Digital Signal Transmission/Approximation of the Error Probability - Revision history

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2026-03-16T13:46:45Z

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	*The task of the decision is to give an estimate for the current received vector  $\boldsymbol{r}$  according to the  [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design\|"MAP decision rule"]].  In the present case,  this rule is with  $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:		*The task of the decision is to give an estimate for the current received vector  $\boldsymbol{r}$  according to the  [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design\|"MAP decision rule"]].  In the present case,  this rule is with  $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:
	:$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}\|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}\|\hspace{0.05cm} m_i )\big ]		:$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}\|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}\|\hspace{0.05cm} m_i )\big ]\hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$
	\hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$
	*In the special case  $N = 2$  and  $M = 2$  considered here,  the decision partitions the two-dimensional space into the two disjoint areas  $I_0$  (highlighted in red)  and  $I_1$  (blue),  as the graphic on the right illustrates.		*In the special case  $N = 2$  and  $M = 2$  considered here,  the decision partitions the two-dimensional space into the two disjoint areas  $I_0$  (highlighted in red)  and  $I_1$  (blue),  as the graphic on the right illustrates.

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	⇒   Always choose message  $m_0$  if the following condition is satisfied:		⇒   Always choose message  $m_0$  if the following condition is satisfied:

	:$${\rm Pr}( m_0) \cdot {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot \vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 \right ]		:$${\rm Pr}( m_0) \cdot {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot \vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 \right ]> {\rm Pr}( m_1) \cdot {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 \right ]\hspace{0.05cm}.$$
	> {\rm Pr}( m_1) \cdot {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 \right ]
	\hspace{0.05cm}.$$

	⇒   The boundary line between the two decision regions  $I_0$  and  $I_1$  is obtained by replacing the  "greater sign"  with the  "equals sign"  in the above equation and transforming the equation slightly:		⇒   The boundary line between the two decision regions  $I_0$  and  $I_1$  is obtained by replacing the  "greater sign"  with the  "equals sign"  in the above equation and transforming the equation slightly:

	:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 - 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_0)\big ] =		:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 - 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_0)\big ] =\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 - 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_1)\big ]$$
	\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 - 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_1)\big ]$$		:$$\Rightarrow \hspace{0.3cm} \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2+ 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_0)}{ {\rm Pr}( m_1)} = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
	:$$\Rightarrow \hspace{0.3cm} \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2
	+ 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_0)}{ {\rm Pr}( m_1)} = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$

	From the plot above one can see:		From the plot above one can see:
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	Let  ${\rm Pr}(m_0) = {\rm Pr}(m_1) = 0.5$.  For AWGN noise with standard deviation  $\sigma_n$,  as already calculated in the section  [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability\|"Definition of the bit error probability"]]  with different nomenclature,  we obtain for the probability of a wrong decision  $(\cal E)$  under the condition that message  $m_0$  was sent:		Let  ${\rm Pr}(m_0) = {\rm Pr}(m_1) = 0.5$.  For AWGN noise with standard deviation  $\sigma_n$,  as already calculated in the section  [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability\|"Definition of the bit error probability"]]  with different nomenclature,  we obtain for the probability of a wrong decision  $(\cal E)$  under the condition that message  $m_0$  was sent:

	:$${\rm Pr}({ \cal E}\hspace{0.05cm} \vert \hspace{0.05cm} m_0) = \int_{-\infty}^{G = 0} p_{r \hspace{0.05cm}\|\hspace{0.05cm}m_0 } ({ \rho } \hspace{0.05cm} \vert \hspace{0.05cm}m_0 ) \,{\rm d} \rho = \int_{-\infty}^{- s_0 } p_{{ n} \hspace{0.05cm}\vert\hspace{0.05cm}m_0 } ({ \eta } \hspace{0.05cm}\|\hspace{0.05cm}m_0 ) \,{\rm d} \eta = \int_{-\infty}^{- s_0 } p_{{ n} } ({ \eta } ) \,{\rm d} \eta =		:$${\rm Pr}({ \cal E}\hspace{0.05cm} \vert \hspace{0.05cm} m_0) = \int_{-\infty}^{G = 0} p_{r \hspace{0.05cm}\|\hspace{0.05cm}m_0 } ({ \rho } \hspace{0.05cm} \vert \hspace{0.05cm}m_0 ) \,{\rm d} \rho = \int_{-\infty}^{- s_0 } p_{{ n} \hspace{0.05cm}\vert\hspace{0.05cm}m_0 } ({ \eta } \hspace{0.05cm}\|\hspace{0.05cm}m_0 ) \,{\rm d} \eta = \int_{-\infty}^{- s_0 } p_{{ n} } ({ \eta } ) \,{\rm d} \eta =\int_{ s_0 }^{\infty} p_{{ n} } ({ \eta } ) \,{\rm d} \eta = {\rm Q} \left ( {s_0 }/{\sigma_n} \right )\hspace{0.05cm}.$$
	\int_{ s_0 }^{\infty} p_{{ n} } ({ \eta } ) \,{\rm d} \eta = {\rm Q} \left ( {s_0 }/{\sigma_n} \right )
	\hspace{0.05cm}.$$

	In deriving the equation,  it was considered that the AWGN noise  $\eta$  is independent of the signal  $(m_0$  or  $m_1)$  and has a symmetric PDF.  The complementary Gaussian error integral was also used:		In deriving the equation,  it was considered that the AWGN noise  $\eta$  is independent of the signal  $(m_0$  or  $m_1)$  and has a symmetric PDF.  The complementary Gaussian error integral was also used:
	:$${\rm Q}(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} {\rm e}^{-u^2/2} \,{\rm d} u		:$${\rm Q}(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} {\rm e}^{-u^2/2} \,{\rm d} u\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	Correspondingly,  for  $m = m_1$   ⇒   $s = s_1 = -2 \cdot E^{1/2}$:		Correspondingly,  for  $m = m_1$   ⇒   $s = s_1 = -2 \cdot E^{1/2}$:
	:$${\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = \int_{0}^{\infty} p_{{ r} \hspace{0.05cm}\vert\hspace{0.05cm}m_1 } ({ \rho } \hspace{0.05cm}\vert\hspace{0.05cm}m_1 ) \,{\rm d} \rho = \int_{- s_1 }^{\infty} p_{{ n} } (\boldsymbol{ \eta } ) \,{\rm d} \eta = {\rm Q} \left ( {- s_1 }/{\sigma_n} \right )		:$${\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = \int_{0}^{\infty} p_{{ r} \hspace{0.05cm}\vert\hspace{0.05cm}m_1 } ({ \rho } \hspace{0.05cm}\vert\hspace{0.05cm}m_1 ) \,{\rm d} \rho = \int_{- s_1 }^{\infty} p_{{ n} } (\boldsymbol{ \eta } ) \,{\rm d} \eta = {\rm Q} \left ( {- s_1 }/{\sigma_n} \right )\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$  With the distance  $d = s_1 - s_0$  of the signal space points, we can summarize the results, still considering  ${\rm Pr}(m_0) + {\rm Pr}(m_1) = 1$:		$\text{Conclusion:}$  With the distance  $d = s_1 - s_0$  of the signal space points, we can summarize the results, still considering  ${\rm Pr}(m_0) + {\rm Pr}(m_1) = 1$:
	:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0) = {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = {\rm Q} \big ( {d}/(2{\sigma_n}) \big )$$		:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0) = {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = {\rm Q} \big ( {d}/(2{\sigma_n}) \big )$$
	:$$\Rightarrow \hspace{0.3cm}{\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_0) + {\rm Pr}(m_1) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1)= \big [ {\rm Pr}(m_0) + {\rm Pr}(m_1) \big ] \cdot		:$$\Rightarrow \hspace{0.3cm}{\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_0) + {\rm Pr}(m_1) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1)= \big [ {\rm Pr}(m_0) + {\rm Pr}(m_1) \big ] \cdot{\rm Q} \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] \hspace{0.05cm}.$$
	{\rm Q} \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] \hspace{0.05cm}.$$

	<u>Notes:</u>		<u>Notes:</u>
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	*The error probability is then		*The error probability is then

	:$${\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Q} \big[ {d}/(2{\sigma_n}) \cdot (1 - \gamma_{\rm opt}) \big ]		:$${\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Q} \big[ {d}/(2{\sigma_n}) \cdot (1 - \gamma_{\rm opt}) \big ]+ {\rm Pr}(m_1) \cdot {\rm Q} \big [ {d}/(2{\sigma_n}) \cdot (1 + \gamma_{\rm opt}) \big ]\hspace{0.05cm}.$$
	+ {\rm Pr}(m_1) \cdot {\rm Q} \big [ {d}/(2{\sigma_n}) \cdot (1 + \gamma_{\rm opt}) \big ]\hspace{0.05cm}.$$

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
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	*With these definitions,  the probability of a correct decision is:		*With these definitions,  the probability of a correct decision is:

	:$${\rm Pr}({ \cal C} ) \hspace{-0.1cm} = \hspace{-0.1cm} \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}\|\hspace{0.05cm} m_i ) = \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}\|\hspace{0.05cm} m_i ) = \sum_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot		:$${\rm Pr}({ \cal C} ) \hspace{-0.1cm} = \hspace{-0.1cm} \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}\|\hspace{0.05cm} m_i ) = \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}\|\hspace{0.05cm} m_i ) = \sum_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot\int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}\|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}\|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho } \hspace{0.05cm}.$$
	\int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}\|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}\|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho } \hspace{0.05cm}.$$

	*For the AWGN channel,  this is according to the section  [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#N-dimensional_Gaussian_noise\| "N–dimensional Gaussian noise"]]:		*For the AWGN channel,  this is according to the section  [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#N-dimensional_Gaussian_noise\| "N–dimensional Gaussian noise"]]:
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	*The following applies to the symbol error probability:		*The following applies to the symbol error probability:
	:$${\rm Pr}({ \cal E} ) = 1 - {\rm Pr}({ \cal C} ) \hspace{0.05cm},$$		:$${\rm Pr}({ \cal E} ) = 1 - {\rm Pr}({ \cal C} ) \hspace{0.05cm},$$
	:$${\rm Pr}({ \cal C} ) = {1}/{3} \cdot		:$${\rm Pr}({ \cal C} ) = {1}/{3} \cdot\big [ {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_1 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_2 ) \big ]\hspace{0.05cm}.$$
	\big [ {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_1 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_2 ) \big ]\hspace{0.05cm}.$$

	*The first term  ${\rm Pr}(\boldsymbol{r} \in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$  in the expression in brackets under the assumption  $m = m_0 \ \Leftrightarrow \ \boldsymbol{s} = \boldsymbol{s}_0$  is visualized in the left graphic by the red region  $I_0$.		*The first term  ${\rm Pr}(\boldsymbol{r} \in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$  in the expression in brackets under the assumption  $m = m_0 \ \Leftrightarrow \ \boldsymbol{s} = \boldsymbol{s}_0$  is visualized in the left graphic by the red region  $I_0$.

	*The complementary region  ${\rm Pr}(\boldsymbol{r} \not\in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$  is marked on the left with either blue or green or blue–green hatching. It applies  ${\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) = 1 - {\rm Pr}({ \cal E}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 )$  with		*The complementary region  ${\rm Pr}(\boldsymbol{r} \not\in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$  is marked on the left with either blue or green or blue–green hatching. It applies  ${\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) = 1 - {\rm Pr}({ \cal E}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 )$  with
	:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) =		:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) ={\rm Pr}(\boldsymbol{ r } \in I_1 \hspace{0.05cm}\cup \hspace{0.05cm} \boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
	{\rm Pr}(\boldsymbol{ r } \in I_1 \hspace{0.05cm}\cup \hspace{0.05cm} \boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$		:$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Pr}(\boldsymbol{ r } \in I_1 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) +{\rm Pr}(\boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
	:$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Pr}(\boldsymbol{ r } \in I_1 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) +		:$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Q} \big [ d_{01}/(2{\sigma_n}) \big ]+{\rm Q} \big [ d_{02}/(2{\sigma_n}) \big ]\hspace{0.05cm}.$$
	{\rm Pr}(\boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
	:$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Q} \big [ d_{01}/(2{\sigma_n}) \big ]+
	{\rm Q} \big [ d_{02}/(2{\sigma_n}) \big ]
	\hspace{0.05cm}.$$

	*The  "less/equal"  sign takes into account that the blue–green hatched area belongs both to the area  "$\boldsymbol{r} \in I_1$"  and to the area  "$\boldsymbol{r} \in I_2$",  so that the sum returns a value that is too large.  This means:   The Union Bound always provides an upper bound.<br>		*The  "less/equal"  sign takes into account that the blue–green hatched area belongs both to the area  "$\boldsymbol{r} \in I_1$"  and to the area  "$\boldsymbol{r} \in I_2$",  so that the sum returns a value that is too large.  This means:   The Union Bound always provides an upper bound.<br>

Maintenance script: Add German interlanguage link

2026-03-16T12:26:39Z

Add German interlanguage link

← Older revision		Revision as of 14:26, 16 March 2026
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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Approximation_der_Fehlerwahrscheinlichkeit]]

Hwang at 10:04, 17 November 2022

2022-11-17T10:04:19Z

← Older revision		Revision as of 12:04, 17 November 2022
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 1:}$  Let's look again at the signal space constellation from the  [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission\|"first chapter ~~page~~"]]  $($lower graphic$)$  with the values		$\text{Example 1:}$  Let's look again at the signal space constellation from the  [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission\|"first chapter section"]]  $($lower graphic$)$  with the values

	[[File:Dig_T_4_3_S2b_version2.png\|right\|frame\|Two signal space constellations\|class=fit]]		[[File:Dig_T_4_3_S2b_version2.png\|right\|frame\|Two signal space constellations\|class=fit]]

Guenter at 11:51, 3 September 2022

2022-09-03T11:51:19Z

Guenter at 11:47, 3 September 2022

2022-09-03T11:47:25Z

← Older revision		Revision as of 13:47, 3 September 2022
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	*The two messages  $m_0 \ \Leftrightarrow \ \boldsymbol{ s_0}$  and  $m_1 \ \Leftrightarrow \ \boldsymbol{ s_1}$  are not necessarily equally probable.<br>		*The two messages  $m_0 \ \Leftrightarrow \ \boldsymbol{ s_0}$  and  $m_1 \ \Leftrightarrow \ \boldsymbol{ s_1}$  are not necessarily equally probable.<br>

	*The task of the decision is to give an estimate for the current ~~reception '''KORREKTUR:~~ received~~'''~~ vector  $\boldsymbol{r}$  according to the  [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design\|"MAP decision rule"]].  In the present case,  this rule is with  $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:		*The task of the decision is to give an estimate for the current received vector  $\boldsymbol{r}$  according to the  [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design\|"MAP decision rule"]].  In the present case,  this rule is with  $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:
	:$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}\|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}\|\hspace{0.05cm} m_i )\big ]		:$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}\|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}\|\hspace{0.05cm} m_i )\big ]
	\hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$		\hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$
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	*The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.  <br>		*The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.  <br>

	*For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the ~~transmitting '''KORREKTUR: transmitted/~~transmission~~'''~~ points:		*For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the transmission points:

	:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$		:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$

	*For  ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,  the decision boundary is shifted toward the less ~~likely '''KORREKTUR:~~ probable~~'''~~ symbol  $\boldsymbol{ s }_1$,  and the more so the larger the AWGN standard deviation  $\sigma_n$.  <br>		*For  ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,  the decision boundary is shifted toward the less probable symbol  $\boldsymbol{ s }_1$,  and the more so the larger the AWGN standard deviation  $\sigma_n$.  <br>

	*The green-dashed decision boundary in the right figure as well as the decision regions  $I_0$  (red)  and  $I_1$  (blue)  are valid for the  (normalized)  standard deviation  $\sigma_n = 1$  and the dashed boundary lines for  $\sigma_n = 0$  resp.  $\sigma_n = 2$. <br>}}		*The green-dashed decision boundary in the right figure as well as the decision regions  $I_0$  (red)  and  $I_1$  (blue)  are valid for the  (normalized)  standard deviation  $\sigma_n = 1$  and the dashed boundary lines for  $\sigma_n = 0$  resp.  $\sigma_n = 2$. <br>}}

Hwang at 10:31, 1 September 2022

2022-09-01T10:31:13Z

← Older revision		Revision as of 12:31, 1 September 2022
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	#$\sigma_n$  specifies the rms value of the AWGN noise.<br>		#$\sigma_n$  specifies the rms value of the AWGN noise.<br>
	#The  "Union Bound"  can only be used for equally probable symbols   ⇒   ${\rm Pr}(m_i) = 1/M$.		#The  "Union Bound"  can only be used for equally probable symbols   ⇒   ${\rm Pr}(m_i) = 1/M$.
	#But also n this case,  an average must be taken over all  $m_i$ in order to calculate the  (average)  error probability.}}		#But also in this case,  an average must be taken over all  $m_i$ in order to calculate the  (average)  error probability.}}

Hwang at 10:25, 1 September 2022

2022-09-01T10:25:07Z

← Older revision		Revision as of 12:25, 1 September 2022
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	*Assuming that  $m = m_i \ \Leftrightarrow \ \boldsymbol{ s } = \boldsymbol{ s }_i$  was sent,  a correct decision is made only if the received value  $\boldsymbol{ r } \in I_i$.  <br>		*Assuming that  $m = m_i \ \Leftrightarrow \ \boldsymbol{ s } = \boldsymbol{ s }_i$  was sent,  a correct decision is made only if the received value  $\boldsymbol{ r } \in I_i$.  <br>

	*The conditional probability  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}\|\hspace{0.05cm}m_2)$  is is  (by far)  largest for  $i = 2$   ⇒   correct decision.		*The conditional probability  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}\|\hspace{0.05cm}m_2)$  is  (by far)  largest for  $i = 2$   ⇒   correct decision.

	*${\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}\|\hspace{0.05cm}m_2)$  is much smaller.  Almost negligible is  ${\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}\|\hspace{0.05cm}m_2)$.		*${\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}\|\hspace{0.05cm}m_2)$  is much smaller.  Almost negligible is  ${\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}\|\hspace{0.05cm}m_2)$.

Hwang at 10:22, 1 September 2022

2022-09-01T10:22:44Z

← Older revision		Revision as of 12:22, 1 September 2022
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	*The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.  <br>		*The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.  <br>

	*For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the transmitting '''KORREKTUR: transmitted''' points:		*For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the transmitting '''KORREKTUR: transmitted/transmission''' points:

	:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$		:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the transmitting '''KORREKTUR: transmitted''' point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.		$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the transmitting '''KORREKTUR: transmitted/transmission''' point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.
	[[File:P ID2026 Dig T 4 3 S5b version1.png\|right\|frame\|To calculate the error probability for AWGN\|class=fit]]		[[File:P ID2026 Dig T 4 3 S5b version1.png\|right\|frame\|To calculate the error probability for AWGN\|class=fit]]

Hwang at 10:12, 1 September 2022

2022-09-01T10:12:44Z

← Older revision		Revision as of 12:12, 1 September 2022
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	*The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.  <br>		*The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.  <br>

	*For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the transmitting points:		*For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the transmitting '''KORREKTUR: transmitted''' points:

	:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$		:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the transmitting point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.		$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the transmitting '''KORREKTUR: transmitted''' point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.
	[[File:P ID2026 Dig T 4 3 S5b version1.png\|right\|frame\|To calculate the error probability for AWGN\|class=fit]]		[[File:P ID2026 Dig T 4 3 S5b version1.png\|right\|frame\|To calculate the error probability for AWGN\|class=fit]]

Hwang at 09:55, 1 September 2022

2022-09-01T09:55:05Z

← Older revision		Revision as of 11:55, 1 September 2022
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	:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$		:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$

	*For  ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,  the decision boundary is shifted toward the less likely symbol  $\boldsymbol{ s }_1$,  and the more so the larger the AWGN standard deviation  $\sigma_n$.  <br>		*For  ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,  the decision boundary is shifted toward the less likely '''KORREKTUR: probable''' symbol  $\boldsymbol{ s }_1$,  and the more so the larger the AWGN standard deviation  $\sigma_n$.  <br>

	*The green-dashed decision boundary in the right figure as well as the decision regions  $I_0$  (red)  and  $I_1$  (blue)  are valid for the  (normalized)  standard deviation  $\sigma_n = 1$  and the dashed boundary lines for  $\sigma_n = 0$  resp.  $\sigma_n = 2$. <br>}}		*The green-dashed decision boundary in the right figure as well as the decision regions  $I_0$  (red)  and  $I_1$  (blue)  are valid for the  (normalized)  standard deviation  $\sigma_n = 1$  and the dashed boundary lines for  $\sigma_n = 0$  resp.  $\sigma_n = 2$. <br>}}
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	<br>		<br>
	If  ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$  holds,  a slightly smaller error probability can be obtained by shifting the decision threshold  $G$.  The following results are derived in detail in the solution to  [[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again\|"Exercise 4.7"]]:		If  ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$  holds,  a slightly smaller error probability can be obtained by shifting the decision threshold  $G$.  The following results are derived in detail in the solution to  [[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again\|"Exercise 4.7"]]:
	*For unequal symbol probabilities, the optimal decision threshold  $G_{\rm opt}$  between regions  $I_0$  and  $I_1$  is closer to the less likely symbol.  The normalized optimal shift with respect to the value  $G = 0$  for equally probable symbols is		*For unequal symbol probabilities, the optimal decision threshold  $G_{\rm opt}$  between regions  $I_0$  and  $I_1$  is closer to the less likely '''KORREKTUR: probable''' symbol.  The normalized optimal shift with respect to the value  $G = 0$  for equally probable symbols is

	::<math>\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot \frac{ \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.</math>		::<math>\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot \frac{ \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.</math>
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	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

	*This is a  $14\%$  shift toward the less likely symbol  $\boldsymbol {s}_1$  (i.e., to the left).  This makes the error probability slightly smaller than for equally likely symbols:		*This is a  $14\%$  shift toward the less likely '''KORREKTUR: probable''' symbol  $\boldsymbol {s}_1$  (i.e., to the left).  This makes the error probability slightly smaller than for equally likely '''KORREKTUR: probable''' symbols:

	::<math>{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.</math>		::<math>{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.</math>

← Older revision		Revision as of 12:04, 17 November 2022
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 1:}$  Let's look again at the signal space constellation from the  [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission\|"first chapter ~~page~~"]]  $($lower graphic$)$  with the values		$\text{Example 1:}$  Let's look again at the signal space constellation from the  [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission\|"first chapter section"]]  $($lower graphic$)$  with the values

	[[File:Dig_T_4_3_S2b_version2.png\|right\|frame\|Two signal space constellations\|class=fit]]		[[File:Dig_T_4_3_S2b_version2.png\|right\|frame\|Two signal space constellations\|class=fit]]

← Older revision		Revision as of 13:51, 3 September 2022
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	We continue to assume a binary system  $(M = 2)$,  but now consider the simple case where this can be described by a single basis function  $(N = 1)$.  The error probability for this has already been calculated in the section  [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability\|"Definition of the bit error probability"]].  <br>		We continue to assume a binary system  $(M = 2)$,  but now consider the simple case where this can be described by a single basis function  $(N = 1)$.  The error probability for this has already been calculated in the section  [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability\|"Definition of the bit error probability"]].  <br>
	With the nomenclature and representation form chosen for the fourth main chapter the following constellation results:		With the nomenclature and representation form chosen for the fourth main chapter the following constellation results:
	[[File:P ID2020 Dig T 4 3 S2 version1.png\|\|right\|frame\|Conditional probability density functions for ~~equal probability '''KORREKTUR:~~ equally probable~~'''~~ symbols\|class=fit]]		[[File:P ID2020 Dig T 4 3 S2 version1.png\|\|right\|frame\|Conditional probability density functions for equally probable symbols\|class=fit]]

	*The received value  $r = s + n$  is now a scalar and is composed of the transmitted signal  $s \in \{s_0, \hspace{0.05cm}s_1\}$  and the noise term  $n$  additively. The abscissa  $\rho$  denotes a realization of  $r$.<br>		*The received value  $r = s + n$  is now a scalar and is composed of the transmitted signal  $s \in \{s_0, \hspace{0.05cm}s_1\}$  and the noise term  $n$  additively. The abscissa  $\rho$  denotes a realization of  $r$.<br>
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	<br>		<br>
	If  ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$  holds,  a slightly smaller error probability can be obtained by shifting the decision threshold  $G$.  The following results are derived in detail in the solution to  [[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again\|"Exercise 4.7"]]:		If  ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$  holds,  a slightly smaller error probability can be obtained by shifting the decision threshold  $G$.  The following results are derived in detail in the solution to  [[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again\|"Exercise 4.7"]]:
	*For unequal symbol probabilities, the optimal decision threshold  $G_{\rm opt}$  between regions  $I_0$  and  $I_1$  is closer to the less ~~likely '''KORREKTUR:~~ probable~~'''~~ symbol.  The normalized optimal shift with respect to the value  $G = 0$  for equally probable symbols is		*For unequal symbol probabilities, the optimal decision threshold  $G_{\rm opt}$  between regions  $I_0$  and  $I_1$  is closer to the less probable symbol.  The normalized optimal shift with respect to the value  $G = 0$  for equally probable symbols is

	::<math>\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot \frac{ \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.</math>		::<math>\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot \frac{ \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.</math>
Line 172:		Line 172:
	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

	*This is a  $14\%$  shift toward the less ~~likely '''KORREKTUR:~~ probable~~'''~~ symbol  $\boldsymbol {s}_1$  (i.e., to the left).  This makes the error probability slightly smaller than for equally ~~likely '''KORREKTUR:~~ probable~~'''~~ symbols:		*This is a  $14\%$  shift toward the less probable symbol  $\boldsymbol {s}_1$  (i.e., to the left).  This makes the error probability slightly smaller than for equally probable symbols:

	::<math>{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.</math>		::<math>{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.</math>
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the ~~transmitting '''KORREKTUR: transmitted/~~transmission~~'''~~ point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.		$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the transmission point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.
	[[File:P ID2026 Dig T 4 3 S5b version1.png\|right\|frame\|To calculate the error probability for AWGN\|class=fit]]		[[File:P ID2026 Dig T 4 3 S5b version1.png\|right\|frame\|To calculate the error probability for AWGN\|class=fit]]

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	$\text{Example 5:}$  The graphic illustrates the  <b>Union Bound</b>  using the example of  $M = 3$  with equally probable symbols:   ${\rm Pr}(m_0) = {\rm Pr}(m_1) = {\rm Pr}(m_2) =1/3$.<br>		$\text{Example 5:}$  The graphic illustrates the  <b>Union Bound</b>  using the example of  $M = 3$  with equally probable symbols:   ${\rm Pr}(m_0) = {\rm Pr}(m_1) = {\rm Pr}(m_2) =1/3$.<br>

	[[File:P ID2041 Dig T 4 3 S6 version1.png\|right\|frame\|To clarify the "Union Bound" ~~'''Korrektur''' schraffiert~~\|class=fit]]		[[File:P ID2041 Dig T 4 3 S6 version1.png\|right\|frame\|To clarify the "Union Bound" \|class=fit]]
	The following should be noted about these representations:		The following should be noted about these representations:
	*The following applies to the symbol error probability:		*The following applies to the symbol error probability: