Difference between revisions of "Modulation Methods/Quadrature Amplitude Modulation"

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{{Header
|Untermenü=Digitale Modulationsverfahren
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|Untermenü=Digital Modulation Methods
|Vorherige Seite=Lineare digitale Modulationsverfahren
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|Vorherige Seite=Linear_Digital_Modulation
|Nächste Seite=Nichtlineare Modulationsverfahren
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|Nächste Seite=Non-Linear_Digital_Modulation
 
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==Allgemeine Beschreibung und Signalraumzuordnung (1)==
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==General description and signal space allocation==
Aufgrund der Orthogonalität von Cosinus und (Minus–)Sinus kann man über einen Übertragungskanal zwei Datenströme unabhängig voneinander übertragen. Die Grafik zeigt das allgemeine Blockschaltbild.  
+
<br>
 +
Due to the orthogonality of cosine and&nbsp; (minus)&nbsp; sine,&nbsp; two data streams can be transmitted independently via the same transmission channel.&nbsp;  The diagram shows the general circuit schematic.
 +
[[File: EN_Mod_T_4_3_S1.png|right|frame| Linear modulator with&nbsp; $\rm I$ and&nbsp; $\rm Q$–components;&nbsp; signal space for&nbsp; $\text{16-QAM}$]]
  
 +
This very general model can be described as follows:
 +
*The binary source symbol sequence&nbsp;$〈q_k〉$&nbsp; with bit rate &nbsp;$R_{\rm B}$&nbsp;is applied to the input.&nbsp;  Thus,&nbsp; the time interval between two symbols is &nbsp;$T_{\rm B} = 1/R_{\rm B}$.
 +
*Two multi-level amplitude coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are derived from each &nbsp;$b$&nbsp; binary input symbols &nbsp;$q_k$,&nbsp; where &nbsp;$\rm I$&nbsp; stands for&nbsp; "inphase  component"&nbsp; and &nbsp;$\rm Q$&nbsp; stands for&nbsp; "quadrature component".
 +
*If &nbsp;$b$&nbsp; is even and the signal space allocation is quadratic, then the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; can each take on one of the &nbsp;$M = 2^{b/2}$&nbsp; amplitude values with equal probability.&nbsp;  This is then referred to as&nbsp; &raquo;'''quadrature amplitude modulation'''&laquo;&nbsp; $\rm (QAM)$.
 +
*The example considered in the graph is for the &nbsp; $\text{16-QAM}$&nbsp; with &nbsp;$b = M = 4$&nbsp; and correspondingly &nbsp;$M^2 =16$&nbsp; signal space points.&nbsp;  For a &nbsp; $\text{256-QAM}$,&nbsp; $b = 8$&nbsp; and &nbsp;$M = 16$&nbsp; would apply: &nbsp; $2^b = M^2 = 256$.
 +
*Next,&nbsp; the coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp; are each applied to a [[Signal_Representation/Discrete-Time_Signal_Representation#Dirac_comb_in_time_and_frequency_domain|$\text{Dirac comb}$]]&nbsp; as pulse weights.&nbsp;  Thus,&nbsp; after pulse shaping with the basic transmission pulse &nbsp;$g_s(t)$,&nbsp; the following holds for both branches of the circuit diagram:
 +
:$$s_{\rm I}(t)  = \sum_{\nu = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm},\hspace{1cm}s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm
 +
Q\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.$$
 +
*Note that because of the redundancy&ndash;free conversion,&nbsp; the symbol duration &nbsp;$T$&nbsp; of these signals is larger by a factor of &nbsp;$b$&nbsp; than the bit duration &nbsp;$T_{\rm B}$&nbsp; of the binary source signal.&nbsp;  In the illustrated &nbsp; $\text{16-QAM}$&nbsp; example, &nbsp;$T = 4 · T_{\rm B}$&nbsp; holds.
 +
*The QAM transmitted signal &nbsp;$s(t)$&nbsp; is then the sum of the two signals multiplied by cosine and minus-sine, respectively:
 +
:$$s_{\rm cos}(t) = s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t), \hspace{1cm} s_{\rm -sin}(t)  = -s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t)$$
 +
:$$\Rightarrow \hspace{0.3cm}s(t) = s_{\rm cos}(t)+ s_{\rm -sin}(t) =
 +
s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t) - s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t)
 +
\hspace{0.05cm}.$$
  
[[File:P_ID1706__Mod_T_4_3_S1_Ganz_neu.png | Blockschaltbild eines linearen Modulators mit I– und Q–Komponente; Signalraumzuordnung 16-QAM]]
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{{BlaueBox|TEXT=
 +
$\text{Conclusion:}$&nbsp; These statements can be summarized as follows:
 +
*The two transmission branches &nbsp; $\rm (I,\ Q)$&nbsp; can be thought of as two completely separate  &nbsp;$M$-level ASK systems <br>that do not interfere with each other as long as all components are optimally designed.
 +
*Quadrature amplitude modulation thus makes it&nbsp; (ideally)&nbsp; possible to double the data rate while maintaining the same quality. }}
  
  
Dieses sehr allgemeine Modell lässt sich wie folgt beschreiben:
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==System description using the equivalent low-pass signal==
*Am Eingang liegt die binäre Quellensymbolfolge $〈q_k〉$ mit der Bitrate $R_{\rm B}$ an. Der zeitliche Abstand zweier Symbole ist damit $T_{\rm B} = 1/R_{\rm B}$.  
+
<br>
*Aus jeweils $b$ binären Eingangssymbolen $q_k$ werden zwei mehrstufige Amplitudenkoeffizienten $a_{\rm }$ und $a_{\rm }$ abgeleitet, wobei „I” für Inphase und „Q” für Quadraturkomponente steht.
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Since the multiplication of &nbsp;$s_{\rm I}(t)$&nbsp; and &nbsp;$s_{\rm Q}(t)$&nbsp; with a cosine or a minus-sine oscillation  only causes a shift in the frequency domain and such a shift is a linear operation,&nbsp; the system description can be greatly simplified using equivalent low-pass signals.
*Ist $b$ geradzahlig und die Signalraumzuordnung quadratisch, so können die Koeffizienten $a_{\rm }$ und $a_{\rm }$ jeweils einen von $M = 2^{b/2}$ Amplitudenwerten mit gleicher Wahrscheinlichkeit annehmen. Man spricht dann von Quadratur–Amplitudenmodulation (QAM).  
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[[File:P_ID1708__Mod_T_4_3_S2_Ganz_neu.png |right|frame| Linear modulator&nbsp; $(\rm I$ and&nbsp; $\rm Q$ components$)$&nbsp; in the equivalent low-pass range]]
*Das in der Grafik betrachtete Beispiel gilt für die 16–QAM mit $b = M =$ 4 und dementsprechend 16 Signalraumpunkten. Bei einer 256–QAM würde $b =$ 8 und $M =$ 16 gelten $(2^b = M^2 =$ 256).  
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#The graph shows the simplified model in the baseband.&nbsp;  This is equivalent to the block diagram considered so far.
 +
#The serial-parallel conversion and the signal space allocation drawn in red in &nbsp;[[Modulation_Methods/Quadrature_Amplitude_Modulation#General_description_and_signal_space_allocation|"the block diagram in the last section"]]&nbsp;are retained.&nbsp; This block is no longer drawn here.
 +
#We also initially disregard the band-pass &nbsp;$H_{\rm BP}(f)$,&nbsp; which is often introduced for technical reasons.
 +
<br clear=all>
 +
Please note the following:
 +
*All double arrows in the baseband model denote complex quantities.&nbsp;  The operations associated with them should also be understood as complex.&nbsp;  For example,&nbsp; the complex amplitude coefficient &nbsp;$a_ν$&nbsp; combines one inphase and one quadrature coefficient:
 +
:$$a_\nu = a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm
 +
Q\hspace{0.03cm}\it \nu} \hspace{0.05cm}.$$
 +
*The equivalent low&ndash;pass representation of the actual, physical and thus per se real transmitted QAM signal &nbsp;$s(t)$&nbsp; is always complex and with the partial signals &nbsp;$s_{\rm I}(t)$&nbsp; and &nbsp;$s_{\rm Q}(t)$ it holds for the equivalent low-pass signal&nbsp; $($German:&nbsp; "äquivalentes Tiefpass&ndash;Signal" &nbsp; &rArr;  &nbsp; subscript:&nbsp; "TP"$)$:
 +
:$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} \cdot s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty} a_\nu \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.$$
 +
*The analytical signal &nbsp;$s_+(t)$&nbsp; is obtained from  the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; by multiplying by the complex exponential function.&nbsp;  The physical signal &nbsp;$s(t)$&nbsp; is then obtained as the real part of &nbsp;$s_+(t)$.
 +
*In order for the signs in the block diagram in the previous section and the sketched baseband model here to match,&nbsp; multiplication by the negative sine wave is required in the quadrature branch,&nbsp; as shown in the following calculation:
 +
:$$s(t)  = {\rm Re}[s_{\rm +}(t)] = {\rm Re}[s_{\rm TP}(t) \cdot{\rm e}^{{\rm j}2\pi f_{\rm T} t}] $$
 +
:$$\Rightarrow \hspace{0.3cm} s(t)  = {\rm Re} \left[\left ( \sum (a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm Q\hspace{0.03cm}\it \nu} ) \cdot g_s (t - \nu \cdot T)\right )\left ( \cos(2 \pi f_{\rm T} t) + {\rm j} \cdot \sin(2 \pi f_{\rm T} t) \right )\right]= s_{\rm I}(t) \cdot \cos(2\pi f_{\rm T} t) -  s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t) \hspace{0.05cm}.$$
 +
*The influence of the band-pass  &nbsp;$H_{\rm BP}(f)$,&nbsp; which in practice often has to be considered at the output of the QAM modulator,&nbsp; can be assigned to the pulse shape filter &nbsp;$g_s(t)$.&nbsp;  If the passband of the band-pass  filter is symmetric about&nbsp;$f_{\rm T}$,&nbsp; its low-pass equivalent&nbsp; (in the time domain) &nbsp; $h_{\rm BP\ →\ TP}(t)$ &nbsp; is purely real and one can replace &nbsp;$g_s(t)$&nbsp; with &nbsp;$g_s(t) \star h_{\rm BP\ →\ TP}(t)$&nbsp; in the model.
  
==Allgemeine Beschreibung und Signalraumzuordnung (2)==
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==Power and energy of QAM signals==
[[File: P_ID1707__Mod_T_4_3_S1_Ganz_neu.png | Blockschaltbild eines linearen Modulators mit I– und Q–Komponente; Signalraumzuordnung 16-QAM]]
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<br>
 +
As shown in the chapter &nbsp;[[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|"Equivalent Low-Pass Signal and its Spectral Function"]]&nbsp; in the book&nbsp; "Signal Representation",&nbsp; the &nbsp; &raquo;'''power'''&laquo;&nbsp; of the transmitted QAM signal &nbsp;$s(t)$&nbsp; can also be calculated from the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$,&nbsp; which is always complex.&nbsp;  Thus,&nbsp; it is equally valid to write:
 +
:$$P  = \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} s^2(t)\,{\rm d}  t = {\rm 1}/{2} \cdot \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} |s_{\rm TP}(t)|^2\,{\rm d}  t \hspace{0.05cm}.$$
  
 +
In contrast,&nbsp; the energy of the unbounded signals&nbsp; $s(t)$&nbsp; and &nbsp; $s_{\rm TP}(t)$&nbsp; is infinite.
  
'''Fortsetzung der Bildbeschreibung zur obigen Grafik:'''  
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However,&nbsp; if we restrict ourselves to a symbol duration&nbsp;$T$,&nbsp; we obtain the&nbsp; &raquo;'''energy per symbol'''&laquo;:
*Anschließend werden die Koeffizienten $a_{\rm Iν}$ und $a_{\rm Qν}$ jeweils einem Diracpuls als Impulsgewichte eingeprägt. Nach der Impulsformung mit dem Sendegrundimpuls $g_s(t)$ gilt somit in den beiden Zweigen des Blockschaltbildes:  
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:$$E_{\rm S} = \frac{{\rm E}[\hspace{0.05cm}|a_{\nu} |^2\hspace{0.05cm}]}{2}\cdot \int_{-\infty}^{+\infty} |g_s(t)|^2\,{\rm d} t = \frac{{\rm E}[|a_{\nu} |^2]}{2}\cdot \int_{-\infty}^{+\infty} |G_s(f)|^2\,{\rm d} \hspace{0.05cm}.$$
$$\begin{align*}s_{\rm I}(t) & = \sum_{\nu = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm},\\s_{\rm Q}(t) & = \sum_{\nu = - \infty}^{+\infty}a_{\rm
 
Q\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.\end{align*}$$
 
*Anzumerken ist, dass wegen der redundanzfreien Umsetzung die Symboldauer $T$ dieser Signale um den Faktor $b$ größer ist als die Bitdauer $T_{\rm B}$ des binären Quellensignals. Im gezeichneten Beispiel (16-QAM) gilt $T = 4 · T_{\rm B}$.
 
*Das QAM–Sendesignal $s(t)$ ist dann die Summe der beiden mit Cosinus bzw. Minus–Sinus multiplizierten Teilsignale:
 
$$\begin{align*}s_{\rm cos}(t) & = s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t),\\ s_{\rm -sin}(t) & = -s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t)\end{align*}$$
 
*Die beiden Übertragungszweige (I, Q) können als zwei völlig getrennte $M$–stufige ASK–Systeme aufgefasst werden, die sich gegenseitig nicht stören, solange alle Komponenten optimal ausgelegt sind. Die Quadratur–Amplitudenmodulation ermöglicht somit (im Idealfall) eine Verdoppelung der Datenrate bei gleichbleibender Qualität.
 
  
==Systembeschreibung durch das äquivalente TP–Signal==
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On the other hand, &nbsp; $E_{\rm B} = E_{\rm S}/b$ &nbsp; gives the&nbsp; &raquo;'''energy per bit'''&laquo;&nbsp; if according to the signal space assignment,&nbsp; $b$&nbsp; binary symbols are combined to the complex coefficient&nbsp; $a_\nu$.
Da die Multiplikation von $s_{\rm I}(t)$ und $s_{\rm Q}(t)$ mit einer Cosinus– bzw. Minus–Sinus–Schwingung nur eine Verschiebung im Frequenzbereich bewirkt und eine solche Verschiebung eine lineare Operation darstellt, lässt sich die Systembeschreibung mit Hilfe der äquivalenten TP–Signale wesentlich vereinfachen.  
 
  
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{{GraueBox|TEXT=
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[[File:P_ID1710__Mod_T_4_3_S3a.png |right|frame| Allocation for $\text{16-QAM}$]]
  
[[File:P_ID1708__Mod_T_4_3_S2_Ganz_neu.png | Linearer Modulator mit I– und Q–Komponente im äquivalenten TP–Bereich]]
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$\text{Example 1:}$ &nbsp; The upper graph shows the signal space allocation for $\text{16-QAM}$,&nbsp; where both the real and imaginary parts of the complex amplitude coefficients &nbsp;$a_ν$&nbsp; can take one of four values &nbsp;$±1$&nbsp; as well as &nbsp;$±1/3$&nbsp;, respectively.
  
 +
Averaging over the $16$ squared distances to the origin yields:
 +
:$${\rm E}\big[\hspace{0.05cm} \vert a_{\nu}\vert^2 \hspace{0.05cm}\big]  \hspace{0.18cm} = \hspace{0.18cm}  \frac{4}{16} \cdot \left [1^2 + 1^2\right ]+ \frac{4}{16} \cdot \left[(1/3 )^2 +(1/3)^2 \right ]
 +
+ \frac{8}{16} \cdot \left [1^2 + (1/3)^2\right ] $$
 +
:$$\Rightarrow \hspace{0.3cm}{\rm E}\big[\hspace{0.05cm} \vert a_{\nu}\vert^2 \hspace{0.05cm}\big]  = {10}/{9}\approx 1.11  \hspace{0.05cm}.$$
  
Die Grafik zeigt das vereinfachte Modell im Basisband. Dieses ist äquivalent zum bisher betrachteten Blockschaltbild. Beachten Sie bitte die folgenden Hinweise:
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In this order,&nbsp; the summands belong to
*Die im Blockschaltbild rot gezeichnete Seriell–Parallel–Wandlung und die Signalraumzuordnung bleibt erhalten, obwohl dieser Block hier nicht mehr eingezeichnet ist. Lassen wir zunächst auch den oft aus schaltungstechnischen Gründen eingebrachten Bandpass $H_{\rm BP}(f)$ außer Betracht.
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*the four red,
*Alle Doppelpfeile in dem obigen Basisbandmodell kennzeichnen komplexe Größen. Die damit verbundenen Operationen sind ebenfalls komplex zu verstehen. Beispielsweise fasst der komplexe Amplitudenkoeffizient $a_ν$ je einen Inphase– und einen Quadraturkoeffizienten zusammen:
+
*the four black,&nbsp; and
$$a_\nu = a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm
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*the eight blue dots.
Q\hspace{0.03cm}\it \nu} \hspace{0.05cm}.$$
 
*Die äquivalente Tiefpass–Repräsentation des tatsächlichen, physikalischen und damit per se reellen Sendesignals $s(t)$ ist bei QAM stets komplex und es gilt mit den Teilsignalen $s_{\rm I}(t)$ und $s_{\rm Q}(t)$:
 
$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} \cdot s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty} a_\nu \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.$$
 
*Zum analytischen Signal  $s_+(t)$ kommt man von $s_{\rm TP}(t)$ durch Multiplikation mit der komplexen Exponentialfunktion. Das physikalische Sendesignal $s(t)$ ergibt sich dann als der Realteil von $s_+(t)$.
 
*Damit die Vorzeichen im Blockschaltbild der letzten Seite und im skizzierten Basisbandmodell übereinstimmen, ist im Quadraturzweig die Multiplikation mit der negativen Sinus–Schwingung erforderlich, wie die nachfolgende Rechnung zeigt:
 
$$\begin{align*}s(t)  & = {\rm Re}[s_{\rm +}(t)] = {\rm Re}[s_{\rm TP}(t) \cdot{\rm e}^{{\rm j}2\pi f_{\rm T} t}] = \\ & = {\rm Re} \left[\left ( \sum (a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm Q\hspace{0.03cm}\it \nu} ) \cdot g_s (t - \nu \cdot T)\right )\left ( \cos(2 \pi f_{\rm T} t) + {\rm j} \cdot \sin(2 \pi f_{\rm T} t) \right )\right]= \\ & = s_{\rm I}(t) \cdot \cos(2\pi f_{\rm T} t) -  s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t) \hspace{0.05cm}.\end{align*}$$
 
*Der Einfluss des Bandpasses $H_{\rm BP}(f)$, der in der Praxis oft am Ausgang des QAM–Modulators zu berücksichtigen ist, kann dem Impulsformfilter $g_s(t)$ beaufschlagt werden. Ist der Durchlassbereich des BP–Filters symmetrisch um $f_{\rm T}$, so ist sein Tiefpass–Äquivalent (im Zeitbereich) $h_{\rm BP→TP}(t)$ rein reell und man kann im Modell $g_s(t)$ durch $g_s(t) \star h_{\rm BP→TP}(t)$ ersetzen.  
 
  
==Leistung und Energie von QAM–Signalen==
 
Wie im Kapitel 4.3 von „Signaldarstellung” gezeigt wird, kann die Leistung des QAM–Sendesignals $s(t)$ auch aus dem äquivalenten TP–Signal $s_{\rm TP}(t)$ berechnet werden, das stets komplex ist:
 
$$P  = \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} s(t)^2\,{\rm d}  t = \frac{\rm 1}{2} \cdot \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} |s_{\rm TP}(t)|^2\,{\rm d}  t \hspace{0.05cm}.$$
 
Dagegen ist die Energie der Signale $s(t)$ und $s_{\rm TP}(t)$ unendlich groß. Beschränkt man sich jedoch auf eine Symboldauer $T$, so erhält man die Energie pro Symbol
 
$$E_{\rm S}  = \frac{{\rm E}[\hspace{0.05cm}|a_{\nu} |^2\hspace{0.05cm}]}{2}\cdot \int_{-\infty}^{+\infty} |g_s(t)|^2\,{\rm d}  t = \frac{{\rm E}[|a_{\nu} |^2]}{2}\cdot \int_{-\infty}^{+\infty} |G_s(f)|^2\,{\rm d}  f  \hspace{0.05cm}.$$
 
Dagegen gibt $E_{\rm B} = E_{\rm S}/b$ die Energie pro Bit an, wenn gemäß der gegebenen Signalraumzuordnung jeweils $b$ Binärsymbole zu einem komplexen Amplitudenkoeffizienten zusammengefasst werden.
 
  
 +
[[File:P_ID2964__Mod_T_4_3_S3b_Ganz_neu.png |right|frame| NRZ rectangular basic pulse]]
 +
For a NRZ rectangular basic transmission pulse &nbsp;$g_s(t)$&nbsp; with amplitude &nbsp;$g_0$&nbsp; and symbol duration &nbsp;$T$,&nbsp; the spectrum &nbsp;$G_s(f)$&nbsp; is $\rm sinc$&ndash;shaped.&nbsp; In this case,&nbsp; the following holds for
  
{{Beispiel}}
+
*the average energy per symbol:
[[File:P_ID1710__Mod_T_4_3_S3a.png | Signalraumzuordnung bei 16-QAM | rechts]]
+
:$$E_{\rm S} = 1/2 \cdot {\rm E}\big[\hspace{0.05cm}\vert a_{\nu} \vert ^2 \big]\hspace{0.05cm}\cdot
Die obere Grafik zeigt die Signalraumzuordnung bei 16–QAM, wobei sowohl der Real– als auch der Imaginärteil der komplexen Amplitudenkoeffizienten $a_ν$ jeweils einen von vier Werten (±1 sowie ±1/3) annehmen kann. Durch Mittelung über die 16 Abstandsquadrate zum Ursprung erhält man:  
+
  g_0^2 \cdot T = {5}/{9}\cdot g_0^2 \cdot T \hspace{0.05cm}\approx 0.555 \cdot g_0^2 \cdot T \hspace{0.05cm},$$
$$\begin{align*}{\rm E}[\hspace{0.05cm}|a_{\nu} |^2 \hspace{0.05cm}] \hspace{-0.18cm} & = \hspace{-0.18cm} \frac{4}{16} \cdot (1^2 + 1^2)+ \frac{4}{16} \cdot \left[({1}/{3} )^2 +({1}/{3})^2 \right ]+\\
+
*the average energy per bit:
\hspace{-0.18cm} & + \hspace{-0.18cm}  \frac{8}{16} \cdot \left [1^2 + ({1}/{3})^2\right ] =  ... \hspace{0.15cm}= \frac{10}{9}\approx 1.11  \hspace{0.05cm}.\end{align*}$$
+
:$$E_{\rm B={E_{\rm S} }/{4}= {5}/{36}\cdot g_0^2 \cdot T \approx 0.139 \cdot g_0^2 \cdot T \hspace{0.05cm}.$$  
  
Die Summanden gehören in dieser Reihenfolge zu den vier roten, den vier schwarzen und den acht blauen Punkten.  
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The&nbsp; "maximum envelope" &nbsp;$s_0$&nbsp; of the transmitted signal &nbsp;$s(t)$&nbsp;  is larger than the amplitude &nbsp;$g_0$&nbsp; of the rectangular pulse by a factor of &nbsp;$\sqrt{2}$ &nbsp; (see bottom sketch)&nbsp; and occurs at one of the four red amplitude coefficients,&nbsp; i.e.,&nbsp; whenever &nbsp;$\vert a_{\rm I \it ν}\vert = \vert a_{\rm Q \it ν}\vert  =1$.}}
  
Bei NRZ–rechteckförmigem Sendegrundimpuls $g_s(t)$ mit der Amplitude $g_0$ und der Symboldauer $T$ ist das Spektrum $G_s(f)$ si–förmig. In diesem Fall gilt für
+
==Signal waveforms for 4–QAM==
[[File:P_ID2964__Mod_T_4_3_S3b_Ganz_neu.png | Rechteckförmiger Grundimpuls | rechts]]
+
<br>
*die mittlere Energie pro Symbol:
+
The following graph shows the signal waveforms of&nbsp; $\rm 4-QAM$,&nbsp; where the coloring corresponds to the signal space allocation defined above.
$$E_{\rm S}  = \frac{{\rm E}[\hspace{0.05cm}|a_{\nu} |^2]\hspace{0.05cm}}{2}\cdot
 
g_0^2 \cdot T = \frac{5}{9}\cdot g_0^2 \cdot T \hspace{0.05cm},$$
 
*die mittlere Energie pro Bit:
 
$$E_{\rm B}  = \frac{E_{\rm S}}{4}\approx 0.139 \cdot g_0^2 \cdot T \hspace{0.05cm}.$$
 
  
 +
[[File:P_ID1711__Mod_T_4_3_S4_neu.png|right|frame| Signal waveforms for&nbsp; $\text{4-QAM}$]]
  
Die maximale Hüllkurve $s_0$ ist um den Faktor „Wurzel aus 2” größer als die Amplitude $g_0$ des Rechteckimpulses (siehe untere Skizze) und tritt bei einem der roten Amplitudenkoeffizienten auf, also immer dann, wenn $|a_{\rm }| = |a_{\rm }| =$ 1 ist.
+
One can see from these plots:
{{end}}  
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*The serial-to-parallel&nbsp; $\rm (S/P)$&nbsp; conversion of the source signal &nbsp;$q(t)$&nbsp; into the two component signals&nbsp; $s_{\rm I}(t)$&nbsp; and &nbsp;$s_{\rm Q}(t)$,&nbsp; each with symbol duration &nbsp;$T = 2T_{\rm B}$&nbsp; and signal values &nbsp;$±g_0$.&nbsp; Here,&nbsp; &nbsp;$T_{\rm B}$&nbsp; denotes the&nbsp; "bit duration".
 +
*The two phase modulated signals &nbsp;$s_{\rm cos}(t)$&nbsp; and &nbsp;$s_{\rm –sin}(t)$&nbsp; with phase jumps around &nbsp;$±π$:
 +
:$$s_{\rm cos} (t)  = s_{\rm I} (t) \cdot \cos(2\pi f_{\rm T}t)\hspace{0.05cm},$$
 +
:$$s_{\rm -sin} (t)  = -s_{\rm Q} (t) \cdot \sin(2\pi f_{\rm T}t)\hspace{0.05cm}. $$
 +
*The QAM signal &nbsp;$s(t) = s_{\rm cos}(t) \ – \ s_{\rm –sin}(t)$&nbsp; with phase jumps by multiples of &nbsp;$±π/2$;&nbsp; their envelope is larger than the two component signals by a factor of &nbsp;$\sqrt{2}$:
 +
:$$s_0 = \sqrt{2} \cdot  g_0 \hspace{0.05cm}.$$
 +
*Here,&nbsp; the basic transmission pulse &nbsp;$g_s(t)$&nbsp; is assumed to be rectangular between  &nbsp;$0$&nbsp; and &nbsp;$T$,&nbsp; i.e.,&nbsp; asymmetric with respect to&nbsp; $t = 0$,&nbsp; for simplicity of representation.&nbsp; The associated spectral function &nbsp;$G_s(f)$&nbsp;of this causal pulse &nbsp; $g_s(t)$&nbsp; is complex,&nbsp; though this has no implications in this context.
 +
<br clear=all>
  
==Signalverläufe der 4–QAM==
+
==Error probabilities of 4–QAM==
Die folgende Grafik zeigt die Signalverläufe der 4–QAM, wobei die Farbgebung mit der eingezeichneten Signalraumzuordnung übereinstimmt.
+
<br>
 +
In the earlier section &nbsp;[[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|"Error probabilities &ndash; a brief overview"]],&nbsp; the bit error probability of&nbsp; "binary phase shift keying" &nbsp; $\rm (BPSK)$&nbsp; was given.&nbsp; Now the results are transferred to&nbsp; $\text{4-QAM}$,&nbsp; where the following conditions still apply:
 +
[[File:P_ID1717__Mod_T_4_3_S5a_ganz_neu.png|right|frame|Phase diagram of&nbsp; $\rm BPSK$]]
 +
*a transmitted signal with the average energy &nbsp;$E_{\rm B}$&nbsp; per bit,
 +
*AWGN noise with the&nbsp; (single-sided)&nbsp; noise power density &nbsp;$N_0$,  
 +
*the best possible receiver realization using the matched-filter principle.
  
  
[[File:P_ID1711__Mod_T_4_3_S4_neu.png | Signalverläufe der 4-QAM]]
+
The upper graph shows the BPSK phase diagram of the detection signal &nbsp;$d(t)$,&nbsp; i.e., including the matched filter.&nbsp; The distance of the useful signal from the threshold&nbsp; $(d_{\rm Q}$–axis$)$&nbsp; is &nbsp;$s_0$ at each of the detection times.
  
 +
Using the additional equations
 +
:$$p_{\rm B} = {\rm Q}\left ( {s_0}/{\sigma_d } \right ), \hspace{0.2cm} E_{\rm B}  =  {1}/{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm} \sigma_d^2 = {N_0}/{T_{\rm B} }$$
 +
the&nbsp; &raquo;'''BPSK error probability'''&laquo; is given by:
 +
$$p_{\rm B, \hspace{0.1cm}BPSK} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$
  
Man erkennt aus diesen Darstellungen:  
+
[[File: P_ID2965__Mod_T_4_3_S5b_neu.png|right|frame|Phase diagram of&nbsp; $\text{4-QAM}$]]
*die Seriell–Parallel–Wandlung des Quellensignals $q(t)$ in die beiden Komponentensignale $s_{\rm I}(t)$ und $s_{\rm Q}(t)$, jeweils mit der Symboldauer $T = 2T_{\rm B}$ und den Signalwerten $±g_0 (T_{\rm B}$ ist die Bitdauer);  
+
In the&nbsp; $\text{4-QAM}$&nbsp; corresponding to the graph below
*die beiden trägerfrequenzmodulierten Signale $s_{\rm cos}(t)$ und $s_{\rm –sin}(t)$ mit Phasensprüngen um $±π$:
+
#there are two thresholds between the areas with lighter/darker background&nbsp; (blue line)&nbsp; and between the dotted/dashed areas&nbsp; (red line),&nbsp; but
$$s_{\rm cos} (t) = s_{\rm I} (t) \cdot \cos(2\pi f_{\rm T}t)\hspace{0.05cm},\hspace{0.2cm}
+
#the distance from each threshold is only &nbsp;$g_0$&nbsp; instead of &nbsp;$s_0$,
s_{\rm -sin} (t)  = -s_{\rm Q} (t) \cdot \sin(2\pi f_{\rm T}t)\hspace{0.05cm}, $$
+
#the noise power &nbsp;$\sigma_d^2$&nbsp; is also only half as large compared to BPSK because of the halved symbol rate in each sub-branch.
*das Sendesignal $s(t) = s_{\rm cos}(t) \ – \ s_{\rm –sin}(t)$ mit Phasensprüngen um Vielfache von $±π/2$; deren Hüllkurve ist gegenüber den beiden Komponentensignalen um den Faktor „Wurzel aus 2” größer:
 
$$s_0 = \sqrt{2} \cdot  g_0 \hspace{0.05cm}.$$
 
  
  
Anzumerken ist, dass hier der Sendegrundimpuls $g_s(t)$ zur Vereinfachung der Darstellung im Bereich von 0 bis $T$ als rechteckförmig (also unsymmetrisch bezüglich $t =$ 0) angenommen wurde. Die zugehörige Spektralfunktion $G_s(f)$ dieses kausalen Impulses ist komplex, was jedoch in diesem Zusammenhang keine Auswirkungen hat.
+
Thus,&nbsp; with the equations
 +
:$$p_{\rm B} = {\rm Q}\left ( {g_0}/{\sigma_d } \right ), \hspace{0.2cm}g_{0}  =  {s_0}/{\sqrt{2}}, \hspace{0.2cm}E_{\rm B}  =  {1}/{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm}
 +
\sigma_d^2 = {N_0}(2 \cdot T_{\rm B} )$$
 +
one obtains the exact same result for the&nbsp; &raquo;'''4-QAM error probability'''&laquo;&nbsp; as for BPSK:
 +
:$$p_{\rm B, \hspace{0.1cm}4-QAM} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$
  
==Fehlerwahrscheinlichkeit der 4–QAM==
+
{{BlaueBox|TEXT=
Im Kapitel 4.2 wurde die Fehlerwahrscheinlichkeit der BPSK angegeben. Nun werden die Ergebnisse auf die 4–QAM übertragen, wobei weiterhin folgende Voraussetzungen gelten:  
+
$\text{Conclusion:}$&nbsp;
*ein Sendesignal mit der mittleren Energie $E_{\rm B}$ pro Bit,  
+
*Under ideal conditions,&nbsp; $\text{4-QAM}$&nbsp; '''has the same error probability as''' &nbsp; $\text{BPSK}$,&nbsp; although twice the amount of information can be transmitted.
*AWGN–Rauschen mit der Rauschleistungsdichte $N_0$,  
+
*However,&nbsp; '''if the conditions are no longer ideal'''&nbsp; &ndash; for example,&nbsp; if there is an unwanted phase offset between transmitter and receiver &ndash;&nbsp; '''there is much more degradation with 4-QAM than with BPSK'''.
*bestmögliche Empfängerrealisierung nach dem Matched–Filter–Prinzip.
+
*This case is considered in more detail in the section &nbsp; [[Digital_Signal_Transmission/Linear_Digital_Modulation_-_Coherent_Demodulation#Error_probabilities_for_4.E2.80.93QAM_and_4.E2.80.93PSK|"Error probabilities for 4–QAM and 4–PSK"]]&nbsp; of the book&nbsp; "Digital Signal Transmission". }}
  
 +
==Quadratic QAM signal space constellations==
 +
<br>
 +
The following figure shows the signal space constellations of&nbsp; $\text{4-QAM}$,&nbsp; $\text{16-QAM}$&nbsp; and&nbsp; $\text{64-QAM}$.
 +
[[File:P_ID1713__Mod_T_4_3_S6a_Ganz_neu.png |right|frame| Signal space constellation and decision regions for &nbsp;$M = 4$, &nbsp;$M = 16$, &nbsp;$M = 64$;<br>Note: &nbsp; The figures refer only to the detection time points.&nbsp; The transitions between the individual points outside the detection times are not shown here.]]
 +
*With the axis labels chosen here, the figures also describe the useful detection signal&nbsp; (at detection times)&nbsp; in the equivalent low-pass range.
 +
*Also plotted are the various decision regions assigned to the noisy detection signal.
 +
*The arrows indicate when decision regions are extended to infinity.
  
[[File:P_ID1717__Mod_T_4_3_S5a_ganz_neu.png | Phasendiagramm von BPSK | rechts]]
 
Die obere Abbildung zeigt das BPSK–Phasendiagramm des Detektionssignals $d(t)$, also inklusive dem Matched–Filter. Der Abstand des Nutzsignals von der Schwelle $(d_{\rm Q}$–Achse) beträgt zu den Detektionszeitpunkten jeweils $s_0$. Mit den weiteren Gleichungen
 
$$p_{\rm B} = {\rm Q}\left ( \frac{s_0}{\sigma_d } \right ), \hspace{0.2cm} E_{\rm B}  = \frac {1}{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm} \sigma_d^2 = \frac{N_0}{T_{\rm B} }$$
 
erhält man für die BPSK–Fehlerwahrscheinlichkeit:
 
$$p_{\rm B, \hspace{0.05cm}BPSK} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = \frac{1}{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$
 
  
 +
The figures  apply to all Nyquist systems
 +
*such as the&nbsp; "rectangular-rectangular"&nbsp; configuration&nbsp; (basic transmission pulse and receiver filter impulse response are both rectangular)
 +
*or a &nbsp;[[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|$\text{root cosine rolloff Nyquist system}$]].
  
[[File: P_ID2965__Mod_T_4_3_S5b_neu.png | Phasendiagramm von 4–QAM | rechts]]
 
Bei der 4–QAM entsprechend der unteren Abbildung
 
*gibt es nun zwei Schwellen zwischen den Bereichen mit hellerem/dunklerem Hintergrund (blaue Linie) sowie zwischen den gepunkteten/gestrichelten Flächen (rot),
 
*ist der Abstand von den Schwellen jeweils nur noch $g_0$ anstelle von $s_0$,
 
*ist aber die Rauschleistung wegen der halb so großen Symbolrate in jedem Teilzweig gegenüber der BPSK auch nur halb so groß.
 
  
 +
However,&nbsp; the transitions between the individual points outside the detection times depend very much on the selected Nyquist system.
  
Mit den Gleichungen
+
Further things to note about these representations:
$$p_{\rm B} = {\rm Q}\left ( \frac{g_0}{\sigma_d } \right ), \hspace{0.2cm}g_{0}  = \frac {s_0}{\sqrt{2}}, \hspace{0.2cm}E_{\rm B}  = \frac {1}{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm}
+
#With a&nbsp; "true QAM structure" &nbsp; &rArr; &nbsp;  a square or at least rectangular signal space configuration: &nbsp; <br>The&nbsp; "two-dimensional detection process"&nbsp; can be solved in a simplified way by two&nbsp; "one-dimensional detection processes".
\sigma_d^2 = \frac{N_0}{2 \cdot T_{\rm B} }$$
+
#$\text{16-QAM}$&nbsp; is thus nothing more than the parallel transmission of two digital signals with &nbsp;$M = 4$&nbsp; amplitude levels each.
erhält man für die 4–QAM–Fehlerwahrscheinlichkeit genau das gleiche Ergebnis wie für die BPSK:
+
#For&nbsp; $\text{64-QAM}$ &nbsp; &rArr; &nbsp; $M = 8$&nbsp; would apply analogously.&nbsp; For&nbsp; $\text{256-QAM}$:&nbsp; the "one-dimensional" number of levels is &nbsp;$M = 16$.
$$p_{\rm B, \hspace{0.05cm}4-QAM} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = \frac{1}{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$
+
#All the properties for redundancy-free multi-level signals mentioned in the later chapter &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding|"Redundancy-free Coding"]]&nbsp; also apply here.
  
  
{{Box}}
+
==Other signal space constellations==
'''Fazit:''' Die 4–QAM weist bei idealen Bedingungen die gleiche Fehlerwahrscheinlichkeit wie die BPSK auf, obwohl die doppelte Informationsmenge übertragen werden kann. Sind allerdings die Bedingungen nicht mehr ideal – zum Beispiel bei einem ungewollten Phasenversatz zwischen Sender und Empfänger – so gibt es bei der 4–QAM eine deutlich stärkere Degradation als bei der BPSK. Dieser Fall wird im Kapitel 1.5 des Buches „Digitalsignalübertragung” noch genauer betrachtet.  
+
<br>
{{end}}
+
The graphic shows further signal space constellations.&nbsp;  On the left,&nbsp; 4-QAM is shown according to the previous description.&nbsp;  The next constellation to the right indicates a four-level phase modulation,&nbsp; which we call&nbsp; "4-PSK"&nbsp; or&nbsp; "QPSK".&nbsp;  A comparison of these two diagrams on the left shows:
 +
[[File:P_ID1714__Mod_T_4_3_S7_neu.png |right|frame|Further signal space constellations of ASK and PSK<br><br>]]
  
==QAM–Signalraumkonstellationen==
+
*The variant referred to here as&nbsp; $\rm QPSK$&nbsp; ("Quaternary Phase Shift Keying")&nbsp;  uses the phase positions&nbsp; $0^\circ$,&nbsp; $90^\circ$,&nbsp; $180^\circ$,&nbsp; $270^\circ$.&nbsp; From the plotted decision regions,&nbsp; it can be seen that the detection cannot be attributed to two binary decisions here.
Die nachfolgende Grafik zeigt die Signalraumkonstellationen von 4–QAM, 16–QAM und 64–QAM. Mit den hier gewählten Achsenbeschriftungen beschreiben die Bilder auch das Detektionsnutzsignal (zu den Detektionszeitpunkten) im äquivalenten Tiefpassbereich. Ebenfalls eingezeichnet sind die verschiedenen Entscheidungsgebiete, die dem verrauschten Detektionssignal zugeordnet werden. Die Pfeile geben an, wenn Entscheidungsgebiete bis ins Unendliche ausgedehnt sind.  
 
  
 +
 +
*$\rm 4–QAM$&nbsp; (left diagram)&nbsp; can also be understood as a four-level phase modulation&nbsp; $($positions &nbsp;$±45^\circ$,&nbsp; $±135^\circ)$.&nbsp; Compared to QPSK,&nbsp; there is a rotation of &nbsp;$±45^\circ$&nbsp; $(π/4)$ &nbsp; &rArr; &nbsp; 4–QAM is also called &nbsp;${\rm π/4}\text{–QPSK}$.
  
[[File:P_ID1713__Mod_T_4_3_S6a_Ganz_neu.png | Signalraumkonstellation und Entscheidungsgebiete bei M = 4, M = 16 und M = 64]]
+
 +
*Similarly to how one arrives at DPSK by pre-coding BPSK,&nbsp; "4-PSK"&nbsp; can also be extended to&nbsp; $\rm 4-DPSK$,&nbsp; thereby facilitating demodulation.&nbsp; This was used e.g. for data transmission over telephone channels &nbsp; &rArr; &nbsp;  CCITT recommendation&nbsp; "V26" $($carrier frequency &nbsp;$\text{1800 Hz}$,&nbsp; data rate &nbsp;$\text{2400 bit/s)}$.
  
  
Zu diesen Darstellungen ist anzumerken:  
+
The two diagrams on the right show higher-level modulation methods:
*Die Bilder beziehen sich nur auf die Detektionszeitpunkte und gelten für alle Nyquistsysteme wie die Rechteck–Rechteck–Konfiguration oder ein Wurzel–Cosinus–Rolloff–Nyquistsystem. 
+
*The&nbsp; "8–PSK" &nbsp;(or&nbsp; "8–DPSK")&nbsp; allows a data rate of up to&nbsp; $\text{4800 bit/s}$&nbsp; for the telephone channel according to CCITT recommendation&nbsp; "V27".
*Die hier nicht dargestellten Übergänge außerhalb der Detektionszeitpunkte zwischen den einzelnen Punkten hängen dagegen sehr wohl vom gewählten Nyquistsystem ab.
+
*Recommendation&nbsp; "V29"&nbsp; enables a hybrid modulation in the form of&nbsp; "16-ASK/PSK",&nbsp; which enables data rates of up to&nbsp; $\text{9600 bit/s}$&nbsp; for permanently connected lines.
*Bei echter QAM–Struktur – das heißt: die Signalraumkonstellation ist quadratisch oder zumindest rechteckförmig – lässt sich die 2D–Detektion durch zwei „eindimensionale” Detektionsvorgänge vereinfacht lösen.
 
*Die 16–QAM ist somit nichts anderes als die parallele Übertragung zweier Digitalsignale mit jeweils $M =$ 4 Amplitudenstufen. Bei der 64–QAM würde entsprechend $M =$ 8 gelten und bei der 256–QAM ist die „eindimensionale” Stufenzahl $M =$ 16.  
 
*Alle im Kapitel 2.2 von Buch „Digitalsignalübertragung” genannten Eigenschaften für mehrstufige Signale gelten auch hier, wobei allerdings die Zusetzung der orthogoalen Trägerfrequenzsignale noch geeignet zu berücksichtigen ist.  
 
  
==Weitere Signalraumkonstellationen==
+
==Nyquist and Root-Nyquist QAM systems==
Die Grafik zeigt weitere Signalraumkonstellationen, wobei links wieder die 4–QAM nach der bisherigen Beschreibung dargestellt ist.  
+
<br>
 +
So far in this chapter,&nbsp; for reasons of easy representation,&nbsp; the rectangular basic transmission pulse has always been assumed.&nbsp;  In practice,&nbsp; however,&nbsp; a&nbsp; &raquo;'''root-Nyquist characteristic'''&laquo;&nbsp; is usually used according to the [[Digital_Signal_Transmission/Optimization_of_Baseband_Transmission_Systems#Root_Nyquist_systems|$\text{description}$]]&nbsp; in the book&nbsp; "Digital Signal Transmission".&nbsp;  Briefly,&nbsp; these systems can be characterized as follows:
  
 +
#The receiver&nbsp; (German:&nbsp; "Empfänger" &nbsp; &rArr; &nbsp; subscript;&nbsp; "E")&nbsp; frequency response &nbsp;$H_{\rm E}(f)$&nbsp; is chosen here to be equal in shape to the (normalized) transmission pulse&nbsp; (German:&nbsp; "Sendeimpuls" &nbsp; &rArr; &nbsp; subscript;&nbsp; "S")&nbsp; spectrum &nbsp;$H_{\rm S}(f)$,&nbsp; which leads to the smallest possible error probability under the constraints of power limitation.
 +
#The overall frequency response &nbsp;$H_{\rm Nyq}(f) = H_{\rm S}(f) · H_{\rm E}(f)$&nbsp; satisfies the [[Digital_Signal_Transmission/Properties_of_Nyquist_Systems#First_Nyquist_criterion_in_the_time_domain|$\text{first Nyquist criterion}$]],&nbsp; so there is no intersymbol interference&nbsp; $\text{(ISI)}$&nbsp; at the receiver.&nbsp; Thus,&nbsp; $H_{\rm S}(f)$&nbsp; and &nbsp;$H_{\rm E}(f)$&nbsp; each have a root-Nyquist characteristic.
 +
#For the frequency response &nbsp;$H_{\rm Nyq}(f)$,&nbsp; one often uses a&nbsp; [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Cosine.E2.80.93square_low-pass_filter|$\text{cosine rolloff low-pass}$]]&nbsp; $H_{\rm CRO}(f)$&nbsp; with equivalent bandwidth &nbsp;$Δf_{\rm CRO} = 1/T$&nbsp; and freely chosen rolloff factor&nbsp;$ (0 ≤ r ≤ 1)$.&nbsp;  The interactive applet &nbsp;[[Applets:Frequency_%26_Impulse_Responses|"Frequency and Impulse Responses"]]&nbsp; illustrates frequency response and impulse response of this low-pass filter.
 +
#The advantage of these root-Nyquist systems is the much smaller bandwidth &nbsp;$(1 + r)/T$&nbsp; compared to the previously considered configuration with rectangular&nbsp; $g_s(t)$&nbsp; and rectangular&nbsp; $h_{\rm E}(t)$,&nbsp; whose spectrum is&nbsp; (theoretically)&nbsp; infinitely extended.
 +
#In terms of error probability,&nbsp; nothing changes compared to the&nbsp; "rectangular-rectangular"&nbsp; configuration because the basic detection pulse &nbsp;$g_d(t)$&nbsp;  before the decision has equidistant zero crossings,&nbsp; thus avoiding intersymbol interference.
  
[[File:P_ID1714__Mod_T_4_3_S7_neu.png | Weitere Signalraumkonstellationen von ASK und PSK]]
 
  
 +
The graph shows phase diagrams for this case, taken from the book &nbsp; [Kam04]<ref name="Kam04">Kammeyer, K.D.:&nbsp; Nachrichtenübertragung.&nbsp; Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref>.&nbsp;  The rolloff factor is&nbsp;$r = 0.5$.&nbsp; From these plots,&nbsp; one can see:
 +
[[File:EN_Mod_T_4_3_S8.png|right|frame|Phase diagram of 4-QAM with a root-Nyquist configuration]]
  
Die Konstellation rechts daneben kennzeichnet eine vierstufige Phasenmodulation (4–PSK bzw. QPSK – ''Quaternary Phase Shift Keying''). Ein Vergleich der beiden linken Diagramme zeigt:
+
*The right plot shows the detection signals in the &nbsp;$\rm I$&nbsp; and &nbsp;$\rm Q$ branches after the root-Nyquist receiver filters in a two-dimensional representation. The corresponding spectra each have cosine-shaped slopes around the Nyquist frequency&nbsp;$f_{\rm Nyq} = 1/(2T)$.  
*Die hier als QPSK bezeichnete Variante verwendet die Phasenlagen 0°, 90°, 180° und 270°. Man erkennt aus den eingezeichneten Entscheidungsgebieten, dass hier die Detektion nicht auf zwei Binärentscheidungen zurückgeführt werden kann.  
+
*At the detection times, only the four yellow points are possible in the phase diagram.&nbsp;  The transitions in between are diverse.&nbsp; It should be noted that only a few lines pass through the coordinate null point.
*Auch die 4–QAM kann als eine vierstufige Phasenmodulation mit den möglichen Phasenlagen ±45° und ±135° aufgefasst werden. Gegenüber dem zweiten Phasendiagramm ergibt sich eine Drehung um 45° $(π$/4), so dass die 4–QAM oft auch als $π$/4–QPSK bezeichnet wird.  
+
*Shown on the left are the two transmitted signals in the equivalent low-pass region, &nbsp; $s_{\rm I}(t) = {\rm Re}\big[s_{\rm TP}(t)\big]$&nbsp; and &nbsp;$s_{\rm Q}(t) = {\rm Im}\big[s_{\rm TP}(t)\big]$.  Due to the root-Nyquist spectral shaping, there is intersymbol interference at the transmitter,&nbsp; which means that the equivalent low-pass signal &nbsp;$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} · s_{\rm Q}(t)$&nbsp; is also not limited to four points at the detection times.
*Ähnlich wie man bei der BPSK durch Vorcodierung zur DPSK kommt, kann auch die 4–PSK zur 4–DPSK erweitert und dadurch die Demodulation erleichtert werden. Diese wird zum Beispiel bei der Datenübertragung über Telefonkanäle gemäß der CCITT–Empfehlung V26 angewendet (Trägerfrequenz 1800 Hz, Datenrate 2400 bit/s).  
+
*The magnitude &nbsp;$|s_{\rm TP}(t)|$&nbsp; &ndash; that is, the distance from zero &ndash;&nbsp; indicates the envelope of the 4-QAM signal. It can be clearly seen from the left diagram that there are strong amplitude dips especially with phase changes around &nbsp;$π$,&nbsp; since then &nbsp;$s_{\rm TP}(t)$&nbsp; also often assumes&nbsp; (complex)&nbsp; values close to zero.
  
  
Die beiden rechten Diagramme zeigen höherstufige Modulationsverfahren:  
+
==Offset–Quadrature amplitude modulation==
*Die 8–PSK bzw. 8–DPSK erlaubt entsprechend der CCITT–Empfehlung V27 beim Telefonkanal eine Datenrate von bis zu 4800 bit/s.
+
<br>
*Die Empfehlung V29 sieht mit der 16–ASK/PSK eine hybride Modulationsform vor, die bei ausgewählten, fest verschalteten Leitungen eine Datenrate von 9600 bit/s ermöglicht.  
+
Taking the equations for the&nbsp; "$\text{4–QAM}$" &nbsp; $($or &nbsp;"$π/4\text{–QPSK}$"$)$&nbsp; as our starting point,&nbsp; we arrive at the &nbsp; "$\text{Offset–4–QAM}$",&nbsp; which we simplify by calling &nbsp; "$\text{Offset–QPSK}$".&nbsp;  Here the transmitted signal is given by:
 +
:$$s(t)  =s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t) -  s_{\rm Q}(t)
 +
\cdot \sin(2 \pi f_{\rm T} t),$$
 +
:$$\hspace{0.3cm}\Rightarrow \hspace{0.3cm}s_{\rm I}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm},\hspace{0.5cm} s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm Q\hspace{0.03cm}\it \nu} \cdot g_s (t -{T}/{2} - \nu \cdot T)\hspace{0.05cm}.$$
  
==Nyquist– und Wurzel–Nyquist–QAM–Systeme (1)==
+
The sole but decisive difference is the time shift of the quadrature component with respect to the in-phase component by half a symbol duration &nbsp;$(T/2)$.&nbsp;  This has the advantage that
Bisher wurde im Kapitel 4.3 aus Darstellungsgründen stets vom rechteckförmigen Sendegrundimpuls ausgegangen. In der Praxis verwendet man aber meist eine Wurzel–Nyquist–Charakteristik entsprechend dem Kapitel 1.4 im Buch „Digitalsignalübertragung”. In aller Kürze lassen sich diese Systeme wie folgt darstellen:
+
*the phase function does not pass through zero,&nbsp; and
*Der Empfängerfrequenzgang $H_{\rm E}(f)$ wird hier formgleich mit dem normierten Sendeimpulsspektrum $H_{\rm S}(f)$ gewählt, was unter der Nebenbedingung der Leistungsbegrenzung (das heißt: konstante mittlere Sendeleistung) zur kleinsten Fehlerwahrscheinlichkeit führt.  
+
*the envelope&nbsp; $|s_{\rm TP}(t)|$&nbsp; therefore fluctuates much less.  
*Der Gesamtfrequenzgang $H_{\rm Nyq}(f) = H_{\rm S}(f) · H_{\rm E}(f)$ erfüllt das erste Nyquistkriterium, so dass es beim Empfänger zu keinen Impulsinterferenzen kommt. $H_{\rm S}(f)$ und $H_{\rm E}(f)$ haben somit jeweils eine Wurzel–Nyquist–Charakteristik.
 
*Für den Frequenzgang $H_{\rm Nyq}(f)$ verwendet man einen Cosinus–Rolloff–Tiefpass $H_{\rm CRO}(f)$ mit der äquivalenten Bandbreite $Δf_{\rm CRO} = 1/T$ und frei wählbarem Rolloff–Faktor $(0 ≤ r ≤ 1)$.  
 
  
  
Mit dem nachfolgenden Interaktionsmodul können Sie sich den Frequenzgang und die Impulsantwort dieses Tiefpasses verdeutlichen:  
+
{{GraueBox|TEXT=
 +
[[File:EN_Mod_T_4_3_S9_neu.png|right|frame|Transmitter phase diagram  and envelope at "$π/4$&ndash;QPSK" (above) and "Offset&ndash;QPSK" (below)]]
 +
$\text{Example 2:}$&nbsp; The upper graph shows
 +
* the phase diagram for &nbsp;$π/4\hspace{-0.05cm}-\hspace{-0.05cm}\text{QPSK}$ on the left,
 +
* on the right,&nbsp; a typical envelope curve,&nbsp; based on a root Nyquist spectrum with rolloff factor &nbsp;$r = 0.5$,&nbsp; as in the last section.
 +
<br><br><br>
 +
The lower graphic shows that the &nbsp; $\text{Offset – QPSK}$&nbsp; has significantly better properties with respect to the envelope&nbsp; (less severe signal dips).
 +
<br><br><br>
  
Frequenzgang und zugehörige Impulsantwort  
+
&rArr; &nbsp; Essential properties of&nbsp; "4-QAM/QPSK"&nbsp; and&nbsp; "Offset-QPSK"&nbsp; can be illustrated with the interactive&nbsp; (German language)&nbsp; SWF applet  &nbsp;[[Applets:QPSK_und_Offset-QPSK_(Applet)|"QPSK and Offset-QPSK"]],&nbsp; where the basic pulse can be alternatively chosen as
 +
*a rectangular pulse,
 +
*a cosine pulse,
 +
*a Nyquist pulse,
 +
*a root Nyquist pulse.}}  
  
Der Vorteil dieser Wurzel–Nyquist–Systeme ist die deutlich kleinere Bandbreite $(1 + r)/T$ gegenüber der bisher betrachteten Konfiguration mit rechteckförmigem $g_s(t)$ und rechteckförmigem $h_{\rm E}(t)$, dessen Spektrum (theoretisch) unendlich weit ausgedehnt ist. Hinsichtlich Fehlerwahrscheinlichkeit ändert sich gegenüber der Rechteck–Rechteck–Konfiguration nichts, da der Grundimpuls $g_d(t)$ vor dem Entscheider äquidistante Nulldurchgänge aufweist und somit Impulsinterferenzen vermieden werden.  
+
==Exercises for the chapter==
 +
<br>
 +
[[Aufgaben:Aufgabe_4.10:_Signalverläufe_der_16–QAM|Exercise 4.10: Signal Waveforms of the 16-QAM]]
  
 +
[[Aufgaben:Aufgabe_4.10Z:_Signalraumkonstellation_der_16–QAM|Exercise 4.10Z: Signal Space Constellation of the 16-QAM]]
  
[[File:P_ID1715__Mod_T_4_3_S8_neu.png | Phasendiagramme der 4-QAM bei Wurzel-Nyquist-Konfiguration]]
+
[[Aufgaben:Aufgabe_4.11:_Frequenzbereichsbetrachtung_der_4–QAM|Exercise 4.11: Frequency Domain Consideration of the 4-QAM]]
  
 +
[[Aufgaben:Aufgabe_4.11Z:_Fehlerwahrscheinlichkeit_bei_QAM|Exercise 4.11Z: Error Probability with QAM]]
  
Die Grafik zeigt die Phasendiagramme für diesen Fall, die dem Buch [Kam04]<ref name="Kam04">Kammeyer, K.D.: ''Nachrichtenübertragung.'' Stuttgart: B.G. Teubner, 4. Auflage, 2004.</ref> entnommen sind. Der Rolloff–Faktor beträgt $r =$ 0.5.
+
[[Aufgaben:Exercise_4.12:_Root-Nyquist_Systems|Exercise 4.12: Root-Nyquist Systems]]
  
 +
[[Aufgaben:Exercise_4.12Z:_4-QAM_Systems_again|Exercise 4.12Z: 4-QAM Systems again]]
  
  
 +
==References==
  
==Quellenverzeichnis==
 
 
<references/>
 
<references/>
  
 
{{Display}}
 
{{Display}}

Latest revision as of 14:50, 13 January 2023

General description and signal space allocation


Due to the orthogonality of cosine and  (minus)  sine,  two data streams can be transmitted independently via the same transmission channel.  The diagram shows the general circuit schematic.

Linear modulator with  $\rm I$ and  $\rm Q$–components;  signal space for  $\text{16-QAM}$

This very general model can be described as follows:

  • The binary source symbol sequence $〈q_k〉$  with bit rate  $R_{\rm B}$ is applied to the input.  Thus,  the time interval between two symbols is  $T_{\rm B} = 1/R_{\rm B}$.
  • Two multi-level amplitude coefficients  $a_{{\rm I}ν}$  and  $a_{{\rm Q}ν}$  are derived from each  $b$  binary input symbols  $q_k$,  where  $\rm I$  stands for  "inphase component"  and  $\rm Q$  stands for  "quadrature component".
  • If  $b$  is even and the signal space allocation is quadratic, then the coefficients  $a_{{\rm I}ν}$  and  $a_{{\rm Q}ν}$  can each take on one of the  $M = 2^{b/2}$  amplitude values with equal probability.  This is then referred to as  »quadrature amplitude modulation«  $\rm (QAM)$.
  • The example considered in the graph is for the   $\text{16-QAM}$  with  $b = M = 4$  and correspondingly  $M^2 =16$  signal space points.  For a   $\text{256-QAM}$,  $b = 8$  and  $M = 16$  would apply:   $2^b = M^2 = 256$.
  • Next,  the coefficients  $a_{{\rm I}ν}$  and  $a_{{\rm Q}ν}$  are each applied to a $\text{Dirac comb}$  as pulse weights.  Thus,  after pulse shaping with the basic transmission pulse  $g_s(t)$,  the following holds for both branches of the circuit diagram:
$$s_{\rm I}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm},\hspace{1cm}s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm Q\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.$$
  • Note that because of the redundancy–free conversion,  the symbol duration  $T$  of these signals is larger by a factor of  $b$  than the bit duration  $T_{\rm B}$  of the binary source signal.  In the illustrated   $\text{16-QAM}$  example,  $T = 4 · T_{\rm B}$  holds.
  • The QAM transmitted signal  $s(t)$  is then the sum of the two signals multiplied by cosine and minus-sine, respectively:
$$s_{\rm cos}(t) = s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t), \hspace{1cm} s_{\rm -sin}(t) = -s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t)$$
$$\Rightarrow \hspace{0.3cm}s(t) = s_{\rm cos}(t)+ s_{\rm -sin}(t) = s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t) - s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t) \hspace{0.05cm}.$$

$\text{Conclusion:}$  These statements can be summarized as follows:

  • The two transmission branches   $\rm (I,\ Q)$  can be thought of as two completely separate  $M$-level ASK systems
    that do not interfere with each other as long as all components are optimally designed.
  • Quadrature amplitude modulation thus makes it  (ideally)  possible to double the data rate while maintaining the same quality.


System description using the equivalent low-pass signal


Since the multiplication of  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$  with a cosine or a minus-sine oscillation only causes a shift in the frequency domain and such a shift is a linear operation,  the system description can be greatly simplified using equivalent low-pass signals.

Linear modulator  $(\rm I$ and  $\rm Q$ components$)$  in the equivalent low-pass range
  1. The graph shows the simplified model in the baseband.  This is equivalent to the block diagram considered so far.
  2. The serial-parallel conversion and the signal space allocation drawn in red in  "the block diagram in the last section" are retained.  This block is no longer drawn here.
  3. We also initially disregard the band-pass  $H_{\rm BP}(f)$,  which is often introduced for technical reasons.


Please note the following:

  • All double arrows in the baseband model denote complex quantities.  The operations associated with them should also be understood as complex.  For example,  the complex amplitude coefficient  $a_ν$  combines one inphase and one quadrature coefficient:
$$a_\nu = a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm Q\hspace{0.03cm}\it \nu} \hspace{0.05cm}.$$
  • The equivalent low–pass representation of the actual, physical and thus per se real transmitted QAM signal  $s(t)$  is always complex and with the partial signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$ it holds for the equivalent low-pass signal  $($German:  "äquivalentes Tiefpass–Signal"   ⇒   subscript:  "TP"$)$:
$$s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} \cdot s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty} a_\nu \cdot g_s (t - \nu \cdot T)\hspace{0.05cm}.$$
  • The analytical signal  $s_+(t)$  is obtained from the equivalent low-pass signal  $s_{\rm TP}(t)$  by multiplying by the complex exponential function.  The physical signal  $s(t)$  is then obtained as the real part of  $s_+(t)$.
  • In order for the signs in the block diagram in the previous section and the sketched baseband model here to match,  multiplication by the negative sine wave is required in the quadrature branch,  as shown in the following calculation:
$$s(t) = {\rm Re}[s_{\rm +}(t)] = {\rm Re}[s_{\rm TP}(t) \cdot{\rm e}^{{\rm j}2\pi f_{\rm T} t}] $$
$$\Rightarrow \hspace{0.3cm} s(t) = {\rm Re} \left[\left ( \sum (a_{\rm I\hspace{0.03cm}\it \nu} + {\rm j} \cdot a_{\rm Q\hspace{0.03cm}\it \nu} ) \cdot g_s (t - \nu \cdot T)\right )\left ( \cos(2 \pi f_{\rm T} t) + {\rm j} \cdot \sin(2 \pi f_{\rm T} t) \right )\right]= s_{\rm I}(t) \cdot \cos(2\pi f_{\rm T} t) - s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t) \hspace{0.05cm}.$$
  • The influence of the band-pass  $H_{\rm BP}(f)$,  which in practice often has to be considered at the output of the QAM modulator,  can be assigned to the pulse shape filter  $g_s(t)$.  If the passband of the band-pass filter is symmetric about $f_{\rm T}$,  its low-pass equivalent  (in the time domain)   $h_{\rm BP\ →\ TP}(t)$   is purely real and one can replace  $g_s(t)$  with  $g_s(t) \star h_{\rm BP\ →\ TP}(t)$  in the model.

Power and energy of QAM signals


As shown in the chapter  "Equivalent Low-Pass Signal and its Spectral Function"  in the book  "Signal Representation",  the   »power«  of the transmitted QAM signal  $s(t)$  can also be calculated from the equivalent low-pass signal  $s_{\rm TP}(t)$,  which is always complex.  Thus,  it is equally valid to write:

$$P = \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} s^2(t)\,{\rm d} t = {\rm 1}/{2} \cdot \lim_{T_{\rm M} \rightarrow \infty} \frac{\rm 1}{T_{\rm M}}\cdot \int_{-T_{\rm M}/2}^{+T_{\rm M}/2} |s_{\rm TP}(t)|^2\,{\rm d} t \hspace{0.05cm}.$$

In contrast,  the energy of the unbounded signals  $s(t)$  and   $s_{\rm TP}(t)$  is infinite.

However,  if we restrict ourselves to a symbol duration $T$,  we obtain the  »energy per symbol«:

$$E_{\rm S} = \frac{{\rm E}[\hspace{0.05cm}|a_{\nu} |^2\hspace{0.05cm}]}{2}\cdot \int_{-\infty}^{+\infty} |g_s(t)|^2\,{\rm d} t = \frac{{\rm E}[|a_{\nu} |^2]}{2}\cdot \int_{-\infty}^{+\infty} |G_s(f)|^2\,{\rm d} f \hspace{0.05cm}.$$

On the other hand,   $E_{\rm B} = E_{\rm S}/b$   gives the  »energy per bit«  if according to the signal space assignment,  $b$  binary symbols are combined to the complex coefficient  $a_\nu$.

Allocation for $\text{16-QAM}$

$\text{Example 1:}$   The upper graph shows the signal space allocation for $\text{16-QAM}$,  where both the real and imaginary parts of the complex amplitude coefficients  $a_ν$  can take one of four values  $±1$  as well as  $±1/3$ , respectively.

Averaging over the $16$ squared distances to the origin yields:

$${\rm E}\big[\hspace{0.05cm} \vert a_{\nu}\vert^2 \hspace{0.05cm}\big] \hspace{0.18cm} = \hspace{0.18cm} \frac{4}{16} \cdot \left [1^2 + 1^2\right ]+ \frac{4}{16} \cdot \left[(1/3 )^2 +(1/3)^2 \right ] + \frac{8}{16} \cdot \left [1^2 + (1/3)^2\right ] $$
$$\Rightarrow \hspace{0.3cm}{\rm E}\big[\hspace{0.05cm} \vert a_{\nu}\vert^2 \hspace{0.05cm}\big] = {10}/{9}\approx 1.11 \hspace{0.05cm}.$$

In this order,  the summands belong to

  • the four red,
  • the four black,  and
  • the eight blue dots.


NRZ rectangular basic pulse

For a NRZ rectangular basic transmission pulse  $g_s(t)$  with amplitude  $g_0$  and symbol duration  $T$,  the spectrum  $G_s(f)$  is $\rm sinc$–shaped.  In this case,  the following holds for

  • the average energy per symbol:
$$E_{\rm S} = 1/2 \cdot {\rm E}\big[\hspace{0.05cm}\vert a_{\nu} \vert ^2 \big]\hspace{0.05cm}\cdot g_0^2 \cdot T = {5}/{9}\cdot g_0^2 \cdot T \hspace{0.05cm}\approx 0.555 \cdot g_0^2 \cdot T \hspace{0.05cm},$$
  • the average energy per bit:
$$E_{\rm B} ={E_{\rm S} }/{4}= {5}/{36}\cdot g_0^2 \cdot T \approx 0.139 \cdot g_0^2 \cdot T \hspace{0.05cm}.$$

The  "maximum envelope"  $s_0$  of the transmitted signal  $s(t)$  is larger than the amplitude  $g_0$  of the rectangular pulse by a factor of  $\sqrt{2}$   (see bottom sketch)  and occurs at one of the four red amplitude coefficients,  i.e.,  whenever  $\vert a_{\rm I \it ν}\vert = \vert a_{\rm Q \it ν}\vert =1$.

Signal waveforms for 4–QAM


The following graph shows the signal waveforms of  $\rm 4-QAM$,  where the coloring corresponds to the signal space allocation defined above.

Signal waveforms for  $\text{4-QAM}$

One can see from these plots:

  • The serial-to-parallel  $\rm (S/P)$  conversion of the source signal  $q(t)$  into the two component signals  $s_{\rm I}(t)$  and  $s_{\rm Q}(t)$,  each with symbol duration  $T = 2T_{\rm B}$  and signal values  $±g_0$.  Here,   $T_{\rm B}$  denotes the  "bit duration".
  • The two phase modulated signals  $s_{\rm cos}(t)$  and  $s_{\rm –sin}(t)$  with phase jumps around  $±π$:
$$s_{\rm cos} (t) = s_{\rm I} (t) \cdot \cos(2\pi f_{\rm T}t)\hspace{0.05cm},$$
$$s_{\rm -sin} (t) = -s_{\rm Q} (t) \cdot \sin(2\pi f_{\rm T}t)\hspace{0.05cm}. $$
  • The QAM signal  $s(t) = s_{\rm cos}(t) \ – \ s_{\rm –sin}(t)$  with phase jumps by multiples of  $±π/2$;  their envelope is larger than the two component signals by a factor of  $\sqrt{2}$:
$$s_0 = \sqrt{2} \cdot g_0 \hspace{0.05cm}.$$
  • Here,  the basic transmission pulse  $g_s(t)$  is assumed to be rectangular between  $0$  and  $T$,  i.e.,  asymmetric with respect to  $t = 0$,  for simplicity of representation.  The associated spectral function  $G_s(f)$ of this causal pulse   $g_s(t)$  is complex,  though this has no implications in this context.


Error probabilities of 4–QAM


In the earlier section  "Error probabilities – a brief overview",  the bit error probability of  "binary phase shift keying"   $\rm (BPSK)$  was given.  Now the results are transferred to  $\text{4-QAM}$,  where the following conditions still apply:

Phase diagram of  $\rm BPSK$
  • a transmitted signal with the average energy  $E_{\rm B}$  per bit,
  • AWGN noise with the  (single-sided)  noise power density  $N_0$,
  • the best possible receiver realization using the matched-filter principle.


The upper graph shows the BPSK phase diagram of the detection signal  $d(t)$,  i.e., including the matched filter.  The distance of the useful signal from the threshold  $(d_{\rm Q}$–axis$)$  is  $s_0$ at each of the detection times.

Using the additional equations

$$p_{\rm B} = {\rm Q}\left ( {s_0}/{\sigma_d } \right ), \hspace{0.2cm} E_{\rm B} = {1}/{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm} \sigma_d^2 = {N_0}/{T_{\rm B} }$$

the  »BPSK error probability« is given by: $$p_{\rm B, \hspace{0.1cm}BPSK} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$

Phase diagram of  $\text{4-QAM}$

In the  $\text{4-QAM}$  corresponding to the graph below

  1. there are two thresholds between the areas with lighter/darker background  (blue line)  and between the dotted/dashed areas  (red line),  but
  2. the distance from each threshold is only  $g_0$  instead of  $s_0$,
  3. the noise power  $\sigma_d^2$  is also only half as large compared to BPSK because of the halved symbol rate in each sub-branch.


Thus,  with the equations

$$p_{\rm B} = {\rm Q}\left ( {g_0}/{\sigma_d } \right ), \hspace{0.2cm}g_{0} = {s_0}/{\sqrt{2}}, \hspace{0.2cm}E_{\rm B} = {1}/{2} \cdot s_0^2 \cdot T_{\rm B} ,\hspace{0.2cm} \sigma_d^2 = {N_0}(2 \cdot T_{\rm B} )$$

one obtains the exact same result for the  »4-QAM error probability«  as for BPSK:

$$p_{\rm B, \hspace{0.1cm}4-QAM} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ).$$

$\text{Conclusion:}$ 

  • Under ideal conditions,  $\text{4-QAM}$  has the same error probability as   $\text{BPSK}$,  although twice the amount of information can be transmitted.
  • However,  if the conditions are no longer ideal  – for example,  if there is an unwanted phase offset between transmitter and receiver –  there is much more degradation with 4-QAM than with BPSK.
  • This case is considered in more detail in the section   "Error probabilities for 4–QAM and 4–PSK"  of the book  "Digital Signal Transmission".

Quadratic QAM signal space constellations


The following figure shows the signal space constellations of  $\text{4-QAM}$,  $\text{16-QAM}$  and  $\text{64-QAM}$.

Signal space constellation and decision regions for  $M = 4$,  $M = 16$,  $M = 64$;
Note:   The figures refer only to the detection time points.  The transitions between the individual points outside the detection times are not shown here.
  • With the axis labels chosen here, the figures also describe the useful detection signal  (at detection times)  in the equivalent low-pass range.
  • Also plotted are the various decision regions assigned to the noisy detection signal.
  • The arrows indicate when decision regions are extended to infinity.


The figures apply to all Nyquist systems


However,  the transitions between the individual points outside the detection times depend very much on the selected Nyquist system.

Further things to note about these representations:

  1. With a  "true QAM structure"   ⇒   a square or at least rectangular signal space configuration:  
    The  "two-dimensional detection process"  can be solved in a simplified way by two  "one-dimensional detection processes".
  2. $\text{16-QAM}$  is thus nothing more than the parallel transmission of two digital signals with  $M = 4$  amplitude levels each.
  3. For  $\text{64-QAM}$   ⇒   $M = 8$  would apply analogously.  For  $\text{256-QAM}$:  the "one-dimensional" number of levels is  $M = 16$.
  4. All the properties for redundancy-free multi-level signals mentioned in the later chapter  "Redundancy-free Coding"  also apply here.


Other signal space constellations


The graphic shows further signal space constellations.  On the left,  4-QAM is shown according to the previous description.  The next constellation to the right indicates a four-level phase modulation,  which we call  "4-PSK"  or  "QPSK".  A comparison of these two diagrams on the left shows:

Further signal space constellations of ASK and PSK

  • The variant referred to here as  $\rm QPSK$  ("Quaternary Phase Shift Keying")  uses the phase positions  $0^\circ$,  $90^\circ$,  $180^\circ$,  $270^\circ$.  From the plotted decision regions,  it can be seen that the detection cannot be attributed to two binary decisions here.


  • $\rm 4–QAM$  (left diagram)  can also be understood as a four-level phase modulation  $($positions  $±45^\circ$,  $±135^\circ)$.  Compared to QPSK,  there is a rotation of  $±45^\circ$  $(π/4)$   ⇒   4–QAM is also called  ${\rm π/4}\text{–QPSK}$.


  • Similarly to how one arrives at DPSK by pre-coding BPSK,  "4-PSK"  can also be extended to  $\rm 4-DPSK$,  thereby facilitating demodulation.  This was used e.g. for data transmission over telephone channels   ⇒   CCITT recommendation  "V26" $($carrier frequency  $\text{1800 Hz}$,  data rate  $\text{2400 bit/s)}$.


The two diagrams on the right show higher-level modulation methods:

  • The  "8–PSK"  (or  "8–DPSK")  allows a data rate of up to  $\text{4800 bit/s}$  for the telephone channel according to CCITT recommendation  "V27".
  • Recommendation  "V29"  enables a hybrid modulation in the form of  "16-ASK/PSK",  which enables data rates of up to  $\text{9600 bit/s}$  for permanently connected lines.

Nyquist and Root-Nyquist QAM systems


So far in this chapter,  for reasons of easy representation,  the rectangular basic transmission pulse has always been assumed.  In practice,  however,  a  »root-Nyquist characteristic«  is usually used according to the $\text{description}$  in the book  "Digital Signal Transmission".  Briefly,  these systems can be characterized as follows:

  1. The receiver  (German:  "Empfänger"   ⇒   subscript;  "E")  frequency response  $H_{\rm E}(f)$  is chosen here to be equal in shape to the (normalized) transmission pulse  (German:  "Sendeimpuls"   ⇒   subscript;  "S")  spectrum  $H_{\rm S}(f)$,  which leads to the smallest possible error probability under the constraints of power limitation.
  2. The overall frequency response  $H_{\rm Nyq}(f) = H_{\rm S}(f) · H_{\rm E}(f)$  satisfies the $\text{first Nyquist criterion}$,  so there is no intersymbol interference  $\text{(ISI)}$  at the receiver.  Thus,  $H_{\rm S}(f)$  and  $H_{\rm E}(f)$  each have a root-Nyquist characteristic.
  3. For the frequency response  $H_{\rm Nyq}(f)$,  one often uses a  $\text{cosine rolloff low-pass}$  $H_{\rm CRO}(f)$  with equivalent bandwidth  $Δf_{\rm CRO} = 1/T$  and freely chosen rolloff factor $ (0 ≤ r ≤ 1)$.  The interactive applet  "Frequency and Impulse Responses"  illustrates frequency response and impulse response of this low-pass filter.
  4. The advantage of these root-Nyquist systems is the much smaller bandwidth  $(1 + r)/T$  compared to the previously considered configuration with rectangular  $g_s(t)$  and rectangular  $h_{\rm E}(t)$,  whose spectrum is  (theoretically)  infinitely extended.
  5. In terms of error probability,  nothing changes compared to the  "rectangular-rectangular"  configuration because the basic detection pulse  $g_d(t)$  before the decision has equidistant zero crossings,  thus avoiding intersymbol interference.


The graph shows phase diagrams for this case, taken from the book   [Kam04][1].  The rolloff factor is $r = 0.5$.  From these plots,  one can see:

Phase diagram of 4-QAM with a root-Nyquist configuration
  • The right plot shows the detection signals in the  $\rm I$  and  $\rm Q$ branches after the root-Nyquist receiver filters in a two-dimensional representation. The corresponding spectra each have cosine-shaped slopes around the Nyquist frequency $f_{\rm Nyq} = 1/(2T)$.
  • At the detection times, only the four yellow points are possible in the phase diagram.  The transitions in between are diverse.  It should be noted that only a few lines pass through the coordinate null point.
  • Shown on the left are the two transmitted signals in the equivalent low-pass region,   $s_{\rm I}(t) = {\rm Re}\big[s_{\rm TP}(t)\big]$  and  $s_{\rm Q}(t) = {\rm Im}\big[s_{\rm TP}(t)\big]$. Due to the root-Nyquist spectral shaping, there is intersymbol interference at the transmitter,  which means that the equivalent low-pass signal  $s_{\rm TP}(t) = s_{\rm I}(t) + {\rm j} · s_{\rm Q}(t)$  is also not limited to four points at the detection times.
  • The magnitude  $|s_{\rm TP}(t)|$  – that is, the distance from zero –  indicates the envelope of the 4-QAM signal. It can be clearly seen from the left diagram that there are strong amplitude dips especially with phase changes around  $π$,  since then  $s_{\rm TP}(t)$  also often assumes  (complex)  values close to zero.


Offset–Quadrature amplitude modulation


Taking the equations for the  "$\text{4–QAM}$"   $($or  "$π/4\text{–QPSK}$"$)$  as our starting point,  we arrive at the   "$\text{Offset–4–QAM}$",  which we simplify by calling   "$\text{Offset–QPSK}$".  Here the transmitted signal is given by:

$$s(t) =s_{\rm I}(t) \cdot \cos(2 \pi f_{\rm T} t) - s_{\rm Q}(t) \cdot \sin(2 \pi f_{\rm T} t),$$
$$\hspace{0.3cm}\Rightarrow \hspace{0.3cm}s_{\rm I}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it \nu} \cdot g_s (t - \nu \cdot T)\hspace{0.05cm},\hspace{0.5cm} s_{\rm Q}(t) = \sum_{\nu = - \infty}^{+\infty}a_{\rm Q\hspace{0.03cm}\it \nu} \cdot g_s (t -{T}/{2} - \nu \cdot T)\hspace{0.05cm}.$$

The sole but decisive difference is the time shift of the quadrature component with respect to the in-phase component by half a symbol duration  $(T/2)$.  This has the advantage that

  • the phase function does not pass through zero,  and
  • the envelope  $|s_{\rm TP}(t)|$  therefore fluctuates much less.


Transmitter phase diagram and envelope at "$π/4$–QPSK" (above) and "Offset–QPSK" (below)

$\text{Example 2:}$  The upper graph shows

  • the phase diagram for  $π/4\hspace{-0.05cm}-\hspace{-0.05cm}\text{QPSK}$ on the left,
  • on the right,  a typical envelope curve,  based on a root Nyquist spectrum with rolloff factor  $r = 0.5$,  as in the last section.




The lower graphic shows that the   $\text{Offset – QPSK}$  has significantly better properties with respect to the envelope  (less severe signal dips).


⇒   Essential properties of  "4-QAM/QPSK"  and  "Offset-QPSK"  can be illustrated with the interactive  (German language)  SWF applet  "QPSK and Offset-QPSK",  where the basic pulse can be alternatively chosen as

  • a rectangular pulse,
  • a cosine pulse,
  • a Nyquist pulse,
  • a root Nyquist pulse.

Exercises for the chapter


Exercise 4.10: Signal Waveforms of the 16-QAM

Exercise 4.10Z: Signal Space Constellation of the 16-QAM

Exercise 4.11: Frequency Domain Consideration of the 4-QAM

Exercise 4.11Z: Error Probability with QAM

Exercise 4.12: Root-Nyquist Systems

Exercise 4.12Z: 4-QAM Systems again


References

  1. Kammeyer, K.D.:  Nachrichtenübertragung.  Stuttgart: B.G. Teubner, 4. Auflage, 2004.