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

Hwang at 10:04, 17 November 2022

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Guenter at 11:51, 3 September 2022

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Hwang at 10:31, 1 September 2022

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Hwang at 09:34, 1 September 2022

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Hwang at 09:23, 1 September 2022

2022-09-01T09:23:16Z

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 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; Let's look again at the signal space constellation from the&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission|"first chapter page"]]&nbsp; $($lower graphic$)$&nbsp; with the values
+$\text{Example 1:}$&nbsp; Let's look again at the signal space constellation from the&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission|"first chapter section"]]&nbsp; $($lower graphic$)$&nbsp; with the values
 [[File:Dig_T_4_3_S2b_version2.png|right|frame|Two signal space constellations|class=fit]]

@@ Line 62: / Line 62: @@
 We continue to assume a binary system&nbsp; $(M = 2)$,&nbsp; but now consider the simple case where this can be described by a single basis function&nbsp;  $(N = 1)$.&nbsp; The error probability for this has already been calculated in the section&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability|"Definition of the bit error probability"]].&nbsp; <br>
 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&nbsp; $r = s + n$&nbsp; is now a scalar and is composed of the transmitted signal&nbsp; $s \in \{s_0, \hspace{0.05cm}s_1\}$&nbsp; and the noise term&nbsp; $n$&nbsp; additively. The abscissa&nbsp; $\rho$&nbsp; denotes a realization of&nbsp; $r$.<br>
@@ Line 146: / Line 146: @@
 <br>
 If&nbsp; ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$&nbsp; holds,&nbsp; a slightly smaller error probability can be obtained by shifting the decision threshold&nbsp; $G$.&nbsp; The following results are derived in detail in the solution to&nbsp; [[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again|"Exercise 4.7"]]:&nbsp;
-*For unequal symbol probabilities, the optimal decision threshold&nbsp; $G_{\rm opt}$&nbsp; between regions&nbsp; $I_0$&nbsp; and&nbsp; $I_1$&nbsp; is closer to the less likely '''KORREKTUR: probable''' symbol.&nbsp; The normalized optimal shift with respect to the value&nbsp; $G = 0$&nbsp; for equally probable symbols is
+*For unequal symbol probabilities, the optimal decision threshold&nbsp; $G_{\rm opt}$&nbsp; between regions&nbsp; $I_0$&nbsp; and&nbsp; $I_1$&nbsp; is closer to the less probable symbol.&nbsp; The normalized optimal shift with respect to the value&nbsp; $G = 0$&nbsp; 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>
@@ Line 172: / Line 172: @@
 \hspace{0.05cm}.</math>
-*This is a&nbsp; $14\%$&nbsp; shift toward the less likely '''KORREKTUR: probable''' symbol&nbsp; $\boldsymbol {s}_1$&nbsp; (i.e., to the left).&nbsp; This makes the error probability slightly smaller than for equally likely '''KORREKTUR: probable''' symbols:
+*This is a&nbsp; $14\%$&nbsp; shift toward the less probable symbol&nbsp; $\boldsymbol {s}_1$&nbsp; (i.e., to the left).&nbsp; 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>
@@ Line 245: / Line 245: @@
 {{GraueBox|TEXT=
-$\text{Example 4:}$&nbsp; For the AWGN channel,&nbsp; there is a two-dimensional Gaussian bell around the transmitting '''KORREKTUR: transmitted/transmission''' point&nbsp; $\boldsymbol{ s }_i$,&nbsp; recognizable in the left graphic by the concentric contour lines.
+$\text{Example 4:}$&nbsp; For the AWGN channel,&nbsp; there is a two-dimensional Gaussian bell around the transmission point&nbsp; $\boldsymbol{ s }_i$,&nbsp; 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]]
@@ Line 329: / Line 329: @@
 $\text{Example 5:}$&nbsp; The graphic illustrates the &nbsp;<b>Union Bound</b>&nbsp; using the example of &nbsp;$M = 3$&nbsp; with equally probable symbols: &nbsp; ${\rm Pr}(m_0) = {\rm Pr}(m_1) =  {\rm Pr}(m_2) =1/3$.<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 applies to the symbol error probability:

@@ Line 22: / Line 22: @@
 *The two messages&nbsp; $m_0 \ \Leftrightarrow \ \boldsymbol{ s_0}$&nbsp; and &nbsp;$m_1 \ \Leftrightarrow \ \boldsymbol{ s_1}$&nbsp; are not necessarily equally probable.<br>
-*The task of the decision is to give an estimate for the current reception '''KORREKTUR: received''' vector&nbsp; $\boldsymbol{r}$&nbsp; according to the&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design|"MAP decision rule"]].&nbsp; In the present case,&nbsp; this rule is with&nbsp; $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:
+*The task of the decision is to give an estimate for the current received vector&nbsp; $\boldsymbol{r}$&nbsp; according to the&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design|"MAP decision rule"]].&nbsp; In the present case,&nbsp; this rule is with&nbsp; $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:
 :$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}|\hspace{0.05cm} m_i )\big ]
 \hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$
@@ Line 50: / Line 50: @@
 *The boundary curve between regions&nbsp; $I_0$&nbsp; and&nbsp; $I_1$&nbsp; is a straight line,&nbsp; since the equation of determination is linear in the received vector&nbsp; $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.&nbsp; <br>
-*For equally probable symbols,&nbsp; the boundary is exactly halfway between&nbsp; $\boldsymbol{ s }_0$&nbsp; and&nbsp; $\boldsymbol{ s }_1$&nbsp; and rotated by &nbsp;$90^\circ$&nbsp; with respect to the line connecting the transmitting '''KORREKTUR: transmitted/transmission''' points:
+*For equally probable symbols,&nbsp; the boundary is exactly halfway between&nbsp; $\boldsymbol{ s }_0$&nbsp; and&nbsp; $\boldsymbol{ s }_1$&nbsp; and rotated by &nbsp;$90^\circ$&nbsp; with respect to the line connecting the transmission points:
 :$$\vert \hspace{-0.05cm} \vert  \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2  - \vert \hspace{-0.05cm} \vert   \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2  = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
-*For&nbsp; ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,&nbsp; the decision boundary is shifted toward the less likely '''KORREKTUR: probable''' symbol&nbsp; $\boldsymbol{ s }_1$,&nbsp; and the more so the larger the AWGN standard deviation&nbsp; $\sigma_n$.&nbsp; <br>
+*For&nbsp; ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,&nbsp; the decision boundary is shifted toward the less probable symbol&nbsp; $\boldsymbol{ s }_1$,&nbsp; and the more so the larger the AWGN standard deviation&nbsp; $\sigma_n$.&nbsp; <br>
 *The green-dashed decision boundary in the right figure as well as the decision regions&nbsp; $I_0$&nbsp; (red)&nbsp; and&nbsp;  $I_1$&nbsp; (blue)&nbsp; are valid for the&nbsp; (normalized)&nbsp; standard deviation&nbsp; $\sigma_n = 1$&nbsp; and the dashed boundary lines for&nbsp; $\sigma_n = 0$&nbsp; resp.&nbsp; $\sigma_n = 2$. <br>}}

@@ Line 323: / Line 323: @@
 #$\sigma_n$&nbsp; specifies the rms value of the AWGN noise.<br>
 #The&nbsp; "Union Bound"&nbsp; can only be used for equally probable symbols &nbsp; &rArr; &nbsp; ${\rm Pr}(m_i) = 1/M$.&nbsp;
-#But also n this case,&nbsp; an average must be taken over all&nbsp; $m_i$&nbsp;in order to calculate the&nbsp; (average)&nbsp; error probability.}}
+#But also in this case,&nbsp; an average must be taken over all&nbsp; $m_i$&nbsp;in order to calculate the&nbsp; (average)&nbsp; error probability.}}

@@ Line 279: / Line 279: @@
 *Assuming that&nbsp; $m = m_i \ \Leftrightarrow \ \boldsymbol{ s } = \boldsymbol{ s }_i$&nbsp; was sent,&nbsp; a correct decision is made only if the received value&nbsp; $\boldsymbol{ r } \in I_i$.&nbsp; <br>
-*The conditional probability&nbsp;  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; is is&nbsp; (by far)&nbsp; largest  for &nbsp;$i = 2$ &nbsp; &#8658; &nbsp; correct decision.&nbsp;
+*The conditional probability&nbsp;  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; is&nbsp; (by far)&nbsp; largest  for &nbsp;$i = 2$ &nbsp; &#8658; &nbsp; correct decision.&nbsp;
 *${\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; is much smaller.&nbsp; Almost negligible is&nbsp;  ${\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm}m_2)$.