Difference between revisions of "Digital Signal Transmission/Signals, Basis Functions and Vector Spaces"

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The following are dealt with in detail:
 
The following are dealt with in detail:
*the meaning of basis functions and finding them using the Gram-Schmidt method,
+
*the meaning of basis functions and finding them using the Gram-Schmidt process,
 
*the structure of the optimal receiver for baseband transmission,
 
*the structure of the optimal receiver for baseband transmission,
 
*the theorem of irrelevance and its importance for the derivation of optimal detectors,
 
*the theorem of irrelevance and its importance for the derivation of optimal detectors,
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<br clear= all>
 
<br clear= all>
  
== The Gram-Schmidt method==
+
== The Gram-Schmidt process==
 
<br>
 
<br>
In &nbsp;$\text{example 1}$&nbsp; in the last section, the specification of the two orthonormal basis functions&nbsp; $\varphi_1(t)$&nbsp; and&nbsp; $\varphi_2(t)$&nbsp; was very easy, because they were of the same form as&nbsp; $s_1(t)$&nbsp; and&nbsp;  $s_2(t)$,&nbsp; respectively. The&nbsp; [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process "Gram-Schmidt method"]&nbsp; finds the basis functions&nbsp; $\varphi_1(t)$, ... , $\varphi_N(t)$&nbsp; for arbitrary predefinable signals&nbsp; $s_1(t)$, ... , $s_M(t)$, as follows:
+
In &nbsp;$\text{Example 1}$&nbsp; in the last section, the specification of the two orthonormal basis functions&nbsp; $\varphi_1(t)$&nbsp; and&nbsp; $\varphi_2(t)$&nbsp; was very easy, because they were of the same form as&nbsp; $s_1(t)$&nbsp; and&nbsp;  $s_2(t)$,&nbsp; respectively. The&nbsp; [https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process "Gram-Schmidt process"]&nbsp; finds the basis functions&nbsp; $\varphi_1(t)$, ... , $\varphi_N(t)$&nbsp; for arbitrary predefinable signals&nbsp; $s_1(t)$, ... , $s_M(t)$, as follows:
  
*Die erste Basisfunktion&nbsp; $\varphi_1(t)$&nbsp; ist stets formgleich mit&nbsp; $s_1(t)$. Es gilt:
+
*The first basis function&nbsp; $\varphi_1(t)$&nbsp; is always equal in form to&nbsp; $s_1(t)$. It holds:
 
:$$\varphi_1(t) = \frac{s_1(t)}{\sqrt{E_1}} = \frac{s_1(t)}{|| s_1(t)||}
 
:$$\varphi_1(t) = \frac{s_1(t)}{\sqrt{E_1}} = \frac{s_1(t)}{|| s_1(t)||}
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_1(t) || = 1, \hspace{0.2cm}s_{11} =|| s_1(t)||,\hspace{0.2cm}s_{1j} = 0 \hspace{0.2cm}{\rm f{\rm \ddot{u}r }}\hspace{0.2cm} j \ge 2
+
\hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_1(t) || = 1, \hspace{0.2cm}s_{11} =|| s_1(t)||,\hspace{0.2cm}s_{1j} = 0 \hspace{0.2cm}{\rm for }}\hspace{0.2cm} j \ge 2
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
*Es wird nun angenommen, dass aus den Signalen&nbsp; $s_1(t)$, ... , $s_{k-1}(t)$&nbsp; bereits die Basisfunktionen&nbsp; $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$&nbsp; berechnet wurden &nbsp;$(n \le k)$. Dann berechnen wir mittels&nbsp; $s_k(t)$&nbsp; die Hilfsfunktion
+
*It is now assumed that from the signals&nbsp; $s_1(t)$, ... , $s_{k-1}(t)$&nbsp; the basis functions&nbsp; $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$&nbsp; have been calculated &nbsp;$(n \le k)$. Then, using&nbsp; $s_k(t)$,&nbsp; we compute the auxiliary function
:$$\theta_k(t) = s_k(t) - \sum\limits_{j = 1}^{n-1}s_{kj} \cdot \varphi_j(t) \hspace{0.4cm}{\rm mit}\hspace{0.4cm}
+
:$$\theta_k(t) = s_k(t) - \sum\limits_{j = 1}^{n-1}s_{kj} \cdot \varphi_j(t) \hspace{0.4cm}{\rm with}\hspace{0.4cm}
 
s_{kj} = \hspace{0.1cm} < \hspace{-0.1cm} s_k(t), \hspace{0.05cm}\varphi_j(t) \hspace{-0.1cm} >, \hspace{0.2cm} j = 1, \hspace{0.05cm} \text{...}\hspace{0.05cm}, n-1\hspace{0.05cm}.$$
 
s_{kj} = \hspace{0.1cm} < \hspace{-0.1cm} s_k(t), \hspace{0.05cm}\varphi_j(t) \hspace{-0.1cm} >, \hspace{0.2cm} j = 1, \hspace{0.05cm} \text{...}\hspace{0.05cm}, n-1\hspace{0.05cm}.$$
  
*Ist&nbsp; $\theta_k(t) \equiv 0$ &nbsp; &#8658; &nbsp; $||\theta_k(t)|| = 0$, so liefert&nbsp; $s_k(t)$&nbsp; keine neue Basisfunktion. Vielmehr lässt sich dann&nbsp; $s_k(t)$&nbsp; durch die&nbsp; $n-1$&nbsp; bereits vorher gefundenen Basisfunktionen &nbsp;$\varphi_1(t)$, ... , $\varphi_{n-1}(t)$&nbsp; ausdrücken:
+
*If&nbsp; $\theta_k(t) \equiv 0$ &nbsp; &#8658; &nbsp; $||\theta_k(t)|| = 0$, then&nbsp; $s_k(t)$&nbsp; does not yield a new basis function. Rather,&nbsp; $s_k(t)$&nbsp; can then be expressed by the&nbsp; $n-1$&nbsp; basis functions &nbsp;$\varphi_1(t)$, ... , $\varphi_{n-1}(t)$&nbsp; already found before:
 
:$$s_k(t) = \sum\limits_{j = 1}^{n-1}s_{kj}\cdot \varphi_j(t)  \hspace{0.05cm}.$$
 
:$$s_k(t) = \sum\limits_{j = 1}^{n-1}s_{kj}\cdot \varphi_j(t)  \hspace{0.05cm}.$$
  
*Eine neue Basisfunktion (nämlich die &nbsp;$n$&ndash;te) ergibt sich, falls &nbsp;$||\theta_k(t)|| \ne 0$&nbsp; ist:
+
*A new basis function (namely, the &nbsp;$n$&ndash;th) results if &nbsp;$||\theta_k(t)|| \ne 0$:&nbsp;  
  
 
:$$\varphi_n(t) =  \frac{\theta_k(t)}{|| \theta_k(t)||}
 
:$$\varphi_n(t) =  \frac{\theta_k(t)}{|| \theta_k(t)||}
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Diese Prozedur wird fortgesetzt, bis alle&nbsp; $M$&nbsp; Signale berücksichtigt wurden. Danach hat man alle&nbsp; $N \le M$&nbsp; orthonormalen Basisfunktionen&nbsp; $\varphi_j(t)$&nbsp; gefunden. Der Sonderfall&nbsp; $N = M$&nbsp; ergibt sich nur dann, wenn alle&nbsp; $M$&nbsp; Signale linear voneinander unabhängig sind.<br>
+
This process is continued until all&nbsp; $M$&nbsp; signals have been considered. Then all&nbsp; $N \le M$&nbsp; orthonormal basis functions&nbsp; $\varphi_j(t)$&nbsp; have been found. The special case&nbsp; $N = M$&nbsp; arises only if all&nbsp; $M$&nbsp; signals are linearly independent.<br>
  
Dieses Verfahren wird nun an einem Beispiel verdeutlicht. Wir verweisen auch auf das interaktive Applet&nbsp; [[Applets:Gram-Schmidt-Verfahren|Gram&ndash;Schmidt&ndash;Verfahren]].
+
This process is now illustrated by an example. We also refer to the interactive applet&nbsp; [[Applets:Gram-Schmidt-Verfahren|"Gram–Schmidt process"]].
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 2:}$&nbsp; Wir betrachten die &nbsp;$M = 4$&nbsp; energiebegrenzten Signale &nbsp;$s_1(t)$, ... , $s_4(t)$&nbsp; entsprechend der Grafik. Zur Vereinfachung der Berechnungen ist hier sowohl die Amplitude als auch die Zeit normiert.  
+
$\text{Example 2:}$&nbsp; We consider the &nbsp;$M = 4$&nbsp; energy-limited signals &nbsp;$s_1(t)$, ... , $s_4(t)$&nbsp; according to the graph. To simplify the calculations, both amplitude and time are normalized here.
  
[[File:P ID1990 Dig T 4 1 S3 version1.png|center|frame|Zum Gram-Schmidt-Verfahren|class=fit]]
+
[[File:P ID1990 Dig T 4 1 S3 version1.png|center|frame|Gram-Schmidt process|class=fit]]
  
Man erkennt aus diesen Skizzen:  
+
One can see from these sketches:
*Die Basisfunktion&nbsp; $\varphi_1(t)$&nbsp; ist formgleich mit&nbsp; $s_1(t)$. Wegen&nbsp; $E_1 = \vert \vert s_1(t) \vert \vert ^3 = 3 \cdot 0.5^2 = 0.75$&nbsp; ergibt sich&nbsp; $s_{11} = \vert \vert s_1(t) \vert \vert = 0.866$. $\varphi_1(t)$&nbsp; selbst besitzt abschnittsweise die Werte&nbsp; $\pm 0.5/0.866 = \pm0.577$.
+
*The basis function&nbsp; $\varphi_1(t)$&nbsp; is equal in form to&nbsp; $s_1(t)$. Because&nbsp; $E_1 = \vert \vert s_1(t) \vert \vert ^3 = 3 \cdot 0.5^2 = 0.75$,&nbsp; we get&nbsp; $s_{11} = \vert \vert s_1(t) \vert \vert = 0.866$. $\varphi_1(t)$&nbsp; itself has section-wise values&nbsp; $\pm 0.5/0.866 = \pm0.577$.
  
*Zur Berechnung der Hilfsfunktion&nbsp; $\theta_2(t)$&nbsp; berechnen wir
+
*To calculate the auxiliary function&nbsp; $\theta_2(t)$,&nbsp; we compute
  
 
:$$s_{21}  = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.05cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0 \cdot (+0.577) + 1 \cdot (-0.577)+ 0 \cdot (-0.577)= -0.577$$
 
:$$s_{21}  = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.05cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0 \cdot (+0.577) + 1 \cdot (-0.577)+ 0 \cdot (-0.577)= -0.577$$
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\varphi_2(t) = \theta_2(t)/s_{22} = (0.408, 0.816, -0.408)\hspace{0.05cm}. $$
 
\varphi_2(t) = \theta_2(t)/s_{22} = (0.408, 0.816, -0.408)\hspace{0.05cm}. $$
  
*Die inneren Produkte zwischen&nbsp; $s_1(t)$&nbsp; mit&nbsp; $\varphi_1(t)$&nbsp; bzw. &nbsp;$\varphi_2(t)$&nbsp; liefern folgende Ergebnisse:
+
*The inner products between&nbsp; $s_1(t)$&nbsp; with&nbsp; $\varphi_1(t)$&nbsp; or &nbsp;$\varphi_2(t)$&nbsp; give the following results:
 
:$$s_{31}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.577) + 0.5 \cdot (-0.577)- 0.5 \cdot (-0.577)= 0.289$$
 
:$$s_{31}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.577) + 0.5 \cdot (-0.577)- 0.5 \cdot (-0.577)= 0.289$$
 
:$$s_{32}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.408) + 0.5 \cdot (+0.816)- 0.5 \cdot (-0.408)= 0.816$$
 
:$$s_{32}  \hspace{0.1cm} =  \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.408) + 0.5 \cdot (+0.816)- 0.5 \cdot (-0.408)= 0.816$$
 
:$$\Rightarrow  \hspace{0.3cm}\theta_3(t) = s_3(t) - 0.289 \cdot \varphi_1(t)- 0.816 \cdot \varphi_2(t) = 0\hspace{0.05cm}.$$
 
:$$\Rightarrow  \hspace{0.3cm}\theta_3(t) = s_3(t) - 0.289 \cdot \varphi_1(t)- 0.816 \cdot \varphi_2(t) = 0\hspace{0.05cm}.$$
  
Das bedeutet: &nbsp; Die grüne Funktion&nbsp; $s_3(t)$&nbsp; liefert keine neue Basisfunktion&nbsp; $\varphi_3(t)$, im Gegensatz zur Funktion&nbsp; $s_4(t)$. Die numerischen Ergebnisse hierfür können der Grafik entnommen werden.}}
+
This means: &nbsp; The green function&nbsp; $s_3(t)$&nbsp; does not yield a new basis function&nbsp; $\varphi_3(t)$, in contrast to the function&nbsp; $s_4(t)$. The numerical results for this can be taken from the graph.}}
  
== Basisfunktionen komplexer Zeitsignale ==
+
== Basis functions of complex time signals ==
 
<br>
 
<br>
In der Nachrichtentechnik hat man es oft mit komplexen Zeitfunktionen zu tun,
+
In communications engineering, one often has to deal with complex time functions,
*nicht etwa, weil es komplexe Signale in der Realität gibt, sondern<br>
+
*not because there are complex signals in reality, but<br>
*weil die Beschreibung eines Bandpass&ndash;Signals im äquivalenten Tiefpass&ndash;Bereich zu komplexen Signalen führt.<br><br>
+
*because the description of a band-pass signal in the equivalent low-pass range leads to complex signals.<br><br>
  
Die Bestimmung der&nbsp; $N \le M$&nbsp; komplexwertigen Basisfunktionen&nbsp; $\xi_k(t)$&nbsp; aus den &nbsp;$M$&nbsp; komplexen Signalen&nbsp; $s_i(t)$&nbsp; kann ebenfalls mit dem&nbsp; [[Digitalsignal%C3%BCbertragung/Signale,_Basisfunktionen_und_Vektorr%C3%A4ume#Das_Verfahren_nach_Gram-Schmidt_.281.29| Gram&ndash;Schmidt&ndash;Verfahren]]&nbsp; erfolgen, doch ist nun zu berücksichtigen, dass das innere Produkt zweier komplexer Signale&nbsp; $x(t)$&nbsp; und&nbsp; $y(t)$&nbsp; wie folgt zu berechnen ist:
+
The determination of the&nbsp; $N \le M$&nbsp; complex-valued basis functions&nbsp; $\xi_k(t)$&nbsp; from the &nbsp;$M$&nbsp; complex signals&nbsp; $s_i(t)$&nbsp; can also be done using the&nbsp; [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#The_Gram-Schmidt_method| "Gram–Schmidt process"]],&nbsp; but it must now be taken into account that the inner product of two complex signals&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp; must be calculated as follows:
 
:$$< \hspace{-0.1cm}x(t), \hspace{0.1cm}y(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}x(t) \cdot y^{\star}(t)\,d \it t
 
:$$< \hspace{-0.1cm}x(t), \hspace{0.1cm}y(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}x(t) \cdot y^{\star}(t)\,d \it t
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
Die entsprechenden Gleichungen lauten nun mit&nbsp; $i = 1, \text{..}. , M$&nbsp; und &nbsp;$k = 1, \text{..}. , N$:
+
The corresponding equations are now with&nbsp; $i = 1, \text{..}. , M$&nbsp; and &nbsp;$k = 1, \text{..}. , N$:
 
:$$s_i(t) = \sum\limits_{k = 1}^{N}s_{ik} \cdot \xi_k(t),\hspace{0.2cm}s_i(t) \in {\cal C},\hspace{0.2cm}s_{ik} \in {\cal C}
 
:$$s_i(t) = \sum\limits_{k = 1}^{N}s_{ik} \cdot \xi_k(t),\hspace{0.2cm}s_i(t) \in {\cal C},\hspace{0.2cm}s_{ik} \in {\cal C}
 
,\hspace{0.2cm}\xi_k(t) \in {\cal C} \hspace{0.05cm},$$
 
,\hspace{0.2cm}\xi_k(t) \in {\cal C} \hspace{0.05cm},$$
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\left\{ \begin{array}{c} 1 \\
 
\left\{ \begin{array}{c} 1 \\
 
  0  \end{array} \right.\quad
 
  0  \end{array} \right.\quad
\begin{array}{*{1}c}{\rm falls}\hspace{0.15cm} k = j
+
\begin{array}{*{1}c}{\rm if}\hspace{0.15cm} k = j
\\ {\rm falls}\hspace{0.15cm} k \ne j \\ \end{array}\hspace{0.05cm}.$$
+
\\ {\rm if}\hspace{0.15cm} k \ne j \\ \end{array}\hspace{0.05cm}.$$
  
Natürlich lässt sich jede komplexe Größe auch durch zwei reelle Größen ausdrücken, nämlich durch Realteil und Imaginärteil. Somit erhält man hier folgende Gleichungen:
+
Of course, any complex quantity can also be expressed by two real quantities, namely real part and imaginary part. Thus, the following equations are obtained here:
 
:$$s_{i}(t)  = s_{{\rm I}\hspace{0.02cm}i}(t) + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}i}(t),
 
:$$s_{i}(t)  = s_{{\rm I}\hspace{0.02cm}i}(t) + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}i}(t),
 
\hspace{0.2cm} s_{{\rm I}\hspace{0.02cm}i}(t) = {\rm Re}\big [s_{i}(t)\big], \hspace{0.2cm} s_{{\rm Q}\hspace{0.02cm}i}(t) = {\rm Im} \big [s_{i}(t)\big ],$$
 
\hspace{0.2cm} s_{{\rm I}\hspace{0.02cm}i}(t) = {\rm Re}\big [s_{i}(t)\big], \hspace{0.2cm} s_{{\rm Q}\hspace{0.02cm}i}(t) = {\rm Im} \big [s_{i}(t)\big ],$$
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\hspace{0.05cm}. $$
 
\hspace{0.05cm}. $$
  
Die Nomenklatur ergibt sich aus der Hauptanwendung für komplexe Basisfunktionen, nämlich der&nbsp; [[Modulationsverfahren/Quadratur–Amplitudenmodulation#Allgemeine_Beschreibung_und_Signalraumzuordnung|Quadratur&ndash;Amplitudenmodulation]]&nbsp; (QAM).  
+
The nomenclature arises from the main application for complex basis functions, namely&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#General_description_and_signal_space_allocation|"quadrature amplitude modulation"]]&nbsp; (QAM).  
*Der Index "I" steht für Inphasekomponente und gibt den Realteil an,  
+
*The subscript "I" stands for inphase component and indicates the real part,
*während die Quadraturkomponente (Imaginärteil) mit dem Index "Q" gekennzeichnet ist.<br>
+
*while the quadrature component (imaginary part) is indicated by the index "Q".<br>
  
  
Um Verwechslungen mit der imaginären Einheit zu vermeiden, wurden hier die komplexen Basisfunktionen&nbsp; $\xi_{k}(t)$&nbsp; mit&nbsp; $k$&nbsp; induziert und nicht mit&nbsp; $j$.<br>
+
To avoid confusion with the imaginary unit, here the complex basis functions&nbsp; $\xi_{k}(t)$&nbsp; were induced with&nbsp; $k$&nbsp; and not with&nbsp; $j$.<br>
  
== Dimension der Basisfunktionen ==
+
== Dimension of the basis functions ==
 
<br>
 
<br>
Bei der Basisbandübertragung sind die möglichen Sendesignale (Betrachtung nur einer Symboldauer)  
+
In baseband transmission, the possible transmitted signals considering only one symbol duration) are
 
:$$s_i(t) = a_i \cdot g_s(t), \hspace{0.2cm} i = 0,  \text{...}\hspace{0.05cm} , M-1,$$
 
:$$s_i(t) = a_i \cdot g_s(t), \hspace{0.2cm} i = 0,  \text{...}\hspace{0.05cm} , M-1,$$
  
wobei&nbsp; $g_s(t)$&nbsp; den ''Sendegrundimpuls'' angibt und die&nbsp; $a_i$&nbsp; in den ersten drei Hauptkapiteln als die ''möglichen Amplitudenkoeffizienten''&nbsp; bezeichnet wurden. Anzumerken ist, dass ab sofort für die Laufvariable&nbsp; $i$&nbsp; die Werte&nbsp; $0$&nbsp; bis &nbsp;$M-1$&nbsp; vorausgesetzt werden.<br>
+
where&nbsp; $g_s(t)$&nbsp; indicates the ''basic transmission pulse'' and the&nbsp; $a_i$&nbsp; were denoted as the possible amplitude coefficients in the first three main chapters. It should be noted that from now on the values&nbsp; $0$&nbsp; to&nbsp;$M-1$&nbsp; are assumed for the indexing variable&nbsp; $i$.&nbsp;<br>
  
Nach der Beschreibung dieses Kapitels handelt es sich unabhängig von der Stufenzahl&nbsp; $M$&nbsp; um ein eindimensionales Modulationsverfahren&nbsp; $(N = 1)$, wobei bei der Basisbandübertragung
+
According to the description of this chapter, regardless of the level number&nbsp; $M$,&nbsp; it is a one-dimensional modulation process&nbsp; $(N = 1)$, where, in the case of baseband transmission
*die Basisfunktion&nbsp; $\varphi_1(t)$&nbsp; gleich dem energienormierten Sendegrundimpuls&nbsp; $g_s(t)$&nbsp; ist:
+
*the basis function&nbsp; $\varphi_1(t)$&nbsp; is equal to the energy normalized basic transmission pulse&nbsp; $g_s(t)$:&nbsp;  
:$$\varphi_1(t) ={g_s(t)}/{\sqrt{E_{gs}}} \hspace{0.3cm}{\rm mit}\hspace{0.3cm}
+
:$$\varphi_1(t) ={g_s(t)}/{\sqrt{E_{gs}}} \hspace{0.3cm}{\rm with}\hspace{0.3cm}
 
E_{gs} = \int_{-\infty}^{+\infty}g_s^2(t)\,d \it t   
 
E_{gs} = \int_{-\infty}^{+\infty}g_s^2(t)\,d \it t   
 
\hspace{0.05cm},$$
 
\hspace{0.05cm},$$
  
*die dimensionslosen Amplitudenkoeffizienten&nbsp; $a_i$&nbsp; in die Signalraumpunkte&nbsp; $s_i$&nbsp; umzurechnen sind, die die Einheit "Wurzel aus Energie" aufweisen.<br>
+
*the dimensionless amplitude coefficients&nbsp; $a_i$&nbsp; are to be converted into the signal space points&nbsp; $s_i$&nbsp; which have the unit "root of energy".<br>
  
  
Die Grafik zeigt eindimensionale  Signalraumkonstellationen&nbsp; $(N=1)$&nbsp; für die Basisbandübertragung, nämlich
+
The graph shows one-dimensional signal space constellations&nbsp; $(N=1)$&nbsp; for baseband transmission, viz.
[[File:P ID1991 Dig T 4 1 S5a version2.png|right|frame|Eindimensionale Modulationsverfahren|class=fit]]
+
[[File:P ID1991 Dig T 4 1 S5a version2.png|right|frame|One-dimensional modulation methods|class=fit]]
:(a) binär unipolar (oben) &nbsp; &rArr; &nbsp; $M = 2$,
+
:(a) binary unipolar (top) &nbsp; &rArr; &nbsp; $M = 2$,
  
:(b) binär bipolar (Mitte) &nbsp; &rArr; &nbsp; $M = 2$, sowie
+
:(b) binary bipolar (center) &nbsp; &rArr; &nbsp; $M = 2$, and
  
:(c) quaternär bipolare (unten) &nbsp; &rArr; &nbsp; $M = 4$.  
+
:(c) quaternary bipolar (bottom) &nbsp; &rArr; &nbsp; $M = 4$.  
  
Die Grafik beschreibt gleichzeitig die eindimensionalen Trägerfrequenzsysteme
+
The graph simultaneously describes the one-dimensional carrier frequency systems
*oben: &nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#On.E2.80.93Off.E2.80.93Keying_.282.E2.80.93ASK.29|Zweistufiges Amplitude Shift Keying]]&nbsp; (2&ndash;ASK), auch bekannt als "On&ndash;Off&ndash;Keying ",  
+
*top: &nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93Off.E2.80.93Keying_.282.E2.80.93ASK.29|"Two-level Amplitude Shift Keying"]]&nbsp; (2&ndash;ASK), also known as "On&ndash;Off&ndash;Keying ",  
*in der Mitte: &nbsp;  [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Binary_Phase_Shift_Keying_.28BPSK.29|Binary Phase Shift Keying]]&nbsp; (BPSK),
+
*in the middle: &nbsp;  [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Binary_Phase_Shift_Keying_.28BPSK.29|"Binary Phase Shift Keying"]]&nbsp; (BPSK),
*unten: &nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#M.E2.80.93stufiges_Amplitude_Shift_Keying_.28M.E2.80.93ASK.29|Vierstufiges Amplitude Shift Keying]]&nbsp; (4&ndash;ASK).<br>
+
*bottom: &nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_Amplitude_Shift_Keying_.28M.E2.80.93ASK.29|"Four-level Amplitude Shift Keying"]]&nbsp; (4&ndash;ASK).<br>
  
  
Die dargestellten Signale&nbsp; $s_i(t)$&nbsp; und die Basisfunktion &nbsp;$\varphi_1(t)$&nbsp; beziehen sich stets auf den äquivalenten Tiefpass&ndash;Bereich.  
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The signals&nbsp; $s_i(t)$&nbsp; and the basis function &nbsp;$\varphi_1(t)$&nbsp; shown always refer to the equivalent low-pass range.
  
Im Bandpass&ndash;Bereich ist $\varphi_1(t)$ eine auf den Zeitbereich $0 \le t \le T$ begrenzte harmonische Schwingung.
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In the band-pass region, $\varphi_1(t)$ is a harmonic oscillation limited to the time domain $0 \le t \le T$.
 
<br clear=all>
 
<br clear=all>
Weitere Anmerkungen:
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Further notes:
*In der Grafik rechts sind am Beispiel "Rechteckimpuls" die zwei bzw. vier möglichen Sendesignale &nbsp;$s_i(t)$&nbsp; angegeben.  
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*In the graph on the right, the two or four possible transmitted signals &nbsp;$s_i(t)$&nbsp; are given for the example "rectangular pulse".
*Man kann daraus den Zusammenhang zwischen Impulsamplitude&nbsp; $A$&nbsp; und Signalenergie&nbsp; $E = A^2 \cdot T$&nbsp;  erkennen.
+
*From this, one can see the relationship between pulse amplitude&nbsp; $A$&nbsp; and signal energy&nbsp; $E = A^2 \cdot T$.&nbsp;   
*Die jeweils linken Darstellungen auf der&nbsp; $\varphi_1(t)$&ndash;Achse gelten aber unabhängig von der&nbsp; $g_s(t)$&ndash;Form, nicht nur für Rechtecke.<br clear =all>
+
*However, the respective left representations on the&nbsp; $\varphi_1(t)$ axis are valid independently of the&nbsp; $g_s(t)$ shape, not only for rectangles.<br clear =all>
  
  
== Zweidimensionale  Modulationsverfahren==
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== Two-dimensional modulation processes==
 
<br>
 
<br>
 
[[File:P ID1992 Dig T 4 1 S5b version1.png|right|frame|Zweidimensionale Signalraumkonstellationen für &nbsp;$M$&ndash;PSK und &nbsp;$M$&ndash;QAM|class=fit]]
 
[[File:P ID1992 Dig T 4 1 S5b version1.png|right|frame|Zweidimensionale Signalraumkonstellationen für &nbsp;$M$&ndash;PSK und &nbsp;$M$&ndash;QAM|class=fit]]

Revision as of 13:38, 13 June 2022

# OVERVIEW OF THE FOURTH MAIN CHAPTER #


The fourth main chapter provides an abstract description of digital signal transmission, which is based on basis functions and signal space constellations. This makes it possible to treat very different configurations - for example band-pass systems and those for the baseband - in a uniform way. The optimal receiver in each case has the same structure in all cases.

The following are dealt with in detail:

  • the meaning of basis functions and finding them using the Gram-Schmidt process,
  • the structure of the optimal receiver for baseband transmission,
  • the theorem of irrelevance and its importance for the derivation of optimal detectors,
  • the optimal receiver for the AWGN channel and implementation aspects,
  • the system description by complex or  $N$–dimensional Gaussian noise,
  • the error probability calculation and approximation under otherwise ideal conditions,
  • the application of the signal space description to carrier frequency systems,
  • the different results for OOK, M-ASK, M-PSK, M-QAM and M-FSK,
  • the different results for coherent and non-coherent demodulation.


Almost all results of this chapter have already been derived in previous sections. However, the approach is fundamentally new:

  • In the $\rm LNTwww$ book "Modulation Methods" and in the first three chapters of this book, the specific system properties were already taken into account in the derivations – for example, whether the digital signal is transmitted in baseband or whether digital amplitude, frequency or phase modulation is present.
  • Here the systems are to be abstracted in such a way that they can be treated uniformly. The optimal receiver in each case has the same structure in all cases, and the error probability can also be specified for non-Gaussian distributed noise.

It should be noted that this rather global approach means that certain system deficiencies can only be recorded very imprecisely, such as

  • the influence of a non-optimal receiver filter on the error probability,
  • an incorrect threshold (threshold drift), or
  • phase jitter (fluctuations in sampling times).

In particular in the presence of intersymbol interference, the procedure should therefore continue in accordance with the  third main chapter

The description is based on the script [KöZ08][1] by  Ralf Kötter  and  Georg Zeitler, which is closely based on the textbook [WJ65][2]. Gerhard Kramer, who has held the chair at the LNT since 2010, treats the same topic with very similar nomenclature in his lecture [Kra17][3].

In order not to make reading unnecessarily difficult for our own students at the TU Munich, we stick to this nomenclature as far as possible, even if it deviates from other $\rm LNTwww$ chapters.

Nomenclature in the fourth chapter


Compared to the other chapters in $\rm LNTwww$, the following nomenclature differences arise here:

  • The  "message"  to be transmitted is an integer value  $m \in \{m_i\}$  with  $i = 0$, ... , $M-1$, where  $M$  specifies the symbol set size. If it simplifies the description,  $i = 1$, ... , $M$    is induced.


  • The result of the decision process at the receiver is also an integer with the same symbol alphabet as at the transmitter. This result is also referred to as the estimated value:
$$\hat{m} \in \{m_i \}, \hspace{0.2cm} i = 0, 1, \text{...}\hspace{0.05cm} , M-1\hspace{0.2cm} ({\rm or}\,\,i = 1, 2, \text{...}\hspace{0.05cm}, M) \hspace{0.05cm}.$$
  • The  "symbol error probability"  $\rm Pr(symbol error)$  or $p_{\rm S}$  is usually referred to as follows in this main chapter:
$${\rm Pr} ({\cal E}) = {\rm Pr} ( \hat{m} \ne m) = 1 - {\rm Pr} ({\cal C}), \hspace{0.4cm}\text{complementary event:}\hspace{0.2cm} {\rm Pr} ({\cal C}) = {\rm Pr} ( \hat{m} = m) \hspace{0.05cm}.$$
  • In a  "probability density function"  (PDF), a distinction is now made between the random variable   ⇒   $r$  and the realization   ⇒   $\rho$  according to   $p_r(\rho)$.   $f_r(r)$  was used for a PDF.


  • With the notation  $p_r(\rho)$,   $r$  and  $\rho$  Sare scalars. On the other hand, if random variable and realization are vectors (of suitable length), this is expressed in bold type:     $p_{ \boldsymbol{ r}}(\boldsymbol{\rho})$  with the vectors  $ \boldsymbol{ r}$  and  $\boldsymbol{\rho}$.


  • In order to avoid confusion with energy values, the threshold value is now called  $G$  instead of  $E$  and this is mainly referred to as the decision limit in this chapter.


  • Based on the two real and energy-limited time functions  $x(t)$  and  $y(t)$,  the  "inner product" is:
$$<\hspace{-0.1cm}x(t), \hspace{0.05cm}y(t) \hspace{-0.1cm}> \hspace{0.15cm}= \int_{-\infty}^{+\infty}x(t) \cdot y(t)\,d \it t \hspace{0.05cm}.$$
$$||x(t) || = \sqrt{<\hspace{-0.1cm}x(t), \hspace{0.05cm}x(t) \hspace{-0.1cm}>} \hspace{0.05cm}.$$


Compared to the script  $\rm [KöZ08]$[1], the naming differs as follows:

  • The probability of the event  $E$  is here  ${\rm Pr}(E)$  instead of  $P(E)$. This nomenclature change was also made because in some equations probabilities and powers appear together.


  • Band–pass signals are still marked with the index "BP" and not with a tilde as in [KöZ08][1]. The corresponding low-pass signal is (usually) provided with the index "TP".

Orthonormal basis functions


In this chapter, we assume a set  $\{s_i(t)\}$  of possible transmitted signals that are uniquely assigned to the possible messages  $m_i$.  With  $i = 1$, ... , $M$  holds:

$$m \in \{m_i \}, \hspace{0.2cm} s(t) \in \{s_i(t) \}\hspace{-0.1cm}: \hspace{0.3cm} m = m_i \hspace{0.1cm} \Leftrightarrow \hspace{0.1cm} s(t) = s_i(t) \hspace{0.05cm}.$$

For what follows, we further assume that the  $M$ signals  $s_i(t)$  are "energy-limited",  which usually means at the same time that they are of finite duration only.

$\text{Theorem:}$  Any set  $\{s_1(t), \hspace{0.05cm} \text{...} \hspace{0.05cm} , s_M(t)\}$  of energy-limited signals can be evolved into  $N \le M$  orthonormal basis functions  $\varphi_1(t), \hspace{0.05cm} \text{...} \hspace{0.05cm} , \varphi_N(t)$.  It holds:

$$s_i(t) = \sum\limits_{j = 1}^{N}s_{ij} \cdot \varphi_j(t) , \hspace{0.3cm}i = 1,\hspace{0.05cm} \text{...}\hspace{0.1cm} , M, \hspace{0.3cm}j = 1,\hspace{0.05cm} \text{...} \hspace{0.1cm}, N \hspace{0.05cm}.$$

In each case, two basis functions  $\varphi_j(t)$  and  $\varphi_k(t)$  must be orthonormal to each other, that is, it must hold  $(\delta_{jk}$  is called the "Kronecker symbol"  or the "Kronecker delta"$)$:

$$<\hspace{-0.1cm}\varphi_j(t), \hspace{0.05cm}\varphi_k(t) \hspace{-0.1cm}> = \int_{-\infty}^{+\infty}\varphi_j(t) \cdot \varphi_k(t)\,d \it t = {\rm \delta}_{jk} = \left\{ \begin{array}{c} 1 \\ 0 \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.1cm}j = k \\ {\rm if}\hspace{0.1cm} j \ne k \\ \end{array} \hspace{0.05cm}.$$


Here, the parameter  $N$  indicates how many basis functions  $\varphi_j(t)$  are needed to represent the  $M$  possible transmitted signals. In other words:   $N$  is the dimension of the vector space spanned by the  $M$  signals. Here, the following holds:

  • If  $N = M$, all transmitted signals are orthogonal to each other. They are not necessarily orthonormal, i.e. the energies  $E_i = <\hspace{-0.1cm}s_i(t), \hspace{0.05cm}s_i(t) \hspace{-0.1cm}>$  may well be unequal to one.
  • The case  $N < M$  arises when at least one signal  $s_i(t)$  can be represented as a linear combination of basis functions  $\varphi_j(t)$  that have resulted from other signals  $s_j(t) \ne s_i(t)$. 


Representation of the three transmitted signals by two basis functions

$\text{Example 1:}$  We consider  $M = 3$  energy-limited signals according to the graph. One recognizes immediately:

  • The signals  $s_1(t)$  and  $s_2(t)$  are orthogonal to each other.
  • The energies are  $E_1 = A^2 \cdot T = E$  and  $E_2 = (A/2)^2 \cdot T = E/4$.
  • The basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$  are equal in form to  $s_1(t)$  and  $s_2(t)$,  respectively, and both have energy one:
$$\varphi_1(t)=\frac{s_1(t)}{\sqrt{E_1} } = \frac{s_1(t)}{\sqrt{A^2 \cdot T} } = \frac{1}{\sqrt{ T} } \cdot \frac{s_1(t)}{A}$$
$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}s_1(t) = s_{11} \cdot \varphi_1(t)\hspace{0.05cm},\hspace{0.1cm}s_{11} = \sqrt{E}\hspace{0.05cm},$$
$$\varphi_2(t) =\frac{s_2(t)}{\sqrt{E_2} } = \frac{s_2(t)}{\sqrt{(A/2)^2 \cdot T} } = \frac{1}{\sqrt{ T} } \cdot \frac{s_2(t)}{A/2}\hspace{0.05cm}$$
$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm}s_2(t) = s_{21} \cdot \varphi_2(t)\hspace{0.05cm},\hspace{0.1cm}s_{21} = {\sqrt{E} }/{2}\hspace{0.05cm}.$$
  • The signal  $s_3(t)$  can be expressed by the previously determined basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$: 
$$s_3(t) =s_{31} \cdot \varphi_1(t) + s_{32} \cdot \varphi_2(t)\hspace{0.05cm},$$
$$\hspace{0.5cm}\Rightarrow \hspace{0.1cm} s_{31} = {A}/{2} \cdot \sqrt {T}= {\sqrt{E} }/{2}\hspace{0.05cm}, \hspace{0.2cm}s_{32} = - A \cdot \sqrt {T} = -\sqrt{E} \hspace{0.05cm}.$$

In the lower right image, the signals are shown in a 2D representation with the basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$  as axes, where  $E = A^2 \cdot T$  and the relation to the other graphs can be seen by the coloring.

The vectorial representatives of the signals  $s_1(t)$,  $s_2(t)$  and  $s_3(t)$  in this two-dimensional vector space can be read from this as follows:

$$\mathbf{s}_1 = (\sqrt{ E}, \hspace{0.1cm}0), $$
$$\mathbf{s}_2 = (0, \hspace{0.1cm}\sqrt{ E}/2), $$
$$\mathbf{s}_3 = (\sqrt{ E}/2,\hspace{0.1cm}-\sqrt{ E} ) \hspace{0.05cm}.$$


The Gram-Schmidt process


In  $\text{Example 1}$  in the last section, the specification of the two orthonormal basis functions  $\varphi_1(t)$  and  $\varphi_2(t)$  was very easy, because they were of the same form as  $s_1(t)$  and  $s_2(t)$,  respectively. The  "Gram-Schmidt process"  finds the basis functions  $\varphi_1(t)$, ... , $\varphi_N(t)$  for arbitrary predefinable signals  $s_1(t)$, ... , $s_M(t)$, as follows:

  • The first basis function  $\varphi_1(t)$  is always equal in form to  $s_1(t)$. It holds:
$$\varphi_1(t) = \frac{s_1(t)}{\sqrt{E_1}} = \frac{s_1(t)}{|| s_1(t)||} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_1(t) || = 1, \hspace{0.2cm}s_{11} =|| s_1(t)||,\hspace{0.2cm}s_{1j} = 0 \hspace{0.2cm}{\rm for }}\hspace{0.2cm} j \ge 2 \hspace{0.05cm}.$$
  • It is now assumed that from the signals  $s_1(t)$, ... , $s_{k-1}(t)$  the basis functions  $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$  have been calculated  $(n \le k)$. Then, using  $s_k(t)$,  we compute the auxiliary function
$$\theta_k(t) = s_k(t) - \sum\limits_{j = 1}^{n-1}s_{kj} \cdot \varphi_j(t) \hspace{0.4cm}{\rm with}\hspace{0.4cm} s_{kj} = \hspace{0.1cm} < \hspace{-0.1cm} s_k(t), \hspace{0.05cm}\varphi_j(t) \hspace{-0.1cm} >, \hspace{0.2cm} j = 1, \hspace{0.05cm} \text{...}\hspace{0.05cm}, n-1\hspace{0.05cm}.$$
  • If  $\theta_k(t) \equiv 0$   ⇒   $||\theta_k(t)|| = 0$, then  $s_k(t)$  does not yield a new basis function. Rather,  $s_k(t)$  can then be expressed by the  $n-1$  basis functions  $\varphi_1(t)$, ... , $\varphi_{n-1}(t)$  already found before:
$$s_k(t) = \sum\limits_{j = 1}^{n-1}s_{kj}\cdot \varphi_j(t) \hspace{0.05cm}.$$
  • A new basis function (namely, the  $n$–th) results if  $||\theta_k(t)|| \ne 0$: 
$$\varphi_n(t) = \frac{\theta_k(t)}{|| \theta_k(t)||} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} || \varphi_n(t) || = 1\hspace{0.05cm}.$$


This process is continued until all  $M$  signals have been considered. Then all  $N \le M$  orthonormal basis functions  $\varphi_j(t)$  have been found. The special case  $N = M$  arises only if all  $M$  signals are linearly independent.

This process is now illustrated by an example. We also refer to the interactive applet  "Gram–Schmidt process".

$\text{Example 2:}$  We consider the  $M = 4$  energy-limited signals  $s_1(t)$, ... , $s_4(t)$  according to the graph. To simplify the calculations, both amplitude and time are normalized here.

Gram-Schmidt process

One can see from these sketches:

  • The basis function  $\varphi_1(t)$  is equal in form to  $s_1(t)$. Because  $E_1 = \vert \vert s_1(t) \vert \vert ^3 = 3 \cdot 0.5^2 = 0.75$,  we get  $s_{11} = \vert \vert s_1(t) \vert \vert = 0.866$. $\varphi_1(t)$  itself has section-wise values  $\pm 0.5/0.866 = \pm0.577$.
  • To calculate the auxiliary function  $\theta_2(t)$,  we compute
$$s_{21} = \hspace{0.1cm} < \hspace{-0.1cm} s_2(t), \hspace{0.05cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0 \cdot (+0.577) + 1 \cdot (-0.577)+ 0 \cdot (-0.577)= -0.577$$
$$ \Rightarrow \hspace{0.3cm}\theta_2(t) = s_2(t) - s_{21} \cdot \varphi_1(t) = (0.333, 0.667, -0.333) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\vert \vert \theta_2(t) \vert \vert^2 = (1/3)^2 + (2/3)^2 + (-1/3)^2 = 0.667$$
$$ \Rightarrow \hspace{0.3cm} s_{22} = \sqrt{0.667} = 0.816,\hspace{0.3cm} \varphi_2(t) = \theta_2(t)/s_{22} = (0.408, 0.816, -0.408)\hspace{0.05cm}. $$
  • The inner products between  $s_1(t)$  with  $\varphi_1(t)$  or  $\varphi_2(t)$  give the following results:
$$s_{31} \hspace{0.1cm} = \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_1(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.577) + 0.5 \cdot (-0.577)- 0.5 \cdot (-0.577)= 0.289$$
$$s_{32} \hspace{0.1cm} = \hspace{0.1cm} < \hspace{-0.1cm} s_3(t), \hspace{0.07cm}\varphi_2(t) \hspace{-0.1cm} > \hspace{0.1cm} = 0.5 \cdot (+0.408) + 0.5 \cdot (+0.816)- 0.5 \cdot (-0.408)= 0.816$$
$$\Rightarrow \hspace{0.3cm}\theta_3(t) = s_3(t) - 0.289 \cdot \varphi_1(t)- 0.816 \cdot \varphi_2(t) = 0\hspace{0.05cm}.$$

This means:   The green function  $s_3(t)$  does not yield a new basis function  $\varphi_3(t)$, in contrast to the function  $s_4(t)$. The numerical results for this can be taken from the graph.

Basis functions of complex time signals


In communications engineering, one often has to deal with complex time functions,

  • not because there are complex signals in reality, but
  • because the description of a band-pass signal in the equivalent low-pass range leads to complex signals.

The determination of the  $N \le M$  complex-valued basis functions  $\xi_k(t)$  from the  $M$  complex signals  $s_i(t)$  can also be done using the  "Gram–Schmidt process",  but it must now be taken into account that the inner product of two complex signals  $x(t)$  and  $y(t)$  must be calculated as follows:

$$< \hspace{-0.1cm}x(t), \hspace{0.1cm}y(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}x(t) \cdot y^{\star}(t)\,d \it t \hspace{0.05cm}.$$

The corresponding equations are now with  $i = 1, \text{..}. , M$  and  $k = 1, \text{..}. , N$:

$$s_i(t) = \sum\limits_{k = 1}^{N}s_{ik} \cdot \xi_k(t),\hspace{0.2cm}s_i(t) \in {\cal C},\hspace{0.2cm}s_{ik} \in {\cal C} ,\hspace{0.2cm}\xi_k(t) \in {\cal C} \hspace{0.05cm},$$
$$< \hspace{-0.1cm}\xi_k(t),\hspace{0.1cm} \xi_j(t)\hspace{-0.1cm} > \hspace{0.1cm} = \int_{-\infty}^{+\infty}\xi_k(t) \cdot \xi_j^{\star}(t)\,d \it t = {\rm \delta}_{ik} = \left\{ \begin{array}{c} 1 \\ 0 \end{array} \right.\quad \begin{array}{*{1}c}{\rm if}\hspace{0.15cm} k = j \\ {\rm if}\hspace{0.15cm} k \ne j \\ \end{array}\hspace{0.05cm}.$$

Of course, any complex quantity can also be expressed by two real quantities, namely real part and imaginary part. Thus, the following equations are obtained here:

$$s_{i}(t) = s_{{\rm I}\hspace{0.02cm}i}(t) + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}i}(t), \hspace{0.2cm} s_{{\rm I}\hspace{0.02cm}i}(t) = {\rm Re}\big [s_{i}(t)\big], \hspace{0.2cm} s_{{\rm Q}\hspace{0.02cm}i}(t) = {\rm Im} \big [s_{i}(t)\big ],$$
$$\xi_{k}(t) = \varphi_k(t) + {\rm j} \cdot \psi_k(t), \hspace{0.2cm} \varphi_k(t) = {\rm Re}\big [\xi_{k}(t)\big ], \hspace{0.2cm} \psi_k(t) = {\rm Im} \big [\xi_{k}(t)\big ],$$
$$\hspace{0.35cm} s_{ik} = s_{{\rm I}\hspace{0.02cm}ik} + {\rm j} \cdot s_{{\rm Q}\hspace{0.02cm}ik}, \hspace{0.2cm} s_{{\rm I}ik} = {\rm Re} \big [s_{ik}\big ], \hspace{0.2cm} s_{{\rm Q}ik} = {\rm Im} \big [s_{ik}\big ],$$
$$ \hspace{0.35cm} s_{{\rm I}\hspace{0.02cm}ik} ={\rm Re}\big [\hspace{0.01cm} < \hspace{-0.1cm} s_i(t), \hspace{0.15cm}\varphi_k(t) \hspace{-0.1cm} > \hspace{0.1cm}\big ], \hspace{0.2cm}s_{{\rm Q}\hspace{0.02cm}ik} = {\rm Re}\big [\hspace{0.01cm} < \hspace{-0.1cm} s_i(t), \hspace{0.15cm}{\rm j} \cdot \psi_k(t) \hspace{-0.1cm} > \hspace{0.1cm}\big ] \hspace{0.05cm}. $$

The nomenclature arises from the main application for complex basis functions, namely  "quadrature amplitude modulation"  (QAM).

  • The subscript "I" stands for inphase component and indicates the real part,
  • while the quadrature component (imaginary part) is indicated by the index "Q".


To avoid confusion with the imaginary unit, here the complex basis functions  $\xi_{k}(t)$  were induced with  $k$  and not with  $j$.

Dimension of the basis functions


In baseband transmission, the possible transmitted signals considering only one symbol duration) are

$$s_i(t) = a_i \cdot g_s(t), \hspace{0.2cm} i = 0, \text{...}\hspace{0.05cm} , M-1,$$

where  $g_s(t)$  indicates the basic transmission pulse and the  $a_i$  were denoted as the possible amplitude coefficients in the first three main chapters. It should be noted that from now on the values  $0$  to $M-1$  are assumed for the indexing variable  $i$. 

According to the description of this chapter, regardless of the level number  $M$,  it is a one-dimensional modulation process  $(N = 1)$, where, in the case of baseband transmission

  • the basis function  $\varphi_1(t)$  is equal to the energy normalized basic transmission pulse  $g_s(t)$: 
$$\varphi_1(t) ={g_s(t)}/{\sqrt{E_{gs}}} \hspace{0.3cm}{\rm with}\hspace{0.3cm} E_{gs} = \int_{-\infty}^{+\infty}g_s^2(t)\,d \it t \hspace{0.05cm},$$
  • the dimensionless amplitude coefficients  $a_i$  are to be converted into the signal space points  $s_i$  which have the unit "root of energy".


The graph shows one-dimensional signal space constellations  $(N=1)$  for baseband transmission, viz.

One-dimensional modulation methods
(a) binary unipolar (top)   ⇒   $M = 2$,
(b) binary bipolar (center)   ⇒   $M = 2$, and
(c) quaternary bipolar (bottom)   ⇒   $M = 4$.

The graph simultaneously describes the one-dimensional carrier frequency systems


The signals  $s_i(t)$  and the basis function  $\varphi_1(t)$  shown always refer to the equivalent low-pass range.

In the band-pass region, $\varphi_1(t)$ is a harmonic oscillation limited to the time domain $0 \le t \le T$.
Further notes:

  • In the graph on the right, the two or four possible transmitted signals  $s_i(t)$  are given for the example "rectangular pulse".
  • From this, one can see the relationship between pulse amplitude  $A$  and signal energy  $E = A^2 \cdot T$. 
  • However, the respective left representations on the  $\varphi_1(t)$ axis are valid independently of the  $g_s(t)$ shape, not only for rectangles.


Two-dimensional modulation processes


Zweidimensionale Signalraumkonstellationen für  $M$–PSK und  $M$–QAM

Zu den zweidimensionalen Modulationsverfahren  $(N = 2)$  gehören

Allgemein ist bei orthogonaler FSK die Anzahl  $N$  der Basisfunktionen  $\varphi_k(t)$  gleich der Anzahl  $M$  möglicher Sendesignale  $s_i(t)$. $N=2$  ist deshalb nur für  $M=2$  möglich.

Die Grafik zeigt Beispiele von Signalraumkonstellationen für Zweidimensionale Modulationsverfahren:

  • Die linke Grafik zeigt die 8–PSK–Konstellation. Beschränkt man sich auf die rot umrandeten Punkte, so liegt eine 4–PSK (Quaternary Phase Shift Keying, QPSK) vor.
  • Die rechte Grafik bezieht sich auf die 16–QAM beziehungsweise – wenn man nur die rot umrandeten Signalraumpunkte betrachtet – auf die 4–QAM.
  • Ein Vergleich der beiden Bilder zeigt, dass die 4–QAM mit der QPSK bei entsprechender Achsenskalierung identisch ist.


Die Grafiken beschreiben die Modulationsverfahren sowohl im Bandpass– als auch im äquivalenten Tiefpassbereich:

  • Bei der Betrachtung als Bandpass–System ist die Basisfunktion  $\varphi_1(t)$  cosinusförmig und   $\varphi_2(t)$  (minus–)sinusförmig – vergleiche  Aufgabe 4.2.
  • Dagegen ist nach der Transformation der QAM–Systeme in den äquivalenten Tiefpassbereich  $\varphi_1(t)$  gleich dem energienormierten (also mit der Energie "1") Sendegrundimpuls  $g_s(t)$, während   $\varphi_2(t)={\rm j} \cdot \varphi_1(t)$  zu setzen ist. Näheres hierzu finden Sie in der  Aufgabe 4.2Z.

Aufgaben zum Kapitel


Aufgabe 4.1: Zum Gram-Schmidt-Verfahren

Aufgabe 4.1Z: Andere Basisfunktionen

Aufgabe 4.2: AM/PM-Schwingungen

Aufgabe 4.2Z: Achtstufiges Phase Shift Keying

Aufgabe 4.3: Unterschiedliche Frequenzen

Quellenverzeichnis

  1. 1.0 1.1 1.2 Kötter, R., Zeitler, G.: Nachrichtentechnik 2. Vorlesungsmanuskript, Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2008.
  2. Wozencraft, J. M.; Jacobs, I. M.: Principles of Communication Engineering. New York: John Wiley & Sons, 1965.
  3. Kramer, G.: Nachrichtentechnik 2. Vorlesungsmanuskript, Lehrstuhl für Nachrichtentechnik, Technische Universität München, 2017.