Aufgaben:Exercise 3.5: Eye Opening with Pseudoternary Coding - Revision history

Guenter at 14:45, 28 June 2022

2022-06-28T14:45:48Z

Hwang at 09:26, 27 June 2022

2022-06-27T09:26:08Z

Hwang at 09:24, 27 June 2022

2022-06-27T09:24:49Z

Guenter at 14:38, 21 June 2022

2022-06-21T14:38:28Z

Guenter at 14:12, 21 June 2022

2022-06-21T14:12:26Z

Guenter at 13:05, 21 June 2022

2022-06-21T13:05:02Z

Hwang at 16:14, 25 May 2022

2022-05-25T16:14:35Z

Hwang at 15:59, 16 May 2022

2022-05-16T15:59:06Z

Guenter at 15:07, 7 May 2022

2022-05-07T15:07:38Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.5: Augenöffnung bei Pseudoternärcodierung to Aufgaben:Exercise 3.5: Eye Opening with Pseudoternary Coding

2022-05-07T15:04:24Z

Guenter moved page Aufgabe 3.5: Augenöffnung bei Pseudoternärcodierung to Exercise 3.5: Eye Opening with Pseudoternary Coding

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1421__Dig_A_3_5.png|right|frame|Eye diagrams with AMI and duobinary code '''KORREKTUR''']]
+[[File:EN_Dig_A_3_5.png|right|frame|Eye diagrams with AMI and duobinary code]]
 Three digital transmission systems are considered,&nbsp; each with the following matching properties:
 * NRZ rectangular pulses with amplitude &nbsp;$s_0 = 2 \, {\rm V}$,

@@ Line 83: / Line 83: @@
 *In contrast,&nbsp; for the lower boundary line of the upper eye:
-:$$d_{\rm bottom}= g_1 \hspace{0.2cm}\text{(associated sequences: }\  "\hspace{-0.1cm}0,\ 0, +1\hspace{-0.1cm}"\hspace{0.2cm}\text{ und } "\hspace{-0.1cm}+1,\ 0,\ 0\hspace{-0.1cm}")\hspace{0.05cm}.$$
+:$$d_{\rm bottom}= g_1 \hspace{0.2cm}\text{(associated sequences: }\  "\hspace{-0.1cm}0,\ 0, +1\hspace{-0.1cm}"\hspace{0.2cm}\text{ and } "\hspace{-0.1cm}+1,\ 0,\ 0\hspace{-0.1cm}")\hspace{0.05cm}.$$
 Thus,&nbsp; for the half eye opening,&nbsp; the following holds true:

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1421__Dig_A_3_5.png|right|frame|Eye diagrams with AMI and duobinary code]]
+[[File:P_ID1421__Dig_A_3_5.png|right|frame|Eye diagrams with AMI and duobinary code '''KORREKTUR''']]
 Three digital transmission systems are considered,&nbsp; each with the following matching properties:
 * NRZ rectangular pulses with amplitude &nbsp;$s_0 = 2 \, {\rm V}$,

@@ Line 18: / Line 18: @@
 The system variants to be investigated in the exercise differ only in terms of the line code:
-$\text{System A}$&nbsp; uses a binary bipolar redundancy-free transmission signal. The following descriptive variables are known:
+&rArr; &nbsp; $\text{System A}$&nbsp; uses a binary bipolar redundancy-free transmission signal. The following descriptive variables are known:
 * Basic detection pulse values &nbsp;$g_0 = 1.56 \, {\rm V}$, $g_1 = g_{\rm &ndash;1} = 0.22 \, {\rm V}$, $g_2 = g_{\rm &ndash;2} = \, \text{ ...} \, \approx 0$
 :$$\Rightarrow \hspace{0.3cm}{\ddot{o}(T_{\rm D})}/{ 2}   = g_{0}
@@ Line 28: / Line 28: @@
 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm U} \approx 15\,{\rm dB}\hspace{0.05cm}.$$
-$\text{System B}$&nbsp; uses AMI coding:
+&rArr; &nbsp; $\text{System B}$&nbsp; uses AMI coding:
 *Here the outer symbols &nbsp;$"+1"$&nbsp; or &nbsp;$"&ndash;1"$&nbsp; occur only in isolation.
 *In the case of three consecutive symbols, the sequences &nbsp;"$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, +1, \, +1, \,\text{ ...}$"&nbsp; and &nbsp;"$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, 0, \, +1, \, \text{ ...} $"&nbsp; among others,&nbsp; are not possible,
@@ Line 34: / Line 34: @@
-$\text{System C}$&nbsp; uses the duobinary code:
+&rArr; &nbsp; $\text{System C}$&nbsp; uses the duobinary code:
 *Here the alternating sequence &nbsp;"$\hspace{-0.1cm} \text{ ...}  \, , \, &ndash;1, \, +1, \, &ndash;1, \, \text{ ...}  $"&nbsp; is excluded by the code,&nbsp; which has a favorable effect on the eye opening.

@@ Line 73: / Line 73: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; 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)'''&nbsp; Since the symbol rate is not changed in the AMI code compared to the redundancy-free binary system,&nbsp; the basic detection pulse values remain unchanged:
 :$$g_0 = 1.56 \, {\rm V}, \ g_1 = g_{\rm &ndash;1} = 0.22 \, {\rm V}, \ g_2 = g_{\rm &ndash;2} \approx 0.$$
-In pseudo-ternary coding, there are always two eye openings:
+In pseudo-ternary coding,&nbsp; 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:
-:$$d_{\rm top}= g_0 - 2 \cdot g_1 \hspace{0.2cm}{\rm (associated} \hspace{0.1cm}{\rm
+:$$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}.$$
-*In contrast, for the lower boundary line of the upper eye:
+*In contrast,&nbsp; for the lower boundary line of the upper eye:
-:$$d_{\rm bottom}= g_1 \hspace{0.2cm}{\rm (associated} \hspace{0.1cm}{\rm
+:$$d_{\rm bottom}= g_1 \hspace{0.2cm}\text{(associated sequences: }\  "\hspace{-0.1cm}0,\ 0, +1\hspace{-0.1cm}"\hspace{0.2cm}\text{ und } "\hspace{-0.1cm}+1,\ 0,\ 0\hspace{-0.1cm}")\hspace{0.05cm}.$$
-Thus, for the half eye opening, the following holds true:
+Thus,&nbsp; for the half eye opening,&nbsp; the following holds true:
 :$${\ddot{o}(T_{\rm D})}/{2}= {1}/{2} \cdot (d_{\rm top} - d_{\rm bottom}) = {1}/{2} \cdot g_0 - {3}/{2} \cdot g_1 \hspace{0.15cm}\underline {=
 .45\,{\rm V}}\hspace{0.05cm}.$$
@@ Line 95: / Line 93: @@
-'''(2)'''&nbsp; 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)'''&nbsp; In terms of noise,&nbsp; there is no difference between the three systems since the same symbol rate is always present.&nbsp; It follows for the AMI code:
 :$$\rho_{\rm U} = \frac{(0.45\,{\rm V})^2}{(0.2\,{\rm V})^2} =
 .06  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
@@ Line 101: / Line 99: @@
 lg}\hspace{0.1cm}\rho_{\rm U} \hspace{0.15cm}\underline {\approx 7\,{\rm dB}} \hspace{0.05cm}.$$
-*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&nbsp; $8 \, {\rm dB}$.
-*The reason for this serious loss of signal-to-noise ratio is that with the AMI code, despite $37\%$ redundancy, the symbol sequence &nbsp;$\text{ ...} , \, &ndash;1, \, +1, \, &ndash;1, \text{ ...} $&nbsp; which is particularly unfavorable with respect to intersymbol interference, is not excluded.
 '''(3)'''&nbsp; The threshold $E_2$ must be in the middle between $d_{\rm top}$ and $d_{\rm bottom}$:
 :$$E_2= {1}/{2} \cdot (d_{\rm top} + d_{\rm bottom}) = {1}/{2} \cdot (g_0 -  g_1 ) \hspace{0.15cm}\underline {=
 .67\,{\rm V}}\hspace{0.05cm}.$$
-The threshold value $E_1$ is symmetrical to this: $E_1 \, \underline {= \, &ndash;0.67 {\rm V}}$.
+*The threshold value $E_1$ is symmetrical to this:&nbsp; $E_1 \, \underline {= \, &ndash;0.67 {\rm V}}$.
 *From this follows:
 :$$d_{\rm top}= g_0, \hspace{0.2cm} d_{\rm bottom} = g_1 \hspace{0.3cm}\Rightarrow
@@ Line 120: / Line 122: @@
-'''(5)'''&nbsp; Using the result from '''(4)''', we obtain analogous to subtask '''(2)''':
+'''(5)'''&nbsp; Using the result from&nbsp; '''(4)''',&nbsp; we obtain analogous to subtask&nbsp; '''(2)''':
 :$$\rho_{\rm U} = \frac{(0.67\,{\rm V})^2}{(0.2\,{\rm V})^2} =
 .2  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
@@ Line 131: / Line 133: @@
 .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&nbsp; $f_{\rm G} \cdot T = 0.5$&nbsp; 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.
 {{ML-Fuß}}
 [[Category:Digital Signal Transmission: Exercises|^3.4 Eye Opening with Multilevel Systems^]]

@@ Line 4: / Line 4: @@
 [[File:P_ID1421__Dig_A_3_5.png|right|frame|Eye diagrams with AMI and duobinary code]]
-Three message transmission systems are considered, each with the following matching properties:
+Three digital transmission systems are considered,&nbsp; each with the following matching properties:
 * NRZ rectangular pulses with amplitude &nbsp;$s_0 = 2 \, {\rm V}$,
-* Coaxial cable with characteristic cable attenuation &nbsp;$a_* = 40 \, {\rm dB}$,
 * AWGN noise with noise power density &nbsp;$N_0$,
-* Receiver filter &nbsp;$H_{\rm E}(f) = 1/H_{\rm K}(f) \cdot H_{\rm G}(f) $ consisting of an ideal channel equalizer &nbsp;$H_{\rm K}(f)^{-1}$&nbsp; and a Gaussian low-pass filter &nbsp;$H_{\rm G}(f)$&nbsp; with normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T \approx 0.5$.
-* Threshold decision with optimal decision thresholds and optimal detection time &nbsp;$T_{\rm D} = 0$.
+* receiver filter &nbsp;$H_{\rm E}(f) = 1/H_{\rm K}(f) \cdot H_{\rm G}(f) $ consisting of an ideal channel equalizer &nbsp;$H_{\rm K}(f)^{-1}$&nbsp; and a Gaussian low-pass filter &nbsp;$H_{\rm G}(f)$&nbsp; with normalized cutoff frequency &nbsp;$f_{\rm G} \cdot T \approx 0.5$,
-The system variants to be investigated in the exercise differ only in terms of the transmission code:
+The system variants to be investigated in the exercise differ only in terms of the line code:
 $\text{System A}$&nbsp; uses a binary bipolar redundancy-free transmission signal. The following descriptive variables are known:
-* Basic pulse values &nbsp;$g_0 = 1.56 \, {\rm V}$, $g_1 = g_{\rm &ndash;1} = 0.22 \, {\rm V}$, $g_2 = g_{\rm &ndash;2} = \, \text{ ...} \, \approx 0$
+* Basic detection pulse values &nbsp;$g_0 = 1.56 \, {\rm V}$, $g_1 = g_{\rm &ndash;1} = 0.22 \, {\rm V}$, $g_2 = g_{\rm &ndash;2} = \, \text{ ...} \, \approx 0$
 :$$\Rightarrow \hspace{0.3cm}{\ddot{o}(T_{\rm D})}/{ 2}   = g_{0}
   -g_{1}-g_{-1} = 1.12\,{\rm V}
@@ Line 26: / Line 30: @@
 $\text{System B}$&nbsp; uses AMI coding:
 *Here the outer symbols &nbsp;$"+1"$&nbsp; or &nbsp;$"&ndash;1"$&nbsp; occur only in isolation.
-*In the case of three consecutive symbols, the sequences &nbsp;"$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, +1, \, +1, \,\text{ ...}$"&nbsp; and &nbsp;"$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, 0, \, +1, \, \text{ ...} $"&nbsp; among others, are not possible,
+*In the case of three consecutive symbols, the sequences &nbsp;"$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, +1, \, +1, \,\text{ ...}$"&nbsp; and &nbsp;"$\hspace{-0.1cm}\text{ ...} \, , \, +1, \, 0, \, +1, \, \text{ ...} $"&nbsp; among others,&nbsp; are not possible,
 * in contrast to the sequence &nbsp;"$\hspace{-0.1cm}\text{ ...}  \, , \, +1, \, &ndash;1, \, +1, \, \text{ ...} $".
 $\text{System C}$&nbsp; uses the duobinary code:
-*Here the alternating sequence &nbsp;"$\hspace{-0.1cm} \text{ ...}  \, , \, &ndash;1, \, +1, \, &ndash;1, \, \text{ ...}  $"&nbsp; is excluded by the code, which has a favorable effect on the eye opening.
+*Here the alternating sequence &nbsp;"$\hspace{-0.1cm} \text{ ...}  \, , \, &ndash;1, \, +1, \, &ndash;1, \, \text{ ...}  $"&nbsp; is excluded by the code,&nbsp; which has a favorable effect on the eye opening.
-''Notes:''
+Notes:
 *The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|"Intersymbol Interference for Multi-Level Transmission"]].
@@ Line 47: / Line 49: @@
 ===Questions===
 <quiz display=simple>
-{Calculate the half eye opening for the '''AMI code'''.
+{Calculate the half eye opening for the&nbsp; '''AMI code'''.
 |type="{}"}
-$\text{System  B:}\hspace{0.4cm} \ddot{o}(T_{\rm D})/2$ = { 0.45 3% } $\ {\rm V}$
+$\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.
@@ Line 60: / Line 62: @@
 $E_2 \ = \ $ { 0.667 3% } $\ {\rm V}$
-{Calculate the half eye opening at the '''duobinary code'''.
+{Calculate the half eye opening at the&nbsp; '''duobinary code'''.
 |type="{}"}
 $\text{System  C:}\hspace{0.4cm}  \ddot{o}(T_{\rm D})/2 \ = \ $ { 0.67 3% } $\ {\rm V}$

@@ Line 39: / Line 39: @@
 ''Notes:''
-*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|Intersymbol Interference for Multi-Level Transmission]].
+*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|"Intersymbol Interference for Multi-Level Transmission"]].
 * Not all of the numerical values given here are necessary to solve this exercise.

← Older revision		Revision as of 15:07, 7 May 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^3.4 ~~Auge bei mehrstufigen Systemen~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^3.4 Eye Opening with Multilevel Systems^]]