Aufgaben:Exercise 4.10: Union Bound - Revision history

Guenter at 15:13, 6 September 2022

2022-09-06T15:13:37Z

Hwang at 12:16, 1 September 2022

2022-09-01T12:16:11Z

Guenter at 13:59, 30 July 2022

2022-07-30T13:59:28Z

Guenter at 13:46, 30 July 2022

2022-07-30T13:46:49Z

Hwang at 08:40, 18 July 2022

2022-07-18T08:40:00Z

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Guenter at 10:12, 23 June 2022

2022-06-23T10:12:40Z

Guenter: Guenter moved page Aufgaben:Exercis 4.10: Union Bound to Aufgaben:Exercise 4.10: Union Bound

2022-06-23T10:06:14Z

Guenter moved page Exercis 4.10: Union Bound to Exercise 4.10: Union Bound

Guenter: Guenter moved page Aufgaben:Aufgabe 4.10: Union Bound to Aufgaben:Exercis 4.10: Union Bound

2022-06-23T10:04:49Z

Guenter moved page Aufgabe 4.10: Union Bound to Exercis 4.10: Union Bound

Javier: Text replacement - "„" to """

2021-05-28T14:36:47Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:36:15Z

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@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Approximation_of_the_Error_Probability}}
-[[File:P_ID2043__Dig_A_4_10.png|right|frame|Signal space constellations with $N=2$&nbsp; and&nbsp; $M=3$ '''KORREKTUR: Konfig.''']]
+[[File:EN_Dig_A_4_10.png|right|frame|Signal space constellations with $N=2$&nbsp; and&nbsp; $M=3$]]
 The so-called&nbsp; "Union Bound"&nbsp; gives an upper bound for the error probability of a non-binary transmission system&nbsp; $(M > 2)$.&nbsp;

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Approximation_of_the_Error_Probability}}
-[[File:P_ID2043__Dig_A_4_10.png|right|frame|Signal space constellations with $N=2$&nbsp; and&nbsp; $M=3$]]
+[[File:P_ID2043__Dig_A_4_10.png|right|frame|Signal space constellations with $N=2$&nbsp; and&nbsp; $M=3$ '''KORREKTUR: Konfig.''']]
 The so-called&nbsp; "Union Bound"&nbsp; gives an upper bound for the error probability of a non-binary transmission system&nbsp; $(M > 2)$.&nbsp;

@@ Line 81: / Line 81: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Points $\boldsymbol{s}_1$ and $\boldsymbol{s}_2$ are the same for all configurations.
+'''(1)'''&nbsp; Points&nbsp; $\boldsymbol{s}_1$&nbsp; and&nbsp; $\boldsymbol{s}_2$&nbsp; are the same for all configurations.
-*The smallest error probability occurs when $\boldsymbol{s}_0$ is farthest from $\boldsymbol{s}_1$ and $\boldsymbol{s}_2$.
+*The smallest error probability occurs when&nbsp; $\boldsymbol{s}_0$&nbsp; is farthest from&nbsp; $\boldsymbol{s}_1$&nbsp; and&nbsp; $\boldsymbol{s}_2$.
-*This is the case for configuration '''C''' &nbsp; &#8658; &nbsp; <u>solution 3</u>.
-'''(2)'''&nbsp; For configuration '''A''', the distance between all points is the same: $d_{01} = d_{02} = d_{12} = 2$.
+'''(2)'''&nbsp; For configuration&nbsp; $\rm A$,&nbsp; the distance between all points is the same:&nbsp; $d_{01} = d_{02} = d_{12} = 2$.
-*Therefore, to calculate the <i>Union Bound</i>, it is not necessary to average over all symbols, and it is valid since, for example, $\boldsymbol{s}_0$ is distorted into symbol $\boldsymbol{s}_1$ or $\boldsymbol{s}_2$ with equal probability:
+*Therefore,&nbsp; to calculate the&nbsp; "Union Bound",&nbsp; it is not necessary to average over all symbols.&nbsp;
 :$${\rm Pr}({ \cal E}) \le p_{\rm UB} = 2 \cdot {\rm Q} \left ( \frac{d_{ik}/2}{\sigma_n} \right ) = 2 \cdot {\rm Q}(2) \approx 2 \cdot 0.023 \hspace{0.1cm}\hspace{0.15cm}\underline {= 4.6\%} \hspace{0.05cm}. $$
-'''(3)'''&nbsp; Here, the falsification probabilities differ for each symbol.
+'''(3)'''&nbsp; Here,&nbsp; the falsification probabilities differ for each symbol.
-*If $\boldsymbol{s}_0$ was sent, with $d_{01} = d_{02} = 2^{0.5}$ and $\sigma = 0.5$:
+*If&nbsp; $\boldsymbol{s}_0$&nbsp; was sent,&nbsp; with $d_{01} = d_{02} = 2^{0.5}$&nbsp; and&nbsp; $\sigma = 0.5$:
 :$$p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_0} = 2 \cdot {\rm Q} \left ( \frac{\sqrt{2}/2}{0.5} \right )
   = 2 \cdot {\rm Q}(\sqrt{2}) = 2 \cdot 0.079 = 0.158 \hspace{0.05cm}. $$
-*In contrast, the other two conditional probabilities are smaller:
+*In contrast,&nbsp; the other two conditional probabilities are smaller:
 :$$p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_1} = p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_2} \hspace{-0.1cm} \ = \ \hspace{-0.1cm}
   {\rm Q} \left ( \frac{{2}/2}{0.5} \right )+{\rm Q} \left ( \frac{\sqrt{2}/2}{0.5} \right )= {\rm Q}(2) +{\rm Q}(\sqrt{2}) = 0.023 + 0.079 = 0.102 \hspace{0.05cm}.$$
-By averaging, we obtain for the Union Bound considering the different distances:
+*By averaging,&nbsp; we obtain for the Union Bound considering the different distances:
 :$$p_{\rm UB} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {1}/{3} \cdot \left [ p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_0} + p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_1} +p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_2}\right ]=
 {1}/{3} \cdot \left [ 2 \cdot  {\rm Q}(\sqrt{2})+ 2 \cdot ({\rm Q}({2}) + {\rm Q}(\sqrt{2})) \right ] =
 {1}/{3} \cdot \left [ 4 \cdot  {\rm Q}(\sqrt{2})+ 2 \cdot {\rm Q}({2})  \right ] $$
 :$$ \Rightarrow \hspace{0.3cm}  p_{\rm UB} =
-{1}/{3} \cdot \left [ 4 \cdot  0.079+ 2 \cdot 0.023  \right ] \hspace{0.1cm}\hspace{0.12cm}\underline {\approx 12.1\% }  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  p_{\rm UB}\ge {\rm Pr}({ \cal E})\hspace{0.05cm}.$$
+{1}/{3} \cdot \big [ 4 \cdot  0.079+ 2 \cdot 0.023  \big ] \hspace{0.1cm}\hspace{0.12cm}\underline {\approx 12.1\% }  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}  p_{\rm UB}\ge {\rm Pr}({ \cal E})\hspace{0.05cm}.$$
@@ Line 113: / Line 116: @@
 :$$d_{01} = d_{02} = \sqrt{2^2 + 1^2}= \sqrt{5} \approx 2.24\hspace{0.2cm}, d_{12} = 2$$
 :$$\Rightarrow \hspace{0.3cm}  p_{\rm UB}  =
-  {1}/{3} \cdot \left [ 4 \cdot  {\rm Q}(\sqrt{5})+ 2 \cdot {\rm Q}({2})  \right ] = {1}/{3} \cdot \left [ 4 \cdot  0.013+ 2 \cdot 0.023  \right ]\hspace{0.1cm}\hspace{0.15cm}\underline {\approx  3.2\%} \hspace{0.05cm}. $$
+  {1}/{3} \cdot \big [ 4 \cdot  {\rm Q}(\sqrt{5})+ 2 \cdot {\rm Q}({2})  \big ] = {1}/{3} \cdot \big [ 4 \cdot  0.013+ 2 \cdot 0.023  \big  ]\hspace{0.1cm}\hspace{0.15cm}\underline {\approx  3.2\%} \hspace{0.05cm}. $$

@@ Line 3: / Line 3: @@
 [[File:P_ID2043__Dig_A_4_10.png|right|frame|Signal space constellations with $N=2$&nbsp; and&nbsp; $M=3$]]
-The so-called "Union Bound" gives an upper bound for the error probability of a non-binary transmission system&nbsp; $(M > 2)$.&nbsp;
+The so-called&nbsp; "Union Bound"&nbsp; gives an upper bound for the error probability of a non-binary transmission system&nbsp; $(M > 2)$.&nbsp;
-The actual (average) error probability is generally given as follows:
+*The actual&nbsp; (average)&nbsp; error probability is generally given as follows:
 :$${\rm Pr}({ \cal E}) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sum\limits_{i = 0 }^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) \hspace{0.05cm},$$
 :$$ {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) \hspace{-0.1cm} \ = \ \hspace{-0.1cm} {\rm Pr} \left [ \bigcup_{k \ne i} { \cal E}_{ik}\right ] \hspace{0.05cm},\hspace{0.2cm}{ \rm where}\hspace{0.2cm}
@@ Line 11: / Line 11: @@
   \hspace{0.05cm}.$$
-The simpler ''Union Bound''&nbsp; provides an upper bound on the falsification probability assuming that message&nbsp; $m_i$&nbsp; $($or signal&nbsp; $\boldsymbol{s}_i)$&nbsp; was sent:
+*The simpler&nbsp; "Union Bound"&nbsp; $\rm (UB)$&nbsp; provides an upper bound on the falsification probability assuming that message&nbsp; $m_i$&nbsp; $($or signal&nbsp; $\boldsymbol{s}_i)$&nbsp; was sent:
 :$$p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_i} \hspace{-0.1cm} \ \ge \ \hspace{-0.1cm} {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} \boldsymbol{ s }_i ) = {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i )\hspace{0.05cm},\ $$
 :$$ p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_i} \hspace{-0.1cm} \ = \ \hspace{-0.2cm}\sum\limits_{k = 0 ,\hspace{0.1cm}  k \ne i}^{M-1}\hspace{-0.1cm}
@@ Line 18: / Line 18: @@
 The following abbreviations are used:
 * ${\rm Q}(x)$&nbsp; is the complementary Gaussian error function;
 * $d_{ik}$&nbsp; denotes the distance between the signal points&nbsp; $\boldsymbol{s}_i$&nbsp; and $\boldsymbol{s}_k$;
-By averaging over all possible signals&nbsp; $\boldsymbol{s}_i$,&nbsp; we then arrive at the actual ''Union Bound''&nbsp;:
 :$$p_{\rm UB} = \sum\limits_{i = 0 }^{M-1} {\rm Pr}(\boldsymbol{ s }_i) \cdot p_{{\rm UB}\hspace{0.05cm}| \hspace{0.05cm}\boldsymbol{ s }_i} \ge {\rm Pr}({ \cal E}) \hspace{0.05cm}.$$
-*The graph shows three different signal space constellations, each with&nbsp; $M = 3$&nbsp; signal space points&nbsp; $\boldsymbol{s}_0$,&nbsp; $\boldsymbol{s}_1$&nbsp; and&nbsp; $\boldsymbol{s}_2$&nbsp; in two-dimensional space &nbsp;$(N = 2)$.
+*The graph shows three different signal space constellations, <br>each with&nbsp; $M = 3$&nbsp; signal space points&nbsp; $\boldsymbol{s}_0$,&nbsp; $\boldsymbol{s}_1$&nbsp; and&nbsp; $\boldsymbol{s}_2$&nbsp; in two-dimensional space &nbsp;$(N = 2)$.
 *The basis functions&nbsp; $\varphi_1(t)$&nbsp; and&nbsp; $\varphi_2(t)$&nbsp; are suitably normalized.
 *Thus, the signal space coordinates are also pure numerical values without unit:
 :$$\boldsymbol{ s }_1 = (-1, \hspace{0.1cm}+1)\hspace{0.05cm}, \hspace{0.2cm}
     \boldsymbol{ s }_2 = (+1, \hspace{0.1cm}+1)\hspace{0.05cm}.$$
-*The signal space point&nbsp; $\boldsymbol{s}_0$&nbsp; in configuration &nbsp;$\rm A$&nbsp; is located such that&nbsp; $\boldsymbol{s}_0$,&nbsp; $\boldsymbol{s}_1$,&nbsp; $\boldsymbol{s}_2$&nbsp; describe an equilateral triangle. *In contrast, in configurations &nbsp;$\rm B$&nbsp; and  &nbsp;$\rm C$,&nbsp; &nbsp; $\boldsymbol{s}_0 = (0, 0)$&nbsp; and&nbsp; $\boldsymbol{s}_0 = (0, \ -1)$, respectively.
+*The signal space point&nbsp; $\boldsymbol{s}_0$&nbsp; in configuration &nbsp;$\rm A$&nbsp; is located such that&nbsp; $\boldsymbol{s}_0$,&nbsp; $\boldsymbol{s}_1$,&nbsp; $\boldsymbol{s}_2$&nbsp; describe an equilateral triangle.
-''Notes:''
-* The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability|"Approximation of the Error Probability"]].
+Notes:
-*Use the AWGN rms value &nbsp;$\sigma_n = 0.5$ for all calculations.
+* The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability|"Approximation of the Error Probability"]].
 * Given the following values of the complementary Gaussian error function:
@@ Line 49: / Line 56: @@
 ===Questions===
 <quiz display=simple>
-{Which of the three configurations leads to the smallest error probability (at least according to the ''Union Bound'' approximation)?
+{Which of the three configurations leads to the smallest error probability&nbsp; $($at least according to the Union Bound approximation$)$?
 |type="()"}
 - Configuration &nbsp;$\rm A$,
@@ Line 55: / Line 62: @@
 + Configuration &nbsp;$\rm C$.
-{Calculate the "averaged Union Bound" &nbsp;$(p_{\rm UB})$&nbsp; for configuration &nbsp;$\rm A$.
+{Calculate the&nbsp; "averaged Union Bound" &nbsp;$(p_{\rm UB})$&nbsp; for configuration &nbsp;$\rm A$.
 |type="{}"}
 $p_{\rm UB} \ = \ ${ 4.6 3% } $\ \%$
-{Calculate the "averaged Union Bound" for configuration &nbsp;$\rm B$.
+{Calculate the&nbsp; "averaged Union Bound"&nbsp; for configuration &nbsp;$\rm B$.
 |type="{}"}
 $p_{\rm UB} \ = \ ${ 12.1 3% } $\ \%$
-{Calculate the "averaged Union Bound" for configuration &nbsp;$\rm C$.
+{Calculate the&nbsp; "averaged Union Bound"&nbsp; for configuration &nbsp;$\rm C$.
 |type="{}"}
 $p_{\rm UB} \ = \ ${ 3.2 3% } $\ \%$
-{How would the noise rms value&nbsp; $\sigma_n$&nbsp; have to be changed for configuration &nbsp;$\rm A$&nbsp; to yield the same <i>Union Bound</i>&nbsp; as in '''(4)'''?
+{How would the noise rms value&nbsp; $\sigma_n$&nbsp; have to be changed for configuration &nbsp;$\rm A$&nbsp; to yield the same&nbsp; "Union Bound"&nbsp; as in subtask&nbsp; '''(4)'''?
 |type="{}"}
 $\sigma_n \ = \ ${ 0.467 3% }

← Older revision		Revision as of 10:12, 23 June 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^4.3 BER-Approximation^]]	+	[[Category:Digital Signal Transmission: Exercises\|^4.3 BER Approximation^]]

← Older revision	Revision as of 10:06, 23 June 2022
(No difference)

@@ Line 3: / Line 3: @@
 [[File:P_ID2043__Dig_A_4_10.png|right|frame|Signalraumkonstellationen mit $N=2$&nbsp; und&nbsp; $M=3$]]
-Die so genannte &bdquo;Union Bound" gibt eine obere Schranke für die Fehlerwahrscheinlichkeit eines nichtbinären Übertragungssystems&nbsp; $(M > 2)$&nbsp; an.
+Die so genannte "Union Bound" gibt eine obere Schranke für die Fehlerwahrscheinlichkeit eines nichtbinären Übertragungssystems&nbsp; $(M > 2)$&nbsp; an.
 Die tatsächliche (mittlere) Fehlerwahrscheinlichkeit ist allgemein wie folgt gegeben:
@@ Line 55: / Line 55: @@
 + Konfiguration &nbsp;$\rm C$.
-{Berechnen Sie die &bdquo;gemittelte Union Bound" &nbsp;$(p_{\rm UB})$&nbsp; für die Konfiguration &nbsp;$\rm A$.
+{Berechnen Sie die "gemittelte Union Bound" &nbsp;$(p_{\rm UB})$&nbsp; für die Konfiguration &nbsp;$\rm A$.
 |type="{}"}
 $p_{\rm UB} \ = \ ${ 4.6 3% } $\ \%$
-{Berechnen Sie die &bdquo;gemittelte Union Bound" für die Konfiguration &nbsp;$\rm B$.
+{Berechnen Sie die "gemittelte Union Bound" für die Konfiguration &nbsp;$\rm B$.
 |type="{}"}
 $p_{\rm UB} \ = \ ${ 12.1 3% } $\ \%$
-{Berechnen Sie die &bdquo;gemittelte Union Bound" für die Konfiguration &nbsp;$\rm C$.
+{Berechnen Sie die "gemittelte Union Bound" für die Konfiguration &nbsp;$\rm C$.
 |type="{}"}
 $p_{\rm UB} \ = \ ${ 3.2 3% } $\ \%$