Aufgaben:Exercise 3.4: Characteristic Function - Revision history

Guenter at 15:48, 6 January 2022

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Guenter at 14:53, 6 January 2022

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Guenter at 16:43, 5 January 2022

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Guenter at 16:35, 5 January 2022

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Noah at 22:08, 28 December 2021

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Noah at 21:29, 26 December 2021

2021-12-26T21:29:55Z

Guenter at 10:07, 20 December 2021

2021-12-20T10:07:10Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.4: Charakteristische Funktion to Aufgaben:Exercise 3.4: Characteristic Function

2021-12-20T10:01:01Z

Guenter moved page Aufgabe 3.4: Charakteristische Funktion to Exercise 3.4: Characteristic Function

Javier: Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

2021-03-23T12:44:46Z

Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

Javier: Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

2020-07-09T09:51:56Z

Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

@@ Line 54: / Line 54: @@
 '''(1)'''&nbsp; Correct are&nbsp; <u>the proposed solutions 2, 3 and 4</u>:
 * $C_x( {\it \Omega} )$&nbsp; is not the Fourier transform to&nbsp; $f_x(x)$,&nbsp; but the inverse Fourier transform:
-:$$C_x( {\it \Omega } ) = \int_{ - \infty }^{ + \infty } {f_x }( x )\cdot {\rm{e}}^{\hspace{0.03cm}{\rm{j}}\hspace{0.03cm}{\it \Omega x}} \hspace{0.1cm}{\rm{d}}x .$$
+:$$C_x( {\it \Omega } ) = \int_{ - \infty }^{ + \infty } {f_x }( x )\cdot {\rm{e}}^{\hspace{0.03cm}{\rm{j}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega\hspace{0.05cm}\cdot \hspace{0.05cm} x}} \hspace{0.1cm}{\rm{d}}x .$$
 *Also for this,&nbsp; the real part is always even and the imaginary part odd.&nbsp; For&nbsp; ${\it \Omega} = 0$&nbsp; holds:
 :$$C_x( {\it \Omega} = 0 ) = \int_{ - \infty }^{ + \infty } {f_x }( x ) \hspace{0.1cm}{\rm{d}}x = 1.$$

@@ Line 62: / Line 62: @@
 '''(2)'''&nbsp; According to the general definition:
-:$$C_y( {\it \Omega  } ) = \int_{ - \infty }^{ + \infty } {f_y }( y )\cdot {\rm{e}}^{{\rm{j}}\hspace{0.01cm}{\it \Omega\hspace{0.03cm} y}} \hspace{0.1cm}{\rm{d}}y   = 0.5\int_1^3 {{\rm{e}}^{{\rm{j}}\hspace{0.01cm}\Omega\hspace{0.03cm} y} \hspace{0.1cm}{\rm{d}}y.} $$
+:$$C_y( {\it \Omega  } ) = \int_{ - \infty }^{ + \infty } {f_y }( y )\cdot {\rm{e}}^{{\rm{j}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega\hspace{0.01cm}\hspace{0.05cm}\cdot \hspace{0.05cm} y}} \hspace{0.1cm}{\rm{d}}y   = 0.5\int_1^3 {{\rm{e}}^{{\rm{j}}\hspace{0.05cm}\cdot \hspace{0.05cm}\Omega\hspace{0.05cm}\cdot \hspace{0.05cm} y} \hspace{0.1cm}{\rm{d}}y.} $$
 *After solving this integral, we get:
-:$$C_y ( {\it \Omega } ) = \frac{{{\rm{e}}^{{\rm{j}}3{\it \Omega } }  - {\rm{e}}^{{\rm{j}}{\it \Omega } } }}{{2{\rm{j}}{\it \Omega } }} = \frac{{{\rm{e}}^{{\rm{j}}{\it \Omega } }  - {\rm{e}}^{{\rm{ - j}}{\it \Omega }} }}{{2{\rm{j}}{\it \Omega } }} \cdot {\rm{e}}^{{\rm{j2}}{\it \Omega } } .$$
+:$$C_y ( {\it \Omega } ) = \frac{{{\rm{e}}^{{\rm{j}}\hspace{0.05cm}\cdot \hspace{0.05cm}3{\it \Omega } }  - {\rm{e}}^{{\rm{j}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega } } }}{{2{\rm{j}}{\it \Omega } }} =  = \frac{{{\rm{e}}^{{\rm{j}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega } }  - {\rm{e}}^{{\rm{ - j}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega }} }}{{2{\rm{j}}{\it \Omega } }} \cdot {\rm{e}}^{{\rm{j\hspace{0.05cm}\cdot \hspace{0.05cm}2}}{\it \Omega } } .$$
 *Using Euler's theorem, this can also be written:
-:$$C_y ( {\it \Omega }  ) = \frac{{\sin ( {\it \Omega }  )}}{{\it \Omega } } \cdot {\rm{e}}^{{\rm{j2}}{\it \Omega } } .$$
+:$$C_y ( {\it \Omega }  ) = \frac{{\sin ( {\it \Omega }  )}}{{\it \Omega } } \cdot {\rm{e}}^{{\rm{j2}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega } } = {\rm si} ( {\it \Omega }  ) \cdot {\rm{e}}^{{\rm{j2}}\hspace{0.05cm}\cdot \hspace{0.05cm}{\it \Omega } }.$$
 *For&nbsp; ${\it \Omega} = \pi/2$&nbsp; we thus obtain a purely real numerical value:
@@ Line 76: / Line 76: @@
-'''(3)'''&nbsp; From the given correspondence it can be read that&nbsp; ${\rm si}(3 {\it \Omega} )$&nbsp; is due to an between&nbsp; $\pm 3$&nbsp; equally distributed random variable and&nbsp; ${\rm si}(2 {\it \Omega} )$&nbsp; gives the transform of a uniform distribution between&nbsp; $\pm 2$&nbsp; .
+'''(3)'''&nbsp; From the given correspondence it can be read that&nbsp; ${\rm si}(3 {\it \Omega} )$&nbsp; is due to an between&nbsp; $\pm 3$&nbsp; equally distributed random variable and&nbsp; ${\rm si}(2 {\it \Omega} )$&nbsp; gives the transform of a uniform distribution between&nbsp; $\pm 2$.
+[[File:P_ID620__Sto_A_3_4_c_neu.png|right|frame|Construction of the trapezoidal PDF]]
-*In the characteristic function, these two proportions are multiplicatively linked. Thus, the resulting PDF&nbsp; $f_z(z)$&nbsp; is the convolution of these two rectangular functions:
+*In the characteristic function, these two proportions are multiplicatively linked.
-[[File:P_ID620__Sto_A_3_4_c_neu.png|center|frame|Construction of trapezoidal PDF]]
+*Thus,&nbsp; the resulting PDF&nbsp; $f_z(z)$&nbsp; is the convolution of these two rectangular functions.
 *The three PDF parameters are thus:
 :$$\hspace{0.15cm}\underline{a = 1},\quad \hspace{0.15cm}\underline{b = 5},

@@ Line 52: / Line 52: @@
 ===Solutions===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct are <u>the proposed solutions 2, 3 and 4</u>:
+'''(1)'''&nbsp; Correct are&nbsp; <u>the proposed solutions 2, 3 and 4</u>:
-* $C_x( {\it \Omega} )$&nbsp; is not the Fourier transform to&nbsp; $f_x(x)$,&nbsp; but the Fourier retransform:
+* $C_x( {\it \Omega} )$&nbsp; is not the Fourier transform to&nbsp; $f_x(x)$,&nbsp; but the inverse Fourier transform:
 :$$C_x( {\it \Omega } ) = \int_{ - \infty }^{ + \infty } {f_x }( x )\cdot {\rm{e}}^{\hspace{0.03cm}{\rm{j}}\hspace{0.03cm}{\it \Omega x}} \hspace{0.1cm}{\rm{d}}x .$$
-*Also for this, the real part is always even and the imaginary part odd.&nbsp; For&nbsp; ${\it \Omega} = 0$&nbsp; holds:
+*Also for this,&nbsp; the real part is always even and the imaginary part odd.&nbsp; For&nbsp; ${\it \Omega} = 0$&nbsp; holds:
 :$$C_x( {\it \Omega} = 0 ) = \int_{ - \infty }^{ + \infty } {f_x }( x ) \hspace{0.1cm}{\rm{d}}x = 1.$$
-*The last alternative does not always hold: &nbsp; A two-point distributed random variable&nbsp; $x \in \{-1, +3\}$&nbsp; with probabilities&nbsp; $0.75$&nbsp; and&nbsp; $0.25$&nbsp; is zero mean&nbsp; $(m_x = 0)$, but still has a complex characteristic function.
+*The last alternative does not always hold: &nbsp; A two-point distributed random variable&nbsp; $x \in \{-1, +3\}$&nbsp; with probabilities&nbsp; $0.75$&nbsp; and&nbsp; $0.25$&nbsp; is zero mean&nbsp; $(m_x = 0)$,&nbsp; but has still a complex characteristic function.
 '''(2)'''&nbsp; According to the general definition:
-:$$C_y( {\it \Omega  } ) = \int_{ - \infty }^{ + \infty } {f_y }( y )\cdot {\rm{e}}^{{\rm{j}}{\it \Omega y}} \hspace{0.1cm}{\rm{d}}y   = 0.5\int_1^3 {{\rm{e}}^{{\rm{j}}\Omega y} \hspace{0.1cm}{\rm{d}}y.} $$
+:$$C_y( {\it \Omega  } ) = \int_{ - \infty }^{ + \infty } {f_y }( y )\cdot {\rm{e}}^{{\rm{j}}\hspace{0.01cm}{\it \Omega\hspace{0.03cm} y}} \hspace{0.1cm}{\rm{d}}y   = 0.5\int_1^3 {{\rm{e}}^{{\rm{j}}\hspace{0.01cm}\Omega\hspace{0.03cm} y} \hspace{0.1cm}{\rm{d}}y.} $$
 *After solving this integral, we get:

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID619__Sto_A_3_4.png|right|frame|Rectangular PDF and trapezoidal PDF]]
+[[File:P_ID619__Sto_A_3_4.png|right|frame|Rectangular and trapezoidal PDF]]
-Given here are the three random variables&nbsp; $x$,&nbsp; $y$&nbsp; and&nbsp; $z$&nbsp;, mostly by their respective probability density functions:
+Given here are three random variables&nbsp; $x$,&nbsp; $y$&nbsp; and&nbsp; $z$,&nbsp; mostly by their respective probability density functions:
-*Nothing else is known about the random variable&nbsp; $x$&nbsp; : &nbsp; This can be both a discrete and a continuous random variable, and can have any PDF&nbsp; $f_x(x)$&nbsp; The mean is generally equal&nbsp; $m_x$.
+*Nothing else is known about the random variable&nbsp; $x$:&nbsp; This can be both a discrete or a continuous random variable,&nbsp; and can have any PDF&nbsp; $f_x(x)$&nbsp; The mean is generally equal&nbsp; $m_x$.
-*The continuous random variable&nbsp; $y$&nbsp; can only take values in the range between&nbsp; $1$&nbsp; to&nbsp; $3$&nbsp; with equal probability.&nbsp; The mean is&nbsp; $$m_y = 2.$$
+*The continuous random variable&nbsp; $y$&nbsp; can take values in the range between&nbsp; $1$&nbsp; to&nbsp; $3$&nbsp; with equal probability.&nbsp; Mean:&nbsp; $m_y = 2.$
 *The random variable&nbsp; $z$&nbsp; has the following characteristic function:
 :$$C_z ({\it \Omega} ) = {\mathop{\rm si}\nolimits}( {3{\it \Omega}} ) \cdot {\mathop{\rm si}\nolimits} ( {2{\it \Omega} } ).$$
-:Besides, the qualitative course of the WDF&nbsp; $f_z(z)$&nbsp; according to the blue sketch is assumed to be known.&nbsp; To be determined are the PDF parameters&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$&nbsp; of this PDF.
+:Besides, the qualitative course of the WDF&nbsp; $f_z(z)$&nbsp; according to the blue sketch is assumed to be known.&nbsp; To be determined are the PDF parameters&nbsp; $a$,&nbsp; $b$,&nbsp; $c$&nbsp; of this PDF.
@@ Line 18: / Line 16: @@
 Hints:
-*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments|expected values and moments]].
+*This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments|Expected values and moments]].
-*Reference is made to the page&nbsp;  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Characteristic_function|charakteristic funcion]]&nbsp;.
+*Reference is made to the section&nbsp;  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments#Characteristic_function|Charakteristic funcion]]&nbsp;.
 *The characteristic function of a between&nbsp; $\pm a$&nbsp; uniformly distributed random variable&nbsp; $z$&nbsp; is:
-:$$C_z ( {\it \Omega} ) = {\mathop{\rm si}\nolimits} ( {a {\it \Omega} } )\quad {\rm{with}}\quad {\mathop{\rm si}\nolimits}( x ) = \sin ( x )/x.$$
+:$$C ( {\it \Omega} ) = {\mathop{\rm si}\nolimits} ( {a {\it \Omega} } )\quad {\rm{with}}\quad {\mathop{\rm si}\nolimits}( x ) = \sin ( x )/x.$$
@@ Line 28: / Line 26: @@
 <quiz display=simple>
-{Which statements are valid with respect to the characteristic function&nbsp; $C_x ( {\it \Omega} )$&nbsp; always &ndash; that is, at any PDF &ndash;?
+{Which statements are valid with respect to the characteristic function&nbsp; $C_x ( {\it \Omega} )$&nbsp; always &ndash; that is, at any PDF&nbsp;?
 |type="[]"}
 - $C_x ( {\it \Omega} )$&nbsp; is the Fourier transform of&nbsp; $f_x(x)$.
@@ Line 34: / Line 32: @@
 + The imaginary part of&nbsp; $C_x ( {\it \Omega} )$&nbsp; is an odd function in&nbsp; ${\it \Omega}$.
 + The value at location&nbsp; ${\it \Omega} = 0$&nbsp; is always&nbsp; $C_x ( {\it \Omega} ) = 1$.
-- For zero mean random variable&nbsp; $(m_x = 0)$&nbsp; is&nbsp; $C_x ( {\it \Omega} )$&nbsp; always real.
+- For a zero mean random variable&nbsp; $(m_x = 0)$&nbsp; &rArr; &nbsp; $C_x ( {\it \Omega} )$&nbsp; is always real.
@@ Line 43: / Line 41: @@
-{Determine the characteristics&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$&nbsp; of the PDF&nbsp; $f_z(z)$.
+{Determine the characteristic parameters&nbsp; $a$,&nbsp; $b$&nbsp; and&nbsp; $c$&nbsp; of the PDF&nbsp; $f_z(z)$.
 |type="{}"}
 $a \ = \ $ { 1 3% }

← Older revision		Revision as of 21:29, 26 December 2021
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−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Kapitel~~	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Expected_Values_and_Moments
	}}		}}

← Older revision		Revision as of 10:07, 20 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^3.3 ~~Erwartungswerte und Momente~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.3 Expected Values and Moments^]]

← Older revision	Revision as of 10:01, 20 December 2021
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← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^3.3 Erwartungswerte und Momente^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.3 Erwartungswerte und Momente^]]

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 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Stochastische_Signaltheorie/Erwartungswerte_und_Momente|Erwartungswerte und Momente]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Erwartungswerte_und_Momente|Erwartungswerte und Momente]].
-*Insbesondere wird auf die Seite&nbsp;  [[Stochastische_Signaltheorie/Erwartungswerte_und_Momente#Charakteristische_Funktion|Charakteristische Funktion]]&nbsp; Bezug genommen.
+*Insbesondere wird auf die Seite&nbsp;  [[Theory_of_Stochastic_Signals/Erwartungswerte_und_Momente#Charakteristische_Funktion|Charakteristische Funktion]]&nbsp; Bezug genommen.
 *Die charakteristische Funktion einer zwischen&nbsp; $\pm a$&nbsp; gleichverteilten Zufallsgröße&nbsp; $z$&nbsp; lautet: