refactor: section mapping
This commit is contained in:
Matthias Jung
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After Width: | Height: | Size: 20 KiB |
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% Author: Prof. Dr. Matthias Jung, DL9MJ
|
||||
% Year: 2026
|
||||
\begin{tikzpicture}
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||||
\draw[-Triangle](0,0) -- ++(0,1.25) node[anchor=south] {$Q$};
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\draw[-Triangle](0,0) -- ++(1.25,0) node[anchor=west] {$I$};
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\draw(0,0) -- ++(-1.25,0);
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\draw(0,0) -- ++(0,-1.25);
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%
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\draw(-0.75,+0.75) node[anchor=center, circle, draw, fill=DARCblue](m0){};
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\draw(-0.75,+0.25) node[anchor=center, circle, draw, fill=DARCblue](m1){};
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%
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\draw(m0.north) node[anchor=south] {$0000$};
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\draw(m1.north) node[anchor=south] {$0001$};
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\draw(m2.north) node[anchor=south] {$0011$};
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\draw(m3.north) node[anchor=south] {$0010$};
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\draw(m4.north) node[anchor=south] {$0100$};
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\draw(m5.north) node[anchor=south] {$0101$};
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\draw(m6.north) node[anchor=south] {$0111$};
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\draw(m7.north) node[anchor=south] {$0110$};
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\draw(m8.north) node[anchor=south] {$1100$};
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\draw(m9.north) node[anchor=south] {$1101$};
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\draw(m10.north) node[anchor=south] {$1111$};
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\draw(m11.north) node[anchor=south] {$1110$};
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\draw(m12.north) node[anchor=south] {$1000$};
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\draw(m13.north) node[anchor=south] {$1001$};
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\draw(m14.north) node[anchor=south] {$1011$};
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\draw(m15.north) node[anchor=south] {$1010$};
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\end{tikzpicture}
|
||||
@@ -1,44 +1,40 @@
|
||||
Im Rahmen der digitalen Signalverarbeitung beschreibt das *Mapping* den Schritt, bei dem digitale Daten in spezifische Signalpunkte (Symbole) umgewandelt werden, die über das Übertragungssystem gesendet werden können. Dies ist ein entscheidender Prozess in der Modulation, insbesondere bei Quadraturamplitudenmodulationen (QAM) und Phasenumtastungen wie QPSK (Quadrature Phase Shift Keying).
|
||||
|
||||
Beispiel für schrittweises Mapping bei QPSK
|
||||
Um die Symbole zu visualisieren, verwenden wir ein *Konstellationsdiagramm*, das die möglichen Signalpunkte in einem zweidimensionalen Raum darstellt. Man bezeichnet die Achsen oft als In-Phase (I) und Quadrature (Q). Jeder Punkt im Diagramm repräsentiert eine bestimmte Amplitude und Phase, die durch das Mapping einer bestimmten Bitkombination zugeordnet ist, wie in Abbildung [ref:a_konstellation] dargestellt.
|
||||
|
||||
1. *Binäre Daten in Symbole umwandeln*:
|
||||
<margin>
|
||||
[picture:1060:a_konstellation:Konstellationsdiagramm]
|
||||
</margin>
|
||||
|
||||
Bei QPSK werden jeweils zwei Bits zu einem Symbol zusammengefasst. Da wir zwei Bits pro Symbol haben, ergeben sich vier mögliche Kombinationen ($\num{00}$, $\num{01}$, $\num{10}$, $\num{11}$). Jede dieser Kombinationen wird einem spezifischen Signalpunkt zugeordnet, der durch eine bestimmte Phase repräsentiert wird.
|
||||
---
|
||||
|
||||
2. *Phasenvergabe*:
|
||||
Schauen wir uns im ersten Schritt QPSK in Abbildung [ref:a_qpsk] an: Bei QPSK werden jeweils zwei Bits zu einem Symbol zusammengefasst. Da wir zwei Bits pro Symbol haben, ergeben sich vier mögliche Kombinationen ($\num{00}$, $\num{01}$, $\num{10}$, $\num{11}$). Jede dieser Kombinationen wird einem spezifischen Signalpunkt zugeordnet, der durch eine bestimmte Phase repräsentiert wird.
|
||||
|
||||
Bei QPSK hat jedes Symbol eine eigene Phase. Die Phasen werden typischerweise in $\qty{90}{\degree}$-Schritten definiert:
|
||||
<margin>
|
||||
[picture:1059:a_qpsk:I-Q-Diagramm für ein QPSK-Mapping]
|
||||
</margin>
|
||||
|
||||
- $\num{00}$ entspricht ($\qty{0}{\degree}$)
|
||||
- $\num{01}$ entspricht ($\qty{90}{\degree}$)
|
||||
- $\num{10}$ entspricht ($\qty{180}{\degree}$)
|
||||
- $\num{11}$ entspricht ($\qty{270}{\degree}$)
|
||||
---
|
||||
|
||||
3. *Mapping auf das Konstellationsdiagramm*:
|
||||
Bei QPSK hat jedes Symbol eine eigene Phase. Die Phasen werden typischerweise in $\qty{90}{\degree}$-Schritten definiert und auf die vier möglichen Bitkombinationen gemapped, zum Beispiel:
|
||||
|
||||
% TODO Bessere Darstellung für das Beispiel mit QPSK
|
||||
[picture:697:a_8qam:I-Q-Diagramm für ein 8QAM-Mapping]
|
||||
Die Darstellung ist für ein 8QAM-Mapping. QPSK im Beispiel entspricht dem äußeren Kreis.
|
||||
- $\num{11}$ entspricht $\qty{45}{\degree}$
|
||||
- $\num{01}$ entspricht $\qty{135}{\degree}$
|
||||
- $\num{00}$ entspricht $\qty{225}{\degree}$
|
||||
- $\num{10}$ entspricht $\qty{315}{\degree}$
|
||||
|
||||
Ein *Konstellationsdiagramm* ist eine grafische Darstellung der möglichen Symbole in einem quadratischen Diagramm, wobei die X-Achse (In-Phase) und die Y-Achse (Quadratur-Phase) die Amplitude der Signalbestandteile darstellen. Bei QPSK befinden sich die vier Signalpunkte an den Enden eines Quadrats im Diagramm:
|
||||
Die Amplitude der Signale bleibt dabei konstant, und die Information wird ausschließlich durch die Phasenlage übertragen. Deshalb liegen die vier Punkte im Konstellationsdiagramm für QPSK auf einem Kreis.
|
||||
|
||||
- Signalpunkt für $\num{00}$: auf der positiven X-Achse ($\qty{0}{\degree}$)
|
||||
- Signalpunkt für $\num{01}$: auf der positiven Y-Achse ($\qty{90}{\degree}$)
|
||||
- Signalpunkt für $\num{10}$: auf der negativen X-Achse ($\qty{180}{\degree}$)
|
||||
- Signalpunkt für $\num{11}$: auf der negativen Y-Achse ($\qty{270}{\degree}$)
|
||||
<indepth>
|
||||
Genau genommen gibt es aber auch noch andere Möglichkeiten, die Phasen den Bitkombinationen zuzuordnen, solange sie eindeutig sind. Das hier gezeigte Mapping ist nur ein Beispiel.
|
||||
</indepth>
|
||||
|
||||
Jeder dieser Punkte repräsentiert ein Symbol, und der Empfänger kann anhand der Phasenlage bestimmen, welche Bitkombination gesendet wurde.
|
||||
---
|
||||
|
||||
Beispielhafte Darstellung im Konstellationsdiagramm bei QPSK
|
||||
Jeder dieser Punkte repräsentiert ein Symbol. Der Empfänger kann anhand der Phasenlage bestimmen, welche Bitkombination gesendet wurde. Das Konstellationsdiagramm bei QPSK zeigt vier Signalpunkte im rechten Winkel zueinander, die den vier verwendeten Phasen entsprechen. Die große Trennung zwischen den einzelnen Phasen ermöglicht eine zuverlässige Decodierung auch unter rauschbehafteten Bedingungen.
|
||||
|
||||
Die vier Punkte im Konstellationsdiagramm bei QPSK lassen sich einfach darstellen:
|
||||
Ändert man neben der Phase auch die Amplitude, so spricht man von einer Quadraturamplitudenmodulation (QAM). Bei QAM werden sowohl die Amplitude als auch die Phase variiert, um mehr Bits pro Symbol zu übertragen. Zum Beispiel kann bei 16-QAM jedes Symbol vier Bits repräsentieren, was zu 16 möglichen Signalpunkten im Konstellationsdiagramm führt. Ein Beispiel für ein 16-QAM-Mapping ist in Abbildung [ref:a_qam] dargestellt.
|
||||
|
||||
- *$\num{00}$* bei $\qty{0}{\degree}$: Repräsentiert einen Punkt auf der positiven X-Achse.
|
||||
- *$\num{01}$* bei $\qty{90}{\degree}$: Ein Punkt auf der positiven Y-Achse.
|
||||
- *$\num{10}$* bei $\qty{180}{\degree}$: Ein Punkt auf der negativen X-Achse.
|
||||
- *$\num{11}$* bei $\qty{270}{\degree}$: Ein Punkt auf der negativen Y-Achse.
|
||||
|
||||
%TODO: Bild zum Mapping am Beispiel von QPSK einfügen und die Punkte im Diagramm im Bezug zum Text setzen.
|
||||
|
||||
Das Konstellationsdiagramm bei QPSK hat also vier Signalpunkte im rechten Winkel zueinander, was die vier Phasen darstellt, die für die Übertragung verwendet werden. Die klare Trennung der Phasen hilft, die Symbole auch bei Rauschen gut auseinanderzuhalten.
|
||||
<margin>
|
||||
[picture:1061:a_qam:I-Q-Diagramm für ein 16-QAM-Mapping]
|
||||
</margin>
|
||||
Reference in New Issue
Block a user