured on the interfering pairs multiplied with the Si ( f ) NEXT coupling functions with the least squares 2 method [11], [12]. Fig. 3 shows the measured and the associated NEXT coupling functions Hi ( f ) . The approximated squared magnitude of H ( f ) on the NEXT PSD on the interfered pair is calculated with: 21 logarithmic scaled frequency axis representatively1.

N 2 S ( f ) = å H ( f ) ⋅ S ( f ) . (2) 1 NEXT i i H ( f ) represents the NEXT transfer function from the twisted i=1 xy pair x to the pair y . 3.2 Estimation of the Crosstalk Function

-20 For the derivation of the parameters ki and xi , the |H (f)|2 NEXT,21 PSDs and are considered as random (|H (f)|2) Si ( f ) SNEXT ( f ) NEXT,21 approx. -40 variables. The transmit PSDs Si ( f ) are statistically independent of each other and are denoted in this -60 context as exogenous variables [9]. The endogenous variable S ( f ) depends on the transmit PSDs. dB NEXT

-80 The realisations Si ( f j ) and SNEXT ( f j ) of the

random variables Si ( f ) as well as SNEXT ( f ) can be -100 measured at a DSLAM at a number of A frequency ()∈{}K points f j j 1, , A . The measurements of the PSDs could occur in an additional module integrated -120 3 4 5 6 7 8 10 10 10 10 10 10 in the DSLAM. The module collects the measuring Frequency in Hz data during operation of the telephone cable and Fig. 3 Measured and approximated transmits the data to a flexible impairment generator or squared magnitude of H 21 ( f ) to an analysis tool for the evaluation of the telephone

loop. Thus, Si ( f j ) and SNEXT ( f j ) are assumed for the following paragraphs. The approximated squared magnitude of H 21 ( f ) is With these assumptions, the equation of the

2 − modified NEXT model can be derived from equation = ⋅ x21 = ⋅ 15 ⋅ 1.4860 (8) H 21 ( f ) approx k21 f 1.2887293 10 f (9) at the point f j as follows:

2 N The function H ( f ) contains a coupling constant x 21 S ( f ) = åk ⋅ f i ⋅ S ( f ) + ε . (10) = − = NEXT j i j i j j k21 1.2887293E 15 and an exponent x21 1.4860. i=1 The correlation coefficient between the measured and the approximated squared magnitude of the NEXT ε The error j (residue) is one possible realisation of a coupling function is ρ = . 2 H 21 ( f ) 0.9437 non-observable random variable . The demands for the The correlation coefficients between the remaining residue ε are: measured squared crosstalk coupling functions and the j approximated squared NEXT coupling are in the same {ε }= ∈[]K (11) magnitude as in the example presented. Therefore, it is E j 0 ; j 1, , A assumed that the potential function defined in equation Var{}ε = σ 2 (12) (3) sufficiently approaches the NEXT transfer j ε ≈ ()σ 2 functions. j N 0, . (13) With the approximation of equation (3) the NEXT power-sum PSD on a twisted pair of the cable bundle In accordance with the equations (11) and (13), the is residues are Gaussian variables with zero mean. The statistical characteristics of the residue are transformed N x ≈ ⋅ i ⋅ (9) to the endogenous variable SNEXT ( f ) to obtain [11], SNEXT ( f ) åki f Si ( f ) . i=1 [12]

The unknown parameters k and x have to be derived {}= σ 2 i i Var SNEXT ( f j ) . (14) from the measured spectra Si ( f ) and SNEXT ( f ) . Equation (10) describes a non-linear multivariate regression model. In these models, the determination

2 In the following, the notation does not distinguish between mistakes and the affiliated random variables. Residue and ε random variable are both denoted with i . of the parameters (regression coefficient) takes place Under these assumptions, the estimated values with iterative algorithms [11]. ˆ K ˆ k1, , kN of the coupling constants are calculated, so The analysis of the coupling constants ki and the that the squared error

exponents xi showed that the exponents xi have less A A æ N ö2 variance than the coupling constants ki . The average ()ε 2 = ç − []⋅ ÷ (18) å ' j ååS'NEXT ( f j ) ki Si ( f j ) exponent for the examined telephone cable is j=1 j==1 è i 1 ø 1.440205. Therefore, the average gradient of the NEXT coupling function is 14.40205 dB/Dec. ANSI between S' ( f ) and the systematic component fixed the gradient of the 1% worst case power-sum NEXT j N model according to measurements on real cable []⋅ å ki Si ( f j ) becomes minimal. bundles on a value of 15 dB/Dec [1], [2]. In i=1 consequence of the relatively low divergence between the actual average exponent of the real cable bundle and the exponent of the standard crosstalk model, the 3.2.2 Matrix Notation standard value of 15 dB/Dec is used in the modified The variables S' ( f ) are summarised in the NEXT crosstalk model. Under this assumption, all NEXT j 2 ()A×1 -matrix squared NEXT coupling functions H i ( f ) possess a constant gradient of 15 dB/Dec. In consequence of the æ S' ( f ) ö constant gradient, the non-linear multiple regression ç NEXT 1 ÷ model from equation (10) can be transferred into a ç S' ( f )÷ S' = NEXT 2 (19) linear multiple regression model. NEXT ç M ÷ ç ÷ ç ÷ è S'NEXT ( f A )ø 3.2.1 Linear Multiple Regression Model With the transformations and the discrete transmission spectra Si ( f j ) are S ( f ) summarised in a ()A× N -matrix = NEXT j (15) S'NEXT ( f j ) 1.5 f j æ L ö ε S1( f1) SN ( f1) ε' = j (16) ç ÷ j 1.5 = ç M M ÷ . (20) f j S ç L ÷ èS1( f A ) SN ( f A )ø the equation of the linear multiple regression model is

N With the definition of the ()N ×1 -vector = ⋅ + ε (17) S'NEXT ( f j ) åki Si ( f j ) ' j . i=1 æ k ö ç 1 ÷ In contrast to the non-linear regression model of ç k ÷ k = 2 (21) equation (10), only the coupling constants (regression ç M ÷ ç ÷ coefficient) k are unknown in equation (17). The ç ÷ i èk ø regression coefficients in equation (17) can be N estimated with the least squares method [11], [12]. For of the regression coefficients (coupling-vector) and the the estimation of the constants ki , the following ()A×1 -error vector prerequisites are fulfilled: 1. The number A of the measured sampling points S ( f ) and S ( f ) is at least equal to the æ ε' ö NEXT j i j ç 1 ÷ ε number of the coupling constants to be ç '2 ÷ ki ε'= (22) ç M ÷ estimated. ç ÷ ç ÷ èε' ø 2. The transmission spectra Si ( f ) are statistically A independent from each other. the linear multiple regression model from equation (17) in matrix notation is -50

= ⋅ + ε -60 S'NEXT S k ' . (23)

-70

The coupling constants ki are estimated with the least -80 squares method as follows: dB

min -90 ()()S' −S ⋅k T S' −S ⋅ k → . (24) NEXT NEXT k -100 |H |2 real 21 |H |2 model 21 After some elementary mathematical transformations, -110 it can be deduced from equation (24) 104 105 106 107 Frequency in Hz Fig. 4 Measured and calculated NEXT T ()− ⋅ = (25) 2 S S'NEXT S k 0 transfer function H 21( f )

ÞST S kˆ = ST S' . (26) NEXT The coupling constant calculated by the modified ˆ = − NEXT crosstalk model amounts to k21 1.5002 E 15 . T Under the above assumptions, the matrix S S is The correlation coefficient between the measured and invertible. The estimated coupling-vector kˆ can be the modelled NEXT coupling function amounts to calculated with ρ = 0.8368. Fig. 5 presents the measured and calculated NEXT − 2 T 1 T ˆ = () (27) coupling function H 91( f ) . k S S S S'NEXT .

-50 4 Experimental and Simulation Results Fig. 4 and Fig. 5 show the squared magnitude of the -60 estimated and measured NEXT coupling function of the cable DIN VDE 0815 J-2Y(ST) 50 x 2 x 0.6 mm -70 STIII. The twisted pair of number 1 represented the -80 reference pair on which the induced near-end crosstalk dB interferences were measured, at 401 frequency-points between 10 kHz and 1.5 MHz. On the pairs 2, 4 and 9 -90 operated SHDSL-systems, which represent the sources for the crosstalk interferences on the reference pair. -100 |H |2 real The transmission spectra of the SHDSL-systems were 91 |H |2 model 91 also measured at 401 frequency-points between 10 -110 kHz and 1.5 MHz. 104 105 106 107 Fig. 4 shows the NEXT transfer function Frequency in Hz 2 Fig. 5 Measured and simulated NEXT H21( f ) model calculated by the modified NEXT transfer function H ( f ) 2 crosstalk model in relation to the measured NEXT 91 2 transfer function H 21( f ) real . The estimated coupling constant of the near-end crosstalk coupling function between the twisted pairs 1 ˆ = − and 9 amounts to k91 8.6859 E 17 . With the gradient of 15 dB/Dec, a correlation coefficient between the measured and valued coupling function is ρ = 0.7708 . 5 Conclusion References: With the modified near-end crosstalk (NEXT) model [1] Starr, T., Cioffi, J. and P. Silverman, proposed in this paper, it is possible to estimate the Understanding Digital Subsciber Line NEXT coupling functions within a cable bundle. For Technologie, 1. Edition, München, Boston, San this the transmit spectra on the interfering pairs and the Francisco, Addison-Wesley, 2000. NEXT spectra on the interfered pair of a cable bundle [2] Network and customer installation interface, are measured. Asymmetric Digital Subscriber Line (ADSL) It was shown that the NEXT coupling functions metallic interface. Alliance for Telecommunica- could be approximated by potential functions. With tions Industry Solutions (ATIS), T1.413-1998 this simplifying assumption, the NEXT-effects within (issue 2). a cable bundle can be modelled by a linear multiple [3] Transmission and Multiplexing; Access regression model. The calculation of the coupling transmission system on metallic access cables; constants (regression-coefficients) is performed via the Symmetrical single pair high bitrate Digital least squares method. Subscriber Line SDSL, ETSI TS 101 524 v1.1.2, The advantages of the modified crosstalk model August, 2001. proposed here, as opposed to the 1% worst case [4] Single-pair high-speed digital subscriber line power-sum model, are: (SHDSL) transceivers, ITU-T G.991.2, February 1. Autonomous determination of the crosstalk 2001. coupling functions during the operation of the [5] Valenti, C., Cable crosstalk parameters models, telephone cable; ANSI T1E1.4/97-302, September 1997. 2. Consideration of the mixed physical and [6] Valenti, C., NEXT and FEXT Models for Twisted- geometrical influential factors for the crosstalk; Pair North American Loop Plant, IEEE Journal on 3. Individual reproduction of the crosstalk Selected Areas in Communications, Vol.20, No.5, interferences of real telephone cables for test June 2002, pp. 893-900 and analysis purposes. [7] Werner, J. J., The HDSL Environment, IEEE Journ. Select. Areas Coummun., Vol.9, No.4, 1991, A flexible impairment generator can realistically pp. 785-800. reproduce the NEXT interferences for xDSL-system [8] Unger, J. H. W., Near-end crosstalk model for line tests with the modified NEXT model. Furthermore, the code studies, Committee T1 Contribution, calculated NEXT coupling functions can be used to T1E1.3/85-244, 12. November 1985. analyse the appearing NEXT interferences in the cable [9] van den Brink, R., Cable reference models for bundle. With the analysis of the NEXT interferences, simulating metallic access networks, ETSI/STC the xDSL-systems could be arranged at the cable TM6 970p02r3, June 1998. bundle in order to minimise the mutual NEXT [10] Söder, G., Modellierung, Simulation und Opti- interferences efficiently. mierung von Nachrichtensystemen, 1. Edition, Berlin, Heidelberg: Springer-Verlag, 1993. [11] Hartung, J. Elpelt, B. and Klösner, K.-H, Statistik, 9. Edition, München, Wien: Oldenburg Verlag, 1989. [12] Fahrmeir, L., Künstler, R., Pigeot, I. und Tutz, G., Statistik, 2. Edition, Berlin, New York, Barcelona, Paris: Springer Verlag, 1999. [13] Bleymüller, J., Gehlert G. and Gülicher, H. Statistik für Wirtschaftswissenschaftler, 8.Edition, München: Verlag Vahlen, 1992.