Die zum Zeitpunkt $i$ am Coder anliegenden Bits werden mit $u_i^{\rm (1)}$, $u_i^{\rm (2)}$ und $u_i^{\rm (3)}$ bezeichnet. Beispielsweise gilt $u_1^{\rm (1)} = 0$, $u_2^{\rm (2)} = 1$ sowie $u_3^{\rm (3)} = 1$.
The bits present at the coder at time $i$ are denoted by $u_i^{\rm (1)}$, $u_i^{\rm (2)}$ and $u_i^{\rm (3)}$ . For example $u_1^{\rm (1)} = 0$, $u_2^{\rm (2)} = 1$, $u_3^{\rm (3)} = 1$.
In dieser Aufgabe sollen ermittelt werden:
The purpose of this exercise is to determine
* die Anzahl $k$ der pro Codierschritt verarbeiteten Informationsbits,
* the number $k$ of information bits processed per coding step,
* die Anzahl $n$ der pro Codierschritt ausgegebenen Codebits,
* die Gedächtnisordnung (oder kurz: das Gedächtnis) $m$,
* die Gesamteinflusslänge (oder kurz: Einflusslänge) $\nu$.
* the number $n$ of code bits output per coding step,
Außerdem sollen Sie für die angegebene Informationssequenz $\underline {u}$ die Codesymbole $x_i^{(1)}$, $x_i^{(2)}$, $x_i^{(3)}$, $x_i^{(4)}$ für die Taktzeitpunkte $i = 1$ und $i = 3$ bestimmen. Dabei ist vorauszusetzen, dass alle Speicherelemente zu Beginn mit Nullen belegt waren.
* the memory order (or, for short, the memory) $m$,
''Hinweise:''
* the total influence length (or for short: influence length) $\nu$.
* Die Aufg.asdöjalf
For the given information sequence $\underline {u}$ you shall determine the code symbols $x_i^{(1)}$, $x_i^{(2)}$, $x_i^{(3)}$ and $x_i^{(4)}$ for the clock instants $i = 1$ and $i = 3$. It must be assumed that all memory elements were filled with zeros at the beginning.
===Fragebogen===
Hints:
* This exercise refers to the chapter [[Channel_Coding/Basics_of_Convolutional_Coding| "Basics of Convolutional Coding"]].
* For completeness, the code bits to the clock step $i = 2$ are also given here:
* However, this last specification is not needed to solve the exercise.
===Questions===
<quiz display=simple>
<quiz display=simple>
{Multiple-Choice
{What are the code parameters $k$ and $n$?
|type="[]"}
|type="{}"}
+ correct
$k \ = \ ${ 3 3% }
- false
$n \ = \ ${ 4 3% }
{What are the memory order $m$ and the total influence length $\nu$?
|type="{}"}
$m \ = \ ${ 2 3% }
$\nu \ = \ ${ 3 3% }
{What are the code bits in the first coding step $(i = 1)$?
|type="{}"}
$x_1^{(1)} \ = \ ${ 0 3% }
$x_1^{(2)} \ = \ ${ 1 3% }
$x_1^{(3)} \ = \ ${ 0 3% }
$x_1^{(4)} \ = \ ${ 0 3% }
{Input-Box Frage
{What are the codebits in the third coding step $(i = 3)$?
|type="{}"}
|type="{}"}
$xyz \ = \ ${ 5.4 3% } $ab$
$x_3^{(1)} \ = \ ${ 1 3% }
$x_3^{(2)} \ = \ ${ 0 3% }
$x_3^{(3)} \ = \ ${ 1 3% }
$x_3^{(4)} \ = \ ${ 1 3% }
</quiz>
</quiz>
===Musterlösung===
===Solution===
{{ML-Kopf}}
{{ML-Kopf}}
'''(1)'''
'''(1)''' At each clock, $\underline {k = 3}$ new information bits are processed and $\underline {n = 4}$ code bits are output.
'''(2)'''
'''(3)'''
'''(4)'''
'''(2)''' We denote here the influence lengths of the information sequences $\underline{u}^{(j)}$ by $\nu_j$, where $1 ≤ j ≤ k = 3$ is to be set. Then holds:
*The memory order $m$ $($or memory for short$)$ of the encoder is given by the longest shift register ⇒ $\underline {m = 2}$.
*The total influence length $($or influence length for short$)$ is equal to the number of memory elements of the entire encoder circuit ⇒ $\underline {\nu = 3}$.
⇒ The code bits in the coding step $i = 2$ have already been mentioned on the information page. Here still follows the derivation:
The bits present at the coder at time $i$ are denoted by $u_i^{\rm (1)}$, $u_i^{\rm (2)}$ and $u_i^{\rm (3)}$ . For example $u_1^{\rm (1)} = 0$, $u_2^{\rm (2)} = 1$, $u_3^{\rm (3)} = 1$.
The purpose of this exercise is to determine
the number $k$ of information bits processed per coding step,
the number $n$ of code bits output per coding step,
the memory order (or, for short, the memory) $m$,
the total influence length (or for short: influence length) $\nu$.
For the given information sequence $\underline {u}$ you shall determine the code symbols $x_i^{(1)}$, $x_i^{(2)}$, $x_i^{(3)}$ and $x_i^{(4)}$ for the clock instants $i = 1$ and $i = 3$. It must be assumed that all memory elements were filled with zeros at the beginning.
(1) At each clock, $\underline {k = 3}$ new information bits are processed and $\underline {n = 4}$ code bits are output.
(2) We denote here the influence lengths of the information sequences $\underline{u}^{(j)}$ by $\nu_j$, where $1 ≤ j ≤ k = 3$ is to be set. Then holds:
The memory order $m$ $($or memory for short$)$ of the encoder is given by the longest shift register ⇒ $\underline {m = 2}$.
The total influence length $($or influence length for short$)$ is equal to the number of memory elements of the entire encoder circuit ⇒ $\underline {\nu = 3}$.
Predefined convolutional encoder
(3) In general, for the $n = 4$ code bits of step $i$:
