4.1. Instituto Nacional de Patrimonio Cultural
4.1.8. Cuy asado
In Chapter 2, the physics of the twisted pair channel was described and the transmission model derived. It was noted that if the wires of a pair are equally coupled to all the other wires in the cable, then by symmetry there would be no leakage into or out of the differential mode. The chapter also pointed out that the randomized twisting of pairs approximates this symmetry rather closely. Whatever crosstalk there is can be considered as shortcomings of the symmetry. Following Hughes [Hughes 1997], one may model crosstalk by assuming the asymmetries are small, random, and independent. Then their effects will add per the central limit theorem, and scalar coupling will be Gaussian.24
Here, the couplings in which asymmetries are most likely to be significant are assumed to be the electrostatic and magnetic linkages between neighboring pairs. Two pairs are
24Hughes then uses measurement to show how small the perturbations are in real cables.
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Z0P cable
Z0D
Z0D Z0P
Z0D
Z0D
Z0C Z0C
FIGURE 3.12
The circuits used for modelling crosstalk in a cable.
considered, starting with the effect of single elements of unbalance. The derivation is in terms of waves and the scattering of waves (where later the scattered waves are added vectorially). The scattering will produce waves in the differential modes of the pairs, and leaked waves to the common modes;25 to support the development, the common modes will be assumed also to have characteristic impedances. To illustrate the modes, the two pairs are considered as normal differential circuits with a notional balanced phantom circuit between them, and a notional common mode ground return phantom circuit, as illustrated in Figure 3.12, which shows the cable section, the circuits (the transformer coupling serves to define the circuits), and the circuits’ nominal characteristic impedances. The two Z0D differential mode circuits are serviceable, their virtues including a consistent character-istic impedance along their length. The phantoms would in practice not have consistent characteristic impedances,26 but for the purposes of this analysis, the absence of consis-tency in the characteristic impedances will not matter. Their impedances need only exist locally to support the egress of waves, and their values can safely vary from place to place.
Inside the cable, the conductors are mutually affected by each others’ changes in voltage (capacitive coupling) and current (inductive coupling). However, while symmetry is main-tained these effects all cancel exactly, and to each circuit the various real capacitive and inductive linkages just appear as the mode’s intrinsic capacitance and inductance per unit length (the C and L of the usual RLCG parameters of the telegrapher’s equation). The effects of a small deviation in a short element of the cable can be considered. Figure 3.13 shows a small capacitive perturbation between two of the conductors27 in the element. There is, of course, capacitance between all of the conductors; the deviation is just the difference between reality and the ideal symmetrical configuration. Also shown is a signal in the top pair’s differential mode, as a wave travelling from left to right. In the symmetrical case, it would pass directly through; in this perturbed case some of it will scatter. Qualitatively, the signal stimulates a current in the capacitor, and this current generates four equal waves radiating away from it in the normal pairs’ modes (see Figure 3.14), and two equal waves radiating from it in the balanced phantom’s modes. There is no coupling into the common phantom. The incident wave also continues as normal, so that which is transmitted is the
25What to do with the common modes is a recurring problem in modelling. Paul Clayton [Clayton 1994] considers the full multidimensional telegrapher’s equations, handling the conductors separately and with channels defined as boundary conditions. Other authors consider a hierarchy of phantom circuits.
26In the past, a shortage of lines has led to practical use of phantoms, but they have always been of low quality even for voice.
27The choice is of one conductor from one pair and one conductor from the other. By symmetry, effects of equal magnitude are obtained by the other choices. Throughout this analysis, the sign convention that positive is “up”
has been used in the figure; this is, of course, arbitrary, and different choices just invert some of the waves.
Z0P Z0D
Z0D
Z0P Z0D
Z0D δC
short incident wave
Z0C Z0C
FIGURE 3.13
Capacitive perturbation.
VI
VD
VD VP
VD VP
VI
VD
iC
FIGURE 3.14 ScatteringδC.
sum of the incident wave and an extra wave. Quantitatively if the incoming wave has scalar voltagevI, then the capacitor current iC is
ic= − jωδC 2
1
1+ jωδCZ
0D
4 + Z20P
vI (3.31)
and the scattered waves have amplitudes:
differential modes vD = Z0D
4 iC, (3.32)
balanced phantom vP = Z0P
2 iC, (3.33)
common phantom vC = 0. (3.34)
The transmitted wave is
v0= vI + vD. (3.35)
Figure 3.15 shows a mutual induction perturbation between two of the conductors in the element. Qualitatively, the signal current stimulates a voltage across the inductance,
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Z0P Z0D
Z0D
Z0P Z0D
Z0D δM
incident wave
Z0C Z0C
vM vM
FIGURE 3.15
Inductive perturbation.
and this voltage generates a pair of complementary waves28 radiating away from it in each differential pair’s mode (see Figure 3.16), and a complementary pair of waves in the common phantom. There is no coupling into the balanced phantom. The incident wave also continues as normal, so that which is transmitted is the sum of the incident wave and an extra wave. Quantitatively, if the incoming wave has scalar voltagevI, then the inductance voltagevMis
vM= jωδM Z0D
1
1+ jωδM
1
Z0D +8Z10DC
vI (3.36)
vI
vD
vD vC
−vD
−vC
vI
−vD
δM
vM vM
FIGURE 3.16 ScatteringδM.
28Equal in magnitude, opposite in sign.
and the scattered waves have amplitudes:
differential modes vD = vM
2 , (3.37)
balanced phantom vP = 0, (3.38)
common phantom vC = vM
4 . (3.39)
The transmitted wave is
v0= vI − vD. (3.40)
In both capacitive and inductive coupling, therefore:
• NEXT, FEXT, and echo are generated simultaneously, and are equal at point of generation.
• All the waves are proportional tovI.29
• For small imbalances, all the couplings are approximately proportional to the frequency and to the imbalance (δC or δM). (A good approximation except when consideringv0; when there is any scattering at all,|v0|2< |vI|2by conservation of energy, and so the higher-order terms are significant here.)
• At higher frequencies, the Z0become nearly resistive and the scattered waves will be approximately±π2 out of phase with the incident wave.
Interestingly, capacitive imbalance seems to not have coupling to the common mode, so common mode currents from external noise sources should be unable to enter the differen-tial modes by this route. External noise ingress, RFI, and so forth must be entering through inductive imbalances only.
Because capacitive and inductive coupling elements have similar effects on crosstalk, they may be treated together. For any given pair of adjacent pairs (and for a study of either FEXT or NEXT), suppose a short cable element of lengthδ has a random coupling variate δX such that the direct coupling from the first pair to the second is
vD
vI
= jωδX. (3.41)
It is assumed thatδX is real (only strictly true when Z0Dis real) and has variance proportional toδ (reasonable if δ is long enough for separate lengths to be independent30). Furthermore, it is assumed thatδX has variance δσx2, which is equivalent to declaring the pairs to have coupling of varianceσx2per unit length.
3.6.3.1 NEXT
Of interest is the total NEXT from the first pair to the second. Along the length of the cable, each perturbationδX contributes a direct NEXT scatter component, and, at the near end of the second pair, their sum emerges with each component attenuated and delayed by the length of the first pair to the perturbation and again by the length of the second pair from the perturbation. Contributions indirectly coupled by more than one act of scattering may be neglected,31although the perturbations are each small, as the indirect contributions
29No surprises here: this circuitry is all linear. That is why analysis may consider one incident wave in isolation without loss of generality.
30This approximation allows integration; physically, at very short lengths one is inside a single physical cause of coupling and so asδ tends to 0 the lengths cease to be independent. For analysis, this effect can be neglected.
31For the same reason, it was safe to neglect the common modes’ transmission properties.
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will be of the order ofδ2or smaller. At any one frequency, the coupling will be a complex number formed as the sum
NEXTcoupling=!∞
0
jωδXsexp(−2γ s), (3.42)
where s is distance along the cable, stepping in increments ofδ, and γ = α + jβ is the propagation coefficient of the pairs.
In this sum of phasors, the succeeding contributions rotate with distance (as well as get smaller). Supposing that the (many) perturbationsδXsare independently random and of comparable size, the conditions for a two-dimensional variation of the central limit theorem are satisfied: a random complex variate of variance32 ω4α2σx2, whose real and imaginary parts are i.i.d Gaussian33with zero mean and varianceω8α2σx2, is expected.
TheδXsdo not change with time,34so the crosstalk between a given pair of pairs is stable.
They also do not change with frequency; however, the phasor rotations (and attenuations) do, so a substantial change of frequency results in a new sum. Above 100 kHz the loss of cables is dominated by the skin effect, so the loss in dB is proportional to
f . Hence, as a function of frequency,α is proportional to
f . For some particular cable type, assumeα = k
f . Hence, this analysis predicts that scalar NEXT coupling between given adjacent pairs at a given frequency is a random complex value with varianceπ2f3/2k σx2. In noise modelling, the power coupling properties are of primary interest. The square of the magnitude of the distribution is a negative exponential distribution whose mean is the variance above.
Hence, this analysis has predicted the form of the FSAN NEXT model, with some additional information: the power coupling is random and has the negative exponential distribution.
The FSAN NEXT model predicts the same coupling for all cables, which seems unlikely according to the above, as k definitely varies between cables, andσx2 seems to be related to the physical processes of manufacture. For the record, one would expect less NEXT from lossier cables. The analysis above also suggests that there is backscatter in telephone cables, and its law has the same form as NEXT, with a bigger constant of proportionality, because it is comparable with all the NEXT couplings to a pair added together. In practice, echo for DSL systems seems dominated by mismatches, particularly at the terminations, so presumably this backscatter is lost in the noise. Incidentally, if the sum above is taken over a finite length of cable (0L), one gets the length-dependent formula of Section 3.6.3.
3.6.3.2 FEXT
One can also consider the total FEXT from the first pair to the second. Along the length of the cable, each perturbationδX contributes a direct FEXT scatter component, and at the far end of the second pair their sum emerges. Each component has travelled the length of the cable and it is reasonable to assume their attenuations are equal. It is tempting to approximate their delays as equal too, but looking ahead this would lead to the false conclusion that the sum of the scattering terms is frequency-independent, because theδXsare frequency-independent. If this were the case then different pairs of pairs would have different coupling functions, but the coupling function for any one pair of pairs would be smooth. However, lab measurements show that each coupling function varies randomly with frequency. Thus, in this analysis it must be assumed there is some variation in delay between the two pairs along their length. Each scatter component can be modelled as having random phase in the
32Which is obtained by integration, takingδ → 0 in the sum above.
33This distribution is radially symmetric, so has phase uniformly distributed over 0 to 2π; the magnitude is independent of phase and has the “Rayleigh” distribution.
34Actually, they change as the rest of the cable properties change: with temperature, physical disturbance of the cable, and age.
sum. Again, contributions coupled by more than one act of scattering are neglected. At any one frequency, the coupling will be a complex number formed as the sum
FEXTcoupling=
!L 0
jωδXsexp(−γ L) = jω exp(−αL)
!L 0
δXsexp(− jβL), (3.43)
where s is distance along the cable, stepping in increments ofδ, γ = α+ jβ is the propagation coefficient of the pairs, andα, which relates to attenuation, may be taken as equal for all paths, whereasβ which relates to delay may not.
In this sum of phasors, the succeeding contributions rotate arbitrarily with distance.
Supposing that the perturbationsδXsare independently random, of comparable size, and many, the conditions for the two-dimensional variation of the central limit theorem are again satisfied: the result should be a random complex variate of varianceω2exp(−2αL)Lσx2 = ω2|H( f )|2Lσx2, whose real and imaginary parts are i.i.d. Gaussian with zero mean and varianceω2|H( f )|2 2Lσx2.
As for NEXT, the crosstalk coupling between a given pair of pairs is stable with time, but a substantial change of frequency results in an independent random value. Again, only the power coupling properties are of interest, and the resulting analysis yields a negative exponential distribution whose mean is the complex variate’s variance. Hence, the analysis has predicted the form of the FSAN NEXT model, with some additional information: the power coupling is random and has the negative exponential distribution. The reservation above about k in the NEXT model does not emerge from this FEXT analysis, suggesting the FSAN FEXT model adequately represents the cable properties in the|H( f )|2term.
References
[ANSI T1.413-2001] ANSI T1.413-2001, Spectrum Management for Loop Transmission Systems, American National Standard, 2001.
[ATIS 2001] ATIS T1E1.4. Spectrum Management For Loop Transmission Systems. T1.417-2001 may be downloaded from https://www.atis.org/atis/docstore/index.asp.
[Berger 1963] J.M. Berger and B. Mandelbrot. A New Model for Error Clustering in Telephone Circuits.
IBM Journal, July 1963.
[Bond 1987] D.J. Bond. A theoretical study of burst noise. BT Technology Journal, Vol. 5, No 4, October 1987.
[Chen 1993] W.Y. Chen and D.L. Waring. DMT ADSL Performance Simulation for CSA. ANSI T1E1.4/93-166, August 1993.
[Clayton 1994] R.P. Clayton. Analysis of Multiconductor Transmission Lines. Wiley, 1994.
[Cook 1993] J.W. Cook. Wideband impulsive noise survey of the access network. BT. Technol. Journal, Vol.
11, No. 3, July 1993.
[Cook 1999] J.W. Cook, R.H. Kirkby, M.G. Booth, K.T. Foster, D.E.A. Clarke, and G. Young. The Noise and Crosstalk Environment for ADSL and VDSL Systems. IEEE Communication Magazine, pp.
73–78, May 1999.
[ETSI ADSL 2002] Transmission and Multiplexing (TM); Access Transmission Systems on Metallic Access Cables; Asymmetric Digital Subscriber Line (ADSL) — European Specific Requirements. ETSI TS 101 388, V1.3.1, May 2002.
[ETSI Spect. Manag. 2002] ETSI TM6. Transmission and Multiplexing (TM); Spectral management on metallic access networks; Part 2: Technical methods for performance evaluations. TM6(01)20, latest version is Dec 02; to eventually be issued as TR 101 830 2.
[ETSI TS 101 388] ETSI TS 101 388 V1.3.1 (2002–05), Asymmetric Digital Subscriber Line (ADSL) — European specific requirements, ETSI Technical Specification, 2002.
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[ETSI TS 101 524] ETSI TS 101 524 V1.2.1 (2003–03), Symmetric single pair high bitrate Digital Subscriber Line (SDSL), ETSI Technical Specification, 2003.
[Ginis 2003] G. Ginis and J. Cioffi. Vectored Transmission for Digital Subscriber Line System. Submitted to IEEE JSAC special issue on twisted pair transmission may be downloaded from http://www-isl.stanford.edu/cioffi/dsm/vectorpap/vector.ps.
[Hagelbarger 1959] D.W. Hagelbarger. Recurrent Codes: Easily Mechanized, Burst-Correcting, Binary Codes. BSTJ, pp. 969–985, July 1959.
[Henkel 1994] W. Henkel and T. Kessler. A Wideband Impulsive Noise Survey in the German Telephone Network: Statistical Description and Modeling. Archiv fur Elekronik und Ubertragungstechnik, Vol. 48, No. 6, 1994.
[Henkel 1999] W. Henkel and T. Kessler. An Impulse-Noise Model — a Proposal for SDSL. ETSI TM6, May 1999, Grenoble meeting, Technical document 45.
[Hughes 1997] H. Hughes. Telecommunications Cables. Wiley, Chichester, 1997.
[Kirkby 1995] R. Kirkby (idea due to John Cook, of BT Laboratories). FEXT Is Not Reciprocal. ANSI T1E1.4/95-141, Orlando meeting, November 1995.
[Kirkby 2001] R. Kirkby. Text for “Realistic Impulsive Noise Model” ETSI TM6 Feb. 2001, Sophia Antipolis meeting, Technical document 20.
[Mandelbrot 1965] B. Mandelbrot. Self-Similar Error Clusters in Communications Systems and the Concept of Conditional Stationarity. IEEE Trans. Communication Technology, March 1965.
[McLaughlin 1999] S. McLaughlin et al. Statistics of Impulse Noise. ETSI TM6, Edinburgh meeting, Sept 1999, Technical documents 18–21.