Wideband Modeling of Twisted-Pair Cables for MIMO Applications

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 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.

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.

...
...
...
...
...
...
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

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

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 cascaded.
Medium inhomogeneities and radiation loss are ig-
nored in the results. The loss of the insulation medium
of the reference cable is too small to be measured with
sufficient precision and is therefore also ignored.
IV. MODEL RESULTS
As one example, the numerical results for the topology
model of a 30 m access cable from Deutsche Telekom [11]
are shown. The statistics of the inductance matrix L has
been chosen such that some reference measures match
the real cable. They are the crosstalk power sum over
frequency, the crosstalk coupling strength cumulative
density function and the average direct channel transfer
function.
0
0.2
0.4
0.6
0.8
1
-60 -50 -40 -30 -20 -10 0 10
F(XdB)
Crosstalk Coupling/dB
Measurement Data
Topology Model
ATIS Model
Fig. 7. Cumulative density functions of intra-quad crosstalk coupling
strength XdB.
Fig. 7 shows the cumulative density functions F(XdB)
of the crosstalk coupling strength in dB XdB of the
ATIS model in comparison to the average crosstalk
coupling strength from the 10-pair Deutsche Telekom
cable and the topology model using the parameters of
the Deutsche Telekom access cable.

The cumulative density functions show that the
crosstalk coupling strength of the measured cable does
not match the ATIS model [5] whereas the topology
model yields a good fit of the crosstalk statistics for
small cable binders. The crosstalk coupling strength XdB
describes the frequency independent part of the crosstalk
transfer functions, as shown in Eq. (4).
For the frequency dependency of the crosstalk, a
comparison of the crosstalk transfer functions and the
crosstalk power sum is shown in Fig. 8.
-90
-80
-70
-60
-50
-40
-30
-20
-10
0
0 50 100 150 200 250 300
transferfunction/dB
f/MHz
Direct channel model
FEXT model
FEXT power sum model
Direct channel ETSI
FEXT power sum ETSI
Fig. 8. Direct channel and crosstalk transfer functions of topology
model for 30m Deutsche Telekom access cable.
In both, the measurement data of the cable in Fig. 1
and the topology model in Fig. 8, the crosstalk trans-
fer functions show a random frequency dependency.
At higher frequencies, the direct channel attenuation
is higher than predicted by the ETSI model. The in-
quad crosstalk is signficantly stronger than the intra-
quad crosstalkers and does not match the prediction
of the ETSI model. It even exceeds the direct channel
transfer function at higher frequencies. This behavior is
also seen in the measurement data shown in Fig. 1.
V. CONCLUSION
The proposed model provides a tool to analyze the
telephony 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 cable
binders. With more measurement data available, the
statistical model of the random coupling elements can
be refined.
The presented model already gives promising results
for FTTdp frequency domain and time domain simula-
tions.
ACKNOWLEDGMENT
The authors would like to thank Peter Muggenthaler
from Deutsche Telekom for providing measurement data
and information about network topologies, as well as
discussions about physical characteristics of twisted pair
cable binders. This valuable information has been used
as basis for the cable model.
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