Aufgaben:Exercise 3.3: Moments for Cosine-square PDF - Revision history

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	[[Category:Theory of Stochastic Signals: Exercises\|^3.3 Expected Values and Moments^]]		[[Category:Theory of Stochastic Signals: Exercises\|^3.3 Expected Values and Moments^]]
	[[de:Aufgaben:3.~~3_Momente_bei_cos²~~-WDF]]		[[de:Aufgaben:Aufgabe 3.3: Momente bei cos²-WDF]]

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	:$$\sigma_x^2=\int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{\rm 2} \hspace{0.1cm}{\rm d}x + \int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{2}\cdot \cos({\pi}/{\rm 2}\cdot\it x) \hspace{0.1cm} {\rm d}x.$$		:$$\sigma_x^2=\int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{\rm 2} \hspace{0.1cm}{\rm d}x + \int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{2}\cdot \cos({\pi}/{\rm 2}\cdot\it x) \hspace{0.1cm} {\rm d}x.$$
	*These two standard integrals can be found in tables. One obtains with  $a = \pi/2$:		*These two standard integrals can be found in tables. One obtains with  $a = \pi/2$:
	:$$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} + \frac{x}{a^2}\cdot {\cos}(a x) + \left( \frac{x^{\rm2}}{{\rm2}a} - \frac{1}{a^3} \right) \cdot \sin(a \cdot x)\right]_{x=0}^{x=2} \hspace{0.5cm}		:$$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} + \frac{x}{a^2}\cdot {\cos}(a x) + \left( \frac{x^{\rm2}}{{\rm2}a} - \frac{1}{a^3} \right) \cdot \sin(a \cdot x)\right]_{x=0}^{x=2} \hspace{0.5cm}\rightarrow \hspace{0.5cm} \sigma_{x}^{\rm 2}=\frac{\rm 4}{\rm 3}-\frac{\rm 8}{\rm \pi^2}\approx 0.524\hspace{0.5cm} \Rightarrow \hspace{0.5cm}\sigma_x \hspace{0.15cm}\underline{\approx 0.722}.$$
	\rightarrow \hspace{0.5cm} \sigma_{x}^{\rm 2}=\frac{\rm 4}{\rm 3}-\frac{\rm 8}{\rm \pi^2}\approx 0.524\hspace{0.5cm} \Rightarrow \hspace{0.5cm}\sigma_x \hspace{0.15cm}\underline{\approx 0.722}.$$

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	[[Category:Theory of Stochastic Signals: Exercises\|^3.3 Expected Values and Moments^]]		[[Category:Theory of Stochastic Signals: Exercises\|^3.3 Expected Values and Moments^]]
			[[de:Aufgaben:3.3_Momente_bei_cos²-WDF]]

Hwang at 12:20, 18 January 2023

2023-01-18T12:20:33Z

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	<quiz display=simple>		<quiz display=simple>
	{Which of the following statements are true for any given PDF  $f_x(x)$ ?  Used are the following quantities:   linear mean  $m_1$,  ~~quadratic mean~~  $m_2$,  variance  $\sigma^2$.		{Which of the following statements are true for any given PDF  $f_x(x)$ ?  Used are the following quantities:   linear mean  $m_1$,  second moment  $m_2$,  variance  $\sigma^2$.
	\|type="[]"}		\|type="[]"}
	- $m_2 = 0$,  if   $m_1 \ne 0$.		- $m_2 = 0$,  if   $m_1 \ne 0$.
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	'''(3)'''  The standard deviation of the signal  $x(t)$  is equal to the standard deviation   $\sigma_x$  or equal to the root of the variance  $\sigma_x^2$.		'''(3)'''  The standard deviation of the signal  $x(t)$  is equal to the standard deviation   $\sigma_x$  or equal to the root of the variance  $\sigma_x^2$.
	*Since the random variable  $x$  has mean  $m_x {= 0}$,  the variance is equal to the ~~quadratic mean~~ according to Steiner's theorem.		*Since the random variable  $x$  has mean  $m_x {= 0}$,  the variance is equal to the second moment according to Steiner's theorem.
	*This is also referred to as the power  $($with respect to  $1 \ \rm \Omega)$  in the context of signals. Thus:		*This is also referred to as the power  $($with respect to  $1 \ \rm \Omega)$  in the context of signals. Thus:
	:$$\sigma_x^{\rm 2}=\int_{-\infty}^{+\infty}x^{\rm 2}\cdot f_x(x)\hspace{0.1cm}{\rm d}x=2 \cdot \int_{\rm 0}^{\rm 2} x^2/2 \cdot \cos^2({\pi}/4\cdot\it x)\hspace{0.1cm} {\rm d}x.$$		:$$\sigma_x^{\rm 2}=\int_{-\infty}^{+\infty}x^{\rm 2}\cdot f_x(x)\hspace{0.1cm}{\rm d}x=2 \cdot \int_{\rm 0}^{\rm 2} x^2/2 \cdot \cos^2({\pi}/4\cdot\it x)\hspace{0.1cm} {\rm d}x.$$

Bene: Text replacement - " rms " to " standard deviation "

2022-02-17T11:34:20Z

Text replacement - " rms " to " standard deviation "

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	'''(3)'''  The standard deviation of the signal  $x(t)$  is equal to the ~~rms~~   $\sigma_x$  or equal to the root of the variance  $\sigma_x^2$.		'''(3)'''  The standard deviation of the signal  $x(t)$  is equal to the standard deviation   $\sigma_x$  or equal to the root of the variance  $\sigma_x^2$.
	*Since the random variable  $x$  has mean  $m_x {= 0}$,  the variance is equal to the quadratic mean according to Steiner's theorem.		*Since the random variable  $x$  has mean  $m_x {= 0}$,  the variance is equal to the quadratic mean according to Steiner's theorem.
	*This is also referred to as the power  $($with respect to  $1 \ \rm \Omega)$  in the context of signals. Thus:		*This is also referred to as the power  $($with respect to  $1 \ \rm \Omega)$  in the context of signals. Thus:
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	'''(6)'''  The mean value does not change the variance and rms.		'''(6)'''  The mean value does not change the variance and rms.
	*By compressing by the factor  $2$  the ~~rms~~ becomes smaller compared to subtask  '''(3)'''  also by this factor:		*By compressing by the factor  $2$  the standard deviation becomes smaller compared to subtask  '''(3)'''  also by this factor:
	:$$\sigma_y=\sigma_x/\rm 2\hspace{0.15cm}\underline{\approx 0.361}.$$		:$$\sigma_y=\sigma_x/\rm 2\hspace{0.15cm}\underline{\approx 0.361}.$$
	{{ML-Fuß}}		{{ML-Fuß}}

Bene: Text replacement - "rms value" to "standard deviation"

2022-02-17T11:11:21Z

Text replacement - "rms value" to "standard deviation"

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	{What is the ~~rms value~~ of the signal  $x(t)$?		{What is the standard deviation of the signal  $x(t)$?
	\|type="{}"}		\|type="{}"}
	$\sigma_x \ = \ $ { 0.722 3% }		$\sigma_x \ = \ $ { 0.722 3% }
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	{What is the ~~rms value~~ of the signal  $y(t)$?		{What is the standard deviation of the signal  $y(t)$?
	\|type="{}"}		\|type="{}"}
	$\sigma_y\ = \ $ { 0.361 3% }		$\sigma_y\ = \ $ { 0.361 3% }

Guenter: Guenter moved page Aufgaben:Exercise 3.3: Moments for cos²-PDF to Aufgaben:Exercise 3.3: Moments for Cosine-square PDF

2022-02-02T15:42:07Z

Guenter moved page Aufgaben:Exercise 3.3: Moments for cos²-PDF to Aufgaben:Exercise 3.3: Moments for Cosine-square PDF

← Older revision		Revision as of 17:42, 2 February 2022
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Guenter at 13:49, 5 January 2022

2022-01-05T13:49:49Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Under all circumstances, <u>statements 3, 4, and 5</u> are correct:		'''(1)'''  Under all circumstances,  <u>statements 3, 4, and 5</u>  are correct:
	*The first statement is never true, as is evident from the  ~~<i>~~Steiner's theorem~~</i> ~~.		*The first statement is never true,  as is evident from the  "Steiner's theorem".
	*The second statement is only valid in the (trivial) special case  $x = 0$.		*The second statement is only valid in the  (trivial)  special case  $x = 0$.


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	'''(3)'''  The ~~rms~~ of the signal  $x(t)$  is equal to the rms   $\sigma_x$  or equal to the root of the variance  $\sigma_x^2$.		'''(3)'''  The standard deviation of the signal  $x(t)$  is equal to the rms   $\sigma_x$  or equal to the root of the variance  $\sigma_x^2$.
	*Since the random variable  $x$  has mean  $m_x {= 0}$ , the variance is equal to the ~~root~~ mean ~~square~~ according to ~~<i>~~Steiner's theorem~~</i>~~.		*Since the random variable  $x$  has mean  $m_x {= 0}$,  the variance is equal to the quadratic mean according to Steiner's theorem.
	*This is also referred to as the power  $($with respect to  $1 \ \rm \Omega)$  in the context of signals. Thus:		*This is also referred to as the power  $($with respect to  $1 \ \rm \Omega)$  in the context of signals. Thus:
	:$$\sigma_x^{\rm 2}=\int_{-\infty}^{+\infty}x^{\rm 2}\cdot f_x(x)\hspace{0.1cm}{\rm d}x=2 \cdot \int_{\rm 0}^{\rm 2} x^2/2 \cdot \cos^2({\pi}/4\cdot\it x)\hspace{0.1cm} {\rm d}x.$$		:$$\sigma_x^{\rm 2}=\int_{-\infty}^{+\infty}x^{\rm 2}\cdot f_x(x)\hspace{0.1cm}{\rm d}x=2 \cdot \int_{\rm 0}^{\rm 2} x^2/2 \cdot \cos^2({\pi}/4\cdot\it x)\hspace{0.1cm} {\rm d}x.$$

	*With the relation  $\cos^2(\alpha) = 0.5 \cdot \big[1 + \cos(2\alpha)\big]$  it follows:		*With the relation  $\cos^2(\alpha) = 0.5 \cdot \big[1 + \cos(2\alpha)\big]$  it follows:
	:$$\sigma_x^2=\int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{\rm 2} \hspace{0.1cm}{\rm d}x + \int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{2}\cdot \cos({\pi}/{\rm 2}\cdot\it x) \hspace{0.1cm} {\rm d}x.$$		:$$\sigma_x^2=\int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{\rm 2} \hspace{0.1cm}{\rm d}x + \int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{2}\cdot \cos({\pi}/{\rm 2}\cdot\it x) \hspace{0.1cm} {\rm d}x.$$

	*These two standard integrals can be found in tables. One obtains with  $a = \pi/2$:		*These two standard integrals can be found in tables. One obtains with  $a = \pi/2$:
	:$$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} + \frac{x}{a^2}\cdot {\cos}(a x) + \left( \frac{x^{\rm2}}{{\rm2}a} - \frac{1}{a^3} \right) \cdot \sin(a \cdot x)\right]_{x=0}^{x=2} \hspace{0.5cm}		:$$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} + \frac{x}{a^2}\cdot {\cos}(a x) + \left( \frac{x^{\rm2}}{{\rm2}a} - \frac{1}{a^3} \right) \cdot \sin(a \cdot x)\right]_{x=0}^{x=2} \hspace{0.5cm}
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	'''(4)'''  Correct is the <u>first mentioned suggestion</u>:		'''(4)'''  Correct is the  <u>first mentioned suggestion</u>:
	*The variant  $y = 2x$  would yield a random ~~size~~ distributed between  $-4$  and  $+4$~~ ~~ .		*The variant  $y = 2x$  would yield a random variable distributed between  $-4$  and  $+4$.
	*In the last proposition  $y = x/2-1$  the mean  $m_y = -1$ ~~would wäre~~.		*In the last proposition  $y = x/2-1$  the mean would be  $m_y = -1$.

Guenter at 13:29, 5 January 2022

2022-01-05T13:29:41Z

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	:$$f_x(x)= {1}/{2}\cdot \cos^2({\pi}/{4}\cdot { x}).$$		:$$f_x(x)= {1}/{2}\cdot \cos^2({\pi}/{4}\cdot { x}).$$

	Next to this, we consider a second random variable  $y$ that returns only values between  $0$  and  $2$  with the following PDF:		Next to this,  we consider a second random variable  $y$  that returns only values between  $0$  and  $2$  with the following PDF:
	:$$f_y(y)=\sin^2({\pi}/{2}\cdot y).$$		:$$f_y(y)=\sin^2({\pi}/{2}\cdot y).$$

	*Both density functions are shown in the graph.		*Both probability density functions are shown in the graph.
	*Outside the ranges  $-2 < x < +2$  resp.  $0 < x < +2$ , the followings holds:   $f_x(x) = 0$  resp.  $f_y(y) = 0$.		*Outside the ranges  $-2 < x < +2$  resp.  $0 < x < +2$,  the followings holds:   $f_x(x) = 0$  resp.  $f_y(y) = 0$.
	*Both random variables can be taken as (normalized) instantaneous values of the associated random signals  $x(t)$  or  $y(t)$  ~~respectively~~.		*Both random variables can be taken as  (normalized)  instantaneous values of the associated random signals  $x(t)$  or  $y(t)$  resp.





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	Hints:		Hints:
	*This exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments\|~~expected~~ values and moments]].		*This exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Expected_Values_and_Moments\|Expected values and moments]].

	*To solve this problem, you can use the following indefinite integral:		*To solve this problem,  you can use the following indefinite integral:
	:$$\int x^{2}\cdot {\cos}(ax)\,{\rm d}x=\frac{2 x}{ a^{ 2}}\cdot \cos(ax)+ \left [\frac{x^{\rm 2}}{\it a} - \frac{\rm 2}{\it a^{\rm 3}} \right ]\cdot \rm sin(\it ax \rm ) .$$		:$$\int x^{2}\cdot {\cos}(ax)\,{\rm d}x=\frac{2 x}{ a^{ 2}}\cdot \cos(ax)+ \left [\frac{x^{\rm 2}}{\it a} - \frac{\rm 2}{\it a^{\rm 3}} \right ]\cdot \rm sin(\it ax \rm ) .$$

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	<quiz display=simple>		<quiz display=simple>
	{Which of the following statements are true for any given PDF  $f_x(x)$ ?		{Which of the following statements are true for any given PDF  $f_x(x)$ ?  Used are the following quantities:   linear mean  $m_1$,  quadratic mean  $m_2$,  variance  $\sigma^2$.
	~~<br>~~Used are the following quantities:   linear mean  $m_1$,  quadratic mean  $m_2$,  variance  $\sigma^2$.
	\|type="[]"}		\|type="[]"}
	- $m_2 = 0$,  if   $m_1 \ne 0$.		- $m_2 = 0$,  if   $m_1 \ne 0$.
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	{~~How large~~ is the DC component (linear mean) of the signal  $x(t)$?		{What is the DC component  ("linear mean")  of the signal  $x(t)$?
	\|type="{}"}		\|type="{}"}
	$m_x \ = \ $ { 0. }		$m_x \ = \ $ { 0. }
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	{The random variable  $y$  can be derived from  $x$ . Which assignment is valid?		{The random variable  $y$  can be derived from  $x$.  Which assignment is valid?
	\|type="()"}		\|type="()"}
	+ $y = 1+x/2.$		+ $y = 1+x/2.$
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	{~~How large~~ is the DC component of the signal $y(t)$?		{What is the DC component of the signal  $y(t)$?
	\|type="{}"}		\|type="{}"}
	$m_y\ = $ { 1 3% }		$m_y\ = $ { 1 3% }

Guenter: Guenter moved page Aufgaben:Exercise 3.3: Moments at cos²-PDF to Aufgaben:Exercise 3.3: Moments for cos²-PDF

2022-01-05T13:22:39Z

Guenter moved page Aufgaben:Exercise 3.3: Moments at cos²-PDF to Aufgaben:Exercise 3.3: Moments for cos²-PDF

← Older revision		Revision as of 15:22, 5 January 2022
(No difference)