Aufgaben:Exercise 1.16: Block Error Probability Bounds for AWGN - Revision history

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← Older revision		Revision as of 15:17, 5 August 2022
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	* The above cited reference  "[Liv10]"  refers to the lecture manuscript "Liva, G.:  Channel Coding.  Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010."		* The above cited reference  "[Liv10]"  refers to the lecture manuscript "Liva, G.:  Channel Coding.  Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010."

−	* Further we refer to the interactive applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\| "Complementary Gaussian error functions"]].	+	* Further we refer to the interactive HTML5/JavaScript applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\| "Complementary Gaussian error functions"]].

@@ Line 53: / Line 53: @@
 Hints:
-* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability|"Bounds for Block Error Probability"]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability|"Bounds for block error probability"]].
 * The above cited reference&nbsp; "[Liv10]"&nbsp; refers to the lecture manuscript "Liva, G.:&nbsp; Channel Coding.&nbsp;  Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010."

@@ Line 101: / Line 101: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig ist die  <u>Antwort 2</u>. Das Distanzspektrum $\{W_i\}$ ist definiert für $i = 0, \ \text{...} \ , \ n$:
+'''(1)'''&nbsp; The correct solution is <u>suggestion 2</u>:
-*$W_{1}$ indicates how often the Hamming weight $w_{\rm H}(\underline{x}_{i}) = 1$ occurs.
+*The distance spectrum&nbsp; $\{W_i\}$&nbsp; is defined for&nbsp; $i = 0, \ \text{...} \ , \ n$:
-With that, the ''Union Bound'' is:
 :$$p_1 = {\rm Pr(Union \hspace{0.15cm}Bound)}= \sum_{i = 1}^{n}\hspace{0.05cm}W_i \cdot {\rm Q}\left ( \sqrt{i/\sigma^2} \right ) \hspace{0.05cm}.$$
-'''(2)'''&nbsp; The distance spectrum of the $(8, 4, 4)$ code was given as $W_{0} = 1 , \ W_{4} = 14, \ W_{8} = 1$. Thus, one obtains for $\sigma = 1$:
+'''(2)'''&nbsp; The distance spectrum of the&nbsp; $(8, 4, 4)$&nbsp; code was given as&nbsp; $W_{0} = 1 , \ W_{4} = 14, \ W_{8} = 1$.&nbsp;
 :$$p_1 =  W_4 \cdot {\rm Q}\left ( 2 \right ) + W_8 \cdot {\rm Q}\left ( 2 \cdot \sqrt{2} \right )
 = 14 \cdot 2.28 \cdot 10^{-2}+ 1 \cdot 0.23 \cdot 10^{-2} \hspace{0.15cm}\underline{\approx 32.15\%}\hspace{0.05cm},$$
-or for $\sigma = 0.5$:
+*For&nbsp; $\sigma = 0.5$:
 :$$p_1 =  14 \cdot {\rm Q}\left ( 4 \right ) + {\rm Q}\left ( 4 \cdot \sqrt{2} \right )
 = 14 \cdot 3.17 \cdot 10^{-5}+ 1.1 \cdot 10^{-8} \hspace{0.15cm}\underline{\approx 0.0444 \%}\hspace{0.05cm}.$$
-'''(3)'''&nbsp; With the minimum distance $d_{\rm min} = 4$ we get:
+'''(3)'''&nbsp; With the minimum distance&nbsp; $d_{\rm min} = 4$&nbsp; we get:
 :$$\sigma = 1.0\text{:} \hspace{0.4cm} p_2 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} W_4 \cdot {\rm Q}\left ( 2 \right ) \hspace{0.15cm}\underline{= 31.92\%}\hspace{0.05cm},$$
@@ Line 127: / Line 130: @@
-'''(4)'''&nbsp; The correct solution is <u>suggestion 1</u>:
+'''(4)'''&nbsp; The correct solution is&nbsp; <u>suggestion 1</u>:
-*The ''Union Bound'' - denoted here by $p_{1}$ - is an upper bound on the block error probability in all cases.
+*The&nbsp; "Union Bound"&nbsp; - denoted here by&nbsp; $p_{1}$ - is an upper bound on the block error probability in all cases.
-*For the bound $p_{2}$ (''Truncated Union Bound'') this is not always true.
-*For example, in the $(7, 4, 3)$ Hamming code &nbsp; ⇒ &nbsp; $W_{3} = W_{4} = 7, \ W_{7} = 1$ is obtained with standard deviation $\sigma = 1$:
+*For the bound&nbsp; $p_{2}$&nbsp; ("Truncated Union Bound")&nbsp; this is not always true.
 :$$p_2 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} 7 \cdot {\rm Q}\left ( \sqrt{3} \right ) = 7 \cdot 4.18 \cdot 10^{-2} \approx 0.293\hspace{0.05cm},$$
 :$$p_1 \hspace{-0.15cm}\ = \ \hspace{-0.15cm} p_2 + 7 \cdot {\rm Q}\left ( \sqrt{4} \right )+ 1 \cdot {\rm Q}\left ( \sqrt{7} \right ) \approx 0.455 \hspace{0.05cm}.$$
-The actual block error probability is likely to be between $p_{2} = 29.3\%$ and $p_{1} = 45.5\%$ (but has not been verified). <br>That is, &nbsp; $p_{2}$ is not an upper bound.
+*The actual block error probability is likely to be between&nbsp; $p_{2} = 29.3\%$&nbsp; and&nbsp; $p_{1} = 45.5\%$&nbsp; (but this has not been verified). <br>That is, &nbsp; $p_{2}$ is not an upper bound.
-'''(5)'''&nbsp; Correct are <u>suggested solutions 1 and 3</u>, as the following calculation for the $(8, 4, 4)$ code shows:
+'''(5)'''&nbsp; Correct are&nbsp; <u>suggested solutions 1 and 3</u>,&nbsp; as the following calculation for the&nbsp; $(8, 4, 4)$&nbsp; code shows:
-*It holds ${\rm Q}(x) ≤ {\rm Q_{CR}}(x) = {\rm e}^{-x^2/2}$. Thus, for the Union Bound
+*It holds&nbsp; ${\rm Q}(x) ≤ {\rm Q_{CR}}(x) = {\rm e}^{-x^2/2}$.&nbsp; Thus,&nbsp; for the Union Bound
 :$$p_1 = W_4 \cdot {\rm Q}\left ( \sqrt{4/\sigma^2} \right ) +W_8 \cdot {\rm Q}\left ( \sqrt{8/\sigma^2} \right )$$
@@ Line 148: / Line 153: @@
 :$$p_1 \le W_4 \cdot {\rm e}^{ - {4}/(2 \sigma^2) } +W_8 \cdot {\rm e}^{ - {8}/(2 \sigma^2) } \hspace{0.05cm}.$$
-*With $\beta = {\rm e}^{-1/(2\sigma^2)}$ can be written for this also (so the given $\beta = 1/\sigma$ is wrong):
+*With&nbsp; $\beta = {\rm e}^{-1/(2\sigma^2)}$&nbsp; can be written for this also&nbsp; (so the given&nbsp; $\beta = 1/\sigma$&nbsp; is wrong):
 :$$p_1 \le W_4 \cdot \beta^4 + W_8 \cdot \beta^8 \hspace{0.05cm}.$$
-*The weight function of the $(8, 4, 4)$ code is:
+*The weight function of the&nbsp; $(8, 4, 4)$&nbsp; code is:
 :$$W(X) = 1 + W_4 \cdot X^4 + W_8 \cdot X^8 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} W(\beta) - 1 = W_4 \cdot \beta^4 + W_8 \cdot \beta^8\hspace{0.3cm}
@@ Line 158: / Line 163: @@
-'''(6)'''&nbsp; With $\sigma = 1$, the Bhattacharyya parameter is $\beta = {\rm e}^{-0.5} = 0.6065$, and thus one obtains for the Bhattacharyya bound:
+'''(6)'''&nbsp; With&nbsp; $\sigma = 1$,&nbsp; the Bhattacharyya parameter is&nbsp; $\beta = {\rm e}^{-0.5} = 0.6065$,&nbsp; and thus one obtains for the Bhattacharyya Bound:
 :$$p_3 = 14 \cdot \beta^4 + \beta^8 = 14 \cdot 0.135 + 0.018= 1.913 \hspace{0.15cm}\underline{= 191.3%}\hspace{0.05cm}.$$
-*Considering that $p_{3}$ is a bound for a probability, $p_{3} = 1.913$ is only a trivial bound.
+*Considering that&nbsp; $p_{3}$&nbsp; is a bound for a probability,&nbsp; $p_{3} = 1.913$&nbsp; is only a trivial bound.
 *For $\sigma = 0.5$, on the other hand, $\beta = {\rm e}^{-2}  \approx 0.135.$ Then holds:
 :$$p_3 = 14 \cdot \beta^4 + \beta^8 = 14 \cdot 3.35 \cdot 10^{-4} + 1.1 \cdot 10^{-7} \hspace{0.15cm}\underline{= 0.47 \%}\hspace{0.05cm}.$$
-A comparison with subtask (2) shows that in the present example the Bhattacharyya bound $p_{3}$ is above the ''union bound'' $p_{1}$ by a factor $(0.47 - 10^{-2})/(0.044 - 10^{-2}) > 10$.
+A comparison with subtask&nbsp; '''(2)'''&nbsp; shows that in the present example the Bhattacharyya Bound&nbsp; $p_{3}$&nbsp; is above the&nbsp; "Union Bound"&nbsp; $p_{1}$&nbsp; by a factor&nbsp;
-*The reason for this large deviation is the Chernoff-Rubin bound, which is well above the ${\rm Q}$ function.
+:$$(0.47 - 10^{-2})/(0.044 - 10^{-2}) > 10.$$
 *In [[Aufgaben:Exercise_1.16Z:_Bounds_for_the_Gaussian_Error_Function|"Exercise 1.16Z"]], the deviation between ${\rm Q}_{\rm CR}$ and ${\rm Q}(x)$ is also calculated quantitatively:
 :$${{\rm Q_{CR}}( x )}/{{\rm Q}( x )} \approx 2.5 \cdot x \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {{\rm Q_{CR}}( x = 4 )}/{{\rm Q}( x = 4)} \approx 10 \hspace{0.05cm}.$$

@@ Line 53: / Line 53: @@
 Hints:
-* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit|"Block error probability bounds"]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Bounds_for_Block_Error_Probability|"Bounds for Block Error Probability"]].
 * The above cited reference&nbsp; "[Liv10]"&nbsp; refers to the lecture manuscript "Liva, G.:&nbsp; Channel Coding.&nbsp;  Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010."

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Limits_for_Block_Error_Probability}}
-[[File:P_ID2414__KC_A_1_15.png|right|frame|Error function&nbsp; ${\rm Q}(x)$&nbsp; and approximations ]]
+[[File:P_ID2414__KC_A_1_15.png|right|frame|Function&nbsp; ${\rm Q}(x)$&nbsp; and approximations;<br>it holds:&nbsp; ${\rm Q_u}(x)\le{\rm Q}(x)\le{\rm Q_o}(x)$ ]]
 We assume the following constellation:
-*a linear block code with code rate&nbsp; $R = k/n$&nbsp; and distance spectrum&nbsp; $\{W_i\}, \ i = 1, \ \text{...} \ , n$,
+*A linear block code with code rate&nbsp; $R = k/n$&nbsp; and distance spectrum&nbsp; $\{W_i\}, \ i = 1, \ \text{...} \ , n$,
 *an AWGN channel characterized by&nbsp; $E_{\rm B}/N_{0}$ &nbsp; ⇒ &nbsp; convertible to noise power&nbsp; $\sigma^2$,
-*a receiver based on ''soft decision'' as well as the ''maximum likelihood criterion''.
-Under the assumption valid for the entire exercise that always the null word&nbsp; $\underline{x}_{1} = (0, 0, \text{... } \ , 0)$&nbsp; is sent, the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|"pairwise error probability"]] with a different codeword&nbsp; $\underline{x}_{l} (l = 2,\ \text{...} \ , 2^k)$:
+Under the assumption valid for the entire exercise that always the zero-word&nbsp; $\underline{x}_{1} = (0, 0, \text{... } \ , 0)$&nbsp; is sent, the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|"pairwise error probability"]]&nbsp; with a different code word&nbsp; $\underline{x}_{l} (l = 2,\ \text{...} \ , 2^k)$:
 :$$ {\rm Pr}[\hspace{0.05cm}\underline{x}_{\hspace{0.02cm}1} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}l}\hspace{0.05cm}] = {\rm Q}\left ( \sqrt{w_{\rm H}(\underline{x}_{\hspace{0.02cm}l})/\sigma^2} \right ) \hspace{0.05cm}.$$
-The derivation of this relation can be found in [Liv10]. Used in this equation are:
+The derivation of this relation can be found in&nbsp; [Liv10].&nbsp; Used in this equation are:
 *the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"complementary Gaussian error function"]]&nbsp; ${\rm Q}(x)$,
-*the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|"Hamming weight"]]&nbsp; $w_{\rm H}(\underline{x}_{l})$&nbsp; of the codeword&nbsp; $\underline{x}_{l}$,
-*the&nbsp; [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation|"AWGN noise power"]]&nbsp; $\sigma^2 = (2 - R - E_{\rm B}/N_{0})^{-1}.$
+*the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|"Hamming weight"]]&nbsp; $w_{\rm H}(\underline{x}_{l})$&nbsp; of the code word&nbsp; $\underline{x}_{l}$,
-Damit lassen sich verschiedene Schranken für die Blockfehlerwahrscheinlichkeit angeben:
+This allows various bounds to be specified for the block error probability:
-*the so called&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|"Union Bound"]]&nbsp; (UB):
+*the so called&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|"Union Bound"]]&nbsp; $\rm (UB)$:
 :$$p_1 = \sum_{l = 2}^{2^k}\hspace{0.05cm}{\rm Pr}[\hspace{0.05cm}\underline{x}_{\hspace{0.02cm}1} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}l}\hspace{0.05cm}] = \sum_{l \hspace{0.05cm}= \hspace{0.05cm}2}^{2^k}\hspace{0.05cm}{\rm Q}\left ( \sqrt{w_{\rm H}(\underline{x}_{\hspace{0.02cm}l})/\sigma^2} \right ) \hspace{0.05cm},$$
-*the so called&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Bounds_for_the_.287.2C_4.2C_3.29_Hamming_code_at_the_AWGN_channel|"Truncated Union Bound"]]&nbsp; (TUB):
+*the so called&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Bounds_for_the_.287.2C_4.2C_3.29_Hamming_code_at_the_AWGN_channel|"Truncated Union Bound"]]&nbsp; $\rm  (TUB)$:
 :$$p_2 = W_{d_{\rm min}} \cdot {\rm Q}\left ( \sqrt{d_{\rm min}/\sigma^2} \right ) \hspace{0.05cm},$$
-*the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya|"Bhattacharyya bound"]]:
+*the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya|"Bhattacharyya Bound"]]:
-:$$p_3 = W(\beta) - 1\hspace{0.05cm},\hspace{0.2cm} {\rm mit}\hspace{0.15cm} \beta = {\rm e}^{ - 1/(2\sigma^2) } \hspace{0.05cm}.$$
+:$$p_3 = W(\beta) - 1\hspace{0.05cm},\hspace{0.2cm} {\rm with}\hspace{0.15cm} \beta = {\rm e}^{ - 1/(2\sigma^2) } \hspace{0.05cm}.$$
-:In this case, replace the distance spectrum&nbsp; $\{W_i\}$&nbsp; with the weight enumerator function:
+:In this case,&nbsp; replace the distance spectrum&nbsp; $\{W_i\}$&nbsp; with the weight enumerator function:
 :$$\left \{ \hspace{0.05cm} W_i \hspace{0.05cm} \right \} \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} W(X) = \sum_{i=0 }^{n} W_i \cdot X^{i} = W_0 + W_1 \cdot X + W_2 \cdot X^{2} + ... \hspace{0.05cm} + W_n \cdot X^{n}\hspace{0.05cm}.$$
-In the transition from the ''Union Bound''&nbsp; $p_{1}$&nbsp; to the more imprecise bound&nbsp; $p_{3}$&nbsp; among others the function&nbsp; ${\rm Q}(x)$&nbsp; is replaced by the&nbsp; [https://en.wikipedia.org/wiki/Chernoff_bound Chernoff-Rubin bound]&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp;. Both functions are shown in the above graph (red and green curve, respectively).
+In the transition from the&nbsp; "Union Bound"&nbsp; $p_{1}$&nbsp; to the more imprecise bound&nbsp; $p_{3}$&nbsp; among others
-In the&nbsp; [[Aufgaben:Exercise_1.16Z:_Bounds_for_the_Gaussian_Error_Function|"Exercise 1.16Z"]]&nbsp; the relationship between these functions is evaluated numerically and referenced to the bounds&nbsp; ${\rm Q}_{o}(x)$ and ${\rm Q}_{u}(x)$&nbsp; which are also drawn in the above graph.
+In the&nbsp; [[Aufgaben:Exercise_1.16Z:_Bounds_for_the_Gaussian_Error_Function|"Exercise 1.16Z"]]&nbsp; the relationship between these functions is evaluated numerically and referenced to the bounds&nbsp; ${\rm Q}_{\rm o}(x)$ and ${\rm Q}_{\rm u}(x)$&nbsp; which are also drawn in the above graph.
@@ Line 46: / Line 54: @@
 Hints:
 * This exercise belongs to the chapter&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit|"Block error probability bounds"]].
-* The above cited reference [Liv10] refers to the lecture manuscript "Liva, G.: ''Channel Coding''.  Chair of Communications Engineering, TU Munich and DLR Oberpfaffenhofen, 2010."
 * Further we refer to the interactive applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen| "Complementary Gaussian error functions"]].
@@ Line 56: / Line 66: @@
 <quiz display=simple>
-{Which equation applies to the ''Union Bound''?
+{Which equation applies to the&nbsp; "Union Bound"?
 |type="[]"}
 - $p_{1} = \sum_{l\hspace{0.05cm}=\hspace{0.05cm}2}^{2^k} W_{l} · {\rm Q}\big[(l/\sigma^2)^{0.5}\big],$
 + $p_{1} = \sum_{i\hspace{0.05cm}=\hspace{0.05cm}1}^{n} W_{i} · {\rm Q}\big[(i/\sigma^2)^{0.5}\big].$
-{Specify the Union Bound for the&nbsp; $(8, 4, 4)$ code and various&nbsp; $\sigma$&nbsp;.
+{Specify the Union Bound for the&nbsp; $(8, 4, 4)$&nbsp; code and various&nbsp; $\sigma$.
 |type="{}"}
 $\sigma = 1.0 \text{:} \hspace{0.4cm} p_{1} \ = \ $ { 32.15 3% } $\ \%$
 $\sigma = 0.5 \text{:} \hspace{0.4cm} p_{1} \ = \ $ { 0.0444 3% } $\ \%$
-{Given the same boundary conditions, what does the ''Truncated Union Bound'' provide?
+{Given the same boundary conditions, what does the&nbsp; "Truncated Union Bound"&nbsp; provide?
 |type="{}"}
 $\sigma = 1.0 \text{:} \hspace{0.4cm} p_{2} \ = \ $ { 31.92 3% } $\ \%$
 $\sigma = 0.5 \text{:} \hspace{0.4cm} p_{2} \ = \ $ { 0.044 3% } $\ \%$
-{Which statement is always true (for all constellations)?
+{Which statement is always true&nbsp; (for all constellations)?
 |type="[]"}
 + The block error probability is never greater than&nbsp; $p_{1}$.
 - The block error probability is never greater than&nbsp; $p_{2}$.
-{How do you get from&nbsp; $p_{1}$&nbsp; to the Bhattacharyya bound&nbsp; $p_{3}$? By using
+{How do you get from&nbsp; $p_{1}$&nbsp; to the&nbsp; "Bhattacharyya Bound"&nbsp; $p_{3}$?&nbsp;
 |type="[]"}
-+ replace the error function&nbsp; ${\rm Q}(x)$&nbsp; with the function&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp;,
++ Replace the error function&nbsp; ${\rm Q}(x)$&nbsp; with the function&nbsp; ${\rm Q}_{\rm CR}(x)$.
-- sets the Bhattacharyya parameter&nbsp; $\beta = 1/\sigma$&nbsp; ,
+- Set the Bhattacharyya parameter&nbsp; $\beta = 1/\sigma$.
-+ instead of&nbsp; $\{W_i\}$&nbsp; uses the weight enumerator function&nbsp; $W(X)$&nbsp;.
++ Instead of&nbsp; $\{W_i\}$&nbsp; uses the weight enumerator function&nbsp; $W(X)$.
-{Specify the Bhattacharyya bound for&nbsp; $\sigma = 1$&nbsp; and&nbsp; $\sigma = 0.5$&nbsp;.
+{Specify the Bhattacharyya Bound for&nbsp; $\sigma = 1$&nbsp; and&nbsp; $\sigma = 0.5$&nbsp;.
 |type="{}"}
 $\sigma = 1.0 \text{:} \hspace{0.4cm} p_{3} \ = \ $ { 191.3 3% } $\ \%$

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Limits_for_Block_Error_Probability}}
-[[File:P_ID2414__KC_A_1_15.png|right|frame|Fehlerfunktion&nbsp; ${\rm Q}(x)$&nbsp; und Näherungen ]]
+[[File:P_ID2414__KC_A_1_15.png|right|frame|Error function&nbsp; ${\rm Q}(x)$&nbsp; and approximations ]]
-Wir gehen von der folgenden Konstellation aus:
+We assume the following constellation:
-*ein linearer Blockcode mit Coderate&nbsp; $R = k/n$&nbsp; und Distanzspektrum&nbsp; $\{W_i\}, \ i = 1, \ \text{...} \ , n$,
+*a linear block code with code rate&nbsp; $R = k/n$&nbsp; and distance spectrum&nbsp; $\{W_i\}, \ i = 1, \ \text{...} \ , n$,
-*ein AWGN–Kanal, gekennzeichnet durch&nbsp; $E_{\rm B}/N_{0}$ &nbsp; ⇒ &nbsp; umrechenbar in die Rauschleistung&nbsp; $\sigma^2$,
+*an AWGN channel characterized by&nbsp; $E_{\rm B}/N_{0}$ &nbsp; ⇒ &nbsp; convertible to noise power&nbsp; $\sigma^2$,
-*ein Empfänger, basierend auf ''Soft Decision'' sowie dem ''Maximum–Likelihood–Kriterium''.
+*a receiver based on ''soft decision'' as well as the ''maximum likelihood criterion''.
-Unter der für die gesamte Aufgabe gültigen Annahme, dass stets das Nullwort&nbsp; $\underline{x}_{1} = (0, 0, \text{...} \ , 0)$&nbsp; gesendet wird, gilt für die&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit#Union_Bound_der_Blockfehlerwahrscheinlichkeit|„paarweise Fehlerwahrscheinlichkeit”]] mit einem anderen Codewort&nbsp; $\underline{x}_{l} (l = 2,\ \text{...} \ , 2^k)$:
+Under the assumption valid for the entire exercise that always the null word&nbsp; $\underline{x}_{1} = (0, 0, \text{... } \ , 0)$&nbsp; is sent, the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|"pairwise error probability"]] with a different codeword&nbsp; $\underline{x}_{l} (l = 2,\ \text{...} \ , 2^k)$:
 :$$ {\rm Pr}[\hspace{0.05cm}\underline{x}_{\hspace{0.02cm}1} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}l}\hspace{0.05cm}] = {\rm Q}\left ( \sqrt{w_{\rm H}(\underline{x}_{\hspace{0.02cm}l})/\sigma^2} \right ) \hspace{0.05cm}.$$
-Die Herleitung dieser Beziehung finden Sie in [Liv10]. In dieser Gleichung werden verwendet:
+The derivation of this relation can be found in [Liv10]. Used in this equation are:
-*die&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen#.C3.9Cberschreitungswahrscheinlichkeit|komplementäre Gaußsche Fehlerfunktion]]&nbsp; ${\rm Q}(x)$,
+*the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|complementary Gaussian error function]]&nbsp; ${\rm Q}(x)$,
-*das&nbsp; [[Channel_Coding/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung|Hamming–Gewicht]]&nbsp; $w_{\rm H}(\underline{x}_{l})$&nbsp; des Codewortes&nbsp; $\underline{x}_{l}$,
+*the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding|Hamming weight]]&nbsp; $w_{\rm H}(\underline{x}_{l})$&nbsp; of the codeword&nbsp; $\underline{x}_{l}$,
-*die&nbsp; [[Digital_Signal_Transmission/Optimierung_der_Basisbandübertragungssysteme#Systemoptimierung_bei_Leistungsbegrenzung|AWGN–Rauschleistung]]&nbsp; $\sigma^2 = (2 · R · E_{\rm B}/N_{0})^{–1}.$
+*the&nbsp; [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#System_optimization_with_power_limitation|AWGN noise power]]&nbsp; $\sigma^2 = (2 - R - E_{\rm B}/N_{0})^{-1}.$
 Damit lassen sich verschiedene Schranken für die Blockfehlerwahrscheinlichkeit angeben:
-*die so genannte&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit#Union_Bound_der_Blockfehlerwahrscheinlichkeit|Union Bound]]&nbsp; (UB):
+*the so called&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Union_Bound_of_the_block_error_probability|Union Bound]]&nbsp; (UB):
 :$$p_1 = \sum_{l = 2}^{2^k}\hspace{0.05cm}{\rm Pr}[\hspace{0.05cm}\underline{x}_{\hspace{0.02cm}1} \hspace{-0.02cm}\mapsto \hspace{-0.02cm}\underline{x}_{\hspace{0.02cm}l}\hspace{0.05cm}] = \sum_{l \hspace{0.05cm}= \hspace{0.05cm}2}^{2^k}\hspace{0.05cm}{\rm Q}\left ( \sqrt{w_{\rm H}(\underline{x}_{\hspace{0.02cm}l})/\sigma^2} \right ) \hspace{0.05cm},$$
-*die so genannte&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit#Schranken_f.C3.BCr_den_.287.2C_4.2C_3.29.E2.80.93Hamming.E2.80.93Code_beim_AWGN.E2.80.93Kanal|Truncated Union Bound]]&nbsp; (TUB):
+*the so called&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#Bounds_for_the_.287.2C_4.2C_3.29_Hamming_code_at_the_AWGN_channel|Truncated Union Bound]]&nbsp; (TUB):
 :$$p_2 = W_{d_{\rm min}} \cdot {\rm Q}\left ( \sqrt{d_{\rm min}/\sigma^2} \right ) \hspace{0.05cm},$$
-*die&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit#Die_obere_Schranke_nach_Bhattacharyya|Bhattacharyya–Schranke]]:
+*the&nbsp; [[Channel_Coding/Limits_for_Block_Error_Probability#The_upper_bound_according_to_Bhattacharyya|Bhattacharyya bound]]:
 :$$p_3 = W(\beta) - 1\hspace{0.05cm},\hspace{0.2cm} {\rm mit}\hspace{0.15cm} \beta = {\rm e}^{ - 1/(2\sigma^2) } \hspace{0.05cm}.$$
-:In diesem Fall ist das Distanzspektrum&nbsp; $\{W_i\}$&nbsp; durch die Gewichtsfunktion zu ersetzen:
+:In this case, replace the distance spectrum&nbsp; $\{W_i\}$&nbsp; with the weight enumerator function:
 :$$\left \{ \hspace{0.05cm} W_i \hspace{0.05cm} \right \} \hspace{0.3cm} \Leftrightarrow \hspace{0.3cm} W(X) = \sum_{i=0 }^{n} W_i \cdot X^{i} = W_0 + W_1 \cdot X + W_2 \cdot X^{2} + ... \hspace{0.05cm} + W_n \cdot X^{n}\hspace{0.05cm}.$$
-Beim Übergang von der ''Union Bound''&nbsp; $p_{1}$&nbsp; zur ungenaueren Schranke&nbsp; $p_{3}$&nbsp; wird unter Anderem die Funktion&nbsp; ${\rm Q}(x)$&nbsp; durch die&nbsp; [https://de.wikipedia.org/wiki/Chernoff-Ungleichung Chernoff–Rubin–Schranke]&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp; ersetzt. Beide Funktionen sind in obigerer Grafik dargestellt (rote bzw. grüne Kurve).
+In the transition from the ''Union Bound''&nbsp; $p_{1}$&nbsp; to the more imprecise bound&nbsp; $p_{3}$&nbsp; among others the function&nbsp; ${\rm Q}(x)$&nbsp; is replaced by the&nbsp; [https://en.wikipedia.org/wiki/Chernoff_bound Chernoff-Rubin bound]&nbsp; ${\rm Q}_{\rm CR}(x)$&nbsp;. Both functions are shown in the above graph (red and green curve, respectively).
-In der&nbsp; [[Aufgaben:Aufgabe_1.16Z:_Schranken_für_die_Gaußsche_Fehlerfunktion|Aufgabe 1.16Z]]&nbsp; wird der Zusammenhang zwischen diesen Funktionen numerisch ausgewertet und zu den Schranken&nbsp; ${\rm Q}_{o}(x)$ und ${\rm Q}_{u}(x)$&nbsp; Bezug genommen, die in obiger Grafik ebenfalls eingezeichnet sind.
+In the&nbsp; [[Aufgaben:Exercise_1.16Z:_Bounds_for_the_Gaussian_Error_Function|Exercise 1.16Z]]&nbsp; the relationship between these functions is evaluated numerically and referenced to the bounds&nbsp; ${\rm Q}_{o}(x)$ and ${\rm Q}_{u}(x)$&nbsp; which are also drawn in the above graph.
-''Hinweise:''
+Hints:
-* Die Aufgabe gehört zum Kapitel&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit|Schranken für die Blockfehlerwahrscheinlichkeit]].
+* This exercise belongs to the chapter&nbsp; [[Channel_Coding/Schranken_für_die_Blockfehlerwahrscheinlichkeit|Schranken für die Blockfehlerwahrscheinlichkeit]].
 * Die oben zitierte Literaturstelle  [Liv10] verweist auf das Vorlesungsmanuskript "Liva, G.: ''Channel Coding''.  Lehrstuhl für Nachrichtentechnik, TU München und DLR Oberpfaffenhofen, 2010."
 * Weiter verweisen wir auf das interaktive Applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen| Komplementäre Gaußsche Fehlerfunktionen]].

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	[[File:P_ID2414__KC_A_1_15.png\|right\|frame\|Fehlerfunktion  ${\rm Q}(x)$  und Näherungen ]]		[[File:P_ID2414__KC_A_1_15.png\|right\|frame\|Fehlerfunktion  ${\rm Q}(x)$  und Näherungen ]]

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	[[File:P_ID2414__KC_A_1_15.png\|right\|frame\|Fehlerfunktion  ${\rm Q}(x)$  und Näherungen ]]		[[File:P_ID2414__KC_A_1_15.png\|right\|frame\|Fehlerfunktion  ${\rm Q}(x)$  und Näherungen ]]

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−	[[Category:Channel Coding: Exercises\|^1.6 ~~Fehlerwahrscheinlichkeitsschranken~~^]]	+	[[Category:Channel Coding: Exercises\|^1.6 Error Probability Bounds^]]