Difference between revisions of "Digital Signal Transmission/Approximation of the Error Probability"

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{{Header
 
{{Header
|Untermenü=Verallgemeinerte Beschreibung digitaler Modulationsverfahren
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|Untermenü=Generalized Description of Digital Modulation Methods
 
|Vorherige Seite=Struktur des optimalen Empfängers
 
|Vorherige Seite=Struktur des optimalen Empfängers
 
|Nächste Seite=Trägerfrequenzsysteme mit kohärenter Demodulation
 
|Nächste Seite=Trägerfrequenzsysteme mit kohärenter Demodulation
 
}}
 
}}
  
== Optimale Entscheidung bei binärer Übertragung==
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== Optimal decision with binary transmission==
 
<br>
 
<br>
Wir gehen hier von einem Übertragungssystem aus, das wie folgt charakterisiert werden kann: &nbsp; $\boldsymbol{r} = \boldsymbol{s} + \boldsymbol{n}$. Dieses System weist folgende Eigenschaften auf:
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We assume here a transmission system which can be characterized as follows: &nbsp; $\boldsymbol{r} = \boldsymbol{s} + \boldsymbol{n}$.&nbsp; This system has the following properties:
*Der das Übertragungssystem vollständig beschreibende Vektorraum wird von &nbsp;$N = 2$&nbsp; zueinander orthogonalen Basisfunktionen&nbsp; $\varphi_1(t)$&nbsp; und&nbsp; $\varphi_2(t)$&nbsp; aufgespannt.<br>
+
*The vector space fully describing the transmission system is spanned by &nbsp;$N = 2$&nbsp; mutually orthogonal basis functions &nbsp; $\varphi_1(t)$ &nbsp; and &nbsp; $\varphi_2(t)$.&nbsp; <br>
  
*Demzufolge ist auch die Wahrscheinlichkeitsdichtefunktion des additiven und weißen Gaußschen Rauschens zweidimensional anzusetzen, gekennzeichnet durch den Vektor&nbsp; $\boldsymbol{ n} = (n_1,\hspace{0.05cm}n_2)$.<br>
+
*Consequently,&nbsp; the probability density function of the additive and white Gaussian noise is also to be set two-dimensional,&nbsp; characterized by the vector&nbsp; $\boldsymbol{ n} = (n_1,\hspace{0.05cm}n_2)$.<br>
  
*Es gibt nur zwei mögliche Sendesignale&nbsp; $(M = 2)$, die durch die beiden Vektoren&nbsp; $\boldsymbol{ s_0} = (s_{01},\hspace{0.05cm}s_{02})$&nbsp; und&nbsp; $\boldsymbol{ s_1} = (s_{11},\hspace{0.05cm}s_{12})$&nbsp; beschrieben werden:
+
*There are only two possible transmitted signals&nbsp; $(M = 2)$,&nbsp; described by the two vectors&nbsp; $\boldsymbol{ s_0} = (s_{01},\hspace{0.05cm}s_{02})$&nbsp; and&nbsp; $\boldsymbol{ s_1} = (s_{11},\hspace{0.05cm}s_{12})$:&nbsp;
:$$s_0(t)= s_{01} \cdot \varphi_1(t) + s_{02} \cdot \varphi_2(t) \hspace{0.05cm},\hspace{1cm}s_1(t) = s_{11} \cdot \varphi_1(t) + s_{12} \cdot \varphi_2(t) \hspace{0.05cm}.$$
+
[[File:P ID2019 Dig T 4 3 S1 version1.png|right|frame|Decision regions for equal&nbsp; (left)&nbsp; and unequal (right)&nbsp; occurrence probabilities|class=fit]]
 +
 +
:$$s_0(t)= s_{01} \cdot \varphi_1(t) + s_{02} \cdot \varphi_2(t) \hspace{0.05cm},$$
 +
:$$s_1(t) = s_{11} \cdot \varphi_1(t) + s_{12} \cdot \varphi_2(t) \hspace{0.05cm}.$$
  
*Die beiden Nachrichten&nbsp; $m_0 \ \Leftrightarrow \ \boldsymbol{ s_0}$&nbsp; und &nbsp;$m_1 \ \Leftrightarrow \ \boldsymbol{ s_1}$&nbsp; sind nicht notwendigermaßen gleichwahrscheinlich.<br>
+
*The two messages&nbsp; $m_0 \ \Leftrightarrow \ \boldsymbol{ s_0}$&nbsp; and &nbsp;$m_1 \ \Leftrightarrow \ \boldsymbol{ s_1}$&nbsp; are not necessarily equally probable.<br>
  
*Aufgabe des Entscheiders ist es, einen Schätzwert für den aktuellen Empfangsvektor&nbsp; $\boldsymbol{r}$&nbsp; nach der&nbsp; [[Digitalsignal%C3%BCbertragung/Struktur_des_optimalen_Empf%C3%A4ngers#Fundamentaler_Ansatz_zum_optimalen_Empf.C3.A4ngerentwurf_.281.29 |MAP&ndash;Entscheidungsregel]]&nbsp; anzugeben. Diese lautet im vorliegenden Fall:
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*The task of the decision is to give an estimate for the current received vector&nbsp; $\boldsymbol{r}$&nbsp; according to the&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Fundamental_approach_to_optimal_receiver_design|"MAP decision rule"]].&nbsp; In the present case,&nbsp; this rule is with&nbsp; $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:
 
:$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}|\hspace{0.05cm} m_i )\big ]  
 
:$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}|\hspace{0.05cm} m_i )\big ]  
\hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}\hspace{0.3cm}{\rm mit}\hspace{0.3cm}
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\hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$
\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)
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*In the special case&nbsp; $N = 2$&nbsp; and&nbsp; $M = 2$&nbsp; considered here,&nbsp; the decision partitions the two-dimensional space into the two disjoint areas&nbsp; $I_0$&nbsp; (highlighted in red)&nbsp; and&nbsp; $I_1$&nbsp; (blue),&nbsp; as the graphic on the right illustrates.
\hspace{0.05cm}.$$
 
  
Im hier betrachteten Sonderfall&nbsp; $N = 2$&nbsp; und&nbsp; $M = 2$&nbsp; partitioniert der Entscheider den zweidimensionalen Raum in die zwei disjunkten Gebiete&nbsp; $I_0$&nbsp; (rot hinterlegt) und&nbsp; $I_1$&nbsp; (blau), wie die folgende Grafik verdeutlicht. Liegt der Empfangswert in&nbsp; $I_0$, so wird als Schätzwert&nbsp; $m_0$&nbsp; ausgegeben, andernfalls&nbsp; $m_1$.
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*If the received value lies in&nbsp; $I_0$, &nbsp; $m_0$&nbsp; is output as the estimated value,&nbsp; otherwise&nbsp; $m_1$.
  
[[File:P ID2019 Dig T 4 3 S1 version1.png|center|frame|Entscheidungsregionen für gleiche (links) bzw. ungleiche (rechts) Auftrittswahrscheinlichkeiten|class=fit]]
 
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Herleitung und Bildbeschreibung:}$&nbsp;
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$\text{Derivation and picture description:}$&nbsp;
Beim AWGN&ndash;Kanal und&nbsp;  $M = 2$&nbsp; lautet somit die Entscheidungsregel:  
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For the AWGN channel and&nbsp;  $M = 2$,&nbsp; the decision rule is thus:
  
Man entscheide sich immer dann für die Nachricht&nbsp; $m_0$, falls folgende Bedingung erfüllt ist:
+
&rArr; &nbsp; Always choose message&nbsp; $m_0$&nbsp; if the following condition is satisfied:
  
 
:$${\rm Pr}( m_0) \cdot  {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot \vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 \right ]
 
:$${\rm Pr}( m_0) \cdot  {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot \vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 \right ]
Line 38: Line 40:
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
Die Grenzlinie zwischen den beiden Entscheidungsregionen&nbsp; $I_0$&nbsp; und&nbsp; $I_1$&nbsp; erhält man, wenn man in obiger Gleichung das Größerzeichen durch das Gleichheitszeichen ersetzt und die Gleichung etwas umformt:
+
&rArr; &nbsp; The boundary line between the two decision regions&nbsp; $I_0$&nbsp; and&nbsp; $I_1$&nbsp; is obtained by replacing the&nbsp; "greater sign"&nbsp; with the&nbsp; "equals sign"&nbsp; in the above equation and transforming the equation slightly:
  
 
:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2  - 2  \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_0)\big ] =  
 
:$$\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2  - 2  \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_0)\big ] =  
Line 45: Line 47:
 
+ 2  \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_0)}{ {\rm Pr}( m_1)} = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$  
 
+ 2  \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_0)}{ {\rm Pr}( m_1)} = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$  
  
Aus dieser Darstellung erkennt man:
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From the plot above one can see:
*Die Grenzkurve zwischen den Regionen&nbsp; $I_0$&nbsp; und&nbsp; $I_1$&nbsp; ist eine Gerade, da die Bestimmungsgleichung linear im Empfangsvektor&nbsp; $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$&nbsp; ist.<br>
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*The boundary curve between regions&nbsp; $I_0$&nbsp; and&nbsp; $I_1$&nbsp; is a straight line,&nbsp; since the equation of determination is linear in the received vector&nbsp; $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$.&nbsp; <br>
  
*Bei gleichwahrscheinlichen Symbolen verläuft die Grenze genau in der Mitte zwischen&nbsp; $\boldsymbol{ s }_0$&nbsp; und&nbsp; $\boldsymbol{ s }_1$&nbsp; und um &nbsp;$90^\circ$&nbsp; verdreht gegenüber der Verbindungslinie zwischen den Sendepunkten (linke Grafik):
+
*For equally probable symbols,&nbsp; the boundary is exactly halfway between&nbsp; $\boldsymbol{ s }_0$&nbsp; and&nbsp; $\boldsymbol{ s }_1$&nbsp; and rotated by &nbsp;$90^\circ$&nbsp; with respect to the line connecting the transmission points:
  
 
:$$\vert \hspace{-0.05cm} \vert  \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2  - \vert \hspace{-0.05cm} \vert  \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2  = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
 
:$$\vert \hspace{-0.05cm} \vert  \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2  - \vert \hspace{-0.05cm} \vert  \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2  = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
  
*Für&nbsp; ${\rm Pr}(m_0) > {\rm Pr}(m_1)$&nbsp; ist die Entscheidungsgrenze in Richtung des unwahrscheinlicheren Symbols&nbsp; $\boldsymbol{ s }_1$&nbsp; verschoben, und zwar um so mehr, je größer die AWGN&ndash;Streuung&nbsp; $\sigma_n$&nbsp; ist.<br>
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*For&nbsp; ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,&nbsp; the decision boundary is shifted toward the less probable symbol&nbsp; $\boldsymbol{ s }_1$,&nbsp; and the more so the larger the AWGN standard deviation&nbsp; $\sigma_n$.&nbsp; <br>
  
*Die grün&ndash;durchgezogene Entscheidungsgrenze im rechten Bild sowie die Entscheidungsregionen&nbsp; $I_0$&nbsp; (rot) und&nbsp;  $I_1$&nbsp; (blau) gelten für die (normierte) Streuung&nbsp; $\sigma_n = 1$&nbsp; und die gestrichelten Grenzlinien für&nbsp; $\sigma_n = 0$&nbsp; bzw.&nbsp; $\sigma_n = 2$.<br>}}
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*The green-dashed decision boundary in the right figure as well as the decision regions&nbsp; $I_0$&nbsp; (red)&nbsp; and&nbsp;  $I_1$&nbsp; (blue)&nbsp; are valid for the&nbsp; (normalized)&nbsp; standard deviation&nbsp; $\sigma_n = 1$&nbsp; and the dashed boundary lines for&nbsp; $\sigma_n = 0$&nbsp; resp.&nbsp; $\sigma_n = 2$. <br>}}
  
==Der Sonderfall gleichwahrscheinlicher Binärsymbolen ==
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==The special case of equally probable binary symbols ==
 
<br>
 
<br>
Wir gehen weiterhin von einem Binärsystem aus&nbsp; $(M = 2)$, betrachten aber nun den einfachen Fall, dass dieses durch eine einzige Basisfunktion beschrieben werden kann&nbsp;  $(N = 1)$. Die Fehlerwahrscheinlichkeit hierfür wurde bereits im Abschnitt&nbsp; [[Digitalsignal%C3%BCbertragung/Fehlerwahrscheinlichkeit_bei_Basisband%C3%BCbertragung#Definition_der_Bitfehlerwahrscheinlichkeit|Definition der Bitfehlerwahrscheinlichkeit]]&nbsp; berechnet.<br>
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We continue to assume a binary system&nbsp; $(M = 2)$,&nbsp; but now consider the simple case where this can be described by a single basis function&nbsp;  $(N = 1)$.&nbsp; The error probability for this has already been calculated in the section&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability|"Definition of the bit error probability"]].&nbsp; <br>
 +
With the nomenclature and representation form chosen for the fourth main chapter the following constellation results:
 +
[[File:P ID2020 Dig T 4 3 S2 version1.png||right|frame|Conditional probability density functions for equally probable symbols|class=fit]]
  
Mit der für das vierte Hauptkapitel gewählten Nomenklatur und Darstellungsform ergibt sich folgende Konstellation:
+
*The received value&nbsp; $r = s + n$&nbsp; is now a scalar and is composed of the transmitted signal&nbsp; $s \in \{s_0, \hspace{0.05cm}s_1\}$&nbsp; and the noise term&nbsp; $n$&nbsp; additively. The abscissa&nbsp; $\rho$&nbsp; denotes a realization of&nbsp; $r$.<br>
*Der Empfangswert&nbsp; $r = s + n$&nbsp; ist nunmehr ein Skalar und setzt sich aus dem Sendesignal&nbsp; $s \in \{s_0, \hspace{0.05cm}s_1\}$&nbsp; und dem Rauschterm&nbsp; $n$&nbsp; additiv zusammen. Die Abszisse&nbsp; $\rho$&nbsp; bezeichnet eine Realisierung von&nbsp; $r$.<br>
 
  
*Die Abszisse ist zudem auf die Bezugsgröße&nbsp; $\sqrt{E}$&nbsp; normiert, wobei hier die Normierungsenergie&nbsp; $E$&nbsp; keine herausgehobene, physikalisch interpretierbare Bedeutung hat.<br>
+
*In addition,&nbsp; the abscissa is normalized to the reference quantity&nbsp; $\sqrt{E}$,&nbsp; whereas here the normalization energy&nbsp; $E$&nbsp; has no prominent,&nbsp; physically interpretable meaning.<br>
  
*Der Rauschterm&nbsp; $n$&nbsp; ist gaußverteilt mit Mittelwert&nbsp; $m_n = 0$&nbsp; und Varianz&nbsp; $\sigma_n^2$. Die Wurzel aus der Varianz&nbsp; $(\sigma_n)$&nbsp; wird als der Effektivwert  oder die Streuung bezeichnet.<br>
+
*The noise term&nbsp; $n$&nbsp; is Gaussian distributed with mean&nbsp; $m_n = 0$&nbsp; and variance&nbsp; $\sigma_n^2$.&nbsp; The root of the variance&nbsp; $(\sigma_n)$&nbsp; is called the&nbsp; "rms value"&nbsp; or the&nbsp; "standard deviation".<br>
  
*Die Entscheidergrenze&nbsp; $G$&nbsp; unterteilt den gesamten Wertebereich von&nbsp; $r$&nbsp; in die zwei Teilbereiche&nbsp; $I_0$&nbsp; $($in dem unter anderem&nbsp; $s_0$&nbsp; liegt$)$ und&nbsp; $I_1$&nbsp; $($mit dem Signalwert &nbsp;$s_1)$.<br>
+
*The decision boundary&nbsp; $G$&nbsp; divides the entire value range of&nbsp; $r$&nbsp; into the two subranges&nbsp; $I_0$&nbsp; $($in which&nbsp; $s_0$&nbsp; lies$)$ and&nbsp; $I_1$&nbsp; $($with the signal value &nbsp;$s_1)$.<br>
  
*Ist&nbsp; $\rho > G$, so liefert der Entscheider den Schätzwert&nbsp; $m_0$, andernfalls&nbsp; $m_1$. Hierbei ist vorausgesetzt, dass die Nachricht&nbsp; $m_i$&nbsp; mit dem Sendesignal&nbsp; $s_i$&nbsp; eineindeutig zusammenhängt: &nbsp; $m_i \Leftrightarrow s_i$.
+
*If&nbsp; $\rho > G$,&nbsp; the decision returns the estimated value&nbsp; $m_0$, otherwise&nbsp; $m_1$.&nbsp; It is assumed that the message&nbsp; $m_i$&nbsp; is uniquely related to the signal&nbsp; $s_i$:&nbsp; &nbsp; $m_i \Leftrightarrow s_i$.
  
[[File:P ID2020 Dig T 4 3 S2 version1.png||center|frame|Bedingte Dichtefunktionen bei gleichwahrscheinlichen Symbolen|class=fit]]
 
  
Die Grafik zeigt die bedingten (eindimensionalen) Wahrscheinlichkeitsdichtefunktionen&nbsp; $p_{\hspace{0.02cm}r\hspace{0.05cm} \vert \hspace{0.05cm}m_0}$&nbsp; und&nbsp;  $p_{\hspace{0.02cm}r\hspace{0.05cm} \vert \hspace{0.05cm}m_1}$&nbsp; für den AWGN&ndash;Kanal, wobei gleiche Symbolwahrscheinlichkeiten vorausgesetzt sind: &nbsp; ${\rm Pr}(m_0) =  {\rm Pr}(m_1)  = 0.5$. Die (optimale) Entscheidergrenze ist somit&nbsp; $G = 0$. Man erkennt aus dieser Darstellung:
+
The graph shows the conditional&nbsp; $($one-dimensional$)$&nbsp; probability density functions &nbsp; $p_{\hspace{0.02cm}r\hspace{0.05cm} \vert \hspace{0.05cm}m_0}$ &nbsp; and &nbsp;  $p_{\hspace{0.02cm}r\hspace{0.05cm} \vert \hspace{0.05cm}m_1}$ &nbsp; for the AWGN channel,&nbsp; assuming equal symbol probabilities: &nbsp; ${\rm Pr}(m_0) =  {\rm Pr}(m_1)  = 0.5$.&nbsp; Thus,&nbsp; the&nbsp; $($optimal$)$&nbsp; decision boundary is&nbsp; $G = 0$.&nbsp; One can see from this plot:
*Ist&nbsp; $m = m_0$&nbsp; und damit&nbsp; $s = s_0 = 2 \cdot E^{1/2}$, so kommt es nur dann zu einer Fehlentscheidung, wenn&nbsp; $\eta$, die Realisierung der Rauschgröße&nbsp; $n$, kleiner ist als&nbsp; $-2 \cdot E^{1/2}$. In diesem Fall ist&nbsp; $\rho < 0$, wobei&nbsp; $\rho$&nbsp; eine Realisierung des Empfangswertes&nbsp; $r$&nbsp; bezeichnet.
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#If&nbsp; $m = m_0$&nbsp; and thus&nbsp; $s = s_0 = 2 \cdot E^{1/2}$,&nbsp; an erroneous decision occurs only if&nbsp; $\eta$,&nbsp; the realization of the noise quantity&nbsp; $n$,&nbsp; is smaller than&nbsp; $-2 \cdot E^{1/2}$.  
*Bei&nbsp; $m = m_1$ &nbsp; &rArr; &nbsp; $s = s_1 = -2 \cdot E^{1/2}$&nbsp; kommt es dagegen immer dann zu einer Fehlentscheidung, wenn&nbsp; $\eta$&nbsp; größer ist als&nbsp; $+2 \cdot E^{1/2}$. In diesem Fall ist&nbsp; $\rho > 0$.
+
#In this case,&nbsp; $\rho < 0$, where&nbsp; $\rho$&nbsp; denotes a realization of the received value&nbsp; $r$.&nbsp;  
 +
#In contrast,&nbsp; for&nbsp; $m = m_1$ &nbsp; &rArr; &nbsp; $s = s_1 = -2 \cdot E^{1/2}$,&nbsp; an erroneous decision occurs whenever&nbsp; $\eta$&nbsp; is greater than&nbsp; $+2 \cdot E^{1/2}$.&nbsp; In this case,&nbsp; $\rho > 0$.
  
  
== Fehlerwahrscheinlichkeit bei gleichwahrscheinlichen Symbolen ==
+
== Error probability for symbols with equal probability ==
 
<br>
 
<br>
Es gelte&nbsp; ${\rm Pr}(m_0) = {\rm Pr}(m_1) = 0.5$. Bei AWGN&ndash;Rauschen mit Effektivwert (Streuung)&nbsp; $\sigma_n$&nbsp; erhält man, wie bereits im Abschnitt&nbsp; [[Digitalsignal%C3%BCbertragung/Fehlerwahrscheinlichkeit_bei_Basisband%C3%BCbertragung#Definition_der_Bitfehlerwahrscheinlichkeit| Definition der Bitfehlerwahrscheinlichkeit]]&nbsp; mit anderer Nomenklatur berechnet wurde, für die Wahrschenlichkeit einer Fehlentscheidung&nbsp; $(\cal E)$&nbsp; unter der Bedingung, dass die Nachricht&nbsp; $m_0$&nbsp; gesendet wurde:
+
Let&nbsp; ${\rm Pr}(m_0) = {\rm Pr}(m_1) = 0.5$.&nbsp; For AWGN noise with standard deviation&nbsp; $\sigma_n$,&nbsp; as already calculated in the section&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission#Definition_of_the_bit_error_probability|"Definition of the bit error probability"]]&nbsp; with different nomenclature,&nbsp; we obtain for the probability of a wrong decision&nbsp; $(\cal E)$&nbsp; under the condition that message&nbsp; $m_0$&nbsp; was sent:
  
 
:$${\rm Pr}({ \cal E}\hspace{0.05cm} \vert \hspace{0.05cm} m_0) = \int_{-\infty}^{G = 0} p_{r \hspace{0.05cm}|\hspace{0.05cm}m_0 } ({ \rho } \hspace{0.05cm} \vert \hspace{0.05cm}m_0 ) \,{\rm d} \rho =  \int_{-\infty}^{-  s_0 } p_{{ n} \hspace{0.05cm}\vert\hspace{0.05cm}m_0 } ({ \eta } \hspace{0.05cm}|\hspace{0.05cm}m_0 ) \,{\rm d} \eta = \int_{-\infty}^{- s_0 } p_{{ n}  } ({ \eta }  ) \,{\rm d} \eta =
 
:$${\rm Pr}({ \cal E}\hspace{0.05cm} \vert \hspace{0.05cm} m_0) = \int_{-\infty}^{G = 0} p_{r \hspace{0.05cm}|\hspace{0.05cm}m_0 } ({ \rho } \hspace{0.05cm} \vert \hspace{0.05cm}m_0 ) \,{\rm d} \rho =  \int_{-\infty}^{-  s_0 } p_{{ n} \hspace{0.05cm}\vert\hspace{0.05cm}m_0 } ({ \eta } \hspace{0.05cm}|\hspace{0.05cm}m_0 ) \,{\rm d} \eta = \int_{-\infty}^{- s_0 } p_{{ n}  } ({ \eta }  ) \,{\rm d} \eta =
Line 86: Line 89:
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
Bei der Herleitung der Gleichung wurde berücksichtigt, dass das AWGN&ndash;Rauschen&nbsp; $\eta$&nbsp; unabhängig vom Signal &nbsp;$(m_0$&nbsp; oder&nbsp; $m_1)$&nbsp; ist und eine symmetrische WDF besitzt. Verwendet wurde zudem das komplementäre Gaußsche Fehlerintegral
+
In deriving the equation,&nbsp; it was considered that the AWGN noise&nbsp; $\eta$&nbsp; is independent of the signal &nbsp;$(m_0$&nbsp; or&nbsp; $m_1)$&nbsp; and has a symmetric PDF.&nbsp; The complementary Gaussian error integral was also used:
 
:$${\rm Q}(x) =  \frac{1}{\sqrt{2\pi}}  \int_{x}^{\infty} {\rm e}^{-u^2/2} \,{\rm d} u  
 
:$${\rm Q}(x) =  \frac{1}{\sqrt{2\pi}}  \int_{x}^{\infty} {\rm e}^{-u^2/2} \,{\rm d} u  
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
Entsprechend gilt für&nbsp; $m = m_1$ &nbsp; &rArr; &nbsp; $s = s_1 = -2 \cdot E^{1/2}$:
+
Correspondingly,&nbsp; for&nbsp; $m = m_1$ &nbsp; &rArr; &nbsp; $s = s_1 = -2 \cdot E^{1/2}$:
 
:$${\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) =  \int_{0}^{\infty} p_{{ r} \hspace{0.05cm}\vert\hspace{0.05cm}m_1 } ({ \rho } \hspace{0.05cm}\vert\hspace{0.05cm}m_1 ) \,{\rm d} \rho =  \int_{- s_1 }^{\infty} p_{{ n}  } (\boldsymbol{ \eta }  ) \,{\rm d} \eta = {\rm Q} \left ( {- s_1 }/{\sigma_n} \right )
 
:$${\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) =  \int_{0}^{\infty} p_{{ r} \hspace{0.05cm}\vert\hspace{0.05cm}m_1 } ({ \rho } \hspace{0.05cm}\vert\hspace{0.05cm}m_1 ) \,{\rm d} \rho =  \int_{- s_1 }^{\infty} p_{{ n}  } (\boldsymbol{ \eta }  ) \,{\rm d} \eta = {\rm Q} \left ( {- s_1 }/{\sigma_n} \right )
 
\hspace{0.05cm}.$$
 
\hspace{0.05cm}.$$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Fazit:}$&nbsp; Mit dem Abstand&nbsp; $d = s_1 - s_0$&nbsp; der Signalraumpunkte kann man die Ergebnisse zusammenfassen, wobei noch&nbsp; ${\rm Pr}(m_0) + {\rm Pr}(m_1) = 1$&nbsp; zu berücksichtigen ist:
+
$\text{Conclusion:}$&nbsp; With the distance&nbsp; $d = s_1 - s_0$&nbsp; of the signal space points, we can summarize the results, still considering&nbsp; ${\rm Pr}(m_0) + {\rm Pr}(m_1) = 1$:&nbsp;  
 
:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0) =  {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = {\rm Q} \big ( {d}/(2{\sigma_n}) \big )$$
 
:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0) =  {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = {\rm Q} \big ( {d}/(2{\sigma_n}) \big )$$
 
:$$\Rightarrow \hspace{0.3cm}{\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_0)  + {\rm Pr}(m_1) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1)= \big [ {\rm Pr}(m_0) + {\rm Pr}(m_1) \big ] \cdot  
 
:$$\Rightarrow \hspace{0.3cm}{\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_0)  + {\rm Pr}(m_1) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1)= \big [ {\rm Pr}(m_0) + {\rm Pr}(m_1) \big ] \cdot  
 
  {\rm Q}  \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] \hspace{0.05cm}.$$
 
  {\rm Q}  \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] \hspace{0.05cm}.$$
  
''Hinweise:''
+
<u>Notes:</u>
*Diese Gleichung gilt unter der Voraussetzung&nbsp; $G = 0$&nbsp; ganz allgemein, also auch für&nbsp; ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$.  
+
#This equation is valid under the condition&nbsp; $G = 0$&nbsp; quite generally,&nbsp; thus also for&nbsp; ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$.  
*Bei&nbsp; [[Digital_Signal_Transmission/Approximation_der_Fehlerwahrscheinlichkeit#Optimale_Schwelle_bei_nicht_gleichwahrscheinlichen_Symbolen|nicht gleichwahrscheinlichen Symbolen]]&nbsp; lässt sich allerdings die Fehlerwahrscheinlichkeit durch eine andere Entscheidergrenze verkleinern.<br>
+
#For&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_threshold_for_non-equally_probable_symbols|"non-equally probable symbols"]],&nbsp; however,&nbsp; the error probability can be reduced by a different decision threshold.<br>
*Die hier genannte Gleichung gilt auch dann, wenn die Signalraumpunkte keine Skalare sind, sondern durch die Vektoren&nbsp; $\boldsymbol{ s}_0$&nbsp; und&nbsp; $\boldsymbol{ s}_1$&nbsp; beschrieben werden.
+
#The equation mentioned here is also valid if the signal space points are not scalars but are described by the vectors&nbsp; $\boldsymbol{ s}_0$&nbsp; and&nbsp; $\boldsymbol{ s}_1$.&nbsp;  
*Der  Abstand&nbsp; $d$&nbsp; ergibt sich dann als die Norm des Differenzvektors: &nbsp; $d = \vert \hspace{-0.05cm} \vert \hspace{0.05cm} \boldsymbol{ s}_1  - \boldsymbol{ s}_0 \hspace{0.05cm} \vert \hspace{-0.05cm} \vert
+
#The distance&nbsp; $d$&nbsp; results then as the norm of the difference vector: &nbsp; $d = \vert \hspace{-0.05cm} \vert \hspace{0.05cm} \boldsymbol{ s}_1  - \boldsymbol{ s}_0 \hspace{0.05cm} \vert \hspace{-0.05cm} \vert
 
\hspace{0.05cm}.$}}
 
\hspace{0.05cm}.$}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 1:}$&nbsp; Betrachten wir nochmals die Signalraumkonstellation von der&nbsp; [[Digitalsignal%C3%BCbertragung/Approximation_der_Fehlerwahrscheinlichkeit#Optimale_Entscheidung_bei_bin.C3.A4rer_.C3.9Cbertragung_.281.29| ersten Kapitelseite]]&nbsp; (untere Grafik) mit den Werten
+
$\text{Example 1:}$&nbsp; Let's look again at the signal space constellation from the&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_decision_with_binary_transmission|"first chapter section"]]&nbsp; $($lower graphic$)$&nbsp; with the values
 +
 
 +
[[File:Dig_T_4_3_S2b_version2.png|right|frame|Two signal space constellations|class=fit]] 
 +
*$\boldsymbol{ s}_0/E^{1/2}  = (3.6, \hspace{0.05cm}0.8)$,
  
[[File:Dig_T_4_3_S2b_version2.png|right|frame|Zwei Signalraumkonstellationen|class=fit]] 
 
*$\boldsymbol{ s}_0/E^{1/2}  = (3.6, \hspace{0.05cm}0.8)$ und
 
 
*$\boldsymbol{ s}_1/E^{1/2}  = (0.4, \hspace{0.05cm}3.2)$.  
 
*$\boldsymbol{ s}_1/E^{1/2}  = (0.4, \hspace{0.05cm}3.2)$.  
  
  
Hier beträgt der Abstand der Signalraumpunkte
+
Here the distance of the signal space points is
  
 
:$$d = \vert \hspace{-0.05cm} \vert s_1 - s_0 \vert \hspace{-0.05cm} \vert = \sqrt{E \cdot (0.4 - 3.6)^2 + E \cdot (3.2 - 0.8)^2} = 4 \cdot \sqrt {E}\hspace{0.05cm}.$$  
 
:$$d = \vert \hspace{-0.05cm} \vert s_1 - s_0 \vert \hspace{-0.05cm} \vert = \sqrt{E \cdot (0.4 - 3.6)^2 + E \cdot (3.2 - 0.8)^2} = 4 \cdot \sqrt {E}\hspace{0.05cm}.$$  
  
Es ergibt sich also der genau gleiche Wert wie für die obere Konstellation mit
+
This results in exactly the same value as for the upper constellation with
*$\boldsymbol{ s}_0/E^{1/2}  = (2, \hspace{0.05cm}0)$ und
+
*$\boldsymbol{ s}_0/E^{1/2}  = (2, \hspace{0.05cm}0)$,
 +
 
*$\boldsymbol{ s}_1/E^{1/2}  = (-2, \hspace{0.05cm}0)$. <br>
 
*$\boldsymbol{ s}_1/E^{1/2}  = (-2, \hspace{0.05cm}0)$. <br>
  
  
Die Abbildungen zeigen diese beiden Konstellationen und lassen folgende Gemeinsamkeiten bzw. Unterschiede erkennen, wobei jeweils von der AWGN&ndash;Rauschvarianz&nbsp; $\sigma_n^2 = N_0/2$&nbsp; ausgegangen wird. Die Kreise in der Grafik veranschaulichen die zirkuläre Symmetrie von 2D&ndash;AWGN&ndash;Rauschen.
+
The figures show these two constellations and reveal the following similarities and differences,&nbsp; assuming the AWGN noise variance&nbsp; $\sigma_n^2 = N_0/2$&nbsp; in each case.&nbsp; The circles in the graph illustrate the circular symmetry of the two-dimensional AWGN noise.
*Wie bereits gesagt, sind sowohl der Abstand der Signalpunkte von der Entscheidungsgeraden&nbsp; $(d/2 = 2 \cdot \sqrt {E})$&nbsp; als auch der AWGN&ndash;Kennwert&nbsp;  $\sigma_n$&nbsp; in beiden Fällen gleich.<br>
+
*As said before,&nbsp; both the distance of the signal points from the decision line&nbsp; $(d/2 = 2 \cdot \sqrt {E})$&nbsp; and the AWGN characteristic value&nbsp;  $\sigma_n$&nbsp; are the same in both cases.<br>
  
*Daraus folgt: &nbsp; Die beiden Anordnungen führen zur gleichen Fehlerwahrscheinlichkeit, wenn man den Parameter&nbsp; $E$&nbsp; (eine Art Normierungsenergie) konstant lässt:
+
*It follows: &nbsp; The two arrangements lead to the same error probability if the parameter&nbsp; $E$&nbsp; $($a kind of normalization energy$)$&nbsp; is kept constant:
  
:$${\rm Pr} ({\rm Symbolfehler}) = {\rm Pr}({ \cal E} ) =  {\rm Q} \big [ {d}/(2{\sigma_n}) \big ]\hspace{0.05cm}.$$
+
:$${\rm Pr} ({\rm symbol\hspace{0.15cm}  error}) = {\rm Pr}({ \cal E} ) =  {\rm Q} \big [ {d}/(2{\sigma_n}) \big ]\hspace{0.05cm}.$$
  
*Die <i>mittlere Energie pro Symbol</i> &nbsp;$(E_{\rm S})$&nbsp; ergibt sich für die obere Konstellation zu
+
*The&nbsp; "mean energy per symbol" &nbsp;$(E_{\rm S})$&nbsp; for the upper constellation is given by
 
:$$E_{\rm S} = 1/2 \cdot \vert \hspace{-0.05cm} \vert s_0 \vert \hspace{-0.05cm} \vert^2 + 1/2 \cdot \vert \hspace{-0.05cm} \vert s_1 \vert \hspace{-0.05cm} \vert^2 = E/2 \cdot \big[(+2)^2 + (-2)^2\big] = 4 \cdot  {E}\hspace{0.05cm}.$$  
 
:$$E_{\rm S} = 1/2 \cdot \vert \hspace{-0.05cm} \vert s_0 \vert \hspace{-0.05cm} \vert^2 + 1/2 \cdot \vert \hspace{-0.05cm} \vert s_1 \vert \hspace{-0.05cm} \vert^2 = E/2 \cdot \big[(+2)^2 + (-2)^2\big] = 4 \cdot  {E}\hspace{0.05cm}.$$  
*Bei der unteren Konstellation erhält man in gleicher Weise:
+
*With the lower constellation one receives in the same way:
 
:$$E_{\rm S} = \ \text{...} \ = E/2 \cdot \big[(3.6)^2 + (0.8)^2\big] + E/2 \cdot \big[(0.4)^2 + (3.2)^2 \big] = 12 \cdot  {E}\hspace{0.05cm}.$$  
 
:$$E_{\rm S} = \ \text{...} \ = E/2 \cdot \big[(3.6)^2 + (0.8)^2\big] + E/2 \cdot \big[(0.4)^2 + (3.2)^2 \big] = 12 \cdot  {E}\hspace{0.05cm}.$$  
*Bei gegebener <i>mittlerer Energie pro Symbol</i> &nbsp;$(E_{\rm S})$&nbsp; ist demnach die obere Konstellation der unteren deutlich überlegen: &nbsp; Die gleiche Fehlerwahrscheinlichkeit ergibt sich mit einem Drittel der aufzuwendenden Energie pro Symbol. Auf diesen Sachverhalt wird in der&nbsp; [[Aufgaben:Aufgabe_4.06Z:_Signalraumkonstellationen|Aufgabe 4.6Z]]&nbsp; noch im Detail eingegangen. }}<br>
+
*For a given mean energy per symbol</i> &nbsp; &rArr; &nbsp; $E_{\rm S}$,&nbsp; the upper constellation is therefore clearly superior to the lower one: &nbsp; The same error probability results with one third of the energy per symbol.&nbsp; This issue will be discussed in detail in&nbsp; [[Aufgaben:Exercise_4.06Z:_Signal_Space_Constellations|"Exercise 4.6Z"]].&nbsp; }}<br>
  
== Optimale Schwelle bei nicht gleichwahrscheinlichen  Symbolen ==
+
== Optimal threshold for non-equally probable symbols ==
 
<br>
 
<br>
Gilt&nbsp; ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$, so kann man durch eine Verschiebung der Entscheidungsgrenze&nbsp; $G$&nbsp; eine etwas kleinere Fehlerwahrscheinlichkeit erreichen. Die nachfolgenden Ergebnisse werden ausführlich in der Musterlösung zur&nbsp; [[Aufgaben:Aufgabe_4.07:_Nochmals_Entscheidungsgrenzen|Aufgabe 4.7]]&nbsp; hergeleitet:
+
If&nbsp; ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$&nbsp; holds,&nbsp; a slightly smaller error probability can be obtained by shifting the decision threshold&nbsp; $G$.&nbsp; The following results are derived in detail in the solution to&nbsp; [[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again|"Exercise 4.7"]]:&nbsp;  
*Bei ungleichen Symbolwahrscheinlichkeiten liegt die optimale Entscheidungsgrenze&nbsp; $G_{\rm opt}$&nbsp; zwischen den Regionen&nbsp; $I_0$&nbsp; und&nbsp; $I_1$&nbsp; näher beim unwahrscheinlicheren Symbol.<br>
+
*For unequal symbol probabilities, the optimal decision threshold&nbsp; $G_{\rm opt}$&nbsp; between regions&nbsp; $I_0$&nbsp; and&nbsp; $I_1$&nbsp; is closer to the less probable symbol.&nbsp; The normalized optimal shift with respect to the value&nbsp; $G = 0$&nbsp; for equally probable symbols is
 
 
*Die normierte optimale Verschiebung gegenüber der Grenze&nbsp; $G = 0$&nbsp; bei gleichwahrscheinlichen Symbolen beträgt
 
  
 
::<math>\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot  \frac{  \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.</math>
 
::<math>\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot  \frac{  \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.</math>
  
*Die Fehlerwahrscheinlichkeit ist dann gleich
+
*The error probability is then
  
 
:$${\rm Pr}({ \cal E} ) =  {\rm Pr}(m_0) \cdot {\rm Q} \big[  {d}/(2{\sigma_n})  \cdot (1 - \gamma_{\rm opt}) \big ]
 
:$${\rm Pr}({ \cal E} ) =  {\rm Pr}(m_0) \cdot {\rm Q} \big[  {d}/(2{\sigma_n})  \cdot (1 - \gamma_{\rm opt}) \big ]
 
  + {\rm Pr}(m_1) \cdot {\rm Q} \big [ {d}/(2{\sigma_n})  \cdot (1 + \gamma_{\rm opt}) \big ]\hspace{0.05cm}.$$
 
  + {\rm Pr}(m_1) \cdot {\rm Q} \big [ {d}/(2{\sigma_n})  \cdot (1 + \gamma_{\rm opt}) \big ]\hspace{0.05cm}.$$
  
[[File:P ID2024 Dig T 4 3 S3 version2.png|right|frame|Dichtefunktionen für gleiche/ungleiche Symbolwahrscheinlichkeiten|class=fit]]
 
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 2:}$&nbsp; Der formale Parameter&nbsp; $\rho$&nbsp; (Abszisse) kennzeichnet wieder eine Realisierung der AWGN&ndash;Zufallsgröße&nbsp; $r = s + n$.
+
$\text{Example 2:}$&nbsp; The formal parameter&nbsp; $\rho$&nbsp; (abscissa)&nbsp; denotes a realization of the AWGN random variable&nbsp; $r = s + n$.&nbsp; For the following  further holds:
 +
[[File:P ID2024 Dig T 4 3 S3 version2.png|right|frame|Density functions for equal/unequal symbol probabilities|class=fit]]
 
   
 
   
Für das Folgende gelte weiter:
 
 
:$$\boldsymbol{ s }_0 = (2 \cdot \sqrt{E},  \hspace{0.1cm} 0), \hspace{0.2cm} \boldsymbol{ s }_1 = (- 2 \cdot \sqrt{E},  \hspace{0.1cm} 0)$$
 
:$$\boldsymbol{ s }_0 = (2 \cdot \sqrt{E},  \hspace{0.1cm} 0), \hspace{0.2cm} \boldsymbol{ s }_1 = (- 2 \cdot \sqrt{E},  \hspace{0.1cm} 0)$$
 
:$$  \Rightarrow \hspace{0.2cm} d = 2 \cdot \sqrt{E},  \hspace{0.2cm} \sigma_n = \sqrt{E} \hspace{0.05cm}.$$  
 
:$$  \Rightarrow \hspace{0.2cm} d = 2 \cdot \sqrt{E},  \hspace{0.2cm} \sigma_n = \sqrt{E} \hspace{0.05cm}.$$  
  
*Bei gleichwahrscheinlichen Symbolen &nbsp; &rArr; &nbsp; ${\rm Pr}( m_0) = {\rm Pr}( m_1) = 1/2$&nbsp; ergibt sich die optimale Entscheidungsgrenze zu&nbsp; $G_{\rm opt} = 0$&nbsp; (siehe obere Skizze). Damit erhält man für die Fehlerwahrscheinlichkeit:
+
*For equally probable symbols &nbsp; &rArr; &nbsp; ${\rm Pr}( m_0) = {\rm Pr}( m_1) = 1/2$,&nbsp; the optimal decision threshold is&nbsp; $G_{\rm opt} = 0$ &nbsp; &rArr; &nbsp; see upper sketch.&nbsp; This gives us for the error probability:
  
 
:$${\rm Pr}({ \cal E} ) =  {\rm Q}  \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} (2) \approx 2.26\% \hspace{0.05cm}.$$
 
:$${\rm Pr}({ \cal E} ) =  {\rm Q}  \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} (2) \approx 2.26\% \hspace{0.05cm}.$$
  
*Nun betrachten wir mit&nbsp; ${\rm Pr}( m_0) = 3/4\hspace{0.05cm},\hspace{0.1cm}{\rm Pr}( m_1) = 1/4\hspace{0.05cm}$&nbsp; ungleiche Symbolwahrscheinlichkeiten (untere Skizze). Die weiteren Systemgrößen seien gegenüber der oberen Grafik unverändert. In diesem Fall beträgt der optimale (normierte) Verschiebungsfaktor
+
*Now let the probabilities be&nbsp; ${\rm Pr}( m_0) = 3/4\hspace{0.05cm},\hspace{0.1cm}{\rm Pr}( m_1) = 1/4\hspace{0.05cm}$ &nbsp; &rArr; &nbsp; see lower sketch.&nbsp; Let the other system variables be unchanged from the upper graph.&nbsp; In this case the optimal&nbsp; (normalized)&nbsp; shift factor is
  
 
::<math>\gamma =  2 \cdot \frac{  \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_1)}{ {\rm Pr}( m_0)} =  2 \cdot
 
::<math>\gamma =  2 \cdot \frac{  \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_1)}{ {\rm Pr}( m_0)} =  2 \cdot
 
\frac{ E}{16  \cdot E} \cdot {\rm ln} \hspace{0.15cm} \frac{1/4}{3/4 } \approx - 0.14
 
\frac{ E}{16  \cdot E} \cdot {\rm ln} \hspace{0.15cm} \frac{1/4}{3/4 } \approx - 0.14
\hspace{0.05cm},</math>
+
\hspace{0.05cm}.</math>
  
:was einer Verschiebung um&nbsp; $14\%$&nbsp; hin zum unwahrscheinlicheren Symbol&nbsp; $\boldsymbol {s}_1$&nbsp; (also nach links) bedeutet. Dadurch wird die Fehlerwahrscheinlichkeit geringfügig kleiner als bei gleichwahrscheinlichen Symbolen:
+
*This is a&nbsp; $14\%$&nbsp; shift toward the less probable symbol&nbsp; $\boldsymbol {s}_1$&nbsp; (i.e., to the left).&nbsp; This makes the error probability slightly smaller than for equally probable symbols:
  
 
::<math>{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.</math>
 
::<math>{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.</math>
  
Man erkennt aus diesen Zahlenwerten:  
+
One recognizes from these numerical values:
*Durch die Schwellenverschiebung wird nun zwar das Symbol $\boldsymbol&nbsp; {s}_1$&nbsp; stärker verfälscht, das wahrscheinlichere Symbol&nbsp; $\boldsymbol {s}_0$&nbsp; jedoch überproportional weniger.<br>
+
#Due to the threshold shift,&nbsp; the symbol $\boldsymbol&nbsp; {s}_1$&nbsp; is now more distorted,&nbsp; but the more probable symbol&nbsp; $\boldsymbol {s}_0$&nbsp; is distorted disproportionately less.<br>
 +
#However,&nbsp; the result should not lead to misinterpretations.&nbsp; In the asymmetrical case &nbsp; &#8658; &nbsp; ${\rm Pr}( m_0) \ne {\rm Pr}( m_1)$&nbsp; there is a smaller error probability than for&nbsp; ${\rm Pr}( m_0) ={\rm Pr}( m_1) = 0.5$,&nbsp; but then only less information can be transmitted with each symbol.
 +
#With the selected numerical values &nbsp; "$0.81 \ \rm bit/symbol$" &nbsp; instead of &nbsp; "$1\ \rm  bit/symbol$".&nbsp;
 +
#From an information theoretic point of view,&nbsp; ${\rm Pr}( m_0) ={\rm Pr}( m_1)$&nbsp; would be optimal.}}
  
*Das Ergebnis sollte aber nicht zu Fehlinterpretationen führen. Im unsymmetrischen Fall &nbsp;&#8658;&nbsp; ${\rm Pr}( m_0) \ne {\rm Pr}( m_1)$&nbsp; ergibt sich zwar eine kleinere Fehlerwahrscheinlichkeit  als für&nbsp; ${\rm Pr}( m_0) ={\rm Pr}( m_1) = 0.5$, aber mit jedem Symbol kann dann auch nur weniger Information übertragen werden.
 
* Bei den gewählten Zahlenwerten&nbsp; $0.81 \ \rm bit/Symbol$&nbsp; statt &nbsp;$1\ \rm  bit/Symbol$. Aus informationstheoretischer Sicht wäre&nbsp; ${\rm Pr}( m_0) ={\rm Pr}( m_1)$&nbsp; optimal.}}
 
  
 +
{{BlaueBox|TEXT= 
 +
$\text{Conclusion:}$&nbsp;
 +
*In the symmetric case &nbsp; &rArr; &nbsp; ${\rm Pr}( m_0) ={\rm Pr}( m_1)$,&nbsp; the conventional conditional PDF values&nbsp; $p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m } ( \rho \hspace{0.05cm}\vert \hspace{0.05cm}m_i )$&nbsp; can be used for decision.
 +
*In the asymmetric case &nbsp; &rArr; &nbsp; ${\rm Pr}( m_0) \ne {\rm Pr}( m_1)$,&nbsp; these functions must be weighted beforehand: &nbsp; ${\rm Pr}(m_i) \cdot p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m_i } ( \rho \hspace{0.05cm}\vert \hspace{0.05cm}m_i )$.
  
{{BlaueBox|TEXT= 
 
$\text{Fazit:}$&nbsp;
 
*Im symmetrischen Fall &nbsp; &rArr; &nbsp; ${\rm Pr}( m_0) ={\rm Pr}( m_1)$&nbsp; können zur Entscheidungsfindung die herkömmlichen bedingten WDF&ndash;Werte&nbsp; $p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m } ( \rho \hspace{0.05cm}\vert \hspace{0.05cm}m_i )$&nbsp; herangezogen werden.
 
*Im unsymmetrischen Fall &nbsp; &rArr; &nbsp;  ${\rm Pr}( m_0) \ne {\rm Pr}( m_1)$&nbsp;  müssen diese Funktionen vorher gewichtet werden: &nbsp; ${\rm Pr}(m_i) \cdot p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m_i } ( \rho \hspace{0.05cm}\vert \hspace{0.05cm}m_i )$.
 
  
Im Folgenden wird dieser Sachverhalt berücksichtigt.}}
+
In the following, we consider this issue.}}
  
== Entscheidungsregionen im nichtbinären Fall ==
+
== Decision regions in the non-binary case ==
 
<br>
 
<br>
Allgemein partitionieren die Entscheidungsregionen&nbsp; $I_i$&nbsp; den &nbsp;$N$&ndash;dimensionalen reellen Raum in&nbsp; $M$&nbsp; zueinander disjunkte Gebiete. $I_i$&nbsp; ist dabei definiert als die Menge aller Punkte, die zum Schätzwert&nbsp; $m_i$&nbsp; führen:
+
In general,&nbsp; the decision regions&nbsp; $I_i$&nbsp; partition the &nbsp;$N$&ndash;dimensional real space into&nbsp; $M$&nbsp; mutually disjoint regions.&nbsp;  
  
::<math>\boldsymbol{ \rho } \in I_i \hspace{0.2cm} \Longleftrightarrow \hspace{0.2cm} \hat{m} = m_i, \hspace{0.3cm}{\rm wobei}\hspace{0.3cm}I_i = \left \{ \boldsymbol{ \rho } \in { \cal R}^N \hspace{0.05cm} | \hspace{0.05cm}
+
*Here,&nbsp; the decision region&nbsp; $I_i$&nbsp; with &nbsp;$i = 0$, ... , $M-1$&nbsp; is defined as the set of all points leading to the estimate&nbsp; $m_i$:&nbsp;
 +
::<math>\boldsymbol{ \rho } \in I_i \hspace{0.2cm} \Leftrightarrow \hspace{0.2cm} \hat{m} = m_i, \hspace{0.3cm}{\rm where}\hspace{0.3cm}I_i = \left \{ \boldsymbol{ \rho } \in { \cal R}^N \hspace{0.05cm} | \hspace{0.05cm}
 
{\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm} | \hspace{0.05cm} m_i ) >  
 
{\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm} | \hspace{0.05cm} m_i ) >  
 
{\rm Pr}( m_k) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm} | \hspace{0.05cm}m_k )\hspace{0.15cm} \forall k \ne i   
 
{\rm Pr}( m_k) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm} | \hspace{0.05cm}m_k )\hspace{0.15cm} \forall k \ne i   
 
\right \} \hspace{0.05cm}.</math>
 
\right \} \hspace{0.05cm}.</math>
  
*Die Form der Entscheidungsregionen&nbsp; $I_i$&nbsp; mit &nbsp;$i = 0$, ... , $M-1$&nbsp; im &nbsp;$N$&ndash;dimensionalen Raum hängen von den bedingten Wahrscheinlichkeitsdichtefunktionen&nbsp; $p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m }$&nbsp; ab, also vom betrachteten Kanal.  
+
*The shape of the decision regions&nbsp; $I_i$&nbsp; in the &nbsp;$N$&ndash;dimensional space depend on the conditional probability density functions&nbsp; $p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m }$,&nbsp; i.e. on the considered channel.
*In vielen Fällen &ndash; so auch beim AWGN&ndash;Kanal &ndash; sind die Entscheidungsgrenzen zwischen je zwei Signalpunkten Gerade, was die weiteren Betrachtungen vereinfacht.<br>
+
 
 +
*In many cases &ndash; including the AWGN channel &ndash; the decision boundaries between every two signal points are straight lines,&nbsp; which simplifies further considerations.<br>
  
  
[[File:P ID2025 Dig T 4 3 S4 version2.png|right|frame|AWGN&ndash;Entscheidungsregionen für <br>$N = 2$, $M = 3$]]
 
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 3:}$&nbsp; Die Grafik zeigt die Entscheidungsregionen&nbsp; $I_0$,&nbsp; $I_1$&nbsp; und&nbsp; $I_2$&nbsp; für ein Übertragungssystem mit den Parametern&nbsp; $N = 2$&nbsp; und&nbsp; $M = 3$. Die normierten Sendevektoren sind dabei
+
$\text{Example 3:}$&nbsp; The graph shows the decision regions&nbsp; $I_0$,&nbsp; $I_1$&nbsp; and&nbsp; $I_2$&nbsp; for a transmission system with the parameters&nbsp; $N = 2$&nbsp; and&nbsp; $M = 3$.  
::<math>\boldsymbol{ s }_0 = (2,\hspace{0.05cm} 2), \hspace{0.2cm} \hspace{0.01cm}
+
[[File:P ID2025 Dig T 4 3 S4 version2.png|right|frame|AWGN decision regions&nbsp; <br>$(N = 2$,&nbsp; $M = 3)$]]
   \boldsymbol{ s }_1 = (1,\hspace{0.05cm} 3), \hspace{0.01cm} \hspace{0.2cm}
+
The normalized transmission vectors here are
   \boldsymbol{ s }_2 = (1,\hspace{0.05cm} -1)
+
 
  \hspace{0.05cm}.</math>
+
:$$\boldsymbol{ s }_0 = (2,\hspace{0.05cm} 2),$$
 +
:$$   \boldsymbol{ s }_1 = (1,\hspace{0.05cm} 3),$$
 +
:$$   \boldsymbol{ s }_2 = (1,\hspace{0.05cm} -1) \hspace{0.05cm}.$$
  
Es sind nun zwei Fälle zu unterscheiden:
+
Now two cases have to be distinguished:
*Bei gleichwahrscheinlichen Symbolen &nbsp; &rArr; &nbsp;  ${\rm Pr}( m_0) =  {\rm Pr}( m_1) ={\rm Pr}( m_2) = 1/3 $ verlaufen die Grenzen zwischen jeweils zwei Regionen stets geradlinig, mittig und rechtwinklig zu den Verbindungsgeraden.<br>
+
*For equally probable symbols &nbsp; &rArr; &nbsp;  ${\rm Pr}( m_0) =  {\rm Pr}( m_1) ={\rm Pr}( m_2) = 1/3 $,&nbsp; the boundaries between two regions are always straight,&nbsp; centered and perpendicular to the connecting lines.<br>
  
  
*Bei ungleichen Symbolwahrscheinlichkeiten sind dagegen die Entscheidungsgrenzen jeweils in Richtung des unwahrscheinlicheren Symbols (parallel) zu verschieben &ndash; umso weiter, je größer die AWGN&ndash;Streuung&nbsp; $\sigma_n$ ist.}}
+
*In the case of unequal symbol probabilities,&nbsp; the decision boundaries are to be shifted&nbsp; $($parallel$)$&nbsp; in the direction of the more improbable symbol in each case&nbsp; &ndash;&nbsp; the further the greater the AWGN standard deviation&nbsp; $\sigma_n$.}}
  
  
  
== Fehlerwahrscheinlichkeitsberechnung im nichtbinären Fall ==
+
== Error probability calculation in the non-binary case ==
 
<br>
 
<br>
Nachdem die Entscheidungsregionen&nbsp; $I_i$&nbsp; festliegen, kann man die Symbolfehlerwahrscheinlichkeit des Gesamtsystems berechnen. Wir benutzen folgende Bezeichnungen, wobei wir aufgrund der Einschränkungen durch unseren Zeichensatz im Fließtext manchmal andere Namen als in Gleichungen verwenden müssen:
+
After the decision regions&nbsp; $I_i$&nbsp; are fixed,&nbsp; we can compute the symbol error probability of the overall system.&nbsp; We use the following names,&nbsp; although we sometimes have to use different names in continuous text than in equations because of the limitations imposed by our character set:
*Symbolfehlerwahrscheinlichkeit: &nbsp; ${\rm Pr}({ \cal E} ) = {\rm Pr(Symbolfehler)} \hspace{0.05cm},$
+
#Symbol error probability: &nbsp; ${\rm Pr}({ \cal E} ) = {\rm Pr(symbol\hspace{0.15cm} error)} \hspace{0.05cm},$
 
+
#Probability of correct decision: &nbsp; ${\rm Pr}({ \cal C} ) = 1 - {\rm Pr}({ \cal E} ) = {\rm Pr(correct \hspace{0.15cm} decision)} \hspace{0.05cm},$
*Wahrscheinlichkeit für eine korrekte Entscheidung: &nbsp; ${\rm Pr}({ \cal C} ) = 1 - {\rm Pr}({ \cal E} ) = {\rm Pr(korrekte \hspace{0.15cm} Entscheidung)} \hspace{0.05cm},$
+
#Conditional probability of a correct decision under the condition&nbsp; $m = m_i$: &nbsp; &nbsp; ${\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = 1 - {\rm Pr}({ \cal E} \hspace{0.05cm}|\hspace{0.05cm} m_i)  \hspace{0.05cm}.$
  
*Bedingte Wahrscheinlichkeit einer korrekten Entscheidung unter der Bedingung &nbsp; $m = m_i$: &nbsp; &nbsp; ${\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = 1 - {\rm Pr}({ \cal E} \hspace{0.05cm}|\hspace{0.05cm} m_i)  \hspace{0.05cm}.$
+
*With these definitions,&nbsp; the probability of a correct decision is:
  
 +
:$${\rm Pr}({ \cal C} ) \hspace{-0.1cm}  =  \hspace{-0.1cm}  \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm} m_i ) =  \sum_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot
 +
\int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho }    \hspace{0.05cm}.$$
  
Mit diesen Definitionen gilt für die Wahrscheinlichkeit einer korrekten Entscheidung:
+
*For the AWGN channel,&nbsp; this is according to the section&nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#N-dimensional_Gaussian_noise| "N&ndash;dimensional Gaussian noise"]]:
 
 
::<math>{\rm Pr}({ \cal C} ) \hspace{-0.1cm}  =  \hspace{-0.1cm}  \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm} m_i ) =  \sum_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot
 
\int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho } 
 
  \hspace{0.05cm}.</math>
 
 
 
Für den AWGN&ndash;Kanal gilt dabei entsprechend dem Abschnitt&nbsp; [[Digital_Signal_Transmission/Struktur_des_optimalen_Empfängers#N.E2.80.93dimensionales_Gau.C3.9Fsches_Rauschen| $N$&ndash;dimensionales Gaußsches Rauschen]]:
 
  
 
::<math>{\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = 1 - {\rm Pr}({ \cal E} \hspace{0.05cm}|\hspace{0.05cm} m_i) = \frac{1}{(\sqrt{2\pi} \cdot \sigma_n)^N} \cdot   
 
::<math>{\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = 1 - {\rm Pr}({ \cal E} \hspace{0.05cm}|\hspace{0.05cm} m_i) = \frac{1}{(\sqrt{2\pi} \cdot \sigma_n)^N} \cdot   
 
  \int_{I_i} {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot || \boldsymbol{ \rho } - \boldsymbol{ s }_i ||^2 \right ] \,{\rm d} \boldsymbol {\rho }\hspace{0.05cm}.</math>
 
  \int_{I_i} {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot || \boldsymbol{ \rho } - \boldsymbol{ s }_i ||^2 \right ] \,{\rm d} \boldsymbol {\rho }\hspace{0.05cm}.</math>
  
*Dieses Integral muss im allgemeinen Fall numerisch berechnet werden.  
+
:#This integral must be calculated numerically in the general case.
*Nur bei einigen wenigen, einfach beschreibbaren Entscheidungsregionen&nbsp; $\{I_i\}$&nbsp; ist eine analytische Lösung möglich.<br>
+
:#Only for a few,&nbsp; easily describable decision regions&nbsp; $\{I_i\}$&nbsp; an analytical solution is possible.<br>
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 4:}$&nbsp; Beim AWGN&ndash;Kanal liegt eine 2D&ndash;Gaußglocke um den Sendepunkt&nbsp; $\boldsymbol{ s }_i$, in der linken Grafik erkennbar an den konzentrischen Höhenlinien.
+
$\text{Example 4:}$&nbsp; For the AWGN channel,&nbsp; there is a two-dimensional Gaussian bell around the transmission point&nbsp; $\boldsymbol{ s }_i$,&nbsp; recognizable in the left graphic by the concentric contour lines.
*Etwas willkürlich ist zudem die Entscheidungsgerade&nbsp; $G$&nbsp; eingezeichnet.  
+
[[File:P ID2026 Dig T 4 3 S5b version1.png|right|frame|To calculate the error probability for AWGN|class=fit]]
*Rechts dargestellt ist in einem anderen Koordinatensystem (verschoben und gedreht) allein die WDF der Rauschkomponente.
 
  
 +
#In addition,&nbsp; the decision line&nbsp; $G$&nbsp; is drawn somewhat arbitrarily.
 +
#Shown alone on the right in a different coordinate system&nbsp; (shifted and rotated)&nbsp; is the PDF of the noise component.
  
[[File:P ID2026 Dig T 4 3 S5b version1.png|center|frame|Zur Berechnung der Fehlerwahrscheinlichkeit bei AWGN|class=fit]]
 
  
Die Grafik kann wie folgt interpretiert werden:
+
The graph can be interpreted as follows:
*Die Wahrscheinlichkeit, dass der Empfangsvektor nicht in das "Sollgebiet"&nbsp; $I_i$&nbsp; fällt, sondern in das rot hinterlegte Gebiet&nbsp; $I_k$, ist&nbsp; $ {\rm Q} (A/\sigma_n)$.
+
*The probability that the received vector does not fall into the blue&nbsp; "target area"&nbsp; $I_i$,&nbsp; but into the red highlighted area&nbsp; $I_k$, is&nbsp; $ {\rm Q} (A/\sigma_n)$;&nbsp; ${\rm Q}(x)$&nbsp; is the Gaussian error function.
*$A$&nbsp; bezeichnet den Abstand zwischen&nbsp; $\boldsymbol{ s }_i$&nbsp; und&nbsp; $G$.
 
* $\sigma_n$&nbsp; gibt den Effektivwert (Wurzel aus der Varianz) des AWGN&ndash;Rauschens an und&nbsp; ${\rm Q}(x)$&nbsp; ist die Gaußsche Fehlerfunktion.<br>
 
  
*Entsprechend ist die Wahrscheinlichkeit für das Ereignis&nbsp; $r \in I_i$&nbsp; gleich dem Komplementärwert
+
*$A$&nbsp; denotes the distance between&nbsp; $\boldsymbol{ s }_i$&nbsp; and&nbsp; $G$.&nbsp;  $\sigma_n$&nbsp; indicates the&nbsp; "rms value"&nbsp; (root of the variance)&nbsp; of the AWGN noise. <br>
 +
 
 +
*Correspondingly,&nbsp; the probability for the event&nbsp; $r \in I_i$&nbsp; is equal to the complementary value
  
 
::<math>{\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_i ) = {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm} \vert \hspace{0.05cm} m_i ) =
 
::<math>{\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_i ) = {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm} \vert \hspace{0.05cm} m_i ) =
 
  1 - {\rm Q} (A/\sigma_n)\hspace{0.05cm}.</math>}}<br>
 
  1 - {\rm Q} (A/\sigma_n)\hspace{0.05cm}.</math>}}<br>
  
Wir betrachten nun die oben angegebenen Gleichungen
+
We now consider the equations given above,
  
::<math>{\rm Pr}({ \cal C} ) =  \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) \hspace{0.3cm}{\rm mit}  
+
::<math>{\rm Pr}({ \cal C} ) =  \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) \hspace{0.3cm}{\rm with}  
 
   \hspace{0.3cm} {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) =  
 
   \hspace{0.3cm} {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) =  
 
   \int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho }  
 
   \int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho }  
   \hspace{0.05cm}</math>
+
   \hspace{0.05cm},</math>
  
etwas genauer, wobei wir wieder von zwei Basisfunktionen&nbsp; $(N = 2)$&nbsp; und den drei Signalraumpunkten&nbsp;  $\boldsymbol{ s }_0$,&nbsp; $\boldsymbol{ s }_1$&nbsp; und&nbsp; $\boldsymbol{ s }_2$ $(M = 3)$&nbsp;  ausgehen.
+
in a little more detail,&nbsp; where we again assume two basis functions&nbsp; $(N = 2)$&nbsp; and three signal space points&nbsp;  $(M = 3)$&nbsp; at&nbsp; $\boldsymbol{ s }_0$,&nbsp; $\boldsymbol{ s }_1$,&nbsp; $\boldsymbol{ s }_2$.
  
[[File:P ID2028 Dig T 4 3 S5 version1.png|right|frame|Fehlerwahrscheinlichkeitsberechnung beim AWGN-Kanal und &nbsp;$M = 3$]]  
+
[[File:P ID2028 Dig T 4 3 S5 version1.png|right|frame|Error probability calculation for AWGN,&nbsp; $M = 3$]]  
*Die Entscheidungsregionen&nbsp; $I_0$,&nbsp; $I_1$&nbsp; und &nbsp;$I_2$&nbsp; sind&nbsp; [[Digital_Signal_Transmission/Approximation_der_Fehlerwahrscheinlichkeit#Optimale_Schwelle_bei_nicht_gleichwahrscheinlichen_Symbolen|bestmöglich]]&nbsp; gewählt.
+
#The decision regions&nbsp; $I_0$,&nbsp; $I_1$&nbsp; and &nbsp;$I_2$&nbsp; are chosen&nbsp; [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Optimal_threshold_for_non-equally_probable_symbols|"best as possible"]].&nbsp;  
*Das AWGN&ndash;Rauschen ist in der Skizze durch jeweils drei kreisförmige Höhenlinien angedeutet.
+
#The AWGN noise is indicated in the sketch by three circular contour lines each.
  
  
Man erkennt aus dieser Darstellung:
+
One can see from this plot:
*Unter der Voraussetzung, dass&nbsp; $m = m_i \ \Leftrightarrow \ \boldsymbol{ s } = \boldsymbol{ s }_i$&nbsp; gesendet wurde, wird nur dann eine richtige Entscheidung getroffen, wenn der Empfangswert&nbsp; $\boldsymbol{ r }$&nbsp; in der Region&nbsp; $I_i$&nbsp; liegt.<br>
+
*Assuming that&nbsp; $m = m_i \ \Leftrightarrow \ \boldsymbol{ s } = \boldsymbol{ s }_i$&nbsp; was sent,&nbsp; a correct decision is made only if the received value&nbsp; $\boldsymbol{ r } \in I_i$.&nbsp; <br>
  
*Die Wahrscheinlichkeit&nbsp;  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; ist für &nbsp;$i = 2$&nbsp; (weitaus) am größten &nbsp; &#8658; &nbsp; richtige Entscheidung. ${\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; ist deutlich kleiner. Nahezu vernachlässigbar ist&nbsp; ${\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm}m_2)$.
+
*The conditional probability&nbsp;  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; is&nbsp; (by far)&nbsp; largest  for &nbsp;$i = 2$ &nbsp; &#8658; &nbsp; correct decision.&nbsp;  
  
 +
*${\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm}m_2)$&nbsp; is much smaller.&nbsp; Almost negligible is&nbsp;  ${\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm}m_2)$.
  
*Die Verfälschungswahrscheinlichkeiten für&nbsp; $m = m_0$&nbsp; bzw.&nbsp; $m = m_1$&nbsp; lauten somit:
+
*Thus,&nbsp; the falsification probabilities for&nbsp; $m = m_0$&nbsp; and&nbsp; $m = m_1$&nbsp; are:
  
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_0 )={\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm} m_0 ) +  {\rm Pr}(\boldsymbol{ r } \in I_2\hspace{0.05cm}|\hspace{0.05cm} m_0 ),</math>
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_0 )={\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm} m_0 ) +  {\rm Pr}(\boldsymbol{ r } \in I_2\hspace{0.05cm}|\hspace{0.05cm} m_0 ),</math>
Line 286: Line 289:
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
*Die größte Verfälschungswahrscheinlichkeit ergibt sich für&nbsp; $m = m_0$. Wegen
+
*The largest falsification probability is obtained for&nbsp; $m = m_0$.&nbsp; Because of
  
 
::<math>{\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm} m_0 ) \approx {\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm} m_1 )
 
::<math>{\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm} m_0 ) \approx {\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm} m_1 )
Line 293: Line 296:
 
  \hspace{0.05cm}</math>
 
  \hspace{0.05cm}</math>
  
:gelten folgende Relationen: &nbsp; ${\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_0 ) > {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_1 ) >{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_2 )\hspace{0.05cm}. $
+
:the following relations hold: &nbsp;  
 +
:$${\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_0 ) > {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_1 ) >{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_2 )\hspace{0.05cm}. $$
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Fazit:}$&nbsp; Diese Ergebnisse können wie folgt zusammengefasst werden:
+
$\text{Conclusion:}$&nbsp; These results can be summarized as follows:
*Zur Berechnung der (mittleren) Fehlerwahrscheinlichkeit muss auch bei gleichwahrscheinlichen Symbolen allgemein über alle&nbsp; $M$&nbsp; Terme gemittelt werden.
+
#To calculate the&nbsp; (average)&nbsp; error probability,&nbsp; it is necessary to average over all&nbsp; $M$&nbsp; terms in general,&nbsp; even in the case of equally probable symbols.
*Im Fall gleichwahrscheinlicher Symbole kann&nbsp; ${\rm Pr}(m_i) = 1/M$&nbsp; vor die Summation gezogen werden, was allerdings den Rechengang nicht sonderlich vereinfacht.  
+
#In the case of equally probable symbols,&nbsp; ${\rm Pr}(m_i) = 1/M$&nbsp; can be drawn in front of the summation,&nbsp; but this does not simplify the calculation very much.
*Nur bei symmetrischer Anordnung kann auf die Mittelung verzichtet werden.<br>}}
+
#Only in the case of symmetrical arrangement the averaging can be omitted.<br>}}
  
== Union Bound - Obere Schranke für die Fehlerwahrscheinlichkeit==
+
== Union Bound - Upper bound for the error probability==
 
<br>
 
<br>
Bei beliebigen Werten von&nbsp; $M$&nbsp; gilt für die Verfälschungswahrscheinlichkeit unter der Voraussetzung, dass die Nachricht&nbsp; $m_i$&nbsp; $($bzw. das Signal &nbsp;$\boldsymbol{s}_i)$&nbsp; gesendet wurde:
+
For arbitrary values of&nbsp; $M$,&nbsp; the following applies to the falsification probability under the condition that the message&nbsp; $m_i$&nbsp; $($or the signal &nbsp;$\boldsymbol{s}_i)$&nbsp; has been sent:
  
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = {\rm Pr} \left [ \bigcup_{k \ne i} { \cal E}_{ik}\right ]   
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = {\rm Pr} \left [ \bigcup_{k \ne i} { \cal E}_{ik}\right ]   
  \hspace{0.05cm},\hspace{0.5cm}{ \cal E}_{ik}\hspace{-0.1cm}: \boldsymbol{ r }{\rm \hspace{0.15cm}liegt \hspace{0.15cm}n\ddot{a}her \hspace{0.15cm}bei \hspace{0.15cm}}\boldsymbol{ s }_k {\rm \hspace{0.15cm}als \hspace{0.15cm}beim \hspace{0.15cm}Sollwert \hspace{0.15cm}}\boldsymbol{ s }_i
+
  \hspace{0.05cm},\hspace{0.5cm}{ \cal E}_{ik}\hspace{-0.1cm}: \boldsymbol{ r }{\rm \hspace{0.15cm}is \hspace{0.15cm}closer \hspace{0.15cm}to \hspace{0.15cm}}\boldsymbol{ s }_k {\rm \hspace{0.15cm}than \hspace{0.15cm}to \hspace{0.15cm}the \hspace{0.15cm}nominal \hspace{0.15cm}value \hspace{0.15cm}}\boldsymbol{ s }_i
 
  \hspace{0.05cm}. </math>
 
  \hspace{0.05cm}. </math>
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Definition:}$&nbsp; Für diesen Ausdruck lässt sich mit einer Booleschen Ungleichung &ndash; der so genannten &nbsp;'''Union Bound''' &ndash; eine obere Schranke angeben:
+
$\text{Definition:}$&nbsp; An upper bound can be specified for this expression with a Boolean inequality &ndash; the so-called &nbsp;'''Union Bound''':
  
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_i ) \le \sum\limits_{k = 0, \hspace{0.1cm}k \ne i}^{M-1}  
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_i ) \le \sum\limits_{k = 0, \hspace{0.1cm}k \ne i}^{M-1}  
 
  {\rm Pr}({ \cal E}_{ik}) =  \sum\limits_{k = 0, \hspace{0.1cm}k \ne i}^{M-1}{\rm Q} \big [ d_{ik}/(2{\sigma_n}) \big ]\hspace{0.05cm}. </math>
 
  {\rm Pr}({ \cal E}_{ik}) =  \sum\limits_{k = 0, \hspace{0.1cm}k \ne i}^{M-1}{\rm Q} \big [ d_{ik}/(2{\sigma_n}) \big ]\hspace{0.05cm}. </math>
  
''Anmerkungen'':
+
<u>Remarks:</u>
*$d_{ik} = \vert \hspace{-0.05cm} \vert \boldsymbol{s}_i - \boldsymbol{s}_k \vert \hspace{-0.05cm} \vert$&nbsp; ist der Abstand der Signalraumpunkte $\boldsymbol{s}_i$ und $\boldsymbol{s}_k$.  
+
#$d_{ik} = \vert \hspace{-0.05cm} \vert \boldsymbol{s}_i - \boldsymbol{s}_k \vert \hspace{-0.05cm} \vert$&nbsp; is the distance between the signal space points $\boldsymbol{s}_i$ and $\boldsymbol{s}_k$.  
*$\sigma_n$&nbsp; gibt den Effektivwert des AWGN&ndash;Rauschens an.<br>
+
#$\sigma_n$&nbsp; specifies the rms value of the AWGN noise.<br>
*Die Union Bound ist nur bei gleichwahrscheinlichen Symbolen &nbsp; &rArr; &nbsp; ${\rm Pr}(m_i) = 1/M$&nbsp; anwendbar.
+
#The&nbsp; "Union Bound"&nbsp; can only be used for equally probable symbols &nbsp; &rArr; &nbsp; ${\rm Pr}(m_i) = 1/M$.&nbsp;  
*Auch dann muss zur Berechnung der (mittleren) Fehlerwahrscheinlichkeit über alle&nbsp; $m_i$&nbsp; gemittelt werden.}}
+
#But also in this case,&nbsp; an average must be taken over all&nbsp; $m_i$&nbsp;in order to calculate the&nbsp; (average)&nbsp; error probability.}}
  
  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 5:}$&nbsp; Die Grafik verdeutlicht die &nbsp;<b>Union Bound</b>&nbsp; am Beispiel &nbsp;$M = 3$&nbsp; mit gleichwahrscheinlichen Symbolen: &nbsp; ${\rm Pr}(m_0) = {\rm Pr}(m_1) =  {\rm Pr}(m_2) =1/3$.<br>
+
$\text{Example 5:}$&nbsp; The graphic illustrates the &nbsp;<b>Union Bound</b>&nbsp; using the example of &nbsp;$M = 3$&nbsp; with equally probable symbols: &nbsp; ${\rm Pr}(m_0) = {\rm Pr}(m_1) =  {\rm Pr}(m_2) =1/3$.<br>
  
[[File:P ID2041 Dig T 4 3 S6 version1.png|center|frame|Zur Verdeutlichung der „Union Bound”|class=fit]]
+
[[File:P ID2041 Dig T 4 3 S6 version1.png|right|frame|To clarify the "Union Bound" |class=fit]]
Zu diesen Darstellungen ist anzumerken:
+
The following should be noted about these representations:
*Für die Symbolfehlerwahrscheinlichkeit gilt:
+
*The following applies to the symbol error probability:
::<math>{\rm Pr}({ \cal E} ) = 1 - {\rm Pr}({ \cal C} )  \hspace{0.05cm},\hspace{0.2cm}{\rm Pr}({ \cal C} ) = {1}/{3} \cdot
+
:$${\rm Pr}({ \cal E} ) = 1 - {\rm Pr}({ \cal C} )  \hspace{0.05cm},$$
  \big [ {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_1 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_2 ) \big ]\hspace{0.05cm}.</math>
+
:$${\rm Pr}({ \cal C} ) = {1}/{3} \cdot
 +
  \big [ {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_1 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_2 ) \big ]\hspace{0.05cm}.$$
  
*Der erste Term&nbsp; ${\rm Pr}(\boldsymbol{r} \in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$&nbsp; im Klammerausdruck unter der Voraussetzung&nbsp; $m = m_0 \  \Leftrightarrow  \ \boldsymbol{s} =  \boldsymbol{s}_0$&nbsp; ist in der linken Grafik durch die rote Region&nbsp; $I_0$&nbsp; visualisiert.
+
*The first term&nbsp; ${\rm Pr}(\boldsymbol{r} \in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$&nbsp; in the expression in brackets under the assumption&nbsp; $m = m_0 \  \Leftrightarrow  \ \boldsymbol{s} =  \boldsymbol{s}_0$&nbsp; is visualized in the left graphic by the red region&nbsp; $I_0$.&nbsp;  
  
*Die Komplementärregion&nbsp; ${\rm Pr}(\boldsymbol{r} \not\in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$&nbsp; ist links entweder blau oder grün oder blau&ndash;grün schraffiert markiert. Es gilt&nbsp; ${\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) = 1 - {\rm Pr}({ \cal E}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 )$&nbsp; mit
+
*The complementary region&nbsp; ${\rm Pr}(\boldsymbol{r} \not\in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$&nbsp; is marked on the left with either blue or green or blue&ndash;green hatching. It applies&nbsp; ${\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) = 1 - {\rm Pr}({ \cal E}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 )$&nbsp; with
 
:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 )  =
 
:$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 )  =
   {\rm Pr}(\boldsymbol{ r } \in I_1  \hspace{0.05cm}\cup \hspace{0.05cm} \boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Pr}(\boldsymbol{ r } \in I_1  \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) +  
+
   {\rm Pr}(\boldsymbol{ r } \in I_1  \hspace{0.05cm}\cup \hspace{0.05cm} \boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
   {\rm Pr}(\boldsymbol{ r } \in I_2  \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) ={\rm Q} \big [ d_{01}/(2{\sigma_n}) \big ]+
+
:$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 )  \le {\rm Pr}(\boldsymbol{ r } \in I_1  \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) +  
 +
   {\rm Pr}(\boldsymbol{ r } \in I_2  \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
 +
:$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 )  \le {\rm Q} \big [ d_{01}/(2{\sigma_n}) \big ]+
 
   {\rm Q} \big [ d_{02}/(2{\sigma_n}) \big ]
 
   {\rm Q} \big [ d_{02}/(2{\sigma_n}) \big ]
 
   \hspace{0.05cm}.$$
 
   \hspace{0.05cm}.$$
  
*Das "$\le$"&ndash;Zeichen berücksichtigt hier, dass die blau&ndash;grün schraffierte Fläche sowohl zum Gebiet &nbsp;"$\boldsymbol{r} \in I_1$"&nbsp; als auch zum Gebiet &nbsp;"$\boldsymbol{r} \in I_2$"&nbsp; gehört, so dass die Summe einen zu großen Wert liefert. Das heißt: &nbsp; Die Union Bound liefert stets eine obere Schranke.<br>
+
*The&nbsp; "less/equal"&nbsp; sign takes into account that the blue&ndash;green hatched area belongs both to the area &nbsp;"$\boldsymbol{r} \in I_1$"&nbsp; and to the area &nbsp;"$\boldsymbol{r} \in I_2$",&nbsp; so that the sum returns a value that is too large.&nbsp; This means: &nbsp; The Union Bound always provides an upper bound.<br>
  
*Die mittlere Grafik verdeutlicht die Berechnung der Union Bound unter der Voraussetzung, dass&nbsp; $m = m_1 \  \Leftrightarrow  \ \boldsymbol{s} =  \boldsymbol{s}_1$&nbsp; gesendet wurde.  
+
*The middle graph illustrates the calculation under the assumption that&nbsp; $m = m_1 \  \Leftrightarrow  \ \boldsymbol{s} =  \boldsymbol{s}_1$&nbsp; was sent.&nbsp; The figure on the right is based on&nbsp; $m = m_2 \  \Leftrightarrow  \ \boldsymbol{s} =  \boldsymbol{s}_2$.&nbsp; }}<br>
*Dem rechten Bild liegt&nbsp; $m = m_2 \  \Leftrightarrow  \ \boldsymbol{s} =  \boldsymbol{s}_2$&nbsp; zugrunde.}}<br>
 
  
  
  
== Weitere Aufwandsreduzierung bei der Union Bound==
+
== Further effort reduction at Union Bound==
 
<br>
 
<br>
[[File:P ID2032 Dig T 4 3 S6b version1.png|right|frame|Zur Definition der Nachbarmengen &nbsp;$N(i)$]]
+
The estimation according to the&nbsp; "Union Bound"&nbsp; can be further improved by considering only those signal space points that are direct neighbors of the current transmitted vector&nbsp; $\boldsymbol{s}_i$:&nbsp;
Die Abschätzung nach der "Union Bound" lässt sich weiter verbessern, indem man nur solche Signalraumpunkte berücksichtigt, die direkte Nachbarn des aktuellen Sendevektors&nbsp; $\boldsymbol{s}_i$&nbsp; sind:
+
[[File:P ID2032 Dig T 4 3 S6b version1.png|right|frame|Definition of the neighboring sets &nbsp;$N(i)$]]
  
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) =  \sum\limits_{k = 0,  \hspace{0.1cm} k \ne i}^{M-1}{\rm Q}\big [ d_{ik}/(2{\sigma_n}) \big ]
 
::<math>{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) =  \sum\limits_{k = 0,  \hspace{0.1cm} k \ne i}^{M-1}{\rm Q}\big [ d_{ik}/(2{\sigma_n}) \big ]
Line 356: Line 362:
 
  \hspace{0.05cm}. </math>
 
  \hspace{0.05cm}. </math>
  
Dazu definieren wir die Nachbarn von&nbsp; $\boldsymbol{s}_i$ als
+
To do this,&nbsp; we define the&nbsp; "neighbors"&nbsp; of&nbsp; $\boldsymbol{s}_i$ as
  
::<math>N(i) = \left \{ k \in \left \{ i = 0, 1, 2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, M-1  \right \}\hspace{0.05cm}|\hspace{0.05cm} I_i  {\rm \hspace{0.15cm}grenzt \hspace{0.15cm}direkt \hspace{0.15cm}an \hspace{0.15cm}}I_k \right \}  
+
::<math>N(i) = \big \{ k \in \left \{ i = 0, 1, 2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, M-1  \big \}\hspace{0.05cm}|\hspace{0.05cm} I_i  {\rm \hspace{0.15cm}is \hspace{0.15cm}directly \hspace{0.15cm}adjacent \hspace{0.15cm}to \hspace{0.15cm}}I_k \right \}  
 
  \hspace{0.05cm}. </math>
 
  \hspace{0.05cm}. </math>
Die Grafik verdeutlicht diese Definiton am Beispiel&nbsp; $M = 5$. Die Regionen&nbsp; $I_0$&nbsp; und&nbsp; $I_3$&nbsp; haben jeweils nur zwei direkte Nachbarn, während&nbsp; $I_4$&nbsp; an alle anderen Entscheidungsregionen angrenzt.
+
The graphic illustrates this definition using&nbsp; $M = 5$&nbsp; as an example.  
 +
*Regions&nbsp; $I_0$&nbsp; and&nbsp; $I_3$&nbsp; each have only two direct neighbors,  
 +
*while&nbsp; $I_4$&nbsp; borders all other decision regions.
 +
 
 +
 
 +
The introduction of the neighboring sets&nbsp; $N(i)$&nbsp; improves the quality of the Union Bound approximation,&nbsp; which means that the limit is then closer to the actual error probability,&nbsp; i.e. it is shifted down.
  
Durch die Einführung der Nachbarmengen&nbsp; $N(i)$&nbsp; wird die Qualität der Union Bound&ndash;Approximation verbessert, das heißt, die Schranke liegt dann näher an der tatsächlichen Fehlerwahrscheinlichkeit, wird also nach unten verschoben.
+
 
<br clear=all>
+
Another and frequently used limit uses only the minimum distance&nbsp; $d_{\rm min}$&nbsp; between two signal space points.  
Eine weitere und häufig verwendete Schranke benutzt nur den minimalen Abstand&nbsp; $d_{\rm min}$&nbsp; zwischen zwei Signalraumpunkten. Im obigen Beispiel tritt dieser zwischen&nbsp; $\boldsymbol{s}_1$&nbsp; und&nbsp; $\boldsymbol{s}_2$&nbsp; auf. Für gleichwahrscheinliche Symbole &nbsp; &#8658; &nbsp; ${\rm Pr}(m_i) =1/M$&nbsp; gilt dann die folgende Abschätzung:
+
*In the above example,&nbsp; this occurs between&nbsp; $\boldsymbol{s}_1$&nbsp; and&nbsp; $\boldsymbol{s}_2$.&nbsp;  
 +
 
 +
*For equally probable symbols &nbsp; &#8658; &nbsp; ${\rm Pr}(m_i) =1/M$&nbsp; the following estimation then applies:
  
 
::<math>{\rm Pr}({ \cal E} )  \le    \sum\limits_{i = 0 }^{M-1} \left [ {\rm Pr}(m_i) \cdot \sum\limits_{k \ne i }{\rm Q} \big [d_{ik}/(2{\sigma_n})\big ] \right ]
 
::<math>{\rm Pr}({ \cal E} )  \le    \sum\limits_{i = 0 }^{M-1} \left [ {\rm Pr}(m_i) \cdot \sum\limits_{k \ne i }{\rm Q} \big [d_{ik}/(2{\sigma_n})\big ] \right ]
Line 371: Line 384:
 
  \hspace{0.05cm}. </math>
 
  \hspace{0.05cm}. </math>
  
Hierzu ist anzumerken:
+
It should be noted here:
*Diese Schranke ist auch für große&nbsp; $M$&ndash;Werte sehr einfach zu berechnen. Bei vielen Anwendungen ergibt sich jedoch damit  für die Fehlerwahrscheinlichkeit ein viel zu großer Wert.<br>
+
#This limit is also very easy to calculate for large&nbsp; $M$ values.&nbsp; In many applications,&nbsp; however,&nbsp; this results in a much too high value for the error probability.<br>
 
+
#The limit is equal to the actual error probability if all regions are directly adjacent to all others and the distances of all&nbsp; $M$&nbsp; signal points from one another are&nbsp; $d_{\rm min}$.&nbsp; <br>
*Die Schranke ist nur dann gleich der tatsächlichen Fehlerwahrscheinlichkeit, wenn alle Regionen an alle anderen direkt angrenzen und die Distanzen aller&nbsp; $M$&nbsp; Signalpunkte zueinander gleich&nbsp; $d_{\rm min}$&nbsp; sind.<br>
+
#In the special case&nbsp; $M = 2$,&nbsp; these two conditions are often met,&nbsp; so that the&nbsp; "Union Bound"&nbsp; corresponds exactly to the actual error probability.<br>
 
 
*Im Sonderfall&nbsp; $M = 2$&nbsp; sind diese beiden Voraussetzungen häufig  erfüllt, so dass die Schranke exakt mit der tatsächlichen Fehlerwahrscheinlichkeit übereinstimmt.<br>
 
  
== Aufgaben zum Kapitel==
+
== Exercises for the chapter==
 
<br>
 
<br>
[[Aufgaben:4.06_Optimale_Entscheidungsgrenzen|Aufgabe 4.6: Optimale Entscheidungsgrenze]]
+
[[Aufgaben:Exercise_4.06:_Optimal_Decision_Boundaries|Exercise 4.6: Optimal Decision Boundaries]]
  
[[Aufgaben:4.06Z_Signalraumkonstellationen|Aufgabe 4.6Z: Signalraumkonstellationen]]
+
[[Aufgaben:Exercise_4.06Z:_Signal_Space_Constellations|Exercise 4.6Z: Signal Space Constellations]]
  
[[Aufgaben:Aufgabe_4.07:_Nochmals_Entscheidungsgrenzen|Aufgabe 4.7: Nochmals Entscheidungsgrenzen]]
+
[[Aufgaben:Exercise_4.07:_Decision_Boundaries_once_again|Exercise 4.7: Decision Boundaries once again]]
  
[[Aufgaben:4.08_Entscheidungsregionen_bei_drei_Symbolen|Aufgabe 4.8: Entscheidungsregionen bei drei Symbolen]]
+
[[Aufgaben:Exercise_4.08:_Decision_Regions_at_Three_Symbols|Exercise 4.8: Decision Regions at Three Symbols]]
  
[[Aufgaben:4.08Z_Fehlerwahrscheinlichkeit_bei_drei_Symbolen|Aufgabe 4.8Z: Fehlerwahrscheinlichkeit bei drei Symbolen]]
+
[[Aufgaben:Exercise_4.08Z:_Error_Probability_with_Three_Symbols|Exercise 4.8Z: Error Probability with Three Symbols]]
  
[[Aufgaben:Aufgabe_4.09:_Entscheidungsregionen_bei_Laplace|Aufgabe 4.9: Entscheidungsregionen bei Laplace]]
+
[[Aufgaben:Exercise_4.09:_Decision_Regions_at_Laplace|Exercise 4.9: Decision Regions at Laplace]]
  
[[Aufgaben:Aufgabe_4.09Z:_Laplace-verteiltes_Rauschen|Aufgabe 4.9Z: Laplace-verteiltes Rauschen]]
+
[[Aufgaben:Exercise_4.09Z:_Laplace_Distributed_Noise|Exercise 4.9Z: Laplace Distributed Noise]]
  
[[Aufgaben:4.10_Union_Bound|Aufgabe 4.10: Union Bound]]
+
[[Aufgaben:Exercise_4.10:_Union_Bound|Exercise 4.10: Union Bound]]
  
  
 
{{Display}}
 
{{Display}}

Latest revision as of 11:04, 17 November 2022


Optimal decision with binary transmission


We assume here a transmission system which can be characterized as follows:   $\boldsymbol{r} = \boldsymbol{s} + \boldsymbol{n}$.  This system has the following properties:

  • The vector space fully describing the transmission system is spanned by  $N = 2$  mutually orthogonal basis functions   $\varphi_1(t)$   and   $\varphi_2(t)$. 
  • Consequently,  the probability density function of the additive and white Gaussian noise is also to be set two-dimensional,  characterized by the vector  $\boldsymbol{ n} = (n_1,\hspace{0.05cm}n_2)$.
  • There are only two possible transmitted signals  $(M = 2)$,  described by the two vectors  $\boldsymbol{ s_0} = (s_{01},\hspace{0.05cm}s_{02})$  and  $\boldsymbol{ s_1} = (s_{11},\hspace{0.05cm}s_{12})$: 
Decision regions for equal  (left)  and unequal (right)  occurrence probabilities
$$s_0(t)= s_{01} \cdot \varphi_1(t) + s_{02} \cdot \varphi_2(t) \hspace{0.05cm},$$
$$s_1(t) = s_{11} \cdot \varphi_1(t) + s_{12} \cdot \varphi_2(t) \hspace{0.05cm}.$$
  • The two messages  $m_0 \ \Leftrightarrow \ \boldsymbol{ s_0}$  and  $m_1 \ \Leftrightarrow \ \boldsymbol{ s_1}$  are not necessarily equally probable.
  • The task of the decision is to give an estimate for the current received vector  $\boldsymbol{r}$  according to the  "MAP decision rule".  In the present case,  this rule is with  $\boldsymbol{ r } = \boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$:
$$\hat{m} = {\rm arg} \max_i \hspace{0.1cm} \big[ {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm}|\hspace{0.05cm} m_i )\big ] \hspace{0.15cm} \in \hspace{0.15cm}\{ m_i\}.$$
  • In the special case  $N = 2$  and  $M = 2$  considered here,  the decision partitions the two-dimensional space into the two disjoint areas  $I_0$  (highlighted in red)  and  $I_1$  (blue),  as the graphic on the right illustrates.
  • If the received value lies in  $I_0$,   $m_0$  is output as the estimated value,  otherwise  $m_1$.


$\text{Derivation and picture description:}$  For the AWGN channel and  $M = 2$,  the decision rule is thus:

⇒   Always choose message  $m_0$  if the following condition is satisfied:

$${\rm Pr}( m_0) \cdot {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot \vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 \right ] > {\rm Pr}( m_1) \cdot {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 \right ] \hspace{0.05cm}.$$

⇒   The boundary line between the two decision regions  $I_0$  and  $I_1$  is obtained by replacing the  "greater sign"  with the  "equals sign"  in the above equation and transforming the equation slightly:

$$\vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 - 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_0)\big ] = \vert \hspace{-0.05cm} \vert \boldsymbol{ \rho } - \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 - 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm}\big [{\rm Pr}( m_1)\big ]$$
$$\Rightarrow \hspace{0.3cm} \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert^2 + 2 \sigma_n^2 \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_0)}{ {\rm Pr}( m_1)} = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$

From the plot above one can see:

  • The boundary curve between regions  $I_0$  and  $I_1$  is a straight line,  since the equation of determination is linear in the received vector  $\boldsymbol{ \rho } = (\rho_1, \hspace{0.05cm}\rho_2)$. 
  • For equally probable symbols,  the boundary is exactly halfway between  $\boldsymbol{ s }_0$  and  $\boldsymbol{ s }_1$  and rotated by  $90^\circ$  with respect to the line connecting the transmission points:
$$\vert \hspace{-0.05cm} \vert \boldsymbol{ s }_1 \vert \hspace{-0.05cm} \vert ^2 - \vert \hspace{-0.05cm} \vert \boldsymbol{ s }_0 \vert \hspace{-0.05cm} \vert ^2 = 2 \cdot \boldsymbol{ \rho }^{\rm T} \cdot (\boldsymbol{ s }_1 - \boldsymbol{ s }_0)\hspace{0.05cm}.$$
  • For  ${\rm Pr}(m_0) > {\rm Pr}(m_1)$,  the decision boundary is shifted toward the less probable symbol  $\boldsymbol{ s }_1$,  and the more so the larger the AWGN standard deviation  $\sigma_n$. 
  • The green-dashed decision boundary in the right figure as well as the decision regions  $I_0$  (red)  and  $I_1$  (blue)  are valid for the  (normalized)  standard deviation  $\sigma_n = 1$  and the dashed boundary lines for  $\sigma_n = 0$  resp.  $\sigma_n = 2$.

The special case of equally probable binary symbols


We continue to assume a binary system  $(M = 2)$,  but now consider the simple case where this can be described by a single basis function  $(N = 1)$.  The error probability for this has already been calculated in the section  "Definition of the bit error probability"
With the nomenclature and representation form chosen for the fourth main chapter the following constellation results:

Conditional probability density functions for equally probable symbols
  • The received value  $r = s + n$  is now a scalar and is composed of the transmitted signal  $s \in \{s_0, \hspace{0.05cm}s_1\}$  and the noise term  $n$  additively. The abscissa  $\rho$  denotes a realization of  $r$.
  • In addition,  the abscissa is normalized to the reference quantity  $\sqrt{E}$,  whereas here the normalization energy  $E$  has no prominent,  physically interpretable meaning.
  • The noise term  $n$  is Gaussian distributed with mean  $m_n = 0$  and variance  $\sigma_n^2$.  The root of the variance  $(\sigma_n)$  is called the  "rms value"  or the  "standard deviation".
  • The decision boundary  $G$  divides the entire value range of  $r$  into the two subranges  $I_0$  $($in which  $s_0$  lies$)$ and  $I_1$  $($with the signal value  $s_1)$.
  • If  $\rho > G$,  the decision returns the estimated value  $m_0$, otherwise  $m_1$.  It is assumed that the message  $m_i$  is uniquely related to the signal  $s_i$:    $m_i \Leftrightarrow s_i$.


The graph shows the conditional  $($one-dimensional$)$  probability density functions   $p_{\hspace{0.02cm}r\hspace{0.05cm} \vert \hspace{0.05cm}m_0}$   and   $p_{\hspace{0.02cm}r\hspace{0.05cm} \vert \hspace{0.05cm}m_1}$   for the AWGN channel,  assuming equal symbol probabilities:   ${\rm Pr}(m_0) = {\rm Pr}(m_1) = 0.5$.  Thus,  the  $($optimal$)$  decision boundary is  $G = 0$.  One can see from this plot:

  1. If  $m = m_0$  and thus  $s = s_0 = 2 \cdot E^{1/2}$,  an erroneous decision occurs only if  $\eta$,  the realization of the noise quantity  $n$,  is smaller than  $-2 \cdot E^{1/2}$.
  2. In this case,  $\rho < 0$, where  $\rho$  denotes a realization of the received value  $r$. 
  3. In contrast,  for  $m = m_1$   ⇒   $s = s_1 = -2 \cdot E^{1/2}$,  an erroneous decision occurs whenever  $\eta$  is greater than  $+2 \cdot E^{1/2}$.  In this case,  $\rho > 0$.


Error probability for symbols with equal probability


Let  ${\rm Pr}(m_0) = {\rm Pr}(m_1) = 0.5$.  For AWGN noise with standard deviation  $\sigma_n$,  as already calculated in the section  "Definition of the bit error probability"  with different nomenclature,  we obtain for the probability of a wrong decision  $(\cal E)$  under the condition that message  $m_0$  was sent:

$${\rm Pr}({ \cal E}\hspace{0.05cm} \vert \hspace{0.05cm} m_0) = \int_{-\infty}^{G = 0} p_{r \hspace{0.05cm}|\hspace{0.05cm}m_0 } ({ \rho } \hspace{0.05cm} \vert \hspace{0.05cm}m_0 ) \,{\rm d} \rho = \int_{-\infty}^{- s_0 } p_{{ n} \hspace{0.05cm}\vert\hspace{0.05cm}m_0 } ({ \eta } \hspace{0.05cm}|\hspace{0.05cm}m_0 ) \,{\rm d} \eta = \int_{-\infty}^{- s_0 } p_{{ n} } ({ \eta } ) \,{\rm d} \eta = \int_{ s_0 }^{\infty} p_{{ n} } ({ \eta } ) \,{\rm d} \eta = {\rm Q} \left ( {s_0 }/{\sigma_n} \right ) \hspace{0.05cm}.$$

In deriving the equation,  it was considered that the AWGN noise  $\eta$  is independent of the signal  $(m_0$  or  $m_1)$  and has a symmetric PDF.  The complementary Gaussian error integral was also used:

$${\rm Q}(x) = \frac{1}{\sqrt{2\pi}} \int_{x}^{\infty} {\rm e}^{-u^2/2} \,{\rm d} u \hspace{0.05cm}.$$

Correspondingly,  for  $m = m_1$   ⇒   $s = s_1 = -2 \cdot E^{1/2}$:

$${\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = \int_{0}^{\infty} p_{{ r} \hspace{0.05cm}\vert\hspace{0.05cm}m_1 } ({ \rho } \hspace{0.05cm}\vert\hspace{0.05cm}m_1 ) \,{\rm d} \rho = \int_{- s_1 }^{\infty} p_{{ n} } (\boldsymbol{ \eta } ) \,{\rm d} \eta = {\rm Q} \left ( {- s_1 }/{\sigma_n} \right ) \hspace{0.05cm}.$$

$\text{Conclusion:}$  With the distance  $d = s_1 - s_0$  of the signal space points, we can summarize the results, still considering  ${\rm Pr}(m_0) + {\rm Pr}(m_1) = 1$: 

$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0) = {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1) = {\rm Q} \big ( {d}/(2{\sigma_n}) \big )$$
$$\Rightarrow \hspace{0.3cm}{\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_0) + {\rm Pr}(m_1) \cdot {\rm Pr}({ \cal E} \hspace{0.05cm}\vert\hspace{0.05cm} m_1)= \big [ {\rm Pr}(m_0) + {\rm Pr}(m_1) \big ] \cdot {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] \hspace{0.05cm}.$$

Notes:

  1. This equation is valid under the condition  $G = 0$  quite generally,  thus also for  ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$.
  2. For  "non-equally probable symbols",  however,  the error probability can be reduced by a different decision threshold.
  3. The equation mentioned here is also valid if the signal space points are not scalars but are described by the vectors  $\boldsymbol{ s}_0$  and  $\boldsymbol{ s}_1$. 
  4. The distance  $d$  results then as the norm of the difference vector:   $d = \vert \hspace{-0.05cm} \vert \hspace{0.05cm} \boldsymbol{ s}_1 - \boldsymbol{ s}_0 \hspace{0.05cm} \vert \hspace{-0.05cm} \vert \hspace{0.05cm}.$


$\text{Example 1:}$  Let's look again at the signal space constellation from the  "first chapter section"  $($lower graphic$)$  with the values

Two signal space constellations
  • $\boldsymbol{ s}_0/E^{1/2} = (3.6, \hspace{0.05cm}0.8)$,
  • $\boldsymbol{ s}_1/E^{1/2} = (0.4, \hspace{0.05cm}3.2)$.


Here the distance of the signal space points is

$$d = \vert \hspace{-0.05cm} \vert s_1 - s_0 \vert \hspace{-0.05cm} \vert = \sqrt{E \cdot (0.4 - 3.6)^2 + E \cdot (3.2 - 0.8)^2} = 4 \cdot \sqrt {E}\hspace{0.05cm}.$$

This results in exactly the same value as for the upper constellation with

  • $\boldsymbol{ s}_0/E^{1/2} = (2, \hspace{0.05cm}0)$,
  • $\boldsymbol{ s}_1/E^{1/2} = (-2, \hspace{0.05cm}0)$.


The figures show these two constellations and reveal the following similarities and differences,  assuming the AWGN noise variance  $\sigma_n^2 = N_0/2$  in each case.  The circles in the graph illustrate the circular symmetry of the two-dimensional AWGN noise.

  • As said before,  both the distance of the signal points from the decision line  $(d/2 = 2 \cdot \sqrt {E})$  and the AWGN characteristic value  $\sigma_n$  are the same in both cases.
  • It follows:   The two arrangements lead to the same error probability if the parameter  $E$  $($a kind of normalization energy$)$  is kept constant:
$${\rm Pr} ({\rm symbol\hspace{0.15cm} error}) = {\rm Pr}({ \cal E} ) = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ]\hspace{0.05cm}.$$
  • The  "mean energy per symbol"  $(E_{\rm S})$  for the upper constellation is given by
$$E_{\rm S} = 1/2 \cdot \vert \hspace{-0.05cm} \vert s_0 \vert \hspace{-0.05cm} \vert^2 + 1/2 \cdot \vert \hspace{-0.05cm} \vert s_1 \vert \hspace{-0.05cm} \vert^2 = E/2 \cdot \big[(+2)^2 + (-2)^2\big] = 4 \cdot {E}\hspace{0.05cm}.$$
  • With the lower constellation one receives in the same way:
$$E_{\rm S} = \ \text{...} \ = E/2 \cdot \big[(3.6)^2 + (0.8)^2\big] + E/2 \cdot \big[(0.4)^2 + (3.2)^2 \big] = 12 \cdot {E}\hspace{0.05cm}.$$
  • For a given mean energy per symbol   ⇒   $E_{\rm S}$,  the upper constellation is therefore clearly superior to the lower one:   The same error probability results with one third of the energy per symbol.  This issue will be discussed in detail in  "Exercise 4.6Z"


Optimal threshold for non-equally probable symbols


If  ${\rm Pr}(m_0) \ne {\rm Pr}(m_1)$  holds,  a slightly smaller error probability can be obtained by shifting the decision threshold  $G$.  The following results are derived in detail in the solution to  "Exercise 4.7"

  • For unequal symbol probabilities, the optimal decision threshold  $G_{\rm opt}$  between regions  $I_0$  and  $I_1$  is closer to the less probable symbol.  The normalized optimal shift with respect to the value  $G = 0$  for equally probable symbols is
\[\gamma_{\rm opt} = \frac{G_{\rm opt}}{s_0 } = 2 \cdot \frac{ \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{{\rm Pr}( m_1)}{{\rm Pr}( m_0)} \hspace{0.05cm}.\]
  • The error probability is then
$${\rm Pr}({ \cal E} ) = {\rm Pr}(m_0) \cdot {\rm Q} \big[ {d}/(2{\sigma_n}) \cdot (1 - \gamma_{\rm opt}) \big ] + {\rm Pr}(m_1) \cdot {\rm Q} \big [ {d}/(2{\sigma_n}) \cdot (1 + \gamma_{\rm opt}) \big ]\hspace{0.05cm}.$$

$\text{Example 2:}$  The formal parameter  $\rho$  (abscissa)  denotes a realization of the AWGN random variable  $r = s + n$.  For the following further holds:

Density functions for equal/unequal symbol probabilities
$$\boldsymbol{ s }_0 = (2 \cdot \sqrt{E}, \hspace{0.1cm} 0), \hspace{0.2cm} \boldsymbol{ s }_1 = (- 2 \cdot \sqrt{E}, \hspace{0.1cm} 0)$$
$$ \Rightarrow \hspace{0.2cm} d = 2 \cdot \sqrt{E}, \hspace{0.2cm} \sigma_n = \sqrt{E} \hspace{0.05cm}.$$
  • For equally probable symbols   ⇒   ${\rm Pr}( m_0) = {\rm Pr}( m_1) = 1/2$,  the optimal decision threshold is  $G_{\rm opt} = 0$   ⇒   see upper sketch.  This gives us for the error probability:
$${\rm Pr}({ \cal E} ) = {\rm Q} \big [ {d}/(2{\sigma_n}) \big ] = {\rm Q} (2) \approx 2.26\% \hspace{0.05cm}.$$
  • Now let the probabilities be  ${\rm Pr}( m_0) = 3/4\hspace{0.05cm},\hspace{0.1cm}{\rm Pr}( m_1) = 1/4\hspace{0.05cm}$   ⇒   see lower sketch.  Let the other system variables be unchanged from the upper graph.  In this case the optimal  (normalized)  shift factor is
\[\gamma = 2 \cdot \frac{ \sigma_n^2}{d^2} \cdot {\rm ln} \hspace{0.15cm} \frac{ {\rm Pr}( m_1)}{ {\rm Pr}( m_0)} = 2 \cdot \frac{ E}{16 \cdot E} \cdot {\rm ln} \hspace{0.15cm} \frac{1/4}{3/4 } \approx - 0.14 \hspace{0.05cm}.\]
  • This is a  $14\%$  shift toward the less probable symbol  $\boldsymbol {s}_1$  (i.e., to the left).  This makes the error probability slightly smaller than for equally probable symbols:
\[{\rm Pr}({ \cal E} )= 0.75 \cdot {\rm Q} \left ( 2 \cdot 1.14 \right ) + 0.25 \cdot {\rm Q} \left ( 2 \cdot 0.86 \right ) = 0.75 \cdot 0.0113 + 0.25 \cdot 0.0427 \approx 1.92\% \hspace{0.05cm}.\]

One recognizes from these numerical values:

  1. Due to the threshold shift,  the symbol $\boldsymbol  {s}_1$  is now more distorted,  but the more probable symbol  $\boldsymbol {s}_0$  is distorted disproportionately less.
  2. However,  the result should not lead to misinterpretations.  In the asymmetrical case   ⇒   ${\rm Pr}( m_0) \ne {\rm Pr}( m_1)$  there is a smaller error probability than for  ${\rm Pr}( m_0) ={\rm Pr}( m_1) = 0.5$,  but then only less information can be transmitted with each symbol.
  3. With the selected numerical values   "$0.81 \ \rm bit/symbol$"   instead of   "$1\ \rm bit/symbol$". 
  4. From an information theoretic point of view,  ${\rm Pr}( m_0) ={\rm Pr}( m_1)$  would be optimal.


$\text{Conclusion:}$ 

  • In the symmetric case   ⇒   ${\rm Pr}( m_0) ={\rm Pr}( m_1)$,  the conventional conditional PDF values  $p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m } ( \rho \hspace{0.05cm}\vert \hspace{0.05cm}m_i )$  can be used for decision.
  • In the asymmetric case   ⇒   ${\rm Pr}( m_0) \ne {\rm Pr}( m_1)$,  these functions must be weighted beforehand:   ${\rm Pr}(m_i) \cdot p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m_i } ( \rho \hspace{0.05cm}\vert \hspace{0.05cm}m_i )$.


In the following, we consider this issue.

Decision regions in the non-binary case


In general,  the decision regions  $I_i$  partition the  $N$–dimensional real space into  $M$  mutually disjoint regions. 

  • Here,  the decision region  $I_i$  with  $i = 0$, ... , $M-1$  is defined as the set of all points leading to the estimate  $m_i$: 
\[\boldsymbol{ \rho } \in I_i \hspace{0.2cm} \Leftrightarrow \hspace{0.2cm} \hat{m} = m_i, \hspace{0.3cm}{\rm where}\hspace{0.3cm}I_i = \left \{ \boldsymbol{ \rho } \in { \cal R}^N \hspace{0.05cm} | \hspace{0.05cm} {\rm Pr}( m_i) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm} | \hspace{0.05cm} m_i ) > {\rm Pr}( m_k) \cdot p_{\boldsymbol{ r} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol{ \rho } \hspace{0.05cm} | \hspace{0.05cm}m_k )\hspace{0.15cm} \forall k \ne i \right \} \hspace{0.05cm}.\]
  • The shape of the decision regions  $I_i$  in the  $N$–dimensional space depend on the conditional probability density functions  $p_{r \hspace{0.05cm}\vert \hspace{0.05cm}m }$,  i.e. on the considered channel.
  • In many cases – including the AWGN channel – the decision boundaries between every two signal points are straight lines,  which simplifies further considerations.


$\text{Example 3:}$  The graph shows the decision regions  $I_0$,  $I_1$  and  $I_2$  for a transmission system with the parameters  $N = 2$  and  $M = 3$.

AWGN decision regions 
$(N = 2$,  $M = 3)$

The normalized transmission vectors here are

$$\boldsymbol{ s }_0 = (2,\hspace{0.05cm} 2),$$
$$ \boldsymbol{ s }_1 = (1,\hspace{0.05cm} 3),$$
$$ \boldsymbol{ s }_2 = (1,\hspace{0.05cm} -1) \hspace{0.05cm}.$$

Now two cases have to be distinguished:

  • For equally probable symbols   ⇒   ${\rm Pr}( m_0) = {\rm Pr}( m_1) ={\rm Pr}( m_2) = 1/3 $,  the boundaries between two regions are always straight,  centered and perpendicular to the connecting lines.


  • In the case of unequal symbol probabilities,  the decision boundaries are to be shifted  $($parallel$)$  in the direction of the more improbable symbol in each case  –  the further the greater the AWGN standard deviation  $\sigma_n$.


Error probability calculation in the non-binary case


After the decision regions  $I_i$  are fixed,  we can compute the symbol error probability of the overall system.  We use the following names,  although we sometimes have to use different names in continuous text than in equations because of the limitations imposed by our character set:

  1. Symbol error probability:   ${\rm Pr}({ \cal E} ) = {\rm Pr(symbol\hspace{0.15cm} error)} \hspace{0.05cm},$
  2. Probability of correct decision:   ${\rm Pr}({ \cal C} ) = 1 - {\rm Pr}({ \cal E} ) = {\rm Pr(correct \hspace{0.15cm} decision)} \hspace{0.05cm},$
  3. Conditional probability of a correct decision under the condition  $m = m_i$:     ${\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = 1 - {\rm Pr}({ \cal E} \hspace{0.05cm}|\hspace{0.05cm} m_i) \hspace{0.05cm}.$
  • With these definitions,  the probability of a correct decision is:
$${\rm Pr}({ \cal C} ) \hspace{-0.1cm} = \hspace{-0.1cm} \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \sum_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot \int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho } \hspace{0.05cm}.$$
\[{\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = 1 - {\rm Pr}({ \cal E} \hspace{0.05cm}|\hspace{0.05cm} m_i) = \frac{1}{(\sqrt{2\pi} \cdot \sigma_n)^N} \cdot \int_{I_i} {\rm exp} \left [ - \frac{1}{2 \sigma_n^2} \cdot || \boldsymbol{ \rho } - \boldsymbol{ s }_i ||^2 \right ] \,{\rm d} \boldsymbol {\rho }\hspace{0.05cm}.\]
  1. This integral must be calculated numerically in the general case.
  2. Only for a few,  easily describable decision regions  $\{I_i\}$  an analytical solution is possible.


$\text{Example 4:}$  For the AWGN channel,  there is a two-dimensional Gaussian bell around the transmission point  $\boldsymbol{ s }_i$,  recognizable in the left graphic by the concentric contour lines.

To calculate the error probability for AWGN
  1. In addition,  the decision line  $G$  is drawn somewhat arbitrarily.
  2. Shown alone on the right in a different coordinate system  (shifted and rotated)  is the PDF of the noise component.


The graph can be interpreted as follows:

  • The probability that the received vector does not fall into the blue  "target area"  $I_i$,  but into the red highlighted area  $I_k$, is  $ {\rm Q} (A/\sigma_n)$;  ${\rm Q}(x)$  is the Gaussian error function.
  • $A$  denotes the distance between  $\boldsymbol{ s }_i$  and  $G$.  $\sigma_n$  indicates the  "rms value"  (root of the variance)  of the AWGN noise.
  • Correspondingly,  the probability for the event  $r \in I_i$  is equal to the complementary value
\[{\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_i ) = {\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm} \vert \hspace{0.05cm} m_i ) = 1 - {\rm Q} (A/\sigma_n)\hspace{0.05cm}.\]


We now consider the equations given above,

\[{\rm Pr}({ \cal C} ) = \sum\limits_{i = 0}^{M-1} {\rm Pr}(m_i) \cdot {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) \hspace{0.3cm}{\rm with} \hspace{0.3cm} {\rm Pr}({ \cal C}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \int_{I_i} p_{{ \boldsymbol{ r }} \hspace{0.05cm}|\hspace{0.05cm}m } (\boldsymbol {\rho } \hspace{0.05cm}|\hspace{0.05cm} m_i ) \,{\rm d} \boldsymbol {\rho } \hspace{0.05cm},\]

in a little more detail,  where we again assume two basis functions  $(N = 2)$  and three signal space points  $(M = 3)$  at  $\boldsymbol{ s }_0$,  $\boldsymbol{ s }_1$,  $\boldsymbol{ s }_2$.

Error probability calculation for AWGN,  $M = 3$
  1. The decision regions  $I_0$,  $I_1$  and  $I_2$  are chosen  "best as possible"
  2. The AWGN noise is indicated in the sketch by three circular contour lines each.


One can see from this plot:

  • Assuming that  $m = m_i \ \Leftrightarrow \ \boldsymbol{ s } = \boldsymbol{ s }_i$  was sent,  a correct decision is made only if the received value  $\boldsymbol{ r } \in I_i$. 
  • The conditional probability  ${\rm Pr}(\boldsymbol{ r } \in I_i\hspace{0.05cm}|\hspace{0.05cm}m_2)$  is  (by far)  largest for  $i = 2$   ⇒   correct decision. 
  • ${\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm}m_2)$  is much smaller.  Almost negligible is  ${\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm}m_2)$.
  • Thus,  the falsification probabilities for  $m = m_0$  and  $m = m_1$  are:
\[{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_0 )={\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm} m_0 ) + {\rm Pr}(\boldsymbol{ r } \in I_2\hspace{0.05cm}|\hspace{0.05cm} m_0 ),\]
\[ {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_1 ) ={\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm} m_1 ) + {\rm Pr}(\boldsymbol{ r } \in I_2\hspace{0.05cm}|\hspace{0.05cm} m_1 ) \hspace{0.05cm}.\]
  • The largest falsification probability is obtained for  $m = m_0$.  Because of
\[{\rm Pr}(\boldsymbol{ r } \in I_1\hspace{0.05cm}|\hspace{0.05cm} m_0 ) \approx {\rm Pr}(\boldsymbol{ r } \in I_0\hspace{0.05cm}|\hspace{0.05cm} m_1 ) \hspace{0.05cm}, \]
\[{\rm Pr}(\boldsymbol{ r } \in I_2\hspace{0.05cm}|\hspace{0.05cm} m_0 ) \gg {\rm Pr}(\boldsymbol{ r } \in I_2\hspace{0.05cm}|\hspace{0.05cm} m_1 ) \hspace{0.05cm}\]
the following relations hold:  
$${\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_0 ) > {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_1 ) >{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_2 )\hspace{0.05cm}. $$

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

  1. To calculate the  (average)  error probability,  it is necessary to average over all  $M$  terms in general,  even in the case of equally probable symbols.
  2. In the case of equally probable symbols,  ${\rm Pr}(m_i) = 1/M$  can be drawn in front of the summation,  but this does not simplify the calculation very much.
  3. Only in the case of symmetrical arrangement the averaging can be omitted.

Union Bound - Upper bound for the error probability


For arbitrary values of  $M$,  the following applies to the falsification probability under the condition that the message  $m_i$  $($or the signal  $\boldsymbol{s}_i)$  has been sent:

\[{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = {\rm Pr} \left [ \bigcup_{k \ne i} { \cal E}_{ik}\right ] \hspace{0.05cm},\hspace{0.5cm}{ \cal E}_{ik}\hspace{-0.1cm}: \boldsymbol{ r }{\rm \hspace{0.15cm}is \hspace{0.15cm}closer \hspace{0.15cm}to \hspace{0.15cm}}\boldsymbol{ s }_k {\rm \hspace{0.15cm}than \hspace{0.15cm}to \hspace{0.15cm}the \hspace{0.15cm}nominal \hspace{0.15cm}value \hspace{0.15cm}}\boldsymbol{ s }_i \hspace{0.05cm}. \]

$\text{Definition:}$  An upper bound can be specified for this expression with a Boolean inequality – the so-called  Union Bound:

\[{\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_i ) \le \sum\limits_{k = 0, \hspace{0.1cm}k \ne i}^{M-1} {\rm Pr}({ \cal E}_{ik}) = \sum\limits_{k = 0, \hspace{0.1cm}k \ne i}^{M-1}{\rm Q} \big [ d_{ik}/(2{\sigma_n}) \big ]\hspace{0.05cm}. \]

Remarks:

  1. $d_{ik} = \vert \hspace{-0.05cm} \vert \boldsymbol{s}_i - \boldsymbol{s}_k \vert \hspace{-0.05cm} \vert$  is the distance between the signal space points $\boldsymbol{s}_i$ and $\boldsymbol{s}_k$.
  2. $\sigma_n$  specifies the rms value of the AWGN noise.
  3. The  "Union Bound"  can only be used for equally probable symbols   ⇒   ${\rm Pr}(m_i) = 1/M$. 
  4. But also in this case,  an average must be taken over all  $m_i$ in order to calculate the  (average)  error probability.


$\text{Example 5:}$  The graphic illustrates the  Union Bound  using the example of  $M = 3$  with equally probable symbols:   ${\rm Pr}(m_0) = {\rm Pr}(m_1) = {\rm Pr}(m_2) =1/3$.

To clarify the "Union Bound"

The following should be noted about these representations:

  • The following applies to the symbol error probability:
$${\rm Pr}({ \cal E} ) = 1 - {\rm Pr}({ \cal C} ) \hspace{0.05cm},$$
$${\rm Pr}({ \cal C} ) = {1}/{3} \cdot \big [ {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_1 ) + {\rm Pr}({ \cal C}\hspace{0.05cm}\vert \hspace{0.05cm} m_2 ) \big ]\hspace{0.05cm}.$$
  • The first term  ${\rm Pr}(\boldsymbol{r} \in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$  in the expression in brackets under the assumption  $m = m_0 \ \Leftrightarrow \ \boldsymbol{s} = \boldsymbol{s}_0$  is visualized in the left graphic by the red region  $I_0$. 
  • The complementary region  ${\rm Pr}(\boldsymbol{r} \not\in I_0\hspace{0.05cm}\vert \hspace{0.05cm} m_0)$  is marked on the left with either blue or green or blue–green hatching. It applies  ${\rm Pr}({ \cal C}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) = 1 - {\rm Pr}({ \cal E}\hspace{0.05cm}\vert \hspace{0.05cm} m_0 )$  with
$${\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) = {\rm Pr}(\boldsymbol{ r } \in I_1 \hspace{0.05cm}\cup \hspace{0.05cm} \boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Pr}(\boldsymbol{ r } \in I_1 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) + {\rm Pr}(\boldsymbol{ r } \in I_2 \hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) $$
$$\Rightarrow \hspace{0.3cm} {\rm Pr}({ \cal E}\hspace{0.05cm}\vert\hspace{0.05cm} m_0 ) \le {\rm Q} \big [ d_{01}/(2{\sigma_n}) \big ]+ {\rm Q} \big [ d_{02}/(2{\sigma_n}) \big ] \hspace{0.05cm}.$$
  • The  "less/equal"  sign takes into account that the blue–green hatched area belongs both to the area  "$\boldsymbol{r} \in I_1$"  and to the area  "$\boldsymbol{r} \in I_2$",  so that the sum returns a value that is too large.  This means:   The Union Bound always provides an upper bound.
  • The middle graph illustrates the calculation under the assumption that  $m = m_1 \ \Leftrightarrow \ \boldsymbol{s} = \boldsymbol{s}_1$  was sent.  The figure on the right is based on  $m = m_2 \ \Leftrightarrow \ \boldsymbol{s} = \boldsymbol{s}_2$. 



Further effort reduction at Union Bound


The estimation according to the  "Union Bound"  can be further improved by considering only those signal space points that are direct neighbors of the current transmitted vector  $\boldsymbol{s}_i$: 

Definition of the neighboring sets  $N(i)$
\[{\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \sum\limits_{k = 0, \hspace{0.1cm} k \ne i}^{M-1}{\rm Q}\big [ d_{ik}/(2{\sigma_n}) \big ] \hspace{0.2cm} \Rightarrow \hspace{0.2cm} {\rm Pr}({ \cal E}\hspace{0.05cm}|\hspace{0.05cm} m_i ) = \sum\limits_{k = 0, \hspace{0.1cm} k \hspace{0.05cm}\in \hspace{0.05cm}N(i)}^{M-1}\hspace{-0.4cm}{\rm Q} \big [ d_{ik}/(2{\sigma_n}) \big ] \hspace{0.05cm}. \]

To do this,  we define the  "neighbors"  of  $\boldsymbol{s}_i$ as

\[N(i) = \big \{ k \in \left \{ i = 0, 1, 2, \hspace{0.05cm}\text{...} \hspace{0.05cm}, M-1 \big \}\hspace{0.05cm}|\hspace{0.05cm} I_i {\rm \hspace{0.15cm}is \hspace{0.15cm}directly \hspace{0.15cm}adjacent \hspace{0.15cm}to \hspace{0.15cm}}I_k \right \} \hspace{0.05cm}. \]

The graphic illustrates this definition using  $M = 5$  as an example.

  • Regions  $I_0$  and  $I_3$  each have only two direct neighbors,
  • while  $I_4$  borders all other decision regions.


The introduction of the neighboring sets  $N(i)$  improves the quality of the Union Bound approximation,  which means that the limit is then closer to the actual error probability,  i.e. it is shifted down.


Another and frequently used limit uses only the minimum distance  $d_{\rm min}$  between two signal space points.

  • In the above example,  this occurs between  $\boldsymbol{s}_1$  and  $\boldsymbol{s}_2$. 
  • For equally probable symbols   ⇒   ${\rm Pr}(m_i) =1/M$  the following estimation then applies:
\[{\rm Pr}({ \cal E} ) \le \sum\limits_{i = 0 }^{M-1} \left [ {\rm Pr}(m_i) \cdot \sum\limits_{k \ne i }{\rm Q} \big [d_{ik}/(2{\sigma_n})\big ] \right ] \le \frac{1}{M} \cdot \sum\limits_{i = 0 }^{M-1} \left [ \sum\limits_{k \ne i } {\rm Q} [d_{\rm min}/(2{\sigma_n})] \right ] = \sum\limits_{k \ne i }{\rm Q} \big [d_{\rm min}/(2{\sigma_n})\big ] = (M-1) \cdot {\rm Q} \big [d_{\rm min}/(2{\sigma_n})\big ] \hspace{0.05cm}. \]

It should be noted here:

  1. This limit is also very easy to calculate for large  $M$ values.  In many applications,  however,  this results in a much too high value for the error probability.
  2. The limit is equal to the actual error probability if all regions are directly adjacent to all others and the distances of all  $M$  signal points from one another are  $d_{\rm min}$. 
  3. In the special case  $M = 2$,  these two conditions are often met,  so that the  "Union Bound"  corresponds exactly to the actual error probability.

Exercises for the chapter


Exercise 4.6: Optimal Decision Boundaries

Exercise 4.6Z: Signal Space Constellations

Exercise 4.7: Decision Boundaries once again

Exercise 4.8: Decision Regions at Three Symbols

Exercise 4.8Z: Error Probability with Three Symbols

Exercise 4.9: Decision Regions at Laplace

Exercise 4.9Z: Laplace Distributed Noise

Exercise 4.10: Union Bound