Aufgaben:Exercise 1.3Z: Rayleigh Fading Revisited: Unterschied zwischen den Versionen
Javier (Diskussion | Beiträge) |
Javier (Diskussion | Beiträge) |
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Zeile 17: | Zeile 17: | ||
\\ {\rm f\ddot{u}r}\hspace{0.15cm} p < 0 \\ \end{array} | \\ {\rm f\ddot{u}r}\hspace{0.15cm} p < 0 \\ \end{array} | ||
.$$ | .$$ | ||
− | * | + | * For channel $\rm B$ ("Blue") only the point cloud is given. It must be estimated whether Rayleigh fading is also present here, and if YES, how large the parameter $\sigma = \sigma_{\rm B}$ is for this channel. |
− | * Finally, subtask '''(3)''' also refers to PDF $f_{\it \phi}(\phi)$ the phase function $\phi(t)$ | + | * Finally, subtask '''(3)''' also refers to PDF $f_{\it \phi}(\phi)$ of the phase function $\phi(t)$ , which is defined as follows: |
$$\phi(t) = \arctan \hspace{0.15cm} \frac{y(t)}{x(t)} | $$\phi(t) = \arctan \hspace{0.15cm} \frac{y(t)}{x(t)} | ||
Zeile 29: | Zeile 29: | ||
''Notes:'' | ''Notes:'' | ||
− | * | + | * This task belongs to chapter [[Mobile_Kommunikation/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings|Wahrscheinlichkeitsdichte des Rayleigh–Fadings]] of this book. |
− | * A similar topic is treated with a different approach in chapter [[ | + | * A similar topic is treated with a different approach in chapter [[Stochastische_Signaltheorie/Weitere_Verteilungen|Weitere Verteilungen]] of the book „Stochastic Signal Theory”. |
* To check your results you can use the interactive applet [[Applets:WDF_VTF|WDF, VTF and Moments]] of the book „Stochastic Signal Theory”. | * To check your results you can use the interactive applet [[Applets:WDF_VTF|WDF, VTF and Moments]] of the book „Stochastic Signal Theory”. | ||
Zeile 37: | Zeile 37: | ||
===Questionnaire=== | ===Questionnaire=== | ||
<quiz display=simple> | <quiz display=simple> | ||
− | Can the channel $\rm B$ also be modeled | + | {Can the channel $\rm B$ also be modeled as Rayleigh fading? |
|type="()"} | |type="()"} | ||
+ Yes. | + Yes. | ||
- No. | - No. | ||
− | Estimate the Rayleigh | + | {Estimate the Rayleigh parameter of channel $\rm B$. Reminder: For channel $\rm R$ this parameter has the value $\sigma_{\rm R} = 0.5$. |
|type="{}"} | |type="{}"} | ||
$\sigma_{\rm B}\ = \ $ { 0.707 3% } | $\sigma_{\rm B}\ = \ $ { 0.707 3% } | ||
− | { | + | {Are the probability density functions (PDFs) $f_{\it \phi}(\phi)$ of the phases different for channel $\rm R$ and $\rm B$ and if YES, how? |
|type="()"} | |type="()"} | ||
− | - | + | - Yes. |
+ No. | + No. | ||
− | {What is the | + | {What is the PDF $f_a(a)$ with $a(t) = |z(t)|$ in both cases? |
|type="()"} | |type="()"} | ||
− | - The amount $a(t)$ is | + | - The amount $a(t)$ is Gaussian distributed. |
− | + The amount $a(t)$ is | + | + The amount $a(t)$ is Rayleigh distributed. |
− | - The amount $a(t)$ is positive | + | - The amount $a(t)$ is positive and exponentially distributed. |
− | {What is the | + | {What is the PDF $f_p(p)$ in both cases with $p(t) = |z(t)|^2$? |
|type="()"} | |type="()"} | ||
− | - The amount $p(t)$ is | + | - The amount $p(t)$ is Gaussian distributed. |
− | - The amount $p(t)$ is | + | - The amount $p(t)$ is Rayleigh distributed. |
− | + The amount $p(t)$ is positive | + | + The amount $p(t)$ is positive and exponentially distributed. |
− | {What is the probability that the | + | {What is the probability that the magnitude $a(t) = |z(t)|$ is greater than a given value $A$? |
|type="()"} | |type="()"} | ||
− | - | + | - ${\rm Pr}(|z(t)|) > A) = {\rm ln}(A/\sigma).$ |
− | + | + | + ${\rm Pr}(|z(t)|) > A) = {\rm e}^{-A^2/2\sigma^2}.$ |
− | - | + | - ${\rm Pr}(|z(t)|) > A) = {\rm e}^{-A/\sigma}.$ |
{Calculate the probability ${\rm Pr}(|z(t)| > 1)$ for both channels. | {Calculate the probability ${\rm Pr}(|z(t)| > 1)$ for both channels. | ||
Zeile 73: | Zeile 73: | ||
Channel $\rm R$:$\hspace{0.4cm} {\rm Pr}(|z(t)| > 1)\ = \ $ { 0.135 3% } | Channel $\rm R$:$\hspace{0.4cm} {\rm Pr}(|z(t)| > 1)\ = \ $ { 0.135 3% } | ||
Channel $\rm B$:$\hspace{0.4cm} {\rm Pr}(|z(t)| > 1)\ = \ $ { 0.368 3% } | Channel $\rm B$:$\hspace{0.4cm} {\rm Pr}(|z(t)| > 1)\ = \ $ { 0.368 3% } | ||
− | </quiz | + | </quiz> |
===Sample solution=== | ===Sample solution=== | ||
{{ML-Kopf}} | {{ML-Kopf}} | ||
− | '''(1)''' <u> | + | '''(1)''' <u>YES</u>: |
*You can see the rotational symmetry here, too, if you consider that only $N = 10\hspace{0.05cm}000$ samples were displayed in the complex plane. | *You can see the rotational symmetry here, too, if you consider that only $N = 10\hspace{0.05cm}000$ samples were displayed in the complex plane. | ||
*Furthermore the following questions would not make sense if you answer NO. | *Furthermore the following questions would not make sense if you answer NO. |
Version vom 25. März 2020, 18:05 Uhr
The graph shows the multiplicative factor $z(t) = x(t) + {\rm j} \cdot y(t)$ of two mobile radio channels (both without multipath propagation) in 2D–representation. The following is assumed:
- The channel $\rm R$ (the designation results from the color „Red” of the point cloud) is Rayleigh distributed with $\sigma_{\rm R} = 0.5$.
- The probability density functions (PDF) of the magnitude $a(t) = |z(t)|$ or the squared magnitude $p(t) = |z(t)|^2$ are $($with $\sigma = \sigma_{\rm R})$:
- $$f_a(a) = \left\{ \begin{array}{c} a/\sigma^2 \cdot {\rm e}^{ -{a^2}/(2\sigma^2)} \\ 0 \end{array} \right.\quad \begin{array}{*{1}c} {\rm f\ddot{u}r}\hspace{0.15cm} a \ge 0 \\ {\rm f\ddot{u}r}\hspace{0.15cm} a < 0 \\ \end{array} \hspace{0.05cm},$$
- $$f_p(p) = \left\{ \begin{array}{c} 1/(2\sigma^2) \cdot {\rm e}^{ -{p}/(2\sigma^2)} \\ 0 \end{array} \right.\quad \begin{array}{*{1}c} {\rm f\ddot{u}r}\hspace{0.15cm} p \ge 0 \\ {\rm f\ddot{u}r}\hspace{0.15cm} p < 0 \\ \end{array} .$$
- For channel $\rm B$ ("Blue") only the point cloud is given. It must be estimated whether Rayleigh fading is also present here, and if YES, how large the parameter $\sigma = \sigma_{\rm B}$ is for this channel.
- Finally, subtask (3) also refers to PDF $f_{\it \phi}(\phi)$ of the phase function $\phi(t)$ , which is defined as follows:
$$\phi(t) = \arctan \hspace{0.15cm} \frac{y(t)}{x(t)} \hspace{0.05cm}.$$
Notes:
- This task belongs to chapter Wahrscheinlichkeitsdichte des Rayleigh–Fadings of this book.
- A similar topic is treated with a different approach in chapter Weitere Verteilungen of the book „Stochastic Signal Theory”.
- To check your results you can use the interactive applet WDF, VTF and Moments of the book „Stochastic Signal Theory”.
Questionnaire
Sample solution
- You can see the rotational symmetry here, too, if you consider that only $N = 10\hspace{0.05cm}000$ samples were displayed in the complex plane.
- Furthermore the following questions would not make sense if you answer NO.
(2) By measuring the two drawn circles you can see that for the „blue” channel the scattering of real– and imaginary part are larger by a factor of about 1.4 (exactly: $\sqrt{2}$) than for the „red” channel: $$\sigma_{\rm B} = \sigma_{\rm R} \cdot \sqrt{2} = 0.5 \cdot \sqrt{2}= {1}/{\sqrt{2}}\hspace{0.15cm} \underline{\approx 0.707} \hspace{0.05cm}.$$
(3) correct is NO:
- In both cases $f_{\it \phi}(\phi)$ describes an equal distribution between $-\pi$ and $+\pi$.
- The larger amplitudes of channel $\rm B$ are not important for the phase function $\phi(t)$.
(4) Correct is the solution 2:
- With Rayleigh–Fading, the real part $x(t)$ and the imaginary part $y(t)$ are each Gaussian distributed.
- The exponential distribution results for the square of the absolute value $p(t) = |z(t)|^2$.
(5) Correct here is the solution 3, as already explained in the sample solution for (4).
(6) The amount $a(t)$ is rayleigh distributed. Therefore, the following applies for the searched probability: $${\rm Pr}(a > A) = \int_{A}^{\infty}\frac{a}{\sigma^2} \cdot {\rm e}^{ -{a^2}/(2\sigma^2)} \hspace{0.15cm}{\rm d}a \hspace{0.05cm}.$ *In some formula collections you can find the solution for this integral, but not in all. *But it is also valid with the one-sided–exponentially distributed random variable $p = a^2$: $${\rm Pr}(a > A) = {\rm Pr}(p > A^2) = \frac{1}{2\sigma^2} \cdot\int_{A^2}^{\infty} {\rm e}^{ -{p}/(2\sigma^2)} \hspace{0.15cm}{\rm d}p \hspace{0.05cm}.$$ *This integral is elementary and gives the result: $${\rm Pr}(a > A) = {\rm e}^{ -{A^2}/(2\sigma^2)} \hspace{0.05cm}.$$ Correct is therefore the <u>solution 2</u>. '''(7)''' For the channel $\rm R$ applies with $\sigma = 0.5$: $${\rm Pr}(|z(t)| > 1) = {\rm e}^{-2} \hspace{0.15cm} \underline{\approx 0.135}
\hspace{0.05cm}.$$
- In the upper graph this corresponds to the number of all points outside the circle drawn, based on the number $N = $10,000$ of all points. *For the channel $\rm B$ because of the double variance $\sigma^2 = 0.5$, ${\rm Pr}(|z(t)|>1) = {\rm e}^{\rm –1} applies. \ \underline {\approx \ 0.368}$. *The (not drawn) reference circle would also have the radius 1 in the lower graph. *The circle drawn in the lower graphic has a larger radius than $A = $1, namely $A = \sqrt{2}\approx $1,414.