Difference between revisions of "Digital Signal Transmission/Carrier Frequency Systems with Non-Coherent Demodulation"

From LNTwww
 
(14 intermediate revisions by 2 users not shown)
Line 8: Line 8:
 
== Rayleigh and Rice Distribution ==
 
== Rayleigh and Rice Distribution ==
 
<br>
 
<br>
The estimation of the phase angle from the incoming signal, which is required for coherent demodulation, is not possible or only possible to a limited extent in many applications. For example, the movement of a mobile subscriber at high speed leads to very rapid temporal changes in the phase angle &nbsp;$\phi$, which makes its sufficiently accurate determination difficult or even impossible.<br>
+
The estimation of the phase angle from the incoming signal,&nbsp; which is required for coherent demodulation,&nbsp; is not possible or only possible to a limited extent in many applications.&nbsp; For example,&nbsp; the movement of a mobile subscriber at high speed leads to very rapid temporal changes in the phase angle &nbsp;$\phi$,&nbsp; which makes its sufficiently accurate determination difficult or even impossible.<br>
  
This fact leads to the&nbsp; '''non-coherent demodulation processes'''&nbsp; mit dem Vorteil reduzierter Komplexität, allerdings mit erhöhter Verfälschungswahrscheinlichkeit. Bei der Herleitung der Gleichungen stößt man auf zwei Wahrscheinlichkeitsdichtefunktionen, die hier vorneweg angegeben werden:
+
This fact leads to the&nbsp; '''non-coherent demodulation processes'''&nbsp; with the advantage of reduced complexity,&nbsp; but with increased error probability.&nbsp; In the derivation of the equations one encounters two probability density functions,&nbsp; which are given here in advance:
  
*Die&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Rayleighverteilung|'''Rayleighverteilung''']]&nbsp; erhält man für die WDF der Zufallsgröße&nbsp; $y$&nbsp; mit Realisierung&nbsp; $\eta$, die sich aus den beiden gaußverteilten und statistisch unabhängigen Komponenten&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; (beide mit der gleichen Streuung&nbsp; $\sigma_n)$&nbsp; wie folgt ergibt:
+
*The&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rayleigh_PDF|'''Rayleigh distribution''']]&nbsp; is obtained for the power density function&nbsp; $\rm (PDF)$&nbsp; of the random variable&nbsp; $y$&nbsp; with realization&nbsp; $\eta$,&nbsp; which is obtained from the two Gaussian distributed and statistically independent components&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; $($both with the same standard deviation&nbsp; $\sigma_n)$&nbsp; as follows:
  
 
::<math>y = \sqrt{u^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) ={\eta}/{\sigma_n^2}
 
::<math>y = \sqrt{u^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) ={\eta}/{\sigma_n^2}
Line 18: Line 18:
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
*Die&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Riceverteilung|'''Riceverteilung''']]&nbsp; erhält man unter gleichen Randbedingungen für den Fall, dass bei einer der Komponenten $($entweder &nbsp;$u$&nbsp; oder &nbsp;$v)$&nbsp; noch eine Konstante&nbsp; $C$&nbsp; addiert wird:
+
*The&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rice_PDF|'''Rice distribution''']]&nbsp; is obtained under the same boundary conditions for the case where a constant &nbsp; $C$&nbsp; is added to one of the components $($either to&nbsp;$u$&nbsp; or &nbsp;$v)$:&nbsp;
  
 
::<math>y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = {\eta}/{\sigma_n^2}
 
::<math>y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = {\eta}/{\sigma_n^2}
Line 24: Line 24:
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
:Die Riceverteilung verwendet die <i>modifizierte Besselfunktion nullter Ordnung</i>, deren Definition und Reihenentwicklung wie folgt lauten:
+
:The Rice distribution uses the&nbsp; "modified zero-order Bessel function",&nbsp; whose definition and series expansion are as follows:
 
::<math>{\rm I }_0 (x) = \frac{1}{ 2\pi} \cdot  \int_{0}^{2\pi} {\rm e }^{-x \hspace{0.03cm}\cdot \hspace{0.03cm}\cos(\alpha)} \,{\rm d} \alpha \hspace{0.2cm} \approx \hspace{0.2cm}
 
::<math>{\rm I }_0 (x) = \frac{1}{ 2\pi} \cdot  \int_{0}^{2\pi} {\rm e }^{-x \hspace{0.03cm}\cdot \hspace{0.03cm}\cos(\alpha)} \,{\rm d} \alpha \hspace{0.2cm} \approx \hspace{0.2cm}
 
  \sum_{k = 0}^{\infty} \frac{(x/2)^{2k}}{k! \cdot \Gamma (k+1)}
 
  \sum_{k = 0}^{\infty} \frac{(x/2)^{2k}}{k! \cdot \Gamma (k+1)}
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
Die Grafik zeigt die Wahrscheinlichkeitsdichtefunktionen von Rayleigh&ndash; und Riceverteilung.  
+
The graph shows the probability density functions of Rayleigh and Rice distributions.&nbsp; To this representation it is to be noted:
 +
[[File:P ID2082 Dig T 4 5 S1 version1_ret.png|right|frame|Rayleigh resp. Rice PDF for $&sigma;_n = 0.5$|class=fit]]
 +
*The Rice distribution is determined by the two parameters&nbsp; $C$&nbsp; and&nbsp; $\sigma_n$.&nbsp; With&nbsp; $C = 0$,&nbsp; the Rice PDF is identical to the Rayleigh PDF.<br>
  
[[File:P ID2082 Dig T 4 5 S1 version1.png|center|frame|Rayleigh- und Rice-WDF, jeweils für $&sigma;_n = 0.5$|class=fit]]
+
*The Rayleigh PDF with larger&nbsp; $\sigma_n$&nbsp; is of the same shape as shown for&nbsp; $\sigma_n = 0.5$,&nbsp; but wider and lower in the ratio of standard deviation.<br>
  
Zu dieser Darstellung ist anzumerken:
+
*$\sigma_n$&nbsp; indicates the standard deviation of the two Gaussian distributed random variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; and not the standard deviation of the Rayleigh distributed random variable&nbsp; $y$.&nbsp; Rather,&nbsp; the following applies to the latter:
*Die Riceverteilung ist durch die beiden Parameter&nbsp; $C$&nbsp; und&nbsp; $\sigma_n$&nbsp; bestimmt. Mit&nbsp; $C = 0$&nbsp; ist die Rice&ndash;WDF identisch mit der Rayleigh&ndash;WDF.<br>
 
 
 
*Die Rayleigh&ndash;WDF mit größerem&nbsp; $\sigma_n$&nbsp; ist formgleich wie für&nbsp; $\sigma_n = 0.5$&nbsp; dargestellt, aber im Verhältnis der Streuungen breiter und niedriger.<br>
 
 
 
*$\sigma_n$&nbsp; gibt die Streuung der beiden gaußverteilten Zufallsgrößen&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; an und nicht die Streuung der rayleighverteilten Zufallsgröße&nbsp; $y$. Für diese gilt vielmehr:
 
  
 
::<math>\sigma_y = \sigma_n  \cdot  \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n   
 
::<math>\sigma_y = \sigma_n  \cdot  \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n   
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
*Die Rayleighverteilung ist extrem unsymmetrisch, erkennbar am (relativ) großen Wert für das&nbsp;  [[Theory_of_Stochastic_Signals/Erwartungswerte_und_Momente#Einige_h.C3.A4ufig_benutzte_Zentralmomente|Zentralmoment 3. Ordnung]] &nbsp; &rArr; &nbsp; ''Charliersche Schiefe'':  
+
*The Rayleigh PDF is extremely asymmetric,&nbsp; recognizable by the large value for the&nbsp;  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Some_common_central_moments|"third order central moment"]] &nbsp; &rArr; &nbsp; "Charlier's skewness":  
 
:$$\mu_3/\sigma_y \approx 0.27.$$
 
:$$\mu_3/\sigma_y \approx 0.27.$$
  
*Die Riceverteilung ist um so symmetrischer, je größer das Verhältnis&nbsp; $C/\sigma_n$&nbsp; ist. Für&nbsp; $C/\sigma_n \ge 4$&nbsp; ist&nbsp; $\mu_3 \approx 0$. Je größer&nbsp; $C/\sigma_n$&nbsp; ist, um so mehr nähert sich die Riceverteilung $($mit &nbsp;$C$, &nbsp;$\sigma_n)$&nbsp; einer&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen#Wahrscheinlichkeitsdichte-_und_Verteilungsfunktion|Gaußverteilung]]&nbsp; mit Mittelwert&nbsp; $C$&nbsp; und Streuung&nbsp; $\sigma_n$&nbsp; an:
+
*The Rice PDF is more symmetrical the larger the quotient&nbsp; $C/\sigma_n$.&nbsp; For&nbsp; $C/\sigma_n \ge 4$:&nbsp; $\mu_3 \approx 0$ &nbsp; &rArr; &nbsp; symmetrical PDF.  
 +
 
 +
*The larger&nbsp; $C/\sigma_n$&nbsp; is,&nbsp; the more the Rice PDF $($with &nbsp;$C$, &nbsp;$\sigma_n)$&nbsp; approaches a&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Probability_density_function_.E2.80.93_Cumulative_density_function|"Gaussian PDF"]]&nbsp; with mean&nbsp; $C$&nbsp; and standard deviation&nbsp; $\sigma_n$:  
  
::<math>p_y (\eta) \approx \frac{1}{\sqrt{2\pi} \cdot \sigma_n}
+
:$$p_y (\eta) \approx \frac{1}{\sqrt{2\pi} \cdot \sigma_n}
  \cdot {\rm exp } \left [ - \frac{(\eta - C)^2}{2 \sigma_n^2}\right ]
+
  \cdot {\rm exp } \left [ - \frac{(\eta - C)^2}{2 \sigma_n^2}\right ]$$
\hspace{0.3cm} \Rightarrow \hspace{0.3cm} m_y = C\hspace{0.05cm},\hspace{0.2cm}\sigma_y = \sigma_n  
+
:$$ \Rightarrow \hspace{0.3cm} m_y = C\hspace{0.05cm},\hspace{0.2cm}\sigma_y = \sigma_n  
  \hspace{0.05cm}.</math>
+
  \hspace{0.05cm}.$$
  
== Nichtkohärente Demodulation von On–Off–Keying==
+
== Non-coherent demodulation of&nbsp; "on–off keying"&nbsp; (OOK)==
 
<br>
 
<br>
Wir betrachten&nbsp;  [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#On.E2.80.93Off.E2.80.93Keying_.282.E2.80.93ASK.29|On&ndash;Off&ndash;Keying]]&nbsp; (bzw. 2&ndash;ASK) im äquivalenten Tiefpassbereich.
+
We consider&nbsp;  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29|"on&ndash;off keying"]]&nbsp; $(\rm OOK$&nbsp; or&nbsp; $\rm 2&ndash;ASK)$&nbsp; in the equivalent low-pass range.
 +
[[File:EN_Dig_A_1_9_neu.png|right|frame|Coherent and non-coherent demodulation of&nbsp; "on-off keying" |class=fit]]
 +
 +
*In the case of coherent demodulation&nbsp; (left graph),&nbsp; the signal space constellation of the received signal&nbsp; $\boldsymbol{r}(t)$&nbsp; is the same as that of the transmitted signal&nbsp; $\boldsymbol{s}(t)$&nbsp; and consists of two points.
 +
 
 +
*The decision boundary&nbsp; $G$&nbsp; lies in the middle between these points&nbsp; $\boldsymbol{r}_0$&nbsp; and&nbsp; $\boldsymbol{r}_1$.
 
   
 
   
*Bei kohärenter Demodulation (linke Grafik) ist die Signalraumkonstellation des Empfangssignals gleich der des Sendesignals und besteht aus zwei Punkten.
+
*The arrows mark the rough direction of noise vectors that may cause transmission errors.
*Die Entscheidungsgrenze&nbsp; $G$&nbsp; liegt in der Mitte zwischen diesen Punkten&nbsp; $\boldsymbol{r}_0$&nbsp; und&nbsp; $\boldsymbol{r}_1$.
 
*Die Pfeile markieren die grobe Richtung von Rauschvektoren, die eventuell zu Übertragungsfehlern führen.
 
  
  
[[File:P ID2083 Dig T 4 5 S2a version1.png|center|frame|Kohärente und nichtkohärente Demodulation von On-Off-Keying|class=fit]]
+
On the other hand, for non-coherent demodulation (right graph):
 +
*The point&nbsp; $\boldsymbol{r}_1 = \boldsymbol{s}_1 = 0$&nbsp; is still preserved.&nbsp; But&nbsp; $\boldsymbol{r}_0 = \boldsymbol{s}_0 \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi}$&nbsp; can lie on any point of the circle around&nbsp; $\boldsymbol{s}_0$,&nbsp; since&nbsp; $\phi$&nbsp; is unknown.<br>
  
Dagegen gilt bei nichtkohärenter Demodulation  (rechte Grafik):
+
*The decision process in consideration of the AWGN noise happens in two dimensions&nbsp; $($indicated by arrows in the right graph$)$.<br>
*Der Punkt&nbsp; $\boldsymbol{r}_1 = \boldsymbol{s}_1 = 0$&nbsp; bleibt weiter erhalten. Dagegen kann&nbsp; $\boldsymbol{r}_0 = \boldsymbol{s}_0 \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi}$&nbsp; auf jeden Punkt des Kreises um&nbsp; $\boldsymbol{s}_0$&nbsp; liegen, da&nbsp; $\phi$&nbsp; unbekannt ist.<br>
 
  
*Der Entscheidungsprozess unter Berücksichtigung des AWGN&ndash;Rauschens ist nun zweidimensional zu interpretieren, wie durch die Pfeile in der rechten Grafik angedeutet.<br>
+
*The decision area&nbsp; $I_1$&nbsp; is a circle whose radius&nbsp; $G$&nbsp; is an optimizable parameter. The decision area&nbsp; $I_0$ &nbsp;lies outside the circle.<br>
  
*Das Entscheidungsgebiet&nbsp; $I_1$&nbsp; ist ein Kreis, dessen Radius&nbsp; $G$&nbsp; ein optimierbarer Parameter ist. Das Entscheidungsgebiet&nbsp; $I_0$ &nbsp;liegt außerhalb des Kreises.<br><br>
 
  
[[File:P ID3147 Dig T 4 5 S2b version1.png|center|frame|Empfänger für nichtkohärente OOK-Demodulation (komplexen Signale sind blau beschriftet)|class=fit]]
+
Thus,&nbsp; the structure of the optimal OOK receiver&nbsp; (in the equivalent low&ndash;pass range)&nbsp; is fixed &nbsp; &rArr; &nbsp;  see second graph:
 +
[[File:EN_Dig_T_4_5_S2b_neu.png|right|frame|Receiver for non-coherent OOK demodulation&nbsp; (complex signals are labeled blue) |class=fit]]
 +
 +
*The input signal&nbsp; $\boldsymbol{r}(t) = \boldsymbol{s}(t) \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi} + \boldsymbol{n}(t)$&nbsp; is generally complex because of the phase angle&nbsp; $\phi$&nbsp; and because of the complex noise term&nbsp; $\boldsymbol{n}(t)$.&nbsp;
 +
 
 +
*Consequently required is now the correlation between the complex received signal&nbsp;  $\boldsymbol{r}(t)$&nbsp; and a complex basis function&nbsp; $\boldsymbol{\xi}_1(t)$.<br>
  
Damit liegt die Strukur des optimalen OOK&ndash;Empfängers (im äquivalenten Tiefpassbereich) fest. Entsprechend dieser zweiten Grafik gilt:
+
*The result is the&nbsp; (complex)&nbsp; detection value&nbsp; $\boldsymbol{r}$,&nbsp; from which the magnitude&nbsp; $y = |\boldsymbol{r}(t)|$&nbsp; is formed as a real decision input variable.<br>
*Das Eingangssignal&nbsp; $\boldsymbol{r}(t) = \boldsymbol{s}(t) \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi} + \boldsymbol{n}(t)$&nbsp; ist aufgrund des Phasenwinkels&nbsp; $\phi$&nbsp; und wegen des komplexen Rauschterms&nbsp; $\boldsymbol{n}(t)$&nbsp; im allgemeinen komplex. Erforderlich ist demzufolge nun die Korrelation zwischen dem komplexen Empfangssignal&nbsp; $\boldsymbol{r}(t)$&nbsp; und einer komplexen Basisfunktion&nbsp; $\boldsymbol{\xi}_1(t)$.<br>
 
  
*Das Ergebnis ist der (komplexe) Detektorwert&nbsp; $\boldsymbol{r}$, woraus als reelle Entscheidereingangsgröße der Betrag&nbsp; $y = |\boldsymbol{r}(t)|$&nbsp; gebildet wird.<br>
+
*If the decision value&nbsp; $y \gt G$,&nbsp;  then&nbsp; $m_0$&nbsp; is output as the estimated value,&nbsp; otherwise&nbsp; $m_1$.  
  
*Ist der Entscheidungswert&nbsp; $y \gt G$, so wird als Schätzwert&nbsp; $m_0$&nbsp; ausgegeben, andernfalls&nbsp; $m_1$. Somit beträgt die Fehlerwahrscheinlichkeit bei gleichwahrscheinlichen Symbolen:
+
*Thus,&nbsp; the symbol error probability for equally probable symbols <br>&rArr; &nbsp; ${\rm Pr}(\boldsymbol{m}_0) = {\rm Pr}(\boldsymbol{m}_1) =1/2$&nbsp; is:
  
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E}}) =  {1}/{ 2} \cdot \int_{0}^{G} p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (\eta \hspace{0.05cm}|\hspace{0.05cm} m_0) \,{\rm d} \eta
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E}}) =  {1}/{ 2} \cdot \int_{0}^{G} p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (\eta \hspace{0.05cm}|\hspace{0.05cm} m_0) \,{\rm d} \eta
Line 83: Line 88:
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
*Aufgrund der Rice&ndash;WDF&nbsp; $p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (\eta\hspace{0.05cm}|\hspace{0.05cm}m_0)$ und der Rayleigh&ndash;WDF $p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (\eta\hspace{0.05cm}|\hspace{0.05cm}m_1)$&nbsp; kann allerdings diese Wahrscheinlichkeit nur numerisch ermittelt werden. Die optimale Entscheidungsgrenze&nbsp; $G$&nbsp; ist vorher als die Lösung der folgenden Gleichung zu bestimmen:
+
*However,&nbsp; due to the Rice PDF &nbsp; $p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (\eta\hspace{0.05cm}|\hspace{0.05cm}m_0)$ &nbsp; and the Rayleigh PDF &nbsp; $p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (\eta\hspace{0.05cm}|\hspace{0.05cm}m_1)$, &nbsp; this probability can only be determined numerically.
 +
 
 +
* The optimal decision limit&nbsp; $G$&nbsp; has to be determined beforehand as the solution of the following equation:
 
::<math>p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (G \hspace{0.05cm}|\hspace{0.05cm}m_0) = p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (G \hspace{0.05cm}|\hspace{0.05cm}m_1)
 
::<math>p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (G \hspace{0.05cm}|\hspace{0.05cm}m_0) = p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (G \hspace{0.05cm}|\hspace{0.05cm}m_1)
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
[[File:P ID3148 Dig T 4 5 S2c version1.png|right|frame|Dichtefunktionen für „OOK, nichtkohärent”]]
+
{{GraueBox|TEXT=
{{GraueBox|TEXT= 
+
[[File:P ID3148 Dig T 4 5 S2c version1.png|right|frame|Density functions for "OOK, non-coherent"]]  
$\text{Beispiel 1:}$&nbsp; Die Grafik zeigt das Ergebnis dieser Gleichung für&nbsp; $\sigma_n = 0.5$&nbsp; und&nbsp; $C = 2$, wobei die (rote) Rice&ndash;WDF durch eine Gauß&ndash;WDF mit Mittelwert&nbsp; $C$&nbsp; und Streuung&nbsp; $\sigma_n$&nbsp; approximiert ist. Man erkennt daraus:
+
$\text{Example 1:}$&nbsp; The graph shows the result of this equation for&nbsp; $\sigma_n = 0.5$&nbsp; and&nbsp; $C = 2$, where the (red) Rice PDF is approximated by a Gaussian PDF with mean&nbsp; $C$&nbsp; and standard deviation&nbsp; $\sigma_n$.&nbsp;  
*Die optimale Entscheidungsgrenze &nbsp;$($hier: &nbsp; $G \approx 1.25)$&nbsp; ergibt sich aus dem Schnittpunkt der beiden Kurven.<br>
 
*Die Symbolfehlerwahrscheinlichkeit&nbsp; $p_{\rm S}$&nbsp; ist die Summe der beiden farblich hinterlegten Flächen. Im Beispiel ergibt sich&nbsp; $p_{\rm S} \approx 5\%$.
 
  
 +
One can see from this sketch:
 +
*The optimal decision value &nbsp;$($here: &nbsp; $G \approx 1.25)$&nbsp; results from the intersection of the two PDF curves.<br>
  
Die Fehlerwahrscheinlichkeit für andere Werte von&nbsp; $C$&nbsp; und&nbsp; $\sigma_n$&nbsp; sowie die optimale Entscheidungsgrenze&nbsp; $G$&nbsp; können Sie mit dem Berechnungstool&nbsp;  [[Applets:On-Off-Keying|Nichtkohärentes On&ndash;Off&ndash;Keying]]&nbsp; bestimmen.}}
+
*The symbol error probability&nbsp; $p_{\rm S}$&nbsp; is the sum of the two colored areas.&nbsp; In the example:&nbsp; $p_{\rm S} \approx 5\%$ results.
  
== Nichtkohärente Demodulation von binärer FSK (2&ndash;FSK)==
+
 
 +
&rArr; &nbsp; You can determine the error probability for other values of&nbsp; $C$&nbsp; and&nbsp; $\sigma_n$&nbsp; as well as the optimal decision value&nbsp; $G$&nbsp; using the&nbsp; HTML5/JavaScript applet&nbsp;  [[Applets:Coherent_and_Non-Coherent_On-Off_Keying|"Coherent and Non-coherent On-Off Keying"]].&nbsp; }}
 +
 
 +
== Non-coherent demodulation of binary FSK (2&ndash;FSK)==
 
<br>
 
<br>
Wie schon im&nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation#Binary_Frequency_Shift_Keying_.282.E2.80.93FSK.29| letzten Kapitel]]&nbsp; gezeigt wurde, lässt sich &nbsp;<i>Binary Frequency Shift Keying</i>&nbsp; (2&ndash;FSK) im äquivalenten Tiefpassbereich durch die Basisfunktionen
+
As already shown in the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_frequency_shift_keying_.282.E2.80.93FSK.29|"last chapter"]],&nbsp; "Binary Frequency Shift Keying"&nbsp; $\rm (2&ndash;FSK)$&nbsp; in the equivalent low-pass range can be described by the basis functions
 
::<math>\xi_1(t) = \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},</math>
 
::<math>\xi_1(t) = \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},</math>
 
::<math> \xi_2(t) = \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}-{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T \hspace{0.05cm}</math>
 
::<math> \xi_2(t) = \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}-{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T \hspace{0.05cm}</math>
  
darstellen. Um Orthogonalität zwischen diesen beiden komplexen Basisfunktionen zu erreichen, muss der&nbsp; [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation#Binary_Frequency_Shift_Keying_.E2.87.92_2.E2.80.93FSK| Modulationsindex]]&nbsp; $h$&nbsp; ganzzahlig gewählt werden:
+
To achieve orthogonality between these two complex basis functions,&nbsp; the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_frequency_shift_keying_.282.E2.80.93FSK.29|"modulation index"]]&nbsp; $h$&nbsp; must be chosen to be integer:
  
 
::<math><  \hspace{-0.05cm}\xi_1(t) \hspace{0.1cm}  \cdot  \hspace{0.1cm} \xi_2(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0  
 
::<math><  \hspace{-0.05cm}\xi_1(t) \hspace{0.1cm}  \cdot  \hspace{0.1cm} \xi_2(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0  
 
   \hspace{0.3cm} \Rightarrow \hspace{0.3cm} h = 2 \cdot \Delta f_{\rm A} \cdot T\hspace{0.05cm}= 1, 2, 3, \text{...}</math>
 
   \hspace{0.3cm} \Rightarrow \hspace{0.3cm} h = 2 \cdot \Delta f_{\rm A} \cdot T\hspace{0.05cm}= 1, 2, 3, \text{...}</math>
  
Die Grafik zeigt die Struktur zur nichtkohärenten orthogonalen Demodulation der binären FSK.<br>
+
The diagram shows the structure for non-coherent orthogonal demodulation of binary FSK.<br>
  
[[File:P ID2087 Dig T 4 5 S3a version2.png|center|frame|Nichtkohärente Demodulation der binären FSK|class=fit]]
+
[[File:EN_Dig_T_4_5_S3a_neu.png|right|frame|Non-coherent demodulation of the binary FSK|class=fit]]
  
Im rauschfreien Fall &nbsp; &#8658; &nbsp;  $n(t) \equiv 0$&nbsp;  gilt für die Ausgänge der beiden Korrelatoren:
+
In the noise-free case &nbsp; &#8658; &nbsp;  $n(t) \equiv 0$&nbsp;  applies to the outputs of the two correlators:
  
::<math>r_1 = \hspace{0.2cm} <  \hspace{-0.05cm}r(t) \hspace{0.1cm}  \cdot  \hspace{0.1cm} \xi_1(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0\hspace{0.05cm},  \hspace{0.2cm} {\rm falls}\hspace{0.15cm} m = m_1\hspace{0.05cm},</math>
+
::<math>r_1 = \hspace{0.2cm} <  \hspace{-0.05cm}r(t) \hspace{0.1cm}  \cdot  \hspace{0.1cm} \xi_1(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0\hspace{0.05cm},  \hspace{0.4cm} {\rm if}\hspace{0.15cm} m = m_1\hspace{0.05cm},</math>
::<math> r_2 = \hspace{0.2cm} <  \hspace{-0.05cm}r(t) \hspace{0.1cm}  \cdot  \hspace{0.1cm} \xi_2(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0\hspace{0.05cm},  \hspace{0.2cm} {\rm falls}\hspace{0.15cm} m = m_0\hspace{0.05cm}.</math>
+
::<math> r_2 = \hspace{0.2cm} <  \hspace{-0.05cm}r(t) \hspace{0.1cm}  \cdot  \hspace{0.1cm} \xi_2(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0\hspace{0.05cm},  \hspace{0.4cm} {\rm if}\hspace{0.15cm} m = m_0\hspace{0.05cm}.</math>
  
Nach jeweiliger Betragsbildung &nbsp; &#8658; &nbsp; $y_1 = |r_1|, \ \ y_2 = |r_2|$&nbsp; ist dann folgende Entscheidungsregel anwendbar:
+
After respective magnitude formation &nbsp; &#8658; &nbsp; $y_1 = |r_1|, \ \ y_2 = |r_2|$&nbsp; the following decision rule is applicable:
 
::<math>\hat{m} =
 
::<math>\hat{m} =
 
\left\{ \begin{array}{c} m_0 \\
 
\left\{ \begin{array}{c} m_0 \\
 
  m_1  \end{array} \right.\quad
 
  m_1  \end{array} \right.\quad
\begin{array}{*{1}c} {\rm falls}\hspace{0.15cm} y_1 > y_2 \hspace{0.05cm},
+
\begin{array}{*{1}c} {\rm if}\hspace{0.15cm} y_1 > y_2 \hspace{0.05cm},
\\  {\rm falls}\hspace{0.15cm} y_1 < y_2  \hspace{0.05cm}.\\ \end{array}</math>
+
\\  {\rm if}\hspace{0.15cm} y_1 < y_2  \hspace{0.05cm}.\\ \end{array}</math>
  
Zur einfacheren Realisierung des Entscheiders kann auch die Differenz&nbsp; $y_1 - y_2$&nbsp; mit der Entscheidungsgrenze&nbsp; $G = 0$&nbsp; ausgewertet werden.<br>
+
For a simpler realization of the decision,&nbsp; the difference&nbsp; $y_1 - y_2$&nbsp; can also be evaluated with the decision boundary&nbsp; $G = 0$.&nbsp; <br>
  
== Fehlerwahrscheinlichkeit bei nichtkohärenter 2&ndash;FSK&ndash;Demodulation==
+
== Error probability with non-coherent 2&ndash;FSK demodulation==
 
<br>
 
<br>
Im Folgenden wird die Fehlerwahrscheinlichkeit unter der Annahme berechnet, dass&nbsp; $m = m_0$&nbsp; gesendet wurde. Unter der weiteren Voraussetzung gleichwahrscheinlicher binärer Nachrichten&nbsp; $m_0$&nbsp; und&nbsp; $m_1$&nbsp; ist die absolute Fehlerwahrscheinlichkeit genau so groß:  
+
In the following,&nbsp; the error probability is calculated under the assumption that&nbsp; $m = m_0$&nbsp; was sent.  
 +
 
 +
*Under the further assumption of equally probable binary messages&nbsp; $m_0$&nbsp; and&nbsp; $m_1$,&nbsp; the absolute error probability is exactly the same:  
 
:$${\rm Pr}({\cal{E}}) =  {\rm Pr}({\cal{E}}\hspace{0.05cm} | \hspace{0.05cm}m_0)
 
:$${\rm Pr}({\cal{E}}) =  {\rm Pr}({\cal{E}}\hspace{0.05cm} | \hspace{0.05cm}m_0)
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
Mit&nbsp; $m = m_0$&nbsp; ergeben sich für die komplexen Korrelationsausgangswerte&nbsp; $r_i$&nbsp; und deren Beträge&nbsp; $y_i$:<br>
+
*With&nbsp; $m = m_0$&nbsp; we get for the complex correlation output values&nbsp; $r_i$&nbsp; and their magnitudes&nbsp; $y_i$:<br>
::<math>r_1 = \sqrt{E} \cdot {\rm e}^{{\rm j}\phi} + n_1\hspace{0.3cm} \Rightarrow \hspace{0.3cm}y_1 = |r_1|\hspace{0.15cm}{\rm ist}\hspace{0.15cm}{\rm riceverteilt}
+
::<math>r_1 = \sqrt{E} \cdot {\rm e}^{{\rm j}\phi} + n_1\hspace{0.3cm} \Rightarrow \hspace{0.3cm}y_1 = |r_1|\hspace{0.15cm}{\rm is}\hspace{0.15cm}{\rm Rice}\hspace{0.15cm}{\rm distributed}
 
  \hspace{0.05cm},</math>
 
  \hspace{0.05cm},</math>
::<math> r_2 =  n_2\hspace{0.3cm} \Rightarrow \hspace{0.3cm}y_2 = |r_2|\hspace{0.15cm}{\rm ist}\hspace{0.15cm}{\rm rayleighverteilt}
+
::<math> r_2 =  n_2\hspace{0.3cm} \Rightarrow \hspace{0.3cm}y_2 = |r_2|\hspace{0.15cm}{\rm is}\hspace{0.15cm}{\rm Rayleigh}\hspace{0.15cm}{\rm distributed}
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
Hierbei steht&nbsp; $E$&nbsp; wegen&nbsp; $M = 2$&nbsp; für die <i>Symbolenergie</i>&nbsp; $(E_{\rm S})$&nbsp; und die <i>Bitenergie</i>&nbsp;  $(E_{\rm B})$&nbsp; gleichermaßen, und&nbsp; $n_1$&nbsp; und&nbsp; $n_2$&nbsp; sind unkorrelierte komplexe Rauschgrößen mit Mittelwert Null und Varianz&nbsp; $2 \cdot \sigma_n^2$. Somit lautet die&nbsp; [[Theory_of_Stochastic_Signals/Zweidimensionale_Zufallsgrößen#Verbundwahrscheinlichkeitsdichtefunktion|Verbundwahrscheinlichkeitsdichtefunktion]]:
+
:Here,&nbsp; $E$&nbsp; due to&nbsp; $M = 2$&nbsp; represents the&nbsp; "average symbol energy"&nbsp; $(E_{\rm S})$&nbsp; and the&nbsp; "average bit energy"&nbsp;  $(E_{\rm B})$&nbsp; equally.&nbsp; $n_1$&nbsp; and&nbsp; $n_2$&nbsp; are uncorrelated complex noise variables with mean zero and variance&nbsp; $2 \cdot \sigma_n^2$.  
 +
 
 +
*Thus,&nbsp; the&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|"joint probability density function"]]&nbsp; is:
 
::<math>p_{y_1,\hspace{0.03cm} y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_1, \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) =  
 
::<math>p_{y_1,\hspace{0.03cm} y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_1, \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) =  
 
   p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) \cdot  
 
   p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) \cdot  
 
   p_{y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0)
 
   p_{y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0)
 
  \hspace{0.05cm},</math>
 
  \hspace{0.05cm},</math>
::<math>\Rightarrow \hspace{0.5cm} p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) = {\eta_1}/{\sigma_n^2}
+
::$$p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) = {\eta_1}/{\sigma_n^2}
 
  \cdot {\rm e }^{  - ({\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [{\eta_1 \cdot  \sqrt{E}}/{ \sigma_n^2}\right ]
 
  \cdot {\rm e }^{  - ({\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [{\eta_1 \cdot  \sqrt{E}}/{ \sigma_n^2}\right ]
  \hspace{0.05cm}, \hspace{0.5cm}
+
  \hspace{0.05cm},$$
p_{y_2 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) = {\eta_2}/{\sigma_n^2}  
+
::$$ p_{y_2 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) = {\eta_2}/{\sigma_n^2}  
  \cdot {\rm e }^{  - \eta_2^2 /({2 \sigma_n^2}) } \hspace{0.05cm}.</math>
+
  \cdot {\rm e }^{  - \eta_2^2 /({2 \sigma_n^2}) } \hspace{0.05cm}.$$
  
Die Fehlerwahrscheinlichkeit ergibt sich allgemein wie folgt:
+
*The error probability is generally obtained as follows:
  
 
::<math>{\rm Pr}({\cal{E}}) = \int_{0}^{\infty} \int_{\eta_1}^{\infty}  
 
::<math>{\rm Pr}({\cal{E}}) = \int_{0}^{\infty} \int_{\eta_1}^{\infty}  
Line 156: Line 170:
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Fazit:}$&nbsp; Für die&nbsp; '''Fehlerwahrscheinlichkeit bei nichtkohärenter Demodulation der binären FSK'''&nbsp; erhält man  nach einigen mathematischen Umformungen  das überraschend einfache Ergebnis
+
$\text{Conclusion:}$&nbsp; For the&nbsp; '''error probability with non-coherent demodulation of the binary FSK''',&nbsp; one obtains after some mathematical transformations the surprisingly simple result
  
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
Zum Vergleich sei nochmals das Ergebnis für die '''kohärente Demodulation''' angegeben:
+
*For comparison,&nbsp; the result for&nbsp; '''coherent demodulation'''&nbsp; is here given again:
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E} }) = {\rm Q}(\sqrt{ E_{\rm S}/N_0})
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E} }) = {\rm Q}(\sqrt{ E_{\rm S}/N_0})
 
  \hspace{0.05cm}.</math>}}
 
  \hspace{0.05cm}.</math>}}
Line 167: Line 181:
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Herleitung:}$&nbsp; Das vorweg genommene Ergebnis soll nun in einigen Rechenschritten hergeleitet werden. Wir gehen dabei von den folgenden Gleichungen aus:
+
$\text{Derivation:}$&nbsp; This result is now to be derived in some calculation steps.&nbsp; We start from the following equations:
  
 
:$${\rm Pr}({\cal{E} }) = \int_{0}^{\infty} p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_0) \cdot \int_{\eta_1}^{\infty}  
 
:$${\rm Pr}({\cal{E} }) = \int_{0}^{\infty} p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_0) \cdot \int_{\eta_1}^{\infty}  
 
   p_{y_2 \hspace{0.03cm}\vert \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2\,\,{\rm d} \eta_1
 
   p_{y_2 \hspace{0.03cm}\vert \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2\,\,{\rm d} \eta_1
  \hspace{0.05cm},\hspace{0.5cm}\text{mit}$$
+
  \hspace{0.05cm},\hspace{0.5cm}\text{with}$$
:$$p_{y_1 \hspace{0.01cm}\vert \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}\vert \hspace{0.05cm}m_0) = {\eta_1}/{\sigma_n^2} \cdot {\rm e }^{  - ({\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [{\eta_1 \cdot  \sqrt{E} }/{ \sigma_n^2}\right ] \hspace{0.05cm},\hspace{0.5cm}p_{y_\hspace{0.01cm}\vert \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0) = {\eta_2}/{\sigma_n^2}  \cdot {\rm e }^{  - \eta_2^2 /({2 \sigma_n^2}) } \hspace{0.05cm}.$$
+
::$$p_{y_1 \hspace{0.01cm}\vert \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}\vert \hspace{0.05cm}m_0) = {\eta_1}/{\sigma_n^2} \cdot {\rm e }^{  - ({\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [{\eta_1 \cdot  \sqrt{E} }/{ \sigma_n^2}\right ] \hspace{0.05cm},$$
 +
::$$p_{y_\hspace{0.01cm}\vert \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0) = {\eta_2}/{\sigma_n^2}  \cdot {\rm e }^{  - \eta_2^2 /({2 \sigma_n^2}) } \hspace{0.05cm}.$$
  
<b>(1)</b> &nbsp; Das innere Integral gibt die Wahrscheinlichkeit an, dass die rayleighverteilte Zufallsgröße&nbsp; $\eta_2$&nbsp; größer ist als&nbsp; $\eta_1$&nbsp; &ndash; siehe Musterlösung zur&nbsp; [[Aufgaben:Aufgabe_4.17Z:_Rayleigh-_und_Riceverteilung|Aufgabe 4.17Z]]:
+
<b>(1)</b> &nbsp; The inner integral gives the probability that the Rayleigh distributed random variable&nbsp; $\eta_2$&nbsp; is larger than&nbsp; $\eta_1$&nbsp; &ndash; see solution to&nbsp; [[Aufgaben:Exercise_4.17Z:_Rayleigh_and_Rice_Distribution|"Exercise 4.17Z"]]:
  
 
:$$\int_{\eta_1}^{\infty}  p_{y_2 \hspace{0.03cm}\vert\hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2 =
 
:$$\int_{\eta_1}^{\infty}  p_{y_2 \hspace{0.03cm}\vert\hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2 =
Line 183: Line 198:
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
<b>(2)</b> &nbsp; Mit den (willkürlichen) Substitutionen&nbsp; $C_0^2  = E/4$&nbsp; und&nbsp; $\sigma_0^2 = \sigma_n^2/2$&nbsp; erhält man daraus:
+
<b>(2)</b> &nbsp; With the&nbsp; (arbitrary)&nbsp; substitutions&nbsp; $C_0^2  = E/4$&nbsp; and&nbsp; $\sigma_0^2 = \sigma_n^2/2$,&nbsp; we obtain:
  
 
:$${\rm Pr}({\cal{E} }) = \int_{0}^{\infty} \frac{\eta_1}{2 \cdot \sigma_0^2}
 
:$${\rm Pr}({\cal{E} }) = \int_{0}^{\infty} \frac{\eta_1}{2 \cdot \sigma_0^2}
Line 192: Line 207:
 
  \hspace{0.05cm}.$$
 
  \hspace{0.05cm}.$$
  
<b>(3)</b> &nbsp; Durch Verschieben von Anteilen vor das Integral gelingt es, dass der Integrand wieder eine&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Riceverteilung|Riceverteilung]]&nbsp; beschreibt:
+
<b>(3)</b> &nbsp; By shifting fractions in front of the integral,&nbsp; we succeed that the integrand again describes a&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rice_PDF|"Rice distribution"]]:&nbsp;  
  
 
::<math>{\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ] \cdot \int\limits_{0}^{\infty} \frac{\eta_1}{ \sigma_0^2}
 
::<math>{\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ] \cdot \int\limits_{0}^{\infty} \frac{\eta_1}{ \sigma_0^2}
Line 199: Line 214:
 
  \hspace{0.05cm}.</math>
 
  \hspace{0.05cm}.</math>
  
<b>(4)</b> &nbsp; Der Integrand beschreibt nun die Rice&ndash;WDF. Das Integral über das gesamte Definitionsgebiet von&nbsp; $0$&nbsp; bis &nbsp;$+\infty$&nbsp; ergibt wie bei jeder WDF den Wert Eins, so dass gilt:
+
<b>(4)</b> &nbsp; The integrand now describes the Rice PDF.&nbsp; The integral over the entire definition area from&nbsp; $0$&nbsp; to &nbsp;$+\infty$&nbsp; yields the value one,&nbsp; as for any PDF,&nbsp; so that holds:
  
 
::<math>{\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ]\hspace{0.05cm}.</math>
 
::<math>{\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ]\hspace{0.05cm}.</math>
  
<b>(5)</b> &nbsp; Mit&nbsp; $C_0^2  = E/4$&nbsp; und&nbsp; $\sigma_0^2 = \sigma_n^2/2$&nbsp; sowie der allgemein gültigen Beziehung&nbsp; $\sigma_n^2 = N_0$&nbsp; erhält man schließlich:
+
<b>(5)</b> &nbsp; Finally, with&nbsp; $C_0^2  = E/4$&nbsp; and&nbsp; $\sigma_0^2 = \sigma_n^2/2$&nbsp; and the generally valid relation&nbsp; $\sigma_n^2 = N_0$,&nbsp; we obtain:
 
:$${\rm Pr}({\cal{E} })  
 
:$${\rm Pr}({\cal{E} })  
 
  = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ] = {1}/{2} \cdot {\rm exp } \left [ - \frac{ E_{\rm S}/4}{N_{\rm 0}/2}\right ] \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}\hspace{0.05cm}.$$
 
  = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ] = {1}/{2} \cdot {\rm exp } \left [ - \frac{ E_{\rm S}/4}{N_{\rm 0}/2}\right ] \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}\hspace{0.05cm}.$$
<b>q.e.d.</b> &nbsp; &nbsp; &nbsp; &nbsp; $E_{\rm S}$&nbsp; gibt hierbei die mittlere Signalenergie pro Symbol an, die bei FSK gleich der Normierungsenergie&nbsp; $E$&nbsp; ist.}}<br>
+
&nbsp; &nbsp; &nbsp; &nbsp; $E_{\rm S}$&nbsp; here indicates the average signal energy per symbol, which is equal to the normalization energy&nbsp; $E$&nbsp; for FSK.
 
+
<div align="right">$\text{q.e.d.}$</div>}}<br>
[[File:P ID2088 Dig T 4 5 S3b version1.png|right|frame|FSK-Fehlerwahrscheinlichkeit bei kohärenter und nichtkohärenter Demodulation|class=fit]]
 
{{GraueBox|TEXT= 
 
$\text{Beispiel 2:}$&nbsp; Die Grafik stellt die Fehlerwahrscheinlichkeitskurven beider Demodulationsverfahren in Abhängigkeit des AWGN&ndash;Qotienten&nbsp; $E_{\rm S}/N_0$&nbsp; vergleichend gegenüber.
 
 
 
Man erkennt:
 
*Die nichtkohärente FSK (rote Kurve) benötigt gegenüber der kohärenten FSK (blaue Kurve) bei&nbsp; $p_{\rm S}= 10^{-5}$&nbsp; ein um &nbsp;$0.8 \ \rm dB$&nbsp; größeres &nbsp;$E_{\rm S}/N_0$.
 
*Bei &nbsp;$p_{\rm S}= 10^{-3}$&nbsp; beträgt der Abstand sogar&nbsp; $1.3 \ \rm dB$.<br>
 
  
*Dagegen beträgt der Abstand zwischen der kohärenten binären FSK von der kohärenten BPSK unabhängig von der Fehlerwahrscheinlichkeit stets&nbsp; $1.3 \ \rm dB$.
+
{{GraueBox|TEXT=
 +
[[File:EN_Dig_T_4_5_S3b.png|right|frame|FSK symbol error probability for coherent and non-coherent demodulation|class=fit]] 
 +
$\text{Example 2:}$&nbsp; The graph compares the error probability curves of both demodulation methods as a function of the AWGN quotient&nbsp; $E_{\rm S}/N_0$.&nbsp;
 +
<br><br><br><br>
 +
One can see:
 +
#The non-coherent BFSK&nbsp; (red curve)&nbsp; requires a &nbsp; $0.8 \ \rm dB$ &nbsp; larger&nbsp;$E_{\rm S}/N_0$&nbsp; compared to the coherent BFSK&nbsp; (blue curve)&nbsp; at&nbsp; $p_{\rm S}= 10^{-5}$.&nbsp;<br><br> 
 +
#At &nbsp; $p_{\rm S}= 10^{-3}$ &nbsp; the distance is even&nbsp; $1.3 \ \rm dB$.<br><br>
 +
#In contrast,&nbsp; the distance between the coherent binary FSK from the coherent BPSK is always&nbsp; $1.3 \ \rm dB$&nbsp; regardless of the error probability.
 
<br clear =all>}}
 
<br clear =all>}}
  
  
== Nichtkohärente Demodulation von mehrstufiger FSK==
+
== Non-coherent demodulation of multi-level FSK==
 
<br>
 
<br>
[[File:P ID2089 Dig T 4 5 S4a version1.png|right|frame|Orthogonale $M$&ndash;stufige FSK für &nbsp;$M= 3$]]
+
[[File:P ID2089 Dig T 4 5 S4a version1.png|right|frame|Orthogonal $M$&ndash;level FSK for &nbsp;$M= 3$]]
  
Wir betrachten nun die Nachrichtenmenge&nbsp; $\{m_1, m_2,\hspace{0.05cm}\text{ ...} \hspace{0.05cm}, m_{M}\}$&nbsp; und bezeichnen&nbsp; $M$&nbsp; als Stufenzahl.
+
We now consider the message set&nbsp; $\{m_1, m_2,\hspace{0.05cm}\text{ ...} \hspace{0.05cm}, m_{M}\}$&nbsp; and denote&nbsp; $M$&nbsp; as the&nbsp; "level number".
*Voraussetzung für die Anwendung des Modulationsverfahrens&nbsp; <i>"Frequency Shift Keying"</i>&nbsp; und zugleich eines nichtkohärenten Demodulators ist wie bei der binären FSK ein ganzzahliger Modulationsindex&nbsp; $h$.<br>
+
#As in the case of binary FSK,&nbsp; a prerequisite for the application of the modulation process&nbsp; "Frequency Shift Keying"&nbsp; and at the same time of a non-coherent demodulator is an integer modulation index&nbsp; $h$.<br><br>
 +
#In this case,&nbsp; the&nbsp; $M$&ndash;level FSK is orthogonal and a signal space constellation results as shown in the adjacent diagram for the special case&nbsp; $M = 3$.&nbsp; <br><br>
  
*In diesem Fall ist die&nbsp; $M$&ndash;stufige FSK orthogonal und es ergibt sich eine Signalraumkonstellation, wie in der nebenstehenden  Grafik für den Sonderfall&nbsp; $M = 3$&nbsp; dargestellt.<br><br>
+
The non-coherent demodulator is sketched below.
 +
*Compared to the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_binary_FSK_.282.E2.80.93FSK.29|"receiver structure for binary FSK"]],&nbsp; this receiver differs only by&nbsp; $M$&nbsp; branches instead of only two,&nbsp; which provide the comparison values&nbsp; $y_1$,&nbsp; $y_2$, ... , $y_M$.<br>
  
Der nichtkohärente Demodulator ist nachfolgend skizziert. Gegenüber der&nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_nichtkohärenter_Demodulation#Nichtkoh.C3.A4rente_Demodulation_von_bin.C3.A4rer_FSK_.282.E2.80.93FSK.29| Empfängerstruktur für binäre FSK]]&nbsp; unterscheidet sich dieser Empfänger lediglich durch&nbsp; $M$&nbsp; Zweige anstelle von nur zweien, welche die Vergleichswerte&nbsp; $y_1$,&nbsp; $y_2$, ... , $y_M$ liefern.<br>
+
*To calculate the error probability,&nbsp; we assume that&nbsp; $m_1$&nbsp; was sent. This means that the decision is correct if the largest detection output value is&nbsp; $y_1$:&nbsp;  
 
 
[[File:P ID2090 Dig T 4 5 S4b version1.png|center|frame|Nichtkohärente Empfängerstruktur für &nbsp;$M$&ndash;stufige FSK|class=fit]]
 
 
 
Zur Berechnung der Fehlerwahrscheinlichkeit gehen wir von der Annahme aus, dass&nbsp; $m_1$&nbsp; gesendet wurde. Das bedeutet, dass die Entscheidung richtig ist, wenn der größte Detektionsausgangswert&nbsp; $y_1$&nbsp; ist:
 
  
 
::<math>{\rm Pr}({\cal{C}}) \hspace{-0.1cm}  =  \hspace{-0.1cm} {\rm Pr} \big [ (y_2 < y_1) \cap (y_3 < y_1) \cap ... \cap (y_{M} < y_1)
 
::<math>{\rm Pr}({\cal{C}}) \hspace{-0.1cm}  =  \hspace{-0.1cm} {\rm Pr} \big [ (y_2 < y_1) \cap (y_3 < y_1) \cap ... \cap (y_{M} < y_1)
Line 239: Line 252:
 
   \hspace{0.05cm}| \hspace{0.05cm}m = m_1\right ]
 
   \hspace{0.05cm}| \hspace{0.05cm}m = m_1\right ]
 
   \hspace{0.01cm}.</math>
 
   \hspace{0.01cm}.</math>
 
+
[[File:EN_Dig_T_4_5_S4b.png|right|frame|Non-coherent receiver structure for &nbsp;$M$&ndash;level FSK|class=fit]]
 +
<br clear=all>
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Fazit:}$&nbsp;
+
$\text{Conclusions:}$&nbsp;
* Die &nbsp;'''Fehlerwahrscheinlichkeit der M&ndash;stufigen FSK bei nichtkohärenter Demodulation'''&nbsp; ist gleich ${\rm Pr}({\cal{E} }) = 1 - {\rm Pr}({\cal{C} })$.  
+
* The &nbsp; '''error probability of M&ndash;level FSK in non-coherent demodulation''' &nbsp; is equal to&nbsp; ${\rm Pr}({\cal{E} }) = 1 - {\rm Pr}({\cal{C} })$.
*Nachfolgend wird gezeigt, dass ${\rm Pr}({\cal{C} })$ wie folgt dargestellt werden kann:
+
 +
*In the following,&nbsp; it is shown that&nbsp; ${\rm Pr}({\cal{C} })$&nbsp; can be represented as follows:
  
 
::<math>{\rm Pr}({\cal{C} }) = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot \frac{1}{i+1} \cdot {\rm exp } \left [ - \frac{i \cdot E_{\rm S} }
 
::<math>{\rm Pr}({\cal{C} }) = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot \frac{1}{i+1} \cdot {\rm exp } \left [ - \frac{i \cdot E_{\rm S} }
 
   {(i+1) \cdot N_0}\right ]  
 
   {(i+1) \cdot N_0}\right ]  
   \hspace{0.05cm}\hspace{0.5cm} \text{mit}\hspace{0.5cm}  E_{\rm S} = E_{\rm B} \cdot {\rm log_2}(M)\hspace{0.05cm}.</math>
+
   \hspace{0.05cm}\hspace{0.5cm} \text{with}\hspace{0.5cm}  E_{\rm S} = E_{\rm B} \cdot {\rm log_2}(M)\hspace{0.05cm}.</math>
  
*Im Sonderfall&nbsp; $M = 2$&nbsp; ergibt sich natürlich wieder das im&nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_kohärenter_Demodulation|letzten Abschnitt]]&nbsp; erhaltene Ergebnis:
+
*In the special case&nbsp; $M = 2$,&nbsp; of course, the result obtained in the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation|"last section"]]&nbsp; is again obtained:
  
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}
 
::<math>p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}
Line 256: Line 271:
  
 
{{BlaueBox|TEXT=   
 
{{BlaueBox|TEXT=   
$\text{Herleitung:}$&nbsp; Das vorweg genommene Ergebnis soll nun in einigen Rechenschritten hergeleitet werden. Wir gehen von der Annahme aus, dass&nbsp; $m_1$&nbsp; gesendet wurde. Das bedeutet, dass die Entscheidung richtig ist, wenn der größte Detektionsausgangswert&nbsp; $y_1$&nbsp; ist. Ansonsten gibt es gewisse Analogien zur Herleitung der&nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_nichtkohärenter_Demodulation#Fehlerwahrscheinlichkeit_bei_nichtkoh.C3.A4renter_2.E2.80.93FSK.E2.80.93Demodulation|BFSK&ndash;Fehlerwahrscheinlichkeit]].<br>
+
$\text{Derivation:}$&nbsp; This result shall now be derived in some calculation steps.  
 +
*We assume that&nbsp; $m_1$&nbsp; has been sent.&nbsp; This means that the decision is correct if the largest detection output value is&nbsp; $y_1$.&nbsp;  
  
<b>(1)</b> &nbsp; Mit der bedingten Wahrscheinlichkeitsdichte&nbsp; $p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_1)$&nbsp; erhält man:
+
*Otherwise,&nbsp; there are certain analogies to the derivation of the&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Error_probability_with_non-coherent_2.E2.80.93FSK_demodulation|"BFSK error probability"]].<br>
 +
 
 +
 
 +
<b>(1)</b> &nbsp; Using the conditional probability density&nbsp; $p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_1)$,&nbsp; we obtain:
  
 
::<math>{\rm Pr}({\cal{C} }) =  \int_{0}^{\infty} {\rm Pr} \left [ \hspace{0.1cm} \bigcap\limits_{k = 2}^M (y_k < y_1)   
 
::<math>{\rm Pr}({\cal{C} }) =  \int_{0}^{\infty} {\rm Pr} \left [ \hspace{0.1cm} \bigcap\limits_{k = 2}^M (y_k < y_1)   
Line 264: Line 283:
 
   \hspace{0.05cm}.</math>
 
   \hspace{0.05cm}.</math>
  
<b>(2)</b> &nbsp; Die Entscheidungswerte&nbsp; $y_2$,&nbsp; $y_3$,&nbsp; ... , $y_M$&nbsp; sind bei gegebenem&nbsp; $y_1$&nbsp; statistisch unabhängig. Deshalb gilt:
+
<b>(2)</b> &nbsp; The decision values&nbsp; $y_2$,&nbsp; $y_3$,&nbsp; ... , $y_M$&nbsp; are statistically independent for given&nbsp; $y_1$.&nbsp; Therefore:
  
 
::<math>{\rm Pr}({\cal{C} }) =  \int_{0}^{\infty}  \left \{ {\rm Pr} \big [ (y_2 < y_1)   
 
::<math>{\rm Pr}({\cal{C} }) =  \int_{0}^{\infty}  \left \{ {\rm Pr} \big [ (y_2 < y_1)   
Line 270: Line 289:
 
   \hspace{0.05cm}.</math>
 
   \hspace{0.05cm}.</math>
  
<b>(3)</b> &nbsp; Der ausgewählte Wert&nbsp; $y_2$&nbsp; konditioniert auf&nbsp; $m_1$&nbsp; besitzt eine Rayleighverteilung mit Parameter&nbsp; $\sigma_n^2$:
+
<b>(3)</b> &nbsp; The selected value&nbsp; $y_2$&nbsp; conditioned on&nbsp; $m_1$&nbsp; has a Rayleigh distribution with parameter&nbsp; $\sigma_n^2$:
  
 
::<math>{\rm Pr} \big [ (y_2 < y_1)   
 
::<math>{\rm Pr} \big [ (y_2 < y_1)   
   \hspace{0.05cm} \vert \hspace{0.05cm}y_1 = \eta_1, m = m_1\big ] \hspace{-0.1cm} = \hspace{-0.1cm} \int_{0}^{\eta_1}  p_{y_2 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm} \vert \hspace{0.05cm}m_1)\,\,{\rm d} \eta_2=  1 - {\rm exp } \big [ - {\eta_1^2 }/({2 \sigma_n^2})\big ] = 1 - a  \hspace{0.2cm}{\rm(Abk\ddot{u}rzung)}
+
   \hspace{0.05cm} \vert \hspace{0.05cm}y_1 = \eta_1, m = m_1\big ] \hspace{-0.1cm} = \hspace{-0.1cm} \int_{0}^{\eta_1}  p_{y_2 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm} \vert \hspace{0.05cm}m_1)\,\,{\rm d} \eta_2=  1 - {\rm exp } \big [ - {\eta_1^2 }/({2 \sigma_n^2})\big ] = 1 - a  \hspace{0.2cm}{\rm(abbreviation)}
 
   \hspace{0.05cm}.</math>
 
   \hspace{0.05cm}.</math>
  
<b>(4)</b> &nbsp; Gesucht ist nun der Ausdruck&nbsp; $(1 -a)^{M-1}$, für den mit der Abkürzung aus <b>(3)</b> gilt:
+
<b>(4)</b> &nbsp; Now we are looking for the expression &nbsp; $(1 -a)^{M-1}$,&nbsp; for which with the abbreviation from&nbsp; <b>(3)</b>&nbsp; holds:
  
 
::<math> (1-a)^{M-1} \hspace{-0.1cm}  =  \hspace{-0.1cm} \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot (-1)^i  \cdot a^i =  \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot {\rm exp } \left [ - \frac{i \cdot \eta_1^2 }{2 \sigma_n^2}\right ]
 
::<math> (1-a)^{M-1} \hspace{-0.1cm}  =  \hspace{-0.1cm} \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot (-1)^i  \cdot a^i =  \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot {\rm exp } \left [ - \frac{i \cdot \eta_1^2 }{2 \sigma_n^2}\right ]
 
   \hspace{0.05cm}.</math>
 
   \hspace{0.05cm}.</math>
  
<b>(5)</b> &nbsp; $y_1$&nbsp; besitzt konditioniert auf&nbsp; $m=m_1$&nbsp; eine&nbsp; [[Digital_Signal_Transmission/Trägerfrequenzsysteme_mit_nichtkohärenter_Demodulation#Rayleigh.E2.80.93_und_Riceverteilung|Riceverteilung]]. Die Wahrscheinlichkeit für eine korrekte Entscheidung lässt sich somit in folgende Form bringen:
+
<b>(5)</b> &nbsp; $y_1$&nbsp; has a&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution|"Rice distribution"]] conditioned on&nbsp; $m=m_1$.&nbsp; Thus,&nbsp; the probability of a correct decision can be expressed in the following form:
  
 
::<math>{\rm Pr}({\cal{C} }) \hspace{-0.1cm}  =  \hspace{-0.1cm} \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot \int_{0}^{\infty} {\rm exp } \left [ - \frac{i \cdot \eta_1^2 }{2 \sigma_n^2}\right ] \cdot \frac{\eta_1}{ \sigma_n^2}\cdot   
 
::<math>{\rm Pr}({\cal{C} }) \hspace{-0.1cm}  =  \hspace{-0.1cm} \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot \int_{0}^{\infty} {\rm exp } \left [ - \frac{i \cdot \eta_1^2 }{2 \sigma_n^2}\right ] \cdot \frac{\eta_1}{ \sigma_n^2}\cdot   
Line 288: Line 307:
 
   \hspace{0.05cm}.</math>
 
   \hspace{0.05cm}.</math>
  
<b>(6)</b> &nbsp; Durch Substitutionen gelingt es, den Integranden gemäß der Riceverteilung zu gestalten. Da sich jede Wahrscheinlichkeitsdichte zu Eins integriert, erhält man:
+
<b>(6)</b> &nbsp; Substitutions succeed in shaping the integrand according to the Rice distribution.&nbsp; Since each probability density integrates to one,&nbsp; we obtain:
  
 
::<math>{\rm Pr}({\cal{C} }) = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot \frac{1}{i+1} \cdot {\rm exp } \left [ - \frac{i \cdot E_{\rm S} }
 
::<math>{\rm Pr}({\cal{C} }) = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i }  \cdot \frac{1}{i+1} \cdot {\rm exp } \left [ - \frac{i \cdot E_{\rm S} }
 
   {(i+1) \cdot N_0}\right ]  
 
   {(i+1) \cdot N_0}\right ]  
   \hspace{0.05cm}\hspace{0.5cm} \text{mit}\hspace{0.5cm}  E_{\rm S} = E_{\rm B} \cdot {\rm log_2}(M)\hspace{0.05cm}.</math>
+
   \hspace{0.05cm}\hspace{0.5cm} \text{with}\hspace{0.5cm}  E_{\rm S} = E_{\rm B} \cdot {\rm log_2}(M)\hspace{0.05cm}.</math>
  
<b>(7)</b> &nbsp; Der Sonderfall&nbsp; $M = 2$&nbsp; führt zum genau gleichen Ergebnis, wie für die binäre FSK berechnet:
+
<b>(7)</b> &nbsp; The special case&nbsp; $M = 2$&nbsp; leads to exactly the same result as calculated for the binary FSK:
  
 
::<math>{\rm Pr}({\cal{C} }) = (-1)^0 \cdot {2-1 \choose 0 }  \cdot \frac{1}{0+1} \cdot {\rm exp } \left [ - \frac{0 \cdot E_{\rm S} }
 
::<math>{\rm Pr}({\cal{C} }) = (-1)^0 \cdot {2-1 \choose 0 }  \cdot \frac{1}{0+1} \cdot {\rm exp } \left [ - \frac{0 \cdot E_{\rm S} }
Line 301: Line 320:
 
::<math>  \Rightarrow \hspace{0.5cm} {\rm Pr}({\cal{C} }) = 1 - {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)}
 
::<math>  \Rightarrow \hspace{0.5cm} {\rm Pr}({\cal{C} }) = 1 - {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)}
 
  \hspace{0.5cm} \Rightarrow \hspace{0.5cm} {\rm Pr}({\cal{E} }) = 1 - {\rm Pr}({\cal{C} })  = {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)}
 
  \hspace{0.5cm} \Rightarrow \hspace{0.5cm} {\rm Pr}({\cal{E} }) = 1 - {\rm Pr}({\cal{C} })  = {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)}
   \hspace{0.05cm}.</math>}}
+
   \hspace{0.05cm}.</math>
 +
<div align="right">$\text{q.e.d.}$</div>}}
  
== Aufgaben zum Kapitel==
+
== Exercises for the chapter==
 
<br>
 
<br>
[[Aufgaben:4.17_Nichtkohärentes_On-Off-Keying|Aufgabe 4.17: Nichtkohärentes On-Off-Keying]]
+
[[Aufgaben:Exercise_4.17:_Non-Coherent_On-Off_Keying|Exercise 4.17: Non-Coherent On-Off Keying]]
  
[[Aufgaben:4.17Z_Rayleigh-_und_Riceverteilung|Aufgabe 4.17Z: Rayleigh- und Riceverteilung]]
+
[[Aufgaben:Exercise_4.17Z:_Rayleigh_and_Rice_Distribution|Exercise 4.17Z: Rayleigh and Rice Distribution]]
  
[[Aufgaben:4.18_Nichtkohärente_FSK–Demodulation|Aufgabe 4.18: Nichtkohärente_FSK–Demodulation]]
+
[[Aufgaben:Exercise_4.18:_Non-Coherent_FSK_Demodulation|Exercise 4.18: Non-Coherent FSK Demodulation]]
  
[[Aufgaben:4.18Z_BER_von_kohärenter_und_nichtkohärenter_FSK|Aufgabe 4.18Z: BER von kohärenter und nichtkohärenter FSK]]
+
[[Aufgaben:Exercise_4.18Z:_BER_of_Coherent_and_Non-Coherent_FSK|Exercise 4.18Z: BER of Coherent and Non-Coherent FSK]]
  
[[Aufgaben:4.19_Orthogonale_mehrstufige_FSK|Aufgabe 4.19: Orthogonale mehrstufige FSK]]
+
[[Aufgaben:Exercise_4.19:_Orthogonal_Multilevel_FSK|Exercise 4.19: Orthogonal Multilevel FSK]]
  
 
{{Display}}
 
{{Display}}

Latest revision as of 13:43, 10 October 2022

Rayleigh and Rice Distribution


The estimation of the phase angle from the incoming signal,  which is required for coherent demodulation,  is not possible or only possible to a limited extent in many applications.  For example,  the movement of a mobile subscriber at high speed leads to very rapid temporal changes in the phase angle  $\phi$,  which makes its sufficiently accurate determination difficult or even impossible.

This fact leads to the  non-coherent demodulation processes  with the advantage of reduced complexity,  but with increased error probability.  In the derivation of the equations one encounters two probability density functions,  which are given here in advance:

  • The  Rayleigh distribution  is obtained for the power density function  $\rm (PDF)$  of the random variable  $y$  with realization  $\eta$,  which is obtained from the two Gaussian distributed and statistically independent components  $u$  and  $v$  $($both with the same standard deviation  $\sigma_n)$  as follows:
\[y = \sqrt{u^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) ={\eta}/{\sigma_n^2} \cdot {\rm exp } \left [ - {\eta^2}/{ (2\sigma_n^2)}\right ] \hspace{0.05cm}.\]
  • The  Rice distribution  is obtained under the same boundary conditions for the case where a constant   $C$  is added to one of the components $($either to $u$  or  $v)$: 
\[y = \sqrt{(u+C)^2 + v^2} \hspace{0.3cm} \Rightarrow \hspace{0.3cm} p_y (\eta) = {\eta}/{\sigma_n^2} \cdot {\rm exp } \left [ - ({\eta^2 + C^2})/(2 \sigma_n^2) \right ] \cdot {\rm I }_0 \left [{\eta \cdot C}/{ \sigma_n^2}\right ] \hspace{0.05cm}.\]
The Rice distribution uses the  "modified zero-order Bessel function",  whose definition and series expansion are as follows:
\[{\rm I }_0 (x) = \frac{1}{ 2\pi} \cdot \int_{0}^{2\pi} {\rm e }^{-x \hspace{0.03cm}\cdot \hspace{0.03cm}\cos(\alpha)} \,{\rm d} \alpha \hspace{0.2cm} \approx \hspace{0.2cm} \sum_{k = 0}^{\infty} \frac{(x/2)^{2k}}{k! \cdot \Gamma (k+1)} \hspace{0.05cm}.\]

The graph shows the probability density functions of Rayleigh and Rice distributions.  To this representation it is to be noted:

Rayleigh resp. Rice PDF for $σ_n = 0.5$
  • The Rice distribution is determined by the two parameters  $C$  and  $\sigma_n$.  With  $C = 0$,  the Rice PDF is identical to the Rayleigh PDF.
  • The Rayleigh PDF with larger  $\sigma_n$  is of the same shape as shown for  $\sigma_n = 0.5$,  but wider and lower in the ratio of standard deviation.
  • $\sigma_n$  indicates the standard deviation of the two Gaussian distributed random variables  $u$  and  $v$  and not the standard deviation of the Rayleigh distributed random variable  $y$.  Rather,  the following applies to the latter:
\[\sigma_y = \sigma_n \cdot \sqrt{2 - {\pi}/{2 }} \hspace{0.2cm} \approx \hspace{0.2cm} 0.655 \cdot \sigma_n \hspace{0.05cm}.\]
  • The Rayleigh PDF is extremely asymmetric,  recognizable by the large value for the  "third order central moment"   ⇒   "Charlier's skewness":
$$\mu_3/\sigma_y \approx 0.27.$$
  • The Rice PDF is more symmetrical the larger the quotient  $C/\sigma_n$.  For  $C/\sigma_n \ge 4$:  $\mu_3 \approx 0$   ⇒   symmetrical PDF.
  • The larger  $C/\sigma_n$  is,  the more the Rice PDF $($with  $C$,  $\sigma_n)$  approaches a  "Gaussian PDF"  with mean  $C$  and standard deviation  $\sigma_n$:
$$p_y (\eta) \approx \frac{1}{\sqrt{2\pi} \cdot \sigma_n} \cdot {\rm exp } \left [ - \frac{(\eta - C)^2}{2 \sigma_n^2}\right ]$$
$$ \Rightarrow \hspace{0.3cm} m_y = C\hspace{0.05cm},\hspace{0.2cm}\sigma_y = \sigma_n \hspace{0.05cm}.$$

Non-coherent demodulation of  "on–off keying"  (OOK)


We consider  "on–off keying"  $(\rm OOK$  or  $\rm 2–ASK)$  in the equivalent low-pass range.

Coherent and non-coherent demodulation of  "on-off keying"
  • In the case of coherent demodulation  (left graph),  the signal space constellation of the received signal  $\boldsymbol{r}(t)$  is the same as that of the transmitted signal  $\boldsymbol{s}(t)$  and consists of two points.
  • The decision boundary  $G$  lies in the middle between these points  $\boldsymbol{r}_0$  and  $\boldsymbol{r}_1$.
  • The arrows mark the rough direction of noise vectors that may cause transmission errors.


On the other hand, for non-coherent demodulation (right graph):

  • The point  $\boldsymbol{r}_1 = \boldsymbol{s}_1 = 0$  is still preserved.  But  $\boldsymbol{r}_0 = \boldsymbol{s}_0 \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi}$  can lie on any point of the circle around  $\boldsymbol{s}_0$,  since  $\phi$  is unknown.
  • The decision process in consideration of the AWGN noise happens in two dimensions  $($indicated by arrows in the right graph$)$.
  • The decision area  $I_1$  is a circle whose radius  $G$  is an optimizable parameter. The decision area  $I_0$  lies outside the circle.


Thus,  the structure of the optimal OOK receiver  (in the equivalent low–pass range)  is fixed   ⇒   see second graph:

Receiver for non-coherent OOK demodulation  (complex signals are labeled blue)
  • The input signal  $\boldsymbol{r}(t) = \boldsymbol{s}(t) \cdot {\rm e}^{\hspace{0.02cm}{\rm j}\hspace{0.03cm}\phi} + \boldsymbol{n}(t)$  is generally complex because of the phase angle  $\phi$  and because of the complex noise term  $\boldsymbol{n}(t)$. 
  • Consequently required is now the correlation between the complex received signal  $\boldsymbol{r}(t)$  and a complex basis function  $\boldsymbol{\xi}_1(t)$.
  • The result is the  (complex)  detection value  $\boldsymbol{r}$,  from which the magnitude  $y = |\boldsymbol{r}(t)|$  is formed as a real decision input variable.
  • If the decision value  $y \gt G$,  then  $m_0$  is output as the estimated value,  otherwise  $m_1$.
  • Thus,  the symbol error probability for equally probable symbols
    ⇒   ${\rm Pr}(\boldsymbol{m}_0) = {\rm Pr}(\boldsymbol{m}_1) =1/2$  is:
\[p_{\rm S} = {\rm Pr}({\cal{E}}) = {1}/{ 2} \cdot \int_{0}^{G} p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (\eta \hspace{0.05cm}|\hspace{0.05cm} m_0) \,{\rm d} \eta + {1}/{ 2} \cdot \int_{G}^{\infty} p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (\eta \hspace{0.05cm}\vert \hspace{0.05cm} m_1) \,{\rm d} \eta \hspace{0.05cm}.\]
  • However,  due to the Rice PDF   $p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (\eta\hspace{0.05cm}|\hspace{0.05cm}m_0)$   and the Rayleigh PDF   $p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (\eta\hspace{0.05cm}|\hspace{0.05cm}m_1)$,   this probability can only be determined numerically.
  • The optimal decision limit  $G$  has to be determined beforehand as the solution of the following equation:
\[p_{y\hspace{0.05cm}\vert \hspace{0.05cm}m} (G \hspace{0.05cm}|\hspace{0.05cm}m_0) = p_{y\hspace{0.05cm}|\hspace{0.05cm}m} (G \hspace{0.05cm}|\hspace{0.05cm}m_1) \hspace{0.05cm}.\]
Density functions for "OOK, non-coherent"

$\text{Example 1:}$  The graph shows the result of this equation for  $\sigma_n = 0.5$  and  $C = 2$, where the (red) Rice PDF is approximated by a Gaussian PDF with mean  $C$  and standard deviation  $\sigma_n$. 

One can see from this sketch:

  • The optimal decision value  $($here:   $G \approx 1.25)$  results from the intersection of the two PDF curves.
  • The symbol error probability  $p_{\rm S}$  is the sum of the two colored areas.  In the example:  $p_{\rm S} \approx 5\%$ results.


⇒   You can determine the error probability for other values of  $C$  and  $\sigma_n$  as well as the optimal decision value  $G$  using the  HTML5/JavaScript applet  "Coherent and Non-coherent On-Off Keying"

Non-coherent demodulation of binary FSK (2–FSK)


As already shown in the  "last chapter",  "Binary Frequency Shift Keying"  $\rm (2–FSK)$  in the equivalent low-pass range can be described by the basis functions

\[\xi_1(t) = \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}+{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T\hspace{0.05cm},\]
\[ \xi_2(t) = \sqrt{1/T} \cdot {\rm e}^{\hspace{0.05cm}-{\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm} \pi \hspace{0.03cm}\cdot \hspace{0.03cm} h \hspace{0.03cm}\cdot \hspace{0.03cm}t/T}\hspace{0.05cm},\hspace{0.2cm} 0 \le t \le T \hspace{0.05cm}\]

To achieve orthogonality between these two complex basis functions,  the  "modulation index"  $h$  must be chosen to be integer:

\[< \hspace{-0.05cm}\xi_1(t) \hspace{0.1cm} \cdot \hspace{0.1cm} \xi_2(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} h = 2 \cdot \Delta f_{\rm A} \cdot T\hspace{0.05cm}= 1, 2, 3, \text{...}\]

The diagram shows the structure for non-coherent orthogonal demodulation of binary FSK.

Non-coherent demodulation of the binary FSK

In the noise-free case   ⇒   $n(t) \equiv 0$  applies to the outputs of the two correlators:

\[r_1 = \hspace{0.2cm} < \hspace{-0.05cm}r(t) \hspace{0.1cm} \cdot \hspace{0.1cm} \xi_1(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0\hspace{0.05cm}, \hspace{0.4cm} {\rm if}\hspace{0.15cm} m = m_1\hspace{0.05cm},\]
\[ r_2 = \hspace{0.2cm} < \hspace{-0.05cm}r(t) \hspace{0.1cm} \cdot \hspace{0.1cm} \xi_2(t) \hspace{-0.05cm}> \hspace{0.2cm}= 0\hspace{0.05cm}, \hspace{0.4cm} {\rm if}\hspace{0.15cm} m = m_0\hspace{0.05cm}.\]

After respective magnitude formation   ⇒   $y_1 = |r_1|, \ \ y_2 = |r_2|$  the following decision rule is applicable:

\[\hat{m} = \left\{ \begin{array}{c} m_0 \\ m_1 \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm} y_1 > y_2 \hspace{0.05cm}, \\ {\rm if}\hspace{0.15cm} y_1 < y_2 \hspace{0.05cm}.\\ \end{array}\]

For a simpler realization of the decision,  the difference  $y_1 - y_2$  can also be evaluated with the decision boundary  $G = 0$. 

Error probability with non-coherent 2–FSK demodulation


In the following,  the error probability is calculated under the assumption that  $m = m_0$  was sent.

  • Under the further assumption of equally probable binary messages  $m_0$  and  $m_1$,  the absolute error probability is exactly the same:
$${\rm Pr}({\cal{E}}) = {\rm Pr}({\cal{E}}\hspace{0.05cm} | \hspace{0.05cm}m_0) \hspace{0.05cm}.$$
  • With  $m = m_0$  we get for the complex correlation output values  $r_i$  and their magnitudes  $y_i$:
\[r_1 = \sqrt{E} \cdot {\rm e}^{{\rm j}\phi} + n_1\hspace{0.3cm} \Rightarrow \hspace{0.3cm}y_1 = |r_1|\hspace{0.15cm}{\rm is}\hspace{0.15cm}{\rm Rice}\hspace{0.15cm}{\rm distributed} \hspace{0.05cm},\]
\[ r_2 = n_2\hspace{0.3cm} \Rightarrow \hspace{0.3cm}y_2 = |r_2|\hspace{0.15cm}{\rm is}\hspace{0.15cm}{\rm Rayleigh}\hspace{0.15cm}{\rm distributed} \hspace{0.05cm}.\]
Here,  $E$  due to  $M = 2$  represents the  "average symbol energy"  $(E_{\rm S})$  and the  "average bit energy"  $(E_{\rm B})$  equally.  $n_1$  and  $n_2$  are uncorrelated complex noise variables with mean zero and variance  $2 \cdot \sigma_n^2$.
\[p_{y_1,\hspace{0.03cm} y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_1, \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) = p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) \cdot p_{y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) \hspace{0.05cm},\]
$$p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) = {\eta_1}/{\sigma_n^2} \cdot {\rm e }^{ - ({\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [{\eta_1 \cdot \sqrt{E}}/{ \sigma_n^2}\right ] \hspace{0.05cm},$$
$$ p_{y_2 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) = {\eta_2}/{\sigma_n^2} \cdot {\rm e }^{ - \eta_2^2 /({2 \sigma_n^2}) } \hspace{0.05cm}.$$
  • The error probability is generally obtained as follows:
\[{\rm Pr}({\cal{E}}) = \int_{0}^{\infty} \int_{\eta_1}^{\infty} p_{y_1,\hspace{0.03cm} y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_1, \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0) \,\,{\rm d} \eta_2\,\,{\rm d} \eta_1 = \int_{0}^{\infty} p_{y_1 \hspace{0.01cm}| \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}| \hspace{0.05cm}m_0) \cdot \int_{\eta_1}^{\infty} p_{y_2 \hspace{0.03cm}| \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}| \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2\,\,{\rm d} \eta_1 \hspace{0.05cm}.\]

$\text{Conclusion:}$  For the  error probability with non-coherent demodulation of the binary FSK,  one obtains after some mathematical transformations the surprisingly simple result

\[p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)} \hspace{0.05cm}.\]
  • For comparison,  the result for  coherent demodulation  is here given again:
\[p_{\rm S} = {\rm Pr}({\cal{E} }) = {\rm Q}(\sqrt{ E_{\rm S}/N_0}) \hspace{0.05cm}.\]


$\text{Derivation:}$  This result is now to be derived in some calculation steps.  We start from the following equations:

$${\rm Pr}({\cal{E} }) = \int_{0}^{\infty} p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_0) \cdot \int_{\eta_1}^{\infty} p_{y_2 \hspace{0.03cm}\vert \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2\,\,{\rm d} \eta_1 \hspace{0.05cm},\hspace{0.5cm}\text{with}$$
$$p_{y_1 \hspace{0.01cm}\vert \hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm}\vert \hspace{0.05cm}m_0) = {\eta_1}/{\sigma_n^2} \cdot {\rm e }^{ - ({\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [{\eta_1 \cdot \sqrt{E} }/{ \sigma_n^2}\right ] \hspace{0.05cm},$$
$$p_{y_\hspace{0.01cm}\vert \hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0) = {\eta_2}/{\sigma_n^2} \cdot {\rm e }^{ - \eta_2^2 /({2 \sigma_n^2}) } \hspace{0.05cm}.$$

(1)   The inner integral gives the probability that the Rayleigh distributed random variable  $\eta_2$  is larger than  $\eta_1$  – see solution to  "Exercise 4.17Z":

$$\int_{\eta_1}^{\infty} p_{y_2 \hspace{0.03cm}\vert\hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm}\vert \hspace{0.05cm}m_0)\,\,{\rm d} \eta_2 = {\rm e }^{ - \eta_1^2 /({2 \sigma_n^2}) } \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {\rm Pr}({\cal{E} }) = \int_{0}^{\infty}{\eta_1}/{\sigma_n^2} \cdot {\rm e }^{ - ({2\eta_1^2 + E})/({2 \sigma_n^2}) }\cdot {\rm I }_0 \left [ {\eta_1 \cdot \sqrt{E} }/{ \sigma_n^2}\right ]\,\,{\rm d} \eta_1 \hspace{0.05cm}.$$

(2)   With the  (arbitrary)  substitutions  $C_0^2 = E/4$  and  $\sigma_0^2 = \sigma_n^2/2$,  we obtain:

$${\rm Pr}({\cal{E} }) = \int_{0}^{\infty} \frac{\eta_1}{2 \cdot \sigma_0^2} \cdot {\rm exp } \left [ - \frac{2 \eta_1^2 + 4 C_0^2}{4 \sigma_0^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta_1 \cdot 2C_0}{ 2 \sigma_0^2}\right ]\,\,{\rm d} \eta_1 = \int_{0}^{\infty} \frac{\eta_1}{2 \cdot \sigma_0^2} \cdot {\rm exp } \left [ - \frac{\eta_1^2 + 2 C_0^2}{2 \sigma_0^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta_1 \cdot C_0}{ \sigma_0^2}\right ]\,\,{\rm d} \eta_1 \hspace{0.05cm}.$$

(3)   By shifting fractions in front of the integral,  we succeed that the integrand again describes a  "Rice distribution"

\[{\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ] \cdot \int\limits_{0}^{\infty} \frac{\eta_1}{ \sigma_0^2} \cdot {\rm exp } \left [ - \frac{\eta_1^2 + C_0^2}{2 \sigma_0^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta_1 \cdot C_0}{ \sigma_0^2}\right ]\,\,{\rm d} \eta_1 \hspace{0.05cm}.\]

(4)   The integrand now describes the Rice PDF.  The integral over the entire definition area from  $0$  to  $+\infty$  yields the value one,  as for any PDF,  so that holds:

\[{\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ]\hspace{0.05cm}.\]

(5)   Finally, with  $C_0^2 = E/4$  and  $\sigma_0^2 = \sigma_n^2/2$  and the generally valid relation  $\sigma_n^2 = N_0$,  we obtain:

$${\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm exp } \left [ - \frac{ C_0^2}{2 \sigma_0^2}\right ] = {1}/{2} \cdot {\rm exp } \left [ - \frac{ E_{\rm S}/4}{N_{\rm 0}/2}\right ] \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)}\hspace{0.05cm}.$$

        $E_{\rm S}$  here indicates the average signal energy per symbol, which is equal to the normalization energy  $E$  for FSK.

$\text{q.e.d.}$


FSK symbol error probability for coherent and non-coherent demodulation

$\text{Example 2:}$  The graph compares the error probability curves of both demodulation methods as a function of the AWGN quotient  $E_{\rm S}/N_0$. 



One can see:

  1. The non-coherent BFSK  (red curve)  requires a   $0.8 \ \rm dB$   larger $E_{\rm S}/N_0$  compared to the coherent BFSK  (blue curve)  at  $p_{\rm S}= 10^{-5}$. 

  2. At   $p_{\rm S}= 10^{-3}$   the distance is even  $1.3 \ \rm dB$.

  3. In contrast,  the distance between the coherent binary FSK from the coherent BPSK is always  $1.3 \ \rm dB$  regardless of the error probability.



Non-coherent demodulation of multi-level FSK


Orthogonal $M$–level FSK for  $M= 3$

We now consider the message set  $\{m_1, m_2,\hspace{0.05cm}\text{ ...} \hspace{0.05cm}, m_{M}\}$  and denote  $M$  as the  "level number".

  1. As in the case of binary FSK,  a prerequisite for the application of the modulation process  "Frequency Shift Keying"  and at the same time of a non-coherent demodulator is an integer modulation index  $h$.

  2. In this case,  the  $M$–level FSK is orthogonal and a signal space constellation results as shown in the adjacent diagram for the special case  $M = 3$. 

The non-coherent demodulator is sketched below.

  • Compared to the  "receiver structure for binary FSK",  this receiver differs only by  $M$  branches instead of only two,  which provide the comparison values  $y_1$,  $y_2$, ... , $y_M$.
  • To calculate the error probability,  we assume that  $m_1$  was sent. This means that the decision is correct if the largest detection output value is  $y_1$: 
\[{\rm Pr}({\cal{C}}) \hspace{-0.1cm} = \hspace{-0.1cm} {\rm Pr} \big [ (y_2 < y_1) \cap (y_3 < y_1) \cap ... \cap (y_{M} < y_1) \hspace{0.05cm}| \hspace{0.05cm} m = m_1\big ] = {\rm Pr} \left [ \hspace{0.1cm} \bigcap\limits_{k = 2}^M (y_k < y_1) \hspace{0.05cm}| \hspace{0.05cm}m = m_1\right ] \hspace{0.01cm}.\]
Non-coherent receiver structure for  $M$–level FSK


$\text{Conclusions:}$ 

  • The   error probability of M–level FSK in non-coherent demodulation   is equal to  ${\rm Pr}({\cal{E} }) = 1 - {\rm Pr}({\cal{C} })$.
  • In the following,  it is shown that  ${\rm Pr}({\cal{C} })$  can be represented as follows:
\[{\rm Pr}({\cal{C} }) = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot \frac{1}{i+1} \cdot {\rm exp } \left [ - \frac{i \cdot E_{\rm S} } {(i+1) \cdot N_0}\right ] \hspace{0.05cm}\hspace{0.5cm} \text{with}\hspace{0.5cm} E_{\rm S} = E_{\rm B} \cdot {\rm log_2}(M)\hspace{0.05cm}.\]
  • In the special case  $M = 2$,  of course, the result obtained in the  "last section"  is again obtained:
\[p_{\rm S} = {\rm Pr}({\cal{E} }) = {1}/{2} \cdot {\rm e}^{-E_{\rm S}/(2 N_0)} \hspace{0.05cm}.\]


$\text{Derivation:}$  This result shall now be derived in some calculation steps.

  • We assume that  $m_1$  has been sent.  This means that the decision is correct if the largest detection output value is  $y_1$. 


(1)   Using the conditional probability density  $p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_1)$,  we obtain:

\[{\rm Pr}({\cal{C} }) = \int_{0}^{\infty} {\rm Pr} \left [ \hspace{0.1cm} \bigcap\limits_{k = 2}^M (y_k < y_1) \hspace{0.05cm}\vert\hspace{0.05cm}y_1 = \eta_1, m = m_1\right ] \cdot p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_1) \,\,{\rm d} \eta_1 \hspace{0.05cm}.\]

(2)   The decision values  $y_2$,  $y_3$,  ... , $y_M$  are statistically independent for given  $y_1$.  Therefore:

\[{\rm Pr}({\cal{C} }) = \int_{0}^{\infty} \left \{ {\rm Pr} \big [ (y_2 < y_1) \hspace{0.05cm}\vert \hspace{0.05cm}y_1 = \eta_1, m = m_1\big ] \right \}^{M-1} \cdot p_{y_1 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_1 \hspace{0.05cm} \vert \hspace{0.05cm}m_1) \,\,{\rm d} \eta_1 \hspace{0.05cm}.\]

(3)   The selected value  $y_2$  conditioned on  $m_1$  has a Rayleigh distribution with parameter  $\sigma_n^2$:

\[{\rm Pr} \big [ (y_2 < y_1) \hspace{0.05cm} \vert \hspace{0.05cm}y_1 = \eta_1, m = m_1\big ] \hspace{-0.1cm} = \hspace{-0.1cm} \int_{0}^{\eta_1} p_{y_2 \hspace{0.01cm} \vert\hspace{0.03cm}m} ( \eta_2 \hspace{0.05cm} \vert \hspace{0.05cm}m_1)\,\,{\rm d} \eta_2= 1 - {\rm exp } \big [ - {\eta_1^2 }/({2 \sigma_n^2})\big ] = 1 - a \hspace{0.2cm}{\rm(abbreviation)} \hspace{0.05cm}.\]

(4)   Now we are looking for the expression   $(1 -a)^{M-1}$,  for which with the abbreviation from  (3)  holds:

\[ (1-a)^{M-1} \hspace{-0.1cm} = \hspace{-0.1cm} \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot (-1)^i \cdot a^i = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot {\rm exp } \left [ - \frac{i \cdot \eta_1^2 }{2 \sigma_n^2}\right ] \hspace{0.05cm}.\]

(5)   $y_1$  has a  "Rice distribution" conditioned on  $m=m_1$.  Thus,  the probability of a correct decision can be expressed in the following form:

\[{\rm Pr}({\cal{C} }) \hspace{-0.1cm} = \hspace{-0.1cm} \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot \int_{0}^{\infty} {\rm exp } \left [ - \frac{i \cdot \eta_1^2 }{2 \sigma_n^2}\right ] \cdot \frac{\eta_1}{ \sigma_n^2}\cdot {\rm exp } \left [ - \frac{\eta_1^2 + E_{\rm S} }{2 \sigma_n^2}\right ] \cdot {\rm I }_0 \left [ \frac{\eta_1 \cdot \sqrt{E_{\rm S} } }{ \sigma_n^2}\right ] \,\,{\rm d} \eta_1 \hspace{0.05cm}.\]

(6)   Substitutions succeed in shaping the integrand according to the Rice distribution.  Since each probability density integrates to one,  we obtain:

\[{\rm Pr}({\cal{C} }) = \sum_{i = 0}^{M-1} (-1)^i \cdot {M-1 \choose i } \cdot \frac{1}{i+1} \cdot {\rm exp } \left [ - \frac{i \cdot E_{\rm S} } {(i+1) \cdot N_0}\right ] \hspace{0.05cm}\hspace{0.5cm} \text{with}\hspace{0.5cm} E_{\rm S} = E_{\rm B} \cdot {\rm log_2}(M)\hspace{0.05cm}.\]

(7)   The special case  $M = 2$  leads to exactly the same result as calculated for the binary FSK:

\[{\rm Pr}({\cal{C} }) = (-1)^0 \cdot {2-1 \choose 0 } \cdot \frac{1}{0+1} \cdot {\rm exp } \left [ - \frac{0 \cdot E_{\rm S} } {(i+1) \cdot N_0}\right ] + (-1)^1 \cdot {2-1 \choose 1 } \cdot \frac{1}{1+1} \cdot {\rm exp } \left [ - \frac{1 \cdot E_{\rm S} } {(i+1) \cdot N_0}\right ] \]
\[ \Rightarrow \hspace{0.5cm} {\rm Pr}({\cal{C} }) = 1 - {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)} \hspace{0.5cm} \Rightarrow \hspace{0.5cm} {\rm Pr}({\cal{E} }) = 1 - {\rm Pr}({\cal{C} }) = {1}/{2} \cdot {\rm e }^{-E_{\rm S}/(2N_0)} \hspace{0.05cm}.\]
$\text{q.e.d.}$

Exercises for the chapter


Exercise 4.17: Non-Coherent On-Off Keying

Exercise 4.17Z: Rayleigh and Rice Distribution

Exercise 4.18: Non-Coherent FSK Demodulation

Exercise 4.18Z: BER of Coherent and Non-Coherent FSK

Exercise 4.19: Orthogonal Multilevel FSK