Transmission properties of pair cables Nils Holte, NTNU NTNU - - PDF document

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Digital Kommunikasjon 2002 1

NTNU

Department of Telecommunications

Transmission properties of pair cables

Nils Holte, NTNU

Digital Kommunikasjon 2002 2

NTNU

Department of Telecommunications

Overview

• Part I:

Physical design of a pair cable

• Part II:

Electrical properties of a single pair

• Part III: Interference between pairs,

crosstalk

• Part IV: Estimates of channel capacity of pair
• cables. How to exploit the existing

cable plant in an optimum way

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Part I Basic design of pair cables

• A single twisted pair
• Binder groups
• The cross-stranding principle
• Building binder groups and cables of

different sizes

• A complete cable

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Cross section of a single pair

Most common conductor diameters: 0.4 mm, 0.6 mm (0.5 mm, 0.9 mm)

Insulation Conductor

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A twisted pair

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Typical pair cables of the Norwegian access network

• 0.4 and 0.6 mm conductor diameter
• Polyethylene insulation (expanded)
• Twisting periods in the interval

50 - 150 mm

• 10 pair cross-stranded binder groups
• 10 - 2000 pairs in a cable

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Pair positions in a 10 pair binder group

Cross-stranding [13]: The positions of all the pairs are alternated randomly along the cable. Interference is thus randomised, and all pairs will be almost uniform

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Cross-stranding

Cross-stranding technique: Each pair runs through a die in the cross-stranding matrix. The positions of the dies are set by a random generator

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Design of binder groups up to 100 pairs

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Design of binder groups up to 1000 pairs

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Typical cable design (”Kombikabel”)

This cable may be used as: 1) overhead cable 2) buried cable 3) in water (fresh water)

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Part II Properties of a single pair

• Equivalent model of a single pair
• Capacitance
• Inductance
• Skin effect
• Resistance
• Per unit length model of a pair
• The telegraph equation - solution
• Propagation constant
• Characteristic impedance
• Reflection coefficient, terminations

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A single pair in uniform insulation

εr

2a 2r

A coarse estimate of the primary parameters R, L, C, G may be found assuming a >> r, uniform insulation, and no surrounding conductors

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Model of a single pair in a cable

The electrical influence of the surrounding pairs in the cable may be modelled as an equivalent

• shield. This model

will give accurate estimates of R, L, C and G [12]

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Capacitance of a single pair

C a r

r

= πε ε 2 ln

The capacitance per unit length of a single pair is given by: The capacitance of cables in the access network is 45 nF/km

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Inductance of a single pair

L a r = µ π 2 ln

The inductance per unit length of a single pair is given by:

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Skin effect

At high frequencies the current will flow in the outer part

• f the conductors, and the skin depth is given by [12]:

δ π µ µ σ = 1 f

r

σ is the conductance of the conductors

For copper the skin depth is given by:

δ = 2 11 , F

kHz

mm

δ δ = = 2 11 0 067 . . mm at 1 kHz mm at 1 MHz

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Resistance of a conductor

The resistance per unit length of a conductor is for r << a given by:

R r r r r

C =

>> <<        1 1 2

2

σπ δ σπ δ δ for (low frequencies) for (high frequencies)

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Resistance of a pair

The resistance per unit length of a pair will be the sum of the resitances of the two conductors and is given by:

R R r r r r

C

= ⋅ = >> <<        2 2 1

2

σπ δ σπ δ δ for (low frequencies) for (high frequencies)

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Conductance of a pair

G C

l

= ⋅ δ ω

The conductance per unit length of a pair is given by:

δl is the dielectric loss factor

A typical value of the loss factor is δl = 0.0003. This means that the conductance is usually negligible for pair cables.

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Per unit length model of a pair

L∆x R∆x G∆x C∆x U+∆U I+∆I ∆x

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Model of a pair of length l

R, L, C, G x=0 x=l I(x) U(x)

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The telegraph equation

d dx U x R j L I x Z I x d dx I x G j C U x Y U x ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = − + = − ⋅ = − + = − ⋅ ω ω

d dx U x Z Y U x

2

( ) ( ) = ⋅ ⋅

Combining the equations: From the circuit diagram:

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Solution of the telegraph equation

U(x) c e c e I(x) c Z e c Z e

1 x 2 x 1 x 2 x

= ⋅ + ⋅ = − ⋅ + ⋅

⋅ − ⋅ ⋅ − ⋅ γ γ γ γ

c1 and c2 are constants γ is propagation constant Z0 is characteristic impedance

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Propagation constant

α is the attenuation constant in Neper/km β is the phase constant in rad/km α α α

dB =

= 20 10 8 69 ln( ) . Neper to dB:

γ ω ω ω ω = ⋅ = + ⋅ + = + ⋅ Z Y R j L G j C R j L j C ( ) ( ) ( ) γ α β = + j

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Characteristic impedance

Z Z Y R j L G j C R j L j C

0 =

= + + = + ( ) ( ) ( ) ω ω ω ω

At high frequencies R << jωL :

Z L C

0 =

The characteristic impedance is approximately 120 ohms at high frequencies for pair cables in the access network

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Attenuation constant

α = + = = ⋅ R C L G L C R C L k f 2 2 2

1

At high frequencies (f > 100 kHz) R <<ωL. By series expansion of γ : At low frequencies (f < 10 kHz) R >>ωL. Hence:

γ ω ω α β ω = ⋅ = + ⋅ ⋅ = = ⋅ ⋅ = j C R j R C R C k f ( ) 1 2 2

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Attenuation constant of pair cables

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Phase constant

At high frequencies (f > 100 kHz):

β ω = ⋅ = ⋅ L C k f

3

Phase velocity:

v x t L C c

r

= = = = = ⋅ = ∆ ∆ wavelength cycle 2 2 1 π β π ω ω β ε

Phase velocity is 200 000 km/s for pair cables at high frequencies (εr = 2.3 for polyethylene)

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Termination of a pair

R, L, C, G x=0 I(l) U(l) ZT x=l

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Reflection coefficient

U(x) V e V e I(x) V Z e V Z e Z U I Z V V V V

x x x x T

= ⋅ + ⋅ = ⋅ − ⋅ = = + −

+ ⋅ − − − ⋅ − + ⋅ − − − ⋅ − + − + − γ γ γ γ ( ) ( ) ( ) ( )

( ) ( )

l l l l

l l

ρ = = − +

− +

V V Z Z Z Z Z

T T

Solution of telegraph equation including termination imp. ZT V+ is voltage of wave in positive direction at x=l V- is voltage of reflected wave in at x=l

Reflection coefficient:

No reflections (ρ =0) for ZT= Z0 . Ideal terminations are assumed in later crosstalk calculations

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Part III Crosstalk in pair cables

• Basic coupling mechanisms
• Crosstalk coupling per unit length
• Near end crosstalk, NEXT
• Far end crosstalk, FEXT
• Statistical crosstalk coupling
• Average NEXT and FEXT
• Crosstalk power sum - crosstalk from many pairs
• Statistical distributions of crosstalk

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Crosstalk mechanisms

Main contributions: Capacitive coupling Inductive coupling

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Ideal and real twisting of a pair

Ideal twisting Actual twisting of a real pair The crosstalk level observed in real cables is caused mainly by deviations from ideal twisting

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Crosstalk coupling per unit length

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Crosstalk coupling per unit length

Normalised NEXT coupling coefficient [11]: Normalised FEXT coupling coefficient [11]:

κ β

N N i j i j i j

x j dU U dx C x C L x L

, , ,

( ) ( ) ( ) = ⋅ ⋅ = ⋅ +       1 1 2

2 1

Ci,j(x) is the mutual capacitance per unit length between pair i and j Li,j(x) is the mutual inductance per unit length between pair i and j β0 is the lossless phase constant,

β ω

0 =

⋅ L C

κ β

F F i j i j i j

x j dU U dx C x C L x L

, , ,

( ) ( ) ( ) = ⋅ ⋅ = ⋅ −       1 1 2

2 1

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Near end crosstalk, NEXT

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Near end crosstalk, NEXT

H f V V j x e dx

NE N x j x

( ) ( ) = =

− −

∫

20 10 2 2

β κ

α β l

Assuming weak coupling, the near end voltage transfer function is given by [11]:

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NEXT between two pairs

10

• 1

10 10

1

30 40 50 60 70 80 90 100

Frequency, MHz Near end crosstalk, dB

pair combination pair combination pair combination average NEXT

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Stochastic model of crosstalk couplings

Crosstalk coupling factors are white Gaussian stocastic processes: NEXT autocorrelation function [6]: FEXT autocorrelation function [6]:

R E x x k

F F F F

( ) ( ) ( ) ( ) τ κ κ τ δ τ = ⋅ +

[ ] =

⋅ R E x x k

N N N N

( ) ( ) ( ) ( ) τ κ κ τ δ τ = ⋅ +

[ ] =

⋅

kN and kF are constants

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Average NEXT

p f E H f E V V k e dx k e k k f

NE N x N N

( ) ( )

.

= [

] =

        = = ⋅ −

( ) ≈

⋅ = ⋅

− −

∫

2 20 10 2 2 4 2 4 2 2 1 5

4 1 4 β β α β α

α l l N

NEXT increases 15 dB/decade with frequency Average NEXT power transfer function between two pairs [6]:

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Far end crosstalk, FEXT

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Far end crosstalk, FEXT

H f V V j x dx

FE F

( ) ( ) = =

∫

2 1 l l l

β κ

Assuming weak coupling, the far end voltage transfer function is given by [11]:

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Average FEXT

q f E H f E V V k dx k k f

FE F F

( ) ( ) = [

] =

        = = ⋅ ⋅ = ⋅ ⋅

∫

2 2 1 2 2 2 2 l l l

l l β β

F2

FEXT increases 20 dB/decade with frequency FEXT increases 10 dB/decade with cable length Average FEXT power transfer function between two pairs [6]:

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Crosstalk from N different pairs crosstalk power sum

• Only crosstalk between identical systems is

considered (self NEXT and self FEXT)

• Crosstalk from different pairs add on a

power basis

• Effective crosstalk is given by the sum of

crosstalk power transfer functions, which is denoted crosstalk power sum

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Crosstalk from N different pairs crosstalk power sum

H f H f

NE ps NE i j j j i N

( ) ( )

, 2 2 1

=

[ ]

= ≠

∑

NEXT crosstalk power sum for pair no i:

H f H f

FE ps FE i j j j i N

( ) ( )

, 2 2 1

=

[ ]

= ≠

∑

FEXT crosstalk power sum for pair no i:

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NEXT power sum

10

• 1

10 10

1

25 30 35 40 45 50 55 60 65 70

Frequency, MHz NEXT power sum, dB

NEXT ps, cable 1 NEXT ps, cable 2 NEXT ps, cable 3 average NEXT ps

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Probability distributions of crosstalk

Crosstalk power transfer function for a single pair combination at a given frequency is gamma-distributed with probability density [6]: ν = 1.0 for NEXT ν = 0.5 for FEXT a is average crosstalk power

p z a z e

z z a

( ) ( ) = ⋅    ⋅ ⋅

− −

1

1

Γ ν ν

ν ν ν

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Probability density of NEXT and FEXT for a single pair combination

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Amplitude relative to average power transfer function Probability density

ny = 0.5 (FEXT) ny = 1.0 (NEXT)

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Probability of crosstalk for an arbitrary pair combination

Different pair combinations will have different levels

• f crosstalk coupling (different kN and kF).

It can be shown that: The crosstalk power transfer function for a random pair combination is approximately gamma-distributed The number of degrees of freedom, ν must be found

• empirically. ν < 1.0 for NEXT and ν < 0.5 for FEXT

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Probability of crosstalk power sum

Crosstalk from different pairs add on a power basis. Hence, crosstalk power sum is approximately gamma-distributed with probability distribution:

p z a z e

ps ps ps ps z a

ps ps ps ps

( ) ( ) = ⋅       ⋅ ⋅

− −

1

1

Γ ν ν

ν ν ν aps=Na, where a is the average crosstalk of one pair combination νps=Nν , where ν is the number of degrees of freedom for a random pair combination N is the number of disturbing pairs

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Probability density of crosstalk power sum

0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Amplitude relative to average power sum Probability density

ny = 0.2 ny = 0.5 ny = 1.0 ny = 2.0 ny = 5.0

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Worst case crosstalk

Crosstalk dimesioning is usually based upon the 99% point

• f NEXT and FEXT power sum, which is given by

(1% of the pairs will have crosstalk that exceeds this limit):

p f N E k f c q f N E k f c

ps ps ps ps 99 2 1 5 99 99 2 2 99

( ) ( ) ( ) ( )

.

= ⋅ [

]⋅

⋅ = ⋅ [

]⋅

⋅ ⋅

N F

for NEXT for FEXT ν ν l

c99(νps) is the ratio between the 99% point and the average power sum in the gamma distribution The expectations E[kN2] and E[kF2] are taken over all pair combinations

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Empirical worst case NEXT model

p F N F

ps 99 4 0 6 1 5

10 49 ( )

. .

= ⋅    ⋅

− N is the number of disturbing pairs in the cable F is the frequency in MHz

International model of 99% point of NEXT power sum based upon 50 pair binder groups: This model fits well with 100% filled Nowegian cables with 10 pair binder groups

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Empirical worst case FEXT model

q F N F L

ps 99 4 0 6 2

3 10 49 ( )

.

= ⋅ ⋅    ⋅ ⋅

− N is the number of disturbing pairs in the cable F is the frequency in MHz L is the cable length in km

International model of 99% point of FEXT power sum based upon 50 pair binder groups: This model fits well with 100% filled Nowegian cables with 10 pair binder groups

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Part IV Channel capacity estimates [1,2,8]

• Shannon’s channel capacity formula
• Signal and noise models
• Realistic estimates of channel capacity
• Channel capacity per bandwidth unit
• One-way transmission
• Two-way transmission
• Crosstalk between different types of systems,

alien NEXT, alien FEXT

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Shannon’s theoretical channel capacity

Maximum theoretical channel capacity in the frequency band [fl,fh]:

C S f N f df

Sh fl fh

= +      

∫ log

( ) ( )

2 1

bit/s

S(f): signal power density N(f): noise power density

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Capacity per bandwidth unit

A realistic estimate of bandwidth efficiency [2]:

η λ ( ) log ( ) ( ) f C f k S f N f

eff

= = ⋅ + ⋅       ∆ ∆

2 1

bit/s/Hz

S(f): signal power density N(f): noise power density λ ≤ 1: factor for margin (safety margin + margin for mod.meth.) keff ≤ 1: factor for overhead (sync bits, RS-code, cyclic prefix)

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Signal transmission

• Attenuation proportional to

due to skin effect (f > 100 kHz)

• Signal transfer function:

f

H f k f

dB

( ) exp = = −

( )

−

10

20 α l

l

αdB: attenuation constant in dB

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Signal and noise models

• Signal:
• Noise models:

N F N F NEXT N F L e FEXT N AWGN

NEXT FEXT L AWGN

( )

.

= = ⋅ = ⋅ ⋅ ⋅ ⋅ =     

− − − −

10 3 10 10

4 1 5 4 2 2 8 α

S F e

L

( ) =

−2α

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Channel capacity vs. frequency

2 4 6 8 10 5 10 15

Frequency, MHz Potential bandwith efficiency, bit/s/Hz

FEXT AWGN NEXT

L=1 km

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Channel capacity vs frequency II

2 4 6 8 10 5 10 15

Frequency, MHz Potential bandwith efficiency, bit/s/Hz

• ne-way transmission

two-way transmission 2*two-way transmission 0.5 1 1.5 2 5 10 15

Frequency, MHz Potential bandwith efficiency, bit/s/Hz

• ne-way transmission

two-way transmission 2*two-way transmission

L=1 km L=3 km

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Total channel capacity

R k S F N N df

• ne way

eff FEXT AWGN fl fh −

= + ⋅ +      

∫ log

( )

2 1

λ R k S F N N N df

two way eff NEXT FEXT AWGN fl fh −

= + ⋅ + +      

∫ log

( )

2 1

λ

One-way transmission: Two-way transmission:

The channel capacity is somewhat greater than this expression for two-way transmission due to uncorrelated NEXT in different frequency bands [9,10]

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Assumptions for estimation of bitrates

• For one-way transmission, the total bitrate found

in the calculations must be divided by downstream and upstream transmission

• Identical systems in all pairs of the cable (only self

NEXT and self FEXT)

• All pairs are used, the cable is 100% filled
• Net bitrate is 90% of total bitrate (keff=0.90)
• Frequency band: f ≥ 100 kHz,

upper limit 11 MHz

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Assumptions II

• Multicarrier modulation [7]
• Adaptive modulation in each sub-band
• M-TCM modulation in each sub-band

4 ≤ M ≤ 16384, 1 - 13 bit/s/Hz

• Distance to Shannon (λ): 9 dB

(6 dB margin + 3 dB for modulation)

• White noise: 80 dB below output signal
• Cable: 0.4 mm, 22.5 dB/km at 1 MHz

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Potential range for .4 mm cable

1 2 3 4 5 10 20 30 40 50 60

Range, Km Bitrate, Mbit/s

• ne-way transmission

two-way transmission 1 2 3 4 5 10 20 30 40 50 60

Range, Km Bitrate, Mbit/s

• ne-way transmission

two-way transmission

VDSL ADSL

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Potential range for .4 mm cable II

2 2.5 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 3 3.5 4

Range, Km Bitrate, Mbit/s

• ne-way transmission

two-way transmission

SHDSL+ SHDSL+ means a multicarrier system using approximately the same frequency band as SHDSL

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Digital Subscriber Line systems, xDSL

• ADSL; Asymmetric Digital Subscriber Lines [5]

– Asymmetric data rates, 256 kbit/s - 8 Mbit/s downstream, range up to 4 - 5 km

• SHDSL; Symmetric High-speed Digital

Subscriber Lines [3]

– Symmetric data rates, 192 kbit/s - 2.3 Mbit/s, range up to 6 - 7 km

• VDSL; Very high-speed Digital Subscriber Lines

– Asymmetric or symmetric data rates (still under standardisation), up to 52 Mbit/s downstream, range typically ≤ 1 km [4]

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Frequency allocations in the access network

Upstream Downstream

ADSL SHDSL low ADSL + VDSL Frequency, kHz Frequency, kHz 2 Mbit/s SHDSL SHDSL low 1104 2 Mbit/s SHDSL 125 211 400 1104 125 276 400

VDSL

SHDSL low ≤ 640 kbit/s

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Conclusions

• Frequency planning in pair cables is very

important

• New systems should be introduced with

great care in order to preserve the potential transmission capacity of the cable

• Full rate SHDSL systems overlaps with

ADSL

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References I

[1] J.-J. Werner, "The HDSL environment," IEEE Journal on Sel. Areas in Commun., August 1991, pp. 785-800 [2]

• T. Starr, J. M. Cioffi, P. J. Silverman, "Understanding digital

subscriber line technology," Prentice Hall, Upper Saddle River, 1999 [3] ITU-T Recommendation G.991.2, " Single-Pair High-Speed Digital Subscriber Line (SHDSL) transceivers", Geneva, 2001 [4] ETSI TS 101 270-1, V1.2.1, "Very high speed Digital Subscriber Line (VDSL); Part 1: Functional requirements", Sophia Antipolis, 1999 [5] ITU-T Recommendation G.992.1, "Asymmetric Digital Subscriber Line (ADSL) transceivers", Geneva, 1999

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References II

[6]

• H. Cravis, T. V. Crater, "Engineering of T1 carrier system

repeatered lines," Bell System Techn. Journal, March 1963,

• pp. 431-486

[7]

• J. A. C. Bingham, "Multicarrier modulation for data transmission:

An idea whose time has come," IEEE Communications Magazine, May 1990, pp. 5-19 [8]

• N. Holte, “Broadband communication in existing copper cables

by means of xDSL systems - a tutorial”, NORSIG, Trondheim, October 2001 [9]

• N. Holte, “A new method for calculating the channel capacity in

pair cables with respect to near end crosstalk”, DSLcon Europe, Munich, November 2001

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References III

[10]

• A. J. Gibbs, R. Addie, "The covariance of near end crosstalk and

its application to PCM system engineering in multipair cable," IEEE Trans. on Commun., Vol. COM-27, No. 2, Feb. 1979, pp.469-477 [11]

• W. Klein, "Die Theorie des Nebensprechens auf Leitungen,"

Springer-Verlag, Berlin, 1955 [12]

• H. Kaden, “Wirbelströme und Schirmüng in der Nachrichten-

technik”, Springer-Verlag, Berlin 1959 [13]

• S. Nordblad, "Multi-paired Cable of nonlayer design for low

capacitance unbalance telecommunication networks," Int. Wire and cable symp., 1971, pp. 156-163