Aufgaben:Exercise 3.10Z: Maximum Likelihood Decoding of Convolutional Codes - Revision history

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Guenter at 12:13, 10 October 2022

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@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Decoding_of_Convolutional_Codes}}
-[[File:P_ID2678__KC_Z_3_10.png|right|frame|Overall system model,&nbsp; given for this exercise]]
+[[File:EN_KC_Z_3_10.png|right|frame|Overall system model,&nbsp; given for this exercise]]
 The Viterbi algorithm represents the best known realization form for the maximum likelihood decoding of a convolutional code.&nbsp; We assume here the following model:
 * The information sequence&nbsp; $\underline{u}$&nbsp; is converted into the code sequence&nbsp; $\underline{x}$&nbsp; by a convolutional code.

@@ Line 82: / Line 82: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct is the <u>proposed solution 3</u>:
+'''(1)'''&nbsp; Correct is the&nbsp; <u>proposed solution 3</u>:
-*Let the two binary sequences be $\underline{x}$ and $\underline{y}$ with $x_i &#8712; \{-1, \, +1\}, \ y_i &#8712; \{-1, \, +1\}$. Let the sequence length be $L$ in each case.
+*Let the two binary sequences be &nbsp; $\underline{x}$ &nbsp; and &nbsp; $\underline{y}$ &nbsp; with &nbsp; $x_i &#8712; \{-1, \, +1\}, \ y_i &#8712; \{-1, \, +1\}$.&nbsp; Let the sequence length be&nbsp; $L$&nbsp; in each case.
 *The Hamming distance $d_{\rm H}(\underline{x}, \, \underline{y})$ gives the number of bits in which $\underline{x}$ and $\underline{y}$ differ, for which thus $x_i \, - y_i = &plusmn;2$ &nbsp; &#8658; &nbsp; $ (x_i \, - y_i)^2 = 4$ holds.
-*Equal symbols $(x_i = y_i)$ do not contribute to the Hamming&ndash;distance and give $(x_i \, &ndash; y_i)^2 = 0$. According to the <u>solution 3</u>, we can therefore write:
+*The Hamming distance &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$ &nbsp; gives the number of bits in which&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{y}$&nbsp; differ,&nbsp; for which thus&nbsp; $x_i \, - y_i = &plusmn;2$ &nbsp; &#8658; &nbsp; $ (x_i \, - y_i)^2 = 4$&nbsp; holds.
 :$$ d_{\rm H}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y}) =
 \frac{1}{4} \cdot \sum_{i=1}^{L} \hspace{0.2cm}(x_i - y_i)^2= \frac{1}{4} \cdot d_{\rm E}^2(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y})\hspace{0.05cm}.$$
-'''(2)'''&nbsp; <u>All proposed solutions</u> are correct:
+'''(2)'''&nbsp; <u>All proposed solutions</u>&nbsp; are correct:
-*In the BSC model, it is common practice to select the code word $\underline{x}$ with the smallest Hamming distance $d_{\rm H}(\underline{x}, \, \underline{y})$ for the given received vector $\underline{y}$:
+*In the BSC model,&nbsp; it is common practice to select the code word&nbsp; $\underline{x}$&nbsp; with the smallest Hamming distance&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$&nbsp; for the given received vector&nbsp; $\underline{y}$:
 :$$\underline{z} = {\rm arg} \min_{\underline{x} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm}
 d_{\rm H}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y})\hspace{0.05cm}.$$
-*But according to the subtask '''(1)''' also applies:
+*But according to the subtask&nbsp; '''(1)'''&nbsp; also applies:
 :$$\underline{z} = {\rm arg} \min_{\underline{x} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm}
 d_{\rm E}^{\hspace{0.15cm}2}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y})/4
@@ Line 103: / Line 105: @@
 \hspace{0.05cm}.$$
-*The factor $1/4$ does not matter for the minimization. Since $d_{\rm E}(\underline{x}, \, \underline{y}) &#8805; 0$, it does not matter whether the minimization is done with respect to $d_{\rm E}(\underline{x}, \, \underline{y})$ or $d_{\rm E}^2(\underline{x}, \, \underline{y})$.
+*The factor&nbsp; $1/4$&nbsp; does not matter for the minimization.&nbsp; Since&nbsp; $d_{\rm E}(\underline{x}, \, \underline{y}) &#8805; 0$,&nbsp; it does not matter whether the minimization is done with respect to &nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$ &nbsp; or &nbsp; $d_{\rm E}^2(\underline{x}, \, \underline{y})$.
-'''(3)'''&nbsp; Correct is the <u>proposed solution 2</u>:
+'''(3)'''&nbsp; Correct is the&nbsp; <u>proposed solution 2</u>:
 *The square of the Euclidean distance can be expressed as follows:
 :$$d_{\rm E}^{\hspace{0.15cm}2}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y}) =
@@ Line 115: / Line 117: @@
 \hspace{0.05cm}.$$
-*The first two summands are each equal to $L$ and need not be considered for minimization.
+*The first two summands are each equal to&nbsp; $L$&nbsp; and need not be considered for minimization.
 *For the last expression in this equation, $&ndash;2 \cdot &#9001; \underline{x}, \, \underline{y} &#9002;$ can be written.
-*Due to the negative sign, minimization becomes maximization &nbsp; &#8658; &nbsp; <u>answer 2</u>.
+*For the last expression in this equation,&nbsp; $&ndash;2 \cdot &#9001; \underline{x}, \, \underline{y} &#9002;$&nbsp; can be written.
-'''(4)'''&nbsp; Correct are <u>proposed solutions 2 and 3</u>:
+'''(4)'''&nbsp; Correct are the&nbsp; <u>proposed solutions 2 and 3</u>:
-*For the AWGN channel, unlike the BSC, no Hamming distance can be specified.
+*For the AWGN channel,&nbsp; unlike the BSC,&nbsp; no Hamming distance can be specified.
 *Based on the equation
 :$$d_{\rm E}^{\hspace{0.15cm}2}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y}) =
@@ Line 128: / Line 133: @@
 \hspace{0.1cm}-2 \cdot \sum_{i=1}^{L} \hspace{0.1cm} x_i \cdot y_i$$
-:the same statements apply for the first and last summands as for the BSC model &ndash; see subtask (3).
+:the same statements apply for the first and last summands as for the BSC model &ndash; see subtask&nbsp; '''(3)'''.
 *For the middle summand, $y_i = x_i + n_i$ and $x_i &#8712; \{&ndash;1, \, +1\}$ hold:
 :$$\sum_{i=1}^{L} \hspace{0.1cm} y_i^{\hspace{0.15cm}2} =
 \hspace{0.1cm}\sum_{i=1}^{L} \hspace{0.1cm} x_i^{\hspace{0.15cm}2} \hspace{0.1cm}+ \hspace{0.1cm}\sum_{i=1}^{L} \hspace{0.1cm} n_i^{\hspace{0.15cm}2}
 \hspace{0.1cm}+2 \cdot \sum_{i=1}^{L} \hspace{0.1cm} x_i \cdot n_i \hspace{0.05cm}.$$
-*The first summand again gives $L$, the second is proportional to the noise power, and the last term vanishes since $\underline{x}$ and $\underline{n}$ are uncorrelated.
+*The first summand gives again&nbsp; $L$,&nbsp; the second is proportional to the noise power,&nbsp; and the last term vanishes since&nbsp; $\underline{x}$&nbsp; and&nbsp; $\underline{n}$&nbsp; are uncorrelated.
 *So for minimizing $d_{\rm E}(\underline{x}, \, \underline{y})$, the sum over $y_i^2$ need not be considered since there is no relation to the code sequences $\underline{x}$.
 {{ML-Fuß}}

@@ Line 44: / Line 44: @@
 * For more information on this topic,&nbsp; see the following sections in this book:
-** [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "MAP and ML criterion"]],
+:* [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "MAP and ML criterion"]],
-** [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_BSC_channel| "ML decision at the BSC channel"]],
+:* [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_BSC_channel| "ML decision at the BSC channel"]],
-** [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_AWGN_channel| "ML decision at the AWGN channel"]],
+:* [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_AWGN_channel| "ML decision at the AWGN channel"]],
-** [[Channel_Coding/Decoding_of_Linear_Block_Codes#Block_diagram_and_requirements| "Decoding linear block codes"]].
+:* [[Channel_Coding/Decoding_of_Linear_Block_Codes#Block_diagram_and_requirements| "Decoding linear block codes"]].
@@ Line 58: / Line 58: @@
 {How are &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$ &nbsp; and &nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$ &nbsp; related in the BSC model?
 |type="()"}
-- &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}(\underline{x}, \, \underline{y})$ is valid.
+- &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}(\underline{x}, \, \underline{y})$&nbsp; is valid.
-- &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}^2(\underline{x}, \, \underline{y})$ is valid.
+- &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}^2(\underline{x}, \, \underline{y})$&nbsp; is valid.
-+ &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}^2(\underline{x}, \, \underline{y})/4$ is valid.
++ &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}^2(\underline{x}, \, \underline{y})/4$&nbsp; is valid.
 {Which of the equations describe the maximum likelihood decoding in the BSC model?&nbsp; The minimization/maximization refers alwaysto all&nbsp; $\underline{x} &#8712;\mathcal{ C}$.

@@ Line 7: / Line 7: @@
 *It is valid&nbsp; $u_i &#8712; \{0, \, 1\}$.&nbsp; In contrast,&nbsp; the code symbols are represented bipolar &nbsp; &#8658; &nbsp; $x_i &#8712; \{&ndash;1, \, +1\}$.
-* Let the channel be given by the models&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{BSC}$]]&nbsp;  &nbsp; &#8658; &nbsp; received values&nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; or&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| $\text{AWGN}$]] &nbsp; &#8658;  $y_i$&nbsp; real-valued.
+* Let the channel be given by the models&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{BSC}$]]&nbsp;  &nbsp; &#8658; &nbsp; received values&nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; or&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input| $\text{AWGN}$]] &nbsp; &#8658;  $y_i$&nbsp; real-valued.
 * Given a received sequence&nbsp; $\underline{y}$&nbsp; the Viterbi algorithm decides for the sequence&nbsp; $\underline{z}$&nbsp; according to

@@ Line 7: / Line 7: @@
 *It is valid&nbsp; $u_i &#8712; \{0, \, 1\}$.&nbsp; In contrast,&nbsp; the code symbols are represented bipolar &nbsp; &#8658; &nbsp; $x_i &#8712; \{&ndash;1, \, +1\}$.
-* Let the channel be given by the models&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80. 93_BSC|$\text{BSC}$]]&nbsp;  &nbsp; &#8658; &nbsp; received values&nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; or&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| $\text{AWGN}$]] &nbsp; &#8658;  $y_i$&nbsp; real-valued.
+* Let the channel be given by the models&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC|$\text{BSC}$]]&nbsp;  &nbsp; &#8658; &nbsp; received values&nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; or&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| $\text{AWGN}$]] &nbsp; &#8658;  $y_i$&nbsp; real-valued.
 * Given a received sequence&nbsp; $\underline{y}$&nbsp; the Viterbi algorithm decides for the sequence&nbsp; $\underline{z}$&nbsp; according to

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Channel_Coding/Decoding_of_Convolutional_Codes}}
-[[File:P_ID2678__KC_Z_3_10.png|right|frame|Overall system model <br>enoder &ndash; channel &ndash; Viterbi]]
+[[File:P_ID2678__KC_Z_3_10.png|right|frame|Overall system model,&nbsp; given for this exercise]]
-The Viterbi algorithm represents the best known realization form for the maximum likelihood decoding of a convolutional code. We assume the following model here:
+The Viterbi algorithm represents the best known realization form for the maximum likelihood decoding of a convolutional code.&nbsp; We assume here the following model:
-* The information sequence&nbsp; $\underline{u}$&nbsp; is converted into the code sequence&nbsp; $\underline{x}$&nbsp; by a convolutional code. It is valid&nbsp; $u_i &#8712; \{0, \, 1\}$. In contrast, the code symbols are represented bipolar &nbsp; &#8658; &nbsp; $x_i &#8712; \{&ndash;1, \, +1\}$.
+* The information sequence&nbsp; $\underline{u}$&nbsp; is converted into the code sequence&nbsp; $\underline{x}$&nbsp; by a convolutional code.
-* Let the channel be given by the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80. 93_BSC|"BSC&ndash;Model"]]&nbsp; given &nbsp; &#8658; &nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; or the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| "AWGN&ndash;channel"]]&nbsp; provided &nbsp; &#8658; &nbsp; real-valued received values&nbsp; $y_i$.
-* Given a receive sequence&nbsp; $\underline{y}$&nbsp; the Viterbi algorithm decides on the code sequence&nbsp; $\underline{z}$&nbsp; according to
+*It is valid&nbsp; $u_i &#8712; \{0, \, 1\}$.&nbsp; In contrast,&nbsp; the code symbols are represented bipolar &nbsp; &#8658; &nbsp; $x_i &#8712; \{&ndash;1, \, +1\}$.
 :$$\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.03cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} |\hspace{0.05cm} \underline{y} ) \hspace{0.05cm}.$$
-*This corresponds to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "Maximum a posteriori"]]&nbsp; (MAP) criterion. If all information sequences&nbsp; $\underline{u}$&nbsp; are equally likely, this transitions to the somewhat simpler&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "Maximum likelihood criterion"]]&nbsp;:
+*This corresponds to the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "Maximum a posteriori"]]&nbsp; $\rm (MAP)$&nbsp; criterion.&nbsp; If all information sequences&nbsp; $\underline{u}$&nbsp; are equally likely,&nbsp; this transitions to the somewhat simpler&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "Maximum likelihood criterion"]]&nbsp; $\rm (ML)$:
 :$$\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{y}  \hspace{0.05cm}|\hspace{0.05cm} \underline{x}_{\hspace{0.03cm}i} ) \hspace{0.05cm}.$$
-*As a further result, the Viterbi&ndash;algorithm additionally outputs the sequence&nbsp; $\underline{v}$&nbsp; as an estimate for the information sequence&nbsp; $\underline{u}$&nbsp;.
+*As a further result,&nbsp; the Viterbi algorithm additionally outputs the sequence&nbsp; $\underline{v}$&nbsp; as an estimate for the information sequence&nbsp; $\underline{u}$.
-In this exercise, the relationship between the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding| "Hamming&ndash;Distance"]]&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$&nbsp; and the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_AWGN_channel| "Euclidean distance"]]
+In this exercise,&nbsp; you should determinethe relationship between the&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding| $\text{Hamming distance}$]]&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$&nbsp; and the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_AWGN_channel|$\text{Euclidean distance}$]]
 :$$d_{\rm E}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y}) =
-\sqrt{\sum_{i=1}^{L} \hspace{0.2cm}(x_i - y_i)^2}\hspace{0.05cm}$$
+\sqrt{\sum_{i=1}^{L} \hspace{0.2cm}(x_i - y_i)^2}\hspace{0.05cm}.$$
-are determined. Then, the above ML criterion is to be formulated with
+Then,&nbsp; the above maximum likelihood criterion is to be formulated with
 * the Hamming distance&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$,
 * the Euclidean distance&nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$, and
-* the&nbsp; [[Channel_Coding/Decoding_of_Convolutional_Codes#Relationship_between_Hamming_distance_and_correlation| "correlation value"]]&nbsp; $&#9001; x \cdot y &#9002;$.
+* the Euclidean distance&nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$,&nbsp; and
@@ Line 29: / Line 36: @@
-Hints:
+* For simplicity,&nbsp; "tilde"&nbsp; and&nbsp; "apostrophe"&nbsp; are omitted.
-* This exercise refers to the chapter&nbsp; [[Channel_Coding/Decoding_of_Convolutional_Codes| "Decoding of Convolutional Codes"]].
-* Reference is made in particular to the page&nbsp; [[Channel_Coding/Decoding_of_Convolutional_Codes#Viterbi_algorithm_based_on_correlation_and_metrics|"Viterbi algorithm &ndash; based on correlation and metrics"]].
+* For more information on this topic,&nbsp; see the following sections in this book:
 ** [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "MAP and ML criterion"]],
 ** [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_BSC_channel| "ML decision at the BSC channel"]],
 ** [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_AWGN_channel| "ML decision at the AWGN channel"]],
 ** [[Channel_Coding/Decoding_of_Linear_Block_Codes#Block_diagram_and_requirements| "Decoding linear block codes"]].
@@ Line 44: / Line 56: @@
 ===Questions===
 <quiz display=simple>
-{How are&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$&nbsp; and&nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$&nbsp; related in the BSC&ndash;model?
+{How are &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$ &nbsp; and &nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$ &nbsp; related in the BSC model?
 |type="()"}
 - &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}(\underline{x}, \, \underline{y})$ is valid.
@@ Line 50: / Line 62: @@
 + &nbsp; $d_{\rm H}(\underline{x}, \, \underline{y}) = d_{\rm E}^2(\underline{x}, \, \underline{y})/4$ is valid.
-{Which of the equations describe the ML decoding in the BSC model? The minimization/maximization refers to all&nbsp; $\underline{x} &#8712;\mathcal{ C}$, respectively.
+{Which of the equations describe the maximum likelihood decoding in the BSC model?&nbsp; The minimization/maximization refers alwaysto all&nbsp; $\underline{x} &#8712;\mathcal{ C}$.
 |type="[]"}
 + $\underline{z} = \arg \min {d_{\rm H}(\underline{x}, \, \underline{y})}$,
@@ Line 56: / Line 68: @@
 + $\underline{z} = \arg \min {d_{\rm E}^2(\underline{x}, \, \underline{y})}$,
-{Which equation describes the ML decision in the BSC model?
+{Which equation describes the maximum likelihood decision in the BSC model?
 |type="()"}
 - $\underline{z} = \arg \min &#9001; \underline{x} \cdot \underline{y} &#9002;$,
 + $\underline{z} = \arg \max &#9001; \underline{x} \cdot \underline{y} &#9002;$.
-{What equations apply to the ML decision in the AWGN model?
+{What equations apply to the maximum likelihood decision in the AWGN model?
 |type="[]"}
 - $\underline{z} = \arg \min {d_{\rm H}(\underline{x}, \, \underline{y})}$,

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Kanalcodierung/Decodierung von Faltungscodes}}
+{{quiz-Header|Buchseite=Channel_Coding/Decoding_of_Convolutional_Codes}}
-[[File:P_ID2678__KC_Z_3_10.png|right|frame|Gesamtsystemmodell <br>Coder &ndash; Kanal &ndash; Viterbi]]
+[[File:P_ID2678__KC_Z_3_10.png|right|frame|Overall system model <br>enoder &ndash; channel &ndash; Viterbi]]
-Der Viterbi&ndash;Algorithmus stellt die bekannteste Realisierungsform für die Maximum&ndash;Likelihood&ndash;Decodierung eines Faltungscodes dar. Wir gehen hier von folgendem Modell aus:
+The Viterbi algorithm represents the best known realization form for the maximum likelihood decoding of a convolutional code. We assume the following model here:
 * Die Informationssequenz&nbsp; $\underline{u}$&nbsp; wird durch einen Faltungscode in die Codesequenz&nbsp; $\underline{x}$&nbsp; umgesetzt. Es gelte&nbsp; $u_i &#8712; \{0, \, 1\}$. Dagegen werden die Codesymbole bipolar dargestellt &nbsp; &#8658; &nbsp; $x_i &#8712; \{&ndash;1, \, +1\}$.
-* Der Kanal sei durch das&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Binary_Symmetric_Channel_.E2.80.93_BSC| BSC&ndash;Modell]]&nbsp; gegeben &nbsp; &#8658; &nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; oder es wird der&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#AWGN.E2.80.93Kanal_bei_bin.C3.A4rem_Eingang| AWGN&ndash;Kanal]]&nbsp; vorausgesetzt &nbsp; &#8658; &nbsp; reellwertige Empfangswerte&nbsp; $y_i$.
+* Der Kanal sei durch das&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Symmetric_Channel_.E2.80.93_BSC| "BSC&ndash;Modell"]]&nbsp; gegeben &nbsp; &#8658; &nbsp; $y_i &#8712; \{&ndash;1, \, +1\}$&nbsp; oder es wird der&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_Binary_Input| "AWGN&ndash;Kanal"]]&nbsp; vorausgesetzt &nbsp; &#8658; &nbsp; reellwertige Empfangswerte&nbsp; $y_i$.
 * Bei gegebener Empfangssequenz&nbsp; $\underline{y}$&nbsp; entscheidet sich der Viterbi&ndash;Algorithmus für die Codesequenz&nbsp; $\underline{z}$&nbsp; gemäß
 :$$\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.03cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{x}_{\hspace{0.03cm}i} |\hspace{0.05cm} \underline{y} ) \hspace{0.05cm}.$$
-*Dies entspricht dem&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Maximum-a-posteriori.E2.80.93_und_Maximum-Likelihood.E2.80.93Kriterium| Maximum&ndash;a&ndash;posteriori]]&nbsp; (MAP)&ndash;Kriterium. Sind alle Informationssequenzen&nbsp; $\underline{u}$&nbsp; gleichwahrscheinlich, so geht dieses in das etwas einfachere&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Maximum-a-posteriori.E2.80.93_und_Maximum-Likelihood.E2.80.93Kriterium| Maximum&ndash;Likelihood&ndash;Kriterium]]&nbsp; über:
+*Dies entspricht dem&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "Maximum&ndash;a&ndash;posteriori"]]&nbsp; (MAP)&ndash;Kriterium. Sind alle Informationssequenzen&nbsp; $\underline{u}$&nbsp; gleichwahrscheinlich, so geht dieses in das etwas einfachere&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Criteria_.C2.BBMaximum-a-posteriori.C2.AB_and_.C2.BBMaximum-Likelihood.C2.AB| "Maximum&ndash;Likelihood&ndash;Kriterium"]]&nbsp; über:
 :$$\underline{z} = {\rm arg} \max_{\underline{x}_{\hspace{0.03cm}i} \hspace{0.05cm} \in \hspace{0.05cm} \mathcal{C}} \hspace{0.1cm} {\rm Pr}( \underline{y}  \hspace{0.05cm}|\hspace{0.05cm} \underline{x}_{\hspace{0.03cm}i} ) \hspace{0.05cm}.$$
@@ Line 14: / Line 14: @@
-In dieser Aufgabe soll der Zusammenhang zwischen der&nbsp; [[Channel_Coding/Zielsetzung_der_Kanalcodierung#Einige_wichtige_Definitionen_zur_Blockcodierung| Hamming&ndash;Distanz]]&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$&nbsp; sowie der&nbsp; [[Channel_Coding/Kanalmodelle_und_Entscheiderstrukturen#Maximum-Likelihood.E2.80.93Entscheidung_beim_AWGN.E2.80.93Kanal| Euklidischen Distanz]]
+In dieser Aufgabe soll der Zusammenhang zwischen der&nbsp; [[Channel_Coding/Objective_of_Channel_Coding#Important_definitions_for_block_coding| "Hamming&ndash;Distanz"]]&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$&nbsp; sowie der&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Maximum-likelihood_decision_at_the_AWGN_channel| "Euklidischen Distanz"]]
 :$$d_{\rm E}(\underline{x}  \hspace{0.05cm}, \hspace{0.1cm}\underline{y}) =
 \sqrt{\sum_{i=1}^{L} \hspace{0.2cm}(x_i - y_i)^2}\hspace{0.05cm}$$
@@ Line 21: / Line 21: @@
 * der Hamming&ndash;Distanz&nbsp; $d_{\rm H}(\underline{x}, \, \underline{y})$,
 * der Euklidischen Distanz&nbsp; $d_{\rm E}(\underline{x}, \, \underline{y})$, und
-* dem&nbsp; [[Channel_Coding/Decodierung_von_Faltungscodes#Zusammenhang_zwischen_Hamming.E2.80.93Distanz_und_Korrelation| Korrelationswert]]&nbsp; $&#9001; x \cdot y &#9002;$.
+* dem&nbsp; [[Channel_Coding/Decoding_of_Convolutional_Codes#Relationship_between_Hamming_distance_and_correlation| "Korrelationswert"]]&nbsp; $&#9001; x \cdot y &#9002;$.

← Older revision		Revision as of 12:13, 10 October 2022
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−	[[Category:Channel Coding: Exercises\|^3.4 ~~Decodierung von Faltungscodes~~^]]	+	[[Category:Channel Coding: Exercises\|^3.4 Decoding of Convolutional Codes^]]