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

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Department of Electrical Engineering and Information Technology

Associate Institute for Signal Processing

Univ.-Prof. Dr.-Ing. Wolfgang Utschick

Wideband Modeling of Twisted-Pair Cables forMIMO 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 technologyshow the demand to extend the used frequency bands upto hundreds of MHz. Access cables are not built for suchhigh frequencies, and measurements of access cables inthis frequency range show a significant change of the cablecharacteristics compared to low frequencies.

The novel modeling approach presented here is designedto be used for evaluation of transmission technologies forfiber-copper hybrid networks, so called FTTdp (Fiber To Thedistribution point), which enables service providers to servecustomers with data rates in the GBit/s range without therequirement to install fiber to the home.

I. INTRODUCTION

As the bandwidth requirements increase, a differentnetwork topology is used for the fourth generation ofbroadband access [1], than in classical ADSL [2] andVDSL [3] networks. A fiber-copper hybrid network al-lows the delivery of data rates in the GBit/s range, whilethe deployment costs are still low compared to purefiber networks. The fourth generation network consistsof distribution points, which are connected to the centraloffice via fiber. The distribution points serve a smallnumber of customers over short distances of copperwires.

With increasing frequency, the idealized modeling ap-proach which was used to design ADSL and VDSLbroadband access, e. g. [4] or [5], can no longer be usedand a more accurate characterization is required for thesystem design.

Therefore, a new modeling approach is proposed inthis paper, which makes it possible to describe thephysical effects of copper access networks for MIMOtransmission at frequencies up to 300 MHz. The modelis furthermore formulated such that it can be fitted tomeasurement data of real cables.

II. RECENT WORK

The work on channel models for the fourth generationbroadband access networks has recently been startedwith measurements of cables under the conditions de-fined for a FTTdp network and comparison of the resultswith previously used models.

A. Differential Single Line Models

Channel models for evaluation of data transmissionon 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, serialresistance R, serial inductance L, parallel capacitance Cand 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)

andγ(ω) =

√

(R + jωL)(G + jωC) (2)

as a function of frequency ω = 2π f and the primary lineconstants.

The matrix description of a transmission line of lengthl is then given by

[

U(0)I(0)

]

=

[

cosh(γl) Z0 sinh(γl)1

Z0sinh(γl) cosh(γl)

]

·

[

U(l)I(l)

]

, (3)

the so called telegrapher’s equations [7], which describethe voltage U and current I at the line input as a functionof voltage and current at the line output.

Due to the high bandwidth which is covered by themodels, 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 specificnoise spectrum that depends on the cable type and thenumber 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, MIMOmodels have been introduced.

The ATIS MIMO model [5] is based on a direct channelmodel description Hchannel( f ) , e. g. on the ETSI model.

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 phaseterm ϕ( f ), the scaling constant κ, the coupling lengthlcoupling and a random coupling strength matrix XdBwhich has been created based on measurements.

This model does not give a complete MIMO descrip-tion of a cable binder, because couplings between thechannels are only described by far-end crosstalk transferfunctions. Therefore, it is not feasible to cascade channelmatrices from the ATIS model.

Several approaches, e. g. [9] have been made to createcable models which are closer to the physical charac-teristics of a cable binder and characterize not only thedifferential mode, but also the phantom mode of a cablebinder [10].

C. Cable Measurements at High Frequencies

Measurement data which is presented in this paperis based on results from a recent study at DeutscheTelekom [11] and fits to measurement data from othermeasurement campaigns, e. g. [12], [13] and [14]. Thedata shows effects which are not covered by the modelswhich are currently in use.

Fig. 1 compares measurements of a short cable binderof a Deutsche Telekom access cable [11] (10-pair cableof 30 m length) with the corresponding results fromthe ETSI model. The measured direct channel shows asignificantly higher attenuation at high frequencies thanis predicted by the ETSI model [4]. This behavior is alsoobserved in the measurement data of [15]. Fig. 1 alsoshows a single dominant crosstalk coupling, which isthe crosstalk between the two pairs of a quad cable. Itdominates the crosstalk power sum in the measurement.

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 50 100 150 200 250 300

tran

sfer

fun

ctio

n/

dB

f/MHz

Direct channel measuredFEXT measuredFEXT power sum measuredDirect channel ETSIFEXT power sum ETSI

Fig. 1. Transfer function of direct line and crosstalkers of 30m line

In the ATIS model, the crosstalk coupling strengthis a function of frequency with the proportionality

|HFEXT( f )|2 ∼ |Hchannel( f ,l)|2 · f 2. Some of the avail-able measurement data, e. g. Fig, 1, [14] or [16] in-dicates that this does not hold for high frequenciesin quad cables where sometimes the proportionality|HFEXT,dualslope( f )|2 ∼ |Hchannel( f ,l)|2 · f 4 is observed.The single line models [4] and [6] and the MIMO model[5] do not distinguish between twisted pair and quadcables and therefore neglect this effect.

In the statistical model of [5], the direct channel char-acteristics are assumed to be constant within the cablebinder and therefore modeled in a deterministic manner,while the available measurement data indicates that thecable characteristics have a random variance over thecable length and over the different pairs in a binder. Ina time domain reflexion measurement of an open-endedcable of 20 m length, as shown in Fig. 2, it can be seenthat a significant amount of energy is reflected at a lengthof less than 20 m. This indicates that the direct channelcharacteristics like line impedance Z0 are not constantover 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 140im

pu

lse

resp

onse

l/m

t/ns

Impulse response 50MHzImpulse response 300MHz

Fig. 2. Reflexion of a 20 m transmission line in time domain

Based on these observations, a channel model forevaluation of FTTdp networks preferably considers acascade of cable binder segments which may includerandom variations of the channel characteristics and canbe fitted to measurement data.

III. TOPOLOGY MODEL

The telephony cable network consists of multiple sec-tions. In many cases, different cable types are used indifferent sections. The proposed topology model usesthree sections as shown in Fig. 3.

1) A drop wire section, running from the distributionpoint to the buildings,

2) an in-building section connecting the drop wirewith the individual subscribers homes,

3) an in-home part with a single quad or pair, possiblywith bridged taps and similar imperfections.

For the topology model, each section is described by amatrix Asec such that the sections can be cascaded. Thisapproach requires an appropriate description for smallcable binder segments.

...

...

...

...

...

...

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 highfrequency circuit design. In twoport theory, each elementis described by a matrix. A chain matrix describes the de-pendency between input voltage and current and outputvoltage and current and a cascade of circuit elements canbe calculated by the product of the chain matrices of theindividual circuit elements.

To extend the chain matrix description to a cablebinder, the multiconductor transmission line theory wasintroduced in [17]. The multiconductor chain matrix A

is defined as[

u(0)i(0)

]

=

[

A11 A12A21 A22

]

·

[

u(l)i(l)

]

(5)

and describes the relation from an input voltage vectoru(0) and an input current vector i(0) to the outputvoltage vector u(l) and output current vector i(l).

The overall chain matrix Aall of multiple cable bindersections Asec i is given by the product of the individualsection 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 elementsof the block matrices A11, A12, A21 and A22. The voltagesare defined as differential voltages between the wires ofeach pair. Fig. 4 shows the circuit corresponding to adifferential element of the multiconductor twisted paircable 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 differentiallines

The crosstalk models as described in Section II-B donot fit to the chain matrix description because thesemodels describe crosstalk transfer functions, only. Fur-thermore, limiting the models to differential mode isnot sufficient to cover some effects of the measurementdata, for example the frequency dependency of far-endcrosstalk.

Therefore, the proposed model does not only describethe differential modes of the pairs, but describes voltageand current of the single wires with respect to a commonreference potential.

This is hereinafter called single-ended description.The methods for conversion from a single-ended chainmatrix to the corresponding differential modes, whichare 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 asreference for the single-ended description, the proposedtopology model uses a separate ground plane as refer-ence for all wires (Fig. 5). For shielded cables, the shieldmay 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 frommulticonductor transmission line theory, which can befound in [20].

From the geometry as shown in Fig. 5(a), the self-inductance and mutual inductance of a short segment aregiven by Eq. (7) [20], which defines the inductance ma-trix L. The following equations hold for the assumptions

of homogeneous media between the conductors and thatthey 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 andon 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 =

µ2π log

(

2hiri

)

for i = k

µ4π log

(

1 + 4hihk

d2ik

)

for i 6= k. (7)

With known permittivity ε and permeability µ of themedia between the conductors, the capacitance matrix Cis 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

εC, (9)

as shown in [20].Finally the resistance matrix R is calculated from the

wire conductivity σwire, the permeability µwire and thewire radius ri.

According to [21], the skin effect can be approximatedby the skin depth δ by

δ =1

√

π f µwireσwire. (10)

The resistance matrix R is then obtained by

rik =

{

12πσriδ

for i = k

0 for i 6= 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, asproposed in [20], gives the definition

YpZs = Tlγ2T−1

l . (12)

Then, γ is a diagonal matrix describing transmissionterm and Z0 is the line impedance matrix defined by

Z0 = ZsTlγ−1T−1

l (13)

and the corresponding admittance matrix Y0 is given by

Y0 = Tlγ−1T−1

l Yp. (14)

The chain matrix Aseg of a cable binder segment offinite length l is then

Aseg =

[

Z0Tl cosh (γl)T−1l Y0 Z0Tl sinh (γl)T−1

lTl sinh (γl)T−1

l Y0 Tl cosh (γl)T−1l

]

.

(15)

Eq. (15) is equivalent to the integration over the dif-ferential elements as shown in Fig. 6. Therefore, thegeometry must not change over the integration lengthl, which means that it is only allowed to integrate overa 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 isobserved in real cables. Most of the crosstalk is causedby imperfections in the cable geometry [9]. In a cablemodel, this requires a random imperfection componentand the statistics of the imperfection must be such thatthe 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 lof each segment is chosen such that the primary lineconstants are approximately constant over the segmentlength.

C. Statistical Model for Primary Line Constants

A major drawback of geometrical models besides com-putational complexity is the fact that relevant parametersto describe geometry imperfections and characteristics ofthe insulation material are difficult to measure.

The proposed model is built from short segmentsaccording to Eq. (15). Each segment is a cascade ofdifferential binder elements as shown in Fig. 6.

The primary line constants can be obtained by elec-trical measurements. Therefore, the statistical model isbased on statistical characteristics of the primary lineconstants.

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 propertiesof an existing cable, some dependencies must be takeninto account.

1) Binder Geometry: The model in [9] describes randomimperfections in cable geometry. The proposed model isbased on primary line constants, but it uses some knowl-edge on the cable binder geometry. Crosstalk couplingstrength does not only depend on random imperfectionsof the twisting of each pair or quad, but also on thedistance between the pairs as shown in Eq. (7) and onthe twist lengths of the individual pairs.

To consider this in a statistical model, it is based onrandom positions of the wires in space, similar to thesingle quad shown in Fig. 5, where an individual twist

length is assigned to each of them. The random variationof coupling inductance lik between lines i and k is thenscaled with respect to the distance dik between the pairsor quads.

This dependency follows from the geometric modeland can be verified by measurement data. It allows toscale the model to arbitrary cable binder sizes, whichis not possible for the ATIS model, where the statisticsmatch only with the measured 100-pair binder.

2) Correlations over Length: The random values of pri-mary line constants are correlated over the length ofthe cable binder. The frequency dependence of crosstalktransfer 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 themeasurement data, as the results in [9] show, whereno correlations were considered. Furthermore, the cor-relation over length is required to guarantee that theresulting transfer functions are independent of the lengthof the cable binder segments used in the model as longas they are sufficiently short. With known correlationlength of the cable, the model segment length can thenbe selected with respect to the sampling theorem. As thecorrelation length in the available measurement data isin the range of meters, this gives a major computationaladvantage in comparison to the geometric model.

3) Homogeneity: Based on the random inductance ma-trix L, conductance and capacitance matrices G andC cannot be created independently. As shown in [17]and [20], Eq. (8) and Eq. (9) hold for the case that theinsulation medium between the wires is homogeneous.The real cable differs from this dependency betweendue to inhomogeneity of the insulation medium, but thedependency 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 theHilbert transform. Therefore, the primary line constantsare divided into a frequency-dependent component anda frequency-independent component. For example, theresistance R(ω) is divided into R̂(ω) and ∆R, whereR(ω) = R̂(ω) + ∆R.

Then,

R(ω)− ∆R =1π

∞∫

−∞

x(L(ω)− ∆L)

ω − xdx (16)

holds for the serial impedance R + jωL.The frequency dependency of the primary line con-

stants originates from the skin effect as described inEq. (10) and (11). The resulting resistance matrix R is adiagonal matrix and therefore, Eq. (16) is only applied to

the diagonal elements of the inductance matrix L, whilethe off-diagonal elements are constant over frequency.

D. Proposed Modeling Steps

The results shown in the next section are based onfollowing modeling steps, which are one method tofulfill the mentioned requirements.

1) In a first step, Eq. (7) is evaluated to calculatethe inductance matrix with respect to perfect cablegeometry.

2) Random variance is added to the inductance matrixusing 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-inductancefrequency 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 segmentchain matrices, which are cascaded.

Medium inhomogeneities and radiation loss are ig-nored in the results. The loss of the insulation mediumof the reference cable is too small to be measured withsufficient precision and is therefore also ignored.

IV. MODEL RESULTS

As one example, the numerical results for the topologymodel of a 30 m access cable from Deutsche Telekom [11]are shown. The statistics of the inductance matrix L hasbeen chosen such that some reference measures matchthe real cable. They are the crosstalk power sum overfrequency, the crosstalk coupling strength cumulativedensity function and the average direct channel transferfunction.

0

0.2

0.4

0.6

0.8

1

-60 -50 -40 -30 -20 -10 0 10

F(X

dB)

Crosstalk Coupling/dB

Measurement DataTopology ModelATIS Model

Fig. 7. Cumulative density functions of intra-quad crosstalk couplingstrength XdB.

Fig. 7 shows the cumulative density functions F(XdB)of the crosstalk coupling strength in dB XdB of theATIS model in comparison to the average crosstalkcoupling strength from the 10-pair Deutsche Telekomcable and the topology model using the parameters ofthe Deutsche Telekom access cable.

The cumulative density functions show that thecrosstalk coupling strength of the measured cable doesnot match the ATIS model [5] whereas the topologymodel yields a good fit of the crosstalk statistics forsmall cable binders. The crosstalk coupling strength XdBdescribes the frequency independent part of the crosstalktransfer functions, as shown in Eq. (4).

For the frequency dependency of the crosstalk, acomparison of the crosstalk transfer functions and thecrosstalk power sum is shown in Fig. 8.

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 50 100 150 200 250 300

tran

sfer

fun

ctio

n/

dB

f/MHz

Direct channel modelFEXT modelFEXT power sum modelDirect channel ETSIFEXT power sum ETSI

Fig. 8. Direct channel and crosstalk transfer functions of topologymodel for 30m Deutsche Telekom access cable.

In both, the measurement data of the cable in Fig. 1and the topology model in Fig. 8, the crosstalk trans-fer functions show a random frequency dependency.At higher frequencies, the direct channel attenuationis higher than predicted by the ETSI model. The in-quad crosstalk is signficantly stronger than the intra-quad crosstalkers and does not match the predictionof the ETSI model. It even exceeds the direct channeltransfer function at higher frequencies. This behavior isalso seen in the measurement data shown in Fig. 1.

V. CONCLUSION

The proposed model provides a tool to analyze thetelephony network for FTTdp applications.

Statistical modeling with respect to the physical char-acteristics guarantees that the resulting channel descrip-tion corresponds to the properties of existing cablebinders. With more measurement data available, thestatistical model of the random coupling elements canbe refined.

The presented model already gives promising resultsfor FTTdp frequency domain and time domain simula-tions.

ACKNOWLEDGMENT

The authors would like to thank Peter Muggenthalerfrom Deutsche Telekom for providing measurement dataand information about network topologies, as well as

discussions about physical characteristics of twisted paircable binders. This valuable information has been usedas basis for the cable model.

REFERENCES

[1] P. Odling, T. Magesacher, S. Host, P. Borjesson, M. Berg, andE. Areizaga, “The fourth generation broadband concept,” Com-munications Magazine, IEEE, vol. 47, no. 1, pp. 62–69, 2009.

[2] ITU-T Rec. G.992.3, “Asymmetric digital subscriber linetransceivers 2 (ADSL2),” 1999.

[3] ITU-T Rec. G.993.2, “Very high speed digital subscriber linetransceivers 3 (VDSL2),” 2006.

[4] ETSI TS 101 270-1, “Transmission and Multiplexing (TM); Accesstransmission systems on metallic access cables; Very high speedDigital Subscriber Line (VDSL); Part 1: Functional requirements,”2003.

[5] J. Maes, M. Guenach, and M. Peeters, “Statistical channel modelfor gain quantification of DSL crosstalk mitigation techniques,” inProceedings of the IEEE International Conference on Communications,2009. IEEE Press, 2009, pp. 2678–2682.

[6] TNO, “G.fast: Wideband modeling of twisted pair cables as two-ports,” 2011, contribution ITU-T SG15/Q4a 11GS3-029.

[7] T. Starr, DSL advances. Prentice Hall, 2003.[8] G. Ginis and J. Cioffi, “Vectored-DMT: A FEXT canceling mod-

ulation scheme for coordinating users,” in IEEE InternationalConference on Communications, 2001. ICC 2001., vol. 1. IEEE, pp.305–309.

[9] B. Lee, J. Cioffi, S. Jagannathan, K. Seong, Y. Kim, M. Mohseni,and M. Brady, “Binder MIMO channels,” IEEE Transactions onCommunications, vol. 55, no. 8, pp. 1617–1628, 2007.

[10] W. Foubert, C. Neus, L. Van Biesen, and Y. Rolain, “Exploitingthe Phantom-Mode Signal in DSL Applications,” Instrumentationand Measurement, IEEE Transactions on, vol. 61, no. 4, pp. 896–902,2012.

[11] P. Muggenthaler, “Ermittlung der Leitungsbeläge und ELFEXT(F)verschiedener Installationskabel in der Kabelversuchsanlage derDeutschen Telekom als Grundlage für das G.fast Kanalmodell,”2012, Internal Report.

[12] TNO, “G.fast: Far-end crosstak in twisted pair cabling: measure-ments and modeling,” 2011, contribution ITU-T SG15/Q4a 11RV-022.

[13] L. Humphrey and T. Morsman, “G.fast: Release of BT cablemeasurements for use in simulations,” 2013, contribution ITU-TSG15/Q4a 2013-1-Q4-066.

[14] A. Carrick and T. Bongard, “G.fast: Additional loop sets forsimulation purposes ,” 2013, contribution ITU-T SG15/Q4a 2013-01-Q4-044.

[15] D. Van Bruyssel and J. Maes, “G.fast: Observations On Chan-nel Characteristics Measurements,” 2011, contribution ITU-TSG15/Q4a 11BM-048.

[16] P. Muggenthaler, C. Tudziers, and E. Berndt, “G.fast: EL-FEXTAnalysis,” 2012, contribution ITU-T SG15/Q4a 2012-06-4A-41.

[17] L. Pipes, “X. matrix theory of multiconductor transmission lines,”The London, Edinburgh, and Dublin Philosophical Magazine and Jour-nal of Science, vol. 24, no. 159, pp. 97–113, 1937.

[18] T. Granberg, Handbook of Digital Techniques for High-Speed Design.Prentice Hall PTR, 2004.

[19] W. Fan, A. Lu, L. Wai, and B. Lok, “Mixed-mode S-parametercharacterization of differential structures,” in Electronics PackagingTechnology, 2003 5th Conference (EPTC 2003). IEEE, 2003, pp. 533–537.

[20] C. Paul, Analysis of multiconductor transmission lines. Wiley-IEEEPress, 2007.

[21] H. Wheeler, “Formulas for the skin effect,” Proceedings of the IRE,vol. 30, no. 9, pp. 412–424, 1942.

[22] F. Lindqvist, P. Börjesson, P. Ödling, S. Höst, K. Ericson, andT. Magesacher, “Low-Order and Causal Twisted-Pair Cable Mod-eling by Means of the Hilbert Transform,” in AIP ConferenceProceedings, vol. 1106, 2009, p. 301.