Aufgaben:Exercise 4.2: Channel Log Likelihood Ratio at AWGN - Revision history

Guenter at 14:21, 29 November 2022

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Guenter at 16:06, 28 November 2022

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Guenter at 15:36, 28 November 2022

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Noah at 16:36, 27 October 2022

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Noah at 16:36, 27 October 2022

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Noah at 16:36, 27 October 2022

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Guenter: Guenter moved page Aufgaben:Exercise 4.2: Channel LLR at AWGN to Aufgaben:Exercise 4.2: Channel Log Likelihood Ratio at AWGN

2022-10-13T08:56:24Z

Guenter moved page Exercise 4.2: Channel LLR at AWGN to Exercise 4.2: Channel Log Likelihood Ratio at AWGN

@@ Line 45: / Line 45: @@
 + They describe the binary transmission under Gaussian noise.
 + The bit error probability without coding is&nbsp; ${\rm Q}(1/\sigma)$.
-+ The channel log likelyhood ratio is given as&nbsp; $L_{\rm K}(y) = K_{\rm L} \cdot y$.
++ The channel log likelihood ratio is given as&nbsp; $L_{\rm K}(y) = K_{\rm L} \cdot y$.
 {Which constant&nbsp; $K_{\rm L}$&nbsp; characterizes the channel&nbsp; $\rm A$?
@@ Line 96: / Line 96: @@
-'''(3)'''&nbsp; The correct <u>solutions are 1 to 4</u>:
+'''(3)'''&nbsp; Correct are the&nbsp; <u>solutions 1 to 4</u>:
-*We first give the respective LLRs of '''Channel A''':
+*We first give the respective log likelihood ratios of&nbsp; '''Channel A''':
 :$$L_{\rm K}(y_1 = +1.0) = +2\hspace{0.05cm},\hspace{0.3cm}
 L_{\rm K}(y_2 = +0.5) = +1\hspace{0.05cm},\hspace{0.3cm}
 L_{\rm K}(y_3 = -1.5) = -3\hspace{0.05cm}. $$
 *This results in the following consequences:
-# The decision for the (most probable) codebit $x_i$ is made based on the sign of $L_{\rm K}(y_i)$: $x_1 = +1, \ x_2 = +1, \ x_3 = \, -1$ &nbsp; &#8658; &nbsp; the <u>proposed solutions 1, 2 and 3</u> are correct.
+# The decision for the&nbsp; $($most probable$)$&nbsp; code bit&nbsp; $x_i$&nbsp; is based on the sign of&nbsp; $L_{\rm K}(y_i)$: <br> &nbsp; &nbsp; $x_1 = +1, \ x_2 = +1, \ x_3 = \, -1$ &nbsp; &#8658; &nbsp; the&nbsp; <u>proposed solutions 1, 2 and 3</u>&nbsp; are correct.
-# The decision "$x_1 = +1$" is more reliable than the decision "$x_2 = +1$" &nbsp; &#8658; &nbsp; <u>Proposition 4</u> is also correct.
+# The decision&nbsp; "$x_1 = +1$"&nbsp; is more reliable than the decision&nbsp; "$x_2 = +1$" &nbsp; &#8658; &nbsp; <u>Proposition 4</u>&nbsp; is also correct.
-# However, the decision "$x_1 = +1$" is less reliable than the decision "$x_3 = \, &ndash;1$" because $|L_{\rm K}(y_1)|$ is smaller than $|L_{\rm K}(y_3)|$ &nbsp; &#8658; &nbsp; Proposed solution 5 is incorrect.
+# However,&nbsp; the decision&nbsp; "$x_1 = +1$"&nbsp; is less reliable than the decision&nbsp; "$x_3 = \, &ndash;1$"&nbsp; because&nbsp; $|L_{\rm K}(y_1)<|L_{\rm K}(y_3)|$ &nbsp; &#8658; &nbsp; proposed solution 5 is incorrect.
-'''(5)'''&nbsp; For '''channel B''', the following applies: &nbsp; $L_{\rm K}(y_1 = +1.0) = +8, \ L_{\rm K}(y_2 = +0.5) = +4$ und $L_{\rm K}(y_3 = \, -1.5) = \, -12$.
+*It is obvious that&nbsp; <u>the first two proposed solutions</u>&nbsp; are true,&nbsp; but not the third,&nbsp; because
 :$$|L_{\rm K}(y_3 = -1.5, {\rm channel\hspace{0.15cm} A)}| = 3
-\hspace{0.5cm} <\hspace{0.5cm}
+\hspace{0.2cm} <\hspace{0.2cm}
 |L_{\rm K}(y_2 = 0.5, {\rm channel\hspace{0.15cm} B)}| = 4\hspace{0.05cm} . $$
 {{ML-Fuß}}

← Older revision		Revision as of 16:06, 28 November 2022
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	:$$\frac{f_{\rm G}( y = \sigma)}{f_{\rm G}( y = 0)} = {\rm e} ^{ - y^2/(2\sigma^2) } \Bigg \|_{\hspace{0.05cm} y \hspace{0.05cm} = \hspace{0.05cm} \sigma} = {\rm e} ^{ -0.5} \approx 0.6065\hspace{0.05cm}.$$		:$$\frac{f_{\rm G}( y = \sigma)}{f_{\rm G}( y = 0)} = {\rm e} ^{ - y^2/(2\sigma^2) } \Bigg \|_{\hspace{0.05cm} y \hspace{0.05cm} = \hspace{0.05cm} \sigma} = {\rm e} ^{ -0.5} \approx 0.6065\hspace{0.05cm}.$$

−	*This means:  At the abscissa value  $y = \sigma$  the mean-free Gaussian function  $f_{\rm G}(y)$  has decayed to  $60.65\%$  of its maximum value.  Thus,  for the constant at  '''channel A''':   $K_{\rm L} = 2/\sigma^2 \ \underline{= 2}$.	+	*This means:  At the abscissa value  $y = \sigma$  the mean-free Gaussian function  $f_{\rm G}(y)$  has decayed to  $60.65\%$  of its maximum value.
		+
		+	*Thus,  for the constant at  '''channel A''':   $K_{\rm L} = 2/\sigma^2 \ \underline{= 2}$.

@@ Line 72: / Line 72: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>All proposed solutions</u> are correct:
+'''(1)'''&nbsp; <u>All proposed solutions</u>&nbsp; are correct:
-* The transfer equation is always $y = x + n$, with $x &#8712; \{+1, \, -1\}$; $n$ gives a Gaussian random variable with variance $\sigma$ &nbsp; &#8658; &nbsp; variance $\sigma^2$ &nbsp; &#8658; &nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| "AWGN Channel"]].
+* The transfer equation is always&nbsp; $y = x + n$,&nbsp; with&nbsp; $x &#8712; \{+1, \, -1\}$.&nbsp;
-'''(2)'''&nbsp; For the AWGN channel, $L_{\rm K}(y) = K_{\rm L} \cdot y$ with constant $K_{\rm L} = 2/\sigma^2$. The standard deviation $\sigma$ can be read from the graph on the data page as the distance of the inflection points within the Gaussian curves from their respective midpoints. For '''channel A''', $\sigma = 1$ results.
+* The&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Error_probability_with_Gaussian_noise|"AWGN bit error probability"]]&nbsp; is calculated using the standard deviation &nbsp; $\sigma$ &nbsp; to &nbsp; ${\rm Q}(1/\sigma)$ &nbsp; where&nbsp; ${\rm Q}(x)$&nbsp; denotes the&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"complementary Gaussian error function"]].
 *The same result is obtained by evaluating the Gaussian function
 :$$\frac{f_{\rm G}( y = \sigma)}{f_{\rm G}( y = 0)} = {\rm e} ^{ -  y^2/(2\sigma^2) } \Bigg |_{\hspace{0.05cm} y \hspace{0.05cm} = \hspace{0.05cm} \sigma} = {\rm e} ^{ -0.5} \approx 0.6065\hspace{0.05cm}.$$
-*This means: At the abscissa value $y = \sigma$ the mean-free Gaussian function $f_{\rm G}(y)$ has decayed to $60.65\%$ of its maximum value. Thus, for the constant at '''channel A''': &nbsp; $K_{\rm L} = 2/\sigma^2 \ \underline{= 2}$.
+*This means:&nbsp; At the abscissa value&nbsp; $y = \sigma$&nbsp; the mean-free Gaussian function&nbsp; $f_{\rm G}(y)$&nbsp; has decayed to&nbsp; $60.65\%$&nbsp; of its maximum value.&nbsp; Thus,&nbsp; for the constant at&nbsp; '''channel A''': &nbsp; $K_{\rm L} = 2/\sigma^2 \ \underline{= 2}$.

@@ Line 43: / Line 43: @@
 {What are the characteristics of the channels shown in the diagram?
 |type="[]"}
-+ They describe the binary transmission under Gaussian interference.
++ They describe the binary transmission under Gaussian noise.
 + The bit error probability without coding is&nbsp; ${\rm Q}(1/\sigma)$.
-+ The channel LLR is given as&nbsp; $L_{\rm K}(y) = K_{\rm L} \cdot y$&nbsp; representable.
++ The channel log likelyhood ratio is given as&nbsp; $L_{\rm K}(y) = K_{\rm L} \cdot y$.
-{Which constant $K_{\rm L}$ characterizes the channel&nbsp; $\rm A$?
+{Which constant&nbsp; $K_{\rm L}$&nbsp; characterizes the channel&nbsp; $\rm A$?
 |type="{}"}
 $K_{\rm L} \ = \ ${ 2 3% }
-{For channel&nbsp; $\rm A$&nbsp; what information do the received values&nbsp; $y_1 = 1, \ y_2 = 0.5$,&nbsp; $y_3 = \, -1.5$&nbsp; provide about the transmitted binary symbols&nbsp; $x_1, \ x_2$&nbsp; and&nbsp; $x_3$, respectively?
+{For channel&nbsp; $\rm A$&nbsp; what information do the received values&nbsp; $y_1 = 1, \ y_2 = 0.5$,&nbsp; $y_3 = \, -1.5$&nbsp; provide about the transmitted binary symbols&nbsp; $x_1, \ x_2$&nbsp; and&nbsp; $x_3$?
 |type="[]"}
 + $y_1 = 1.0$&nbsp; states that probably&nbsp; $x_1 = +1$&nbsp; was sent.
 + $y_2 = 0.5$&nbsp; states that probably&nbsp; $x_2 = +1$&nbsp; was sent.
 + $y_3 = \, -1.5$&nbsp; states that probably&nbsp; $x_3 = \, -1$&nbsp; was sent.
-+ The decision&nbsp; "$y_1 &#8594; x_1$"&nbsp; is more certain than&nbsp; "$y_2 &#8594; x_2$".
++ The decision&nbsp; "$y_1 &#8594; x_1$"&nbsp; is safer than&nbsp; "$y_2 &#8594; x_2$".
 - The decision&nbsp; "$y_1 &#8594; x_1$"&nbsp; is safer than&nbsp; "$y_3 &#8594; x_3$".
@@ Line 63: / Line 63: @@
 $K_{\rm L} \ = \ ${ 8 3% }
-{What information does channel&nbsp; $\rm B$&nbsp; provide about the received values&nbsp; $y_1 = 1, \ y_2 = 0.5$,&nbsp; $y_3 = -1.5$&nbsp; about the transmitted binary symbols&nbsp; $x_1, \ x_2$&nbsp; respectively.&nbsp; $x_3$?
+{What information does channel&nbsp; $\rm B$&nbsp; provide about the received values&nbsp; $y_1 = 1, \ y_2 = 0.5$,&nbsp; $y_3 = -1.5$&nbsp; about the transmitted binary symbols&nbsp; $x_1, \ x_2$&nbsp; and&nbsp; $x_3$?
 |type="[]"}
 + For&nbsp; $x_1, \ x_2, \ x_3$&nbsp; is decided the same as for channel&nbsp; $\rm A$.
 + The estimate&nbsp; "$x_2 = +1$"&nbsp; is four times more certain than for channel&nbsp; $\rm A$.
-- The estimate&nbsp; "$x_3 = \, -1$"&nbsp; at channel&nbsp; $\rm A$&nbsp; is more reliable than the estimate&nbsp; "$x_2 = +1$" at channel&nbsp; $\rm B$.
+- The estimate&nbsp; "$x_3 = \, -1$"&nbsp; at channel&nbsp; $\rm A$&nbsp; is more reliable than the estimate&nbsp; "$x_2 = +1$"&nbsp; at channel&nbsp; $\rm B$.
 </quiz>

← Older revision		Revision as of 15:37, 28 November 2022
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	* This exercise belongs to the chapter  [[Channel_Coding/Soft-in_Soft-Out_Decoder\| "Soft–in Soft–out Decoder"]].		* This exercise belongs to the chapter  [[Channel_Coding/Soft-in_Soft-Out_Decoder\| "Soft–in Soft–out Decoder"]].

−	* Reference is made in particular to the sections  [[Channel_Coding/Soft-in_Soft-Out_Decoder#Reliability_information_-_Log_Likelihood_Ratio\|"Reliability Information – Log Likelihood Ratio"]]  ~~and ~~ [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input\|"AWGN–Channel at Binary Input"]].	+	* Reference is made in particular to the sections
		+	:*[[Channel_Coding/Soft-in_Soft-Out_Decoder#Reliability_information_-_Log_Likelihood_Ratio\|"Reliability Information – Log Likelihood Ratio"]],
		+
		+	:* [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input\|"AWGN–Channel at Binary Input"]].

@@ Line 2: / Line 2: @@
 [[File:P_ID2980__KC_A_4_2_v2.png|right|frame|Conditional Gaussian functions]]
-We consider two channels&nbsp; $\rm A$&nbsp; and&nbsp; $\rm B$&nbsp;, each with.
+We consider two channels&nbsp; $\rm A$&nbsp; and&nbsp; $\rm B$,&nbsp; each with
-* binary bipolar input&nbsp; $x &#8712; \{+1, \, -1\}$, and
+* binary bipolar input&nbsp; $x &#8712; \{+1, \, -1\}$,&nbsp; and
 * continuous-valued output&nbsp; $y &#8712; {\rm \mathcal{R}}$&nbsp; (real number).
 The graph shows for both channels
-* as blue curve the density functions&nbsp; $f_{y\hspace{0.05cm}|\hspace{0.05cm}x=+1}$,
+* as blue curve the probability density functions&nbsp; $f_{y\hspace{0.05cm}|\hspace{0.05cm}x=+1}$,
-In&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Reliability_information_-_Log_Likelihood_Ratio| "theory section"]]&nbsp; the channel LLR was derived for this AWGN constellation as follows:
 :$$L_{\rm K}(y) = L(y\hspace{0.05cm}|\hspace{0.05cm}x) =  {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}(y \hspace{0.05cm}|\hspace{0.05cm}x=+1) }{{\rm Pr}(y \hspace{0.05cm}|\hspace{0.05cm}x = -1)}
 \hspace{0.05cm}.$$
-Evaluating this equation analytically, we obtain with the proportionality constant&nbsp; $K_{\rm L} = 2/\sigma^2$:
+Evaluating this equation analytically,&nbsp; we obtain with the proportionality constant&nbsp; $K_{\rm L} = 2/\sigma^2$:
 :$$L_{\rm K}(y) =
 K_{\rm L} \cdot y
@@ Line 25: / Line 27: @@
+* Reference is made in particular to the sections&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Reliability_information_-_Log_Likelihood_Ratio|"Reliability Information &ndash; Log Likelihood Ratio"]]&nbsp; and&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"AWGN&ndash;Channel at Binary Input"]].

@@ Line 30: / Line 30: @@
 Hints:
 * This exercise belongs to the chapter&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder| "Soft&ndash;in Soft&ndash;out Decoder"]].
-* Reference is made in particular to the pages&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Reliability_information_-_Log_Likelihood_Ratio|"Reliability Information &ndash; Log Likelihood Ratio"]]&nbsp; and&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"AWGN&ndash;Channel at Binary Input"].
+* Reference is made in particular to the pages&nbsp; [[Channel_Coding/Soft-in_Soft-Out_Decoder#Reliability_information_-_Log_Likelihood_Ratio|"Reliability Information &ndash; Log Likelihood Ratio"]]&nbsp; and&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input|"AWGN&ndash;Channel at Binary Input"]].

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