Difference between revisions of "Theory of Stochastic Signals/Further Distributions"

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$\text{Definition:}$&nbsp; A continuous random variable&nbsp; $x$&nbsp; is called&nbsp; '''Rayleigh distributed''' if it cannot take negative values and the probability density function (PDF) for&nbsp; $x \ge 0$&nbsp; with the distribution parameter&nbsp; $λ$&nbsp; has the following shape:
+
$\text{Definition:}$&nbsp; A continuous-valued random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rayleigh distributed'''&laquo;&nbsp; if it cannot take negative values and the probability density function&nbsp; $\rm (PDF)$&nbsp; for&nbsp; $x \ge 0$&nbsp; has the following shape with the distribution parameter&nbsp; $λ$:
 
:$$f_{x}(x)=\frac{x}{\lambda^2}\cdot {\rm e}^{-x^2 / ( 2 \hspace{0.05cm}\cdot \hspace{0.05cm}\lambda^2) } .$$}}
 
:$$f_{x}(x)=\frac{x}{\lambda^2}\cdot {\rm e}^{-x^2 / ( 2 \hspace{0.05cm}\cdot \hspace{0.05cm}\lambda^2) } .$$}}
  
  
The name goes back to the English physicist&nbsp; [https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh John William Strutt]&nbsp; the "third Baron Rayleigh".&nbsp; In 1904 he received the physics&ndash;Nobel Prize.  
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The name goes back to the English physicist&nbsp; [https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh $\text{John William Strutt}$]&nbsp; the&nbsp; "third Baron Rayleigh".&nbsp; In 1904 he received the Physics Nobel Prize.  
  
*The Rayleigh distribution plays a central role in the description of time-varying channels.&nbsp; Such channels are described in the book&nbsp; [[Mobile Communications]]&nbsp; .  
+
*The Rayleigh distribution plays a central role in the description of time-varying channels.&nbsp; Such channels are described in the book&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading|"Mobile Communications"]].  
*For example, ''non-frequency selective fading'' exhibits such a distribution when there is no line-of-sight between the base station and the mobile user.  
+
*For example,&nbsp; "non-frequency selective fading"&nbsp; exhibits such a distribution when there is no&nbsp; "line-of-sight"&nbsp; between the base station and the mobile user.  
  
  
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:$$m_k=(2\cdot \lambda^{\rm 2})^{\it k/\rm 2}\cdot {\rm \Gamma}( 1+ {\it k}/{\rm 2}) \hspace{0.3cm}{\rm with }\hspace{0.3cm}{\rm \Gamma}(x)= \int_{0}^{\infty} t^{x-1} \cdot  
 
:$$m_k=(2\cdot \lambda^{\rm 2})^{\it k/\rm 2}\cdot {\rm \Gamma}( 1+ {\it k}/{\rm 2}) \hspace{0.3cm}{\rm with }\hspace{0.3cm}{\rm \Gamma}(x)= \int_{0}^{\infty} t^{x-1} \cdot  
 
{\rm e}^{-t} \hspace{0.1cm}{\rm d}t.$$
 
{\rm e}^{-t} \hspace{0.1cm}{\rm d}t.$$
*From this, the mean&nbsp; $m_1$&nbsp; and the rms&nbsp; $\sigma_1$&nbsp; can be calculated as follows:
+
*From this,&nbsp; the mean&nbsp; $m_1$&nbsp; and the standard deviation&nbsp; $\sigma$&nbsp; can be calculated as follows:
 
:$$m_1=\sqrt{2}\cdot \lambda\cdot {\rm \Gamma}(1.5) =  
 
:$$m_1=\sqrt{2}\cdot \lambda\cdot {\rm \Gamma}(1.5) =  
 
\sqrt{2}\cdot \lambda\cdot {\sqrt{\pi}}/{2} =\lambda\cdot\sqrt{{\pi}/{2}},$$
 
\sqrt{2}\cdot \lambda\cdot {\sqrt{\pi}}/{2} =\lambda\cdot\sqrt{{\pi}/{2}},$$
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= \sqrt{m_2 - m_1^2}
 
= \sqrt{m_2 - m_1^2}
 
=\lambda\cdot\sqrt{2-{\pi}/{2}}.$$
 
=\lambda\cdot\sqrt{2-{\pi}/{2}}.$$
*To model a Rayleigh distributed random variable&nbsp; $x$&nbsp; one uses, for example, two Gaussian distributed, mean-free, and statistically independent random variables&nbsp; $u$&nbsp; and&nbsp; $v$, both of which have rms&nbsp; $σ = λ$&nbsp; The variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are then linked as follows:  
+
*To model a Rayleigh distributed random variable&nbsp; $x$&nbsp; one uses,&nbsp; for example,&nbsp; two Gaussian distributed zero mean and statistically independent random variables&nbsp; $u$&nbsp; and&nbsp; $v$,&nbsp; both of which have rms&nbsp; $σ = λ$.&nbsp; The variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; are then linked as follows:  
 
:$$x=\sqrt{u^2+v^2}.$$
 
:$$x=\sqrt{u^2+v^2}.$$
  
[[File:P_ID62__Sto_T_3_7_S1_neu.png |right|frame|Sample signal and PDF of a Rayleigh distributed random variable|class=fit]]
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[[File:P_ID62__Sto_T_3_7_S1_neu.png |right|frame|Sample signal and PDF of a Rayleigh distributed random variable|class=fit]]   
 
$\text{Example 1:}$&nbsp; The graph shows:  
 
$\text{Example 1:}$&nbsp; The graph shows:  
*the time course&nbsp; $x(t)$&nbsp; of a Rayleigh distributed random variable as well as.
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*the time course&nbsp; $x(t)$&nbsp; of a Rayleigh distributed random variable,&nbsp; as well as,
*the associated density function&nbsp; $f_{x}(x)$.  
+
*the associated probability  density function&nbsp; $f_{x}(x)$.  
  
  
 
One can see from this representation:  
 
One can see from this representation:  
*The Rayleigh&ndash;PDF is always asymmetric.  
+
*The Rayleigh PDF is always asymmetric.  
 
*The mean&nbsp; $m_1$&nbsp; lies about&nbsp; $25\%$&nbsp; above the PDF maximum.
 
*The mean&nbsp; $m_1$&nbsp; lies about&nbsp; $25\%$&nbsp; above the PDF maximum.
*The PDF maximum occurs at&nbsp; $x = λ$&nbsp;. }}
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*The PDF maximum occurs at&nbsp; $x = λ$.
 +
 
 +
 
 +
With the HTML 5/JavaScript applet&nbsp; [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions|"PDF, CDF and Moments of Special Distributions"]]&nbsp; you can display,&nbsp; among others,&nbsp; the characteristics of the&nbsp; Rayleigh distribution.}}
  
  
 
==Rice PDF==
 
==Rice PDF==
 
<br>
 
<br>
Auch die Riceverteilung spielt für die Beschreibung zeitvarianter Kanäle eine wichtige Rolle, unter anderem auch deshalb,  
+
Rice distribution also plays an important role in the description of time-varying channels,
*weil&nbsp; ''nichtfrequenzselektives Fading''&nbsp; dann riceverteilt ist,  
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*because "non-frequency selective fading"&nbsp; is Rice distributed,  
*wenn zwischen der Basisstation und dem Mobilteilnehmer eine&nbsp; ''Sichtverbindung''&nbsp; besteht.  
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*if there is&nbsp; "line-of-sight"&nbsp; between the base station and the mobile subscriber.  
  
  
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$\text{Definition:}$&nbsp; Eine kontinuierliche Zufallsgröße&nbsp; $x$&nbsp; nennt man&nbsp; '''riceverteilt''', wenn sie keine negativen Werte annehmen kann und die Wahrscheinlichkeitsdichtefunktion (WDF) für&nbsp; $x > 0$&nbsp; den folgenden Verlauf hat:  
+
$\text{Definition:}$&nbsp; A continuous-valued random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Rice distributed'''&laquo;&nbsp;  if it cannot take negative values and the probability density function&nbsp;  $\rm (PDF)$&nbsp; for&nbsp; $x > 0$&nbsp; has the following shape:  
:$$f_{\rm x}(x)=\frac{x}{\lambda^2}\cdot{\rm e}^{-({C^2+\it x^{\rm 2} })/ ({\rm 2 \it \lambda^{\rm 2} })}\cdot {\rm I_0}(\frac{\it x\cdot C}{\lambda^{\rm 2} }) \hspace{0.4cm}{\rm mit} \hspace{0.4cm} {\rm I_0}(x) = \sum_{k=0}^{\infty}\frac{(x/2)^{2k} }{k! \cdot {\rm \Gamma ({\it k}+1)} }.$$
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:$$f_{\rm x}(x)=\frac{x}{\lambda^2}\cdot{\rm e}^{-({C^2+\it x^{\rm 2} })/ ({\rm 2 \it \lambda^{\rm 2} })}\cdot {\rm I_0}(\frac{\it x\cdot C}{\lambda^{\rm 2} }) \hspace{0.4cm}{\rm with} \hspace{0.4cm} {\rm I_0}(x) = \sum_{k=0}^{\infty}\frac{(x/2)^{2k} }{k! \cdot {\rm \gamma ({\it k}+1)} }.$$
${\rm I_0}( ... )$ bezeichnet die&nbsp; [https://de.wikipedia.org/wiki/Besselsche_Differentialgleichung modifizierte Besselfunktion nullter Ordnung].}}
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${\rm I_0}( ... )$ denotes the&nbsp; [https://en.wikipedia.org/wiki/Bessel_function $\text{modified zero-order Bessel function}$].}}
 +
 
 +
 
 +
The name is due to the mathematician and logician&nbsp; [https://en.wikipedia.org/wiki/Henry_Gordon_Rice $\text{Henry Gordon Rice}$].&nbsp; He taught as a mathematics professor at the University of New Hampshire.
 +
 
  
 +
'''Characteristic properties of the Rice distribution''':
 +
*The additional parameter&nbsp; $C$&nbsp; compared to the Rayleigh distribution is a measure of the&nbsp; "strength"&nbsp; of the direct component.&nbsp; The larger the quotient&nbsp; $C/λ$,&nbsp;  the more the&nbsp; Rice channel approximates the Gaussian channel.&nbsp; For&nbsp; $C = 0$&nbsp; the Rice distribution transitions to the&nbsp; [[Theory_of_Stochastic_Signals/Further_Distributions#Rayleigh_PDF|$\text{Rayleigh distribution}$]].
 +
*In the Rice distribution,&nbsp; the expression for the moment&nbsp; $m_k$&nbsp; is much more complicated and can only be specified using hypergeometric functions.&nbsp;
 +
*However,&nbsp; if&nbsp; $λ \ll C$,&nbsp; then&nbsp; $m_1 ≈ C$&nbsp; and&nbsp; $σ ≈ λ$&nbsp; holds.
 +
*Under these conditions,&nbsp; the Rice distribution can be approximated by a&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|$\text{Gaussian distribution}$]]&nbsp; with mean&nbsp; $C$&nbsp; and rms&nbsp; $λ$.
 +
*To model a Rice distributed random variable&nbsp; $x$&nbsp; we use a similar model as for the Rayleigh distribution,&nbsp; except that now at least one of the two Gaussian distributed and statistically independent random variables&nbsp; $(u$&nbsp; and/or &nbsp;$v$&nbsp;)&nbsp; must have a non-zero mean.
 +
:$$x=\sqrt{u^2+v^2}\hspace{0.5cm}{\rm with}\hspace{0.5cm}|m_u| + |m_v| > 0 .$$
  
Der Name geht auf den Mathematiker und Logiker&nbsp; [https://de.wikipedia.org/wiki/Henry_Gordon_Rice Henry Gordon Rice]&nbsp; zurück. Er lehrte als  Mathematikprofessor an der University of New Hampshire.  
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{{GraueBox|TEXT=
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[[File:P_ID63__Sto_T_3_7_S2_neu.png |right|frame|Sample signal and PDF of a Rice distributed random variable]]
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$\text{Example 2:}$&nbsp; The graph shows the time course of a Rice distributed random variable&nbsp; $x$&nbsp; and its probability density function&nbsp; $f_{\rm x}(x)$,&nbsp; where&nbsp; $C/λ = 2$&nbsp; holds.  
  
 +
*Somewhat casually put: &nbsp; The Rice distribution is a compromise between the Rayleigh and the Gaussian distributions.
 +
*Here the mean&nbsp; $m_1$&nbsp; is slightly larger than&nbsp; $C$.
  
'''Charakteristische Eigenschaften der Riceverteilung''':
 
*Der gegenüber der Rayleighverteilung zusätzliche Parameter&nbsp; $C$&nbsp; ist ein Maß für die „Stärke” der Direktkomponente.&nbsp; Je größer der Quotient&nbsp; $C/λ$&nbsp; ist, desto mehr nähert sich der&nbsp; Rice&ndash;Kanal dem Gauß&ndash;Kanal an.&nbsp; Für&nbsp; $C = 0$&nbsp; geht die Riceverteilung in die&nbsp; [[Theory_of_Stochastic_Signals/Weitere_Verteilungen#Rayleighverteilung|Rayleighverteilung]] über.
 
*Bei der Riceverteilung ist der Ausdruck für das Moment&nbsp; $m_k$&nbsp; deutlich komplizierter und nur mit Hilfe hypergeometrischer Funktionen angebbar.&nbsp; Ist jedoch&nbsp; $λ$&nbsp; sehr viel kleiner als&nbsp; $C$, so gilt&nbsp; $m_1 ≈ C$&nbsp; und&nbsp; $σ ≈ λ$.
 
*Unter diesen Voraussetzungen kann die Riceverteilung durch eine&nbsp; [[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|Gaußverteilung]]&nbsp; mit Mittelwert&nbsp; $C$&nbsp; und Streuung&nbsp; $λ$&nbsp; angenähert werden.
 
*Zur Modellierung einer riceverteilten Zufallsgröße&nbsp; $x$&nbsp; verwenden wir ein ähnliches Modell wie für die Rayleighverteilung, nur muss nun zumindest eine der beiden gaußverteilten und statistisch voneinander unabhängigen Zufallsgrößen&nbsp; $(u$&nbsp; und/oder &nbsp;$v$&nbsp;)&nbsp; einen Mittelwert ungleich Null aufweisen.
 
:$$x=\sqrt{u^2+v^2}\hspace{0.5cm}{\rm mit}\hspace{0.5cm}|m_u| + |m_v| > 0 .$$
 
  
[[File:P_ID63__Sto_T_3_7_S2_neu.png |right|frame| Mustersignal und WDF einer riceverteilten Zufallsgröße]]
 
{{GraueBox|TEXT= 
 
$\text{Beispiel 2:}$&nbsp; Die Grafik zeigt den zeitlichen Verlauf einer riceverteilten Zufallsgröße&nbsp; $x$&nbsp; sowie deren Dichtefunktion&nbsp; $f_{\rm x}(x)$, wobei&nbsp; $C/λ = 2$&nbsp; gilt.
 
  
*Etwas salopp ausgedrückt: &nbsp; Die Riceverteilung ist ein Kompromiss zwischen der Rayleigh&ndash; und der Gaußverteilung.
 
*Der Mittelwert&nbsp; $m_1$&nbsp; ist hier etwas größer als&nbsp; $C$. }}
 
  
  
Mit dem interaktiven Applet&nbsp; [[Applets:WDF,_VTF_und_Momente_spezieller_Verteilungen_(Applet)|WDF, VTF und Momente spezieller Verteilungen]]&nbsp; können Sie sich unter anderem die Kenngrößen&nbsp; $($WDF, VTF, Momente$)$&nbsp; der Rayleigh&ndash; sowie der Riceverteilung anzeigen lassen.
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With the HTML 5/JavaScript applet&nbsp; [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions|"PDF, CDF and Moments of Special Distributions"]]&nbsp; you can display,&nbsp; among others,&nbsp; the characteristics of the&nbsp; Rice distribution.}}
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$\text{Definition:}$&nbsp; Eine kontinuierliche Zufallsgröße&nbsp; $x$&nbsp; nennt man&nbsp; '''cauchyverteilt''', wenn die Wahrscheinlichkeitsdichtefunktion (WDF) und die Verteilungsfunktion (VTF)   mit dem Verteilungsparameter&nbsp; $λ$&nbsp; folgende Form haben:  
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$\text{Definition:}$&nbsp; A continuous-valued random variable&nbsp; $x$&nbsp; is called&nbsp; &raquo;'''Cauchy distributed'''&laquo;&nbsp; if the probability density function&nbsp; $\rm (PDF)$&nbsp; and the cumulative distribution function&nbsp; $\rm  (CDF)$&nbsp; with parameter&nbsp; $λ$&nbsp; have the following form:  
 
:$$f_{x}(x)=\frac{1}{\pi}\cdot\frac{\lambda}{\lambda^2+x^2},$$
 
:$$f_{x}(x)=\frac{1}{\pi}\cdot\frac{\lambda}{\lambda^2+x^2},$$
 
:$$F_{x}(r)={\rm 1}/{2}+{\rm arctan}({r}/{\lambda}).$$
 
:$$F_{x}(r)={\rm 1}/{2}+{\rm arctan}({r}/{\lambda}).$$
Manchmal wird in der Literatur auch noch ein Mittelwert&nbsp; $m_1$&nbsp; berücksichtigt. }}
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Sometimes in the literature,&nbsp; a mean&nbsp; $m_1$&nbsp; is also considered. }}
 +
 
 +
 
 +
The name derives from the French mathematician&nbsp; [https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy $\text{Augustin-Louis Cauchy}$],&nbsp; a pioneer of calculus who further developed the foundations established by&nbsp; [https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz $\text{Gottfried Wilhelm Leibniz}$]&nbsp; and&nbsp; [https://en.wikipedia.org/wiki/Isaac_Newton $\text{Sir Isaac Newton}$]&nbsp; and formally proved fundamental propositions.&nbsp; In particular,&nbsp; many central theorems of&nbsp; "Function Theory"&nbsp; derive from Cauchy.
 +
 
 +
The Cauchy distribution has less practical significance for communications engineering,&nbsp; but is mathematically very interesting.&nbsp; It has the following properties in the symmetric form&nbsp; $($with mean&nbsp; $m_1 = 0)$:
 +
*For the Cauchy distribution, all moments&nbsp; $m_k$&nbsp; for even&nbsp; $k$&nbsp; have an infinitely large value, and this is independent of the parameter&nbsp; $λ$.
 +
*Thus,&nbsp; this distribution also has an infinitely large variance&nbsp; $\sigma^2 = m_2$ &nbsp; ⇒ &nbsp; "power" &nbsp; &rArr; &nbsp; it is obvious that no physical variable can be Cauchy distributed.
 +
*The quotient&nbsp; $u/v$&nbsp; of two independent Gaussian distributed zero mean variables&nbsp; $u$&nbsp; and&nbsp; $v$&nbsp; is Cauchy distributed with the distribution parameter&nbsp; $λ = σ_u/σ_v$&nbsp; . 
 +
*A Cauchy distributed random variable&nbsp; $x$&nbsp; can be generated from a random variable&nbsp; $\pm1$&nbsp; uniformly distributed between&nbsp; $u$&nbsp; by the following&nbsp; [[Theory_of_Stochastic_Signals/Exponentially_Distributed_Random_Variables#Transformation_of_random_variables|$\text{nonlinear transformation}$]]:
 +
:$$x=\lambda \cdot {\tan}( {\pi}/{2}\cdot u).$$
 +
*Because of symmetry,&nbsp; for odd&nbsp; $k$&nbsp; all moments&nbsp; $m_k = 0$,&nbsp; assuming the&nbsp; "Cauchy Principal Value".
 +
*So the mean value&nbsp; $m_X = 0$&nbsp; and the Charlier skewness&nbsp; $S_X = 0$&nbsp; also hold.
 +
 
  
 +
{{GraueBox|TEXT=
 +
[[File:EN_Sto_T_3_7_S3.png |right|frame| PDF of a Cauchy distributed random variable]]
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$\text{Example 3:}$&nbsp; The graph shows the typical course of the Cauchy PDF.
  
Der Name geht auf den französischen Mathematiker&nbsp; [https://de.wikipedia.org/wiki/Augustin-Louis_Cauchy Augustin-Louis Cauchy]&nbsp; zurück, ein Pionier der Analysis, der die von&nbsp; [https://de.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz Gottfried Wilhelm Leibniz]&nbsp; und&nbsp; [https://de.wikipedia.org/wiki/Isaac_Newton Sir Isaac Newton]&nbsp; aufgestellten Grundlagen weiterentwickelte und fundamentale Aussagen formal bewies.&nbsp; Insbesondere stammen viele zentrale Sätze der Funktionentheorie von Cauchy.
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*The slow decline of this function towards the edges can be seen.  
 +
*As this occurs asymptotically with&nbsp; $1/x^2$&nbsp; the variance and all higher order moments (with even index) are infinite.  
  
  
Die Cauchyverteilung hat weniger praktische Bedeutung für die Nachrichtentechnik, ist mathematisch aber sehr interessant.
 
  
Sie weist in der symmetrischen Form&nbsp; $($mit Mittelwert&nbsp;  $m_1 = 0)$&nbsp; folgende Eigenschaften auf:
 
*Bei der Cauchyverteilung besitzen alle Momente&nbsp; $m_k$&nbsp; für gerades&nbsp; $k$&nbsp; einen unendlich großen Wert, und zwar unabhängig vom Parameter&nbsp; $λ$.
 
  
*Damit besitzt diese Verteilung auch eine unendlich große Varianz&nbsp;  $\sigma^2 = m_2$ &nbsp; ⇒  &nbsp; Leistung.
 
*Deshalb ist es offensichtlich, dass keine physikalische Größe cauchyverteilt sein kann.
 
*Der Quotient&nbsp; $u/v$&nbsp; zweier unabhängiger gaußverteilter mittelwertfreier Größen&nbsp; $u$&nbsp; und&nbsp; $v$&nbsp; ist mit dem Verteilungsparameter&nbsp; $λ = σ_u/σ_v$&nbsp; cauchyverteilt. 
 
*Eine cauchyverteilte Zufallsgröße&nbsp; $x$&nbsp; kann aus einer zwischen&nbsp; $\pm1$&nbsp; gleichverteilten Größe&nbsp; $u$&nbsp; durch  folgende&nbsp;  [[Theory_of_Stochastic_Signals/Exponentialverteilte_Zufallsgrößen#Transformation_von_Zufallsgr.C3.B6.C3.9Fen|nichtlineare Transformation]]&nbsp;erzeugt werden:
 
:$$x=\lambda \cdot {\tan}( {\pi}/{2}\cdot u).$$
 
*Aufgrund der Symmetrie sind für ungerades&nbsp; $k$&nbsp; alle Momente&nbsp;  $m_k = 0$, wenn man vom "Cauchy Principal Value" ausgeht.
 
*Somit gilt auch für den Mittelwert&nbsp; $m_X = 0$&nbsp; und die Charliersche Schiefe&nbsp; $S_X = 0$.
 
  
  
[[File:EN_Sto_T_3_7_S3.png |right|frame| WDF einer cauchyverteilten Zufallsgröße]]
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With the HTML 5/JavaScript applet&nbsp; [[Applets:PDF,_CDF_and_Moments_of_Special_Distributions|"PDF, CDF and Moments of Special Distributions"]]&nbsp; you can display,&nbsp; among others,&nbsp; the characteristics of the&nbsp; Cauchy distribution.}}
{{GraueBox|TEXT= 
 
$\text{Beispiel 3:}$&nbsp; Die Grafik zeigt den typischen Verlauf der Cauchy-WDF.  
 
  
*Zu erkennen ist der langsame Abfall dieser Funktion zu den Rändern hin.
 
*Da dieser asymptotisch mit&nbsp; $1/x^2$&nbsp; erfolgt, sind die Varianz und alle Momente höherer Ordnung (mit geradzahligem Index) unendlich groß. }}
 
  
  
==Tschebyscheffsche Ungleichung==
+
==Chebyshev's inequality==
 
<br>
 
<br>
Bei einer Zufallsgröße&nbsp; $x$&nbsp; mit bekannter Wahrscheinlichkeitsdichtefunktion&nbsp; $f_{x}(x)$&nbsp; kann die Wahrscheinlichkeit, dass die Zufallsgröße&nbsp; $x$&nbsp; betragsmäßig um mehr als einen Wert&nbsp; $ε$&nbsp; von ihrem Mittelwert&nbsp; $m_{x}$&nbsp; abweicht, entsprechend der in diesem Kapitel allgemein beschriebenen Weise exakt berechnet werden.  
+
[[File:EN_Sto_T_3_7_S4_neu.png |frame| Chebyshev's inequality | right]]
 +
Given a random variable&nbsp; $x$&nbsp; with known probability density function&nbsp; $f_{x}(x)$&nbsp; the probability that the random variable&nbsp; $x$&nbsp; deviates in magnitude by more than the value&nbsp; $ε$&nbsp; from its mean&nbsp; $m_{x}$&nbsp; can be calculated exactly according to the way generally described in this chapter.  
  
[[File:EN_Sto_T_3_7_S4.png |frame| Tschebyscheffsche Ungleichung | rechts]]
+
*If besides the mean&nbsp; $m_{x}$&nbsp; the standard deviation&nbsp; $σ_{x}$&nbsp; is known,&nbsp; but not the exact&nbsp; $\rm PDF$&nbsp; course&nbsp; $f_{x}(x)$,&nbsp; at least an upper bound can be given for this probability:  
*Ist neben dem Mittelwert&nbsp; $m_{x}$&nbsp; zwar noch die Streuung&nbsp; $σ_{x}$&nbsp; bekannt, nicht jedoch der exakte WDF-Verlauf&nbsp; $f_{x}(x)$, so lässt sich für diese Wahrscheinlichkeit zumindest eine obere Schranke angeben:  
 
 
:$${\rm Pr}(|x - m_{\rm x}|\ge\varepsilon)\le\frac{\sigma_{x}^{\rm 2}}{\varepsilon^{\rm 2}}. $$
 
:$${\rm Pr}(|x - m_{\rm x}|\ge\varepsilon)\le\frac{\sigma_{x}^{\rm 2}}{\varepsilon^{\rm 2}}. $$
*Diese von&nbsp; [https://de.wikipedia.org/wiki/Pafnuti_Lwowitsch_Tschebyschow Pafnuti L. Tschebyscheff]&nbsp; angegebene Schranke – bekannt als „Tschebyscheffsche Ungleichung” – ist im Allgemeinen allerdings nur eine sehr grobe Näherung für die tatsächliche Überschreitungswahrscheinlichkeit.&nbsp; Sie sollte deshalb nur bei unbekanntem Verlauf der WDF&nbsp; $f_{x}(x)$&nbsp; angewendet werden.
+
*This bound given by&nbsp; [https://en.wikipedia.org/wiki/Pafnuty_Chebyshev $\text{Pafnuti L. Chebyshev}$]&nbsp; - known as&nbsp; "Chebyshev's inequality"&nbsp; - is in general only a very rough approximation for the actual exceeding probability.&nbsp; It should therefore be applied only in the case of an unknown course of the PDF&nbsp; $f_{x}(x)$.
 
<br clear=all>  
 
<br clear=all>  
 
{{GraueBox|TEXT=   
 
{{GraueBox|TEXT=   
$\text{Beispiel 4:}$&nbsp; Wir gehen von einer gaußverteilten und mittelwertfreien Zufallsgröße&nbsp; $x$&nbsp; aus.  
+
$\text{Example 4:}$&nbsp; We assume a Gaussian distributed and zero mean random variable&nbsp; $x$.  
*Damit ist die Wahrscheinlichkeit, dass deren Betrag&nbsp; $\vert x \vert $&nbsp; größer als die dreifache Streuung&nbsp; $(3 · σ_{x})$&nbsp; ist, einfach berechenbar.&nbsp; Ergebnis:&nbsp; ${\rm 2 · Q(3) ≈ 2.7 · 10^{-3} }.$  
+
*Thus,&nbsp; the probability that its absolute value&nbsp; $\vert x \vert $&nbsp; is greater than three times the rms&nbsp; $(3 \cdot σ_{x})$&nbsp; is easily computable.&nbsp; Result:&nbsp; ${\rm 2 - Q(3) ≈ 2.7 \cdot 10^{-3} }.$  
*Die Tschebyscheffsche Ungleichung liefert hier als eine obere Schranke den deutlich zu großen Wert&nbsp; $1/9 ≈ 0.111$.
+
*Chebyshev's inequality yields here as an upper bound the clearly too large value&nbsp; $1/9 ≈ 0.111$.
* Diese Schranke nach Tschebyscheff  würde für jede beliebige WDF–Form ebenfalls gelten.}}
+
*This Chebyshev bound would hold for any PDF form.}}
  
==Aufgaben zum Kapitel==
+
==Exercises for the chapter==
  
[[Aufgaben:3.10 Rayleighfading|Aufgabe 3.10: Rayleighfading]]
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[[Aufgaben:Exercise_3.10:_Rayleigh_Fading|Exercise 3.10: Rayleigh Fading]]
  
[[Aufgaben:3.10Z Rayleigh? Oder Rice?|Aufgabe 3.10Z: Rayleigh? Oder Rice?]]
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[[Aufgaben:Exercise_3.10Z:_Rayleigh%3F_Or_Rice%3F|Exercise 3.10Z: Rayleigh? Or Rice?]]
  
[[Aufgaben:3.11 Tschebyscheffsche Ungleichung|Aufgabe 3.11: Tschebyscheffsche Ungleichung]]
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[[Aufgaben:Exercise_3.11:_Chebyshev%27s_Inequality|Exercise 3.11: Chebyshev's Inequality]]
  
[[Aufgaben:3.12 Cauchyverteilung|Aufgabe 3.12: Cauchyverteilung]]
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[[Aufgaben:Exercise_3.12:_Cauchy_Distribution|Exercise 3.12: Cauchy Distribution]]
  
 
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Latest revision as of 10:03, 22 December 2022

Rayleigh PDF


$\text{Definition:}$  A continuous-valued random variable  $x$  is called  »Rayleigh distributed«  if it cannot take negative values and the probability density function  $\rm (PDF)$  for  $x \ge 0$  has the following shape with the distribution parameter  $λ$:

$$f_{x}(x)=\frac{x}{\lambda^2}\cdot {\rm e}^{-x^2 / ( 2 \hspace{0.05cm}\cdot \hspace{0.05cm}\lambda^2) } .$$


The name goes back to the English physicist  $\text{John William Strutt}$  the  "third Baron Rayleigh".  In 1904 he received the Physics Nobel Prize.

  • The Rayleigh distribution plays a central role in the description of time-varying channels.  Such channels are described in the book  "Mobile Communications".
  • For example,  "non-frequency selective fading"  exhibits such a distribution when there is no  "line-of-sight"  between the base station and the mobile user.


Characteristic properties of Rayleigh distribution:

  • A Rayleigh distributed random variable  $x$  cannot take negative values.
  • The theoretically possible value  $x = 0$  also occurs only with probability "zero".
  • The  $k$-th moment of a Rayleigh distributed random variable  $x$  results in general to
$$m_k=(2\cdot \lambda^{\rm 2})^{\it k/\rm 2}\cdot {\rm \Gamma}( 1+ {\it k}/{\rm 2}) \hspace{0.3cm}{\rm with }\hspace{0.3cm}{\rm \Gamma}(x)= \int_{0}^{\infty} t^{x-1} \cdot {\rm e}^{-t} \hspace{0.1cm}{\rm d}t.$$
  • From this,  the mean  $m_1$  and the standard deviation  $\sigma$  can be calculated as follows:
$$m_1=\sqrt{2}\cdot \lambda\cdot {\rm \Gamma}(1.5) = \sqrt{2}\cdot \lambda\cdot {\sqrt{\pi}}/{2} =\lambda\cdot\sqrt{{\pi}/{2}},$$
$$m_2=2 \lambda^2 \cdot {\rm \Gamma}(2) = 2 \lambda^2 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}\sigma = \sqrt{m_2 - m_1^2} =\lambda\cdot\sqrt{2-{\pi}/{2}}.$$
  • To model a Rayleigh distributed random variable  $x$  one uses,  for example,  two Gaussian distributed zero mean and statistically independent random variables  $u$  and  $v$,  both of which have rms  $σ = λ$.  The variables  $u$  and  $v$  are then linked as follows:
$$x=\sqrt{u^2+v^2}.$$
Sample signal and PDF of a Rayleigh distributed random variable

$\text{Example 1:}$  The graph shows:

  • the time course  $x(t)$  of a Rayleigh distributed random variable,  as well as,
  • the associated probability density function  $f_{x}(x)$.


One can see from this representation:

  • The Rayleigh PDF is always asymmetric.
  • The mean  $m_1$  lies about  $25\%$  above the PDF maximum.
  • The PDF maximum occurs at  $x = λ$.


With the HTML 5/JavaScript applet  "PDF, CDF and Moments of Special Distributions"  you can display,  among others,  the characteristics of the  Rayleigh distribution.


Rice PDF


Rice distribution also plays an important role in the description of time-varying channels,

  • because "non-frequency selective fading"  is Rice distributed,
  • if there is  "line-of-sight"  between the base station and the mobile subscriber.


$\text{Definition:}$  A continuous-valued random variable  $x$  is called  »Rice distributed«  if it cannot take negative values and the probability density function  $\rm (PDF)$  for  $x > 0$  has the following shape:

$$f_{\rm x}(x)=\frac{x}{\lambda^2}\cdot{\rm e}^{-({C^2+\it x^{\rm 2} })/ ({\rm 2 \it \lambda^{\rm 2} })}\cdot {\rm I_0}(\frac{\it x\cdot C}{\lambda^{\rm 2} }) \hspace{0.4cm}{\rm with} \hspace{0.4cm} {\rm I_0}(x) = \sum_{k=0}^{\infty}\frac{(x/2)^{2k} }{k! \cdot {\rm \gamma ({\it k}+1)} }.$$

${\rm I_0}( ... )$ denotes the  $\text{modified zero-order Bessel function}$.


The name is due to the mathematician and logician  $\text{Henry Gordon Rice}$.  He taught as a mathematics professor at the University of New Hampshire.


Characteristic properties of the Rice distribution:

  • The additional parameter  $C$  compared to the Rayleigh distribution is a measure of the  "strength"  of the direct component.  The larger the quotient  $C/λ$,  the more the  Rice channel approximates the Gaussian channel.  For  $C = 0$  the Rice distribution transitions to the  $\text{Rayleigh distribution}$.
  • In the Rice distribution,  the expression for the moment  $m_k$  is much more complicated and can only be specified using hypergeometric functions. 
  • However,  if  $λ \ll C$,  then  $m_1 ≈ C$  and  $σ ≈ λ$  holds.
  • Under these conditions,  the Rice distribution can be approximated by a  $\text{Gaussian distribution}$  with mean  $C$  and rms  $λ$.
  • To model a Rice distributed random variable  $x$  we use a similar model as for the Rayleigh distribution,  except that now at least one of the two Gaussian distributed and statistically independent random variables  $(u$  and/or  $v$ )  must have a non-zero mean.
$$x=\sqrt{u^2+v^2}\hspace{0.5cm}{\rm with}\hspace{0.5cm}|m_u| + |m_v| > 0 .$$
Sample signal and PDF of a Rice distributed random variable

$\text{Example 2:}$  The graph shows the time course of a Rice distributed random variable  $x$  and its probability density function  $f_{\rm x}(x)$,  where  $C/λ = 2$  holds.

  • Somewhat casually put:   The Rice distribution is a compromise between the Rayleigh and the Gaussian distributions.
  • Here the mean  $m_1$  is slightly larger than  $C$.



With the HTML 5/JavaScript applet  "PDF, CDF and Moments of Special Distributions"  you can display,  among others,  the characteristics of the  Rice distribution.



Cauchy PDF


$\text{Definition:}$  A continuous-valued random variable  $x$  is called  »Cauchy distributed«  if the probability density function  $\rm (PDF)$  and the cumulative distribution function  $\rm (CDF)$  with parameter  $λ$  have the following form:

$$f_{x}(x)=\frac{1}{\pi}\cdot\frac{\lambda}{\lambda^2+x^2},$$
$$F_{x}(r)={\rm 1}/{2}+{\rm arctan}({r}/{\lambda}).$$

Sometimes in the literature,  a mean  $m_1$  is also considered.


The name derives from the French mathematician  $\text{Augustin-Louis Cauchy}$,  a pioneer of calculus who further developed the foundations established by  $\text{Gottfried Wilhelm Leibniz}$  and  $\text{Sir Isaac Newton}$  and formally proved fundamental propositions.  In particular,  many central theorems of  "Function Theory"  derive from Cauchy.

The Cauchy distribution has less practical significance for communications engineering,  but is mathematically very interesting.  It has the following properties in the symmetric form  $($with mean  $m_1 = 0)$:

  • For the Cauchy distribution, all moments  $m_k$  for even  $k$  have an infinitely large value, and this is independent of the parameter  $λ$.
  • Thus,  this distribution also has an infinitely large variance  $\sigma^2 = m_2$   ⇒   "power"   ⇒   it is obvious that no physical variable can be Cauchy distributed.
  • The quotient  $u/v$  of two independent Gaussian distributed zero mean variables  $u$  and  $v$  is Cauchy distributed with the distribution parameter  $λ = σ_u/σ_v$  .
  • A Cauchy distributed random variable  $x$  can be generated from a random variable  $\pm1$  uniformly distributed between  $u$  by the following  $\text{nonlinear transformation}$:
$$x=\lambda \cdot {\tan}( {\pi}/{2}\cdot u).$$
  • Because of symmetry,  for odd  $k$  all moments  $m_k = 0$,  assuming the  "Cauchy Principal Value".
  • So the mean value  $m_X = 0$  and the Charlier skewness  $S_X = 0$  also hold.


PDF of a Cauchy distributed random variable

$\text{Example 3:}$  The graph shows the typical course of the Cauchy PDF.

  • The slow decline of this function towards the edges can be seen.
  • As this occurs asymptotically with  $1/x^2$  the variance and all higher order moments (with even index) are infinite.




With the HTML 5/JavaScript applet  "PDF, CDF and Moments of Special Distributions"  you can display,  among others,  the characteristics of the  Cauchy distribution.


Chebyshev's inequality


Chebyshev's inequality

Given a random variable  $x$  with known probability density function  $f_{x}(x)$  the probability that the random variable  $x$  deviates in magnitude by more than the value  $ε$  from its mean  $m_{x}$  can be calculated exactly according to the way generally described in this chapter.

  • If besides the mean  $m_{x}$  the standard deviation  $σ_{x}$  is known,  but not the exact  $\rm PDF$  course  $f_{x}(x)$,  at least an upper bound can be given for this probability:
$${\rm Pr}(|x - m_{\rm x}|\ge\varepsilon)\le\frac{\sigma_{x}^{\rm 2}}{\varepsilon^{\rm 2}}. $$
  • This bound given by  $\text{Pafnuti L. Chebyshev}$  - known as  "Chebyshev's inequality"  - is in general only a very rough approximation for the actual exceeding probability.  It should therefore be applied only in the case of an unknown course of the PDF  $f_{x}(x)$.


$\text{Example 4:}$  We assume a Gaussian distributed and zero mean random variable  $x$.

  • Thus,  the probability that its absolute value  $\vert x \vert $  is greater than three times the rms  $(3 \cdot σ_{x})$  is easily computable.  Result:  ${\rm 2 - Q(3) ≈ 2.7 \cdot 10^{-3} }.$
  • Chebyshev's inequality yields here as an upper bound the clearly too large value  $1/9 ≈ 0.111$.
  • This Chebyshev bound would hold for any PDF form.

Exercises for the chapter

Exercise 3.10: Rayleigh Fading

Exercise 3.10Z: Rayleigh? Or Rice?

Exercise 3.11: Chebyshev's Inequality

Exercise 3.12: Cauchy Distribution