Aufgaben:Exercise 3.5Z: Antenna Areas - Revision history

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

2022-02-17T11:11:24Z

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

Guenter at 09:09, 13 January 2022

2022-01-13T09:09:15Z

Guenter at 08:56, 13 January 2022

2022-01-13T08:56:34Z

Noah at 21:55, 29 December 2021

2021-12-29T21:55:02Z

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Noah at 23:21, 28 December 2021

2021-12-28T23:21:07Z

Guenter at 10:11, 20 December 2021

2021-12-20T10:11:34Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.5Z: Antennengebiete to Aufgaben:Exercise 3.5Z: Antenna Areas

2021-12-20T10:09:46Z

Guenter moved page Aufgabe 3.5Z: Antennengebiete to Exercise 3.5Z: Antenna Areas

Javier: Text replacement - "„" to """

2021-05-28T14:32:00Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:30:45Z

Text replacement - "”" to """

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

2021-03-23T12:44:14Z

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

@@ Line 41: / Line 41: @@
-{What value results for the rms value of the random variable&nbsp; $\alpha$&nbsp; in the area&nbsp; $K$?
+{What value results for the standard deviation of the random variable&nbsp; $\alpha$&nbsp; in the area&nbsp; $K$?
 |type="{}"}
 $\sigma_\alpha \ = \ $ { 1.814 3% }

@@ Line 90: / Line 90: @@
-'''(3)'''&nbsp; F&uuml;r the variance or scatter of the angle of incidence&nbsp; $\alpha$&nbsp; holds:
+'''(3)'''&nbsp; For the variance of the angle of incidence&nbsp; $\alpha$&nbsp; holds:
 :$$\sigma_{\alpha}^{\rm 2}=\int_{-\rm\pi}^{\rm\pi}\hspace{-0.1cm}\it\alpha^{\rm 2}\cdot \it f_{\alpha}(\alpha)\,\,{\rm d} \alpha=\frac{\rm 1}{\rm 2\cdot\it \pi}\cdot \frac{\alpha^{\rm 3}}{\rm 3}\Bigg|_{\rm -\pi}^{\rm\pi}=\frac{\rm 2\cdot\pi^{3}}{\rm 2\cdot\rm \pi\cdot \rm 3}=\frac{\rm \pi^2}{\rm 3} = \rm 3. 29. \hspace{0.5cm}\Rightarrow \hspace{0.5cm}\sigma_{\alpha}\hspace{0.15cm}\underline{=1.814}.$$
@@ Line 106: / Line 106: @@
-'''(6)'''&nbsp; Correct <u>Suggested solution 3</u>:
+'''(6)'''&nbsp; Correct is&nbsp; <u>the suggested solution 3</u>:
-*The probability density function (PDF)&nbsp; $f_\alpha(\alpha)$&nbsp; is f&nbsp; for a given angle&nbsp; $\alpha$&nbsp; directly proportional to the distance&nbsp; $A$&nbsp; between the antenna and the boundary line.
+*The PDF&nbsp; $f_\alpha(\alpha)$&nbsp; is f&nbsp; for a given angle&nbsp; $\alpha$&nbsp; directly proportional to the distance&nbsp; $A$&nbsp; between the antenna and the boundary line.
 *For&nbsp; $\alpha = \pm 2\pi/3 = \pm 120^\circ$&nbsp; against&nbsp; $A = R$,&nbsp; for&nbsp; $\alpha \pm \pi = \pm 180^\circ$&nbsp; against&nbsp; $A = R/2$.
 *In between the distance becomes successively smaller.&nbsp; This means: &nbsp; The PDF decreases towards the boundary.
@@ Line 125: / Line 125: @@
-'''(8)'''&nbsp; Since the overall PDF area is constantly equal&nbsp; $1$&nbsp; and the PDF decreases at the boundaries, it must have a larger value in the range&nbsp; $|\alpha| < 2\pi/3$&nbsp; than calculated in&nbsp; '''(1)'''&nbsp; .
+'''(8)'''&nbsp; Since the overall PDF area is constantly equal&nbsp; $1$&nbsp; and the PDF decreases at the boundaries, it must have a larger value in the range&nbsp; $|\alpha| < 2\pi/3$&nbsp; than in&nbsp; '''(1)'''.
 * With the results from&nbsp; '''(1)'''&nbsp; and&nbsp; '''(7)'''&nbsp; holds:
 :$$f_{\alpha}(\alpha = 0)=\frac{1/(2\pi)}{2/3+{\rm sin(\rm 60^{\circ})}/({\rm 2\pi})} = \frac{\rm 1}{{\rm 4\cdot\pi}/{\rm 3}+\rm sin(60^{\circ})}\hspace{0.15cm}\underline{\approx \rm 0.198}.$$
-*Like the probability calculated in item&nbsp; '''(7)'''&nbsp; also simultaneously the PDF value in the range&nbsp; $|\alpha| < 2\pi/3$&nbsp; increases by a factor&nbsp; $1.242$&nbsp; as the coverage area becomes smaller.
+*Like the probability in&nbsp; '''(7)'''&nbsp; also simultaneously the PDF value in the range&nbsp; $|\alpha| < 2\pi/3$&nbsp; increases by a factor&nbsp; $1.242$&nbsp; as the coverage area becomes smaller.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID188__Sto_Z_3_5.png|right|frame|Two antenna areas: &nbsp;$K$&nbsp; and&nbsp; $G$]]
-We first consider &ndash; as sketched in the image above &ndash; a receiving antenna serving a circular area&nbsp; $K$&nbsp; . It is assumed that this antenna can detect all signals incident at different angles&nbsp; $\alpha$&nbsp; equally well:
+We first consider &ndash; as sketched in the image above &ndash; a receiving antenna serving a circular area&nbsp; $K$.&nbsp;  It is assumed that this antenna can detect all signals incident at different angles&nbsp; $\alpha$&nbsp; equally well:
-*According to the sketch, the angle&nbsp; $\alpha$&nbsp; refers to the&nbsp; $x$&ndash;axis.
+*According to the sketch,&nbsp; the angle&nbsp; $\alpha$&nbsp; refers to the&nbsp; $x$&ndash;axis.
-*The value $\alpha = 0$ therefore means that the signal is moving towards the antenna in the direction of the negative $x$&ndash;axis.
+*The value&nbsp; $\alpha = 0$&nbsp; therefore means that the signal is moving towards the antenna in the direction of the negative&nbsp; $x$&ndash;axis.
 Further we assume:
 *The range of values of the angle of incidence&nbsp; $\alpha$&nbsp; with this definition&nbsp; $-\pi < \alpha \le +\pi$.
-*There are very many users in the coverage area whose positions&nbsp; $(x, y)$&nbsp; are "statistically" distributed over the area&nbsp; $K$&nbsp;.
+*There are very many users in the coverage area whose positions&nbsp; $(x, y)$&nbsp; are&nbsp; "statistically distributed"&nbsp; over the area&nbsp; $K$.&nbsp;
 From subtask&nbsp; '''(5)'''&nbsp; we assume the coverage area&nbsp; $G$&nbsp; outlined below.
-*Because of an obstacle, the&nbsp; $x$&ndash;coordinate of all participants must now be greater&ouml;&space;than&nbsp; $-R/2$&nbsp;.
+*Because of an obstacle,&nbsp; the&nbsp; $x$&ndash;coordinate of all participants must now be greater&ouml;&space;than&nbsp; $-R/2$.&nbsp;
-*Also in the coverage area&nbsp; $G$&nbsp; the subscribers would again be "statistically distributed".
+*Also in the coverage area&nbsp; $G$&nbsp; the subscribers would again be&nbsp; "statistically distributed".
@@ Line 44: / Line 41: @@
-{What value results for the rms of the random size&nbsp; $\alpha$&nbsp; in the area&nbsp; $K$?
+{What value results for the rms value of the random variable&nbsp; $\alpha$&nbsp; in the area&nbsp; $K$?
 |type="{}"}
 $\sigma_\alpha \ = \ $ { 1.814 3% }
-{What is the probability that in area&nbsp; $K$&nbsp; the antenna locates a user at an angle between&nbsp; $\pm45^\circ$&nbsp; ?
+{What is the probability that in area&nbsp; $K$&nbsp; the antenna locates a user at an angle between&nbsp; $\pm45^\circ$ ?
 |type="{}"}
 ${\rm Pr}(-π/4 ≤ α ≤ +π/4) \ = \ $ { 25 3% } $\ \%$
@@ Line 62: / Line 59: @@
 {What statements are now valid with respect to&nbsp; $f_\alpha(\alpha)$&nbsp; in the range&nbsp; $|\alpha| > \alpha_0$&nbsp; ?
 |type="[]"}
-- The PDF has the same course "outside" as "inside".
+- The PDF has the same course&nbsp; "outside"&nbsp; as&nbsp; "inside".
-- The PDF is "outside" identically zero.
+- The PDF is&nbsp; "outside"&nbsp; identically zero.
 + The PDF decreases towards the edges in this area.
 - The PDF increases towards the edges in this area.

← Older revision		Revision as of 23:21, 28 December 2021
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−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Gleichverteilte Zufallsgröße~~	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Uniformly_Distributed_Random_Variables
	}}		}}

← Older revision		Revision as of 10:11, 20 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^3.4 ~~Gleichverteilte Zufallsgröße~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.4 Uniformly Distributed Random Variable^]]

← Older revision	Revision as of 10:09, 20 December 2021
(No difference)

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 Weiter setzen wir voraus:
 *Der Wertebereich des Einfallswinkels&nbsp; $\alpha$&nbsp; betr&auml;gt mit dieser Definition&nbsp; $-\pi < \alpha \le +\pi$.
-*Es halten sich sehr viele Teilnehmer im Versorgungsgebiet auf, deren Positionen&nbsp; $(x, y)$&nbsp; &bdquo;statistisch" &uuml;ber das Gebiet&nbsp; $K$&nbsp; verteilt sind.
+*Es halten sich sehr viele Teilnehmer im Versorgungsgebiet auf, deren Positionen&nbsp; $(x, y)$&nbsp; "statistisch" &uuml;ber das Gebiet&nbsp; $K$&nbsp; verteilt sind.
 Ab der Teilaufgabe&nbsp; '''(5)'''&nbsp; gehen wir von dem unten skizzierten Versorgungsgebiet&nbsp; $G$&nbsp; aus.
 *Wegen eines Hindernisses muss nun die&nbsp; $x$&ndash;Koordinate aller Teilnehmer gr&ouml;&szlig;er als&nbsp; $-R/2$&nbsp;  sein.
-*Auch im Versorgungsgebiet&nbsp; $G$&nbsp; seien die Teilnehmer wieder &bdquo;statistisch verteilt".
+*Auch im Versorgungsgebiet&nbsp; $G$&nbsp; seien die Teilnehmer wieder "statistisch verteilt".
@@ Line 62: / Line 62: @@
 {Welche Aussagen sind nun hinsichtlich&nbsp; $f_\alpha(\alpha)$&nbsp; im Bereich&nbsp; $|\alpha| > \alpha_0$&nbsp; g&uuml;ltig?
 |type="[]"}
-- Die WDF hat &bdquo;außen" den gleichen Verlauf wie &bdquo;innen".
+- Die WDF hat "außen" den gleichen Verlauf wie "innen".
-- Die WDF ist &bdquo;außen" identisch Null.
+- Die WDF ist "außen" identisch Null.
 + Die WDF f&auml;llt in diesem Bereich zu den R&auml;ndern hin ab.
 - Die WDF steigt in diesem Bereich zu den R&auml;ndern hin an.

← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^3.4 Gleichverteilte Zufallsgröße^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.4 Gleichverteilte Zufallsgröße^]]