Channel Coding/Algebraic and Polynomial Description - Revision history

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2026-03-16T14:15:13Z

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	:$${ \boldsymbol{\rm G}}_0=\begin{pmatrix}1 & 1\end{pmatrix}\hspace{0.05cm},\hspace{0.3cm}{ \boldsymbol{\rm G}}_1=\begin{pmatrix}0 & 1\end{pmatrix}\hspace{0.3cm} \Rightarrow \hspace{0.3cm}$$		:$${ \boldsymbol{\rm G}}_0=\begin{pmatrix}1 & 1\end{pmatrix}\hspace{0.05cm},\hspace{0.3cm}{ \boldsymbol{\rm G}}_1=\begin{pmatrix}0 & 1\end{pmatrix}\hspace{0.3cm} \Rightarrow \hspace{0.3cm}$$
	*Thus,  the resulting generator matrix is:		*Thus,  the resulting generator matrix is:
	:$${ \boldsymbol{\rm G}}=\begin{pmatrix}		:$${ \boldsymbol{\rm G}}=\begin{pmatrix}11 & 01 & 00 & 00 & 00 & \cdots & \\00 & 11 & 01 & 00 & 00 & \cdots & \\00 & 00 & 11 & 01 & 00 & \cdots & \\00 & 00 & 00 & 11 & 01 & \cdots & \\\cdots & \cdots & \cdots & \cdots & \cdots & \cdots\end{pmatrix}\hspace{0.05cm}.$$
	11 & 01 & 00 & 00 & 00 & \cdots & \\
	00 & 11 & 01 & 00 & 00 & \cdots & \\
	00 & 00 & 11 & 01 & 00 & \cdots & \\
	00 & 00 & 00 & 11 & 01 & \cdots & \\
	\cdots & \cdots & \cdots & \cdots & \cdots & \cdots
	\end{pmatrix}\hspace{0.05cm}.$$

	*For the input sequence   $\underline{u} = (1, 0, 1, 1)$,  the encoded sequence starts with   $\underline{x} = (1, 1, 0, 1, 1, 1, 1, 0, \ \text{...})$.		*For the input sequence   $\underline{u} = (1, 0, 1, 1)$,  the encoded sequence starts with   $\underline{x} = (1, 1, 0, 1, 1, 1, 1, 0, \ \text{...})$.

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2026-03-16T13:46:41Z

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	*From the adjacent sketch can be derived:		*From the adjacent sketch can be derived:

	:$${ \boldsymbol{\rm G}}_0=\begin{pmatrix}		:$${ \boldsymbol{\rm G}}_0=\begin{pmatrix}1 & 1\end{pmatrix}\hspace{0.05cm},\hspace{0.3cm}{ \boldsymbol{\rm G}}_1=\begin{pmatrix}0 & 1\end{pmatrix}\hspace{0.3cm} \Rightarrow \hspace{0.3cm}$$
	1 & 1
	\end{pmatrix}\hspace{0.05cm},\hspace{0.3cm}
	{ \boldsymbol{\rm G}}_1=\begin{pmatrix}
	0 & 1
	\end{pmatrix}\hspace{0.3cm} \Rightarrow \hspace{0.3cm}$$
	*Thus,  the resulting generator matrix is:		*Thus,  the resulting generator matrix is:
	:$${ \boldsymbol{\rm G}}=\begin{pmatrix}		:$${ \boldsymbol{\rm G}}=\begin{pmatrix}

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2026-03-16T12:27:31Z

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	{{Display}}		{{Display}}
			[[de:Kanalcodierung/Algebraische_und_polynomische_Beschreibung]]

Guenter at 17:50, 2 December 2022

2022-12-02T17:50:29Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 8:}$		$\text{Example 8:}$
	The ~~coder '''KORREKTUR:~~ encoder~~'''~~ of rate   $2/3$   considered often in the last sections is not systematic because  e.g.  $\underline{x}^{(1)} ≠ \underline{u}^{(1)}, \ \underline{x}^{(2)} ≠ \underline{u}^{(2)}$   holds  $($see adjacent graphic$)$.<br>		The encoder of rate   $2/3$   considered often in the last sections is not systematic because  e.g.  $\underline{x}^{(1)} ≠ \underline{u}^{(1)}, \ \underline{x}^{(2)} ≠ \underline{u}^{(2)}$   holds  $($see adjacent graphic$)$.<br>
	[[File:P ID2613 KC T 3 2 S1 neu.png\|right\|frame\|Convolutional encoder of rate  $2/3$]]		[[File:P ID2613 KC T 3 2 S1 neu.png\|right\|frame\|Convolutional encoder of rate  $2/3$]]

Hwang at 14:28, 30 November 2022

2022-11-30T14:28:29Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 8:}$		$\text{Example 8:}$
	The coder of rate   $2/3$   considered often in the last sections is not systematic because  e.g.  $\underline{x}^{(1)} ≠ \underline{u}^{(1)}, \ \underline{x}^{(2)} ≠ \underline{u}^{(2)}$   holds  $($see adjacent graphic$)$.<br>		The coder '''KORREKTUR: encoder''' of rate   $2/3$   considered often in the last sections is not systematic because  e.g.  $\underline{x}^{(1)} ≠ \underline{u}^{(1)}, \ \underline{x}^{(2)} ≠ \underline{u}^{(2)}$   holds  $($see adjacent graphic$)$.<br>
	[[File:P ID2613 KC T 3 2 S1 neu.png\|right\|frame\|Convolutional encoder of rate  $2/3$]]		[[File:P ID2613 KC T 3 2 S1 neu.png\|right\|frame\|Convolutional encoder of rate  $2/3$]]

Hwang at 14:24, 30 November 2022

2022-11-30T14:24:42Z

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	Now we extend the result to convolutional encoders with more than one input   ⇒   $k ≥ 2$  $($see graph$)$.<br>		Now we extend the result to convolutional encoders with more than one input   ⇒   $k ≥ 2$  $($see graph$)$.<br>

	In order to map a convolutional code of rate  $k/n$  in the D&~~ndashdomain~~,  the dimension of the above vector equation must be increased with respect to input and transfer function:		In order to map a convolutional code of rate  $k/n$  in the D–domain,  the dimension of the above vector equation must be increased with respect to input and transfer function:

	::<math>\underline{X}(D) = \underline{U}(D) \cdot { \boldsymbol{\rm G}}(D)\hspace{0.05cm}.</math>		::<math>\underline{X}(D) = \underline{U}(D) \cdot { \boldsymbol{\rm G}}(D)\hspace{0.05cm}.</math>

Hwang at 14:22, 30 November 2022

2022-11-30T14:22:19Z

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	*The  $($rather simple$)$  $\rm proof$  of this important result can be found in the specification for  [[Aufgaben:Exercise_3.3Z:_Convolution_and_D-Transformation\|"Exercise 3.3Z"]].		*The  $($rather simple$)$  $\rm proof$  of this important result can be found in the specification for  [[Aufgaben:Exercise_3.3Z:_Convolution_and_D-Transformation\|"Exercise 3.3Z"]].

	*As in  »[[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80. 93_Transfer_function\|$\text{system theory}$]]«  commonly,  the  D–transform  $G(D)$  of the impulse response  $\underline{g}$  is also called  "transfer function".}}		*As in  »[[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|$\text{system theory}$]]«  commonly,  the  D–transform  $G(D)$  of the impulse response  $\underline{g}$  is also called  "transfer function".}}

Hwang at 14:21, 30 November 2022

2022-11-30T14:21:27Z

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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Theorem:}$  As with all spectral transformations,  the  »<b>multiplication</b>«  applies to the  D–transform in the image domain,  since the  $($discrete$)$  time ~~functionss~~  $\underline{u}$  and  $\underline{g}$  are interconnected by the  »<b>convolution</b>«:		$\text{Theorem:}$  As with all spectral transformations,  the  »<b>multiplication</b>«  applies to the  D–transform in the image domain,  since the  $($discrete$)$  time functions  $\underline{u}$  and  $\underline{g}$  are interconnected by the  »<b>convolution</b>«:

	::<math>\underline{x} = \underline{u} * \underline{g} \quad		::<math>\underline{x} = \underline{u} * \underline{g} \quad

Hwang at 14:19, 30 November 2022

2022-11-30T14:19:54Z

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	X(D) = \sum_{i = 0}^{\infty} x_i \cdot D\hspace{0.05cm}^i \hspace{0.05cm}.</math>}}<br>		X(D) = \sum_{i = 0}^{\infty} x_i \cdot D\hspace{0.05cm}^i \hspace{0.05cm}.</math>}}<br>

	<u>Note</u>:   In the literature,  sometimes  $x(D)$  is used instead of  $X(D)$.  However,  we write in our learning tutorial all image domain functions   ⇒   "spectral domain functions"  with capital letters, &nbsp: for example the Fourier transform, the Laplace transform and the D–transform:		<u>Note</u>:   In the literature,  sometimes  $x(D)$  is used instead of  $X(D)$.  However,  we write in our learning tutorial all image domain functions   ⇒   "spectral domain functions"  with capital letters,   for example the Fourier transform, the Laplace transform and the D–transform:

	::<math>x(t) \hspace{0.15cm}		::<math>x(t) \hspace{0.15cm}

Hwang at 14:19, 30 November 2022

2022-11-30T14:19:30Z

← Older revision		Revision as of 16:19, 30 November 2022
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	X(D) = \sum_{i = 0}^{\infty} x_i \cdot D\hspace{0.05cm}^i \hspace{0.05cm}.</math>}}<br>		X(D) = \sum_{i = 0}^{\infty} x_i \cdot D\hspace{0.05cm}^i \hspace{0.05cm}.</math>}}<br>

	<u>Note</u>:   In the literature,  sometimes  $x(D)$  is used instead of  $X(D)$.  However,  we write in our learning tutorial all image domain functions   ⇒   "spectral domain functions"  with capital letters,&nbsp: for example the Fourier transform, the Laplace transform and the D–transform:		<u>Note</u>:   In the literature,  sometimes  $x(D)$  is used instead of  $X(D)$.  However,  we write in our learning tutorial all image domain functions   ⇒   "spectral domain functions"  with capital letters, &nbsp: for example the Fourier transform, the Laplace transform and the D–transform:

	::<math>x(t) \hspace{0.15cm}		::<math>x(t) \hspace{0.15cm}