Aufgaben:Exercise 2.8: COST Delay Models - Revision history

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	[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]		[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]
	[[de:~~Exercises~~:~~Exercise_2~~.8:~~_COST_Delay_Models~~]]		[[de:Aufgaben:Exercise 2.8: COST Delay Models]]

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	[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]		[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]
			[[de:Exercises:Exercise_2.8:_COST_Delay_Models]]

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

2022-02-17T11:41:20Z

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

← Older revision		Revision as of 13:41, 17 February 2022
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	For the models  ${\rm RA}$,  ${\rm TU}$  and  ${\rm BU}$  the following parameters are to be determined:		For the models  ${\rm RA}$,  ${\rm TU}$  and  ${\rm BU}$  the following parameters are to be determined:
	* The  <b>delay spread</b>  $T_{\rm V}$  is the standard deviation of the delay  $\tau$. <br>If the delay power spectral density  ${\it \Phi}_{\rm V}(\tau)$  has an exponential course as with the profiles  ${\rm RA}$  and  ${\rm TU}$, then  $T_{\rm V} = \tau_0$, see  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth\|Exercise 2.7]].		* The  <b>delay spread</b>  $T_{\rm V}$  is the standard deviation of the delay  $\tau$. <br>If the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau)$  has an exponential course as with the profiles  ${\rm RA}$  and  ${\rm TU}$, then  $T_{\rm V} = \tau_0$, see  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth\|Exercise 2.7]].

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

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

2022-02-17T11:28:48Z

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

← Older revision		Revision as of 13:28, 17 February 2022
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	:$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$		:$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$

	The value of the parameter  $\tau_0$  (time constant of the ~~autocorrelation~~ function) should be determined from the graphic in subtask  '''(1)'''.  Note the specified values of   $\tau_{-30}$ for  ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:		The value of the parameter  $\tau_0$  (time constant of the auto-correlation function) should be determined from the graphic in subtask  '''(1)'''.  Note the specified values of   $\tau_{-30}$ for  ${\it \Phi}_{\rm V}(\tau_{-30})=-30 \ \rm dB$:
	:$${\rm RA}\text{:}\hspace{0.15cm}\tau_{-30} = 0.75\,{\rm µ s} \hspace{0.05cm},\hspace{0.2cm}		:$${\rm RA}\text{:}\hspace{0.15cm}\tau_{-30} = 0.75\,{\rm µ s} \hspace{0.05cm},\hspace{0.2cm}
	{\rm TU}\text{:}\hspace{0.15cm}\tau_{-30} = 6.9\,{\rm µ s} \hspace{0.05cm}. $$		{\rm TU}\text{:}\hspace{0.15cm}\tau_{-30} = 6.9\,{\rm µ s} \hspace{0.05cm}. $$

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

2022-02-17T11:20:13Z

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

← Older revision		Revision as of 13:20, 17 February 2022
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	For the models  ${\rm RA}$,  ${\rm TU}$  and  ${\rm BU}$  the following parameters are to be determined:		For the models  ${\rm RA}$,  ${\rm TU}$  and  ${\rm BU}$  the following parameters are to be determined:
	* The  <b>delay spread</b>  $T_{\rm V}$  is the standard deviation of the delay  $\tau$. <br>If the delay power density ~~spectrum~~  ${\it \Phi}_{\rm V}(\tau)$  has an exponential course as with the profiles  ${\rm RA}$  and  ${\rm TU}$, then  $T_{\rm V} = \tau_0$, see  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth\|Exercise 2.7]].		* The  <b>delay spread</b>  $T_{\rm V}$  is the standard deviation of the delay  $\tau$. <br>If the delay power spectral density  ${\it \Phi}_{\rm V}(\tau)$  has an exponential course as with the profiles  ${\rm RA}$  and  ${\rm TU}$, then  $T_{\rm V} = \tau_0$, see  [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth\|Exercise 2.7]].

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

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"

← Older revision		Revision as of 14:37, 23 March 2021
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	{{ML-Fuß}}		{{ML-Fuß}}

	[[Category:~~Exercises for~~ Mobile Communications\|^2.3 The GWSSUS Channel Model^]]		[[Category:Mobile Communications: Exercises\|^2.3 The GWSSUS Channel Model^]]

Guenter at 11:17, 16 February 2021

2021-02-16T11:17:33Z

← Older revision		Revision as of 13:17, 16 February 2021
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	'''(3)'''  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)'''  In Exercise 2.7 it was also shown that for the coherence bandwidth  $B_{\rm K} \approx 0.276/\tau_0$  applies.  It follows:
	*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ ("Rural Area"),		*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ ("Rural Area"),
	*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ ("Typical Urban")		*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ ("Typical Urban")

Guenter at 11:16, 16 February 2021

2021-02-16T11:16:25Z

← Older revision		Revision as of 13:16, 16 February 2021
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	Here  $\tau_{-30}$  denotes the delay that leads to the logarithmic ordinate value  $-30 \ \rm dB$.  Thus one obtains		Here  $\tau_{-30}$  denotes the delay that leads to the logarithmic ordinate value  $-30 \ \rm dB$.  Thus one obtains
	* for ~~rural area~~  (~~<b>~~RA~~</b>~~)  with  $\tau_{–30} = 0.75 \ \rm µ s$:		* for "Rural Area"  $\rm (RA)$  with  $\tau_{–30} = 0.75 \ \rm µ s$:
	:$$\tau_{\rm 0} = \frac{0.75\,{\rm µ s}}{ 6.9} \hspace{0.1cm}\underline {\approx 0.109\,{\rm µ s}}		:$$\tau_{\rm 0} = \frac{0.75\,{\rm µ s}}{ 6.9} \hspace{0.1cm}\underline {\approx 0.109\,{\rm µ s}}
	\hspace{0.05cm},$$		\hspace{0.05cm},$$
	* for urban and suburban areas  ~~(<i>~~Typical Urban~~</i>, <b>~~TU~~</b>~~)  with  $\tau_{–30} = 6.9 \ \rm µ s$:		* for urban and suburban areas  ⇒   "Typical Urban" $\rm (TU)$  with  $\tau_{–30} = 6.9 \ \rm µ s$:
	:$$\tau_{\rm 0} = \frac{6.9\,{\rm µ s}}{ 6.9} \hspace{0.1cm}\underline {\approx 1\,{\rm µ s}}		:$$\tau_{\rm 0} = \frac{6.9\,{\rm µ s}}{ 6.9} \hspace{0.1cm}\underline {\approx 1\,{\rm µ s}}
	\hspace{0.05cm},$$		\hspace{0.05cm},$$
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	'''(2)'''  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)'''  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:
	* for ~~&bdquo;~~Rural Area~~”~~:  $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm µ s}$,		* for "Rural Area":  $\hspace{0.4cm} T_{\rm V} \ \underline {= 0.109 \ \rm µ s}$,
	* for ~~&bdquo;~~Typical Urban~~”~~:  $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm µ s}$.		* for "Typical Urban":  $\hspace{0.4cm} T_{\rm V} \ \underline {= 1 \ \rm µ s}$.



	'''(3)'''  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)'''  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:
	*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ (~~&bdquo;~~Rural Area~~”~~),		*$B_{\rm K} \ \underline {\approx 2500 \ \rm kHz}$ ("Rural Area"),
	*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ (~~&bdquo;~~Typical Urban~~”~~)		*$B_{\rm K} \ \underline {\approx 276 \ \ \rm kHz}$ ("Typical Urban")

Guenter at 11:09, 16 February 2021

2021-02-16T11:09:01Z

← Older revision		Revision as of 13:09, 16 February 2021
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	:$$10 \cdot {\rm lg}\hspace{0.15cm} ({{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0}) \hspace{0.05cm}.$$		:$$10 \cdot {\rm lg}\hspace{0.15cm} ({{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0}) \hspace{0.05cm}.$$

	Here the abbreviation  $\phi_0 = \phi_{\rm V}(\tau = 0)$  is used.  These are the so-called ~~<i>~~COST~~–~~delay models~~</i>~~.		Here the abbreviation  $\phi_0 = \phi_{\rm V}(\tau = 0)$  is used.  These are the so-called COST delay models.

	The upper sketch contains the two profiles  ${\rm RA}$  (~~<i>~~Rural Area~~</i>~~) and  ${\rm TU}$  (~~<i>~~Typical Urban~~</i>~~).  Both of these are exponential:		The upper sketch contains the two profiles  ${\rm RA}$  ("Rural Area") and  ${\rm TU}$  ("Typical Urban").  Both of these are exponential:
	:$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$		:$${{\it \Phi}_{\rm V}(\tau)}/{\it \Phi}_{\rm 0} = {\rm e}^{ -\tau / \tau_0} \hspace{0.05cm}.$$

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	The lower graph applies to less favourable conditions in		The lower graph applies to less favourable conditions in
	* urban areas (~~<i>~~Bad Urban~~</i>~~,  ${\rm BU}$):		* urban areas ("Bad Urban",  ${\rm BU}$):
	:$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}		:$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}
	= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\		= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\
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	\\ \hspace{-0.15cm} {\,\, \,\, \rm if}\hspace{0.15cm}5\,{\rm µ s} < \tau < 10\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}, \end{array}$$		\\ \hspace{-0.15cm} {\,\, \,\, \rm if}\hspace{0.15cm}5\,{\rm µ s} < \tau < 10\,{\rm µ s}\hspace{0.05cm},\hspace{0.15cm}\tau_0 = 1\,{\rm µ s} \hspace{0.05cm}, \end{array}$$

	* in rural areas (~~<i>~~Hilly Terrain~~</i>~~,  ${\rm HT}$):		* in rural areas ("Hilly Terrain",  ${\rm HT}$):
	:$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}		:$${{\it \Phi}_{\rm V}(\tau)}/{{\it \Phi}_{\rm 0}}
	= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\		= \left\{ \begin{array}{c} {\rm e}^{ -\tau / \tau_0} \\