Difference between revisions of "Aufgaben:Exercise 3.5: Eye Opening with Pseudoternary Coding"
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| − | [[File: | + | [[File:EN_Dig_A_3_5.png|right|frame|Eye diagrams with AMI and duobinary code]] |
| − | Three | + | Three digital transmission systems are considered, each with the following matching properties: |
* NRZ rectangular pulses with amplitude $s_0 = 2 \, {\rm V}$, | * NRZ rectangular pulses with amplitude $s_0 = 2 \, {\rm V}$, | ||
| − | * | + | |
| + | * coaxial cable with characteristic cable attenuation $a_* = 40 \, {\rm dB}$, | ||
| + | |||
* AWGN noise with noise power density $N_0$, | * AWGN noise with noise power density $N_0$, | ||
| − | * | + | |
| − | * | + | * receiver filter $H_{\rm E}(f) = 1/H_{\rm K}(f) \cdot H_{\rm G}(f) $ consisting of an ideal channel equalizer $H_{\rm K}(f)^{-1}$ and a Gaussian low-pass filter $H_{\rm G}(f)$ with normalized cutoff frequency $f_{\rm G} \cdot T \approx 0.5$, |
| + | |||
| + | * threshold decision with optimal decision thresholds and optimal detection time $T_{\rm D} = 0$. | ||
| − | The system variants to be investigated in the exercise differ only in terms of the | + | The system variants to be investigated in the exercise differ only in terms of the line code: |
| − | $\text{System A}$ uses a binary bipolar redundancy-free transmission signal. The following descriptive variables are known: | + | ⇒ $\text{System A}$ uses a binary bipolar redundancy-free transmission signal. The following descriptive variables are known: |
| − | * Basic pulse values $g_0 = 1.56 \, {\rm V}$, $g_1 = g_{\rm –1} = 0.22 \, {\rm V}$, $g_2 = g_{\rm –2} = \, \text{ ...} \, \approx 0$ | + | * Basic detection pulse values $g_0 = 1.56 \, {\rm V}$, $g_1 = g_{\rm –1} = 0.22 \, {\rm V}$, $g_2 = g_{\rm –2} = \, \text{ ...} \, \approx 0$ |
:$$\Rightarrow \hspace{0.3cm}{\ddot{o}(T_{\rm D})}/{ 2} = g_{0} | :$$\Rightarrow \hspace{0.3cm}{\ddot{o}(T_{\rm D})}/{ 2} = g_{0} | ||
-g_{1}-g_{-1} = 1.12\,{\rm V} | -g_{1}-g_{-1} = 1.12\,{\rm V} | ||
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10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm U} \approx 15\,{\rm dB}\hspace{0.05cm}.$$ | 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm U} \approx 15\,{\rm dB}\hspace{0.05cm}.$$ | ||
| − | $\text{System B}$ uses AMI coding: | + | ⇒ $\text{System B}$ uses AMI coding: |
*Here the outer symbols $"+1"$ or $"–1"$ occur only in isolation. | *Here the outer symbols $"+1"$ or $"–1"$ occur only in isolation. | ||
| − | *In the case of three consecutive symbols, the sequences "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, +1, \, +1, \,\text{ ...}$" and "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, 0, \, +1, \, \text{ ...} $" among others, are not possible, | + | *In the case of three consecutive symbols, the sequences "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, +1, \, +1, \,\text{ ...}$" and "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, 0, \, +1, \, \text{ ...} $" among others, are not possible, |
* in contrast to the sequence "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, –1, \, +1, \, \text{ ...} $". | * in contrast to the sequence "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, –1, \, +1, \, \text{ ...} $". | ||
| − | $\text{System C}$ uses the duobinary code: | + | ⇒ $\text{System C}$ uses the duobinary code: |
| − | *Here the alternating sequence "$\hspace{-0.1cm} \text{ ...} \, , \, –1, \, +1, \, –1, \, \text{ ...} $" is excluded by the code, which has a favorable effect on the eye opening. | + | *Here the alternating sequence "$\hspace{-0.1cm} \text{ ...} \, , \, –1, \, +1, \, –1, \, \text{ ...} $" is excluded by the code, which has a favorable effect on the eye opening. |
| − | + | Notes: | |
| − | |||
| − | |||
*The exercise belongs to the chapter [[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|"Intersymbol Interference for Multi-Level Transmission"]]. | *The exercise belongs to the chapter [[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|"Intersymbol Interference for Multi-Level Transmission"]]. | ||
| Line 47: | Line 49: | ||
===Questions=== | ===Questions=== | ||
<quiz display=simple> | <quiz display=simple> | ||
| − | {Calculate the half eye opening for the '''AMI code'''. | + | {Calculate the half eye opening for the '''AMI code'''. |
|type="{}"} | |type="{}"} | ||
| − | $\text{System B:}\hspace{0.4cm} \ddot{o}(T_{\rm D})/2$ | + | $\text{System B:}\hspace{0.4cm} \ddot{o}(T_{\rm D})/2 \ = \ $ { 0.45 3% } $\ {\rm V}$ |
{Calculate the worst-case signal-to-noise ratio for this system. | {Calculate the worst-case signal-to-noise ratio for this system. | ||
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$E_2 \ = \ $ { 0.667 3% } $\ {\rm V}$ | $E_2 \ = \ $ { 0.667 3% } $\ {\rm V}$ | ||
| − | {Calculate the half eye opening at the '''duobinary code'''. | + | {Calculate the half eye opening at the '''duobinary code'''. |
|type="{}"} | |type="{}"} | ||
$\text{System C:}\hspace{0.4cm} \ddot{o}(T_{\rm D})/2 \ = \ $ { 0.67 3% } $\ {\rm V}$ | $\text{System C:}\hspace{0.4cm} \ddot{o}(T_{\rm D})/2 \ = \ $ { 0.67 3% } $\ {\rm V}$ | ||
| Line 71: | Line 73: | ||
===Solution=== | ===Solution=== | ||
{{ML-Kopf}} | {{ML-Kopf}} | ||
| − | '''(1)''' Since the symbol rate is not changed in the AMI code compared to the redundancy-free binary system, the basic pulse values remain unchanged: | + | '''(1)''' Since the symbol rate is not changed in the AMI code compared to the redundancy-free binary system, the basic detection pulse values remain unchanged: |
:$$g_0 = 1.56 \, {\rm V}, \ g_1 = g_{\rm –1} = 0.22 \, {\rm V}, \ g_2 = g_{\rm –2} \approx 0.$$ | :$$g_0 = 1.56 \, {\rm V}, \ g_1 = g_{\rm –1} = 0.22 \, {\rm V}, \ g_2 = g_{\rm –2} \approx 0.$$ | ||
| − | In pseudo ternary coding, there are always two eye openings: | + | In pseudo-ternary coding, there are always two eye openings: |
*The upper boundary line of the upper eye results in the AMI code as in the redundancy-free binary system: | *The upper boundary line of the upper eye results in the AMI code as in the redundancy-free binary system: | ||
| − | :$$d_{\rm top}= g_0 - 2 \cdot g_1 \hspace{0.2cm}{ | + | :$$d_{\rm top}= g_0 - 2 \cdot g_1 \hspace{0.2cm}\text{(associated sequence: } "\hspace{-0.1cm}-1, +1, -1\hspace{-0.1cm}") |
| − | |||
\hspace{0.05cm}.$$ | \hspace{0.05cm}.$$ | ||
| − | *In contrast, for the lower boundary line of the upper eye: | + | *In contrast, for the lower boundary line of the upper eye: |
| − | :$$d_{\rm bottom}= g_1 \hspace{0.2cm}{ | + | :$$d_{\rm bottom}= g_1 \hspace{0.2cm}\text{(associated sequences: }\ "\hspace{-0.1cm}0,\ 0, +1\hspace{-0.1cm}"\hspace{0.2cm}\text{ and } "\hspace{-0.1cm}+1,\ 0,\ 0\hspace{-0.1cm}")\hspace{0.05cm}.$$ |
| − | |||
| − | Thus, for the half eye opening, the following holds true: | + | Thus, for the half eye opening, the following holds true: |
:$${\ddot{o}(T_{\rm D})}/{2}= {1}/{2} \cdot (d_{\rm top} - d_{\rm bottom}) = {1}/{2} \cdot g_0 - {3}/{2} \cdot g_1 \hspace{0.15cm}\underline {= | :$${\ddot{o}(T_{\rm D})}/{2}= {1}/{2} \cdot (d_{\rm top} - d_{\rm bottom}) = {1}/{2} \cdot g_0 - {3}/{2} \cdot g_1 \hspace{0.15cm}\underline {= | ||
0.45\,{\rm V}}\hspace{0.05cm}.$$ | 0.45\,{\rm V}}\hspace{0.05cm}.$$ | ||
| Line 93: | Line 93: | ||
| − | '''(2)''' In terms of noise, there is no difference between the three systems since the same symbol rate is always present. It follows for the AMI code: | + | '''(2)''' In terms of noise, there is no difference between the three systems since the same symbol rate is always present. It follows for the AMI code: |
:$$\rho_{\rm U} = \frac{(0.45\,{\rm V})^2}{(0.2\,{\rm V})^2} = | :$$\rho_{\rm U} = \frac{(0.45\,{\rm V})^2}{(0.2\,{\rm V})^2} = | ||
5.06 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} | 5.06 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} | ||
| Line 99: | Line 99: | ||
lg}\hspace{0.1cm}\rho_{\rm U} \hspace{0.15cm}\underline {\approx 7\,{\rm dB}} \hspace{0.05cm}.$$ | lg}\hspace{0.1cm}\rho_{\rm U} \hspace{0.15cm}\underline {\approx 7\,{\rm dB}} \hspace{0.05cm}.$$ | ||
| − | *The loss compared to the redundancy-free binary system is thus almost $8 \, {\rm dB}$. | + | *The loss compared to the redundancy-free binary system is thus almost $8 \, {\rm dB}$. |
| − | *The reason for this serious loss of signal-to-noise ratio is that with the AMI code, despite $37\%$ redundancy, the symbol sequence $\text{ ...} , \, –1, \, +1, \, –1, \text{ ...} $ which is particularly unfavorable with respect to intersymbol interference | + | |
| + | *The reason for this serious loss of signal-to-noise ratio is that with the AMI code, despite $37\%$ redundancy, the symbol sequence "$\text{ ...} , \, –1, \, +1, \, –1, \text{ ...} $" is not excluded, which is particularly unfavorable with respect to intersymbol interference. | ||
| − | '''(3)''' The threshold $E_2$ must be in the middle between $d_{\rm top}$ and $d_{\rm bottom}$: | + | |
| + | '''(3)''' The threshold $E_2$ must be in the middle between $d_{\rm top}$ and $d_{\rm bottom}$: | ||
:$$E_2= {1}/{2} \cdot (d_{\rm top} + d_{\rm bottom}) = {1}/{2} \cdot (g_0 - g_1 ) \hspace{0.15cm}\underline {= | :$$E_2= {1}/{2} \cdot (d_{\rm top} + d_{\rm bottom}) = {1}/{2} \cdot (g_0 - g_1 ) \hspace{0.15cm}\underline {= | ||
0.67\,{\rm V}}\hspace{0.05cm}.$$ | 0.67\,{\rm V}}\hspace{0.05cm}.$$ | ||
| − | The threshold value $E_1$ is symmetrical to this: $E_1 \, \underline {= \, –0.67 {\rm V}}$. | + | *The threshold value $E_1$ is symmetrical to this: $E_1 \, \underline {= \, –0.67 {\rm V}}$. |
| − | '''(4)''' We again assume the same basic pulse values. | + | |
| − | *The worst-case sequence with respect to the upper boundary line of the upper eye is $\text{ ...} , 0, \, +1, \, 0, \text{ ...} $, | + | '''(4)''' We again assume the same basic detection pulse values. |
| − | *while the lower boundary line is defined by $\text{ ...} , 0, \, 0, \, +1, \text{ ...} $ or $\text{ ...} , +1, \, 0, \, 0, \text{ ...} $ respectively. | + | *The worst-case sequence with respect to the upper boundary line of the upper eye is "$\text{ ...} , 0, \, +1, \, 0, \text{ ...} $", |
| + | *while the lower boundary line is defined by "$\text{ ...} , 0, \, 0, \, +1, \text{ ...} $" or "$\text{ ...} , +1, \, 0, \, 0, \text{ ...} $" respectively. | ||
| + | |||
*From this follows: | *From this follows: | ||
:$$d_{\rm top}= g_0, \hspace{0.2cm} d_{\rm bottom} = g_1 \hspace{0.3cm}\Rightarrow | :$$d_{\rm top}= g_0, \hspace{0.2cm} d_{\rm bottom} = g_1 \hspace{0.3cm}\Rightarrow | ||
| Line 118: | Line 122: | ||
| − | '''(5)''' Using the result from '''(4)''', we obtain analogous to subtask '''(2)''': | + | '''(5)''' Using the result from '''(4)''', we obtain analogous to subtask '''(2)''': |
:$$\rho_{\rm U} = \frac{(0.67\,{\rm V})^2}{(0.2\,{\rm V})^2} = | :$$\rho_{\rm U} = \frac{(0.67\,{\rm V})^2}{(0.2\,{\rm V})^2} = | ||
11.2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} | 11.2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} | ||
| Line 129: | Line 133: | ||
0.89\,{\rm V}, \hspace{0.2cm}E_1 = - 0.89\,{\rm V}\hspace{0.05cm}.$$ | 0.89\,{\rm V}, \hspace{0.2cm}E_1 = - 0.89\,{\rm V}\hspace{0.05cm}.$$ | ||
| − | *It should be noted that the same cutoff frequency $f_{\rm G} \cdot T = 0.5$ was always assumed here. | + | *It should be noted that the same cutoff frequency $f_{\rm G} \cdot T = 0.5$ was always assumed here. |
| − | *If the cutoff frequency is optimized, it may well be that the duobinary code is superior to the redundancy-free binary code if the characteristic cable attenuation is sufficiently large. | + | |
| + | *If the cutoff frequency is optimized, it may well be that the duobinary code is superior to the redundancy-free binary code if the characteristic cable attenuation is sufficiently large. | ||
{{ML-Fuß}} | {{ML-Fuß}} | ||
[[Category:Digital Signal Transmission: Exercises|^3.4 Eye Opening with Multilevel Systems^]] | [[Category:Digital Signal Transmission: Exercises|^3.4 Eye Opening with Multilevel Systems^]] | ||
Latest revision as of 15:45, 28 June 2022
Eye diagrams with AMI and duobinary code
Three digital transmission systems are considered, each with the following matching properties:
- NRZ rectangular pulses with amplitude $s_0 = 2 \, {\rm V}$,
- coaxial cable with characteristic cable attenuation $a_* = 40 \, {\rm dB}$,
- AWGN noise with noise power density $N_0$,
- receiver filter $H_{\rm E}(f) = 1/H_{\rm K}(f) \cdot H_{\rm G}(f) $ consisting of an ideal channel equalizer $H_{\rm K}(f)^{-1}$ and a Gaussian low-pass filter $H_{\rm G}(f)$ with normalized cutoff frequency $f_{\rm G} \cdot T \approx 0.5$,
- threshold decision with optimal decision thresholds and optimal detection time $T_{\rm D} = 0$.
The system variants to be investigated in the exercise differ only in terms of the line code:
⇒ $\text{System A}$ uses a binary bipolar redundancy-free transmission signal. The following descriptive variables are known:
- Basic detection pulse values $g_0 = 1.56 \, {\rm V}$, $g_1 = g_{\rm –1} = 0.22 \, {\rm V}$, $g_2 = g_{\rm –2} = \, \text{ ...} \, \approx 0$
- $$\Rightarrow \hspace{0.3cm}{\ddot{o}(T_{\rm D})}/{ 2} = g_{0} -g_{1}-g_{-1} = 1.12\,{\rm V} \hspace{0.05cm}.$$
- Noise rms value $\sigma_d \approx 0.2 \, {\rm V}$
- $$\Rightarrow \hspace{0.3cm}\rho_{\rm U} = \frac{\big[\ddot{o}(T_{\rm D})/2\big]^2}{ \sigma_d^2}\approx 31.36\,{\rm dB} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm U} \approx 15\,{\rm dB}\hspace{0.05cm}.$$
⇒ $\text{System B}$ uses AMI coding:
- Here the outer symbols $"+1"$ or $"–1"$ occur only in isolation.
- In the case of three consecutive symbols, the sequences "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, +1, \, +1, \,\text{ ...}$" and "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, 0, \, +1, \, \text{ ...} $" among others, are not possible,
- in contrast to the sequence "$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, –1, \, +1, \, \text{ ...} $".
⇒ $\text{System C}$ uses the duobinary code:
- Here the alternating sequence "$\hspace{-0.1cm} \text{ ...} \, , \, –1, \, +1, \, –1, \, \text{ ...} $" is excluded by the code, which has a favorable effect on the eye opening.
Notes:
- The exercise belongs to the chapter "Intersymbol Interference for Multi-Level Transmission".
- Not all of the numerical values given here are necessary to solve this exercise.
Questions
Solution
(1) Since the symbol rate is not changed in the AMI code compared to the redundancy-free binary system, the basic detection pulse values remain unchanged:
- $$g_0 = 1.56 \, {\rm V}, \ g_1 = g_{\rm –1} = 0.22 \, {\rm V}, \ g_2 = g_{\rm –2} \approx 0.$$
In pseudo-ternary coding, there are always two eye openings:
- The upper boundary line of the upper eye results in the AMI code as in the redundancy-free binary system:
- $$d_{\rm top}= g_0 - 2 \cdot g_1 \hspace{0.2cm}\text{(associated sequence: } "\hspace{-0.1cm}-1, +1, -1\hspace{-0.1cm}") \hspace{0.05cm}.$$
- In contrast, for the lower boundary line of the upper eye:
- $$d_{\rm bottom}= g_1 \hspace{0.2cm}\text{(associated sequences: }\ "\hspace{-0.1cm}0,\ 0, +1\hspace{-0.1cm}"\hspace{0.2cm}\text{ and } "\hspace{-0.1cm}+1,\ 0,\ 0\hspace{-0.1cm}")\hspace{0.05cm}.$$
Thus, for the half eye opening, the following holds true:
- $${\ddot{o}(T_{\rm D})}/{2}= {1}/{2} \cdot (d_{\rm top} - d_{\rm bottom}) = {1}/{2} \cdot g_0 - {3}/{2} \cdot g_1 \hspace{0.15cm}\underline {= 0.45\,{\rm V}}\hspace{0.05cm}.$$
The corresponding equation for the redundancy-free binary system is:
- $${\ddot{o}(T_{\rm D})}/{2}= g_0 - 2 \cdot g_1 \hspace{0.05cm}.$$
(2) In terms of noise, there is no difference between the three systems since the same symbol rate is always present. It follows for the AMI code:
- $$\rho_{\rm U} = \frac{(0.45\,{\rm V})^2}{(0.2\,{\rm V})^2} = 5.06 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm U} \hspace{0.15cm}\underline {\approx 7\,{\rm dB}} \hspace{0.05cm}.$$
- The loss compared to the redundancy-free binary system is thus almost $8 \, {\rm dB}$.
- The reason for this serious loss of signal-to-noise ratio is that with the AMI code, despite $37\%$ redundancy, the symbol sequence "$\text{ ...} , \, –1, \, +1, \, –1, \text{ ...} $" is not excluded, which is particularly unfavorable with respect to intersymbol interference.
(3) The threshold $E_2$ must be in the middle between $d_{\rm top}$ and $d_{\rm bottom}$:
- $$E_2= {1}/{2} \cdot (d_{\rm top} + d_{\rm bottom}) = {1}/{2} \cdot (g_0 - g_1 ) \hspace{0.15cm}\underline {= 0.67\,{\rm V}}\hspace{0.05cm}.$$
- The threshold value $E_1$ is symmetrical to this: $E_1 \, \underline {= \, –0.67 {\rm V}}$.
(4) We again assume the same basic detection pulse values.
- The worst-case sequence with respect to the upper boundary line of the upper eye is "$\text{ ...} , 0, \, +1, \, 0, \text{ ...} $",
- while the lower boundary line is defined by "$\text{ ...} , 0, \, 0, \, +1, \text{ ...} $" or "$\text{ ...} , +1, \, 0, \, 0, \text{ ...} $" respectively.
- From this follows:
- $$d_{\rm top}= g_0, \hspace{0.2cm} d_{\rm bottom} = g_1 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\ddot{o}(T_{\rm D})}/{2} = {g_0}/{2} - {g_1}/{2}\hspace{0.15cm}\underline { = 0.667\,{\rm V}} \hspace{0.05cm}.$$
(5) Using the result from (4), we obtain analogous to subtask (2):
- $$\rho_{\rm U} = \frac{(0.67\,{\rm V})^2}{(0.2\,{\rm V})^2} = 11.2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm U} \hspace{0.15cm}\underline {\approx 10.5\,{\rm dB}} \hspace{0.05cm}.$$
- Prerequisite for this result are thresholds at
- $$E_2= {1}/{2} \cdot (g_0 + g_1 ) = 0.89\,{\rm V}, \hspace{0.2cm}E_1 = - 0.89\,{\rm V}\hspace{0.05cm}.$$
- It should be noted that the same cutoff frequency $f_{\rm G} \cdot T = 0.5$ was always assumed here.
- If the cutoff frequency is optimized, it may well be that the duobinary code is superior to the redundancy-free binary code if the characteristic cable attenuation is sufficiently large.
