Aufgaben:Exercise 3.7Z: Regenerator Field Length

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Results of a system simulation

By simulation,  it was shown that there is approximately a linear relationship between the so-called  "system efficiency"   $\eta$  and the  "characteristic cable attenuation"   $a_*$  of a coaxial cable – both plotted in  $\rm dB$  – if the characteristic cable attenuation is sufficiently large  $(a_* ≥ 40 \ \rm dB)$:

$$10 \cdot {\rm lg}\hspace{0.1cm}\eta \hspace{0.15cm} {\rm (in \hspace{0.15cm}dB)}= A + B \cdot a_{\star}\hspace{0.05cm}.$$

In the table,  the empirically found coefficients  $A$  and  $B$  are given for four exemplary system variants:

• for the intersymbol interference binary system  $(M = 2)$  with Gaussian low-pass filter  $\rm (GLP)$,  see chapter 
  "Error Probability with Intersymbol Interference",

• for the intersymbol interference octal system  $(M = 8)$  with Gaussian low-pass filter  $\rm (GLP)$,  see chapter 
  "Intersymbol Interference for Multi-Level Transmission",

• for optimal ISI–free systems  $\rm (ONE)$,  see chapter  "Linear Nyquist Equalization";  $M = 2$  and  $M = 8$.

The larger the system efficiency  $\eta$,  the better a system is for a given value  $a_*$  (and thus a fixed cable length).

For the calculation of the  regenerator field length  (distance between two repeaters),  it should be noted:

• The worst-case error probability should not be larger than  $10^{-10}$, which results in the minimum sink SNR:

$$10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm min} \approx 16.1\,{\rm dB} \hspace{0.05cm},$$

• The logarithmized ratio of transmit energy  (per bit)  and AWGN noise power density is about  $100 \ \rm dB$,  for example:

$$s_0 = 3\,{\rm V},\hspace{0.2cm}R_{\rm B} = 1\,{\rm Gbit/s},\hspace{0.2cm}N_{\rm 0} = 9 \cdot 10^{-19}\,{\rm V^2/Hz}$$

$$\Rightarrow \hspace{0.3cm}10 \cdot {\rm lg}\hspace{0.1cm}\frac{s_0^2 }{N_0 \cdot R_{\rm B}}= 10 \cdot {\rm lg} \hspace{0.1cm} \frac{9\,{\rm V^2} } {9 \cdot 10^{-19}\,{\rm V^2/Hz}\cdot 10^{-9}\,{\rm 1/s}}= 100\,{\rm dB} \hspace{0.05cm}.$$

• A standard coaxial cable with dimensions  $2.6 \ \rm mm$  (inside)  and  $9.5 \ \rm mm$  (outside)  is to be used,  for which the following relationship is valid:

$$a_{\star} = \frac{2.36\,{\rm dB} } {{\rm km} \cdot \sqrtTemplate:\rm MHz} \cdot l \cdot \sqrt{{R_{\rm B}}/{2}}\hspace{0.05cm}.$$

Here,  $a_*$  denotes the characteristic attenuation at half the bit rate – in the example at  $500 \ \rm MHz$  – and  $l$  denotes the cable length.

Note:  The exercise belongs to the chapter  "Linear Nyquist Equalization".

Questions

1 Which of the following statements are true?

	The system  $({\rm ONE}, \ M = 8)$  is best for any  $a_*$.
	The system  $({\rm GLP}, \ M = 2)$  is worst for  $a_* ≥ 40 \ \rm dB$.

2 Starting from which cable attenuation is the system  $({\rm GLP}, \ M = 8)$  better than the system  $({\rm ONE}, \ M = 2)$?

$a_{\rm *, \ limit}\ = \ $ $\ \rm dB$

3 What is the minimum value  $\eta_{\hspace{0.05cm}\rm min}$  that the system efficiency must never fall below?

$10 \cdot {\rm lg} \ \eta_{\hspace{0.05cm}\rm min} \ = \ $ $\ \rm dB$

4 What is the maximum length of the coaxial cable for the system  $({\rm ONE}, \ M = 8)$? 

$l_{\hspace{0.05cm}\rm max}\ = \ $ $\ \rm km$

5 What is the maximum length of the coaxial cable for the system  $({\rm GTP}, \ M = 2)$? 

$l_{\hspace{0.05cm}\rm max}\ = \ $ $\ \rm km$

Solution

Solution

(1)  Calculating the system efficiency under the assumption  $a_* = 40 \ \rm dB$,  we obtain for the four system variants:

$$({\rm GLP},\hspace{0.1cm}M=2) \text{:}\hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\eta= +9.4\,{\rm dB} -1.10 \cdot 40\,{\rm dB} = -34.6\,{\rm dB}\hspace{0.05cm},$$

$$({\rm GLP},\hspace{0.1cm}M=8) \text{:}\hspace{0.3cm}10 \cdot {\rm lg}\hspace{0.1cm}\eta= -1.3\,{\rm dB} -0.91 \cdot 40\,{\rm dB} = -37.7\,{\rm dB}\hspace{0.05cm},$$

$$({\rm ONE},\hspace{0.1cm}M=2) \text{:}\hspace{0.3cm}10 \cdot {\rm lg}\hspace{0.1cm}\eta= +4.5\,{\rm dB} -0.96 \cdot 40 \,{\rm dB}= -33.9\,{\rm dB}\hspace{0.05cm},$$

$$({\rm ONE},\hspace{0.1cm}M=8) \text{:}\hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\eta = -9.3\,{\rm dB} -0.54 \cdot 40\,{\rm dB} = -30.9\,{\rm dB}\hspace{0.05cm}.$$

From this it follows:

• The  first statement  is true because the system  $({\rm ONE},\hspace{0.1cm} M = 8)$  is already best at  $40 \ \rm dB$  cable attenuation and has the most favorable  $\rm B$  coefficient.

• In contrast,  the second statement is false because,  for example,  at $40 \ \rm dB$ cable attenuation,  the octal $\rm GLP$ system is worse than the binary one.

(2)  As a determination equation,  we use here:

$$-1.3\,{\rm dB} -0.91 \cdot a_{\star} = +4.5 \,{\rm dB}-0.96 \cdot a_{\star}\hspace{0.3cm}\Rightarrow \hspace{0.3cm} 0.05 \cdot a_{\star} = 5.8\,{\rm dB}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}a_{\star,\hspace{0.05cm}{\rm limit}} \hspace{0.15cm}\underline {= 116\,{\rm dB}}\hspace{0.05cm}.$$

That is:

• Up to the characteristic cable attenuation  $a_* = 116 \ \rm dB$  $($note:  this is an unrealistically large value for currently realized systems$)$,  the binary Nyquist system is superior to the system $({\rm GLP},\hspace{0.1cm} M = 8)$.

• Only for larger values than  $a_{\rm *, \ limit} = 116 \ \rm dB$  does the advantage of the latter  $(M = 8$  and thus significantly lower symbol rate$)$ outweigh the disadvantage  $($octal decision and thus greater weight of intersymbol interference$)$.

(3)  The sink SNR should be at least  $16.1 \ \rm dB$,  which means that it must be valid:

$$10 \cdot {\rm lg}\hspace{0.1cm}\rho = 10 \cdot {\rm lg}\hspace{0.1cm}\frac{s_0^2 }{N_0 \cdot R_{\rm B}} + 10 \cdot {\rm lg}\hspace{0.1cm}\eta \hspace{0.3cm}\Rightarrow \hspace{0.3cm}10 \cdot {\rm lg}\hspace{0.1cm}\eta \ >\ 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm min} - 10 \cdot {\rm lg}\hspace{0.1cm}\frac{s_0^2 }{N_0 \cdot R_{\rm B}} =\ 16.1- 100\hspace{0.15cm}\underline {= -83.9\,{\rm dB} = 10 \cdot {\rm lg}\hspace{0.1cm}\eta_{\hspace{0.05cm} \rm min}}\hspace{0.05cm}.$$

(4)  For the system considered here:   $10 \cdot {\rm lg}\hspace{0.1cm}\eta = -9.3\,{\rm dB} -0.54 \cdot a_{\star}.$

• Thus,  from the system efficiency condition   ⇒   $10 \cdot {\rm lg} \, \eta > \hspace{0.1cm}–83.9 \ \rm dB $,  the characteristic cable attenuation condition is:

$$a_{\star} < \frac{-83.9\,{\rm dB} + 9.3\,{\rm dB}} {-0.54} \approx138.1\,{\rm dB} \hspace{0.05cm}.$$

• With the given equation

$$a_{\star} = \frac{2.36\,{\rm dB} } {{\rm km} \cdot \sqrtTemplate:\rm MHz} \cdot l \cdot \sqrt{{R_{\rm B}}/{2}}\hspace{0.05cm}.$$

thus the maximum cable length  ("regenerator field length")  can be specified:

$$l_{\rm max} = \frac{138.1\,{\rm dB} } {2.36\,{\rm dB}/{\rm km} \cdot \sqrt{\rm MHz})\cdot \sqrt{500\,{\rm MHz}}} \hspace{0.15cm}\underline {\approx 2.62\,{\rm km}} \hspace{0.05cm}.$$

(5)  Following the same procedure,  but in a more compact notation,  results in a smaller  "regenerator field length"  for this  "worse"  system:

$$l_{\rm max} = \frac{-(83.9\,{\rm dB}+A)/B } {2.36\,{\rm dB}/{\rm km} \cdot \sqrt{500}} = \frac{+(83.9+9.4)/1.10 } {2.36\cdot\sqrt{500}}\hspace{0.1cm}{\rm km}\hspace{0.15cm}\underline {\approx 1.61\, {\rm km}}\hspace{0.05cm}.$$