Exercise 2.5Z: Range and Bit Rate with ADSL

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"Range vs. bit rate" diagram

The development of xDSL technology began in 1995 with the first standard for $\rm ADSL$  (Asymmetric Digital Subscriber Line). From 2006, the faster  $\rm VDSL$  (Very High Data Rate Digital Subscriber Line) also came into use in Germany.

The graph shows five system variants in a diagram in which the achievable cable length  $l_{\rm max}$  is plotted as a function of the total bit rate  $R_{\rm ges}$ :

• $\boldsymbol{\rm A}\text{:} \hspace{0.3cm} \text{ Bitraten} \ 0.2 \ {\rm Mbit/s} + 2 \ {\rm Mbit/s}; \hspace{0.2cm} \text{ cable length } l_{\rm max} \approx 3.5 \ {\rm km},$
• $\boldsymbol{\rm B}\text{:} \hspace{0.3cm} \text{ Bitraten} \ 0.2 \ {\rm Mbit/s} + 6 \ {\rm Mbit/s}; \hspace{0.2cm} \text{ cable length } l_{\rm max} \approx 2 \ {\rm km},$
• $\boldsymbol{\rm C}\text{:} \hspace{0.3cm} \text{ Bitraten} \ 2 \ {\rm Mbit/s} + 13 \ {\rm Mbit/s}; \hspace{0.3cm} \text{ cable length } l_{\rm max} \approx 1 \ {\rm km},$
• $\boldsymbol{\rm D}\text{:} \hspace{0.3cm} \text{ Bitraten} \ 2 \ {\rm Mbit/s} + 26 \ {\rm Mbit/s}; \hspace{0.3cm} \text{ cable length } \ l_{\rm max} \approx 0.8 \ {\rm km},$
• $\boldsymbol{\rm E}\text{:} \hspace{0.3cm}\text{ Bitraten} \ 2 \ {\rm Mbit/s} + 51 \ {\rm Mbit/s}; \hspace{0.35cm} \text{ cable length } l_{\rm max} \approx 0.4 \ {\rm km}.$

To this graphic is to be noted further:

• All data applies to a balanced copper pair with  $\text{0.4 mm}$  diameter.
• One of the bit rates given here is for upstream, the other is for downstream.
• The total bitrate is the sum of the two portions.
• Which bit rate refers to the upstream and which to the downstream is asked in the subtask (1).
• The colored differentiation of the drawn points refers to the subdivision into ADSL and VDSL. This is referred to in the subtask (2).
• The curve drawn in blue shows a rule of thumb that approximates the relationship between range and total bit rate:

$$l_{\rm max}\,{\rm \big [in}\,\,{\rm km \big]} = \frac {20}{4 + R_{\rm ges}\,{\rm \big[in}\,\,{\rm Mbit/s\big]}} \hspace{0.05cm}.$$

• Dashed are deviations from this by  $\pm 25\%$.

One often characterizes a wireline transmission system by the cable attenuation at half the bit rate (note the "a" in the attenuation):

$${\rm a}_{\rm \star} = {\rm a}_{\rm K}(f = {R_{\rm B}}/{2}) = \alpha_{\rm K}(f = {R_{\rm B}}/{2}) \cdot l\hspace{0.05cm}.$$

The attenuation coefficient (noted as "alpha") is given for a $\text{0.4 mm}$ balanced copper pair as follows:

$$\alpha_{\rm K}(f ) = \left [ 5.1 + 14.3 \cdot \left ({f}/({\rm 1\,MHz})\right )^{0.59} \right ] {\rm dB}/{\rm km} \hspace{0.05cm}.$$

For the downlink of variant  $\boldsymbol{\rm A}$  with  $R_{\rm B} = 2 \ \rm Mbit/s$  thus results with  $l = l_{\rm max} = 3.5 \ \rm km$:

$$\alpha_{\rm K}(f = {\rm 1\,MHz}) = \left [ 5.1 + 14.3 \right ] {\rm dB}/{\rm km} = 19.4\,{\rm dB}/{\rm km}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} {\rm a}_{\rm \star} = 19.4\,{\rm dB}/{\rm km} \cdot 3.5\,{\rm km} = 67.9\,{\rm dB}\hspace{0.05cm}.$$

The values for the other system variants are to be determined in subtask (4).

Hint:

• This exercise belongs to the chapter  "Methods to Reduce the Bit Error Rate in DSL".

Questions

1 For example, consider the system variant  $\boldsymbol{\rm C}$. Which applies?

	The upstream bit rate is  $13 \ \rm Mbit/s$.
	The downstream bit rate is  $13 \rm Mbit/s$.

2 Which of the drawn variants are ADSL or VDSL systems?

	The red dots indicate VDSL systems.
	The green dots indicate VDSL systems.

3 What range would result from the given rule of thumb for  $R_{\rm ges} = 1 \ \rm Gbit/s$.

$l_{\rm max} \ = \ $ $ \ \rm m$.

4 Calculate the characteristic cable attenuations for the variant

$\boldsymbol{\rm B}\text{:} \hspace{0.4cm} {\rm a}_{\ast} \ = \ $ $ \ \rm dB$

$\boldsymbol{\rm C}\text{:} \hspace{0.4cm} {\rm a}_{\ast} \ = \ $ $ \ \rm dB$

$\boldsymbol{\rm D}\text{:} \hspace{0.4cm} {\rm a}_{\ast} \ = \ $ $ \ \rm dB$

$\boldsymbol{\rm E}\text{:} \hspace{0.4cm} {\rm a}_{\ast} \ = \ $ $ \ \rm dB$

Solution

Solution

(1)  Correct is the second proposed solution:

• In all xDSL variants, the downstream is operated at a higher bit rate than the upstream.
• This principle results from user behavior. One fetches much more data to the computer (downstream) than in the reverse direction.

(2)  Correct is the second proposed solution:

• VDSL offers higher data rates.
• High data rate, however, is only possible with relatively short line lengths.

(3)  The range of such a Gbit/s system over two-wire line would be about $20/1000

\ {\rm km} \underline{= 20 \ \rm meters}$. 

• Consider this subtask rather academic.

(4)  Here the following characteristic cable attenuations result. For

• variant $\boldsymbol{\rm B} \ (R_{\rm B}/2 = 3 {\rm \ Mbit/s}, \ l_{\rm max} = 2 {\rm \ km})\text{:}$

$${\rm a}_{\rm \star} = \left [ 5.1 + 14.3 \cdot 3^{0.59} \right ] \cdot 2\,{\rm dB}\hspace{0.15cm}\underline{ \approx 64.9\,{\rm dB}}\hspace{0.05cm},$$

• variant $\boldsymbol{\rm C} \ (R_{\rm B}/2 = 6.5 {\rm \ Mbit/s}, \ l_{\rm max} = 1 {\rm \ km})\text{:}$

$${\rm a}_{\rm \star} = \left [ 5.1 + 14.3 \cdot 6.5^{0.59} \right ] \cdot 1\,{\rm dB} \hspace{0.15cm}\underline{\approx 48.2\,{\rm dB}}\hspace{0.05cm},$$

• variant $\boldsymbol{\rm D} \ (R_{\rm B}/2 = 13 {\rm \ Mbit/s}, \ l_{\rm max} = 0.8 {\rm \ km})\text{:}$

$${\rm a}_{\rm \star} = \left [ 5.1 + 14.3 \cdot 13^{0.59} \right ] \cdot 0.8\,{\rm dB}\hspace{0.15cm}\underline{ \approx 56\,{\rm dB}}\hspace{0.05cm},$$

• variant $\boldsymbol{\rm E} \ (R_{\rm B}/2 = 25.5 {\rm \ Mbit/s}, \ l_{\rm max} = 0.4 {\rm \ km})\text{:}$

$${\rm a}_{\rm \star} = \left [ 5.1 + 14.3 \cdot 25.5^{0.59} \right ] \cdot 0.4\,{\rm dB}\hspace{0.15cm}\underline{ \approx 40.7\,{\rm dB}}\hspace{0.05cm}.$$

Further, it should be noted:

• The characteristic cable attenuation ${\rm a}_{\ast}$ of ADSL systems is in the range $65 \ \rm dB$ ... $68 \ \rm dB$.
• The VDSL variants provide characteristic cable attenuations between $40 \ \rm dB$ and $56 \ \rm dB$.
• It should be noted, however, that this system parameter ${\rm a}_{\ast}$, which is important in conventional binary baseband transmission, does not reflect the conditions in OFDM or Discrete Multitone Transmission sufficiently well.