Difference between revisions of "Aufgaben:Exercise 3.3: Moments for Cosine-square PDF"
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}} | }} | ||
| − | [[File:P_ID119__Sto_A_3_3.png|right|frame| | + | [[File:P_ID119__Sto_A_3_3.png|right|frame|Squared cosine PDF <br>and a similar PDF]] |
As in [[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1]] and [[Aufgaben:Exercise_3.2:_cos²-CDF_and_CDF_with_Step_Functions|Exercise 3. 2]] we consider the random variable restricted to the range of values from $-2$ to $+2$ with the following PDF in this section: | As in [[Aufgaben:Exercise_3.1:_cos²-PDF_and_PDF_with_Dirac_Functions|Exercise 3.1]] and [[Aufgaben:Exercise_3.2:_cos²-CDF_and_CDF_with_Step_Functions|Exercise 3. 2]] we consider the random variable restricted to the range of values from $-2$ to $+2$ with the following PDF in this section: | ||
:$$f_x(x)= {1}/{2}\cdot \cos^2({\pi}/{4}\cdot { x}).$$ | :$$f_x(x)= {1}/{2}\cdot \cos^2({\pi}/{4}\cdot { x}).$$ | ||
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*Both density functions are shown in the graph. | *Both density functions are shown in the graph. | ||
| − | *Outside the ranges $-2 < x < +2$ resp. $0 < x < +2$ | + | *Outside the ranges $-2 < x < +2$ resp. $0 < x < +2$ , the followings holds: $f_x(x) = 0$ resp. $f_y(y) = 0$. |
| − | *Both random | + | *Both random variables can be taken as (normalized) instantaneous values of the associated random signals $x(t)$ or $y(t)$ respectively. |
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| − | === | + | ===Questions=== |
<quiz display=simple> | <quiz display=simple> | ||
| − | { | + | {Which of the following statements are true for any given PDF $f_x(x)$ ? |
| − | <br> | + | <br>Used are the following quantities: linear mean $m_1$, root mean square $m_2$, variance $\sigma^2$. |
|type="[]"} | |type="[]"} | ||
| − | - $m_2 = 0$, | + | - $m_2 = 0$, if $m_1 \ne 0$. |
| − | - $m_2 = 0$, | + | - $m_2 = 0$, if $m_1 = 0$. |
| − | + $m_1 = 0$, | + | + $m_1 = 0$, if $m_2 = 0$. |
| − | + $m_2 > \sigma^2$, | + | + $m_2 > \sigma^2$, if $m_1 \ne 0$. |
| − | + $m_1 = 0$, | + | + $m_1 = 0$, if $f_x(-x) = f_x(x)$. |
| − | - $f_x(-x) = f_x(x)$, | + | - $f_x(-x) = f_x(x)$, if $m_1 = 0$. |
| − | { | + | {How large is the DC component (linear mean) of the signal $x(t)$? |
|type="{}"} | |type="{}"} | ||
$m_x \ = \ $ { 0. } | $m_x \ = \ $ { 0. } | ||
| − | { | + | {What is the rms value (Effektivwert?) of the signal $x(t)$? |
|type="{}"} | |type="{}"} | ||
| − | $\sigma_x \ = | + | $\sigma_x \ = \ $ { 0.722 3% } |
| − | { | + | {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 $y(t)$? |
|type="{}"} | |type="{}"} | ||
| − | $m_y\ = | + | $m_y\ = $ { 1 3% } |
| − | { | + | {What is the rms value (Effektivwert) of the signal $y(t)$? |
|type="{}"} | |type="{}"} | ||
| − | $\sigma_y\ = | + | $\sigma_y\ = \ $ { 0.361 3% } |
| − | |||
</quiz> | </quiz> | ||
| − | === | + | ===Solution=== |
{{ML-Kopf}} | {{ML-Kopf}} | ||
'''(1)''' Unter allen Umständen richtig sind <u>die Aussagen 3, 4 und 5</u>: | '''(1)''' Unter allen Umständen richtig sind <u>die Aussagen 3, 4 und 5</u>: | ||
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*Mit der Beziehung $\cos^2(\alpha) = 0.5 \cdot \big[1 + \cos(2\alpha)\big]$ folgt daraus: | *Mit der Beziehung $\cos^2(\alpha) = 0.5 \cdot \big[1 + \cos(2\alpha)\big]$ folgt daraus: | ||
| − | :$$\sigma_x^2=\int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{\rm 2} \hspace{0.1cm}{\rm d}x | + | :$$\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.$$ |
| − | * | + | *These two standard integrals can be found in tables. One obtains with $a = \pi/2$: |
| − | :$$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} | + | :$$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} + \frac{x}{a^2}\cdot {\cos}(a x) + \left( \frac{x^{\rm2}}{{\rm2}a} - \frac{1}{a^3} \right) \cdot \sin(a \cdot x)\right]_{x=0}^{x=2} \hspace{0.5cm} |
| − | \ | + | \rightarrow \hspace{0.5cm} \sigma_{x}^{\rm 2}=\frac{\rm 4}{\rm 3}-\frac{\rm 8}{\rm \pi^2}\approx 0.524\hspace{0.5cm} \Rightarrow \hspace{0.5cm}\sigma_x \hspace{0.15cm}\underline{\approx 0.722}.$$ |
| − | '''(4)''' | + | '''(4)''' Correct is the <u>first mentioned suggestion</u>: |
| − | * | + | *The variant $y = 2x$ would yield a random size distributed between $-4$ and $+4$ . |
| − | * | + | *In the last proposition $y = x/2-1$ the mean $m_y = -1$ would wäre. |
| − | '''(5)''' | + | '''(5)''' From the graph on the specification sheet it is already obvious that $m_y \hspace{0.15cm}\underline{=+1}$ must hold. |
| − | '''(6)''' | + | '''(6)''' The mean value ädoes not change the variance and dispersion. |
| − | * | + | *By compressing by the factor $2$ the scatter becomes smaller againstüber Teilaufgabe '''(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}} | ||
Revision as of 18:11, 27 December 2021
Squared cosine PDF
and a similar PDF
and a similar PDF
As in Exercise 3.1 and Exercise 3. 2 we consider the random variable restricted to the range of values from $-2$ to $+2$ with the following PDF in this section:
- $$f_x(x)= {1}/{2}\cdot \cos^2({\pi}/{4}\cdot { x}).$$
Next to this, we consider a second random variable $y$ that returns only values between $0$ and $2$ with the following PDF:
- $$f_y(y)=\sin^2({\pi}/{2}\cdot y).$$
- Both 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. $f_y(y) = 0$.
- Both random variables can be taken as (normalized) instantaneous values of the associated random signals $x(t)$ or $y(t)$ respectively.
Hints:
- This exercise belongs to the chapter expected values and moments.
- To solve this problem, you can use the following indefinite integral:
- $$\int x^{2}\cdot {\cos}(ax)\,{\rm d}x=\frac{2 x}{ a^{ 2}}\cdot \cos(ax)+ \left [\frac{x^{\rm 2}}{\it a} - \frac{\rm 2}{\it a^{\rm 3}} \right ]\cdot \rm sin(\it ax \rm ) .$$
Questions
Solution
(1) Unter allen Umständen richtig sind die Aussagen 3, 4 und 5:
- Die erste Aussage ist nie erfüllt, wie aus dem Satz von Steiner ersichtlich ist.
- Die zweite Aussage gilt nur im (trivialen) Sonderfall $x = 0$.
Es gibt aber auch mittelwertfreie Zufallsgrößen mit unsymmetrischer WDF.
- Das bedeutet: Die Aussage 6 trifft nicht immer zu.
(2) Aufgrund der WDF-Symmetrie bezüglich $x = 0$ ergibt sich für den linearen Mittelwert $m_x \hspace{0.15cm}\underline{= 0}$.
(3) Der Effektivwert des Signals $x(t)$ ist gleich der Streuung $\sigma_x$ bzw. gleich der Wurzel aus der Varianz $\sigma_x^2$.
- Da die Zufallsgröße $x$ den Mittelwert $m_x {= 0}$ aufweist, ist die Varianz nach dem Satz von Steiner gleich dem quadratischen Mittelwert.
- Dieser wird in Zusammenhang mit Signalen auch als die Leistung $($bezogen auf $1 \ \rm \Omega)$ bezeichnet. Somit gilt:
- $$\sigma_x^{\rm 2}=\int_{-\infty}^{+\infty}x^{\rm 2}\cdot f_x(x)\hspace{0.1cm}{\rm d}x=2 \cdot \int_{\rm 0}^{\rm 2} x^2/2 \cdot \cos^2({\pi}/4\cdot\it x)\hspace{0.1cm} {\rm d}x.$$
- Mit der Beziehung $\cos^2(\alpha) = 0.5 \cdot \big[1 + \cos(2\alpha)\big]$ folgt daraus:
- $$\sigma_x^2=\int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{\rm 2} \hspace{0.1cm}{\rm d}x + \int_{\rm 0}^{\rm 2}{x^{\rm 2}}/{2}\cdot \cos({\pi}/{\rm 2}\cdot\it x) \hspace{0.1cm} {\rm d}x.$$
- These two standard integrals can be found in tables. One obtains with $a = \pi/2$:
- $$\sigma_x^{\rm 2}=\left[\frac{x^{\rm 3}}{\rm 6} + \frac{x}{a^2}\cdot {\cos}(a x) + \left( \frac{x^{\rm2}}{{\rm2}a} - \frac{1}{a^3} \right) \cdot \sin(a \cdot x)\right]_{x=0}^{x=2} \hspace{0.5cm} \rightarrow \hspace{0.5cm} \sigma_{x}^{\rm 2}=\frac{\rm 4}{\rm 3}-\frac{\rm 8}{\rm \pi^2}\approx 0.524\hspace{0.5cm} \Rightarrow \hspace{0.5cm}\sigma_x \hspace{0.15cm}\underline{\approx 0.722}.$$
(4) Correct is the first mentioned suggestion:
- The variant $y = 2x$ would yield a random size distributed between $-4$ and $+4$ .
- In the last proposition $y = x/2-1$ the mean $m_y = -1$ would wäre.
(5) From the graph on the specification sheet it is already obvious that $m_y \hspace{0.15cm}\underline{=+1}$ must hold.
(6) The mean value ädoes not change the variance and dispersion.
- By compressing by the factor $2$ the scatter becomes smaller againstüber Teilaufgabe (3) also by this factor:
- $$\sigma_y=\sigma_x/\rm 2\hspace{0.15cm}\underline{\approx 0.361}.$$
