[[File:P_ID2062__Dig_A_4_12.png|right|frame|Signalraumkonstellation der 16–QAM]]
[[File:P_ID2062__Dig_A_4_12.png|right|frame|Signal space constellation of $\rm 16–QAM$]]
Beigefügte Grafik zeigt die Signalraumkonstellation der [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation#Quadraturamplitudenmodulation_.28M.E2.80.93QAM.29| Quadraturamplitudenmodulation]] mit $M = 16$ Signalraumpunkten. Für dieses Modulationsverfahren sollen berechnet werden:
The graph shows the signal space constellation of [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"quadrature amplitude modulation"]] with $M = 16$ signal space points. The following should be calculated for this modulation method:
* die mittlere Energie pro Symbol bzw. pro Bit,
* the average energy per symbol or per bit,
* die mittlere Symbolfehlerwahrscheinlichkeit $p_{\rm S}$ sowie die [[Digitalsignal%C3%BCbertragung/Approximation_der_Fehlerwahrscheinlichkeit#Union_Bound_-_Obere_Schranke_f.C3.BCr_die_Fehlerwahrscheinlichkeit| Union Bound]] als obere Schranke,
* die mittlere Bitfehlerwahrscheinlichkeit $p_{\rm B}$ bei Graycodierung. Die Gray–Zuordnung ist in der Grafik angegeben (rote Beschriftung).
* the average symbol error probability $p_{\rm S}$,
''Hinweise:''
*the [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Union_Bound_-_Upper_bound_for_the_error_probability|"Union Bound"]] $p_{\rm UB}$ as upper bound,
* Die Aufgabe behandelt einen Teilaspekt des Kapitels [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation| Trägerfrequenzsysteme mit kohärenter Demodulation]].
* Die Wahrscheinlichkeit, dass das linke obere Symbol in eines der benachbarten Symbole verfälscht wird, wird mit $p$ abgekürzt (blaue Pfeile in der Grafik).
* Eine diagonale Verfälschung ⇒ zwei Bit verfälscht (grüner Pfeil) wird ausgeschlossen.
* Für den AWGN–Kanal gilt mit dem komplementären Gaußschen Fehlerintegrale für diese Hilfsgröße:
* Verwenden Sie für numerische Berechnungen $E = 1 \ \rm mWs$ und $p = 0.004$. Aus diesen Werten kann die AWGN–Rauschleistungsdichte $N_0$ näherungsweise berechnet werden:
* the average bit error probability $p_{\rm B}$ with Gray coding.
===Fragebogen===
Notes:
# The exercise deals with a partial aspect of the chapter [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation|"Carrier Frequency Systems with Coherent Demodulation"]].
#The Gray assignment is given in the graphic $($red lettering$)$.
#The probability that the upper left symbol is falsified into one of the neighboring symbols is abbreviated to $p$ $($blue arrows in the graph$)$.
#A diagonal falsification ⇒ two bit falsified $($green arrow$)$ is excluded.
#For the AWGN channel, with the complementary Gaussian error integral for this auxiliary variable, the following applies: $p = {\rm Q} \left ( \sqrt{ { 2E}/{ N_0} }\right )\hspace{0.05cm}.$
# For numerical calculations, use $E = 1 \ \rm mWs$ and $p = 0.4\%$.
#The AWGN noise power density $N_0$ can be calculated approximately from these values:
{Calculate the actual symbol error probability $p_{\rm S} < p_{\rm UB}$.
|type="{}"}
$p_{\rm S} \ = \ $ { 1.2 3% } $\ \%$
{Input-Box Frage
{Calculate the actual bit error probability $p_{\rm B}$ for Gray coding.
|type="{}"}
|type="{}"}
$xyz$ = { 5.4 3% } $ab$
$p_{\rm B} \ = \ $ { 0.3 3% } $\ \%$
</quiz>
</quiz>
===Musterlösung===
===Solution===
{{ML-Kopf}}
{{ML-Kopf}}
'''(1)'''
'''(1)''' The quotient $E_{\rm S}/E$ is obtained as the mean square distance of the $M = 16$ signal space points $\boldsymbol{s}_i$ from the origin.
'''(2)'''
'''(3)'''
*With the given signal space constellation of the $\rm 16–QAM$ we obtain:
*The same result is obtained with the equation given in the [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation| "theory section"]]:
'''(3)''' The "Union Bound" is an upper bound on the symbol error probability.
*It only takes into account the transition to adjacent decision regions due to AWGN noise.
*From the graph, it can be seen that the corner symbols (filled in yellow) can only be biased towards two other symbols and the remaining edge symbols (filled in green) can be biased in three directions.
*The "worst case" are the four inner symbols (with blue filling) with four falsification possibilities each.
*The same result is obtained with the equation given in the [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation| "theory section"]]:
The graph shows the signal space constellation of "quadrature amplitude modulation" with $M = 16$ signal space points. The following should be calculated for this modulation method:
The Gray assignment is given in the graphic $($red lettering$)$.
The probability that the upper left symbol is falsified into one of the neighboring symbols is abbreviated to $p$ $($blue arrows in the graph$)$.
A diagonal falsification ⇒ two bit falsified $($green arrow$)$ is excluded.
For the AWGN channel, with the complementary Gaussian error integral for this auxiliary variable, the following applies: $p = {\rm Q} \left ( \sqrt{ { 2E}/{ N_0} }\right )\hspace{0.05cm}.$
For numerical calculations, use $E = 1 \ \rm mWs$ and $p = 0.4\%$.
The AWGN noise power density $N_0$ can be calculated approximately from these values:
(3) The "Union Bound" is an upper bound on the symbol error probability.
It only takes into account the transition to adjacent decision regions due to AWGN noise.
From the graph, it can be seen that the corner symbols (filled in yellow) can only be biased towards two other symbols and the remaining edge symbols (filled in green) can be biased in three directions.
The "worst case" are the four inner symbols (with blue filling) with four falsification possibilities each.
