Aufgaben:Exercise 2.8: COST Delay Models - Revision history

Bene: Text replacement - "power spectral density" to "power-spectral density"

2022-02-17T11:41:20Z

Text replacement - "power spectral density" to "power-spectral density"

Bene: Text replacement - "autocorrelation" to "auto-correlation"

2022-02-17T11:28:48Z

Text replacement - "autocorrelation" to "auto-correlation"

Bene: Text replacement - "power density spectrum" to "power spectral density"

2022-02-17T11:20:13Z

Text replacement - "power density spectrum" to "power spectral density"

Javier: Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

2021-03-23T12:37:43Z

Text replacement - "Category:Exercises for Mobile Communications" to "Category:Mobile Communications: Exercises"

Guenter at 11:17, 16 February 2021

2021-02-16T11:17:33Z

Guenter at 11:16, 16 February 2021

2021-02-16T11:16:25Z

Guenter at 11:09, 16 February 2021

2021-02-16T11:09:01Z

Guenter at 17:37, 15 January 2021

2021-01-15T17:37:49Z

Guenter at 16:36, 15 January 2021

2021-01-15T16:36:55Z

Javier: Text replacement - "===Sample solution===" to "===Solution==="

2020-12-16T13:16:41Z

Text replacement - "===Sample solution===" to "===Solution==="

@@ Line 31: / Line 31: @@
 For the models&nbsp; ${\rm RA}$, &nbsp;${\rm TU}$&nbsp; and&nbsp; ${\rm BU}$&nbsp; the following parameters are to be determined:
-* The&nbsp; <b>delay spread</b>&nbsp; $T_{\rm V}$&nbsp; is the standard deviation of the delay&nbsp; $\tau$. <br>If the delay power spectral density&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; has an exponential course as with the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$, then&nbsp; $T_{\rm V} = \tau_0$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
+* The&nbsp; <b>delay spread</b>&nbsp; $T_{\rm V}$&nbsp; is the standard deviation of the delay&nbsp; $\tau$. <br>If the delay power-spectral density&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; has an exponential course as with the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$, then&nbsp; $T_{\rm V} = \tau_0$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
 * The <b>coherence bandwidth</b>&nbsp; $B_{\rm K}$&nbsp; is the value of &nbsp;$\Delta f$ at which the magnitude of the frequency correlation function&nbsp; $\varphi_{\rm F}(\Delta f)$&nbsp; has dropped to half its value for the first time. With exponential&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; as with &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp; the product is&nbsp; $T_{\rm V}  \cdot B_{\rm K} \approx 0.276$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
@@ Line 53: / Line 53: @@
 ===Questionnaire===
 <quiz display=simple>
-{Specify the parameter&nbsp; $\tau_0$&nbsp; of the delay power spectral density for the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp;.
+{Specify the parameter&nbsp; $\tau_0$&nbsp; of the delay power-spectral density for the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp;.
 |type="{}"}
 ${\rm RA} \text{:} \ \hspace{0.4cm} \tau_0 \ = \ ${ 0.109 3% } $\ \rm &micro; s$
@@ Line 108: / Line 108: @@
-'''(2)'''&nbsp; In&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is&nbsp; $T_{\rm V} =\tau_0$&nbsp; when the delay power spectral density decreases exponentially according to&nbsp; ${\rm e}^{-\tau/\tau_0}$.&nbsp; Thus the following applies:
+'''(2)'''&nbsp; In&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is&nbsp; $T_{\rm V} =\tau_0$&nbsp; when the delay power-spectral density decreases exponentially according to&nbsp; ${\rm e}^{-\tau/\tau_0}$.&nbsp; Thus the following applies:
 * for "Rural Area":&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm &micro; s}$,
 * for "Typical Urban":&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm &micro; s}$.
@@ Line 152: / Line 152: @@
 *For the latter, the power percentage of all later echoes&nbsp; $($later than&nbsp; $15 \ \rm &micro; s)$&nbsp; is only about&nbsp; $12\%$.
 <br clear=all>
-'''(7)'''&nbsp; The area of the entire power spectral density gives&nbsp; $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
+'''(7)'''&nbsp; The area of the entire power-spectral density gives&nbsp; $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
 [[File:EN_Mob_A_2_8.png|right|frame|Delay PDF of profile &nbsp;${\rm BU}$ ]]

@@ Line 11: / Line 11: @@
 :$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$
-The value of the parameter&nbsp; $\tau_0$&nbsp; (time constant of the autocorrelation function) should be determined from the graphic in subtask&nbsp; '''(1)'''.&nbsp; Note the specified values of &nbsp; $\tau_{-30}$ for&nbsp; ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:
+The value of the parameter&nbsp; $\tau_0$&nbsp; (time constant of the auto-correlation function) should be determined from the graphic in subtask&nbsp; '''(1)'''.&nbsp; Note the specified values of &nbsp; $\tau_{-30}$ for&nbsp; ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:
 :$${\rm RA}\text{:}\hspace{0.15cm}\tau_{-30} = 0.75\,{\rm &micro; s} \hspace{0.05cm},\hspace{0.2cm}
   {\rm TU}\text{:}\hspace{0.15cm}\tau_{-30} = 6.9\,{\rm &micro; s} \hspace{0.05cm}.  $$

@@ Line 31: / Line 31: @@
 For the models&nbsp; ${\rm RA}$, &nbsp;${\rm TU}$&nbsp; and&nbsp; ${\rm BU}$&nbsp; the following parameters are to be determined:
-* The&nbsp; <b>delay spread</b>&nbsp; $T_{\rm V}$&nbsp; is the standard deviation of the delay&nbsp; $\tau$. <br>If the delay power density spectrum&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; has an exponential course as with the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$, then&nbsp; $T_{\rm V} = \tau_0$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
+* The&nbsp; <b>delay spread</b>&nbsp; $T_{\rm V}$&nbsp; is the standard deviation of the delay&nbsp; $\tau$. <br>If the delay power spectral density&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; has an exponential course as with the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$, then&nbsp; $T_{\rm V} = \tau_0$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
 * The <b>coherence bandwidth</b>&nbsp; $B_{\rm K}$&nbsp; is the value of &nbsp;$\Delta f$ at which the magnitude of the frequency correlation function&nbsp; $\varphi_{\rm F}(\Delta f)$&nbsp; has dropped to half its value for the first time. With exponential&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; as with &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp; the product is&nbsp; $T_{\rm V}  \cdot B_{\rm K} \approx 0.276$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
@@ Line 53: / Line 53: @@
 ===Questionnaire===
 <quiz display=simple>
-{Specify the parameter&nbsp; $\tau_0$&nbsp; of the delay power density spectrum for the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp;.
+{Specify the parameter&nbsp; $\tau_0$&nbsp; of the delay power spectral density for the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp;.
 |type="{}"}
 ${\rm RA} \text{:} \ \hspace{0.4cm} \tau_0 \ = \ ${ 0.109 3% } $\ \rm &micro; s$
@@ Line 108: / Line 108: @@
-'''(2)'''&nbsp; In&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is&nbsp; $T_{\rm V} =\tau_0$&nbsp; when the delay power density spectrum decreases exponentially according to&nbsp; ${\rm e}^{-\tau/\tau_0}$.&nbsp; Thus the following applies:
+'''(2)'''&nbsp; In&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is&nbsp; $T_{\rm V} =\tau_0$&nbsp; when the delay power spectral density decreases exponentially according to&nbsp; ${\rm e}^{-\tau/\tau_0}$.&nbsp; Thus the following applies:
 * for "Rural Area":&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm &micro; s}$,
 * for "Typical Urban":&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm &micro; s}$.
@@ Line 152: / Line 152: @@
 *For the latter, the power percentage of all later echoes&nbsp; $($later than&nbsp; $15 \ \rm &micro; s)$&nbsp; is only about&nbsp; $12\%$.
 <br clear=all>
-'''(7)'''&nbsp; The area of the entire power density spectrum gives&nbsp; $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
+'''(7)'''&nbsp; The area of the entire power spectral density gives&nbsp; $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
 [[File:EN_Mob_A_2_8.png|right|frame|Delay PDF of profile &nbsp;${\rm BU}$ ]]

← Older revision		Revision as of 12:37, 23 March 2021
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−	[[Category:~~Exercises for~~ Mobile Communications\|^2.3 The GWSSUS Channel Model^]]	+	[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]

@@ Line 114: / Line 114: @@
-'''(3)'''&nbsp; In [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]] it was also shown that for the coherence bandwidth&nbsp; $B_{\rm K} \approx 0.276/\tau_0$&nbsp; applies.&nbsp; It follows:
+'''(3)'''&nbsp; In Exercise 2.7 it was also shown that for the coherence bandwidth&nbsp; $B_{\rm K} \approx 0.276/\tau_0$&nbsp; applies.&nbsp; It follows:
 *$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ ("Rural Area"),
 *$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ ("Typical Urban")

@@ Line 100: / Line 100: @@
 Here&nbsp; $\tau_{-30}$&nbsp; denotes the delay that leads to the logarithmic ordinate value&nbsp; $-30 \ \rm dB$.&nbsp; Thus one obtains
-* for rural area&nbsp; (<b>RA</b>)&nbsp; with&nbsp; $\tau_{&ndash;30} = 0.75 \ \rm &micro; s$:
+* for "Rural Area"&nbsp; $\rm (RA)$&nbsp; with&nbsp; $\tau_{&ndash;30} = 0.75 \ \rm &micro; s$:
 :$$\tau_{\rm 0} = \frac{0.75\,{\rm &micro; s}}{ 6.9} \hspace{0.1cm}\underline {\approx 0.109\,{\rm &micro; s}}
   \hspace{0.05cm},$$
-* for urban and suburban areas&nbsp; (<i>Typical Urban</i>, <b>TU</b>)&nbsp; with&nbsp; $\tau_{&ndash;30} = 6.9 \ \rm &micro; s$:
+* for urban and suburban areas&nbsp; &rArr; &nbsp; "Typical Urban" $\rm (TU)$&nbsp; with&nbsp; $\tau_{&ndash;30} = 6.9 \ \rm &micro; s$:
 :$$\tau_{\rm 0} = \frac{6.9\,{\rm &micro; s}}{ 6.9} \hspace{0.1cm}\underline {\approx 1\,{\rm &micro; s}}
   \hspace{0.05cm},$$
@@ Line 109: / Line 109: @@
 '''(2)'''&nbsp; In&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is&nbsp; $T_{\rm V} =\tau_0$&nbsp; when the delay power density spectrum decreases exponentially according to&nbsp; ${\rm e}^{-\tau/\tau_0}$.&nbsp; Thus the following applies:
-* for &bdquo;Rural Area&rdquo;:&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm &micro; s}$,
+* for "Rural Area":&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm &micro; s}$,
-* for &bdquo;Typical Urban&rdquo;:&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm &micro; s}$.
+* for "Typical Urban":&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm &micro; s}$.
 '''(3)'''&nbsp; In [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]] it was also shown that for the coherence bandwidth&nbsp; $B_{\rm K} \approx 0.276/\tau_0$&nbsp; applies.&nbsp; It follows:
-*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ (&bdquo;Rural Area&rdquo;),
+*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ ("Rural Area"),
-*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ (&bdquo;Typical Urban&rdquo;)
+*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ ("Typical Urban")

@@ Line 6: / Line 6: @@
 :$$10 \cdot {\rm lg}\hspace{0.15cm} ({{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0}) \hspace{0.05cm}.$$
-Here the abbreviation&nbsp; $\phi_0 = \phi_{\rm V}(\tau = 0)$&nbsp; is used.&nbsp; These are the so-called <i>COST&ndash;delay models</i>.
+Here the abbreviation&nbsp; $\phi_0 = \phi_{\rm V}(\tau = 0)$&nbsp; is used.&nbsp; These are the so-called COST delay models.
-The upper sketch contains the two profiles &nbsp;${\rm RA}$&nbsp; (<i>Rural Area</i>) and &nbsp;${\rm TU}$&nbsp; (<i>Typical Urban</i>).&nbsp; Both of these are exponential:
+The upper sketch contains the two profiles &nbsp;${\rm RA}$&nbsp; ("Rural Area") and &nbsp;${\rm TU}$&nbsp; ("Typical Urban").&nbsp; Both of these are exponential:
 :$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$
@@ Line 16: / Line 16: @@
 The lower graph applies to less favourable conditions in
-* urban areas (<i>Bad Urban</i>, &nbsp;${\rm BU}$):
+* urban areas ("Bad Urban", &nbsp;${\rm BU}$):
 :$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}
   = \left\{ \begin{array}{c}  {\rm e}^{ -\tau / \tau_0} \\
@@ Line 23: / Line 23: @@
 \\  \hspace{-0.15cm} {\,\, \,\, \rm if}\hspace{0.15cm}5\,{\rm &micro; s} < \tau < 10\,{\rm &micro; s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm &micro; s} \hspace{0.05cm}, \end{array}$$
-* in rural areas (<i>Hilly Terrain</i>, &nbsp;${\rm HT}$):
+* in rural areas ("Hilly Terrain", &nbsp;${\rm HT}$):
 :$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}
   = \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\

@@ Line 99: / Line 99: @@
   \hspace{0.05cm}.$$
-Here $\tau_{-30}$ denotes the delay that leads to the logarithmic ordinate value $-30 \ \rm dB$. Thus one obtains
+Here&nbsp; $\tau_{-30}$&nbsp; denotes the delay that leads to the logarithmic ordinate value&nbsp; $-30 \ \rm dB$.&nbsp; Thus one obtains
-* for rural area (<b>RA</b>) with $\tau_{&ndash;30} = 0.75 \ \rm &micro; s$:
+* for rural area&nbsp; (<b>RA</b>)&nbsp; with&nbsp; $\tau_{&ndash;30} = 0.75 \ \rm &micro; s$:
-:$$\tau_{\rm 0} = \frac{0.75\,{\rm \mu s}}{ 6.9} \hspace{0.1cm}\underline {\approx 0.109\,{\rm &micro; s}}
+:$$\tau_{\rm 0} = \frac{0.75\,{\rm &micro; s}}{ 6.9} \hspace{0.1cm}\underline {\approx 0.109\,{\rm &micro; s}}
   \hspace{0.05cm},$$
-* for urban and suburban areas (<i>Typical Urban</i>, <b>TU</b>) with $\tau_{&ndash;30} = 6.9 \ \rm &micro; s$:
+* for urban and suburban areas&nbsp; (<i>Typical Urban</i>, <b>TU</b>)&nbsp; with&nbsp; $\tau_{&ndash;30} = 6.9 \ \rm &micro; s$:
-:$$\tau_{\rm 0} = \frac{6.9\,{\rm \mu s}}{ 6.9} \hspace{0.1cm}\underline {\approx 1\,{\rm &micro; s}}
+:$$\tau_{\rm 0} = \frac{6.9\,{\rm &micro; s}}{ 6.9} \hspace{0.1cm}\underline {\approx 1\,{\rm &micro; s}}
   \hspace{0.05cm},$$
-'''(2)'''&nbsp; In [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is $T_{\rm V} =\tau_0$ when the delay power density spectrum decreases exponentially according to ${\rm e}^{-\tau/\tau_0}$. Thus the following applies:
+'''(2)'''&nbsp; In&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]], it was shown that the delay spread is&nbsp; $T_{\rm V} =\tau_0$&nbsp; when the delay power density spectrum decreases exponentially according to&nbsp; ${\rm e}^{-\tau/\tau_0}$.&nbsp; Thus the following applies:
-* for &bdquo;Rural Area&rdquo;: $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm &micro; s}$,
+* for &bdquo;Rural Area&rdquo;:&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm &micro; s}$,
-* for &bdquo;Typical Urban&rdquo;: $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm &micro; s}$.
+* for &bdquo;Typical Urban&rdquo;:&nbsp; $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm &micro; s}$.
-'''(3)'''&nbsp; In [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]] it was also shown that for the coherence bandwidth $B_{\rm K} \approx 0.276/\tau_0$ applies. It follows:
+'''(3)'''&nbsp; In [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]] it was also shown that for the coherence bandwidth&nbsp; $B_{\rm K} \approx 0.276/\tau_0$&nbsp; applies.&nbsp; It follows:
 *$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ (&bdquo;Rural Area&rdquo;),
 *$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ (&bdquo;Typical Urban&rdquo;)
@@ Line 122: / Line 122: @@
 '''(4)'''&nbsp;The <u>second solution</u> is correct:
-*Frequency selectivity of the mobile radio channel is present if the signal bandwidth $B_{\rm S}$ is larger than the coherence bandwidth $B_{\rm K}$ (or at least of the same order of magnitude).
+*Frequency selectivity of the mobile radio channel is present if the signal bandwidth&nbsp; $B_{\rm S}$&nbsp; is larger than the coherence bandwidth&nbsp; $B_{\rm K}$&nbsp; (or at least of the same order of magnitude).
-*The smaller $B_{\rm K}$ is, the more often this happens.
+*The smaller&nbsp; $B_{\rm K}$&nbsp; is, the more often this happens.
-'''(5)'''&nbsp; According to the given equation, we have ${\it \Phi}_{\rm V}(\tau = 5.001 \ \rm &micro; s)/{\it \Phi}_0 \hspace{0.15cm}\underline{\approx0.5}$.
+'''(5)'''&nbsp; According to the given equation, we have&nbsp; ${\it \Phi}_{\rm V}(\tau = 5.001 \ \rm &micro; s)/{\it \Phi}_0 \hspace{0.15cm}\underline{\approx0.5}$.
-*On the other hand, for slightly smaller $\tau$ (for example $\tau = 4,999 \ \rm &micro; s$) we have approximately
+*On the other hand, for slightly smaller&nbsp; $\tau$&nbsp; $($for example $\tau = 4.999 \ \rm &micro; s)$&nbsp; we have approximately
 :$${{\it \Phi}_{\rm V}(\tau = 4.999\,{\rm \mu s})}/{{\it \Phi}_{\rm 0}} = {\rm e}^{ -{4.999\,{\rm &micro; s}}/{ 1\,{\rm \mu s}}}
   \approx {\rm e}^{-5} \hspace{0.1cm}\underline {= 0.00674 }\hspace{0.05cm}.$$
-'''(6)'''&nbsp; The power $P_1$ of all signal components with delays between $0$ and $5 \ \mu\rm &nbsp; s$ is:
+'''(6)'''&nbsp; The power&nbsp; $P_1$&nbsp; of all signal components with delays between&nbsp; $0$&nbsp; and&nbsp; $5 \ &micro;\rm &nbsp; s$&nbsp; is:
 :$$P_1 =  {\it \Phi}_{\rm 0} \cdot \int_{0}^{5\,{\rm \mu s}} {\rm exp}[ -{\tau}/{ \tau_0}] \hspace{0.15cm}{\rm d} \tau \hspace{0.15cm} \approx \hspace{0.15cm}
   {\it \Phi}_{\rm 0} \cdot \int_{0}^{\infty} {\rm e}^{ -{\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau
   = {\it \Phi}_{\rm 0} \cdot \tau_0 \hspace{0.05cm}.$$
-*The power outside $[0\;\mu \mathrm{s}, 5\;\mu \mathrm{s}]$ is
+*The power outside&nbsp; $[0\;&micro; \mathrm{s}, 5\;&micro; \mathrm{s}]$&nbsp; is
 :$$P_2 =  \frac{{\it \Phi}_{\rm 0}}{2} \cdot \int_{5\,{\rm &micro; s}}^{\infty} {\rm exp}[ \frac{5\,{\rm &micro; s} -\tau}{ \tau_0}] \hspace{0.15cm}{\rm d} \tau \hspace{0.15cm} \approx \hspace{0.15cm}
   \frac{{\it \Phi}_{\rm 0}}{2} \cdot \int_{0}^{\infty} {\rm exp}[ -{\tau}/{ \tau_0}] \hspace{0.15cm}{\rm d} \tau
   = \frac{{\it \Phi}_{\rm 0} \cdot \tau_0}{2} \hspace{0.05cm}. $$
-*Correspondingly, the percentage of power between $0$ and $5 \  \mu\rm &nbsp; s$ is
+[[File:P_ID2184__Mob_A_2_8f.png|right|frame|Delay power density of the COST profiles &nbsp;${\rm BU}$&nbsp; and &nbsp;${\rm HT}$]]
-[[File:EN_Mob_A_2_8.png|right|frame|Delay power density of the COST profiles &nbsp;${\rm BU}$&nbsp; and &nbsp;${\rm HT}$]]
 :$$\frac{P_1}{P_1+ P_2} =  \frac{2}{3} \hspace{0.15cm}\underline {\approx 66.7\%}\hspace{0.05cm}.$$
-The figure shows ${\it \Phi}_{\rm V}(\tau)$ in linear scale:
+The figure shows&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; in linear scale.&nbsp; The areas&nbsp; $P_1$&nbsp; and&nbsp; $P_2$&nbsp; are labeled.
 *The left graph is for &nbsp;${\rm BU}$, the right graph is for &nbsp;${\rm HT}$.
-*For the latter, the power percentage of all later echoes (later than $15 \ \rm &micro; s$) is only about $12\%$.
+*For the latter, the power percentage of all later echoes&nbsp; $($later than&nbsp; $15 \ \rm &micro; s)$&nbsp; is only about&nbsp; $12\%$.
 <br clear=all>
-'''(7)'''&nbsp; The area of the entire power density spectrum gives $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
+'''(7)'''&nbsp; The area of the entire power density spectrum gives&nbsp; $P = 1.5 \cdot \phi_0 \cdot \tau_0$.
-*Normalizing ${\it \Phi}_{\rm V}(\tau)$ to this value yields the probability density function $f_{\rm V}(\tau)$, as shown in the next graph (left diagram).
+[[File:EN_Mob_A_2_8.png|right|frame|Delay PDF of profile &nbsp;${\rm BU}$ ]]
-[[File:P_ID2185__Mob_A_2_8g.png|right|frame|Delay PDF of profile &nbsp;${\rm BU}$]]
+*Normalizing&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; to this value yields the probability density function&nbsp; $f_{\rm V}(\tau)$, as shown in the graph on the right (left diagram).
-*With $\tau_0 = 1 \ \ \rm &micro; s$ and &nbsp; $\tau_5 = 5 \ \ \rm &micro; s$, the mean is:
+*With&nbsp; $\tau_0 = 1 \ \ \rm &micro; s$&nbsp; and&nbsp; $\tau_5 = 5 \ \ \rm &micro; s$, the mean is:
 :$$m_{\rm V}=   \int_{0}^{\infty} f_{\rm V}(\tau) \hspace{0.15cm}{\rm d} \tau$$
 :$$\Rightarrow \hspace{0.3cm}m_{\rm V}=  \frac{2}{3\tau_0} \cdot  \int_{0}^{\tau_5} \tau \cdot {\rm e}^{ - {\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau \  + $$
 :$$ \hspace{1.7cm}+\  \frac{1}{3\tau_0} \cdot  \int_{\tau_5}^{\infty} \tau \cdot {\rm e}^{ (\tau_5 -\tau)/\tau_0}\hspace{0.15cm}{\rm d} \tau \hspace{0.05cm}. $$
-*The first integral is equal to $2\tau_0/3$ according to the provided expression.
+*The first integral is equal to&nbsp; $2\tau_0/3$&nbsp; according to the provided expression.
-*With the substitution $\tau' = \tau \, -\tau_5$ you finally obtain using the integral solutions given above:
+*With the substitution&nbsp; $\tau' = \tau \, -\tau_5$&nbsp; you finally obtain using the integral solutions given above:
 :$$m_{\rm V} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \frac{2\tau_0}{3} +  \frac{1}{3\tau_0}  \cdot  \int_{0}^{\infty} (\tau_5 + \tau') \cdot{\rm e}^{ - {\tau}'/{ \tau_0}}  \hspace{0.15cm}{\rm d} \tau ' =  \frac{2\tau_0}{3} +
   \frac{\tau_5}{3\tau_0}  \cdot  \int_{0}^{\infty} \cdot{\rm e}^{ - {\tau}'/{ \tau_0}}  \hspace{0.15cm}{\rm d} \tau ' +
@@ Line 172: / Line 172: @@
   \hspace{0.05cm}. $$
-*The variance $\sigma_{\rm V}^2$ is equal to the second moment (mean of the square) of the zero-mean random variable $\theta = \tau \, &ndash;m_{\rm V}$, whose PDF is shown in the right graph
+*The variance&nbsp; $\sigma_{\rm V}^2$&nbsp; is equal to the second moment (mean of the square) of the zero-mean random variable&nbsp; $\theta = \tau \, &ndash;m_{\rm V}$, whose PDF is shown in the right graph
-*From this $T_{\rm V} = \sigma_{\rm V}$ can be specified.
+*From this&nbsp; $T_{\rm V} = \sigma_{\rm V}$&nbsp; can be specified.
-*A second possibility is to first calculate the mean square value of the random variable $\tau$ and from this the variance $\sigma_{\rm V}^2$ using Steiner's theorem.
+A second possibility is to first calculate the mean square value of the random variable&nbsp; $\tau$&nbsp; and from this the variance&nbsp; $\sigma_{\rm V}^2$&nbsp; using Steiner's theorem.
 *With the substitutions and approximations already described above, one obtains
 :$$m_{\rm V2} \hspace{-0.1cm} \ \approx \ \hspace{-0.1cm}    \frac{2}{3\tau_0} \cdot  \int_{0}^{\infty} \tau^2 \cdot {\rm e}^{ - {\tau}/{ \tau_0}} \hspace{0.15cm}{\rm d} \tau  +  \frac{1}{3\tau_0} \cdot  \int_{0}^{\infty} (\tau_5 + \tau')^2 \cdot {\rm e}^{ - {\tau}'/{ \tau_0}}  \hspace{0.15cm}{\rm d} \tau ' $$
@@ Line 188: / Line 189: @@
 :$$\Rightarrow \hspace{0.3cm} T_{\rm V} = \sigma_{\rm V} \hspace{0.15cm}\underline {\approx 2.56\,{\rm &micro; s}}\hspace{0.05cm}.$$
-The above graph shows the parameters $T_{\rm V}$ and $\sigma_{\rm V}$.
+The above graph shows the parameters&nbsp; $T_{\rm V}$ and $\sigma_{\rm V}$.
 {{ML-Fuß}}
 [[Category:Exercises for Mobile Communications|^2.3 The GWSSUS Channel Model^]]

@@ Line 4: / Line 4: @@
 [[File:Mob_A_2_8_version2.png|right|frame|COST delay models]]
 On the right, four delay power density spectra are plotted logarithmically as a function of the delay time&nbsp; $\tau$&nbsp;
-:$$10 \cdot {\rm lg}\hspace{0.15cm} ({{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0}) \hspace{0.05cm},$$
+:$$10 \cdot {\rm lg}\hspace{0.15cm} ({{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0}) \hspace{0.05cm}.$$
-Here the abbreviation&nbsp; $\phi_0 = \phi_{\rm V}(\tau = 0)$&nbsp; is used. These are the so-called <i>COST&ndash;delay models</i>.
+Here the abbreviation&nbsp; $\phi_0 = \phi_{\rm V}(\tau = 0)$&nbsp; is used.&nbsp; These are the so-called <i>COST&ndash;delay models</i>.
-The upper sketch contains the two profiles &nbsp;${\rm RA}$&nbsp; (<i>Rural Area</i>) and &nbsp;${\rm TU}$&nbsp; (<i>Typical Urban</i>). Both of these are exponential:
+The upper sketch contains the two profiles &nbsp;${\rm RA}$&nbsp; (<i>Rural Area</i>) and &nbsp;${\rm TU}$&nbsp; (<i>Typical Urban</i>).&nbsp; Both of these are exponential:
 :$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$
-The value of the parameter&nbsp; $\tau_0$&nbsp; (time constant of the ACF) should be determined from the graphic in subtask '''(1)'''. Note the specified values of &nbsp; $\tau_{-30}$ for&nbsp; ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:
+The value of the parameter&nbsp; $\tau_0$&nbsp; (time constant of the autocorrelation function) should be determined from the graphic in subtask&nbsp; '''(1)'''.&nbsp; Note the specified values of &nbsp; $\tau_{-30}$ for&nbsp; ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:
 :$${\rm RA}\text{:}\hspace{0.15cm}\tau_{-30} = 0.75\,{\rm &micro; s} \hspace{0.05cm},\hspace{0.2cm}
   {\rm TU}\text{:}\hspace{0.15cm}\tau_{-30} = 6.9\,{\rm &micro; s} \hspace{0.05cm}.  $$
@@ Line 30: / Line 30: @@
 \\  \hspace{-0.35cm} {\rm if}\hspace{0.15cm}15\,{\rm &micro; s} < \tau < 20\,{\rm &micro; s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm &micro; s} \hspace{0.05cm}. \end{array}$$
-For the models&nbsp; &nbsp;${\rm RA}$, &nbsp;${\rm TU}$&nbsp; and&nbsp; &nbsp; &nbsp;${\rm BU}$&nbsp; the following parameters are to be determined:
+For the models&nbsp; ${\rm RA}$, &nbsp;${\rm TU}$&nbsp; and&nbsp; ${\rm BU}$&nbsp; the following parameters are to be determined:
-* The&nbsp; <b>delay spread</b>&nbsp; $T_{\rm V}$&nbsp; is the standard deviation of the delay&nbsp; $\tau$. <br>If the delaypower density spectrum&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; has an exponential course as with the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$, then&nbsp; $T_{\rm V} = \tau_0$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
+* The&nbsp; <b>delay spread</b>&nbsp; $T_{\rm V}$&nbsp; is the standard deviation of the delay&nbsp; $\tau$. <br>If the delay power density spectrum&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; has an exponential course as with the profiles &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$, then&nbsp; $T_{\rm V} = \tau_0$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
-* The <b>coherence bandwidth</b>&nbsp; $B_{\rm K}$&nbsp; is the value of &nbsp;$\Delta f$ at which the magnitude of the frequency correlation function&nbsp; $\varphi_{\rm F}(\Delta f)$&nbsp; has dropped to half its value for the first time. With exponential&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; as with &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp; the product&nbsp; $T_{\rm V} is \cdot B_{\rm K} \approx 0.276$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
+* The <b>coherence bandwidth</b>&nbsp; $B_{\rm K}$&nbsp; is the value of &nbsp;$\Delta f$ at which the magnitude of the frequency correlation function&nbsp; $\varphi_{\rm F}(\Delta f)$&nbsp; has dropped to half its value for the first time. With exponential&nbsp; ${\it \Phi}_{\rm V}(\tau)$&nbsp; as with &nbsp;${\rm RA}$&nbsp; and &nbsp;${\rm TU}$&nbsp; the product is&nbsp; $T_{\rm V}  \cdot B_{\rm K} \approx 0.276$, see&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7]].
@@ Line 41: / Line 41: @@
 ''Notes:''
-* This task belongs to chapter&nbsp; [[Mobile_Communications/Das_GWSSUS%E2%80%93Kanalmodell| GWSSUS&ndash;Kanalmodell]].
+*This exercise belongs to the chapter [[Mobile_Communications/The_GWSSUS_Channel_Model|The GWSSUS Channel Model]].
 * The following integrals are given:
 :$$\frac{1}{\tau_0} \cdot \int_{0}^{\infty}\hspace{-0.15cm} {\rm e}^{ -\tau / \tau_0} \hspace{0.15cm}{\rm d} \tau = 1
@@ Line 69: / Line 69: @@
 {For which channel does frequency selectivity play a greater role?
-|type="[]"}
+|type="()"}
 - Rural Area &nbsp;$({\rm RA})$.
 + Typical urban &nbsp;$({\rm TU})$.
-{How large is the (normalized) power density for &bdquo;Bad Urban&rdquo;&nbsp; $({\rm BU})$ &nbsp; with &nbsp; $\tau = 5,001 \ \rm &micro; s$&nbsp; and with &nbsp; $\tau = 4,999 \ \rm &micro; s$?
+{How large is the (normalized) power density for&nbsp; "Bad Urban"&nbsp; $({\rm BU})$ &nbsp; with &nbsp; $\tau = 5.001 \ \rm &micro; s$&nbsp; and with &nbsp; $\tau = 4.999 \ \rm &micro; s$?
 |type="{}"}
 ${\it \Phi}_{\rm V}(\tau = 5.001 \ \rm &micro; s) \ = \ ${ 0.5 3% } $\ \cdot {\it \Phi}_0$
-${\it \Phi}_{\rm V}(\tau = 4,999 \ \rm &micro; s) \ = \ ${ 0.00674 3% } $\ \cdot {\it \Phi}_0$
+${\it \Phi}_{\rm V}(\tau = 4.999 \ \rm &micro; s) \ = \ ${ 0.00674 3% } $\ \cdot {\it \Phi}_0$
-{We consider&nbsp; ${\rm BU}$ again. Let $P_1$ be the power of the signal between $0$&nbsp; and&nbsp; $5 \ \rm &micro; s$, and let $P_2$ be the remaining signal power. What percentage of the total signal power is $0$&nbsp; and&nbsp; $5 \ \rm &micro; s$?
+{We consider&nbsp; ${\rm BU}$ again. Let $P_1$ be the power of the signal between $0$&nbsp; and&nbsp; $5 \ \rm &micro; s$, and let $P_2$ be the remaining signal power. <br>What percentage of the total signal power comes from the interval
 |type="{}"}
 $P_1/(P_1 + P_2) \ = \ ${ 66.7 3% } $\ \rm \%$
-{Calculate the delay spread&nbsp; $T_{\rm V}$&nbsp; of the profile&nbsp; ${\rm BU}$. ''Note'': The average delay is&nbsp; $m_{\rm V} = E[\tau] = 2.667 \ \rm &micro; s$.
+{Calculate the delay spread&nbsp; $T_{\rm V}$&nbsp; of the profile&nbsp; ${\rm BU}$. <br>''Note'':&nbsp; The average delay is&nbsp; $m_{\rm V} = E[\tau] = 2.667 \ \rm &micro; s$.
 |type="{}"}
 $T_{\rm V} \ = \ ${ 2.56 3% } $\ \rm &micro; s$

@@ Line 87: / Line 87: @@
 </quiz>
-===Sample solution===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; The following property can be seen from the graph: