Wideband Modeling of Twisted-Pair Cables for MIMO Applications

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  • 1. Technische Universität München Wideband Modeling of Twisted-Pair Cables for MIMO Appli- cations Globecom 2013 - Symposium on Selected Areas in Communications (GC13 SAC) Rainer Strobel, Reinhard Stolle, and Wolfgang Utschick c 2013 IEEE. Personal use of this material is permitted. However, permission to reprint/re- publish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. Department of Electrical Engineering and Information Technology Associate Institute for Signal Processing Univ.-Prof. Dr.-Ing. Wolfgang Utschick
  • 2. Wideband Modeling of Twisted-Pair Cables for MIMO Applications Rainer Strobel∗‡ , Reinhard Stolle† , Wolfgang Utschick∗ ∗Associate Institute for Signal Processing, Technische Universität München, 80333 München {rainer.strobel,utschick}@tum.de †Hochschule Augsburg, 86161 Augsburg, reinhard.stolle@hs-augsburg.de ‡Lantiq Deutschland GmbH, 85579 Neubiberg, rainer.strobel@lantiq.com Abstract—Recent trends in broadband access technology show the demand to extend the used frequency bands up to hundreds of MHz. Access cables are not built for such high frequencies, and measurements of access cables in this frequency range show a significant change of the cable characteristics compared to low frequencies. The novel modeling approach presented here is designed to be used for evaluation of transmission technologies for fiber-copper hybrid networks, so called FTTdp (Fiber To The distribution point), which enables service providers to serve customers with data rates in the GBit/s range without the requirement to install fiber to the home. I. INTRODUCTION As the bandwidth requirements increase, a different network topology is used for the fourth generation of broadband access [1], than in classical ADSL [2] and VDSL [3] networks. A fiber-copper hybrid network al- lows the delivery of data rates in the GBit/s range, while the deployment costs are still low compared to pure fiber networks. The fourth generation network consists of distribution points, which are connected to the central office via fiber. The distribution points serve a small number of customers over short distances of copper wires. With increasing frequency, the idealized modeling ap- proach which was used to design ADSL and VDSL broadband access, e. g. [4] or [5], can no longer be used and a more accurate characterization is required for the system design. Therefore, a new modeling approach is proposed in this paper, which makes it possible to describe the physical effects of copper access networks for MIMO transmission at frequencies up to 300 MHz. The model is furthermore formulated such that it can be fitted to measurement data of real cables. II. RECENT WORK The work on channel models for the fourth generation broadband access networks has recently been started with measurements of cables under the conditions de- fined for a FTTdp network and comparison of the results with previously used models. A. Differential Single Line Models Channel models for evaluation of data transmission on twisted pair cables are mainly based on a character- ization of the differential mode of a single twisted pair. The models describe the primary line constants, serial resistance R, serial inductance L, parallel capacitance C and parallel conductance G per unit length. Popular models for access cables are the ETSI model [4] used for VDSL up to 30 MHz or the recently intro- duced ITU model [6] for the approximation of differen- tial mode transfer functions up to 300 MHz. The secondary line constants, line impedance Z0(ω) and propagation constant γ(ω) are given by Z0(ω) = R + jωL G + jωC (1) and γ(ω) = (R + jωL)(G + jωC) (2) as a function of frequency ω = 2π f and the primary line constants. The matrix description of a transmission line of length l is then given by U(0) I(0) = cosh(γl) Z0 sinh(γl) 1 Z0 sinh(γl) cosh(γl) · U(l) I(l) , (3) the so called telegrapher’s equations [7], which describe the voltage U and current I at the line input as a function of voltage and current at the line output. Due to the high bandwidth which is covered by the models, the primary line constants are no longer con- stant over frequency. Therefore, models like [4] or [6] approximate them with nonlinear functions. Crosstalk is modeled in [4] as noise with a specific noise spectrum that depends on the cable type and the number of lines in a binder. B. Crosstalk Models For the analysis of crosstalk cancelation for VDSL [8], this model is no longer appropriate and therefore, MIMO models have been introduced. The ATIS MIMO model [5] is based on a direct channel model description Hchannel( f ) , e. g. on the ETSI model.
  • 3. Additionally, crosstalk coupling paths HFEXT( f ) accord- ing to HFEXT ik( f ) = |Hchannel( f )| f ejϕ( f ) κ lcoupling10xdB ik/20 (4) are added. Additional parameters are a random phase term ϕ( f ), the scaling constant κ, the coupling length lcoupling and a random coupling strength matrix XdB which has been created based on measurements. This model does not give a complete MIMO descrip- tion of a cable binder, because couplings between the channels are only described by far-end crosstalk transfer functions. Therefore, it is not feasible to cascade channel matrices from the ATIS model. Several approaches, e. g. [9] have been made to create cable models which are closer to the physical charac- teristics of a cable binder and characterize not only the differential mode, but also the phantom mode of a cable binder [10]. C. Cable Measurements at High Frequencies Measurement data which is presented in this paper is based on results from a recent study at Deutsche Telekom [11] and fits to measurement data from other measurement campaigns, e. g. [12], [13] and [14]. The data shows effects which are not covered by the models which are currently in use. Fig. 1 compares measurements of a short cable binder of a Deutsche Telekom access cable [11] (10-pair cable of 30 m length) with the corresponding results from the ETSI model. The measured direct channel shows a significantly higher attenuation at high frequencies than is predicted by the ETSI model [4]. This behavior is also observed in the measurement data of [15]. Fig. 1 also shows a single dominant crosstalk coupling, which is the crosstalk between the two pairs of a quad cable. It dominates the crosstalk power sum in the measurement. -90 -80 -70 -60 -50 -40 -30 -20 -10 0 0 50 100 150 200 250 300 transferfunction/dB f/MHz Direct channel measured FEXT measured FEXT power sum measured Direct channel ETSI FEXT power sum ETSI Fig. 1. Transfer function of direct line and crosstalkers of 30m line In the ATIS model, the crosstalk coupling strength is a function of frequency with the proportionality |HFEXT( f )|2 ∼ |Hchannel( f,l)|2 · f2. Some of the avail- able measurement data, e. g. Fig, 1, [14] or [16] in- dicates that this does not hold for high frequencies in quad cables where sometimes the proportionality |HFEXT,dualslope( f )|2 ∼ |Hchannel( f,l)|2 · f4 is observed. The single line models [4] and [6] and the MIMO model [5] do not distinguish between twisted pair and quad cables and therefore neglect this effect. In the statistical model of [5], the direct channel char- acteristics are assumed to be constant within the cable binder and therefore modeled in a deterministic manner, while the available measurement data indicates that the cable characteristics have a random variance over the cable length and over the different pairs in a binder. In a time domain reflexion measurement of an open-ended cable of 20 m length, as shown in Fig. 2, it can be seen that a significant amount of energy is reflected at a length of less than 20 m. This indicates that the direct channel characteristics like line impedance Z0 are not constant over the cable length. -0.05 0 0.05 0.1 0.15 0 5 10 15 20 25 30 0 20 40 60 80 100 120 140 impulseresponse l/m t/ns Impulse response 50MHz Impulse response 300MHz Fig. 2. Reflexion of a 20 m transmission line in time domain Based on these observations, a channel model for evaluation of FTTdp networks preferably considers a cascade of cable binder segments which may include random variations of the channel characteristics and can be fitted to measurement data. III. TOPOLOGY MODEL The telephony cable network consists of multiple sec- tions. In many cases, different cable types are used in different sections. The proposed topology model uses three sections as shown in Fig. 3. 1) A drop wire section, running from the distribution point to the buildings, 2) an in-building section connecting the drop wire with the individual subscribers homes, 3) an in-home part with a single quad or pair, possibly with bridged taps and similar imperfections. For the topology model, each section is described by a matrix Asec such that the sections can be cascaded. This approach requires an appropriate description for small cable binder segments.
  • 4. ... ... ... ... ... ... drop wire in house in home Asec 1 Asec 2 Asec 3 Fig. 3. Telephony network topology A. Multiconductor Models Cascades of circuit elements are widely used in high frequency circuit design. In twoport theory, each element is described by a matrix. A chain matrix describes the de- pendency between input voltage and current and output voltage and current and a cascade of circuit elements can be calculated by the product of the chain matrices of the individual circuit elements. To extend the chain matrix description to a cable binder, the multiconductor transmission line theory was introduced in [17]. The multiconductor chain matrix A is defined as u(0) i(0) = A11 A12 A21 A22 · u(l) i(l) (5) and describes the relation from an input voltage vector u(0) and an input current vector i(0) to the output voltage vector u(l) and output current vector i(l). The overall chain matrix Aall of multiple cable binder sections Asec i is given by the product of the individual section matrices, as shown in Aall = ∏ i Asec i. (6) If the models from Sec. II-A are used in a multiconduc- tor description, they describe only the diagonal elements of the block matrices A11, A12, A21 and A22. The voltages are defined as differential voltages between the wires of each pair. Fig. 4 shows the circuit corresponding to a differential element of the multiconductor twisted pair cable binder with two pairs in differential mode. l l+dl u1(l) i1(l) u2(l) i2(l) u1(l + dl) i1(l + dl) u2(l + dl) i2(l + dl) Fig. 4. Cable binder segment with multiple uncoupled differential lines The crosstalk models as described in Section II-B do not fit to the chain matrix description because these models describe crosstalk transfer functions, only. Fur- thermore, limiting the models to differential mode is not sufficient to cover some effects of the measurement data, for example the frequency dependency of far-end crosstalk. Therefore, the proposed model does not only describe the differential modes of the pairs, but describes voltage and current of the single wires with respect to a common reference potential. This is hereinafter called single-ended description. The methods for conversion from a single-ended chain matrix to the corresponding differential modes, which are of interest for signal transmission, are described in [18] and in [19]. B. Single-Ended Geometry Model Alternatively to [9], where one of the wires is used as reference for the single-ended description, the proposed topology model uses a separate ground plane as refer- ence for all wires (Fig. 5). For shielded cables, the shield may be used as reference potential. dik hi 2ri k i (a) Single segment (b) Twisted quad Fig. 5. Geometrical model of quad cable The following derivations are based on results from multiconductor transmission line theory, which can be found in [20]. From the geometry as shown in Fig. 5(a), the self- inductance and mutual inductance of a short segment are given by Eq. (7) [20], which defines the inductance ma- trix L. The following equations hold for the assumptions
  • 5. of homogeneous media between the conductors and that they are widely separated in space. The self inductance lii of wire i depends on the dis- tance hi between the wire and the ground plane and on the radius ri of the wire. The mutual inductance lik between wires i and k also depends on the distance dik between the wires lik =    µ 2π log 2hi ri for i = k µ 4π log 1 + 4hihk d2 ik for i = k . (7) With known permittivity ε and permeability µ of the media between the conductors, the capacitance matrix C is obtained by matrix inversion [20] C = µεL−1 (8) from the inductance matrix. With the conductivity σ of the insulation medium, the conductance matrix G is given by G = σ ε C, (9) as shown in [20]. Finally the resistance matrix R is calculated from the wire conductivity σwire, the permeability µwire and the wire radius ri. According to [21], the skin effect can be approximated by the skin depth δ by δ = 1 π f µwireσwire . (10) The resistance matrix R is then obtained by rik = 1 2πσriδ for i = k 0 for i = k (11) which is a diagonal matrix [20]. To calculate the secondary line constant matrices γ and Z0 from the serial impedance matrix Zs = R( f ) + jωL and the parallel admittance matrix Yp = G + jωC, diagonalization of the product matrix Yp · Zs is needed. Eigenvalue decomposition on the product matrix, as proposed in [20], gives the definition YpZs = Tlγ2 T−1 l . (12) Then, γ is a diagonal matrix describing transmission term and Z0 is the line impedance matrix defined by Z0 = ZsTlγ−1 T−1 l (13) and the corresponding admittance matrix Y0 is given by Y0 = Tlγ−1 T−1 l Yp. (14) The chain matrix Aseg of a cable binder segment of finite length l is then Aseg = Z0Tl cosh (γl) T−1 l Y0 Z0Tl sinh (γl) T−1 l Tl sinh (γl) T−1 l Y0 Tl cosh (γl) T−1 l . (15) Eq. (15) is equivalent to the integration over the dif- ferential elements as shown in Fig. 6. Therefore, the geometry must not change over the integration length l, which means that it is only allowed to integrate over a fraction of the twist-length of the twisted pair cable. On a cable with perfect twisted pair geometry, crosstalk coupling would be much weaker than it is observed in real cables. Most of the crosstalk is caused by imperfections in the cable geometry [9]. In a cable model, this requires a random imperfection component and the statistics of the imperfection must be such that the crosstalk statistics match the measurement data. If the statistical model is based on primary line con- stants, as described in the next section, the length l of each segment is chosen such that the primary line constants are approximately constant over the segment length. C. Statistical Model for Primary Line Constants A major drawback of geometrical models besides com- putational complexity is the fact that relevant parameters to describe geometry imperfections and characteristics of the insulation material are difficult to measure. The proposed model is built from short segments according to Eq. (15). Each segment is a cascade of differential binder elements as shown in Fig. 6. The primary line constants can be obtained by elec- trical measurements. Therefore, the statistical model is based on statistical characteristics of the primary line constants. l l+dl u1(l) i1(l) u2(l) u3(l) u4(l) i2(l) i3(l) i4(l) u1(l + dl) i1(l + dl) u2(l + dl) u3(l + dl) u4(l + dl) i2(l + dl) i3(l + dl) i4(l + dl) Fig. 6. Cable binder segment for common mode model However, the primary line constants cannot be de- scribed independently. To match the physical properties of an existing cable, some dependencies must be taken into account. 1) Binder Geometry: The model in [9] describes random imperfections in cable geometry. The proposed model is based on primary line constants, but it uses some knowl- edge on the cable binder geometry. Crosstalk coupling strength does not only depend on random imperfections of the twisting of each pair or quad, but also on the distance between the pairs as shown in Eq. (7) and on the twist lengths of the individual pairs. To consider this in a statistical model, it is based on random positions of the wires in space, similar to the single quad shown in Fig. 5, where an individual twist
  • 6. length is assigned to each of them. The random variation of coupling inductance lik between lines i and k is then scaled with respect to the distance dik between the pairs or quads. This dependency follows from the geometric model and can be verified by measurement data. It allows to scale the model to arbitrary cable binder sizes, which is not possible for the ATIS model, where the statistics match only with the measured 100-pair binder. 2) Correlations over Length: The random values of pri- mary line constants are correlated over the length of the cable binder. The frequency dependence of crosstalk transfer functions observed in measurements, e. g. in Fig. 1 depends on the correlation over cable length. If correlation over cable length is neglected, the re- sulting crosstalk transfer functions do not match the measurement data, as the results in [9] show, where no correlations were considered. Furthermore, the cor- relation over length is required to guarantee that the resulting transfer functions are independent of the length of the cable binder segments used in the model as long as they are sufficiently short. With known correlation length of the cable, the model segment length can then be selected with respect to the sampling theorem. As the correlation length in the available measurement data is in the range of meters, this gives a major computational advantage in comparison to the geometric model. 3) Homogeneity: Based on the random inductance ma- trix L, conductance and capacitance matrices G and C cannot be created independently. As shown in [17] and [20], Eq. (8) and Eq. (9) hold for the case that the insulation medium between the wires is homogeneous. The real cable differs from this dependency between due to inhomogeneity of the insulation medium, but the dependency still holds approximately. 4) Causality: Furthermore, there is a dependency be- tween resistance R(ω) and inductance L(ω) over fre- quency as well as between capacitance C(ω) and con- ductance G(ω). The dependency is given by the require- ment that each segment transfer function must be causal. According to [22], this can be achieved by applying the Hilbert transform. Therefore, the primary line constants are divided into a frequency-dependent component and a frequency-independent component. For example, the resistance R(ω) is divided into ˆR(ω) and ∆R, where R(ω) = ˆR(ω) + ∆R. Then, R(ω) − ∆R = 1 π ∞ −∞ x(L(ω) − ∆L) ω − x dx (16) holds for the serial impedance R + jωL. The frequency dependency of the primary line con- stants originates from the skin effect as described in Eq. (10) and (11). The resulting resistance matrix R is a diagonal matrix and therefore, Eq. (16) is only applied to the diagonal elements of the inductance matrix L, while the off-diagonal elements are constant over frequency. D. Proposed Modeling Steps The results shown in the next section are based on following modeling steps, which are one method to fulfill the mentioned requirements. 1) In a first step, Eq. (7) is evaluated to calculate the inductance matrix with respect to perfect cable geometry. 2) Random variance is added to the inductance matrix using correlated Gaussian random values. 3) The resistance matrix is calculated based on Eq. (10) and Eq. (11) with respect to cable characteristics. 4) Based on the resistance matrix, the self-inductance frequency dependency is corrected using Eq. (16). 5) Capacitance and conductance matrices are calcu- lated using matrix inversion (Eq. (8) and Eq. (9)). 6) Evaluation of Eq. (12) to (15) gives the segment chain matrices, which are cascad
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