Aufgaben:Exercise 2.4: Dual Code and Gray Code - Revision history

Hwang at 15:46, 16 May 2022

2022-05-16T15:46:41Z

Guenter at 14:28, 16 May 2022

2022-05-16T14:28:11Z

Guenter at 13:17, 16 May 2022

2022-05-16T13:17:00Z

Guenter: Guenter moved page Aufgaben:Exercise 2.4: Dualcode and Graycode to Aufgaben:Exercise 2.4: Dual Code and Gray Code

2022-05-15T11:54:26Z

Guenter moved page Exercise 2.4: Dualcode and Graycode to Exercise 2.4: Dual Code and Gray Code

Hwang at 12:31, 19 April 2022

2022-04-19T12:31:40Z

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Guenter at 12:56, 4 April 2022

2022-04-04T12:56:41Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.4: Dualcodierung und Graycodierung to Aufgaben:Exercise 2.4: Dualcode and Graycode

2022-04-04T12:52:40Z

Guenter moved page Aufgabe 2.4: Dualcodierung und Graycodierung to Exercise 2.4: Dualcode and Graycode

Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

2021-03-23T12:53:12Z

Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Javier: Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

2020-07-09T12:55:28Z

Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

Tasnad: Text replacement - "Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)" to "Komplementäre_Gaußsche_Fehlerfunktionen"

2020-06-10T15:55:34Z

Text replacement - "Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)" to "Komplementäre_Gaußsche_Fehlerfunktionen"

← Older revision		Revision as of 15:46, 16 May 2022
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	*Reference is also made to the chapter  [[Digital_Signal_Transmission/Redundanzfreie_Codierung\|"Redundancy-Free Coding"]].		*Reference is also made to the chapter  [[Digital_Signal_Transmission/Redundanzfreie_Codierung\|"Redundancy-Free Coding"]].

−	*For numerical evaluation of the Q–function you can use the HTML5/JavaScript applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|Complementary Gaussian Error Functions]].	+	*For numerical evaluation of the Q–function you can use the HTML5/JavaScript applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]].

@@ Line 71: / Line 71: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; In the signal $s_{2}(t)$ one recognizes the realization of the dual code indicated at the beginning. On the other hand, in the signal $s_{2}(t)$ a gray code  $\Rightarrow$  <u>solution 1</u> with the following mapping was used:
+'''(1)'''&nbsp; In the signal&nbsp; $s_{2}(t)$&nbsp; one recognizes the realization of the dual code indicated at the beginning.&nbsp; On the other hand,&nbsp; in the signal&nbsp; $s_{2}(t)$&nbsp; a Gray code &nbsp;  $\Rightarrow$ &nbsp; <u>solution 1</u> with the following mapping was used:
 :$$\mathbf{HH}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} -1, \hspace{0.35cm} \mathbf{HL}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} -1/3, \hspace{0.35cm} \mathbf{LL}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} +1/3, \hspace{0.35cm} \mathbf{LH}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} +1 \hspace{0.05cm}.$$
-'''(2)'''&nbsp; Let the probability $p$ that the amplitude value $3 \, \rm V$ falls below the adjacent decision threshold $2\,  \rm V$ due to the Gaussian distributed noise with standard deviation $\sigma_{d}$ be $1\,  \%$. It follows that:
+'''(2)'''&nbsp; Let the probability&nbsp; $p$&nbsp; that the amplitude value&nbsp; $3 \, \rm V$&nbsp; falls below the adjacent decision threshold&nbsp; $2\,  \rm V$&nbsp; due to the Gaussian distributed noise with standard deviation&nbsp; $\sigma_{d}$&nbsp; be $1\,  \%$.&nbsp; It follows that:
 :$$ p = {\rm Q} \left ( \frac{3\,{\rm V} - 2\,{\rm V}} { \sigma_d}\right ) = 1 \%\hspace{0.3cm}\Rightarrow \hspace{0.3cm} {1\,{\rm V} }/ { \sigma_d} \approx 2.33 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} { \sigma_d}\hspace{0.15cm}\underline {\approx 0.43\,{\rm V}}\hspace{0.05cm}.$$
-'''(3)'''&nbsp;  The two outer symbols are each distorted with probability $p$, the two inner symbols with double probability $(2p)$. By averaging considering equal symbol occurrence probabilities, we obtain
+'''(3)'''&nbsp;  The two outer symbols are each falsified with probability&nbsp; $p$,&nbsp; the two inner symbols with double probability&nbsp; $(2p)$.&nbsp; By averaging considering equal symbol occurrence probabilities,&nbsp; we obtain
 :$$p_{\rm S} = 1.5 \cdot p \hspace{0.15cm}\underline { = 1.5 \,\%} \hspace{0.05cm}.$$
-'''(4)'''&nbsp; Each symbol error results in exactly one bit error. However, since each quaternary symbol contains exactly two binary symbols, the bit error probability is obtained:
+'''(4)'''&nbsp; Each symbol error results in exactly one bit error.&nbsp; However,&nbsp; since each quaternary symbol contains exactly two binary symbols,&nbsp; the bit error probability is obtained:
 :$$p_{\rm B} = {p_{\rm S}}/ { 2}\hspace{0.15cm}\underline { = 0.75 \,\%} \hspace{0.05cm}.$$
-'''(5)'''&nbsp; When calculating the symbol error probability $p_{\rm S}$, the mapping used is not taken into account. As in subtask '''(3)''', we obtain  $p_{\rm S} \hspace{0.15cm}\underline{ = 1.5 \, \%}$.
+'''(5)'''&nbsp; When calculating the symbol error probability&nbsp; $p_{\rm S}$,&nbsp; the mapping used is not taken into account.&nbsp; As in subtask&nbsp; '''(3)''',&nbsp; we obtain&nbsp;  $p_{\rm S} \hspace{0.15cm}\underline{ = 1.5 \, \%}$.
-'''(6)'''&nbsp; The two outer symbols are distorted with $p$ and lead to only one bit error each even with dual code.
+'''(6)'''&nbsp; The two outer symbols are falsified with&nbsp; $p$&nbsp; and lead to only one bit error each even with dual code.
-* The inner symbols are distorted with $2p$ and now lead to $1.5$ bit errors on average.
+* The inner symbols are falsified with&nbsp; $2p$&nbsp; and now lead to&nbsp; $1.5$&nbsp; bit errors on average.
-*Taking into account the factor $2$ in the denominator – see subtask '''(2)''' – we thus obtain for the bit error probability of the dual code:
+*Taking into account the factor&nbsp; $2$&nbsp; in the denominator – see subtask&nbsp; '''(2)'''&nbsp; – we thus obtain for the bit error probability of the dual code:
 :$$p_{\rm B} = \frac{1} { 4} \cdot \frac{p + 2p \cdot 1.5 + 2p \cdot 1.5 + p} { 2} = p \hspace{0.15cm}\underline { = 1 \,\%} \hspace{0.05cm}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1325__Dig_A_2_4.png|right|frame|Quaternary signals with dual and gray coding]]
+[[File:P_ID1325__Dig_A_2_4.png|right|frame|Quaternary signals with dual and Gray coding]]
-The two shown signals &nbsp;$s_{1}(t)$&nbsp; and &nbsp;$s_{2}(t)$&nbsp; are two different realizations of a redundancy-free quaternary transmit signal, both derived from the blue drawn source signal &nbsp;$q(t)$.&nbsp;
+The two shown signals &nbsp;$s_{1}(t)$&nbsp; and &nbsp;$s_{2}(t)$&nbsp; are two different realizations of a redundancy-free quaternary transmitted signal,&nbsp; both derived from the blue drawn binary source signal &nbsp;$q(t)$.&nbsp;
-For one of the transmitted signals, the so-called '''dual code''' with mapping
+For one of the transmitted signals,&nbsp; the so-called&nbsp; '''dual code'''&nbsp; with mapping
 :$$\mathbf{LL}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} -s_0, \hspace{0.35cm} \mathbf{LH}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} -s_0/3,\hspace{0.35cm}
 \mathbf{HL}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} +s_0/3, \hspace{0.35cm} \mathbf{HH}\hspace{0.1cm}\Leftrightarrow \hspace{0.1cm} +s_0$$
-was used, for the other one a certain form of a '''gray code'''. This is characterized by the fact that the binary representation of adjacent amplitude values always differ only in a single bit.
+was used,&nbsp; for the other one a certain form of a&nbsp; '''Gray code'''.&nbsp; This is characterized by the fact that the binary representation of adjacent amplitude values always differ only in a single bit.
 The solution of the exercise should be based on the following assumptions:
-*The amplitude levels are &nbsp;$±3\, \rm V$ and $±1 \, \rm V$.
+*The amplitude levels are &nbsp;$±3\, \rm V$&nbsp; and&nbsp; $±1 \, \rm V$.
 *The decision thresholds lie in the middle between two adjacent amplitude values, i.e. at &nbsp;$–2\, \rm V$, $0\, \rm V$&nbsp; and &nbsp;$+2\, \rm V$.
-*The noise rms value &nbsp;$\sigma_{d}$&nbsp; is to be chosen so that the distortion probability from the outer symbol &nbsp;$(+s_0)$&nbsp; to the nearest symbol &nbsp;$(+s_{0}/3)$&nbsp; is exactly &nbsp;$p = 1\%$.
+*The decision thresholds lie in the middle between two adjacent amplitude values,&nbsp; i.e. at &nbsp;$–2\, \rm V$,&nbsp; $0\, \rm V$&nbsp; and &nbsp;$+2\, \rm V$.
-*Distortions to non-adjacent symbols can be excluded; in the case of Gaussian perturbations, this simplification is always allowed in practice.
 One distinguishes in principle between
-*the &nbsp;''symbol error probability''&nbsp; $p_{\rm S}$&nbsp; (related to the quaternary signal) and
+*the &nbsp;"symbol error probability"&nbsp; $p_{\rm S}$&nbsp; (related to the quaternary signal)&nbsp; and
-*the &nbsp;''bit error probability''&nbsp; $p_{B}$&nbsp; (related to the binary source signal).
+*the &nbsp;"bit error probability"&nbsp; $p_{B}$&nbsp; (related to the binary source signal).
@@ Line 28: / Line 30: @@
-''Notes:''
+*Reference is also made to the chapter&nbsp; [[Digital_Signal_Transmission/Redundanzfreie_Codierung|"Redundancy-Free Coding"]].
-*For numerical evaluation of the Q–function you can use the interactive applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Complementary Gaussian Error Functions]]&nbsp; benutzen.
+*For numerical evaluation of the Q–function you can use the HTML5/JavaScript applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Complementary Gaussian Error Functions]].
@@ Line 40: / Line 42: @@
 <quiz display=simple>
-{Which of the signals &nbsp;$s_{1}(t)$&nbsp; or &nbsp;$s_{2}(t)$&nbsp; uses a '''gray coding'''?
+{Which of the signals &nbsp;$s_{1}(t)$&nbsp; or &nbsp;$s_{2}(t)$&nbsp; uses&nbsp; '''Gray coding'''?
-|type="()"}
+|type="[]"}
-+$s_{1}(t)$&nbsp; uses a gray coding.
++$s_{1}(t)$&nbsp; uses Gray coding.
--$s_{2}(t)$&nbsp; uses a gray coding.
+-$s_{2}(t)$&nbsp; uses Gray coding.
 {Determine the noise rms value from the given condition.
@@ Line 49: / Line 51: @@
 $\sigma_{d} \ = \ $ { 0.43 3% } $\ \rm V$
-{What is the symbol error probability using the '''gray code'''?
+{What is the symbol error probability using the&nbsp; '''Gray code'''?
 |type="{}"}
 $p_{\rm S} \ = \ $ { 1.5 3% } $\ \%$
-{What is the bit error probability with the gray code?
+{What is the bit error probability with the Gray code?
 |type="{}"}
 $p_{\rm B} \ = \ $ { 0.75 3% } $\ \%$
-{What is the symbol error probability with the '''dual code'''?
+{What is the symbol error probability with the&nbsp; '''dual code'''?
 |type="{}"}
 $p_{\rm S} \ = \ $ { 1.5 3% } $\ \%$

← Older revision		Revision as of 12:56, 4 April 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^2.2 ~~Redundanzfreie Codierung~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^2.2 Redundancy-Free Coding^]]

← Older revision		Revision as of 12:53, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^2.2 Redundanzfreie Codierung^]]	+	[[Category:Digital Signal Transmission: Exercises\|^2.2 Redundanzfreie Codierung^]]

← Older revision	Revision as of 11:54, 15 May 2022
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← Older revision	Revision as of 12:52, 4 April 2022
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 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp;  [[Digitalsignalübertragung/Grundlagen_der_codierten_Übertragung|Grundlagen der codierten Übertragung]].
+*Die Aufgabe gehört zum  Kapitel&nbsp;  [[Digital_Signal_Transmission/Grundlagen_der_codierten_Übertragung|Grundlagen der codierten Übertragung]].
-*Bezug genommen wird auch auf das Kapitel&nbsp; [[Digitalsignalübertragung/Redundanzfreie_Codierung|Redundanzfreie Codierung]].
+*Bezug genommen wird auch auf das Kapitel&nbsp; [[Digital_Signal_Transmission/Redundanzfreie_Codierung|Redundanzfreie Codierung]].
 *Zur numerischen Auswertung der Q–Funktion können Sie das interaktive Applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen]]&nbsp; benutzen.

← Older revision		Revision as of 15:55, 10 June 2020
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	*Bezug genommen wird auch auf das Kapitel  [[Digitalsignalübertragung/Redundanzfreie_Codierung\|Redundanzfreie Codierung]].		*Bezug genommen wird auch auf das Kapitel  [[Digitalsignalübertragung/Redundanzfreie_Codierung\|Redundanzfreie Codierung]].

−	*Zur numerischen Auswertung der Q–Funktion können Sie das interaktive Applet  [[Applets:~~Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)~~\|Komplementäre Gaußsche Fehlerfunktionen]]  benutzen.	+	*Zur numerischen Auswertung der Q–Funktion können Sie das interaktive Applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|Komplementäre Gaußsche Fehlerfunktionen]]  benutzen.