Aufgaben:Exercise 2.6: Dimensions in GWSSUS - Revision history

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2026-03-16T13:47:54Z

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	The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:		The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:
	* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#~~The_First_Fourier_Integral~~\|Fourier transform]].		* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|Fourier transform]].
	* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#~~The_Second_Fourier_Integral~~\|inverse Fourier transform]]  (opposite direction).		* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|inverse Fourier transform]]  (opposite direction).


	For example:		For example:
	:$$\eta_{\rm VZ}(\tau, t)		:$$\eta_{\rm VZ}(\tau, t)\hspace{0.2cm} \stackrel{\tau, \hspace{0.02cm}f}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} \eta_{\rm FZ}(f,t)\hspace{0.05cm},\hspace{0.4cm}\eta_{\rm FZ}(f,t)\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$
	\hspace{0.2cm} \stackrel{\tau, \hspace{0.02cm}f}{\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet} \hspace{0.2cm} \eta_{\rm FZ}(f,t)\hspace{0.05cm},
	\hspace{0.4cm}\eta_{\rm FZ}(f,t)
	\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$

	*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power-spectral density  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.		*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power-spectral density  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
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	Let us consider here the system function  $\eta_{\rm VZ}(\tau, t)$, i.e. the time–variant impulse response  $h(\tau, t)$.  We define:		Let us consider here the system function  $\eta_{\rm VZ}(\tau, t)$, i.e. the time–variant impulse response  $h(\tau, t)$.  We define:
	:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot		:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm},$$
	\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm},$$		:$$\Delta \tau = \tau_2 - \tau_1 \hspace{0.05cm}, \hspace{0.2cm} \Delta t = t_2 - t_1\hspace{0.3cm} \Rightarrow \hspace{0.3cm}\varphi_{\rm VZ}(\Delta \tau, \Delta t) \hspace{0.05cm}, $$
	:$$\Delta \tau = \tau_2 - \tau_1 \hspace{0.05cm}, \hspace{0.2cm} \Delta t = t_2 - t_1
	\hspace{0.3cm} \Rightarrow \hspace{0.3cm}
	\varphi_{\rm VZ}(\Delta \tau, \Delta t) \hspace{0.05cm}, $$
	:$$\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.$$		:$$\varphi_{\rm VZ}(\Delta \tau, \Delta t) = \delta(\Delta \tau) \cdot {\it \Phi}_{\rm VZ}(\tau, \Delta t) \hspace{0.05cm}.$$
	:$${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)\hspace{0.05cm}. $$		:$${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)\hspace{0.05cm}. $$
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	*$\eta_{\rm VZ}(\tau, t)$  is the time-variant impulse response, for which the term  $h(\tau, t)$  is also common.  Like every impulse response,  $h(\tau, t)$  has the unit  $[1/\rm s]$.		*$\eta_{\rm VZ}(\tau, t)$  is the time-variant impulse response, for which the term  $h(\tau, t)$  is also common.  Like every impulse response,  $h(\tau, t)$  has the unit  $[1/\rm s]$.
	*By Fourier transform of the function  $\eta_{\rm VZ}(\tau, t)$  with respect to the delay  $\tau$  one obtains		*By Fourier transform of the function  $\eta_{\rm VZ}(\tau, t)$  with respect to the delay  $\tau$  one obtains
	:$$\eta_{\rm FZ}(f, t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau		:$$\eta_{\rm FZ}(f, t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau\hspace{0.05cm}. $$
	\hspace{0.05cm}. $$

	*Due to the integration over  $\tau$  $($unit:  $\rm s)$, the time-variant transfer function  $\eta_{\rm FZ}(f, t)$  is dimensionless.  <br>In some literature,  $H(f, t)$  is also used instead of  $\eta_{\rm FZ}(f, t)$.		*Due to the integration over  $\tau$  $($unit:  $\rm s)$, the time-variant transfer function  $\eta_{\rm FZ}(f, t)$  is dimensionless.  <br>In some literature,  $H(f, t)$  is also used instead of  $\eta_{\rm FZ}(f, t)$.

	*The delay–Doppler representation  $\eta_{\rm VD}(\tau, f_{\rm D})$  is dimensionless, too.  This function results from the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  by Fourier transform with respect to  $t$:		*The delay–Doppler representation  $\eta_{\rm VD}(\tau, f_{\rm D})$  is dimensionless, too.  This function results from the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  by Fourier transform with respect to  $t$:
	:$$\eta_{\rm VD}(\tau, f_{\rm D}) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} t}\hspace{0.15cm}{\rm d}t		:$$\eta_{\rm VD}(\tau, f_{\rm D}) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} t}\hspace{0.15cm}{\rm d}t\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	*The function  $\eta_{\rm FD}(t, f_{\rm D})$  is obtained from the dimensionless functions  $\eta_{\rm VD}(\tau, f_{\rm D})$  and  $\eta_{\rm FZ}(f, t)$  respectively by a Fourier transform, which results in the unit  $[\rm s] = [1/\rm Hz]$.		*The function  $\eta_{\rm FD}(t, f_{\rm D})$  is obtained from the dimensionless functions  $\eta_{\rm VD}(\tau, f_{\rm D})$  and  $\eta_{\rm FZ}(f, t)$  respectively by a Fourier transform, which results in the unit  $[\rm s] = [1/\rm Hz]$.
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	'''(2)'''  <u>Solutions 2 and 3</u> are correct:		'''(2)'''  <u>Solutions 2 and 3</u> are correct:
	*The auto-correlation  $\rm (ACF)$  function is by definition the following expected value:		*The auto-correlation  $\rm (ACF)$  function is by definition the following expected value:
	:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot		:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$
	\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$

	*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.		*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.

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2026-03-16T12:27:07Z

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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_2.6:_Units_in_GWSSUS]]

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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	\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$		\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$

	*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power spectral density  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.		*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power-spectral density  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
	*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra are labeled in blue. The GWSSUS model is always assumed.		*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra are labeled in blue. The GWSSUS model is always assumed.

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	*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.		*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.

	*The Dirac function  ${\rm \delta}(\Delta \tau)$  has the unit  $[1/\rm s]$, since the integral over all  $\tau$  $($with unit  $[\rm s])$ must be  $1$.  Therefore, both the delay–time cross power spectral density  ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$  and the delay power spectral density  ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$  have unit  $[1/\rm s]$.		*The Dirac function  ${\rm \delta}(\Delta \tau)$  has the unit  $[1/\rm s]$, since the integral over all  $\tau$  $($with unit  $[\rm s])$ must be  $1$.  Therefore, both the delay–time cross power-spectral density  ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$  and the delay power-spectral density  ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$  have unit  $[1/\rm s]$.


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	*The function  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  has unit  $[1/\rm s]$.  Its Fourier transform with respect to  $\tau$  is  $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to  $t$  is  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.  Both  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  are therefore without unit.		*The function  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  has unit  $[1/\rm s]$.  Its Fourier transform with respect to  $\tau$  is  $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to  $t$  is  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.  Both  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  are therefore without unit.

	*The frequency–Doppler cross power spectral density has the unit  $[\rm s] = [1/\rm Hz]$, because		*The frequency–Doppler cross power-spectral density has the unit  $[\rm s] = [1/\rm Hz]$, because
	:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$		:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
	{{ML-Fuß}}		{{ML-Fuß}}

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

2022-02-17T11:29:04Z

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

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	'''(2)'''  <u>Solutions 2 and 3</u> are correct:		'''(2)'''  <u>Solutions 2 and 3</u> are correct:
	*The ~~autocorrelation~~  $\rm (ACF)$  function is by definition the following expected value:		*The auto-correlation  $\rm (ACF)$  function is by definition the following expected value:
	:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot		:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
	\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$		\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$

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

2022-02-17T11:20:08Z

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

← Older revision		Revision as of 13:20, 17 February 2022
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	\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$		\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$

	*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power density ~~spectrum~~  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.		*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power spectral density  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
	*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra are labeled in blue. The GWSSUS model is always assumed.		*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra are labeled in blue. The GWSSUS model is always assumed.

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	*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.		*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.

	*The Dirac function  ${\rm \delta}(\Delta \tau)$  has the unit  $[1/\rm s]$, since the integral over all  $\tau$  $($with unit  $[\rm s])$ must be  $1$.  Therefore, both the delay–time cross power density ~~spectrum~~  ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$  and the delay power density ~~spectrum~~  ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$  have unit  $[1/\rm s]$.		*The Dirac function  ${\rm \delta}(\Delta \tau)$  has the unit  $[1/\rm s]$, since the integral over all  $\tau$  $($with unit  $[\rm s])$ must be  $1$.  Therefore, both the delay–time cross power spectral density  ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$  and the delay power spectral density  ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$  have unit  $[1/\rm s]$.


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	*The function  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  has unit  $[1/\rm s]$.  Its Fourier transform with respect to  $\tau$  is  $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to  $t$  is  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.  Both  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  are therefore without unit.		*The function  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  has unit  $[1/\rm s]$.  Its Fourier transform with respect to  $\tau$  is  $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to  $t$  is  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.  Both  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  are therefore without unit.

	*The frequency–Doppler cross power density ~~spectrum~~ has the unit  $[\rm s] = [1/\rm Hz]$, because		*The frequency–Doppler cross power spectral density has the unit  $[\rm s] = [1/\rm Hz]$, because
	:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$		:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
	{{ML-Fuß}}		{{ML-Fuß}}

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

2021-03-23T12:37:21Z

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

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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^]]

Guenter at 13:32, 14 January 2021

2021-01-14T13:32:38Z

← Older revision		Revision as of 15:32, 14 January 2021
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	{Which of the following specified dimensions of the system functions are correct?		{Which of the following specified dimensions of the system functions are correct?
	\|type="[]"}		\|type="[]"}
	+ $\eta_{\rm VZ}(\tau, t)$  has the ~~dimension~~  $[1/\rm s]$.		+ $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$.
	+ $\eta_{\rm FZ}(f, t)$  is ~~dimensionless~~.		+ $\eta_{\rm FZ}(f, t)$  is without unit.
	+ $\eta_{\rm VD}(\tau, f_{\rm D})$  is ~~dimensionless~~.		+ $\eta_{\rm VD}(\tau, f_{\rm D})$  is without unit.
	+ $\eta_{\rm FD}(f, f_{\rm D})$  has the ~~dimension~~  $[1/\rm Hz]$.		+ $\eta_{\rm FD}(f, f_{\rm D})$  has the unit  $[1/\rm Hz]$.

	{Which of the following statements are correct?		{Which of the following statements are correct?
	\|type="[]"}		\|type="[]"}
	- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the ~~dimension~~  $[1/\rm s]$.		- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s]$.
	+ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$  has the ~~dimension~~  $[1/\rm s]$.		+ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$  has the unit  $[1/\rm s]$.
	+ ${\it \phi}_{\rm V}(\tau)$  has the ~~dimension~~  $[1/\rm s]$.		+ ${\it \phi}_{\rm V}(\tau)$  has the unit  $[1/\rm s]$.

	{Which of the following statements are correct?		{Which of the following statements are correct?
	\|type="[]"}		\|type="[]"}
	+ $\varphi_{\rm FZ}(\Delta f, \Delta t),  \varphi_{\rm F}(\Delta f)$  and  $\varphi_{\rm Z}(\Delta t)$  have no ~~dimension~~.		+ $\varphi_{\rm FZ}(\Delta f, \Delta t),  \varphi_{\rm F}(\Delta f)$  and  $\varphi_{\rm Z}(\Delta t)$  have no unit.
	- ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  has the ~~dimension~~  $[1/\rm s]$.		- ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  has the unit  $[1/\rm s]$.
	+ ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  and  ${\it \Phi}_{\rm D}(f_{\rm D})$  have the ~~dimension~~ $[1/\rm Hz]$ each.		+ ${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D})$  and  ${\it \Phi}_{\rm D}(f_{\rm D})$  have the unit  $[1/\rm Hz]$ each.
	</quiz>		</quiz>

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	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  <u>All statements are correct</u>:		'''(1)'''  <u>All statements are correct</u>:
	*$\eta_{\rm VZ}(\tau, t)$ is the time-variant impulse response, for which the term $h(\tau, t)$ is also common. Like every impulse response, $h(\tau, t)$ has the unit $[1/\rm s]$.		*$\eta_{\rm VZ}(\tau, t)$  is the time-variant impulse response, for which the term  $h(\tau, t)$  is also common.  Like every impulse response,  $h(\tau, t)$  has the unit  $[1/\rm s]$.
	*By Fourier ~~transformation~~ of the function $\eta_{\rm VZ}(\tau, t)$ with respect to the delay $\tau$ one obtains		*By Fourier transform of the function  $\eta_{\rm VZ}(\tau, t)$  with respect to the delay  $\tau$  one obtains
	:$$\eta_{\rm FZ}(f, t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau		:$$\eta_{\rm FZ}(f, t) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \tau}\hspace{0.15cm}{\rm d}\tau
	\hspace{0.05cm}. $$		\hspace{0.05cm}. $$

	*Due to the integration over $\tau$ (unit: $\rm s$), the time-variant transfer function $\eta_{\rm FZ}(f, t)$ is dimensionless. In some literature, $H(f, t)$ is also used instead of $\eta_{\rm FZ}(f, t)$.		*Due to the integration over  $\tau$  $($unit:  $\rm s)$, the time-variant transfer function  $\eta_{\rm FZ}(f, t)$  is dimensionless.  <br>In some literature,  $H(f, t)$  is also used instead of  $\eta_{\rm FZ}(f, t)$.

	*The delay–Doppler representation $\eta_{\rm VD}(\tau, f_{\rm D})$ is ~~also~~ dimensionless. This function results from the time-variant impulse response $\eta_{\rm VZ}(\tau, t)$ by Fourier ~~transformation~~ with respect to $t$:		*The delay–Doppler representation  $\eta_{\rm VD}(\tau, f_{\rm D})$  is dimensionless, too.  This function results from the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  by Fourier transform with respect to  $t$:
	:$$\eta_{\rm VD}(\tau, f_{\rm D}) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} t}\hspace{0.15cm}{\rm d}t		:$$\eta_{\rm VD}(\tau, f_{\rm D}) = \int_{-\infty}^{+\infty} \eta_{\rm VZ}(\tau, t) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} t}\hspace{0.15cm}{\rm d}t
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*The function $\eta_{\rm FD}(t, f_{\rm D})$ is obtained from the dimensionless functions $\eta_{\rm VD}(\tau, f_{\rm D})$ and $\eta_{\rm FZ}(f, t)$ respectively by a Fourier ~~transformation~~, which results in the unit $[\rm s] = [1/\rm Hz]$.		*The function  $\eta_{\rm FD}(t, f_{\rm D})$  is obtained from the dimensionless functions  $\eta_{\rm VD}(\tau, f_{\rm D})$  and  $\eta_{\rm FZ}(f, t)$  respectively by a Fourier transform, which results in the unit  $[\rm s] = [1/\rm Hz]$.



	'''(2)'''  <u>Solutions 2 and 3</u> are correct:		'''(2)'''  <u>Solutions 2 and 3</u> are correct:
	*The autocorrelation (ACF) function is by definition the following expected value:		*The autocorrelation  $\rm (ACF)$  function is by definition the following expected value:
	:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot		:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
	\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$		\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm}.$$

	*Since the time-variant impulse response $\eta_{\rm VZ}(\tau, t)$ has the unit $[1/\rm s]$, its ACF $\varphi_{\rm VZ}$ has the unit $[1/\rm s^2]$, both in the general case $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$ and with the GWSSUS case $\varphi_{\rm VZ}(\~~delta~~ \tau, \ \~~delta~~ t)$.		*Since the time-variant impulse response  $\eta_{\rm VZ}(\tau, t)$  has the unit  $[1/\rm s]$, its ACF  $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the unit  $[1/\rm s^2]$, both in the general case  $\varphi_{\rm VZ}(\tau_1, l_1, \tau_2, t_2)$  and with the GWSSUS case  $\varphi_{\rm VZ}(\Delta \tau, \ \Delta t)$.

	*The Dirac function $\delta(\Delta \tau)$ has the ~~dimension~~ $[1/\rm s]$, since the integral over all $\tau$ (with ~~dimension~~ $[\rm s]$) must be $1$. Therefore, both the delay–time cross power density spectrum ${\it \Phi}_{\rm VZ}(\tau, \~~delta~~ \tau)$ and the delay power density spectrum ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \~~delta~~ t = 0)$ have ~~dimension~~ $[1/\rm s]$.		*The Dirac function  ${\rm \delta}(\Delta \tau)$  has the unit  $[1/\rm s]$, since the integral over all  $\tau$  $($with unit  $[\rm s])$ must be  $1$.  Therefore, both the delay–time cross power density spectrum  ${\it \Phi}_{\rm VZ}(\tau, \Delta \tau)$  and the delay power density spectrum  ${\it \Phi}_{\rm V}(\tau) = {\it \Phi}_{\rm VZ}(\tau, \Delta t = 0)$  have unit  $[1/\rm s]$.



	'''(3)'''  <u>Statements 1 and 3</u> are correct:		'''(3)'''  <u>Statements 1 and 3</u> are correct:
	*The function ${\it \Phi}_{\rm VZ}(\tau, \~~delta~~ t)$ has ~~dimension~~ $[1/\rm s]$. Its Fourier transform with respect to $\tau$ is $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to $t$ is ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$. Both $\varphi_{\rm FZ}(\Delta f, \Delta t)$ and ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$ are therefore ~~dimensionless~~.		*The function  ${\it \Phi}_{\rm VZ}(\tau, \Delta t)$  has unit  $[1/\rm s]$.  Its Fourier transform with respect to  $\tau$  is  $\varphi_{\rm FZ}(\Delta f, \Delta t)$, while its Fourier transform with respect to  $t$  is  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$.  Both  $\varphi_{\rm FZ}(\Delta f, \Delta t)$  and  ${\it \Phi}_{\rm VD}(\tau, f_{\rm D})$  are therefore without unit.

	*The frequency–Doppler cross power density spectrum has the unit $[\rm s] = [1/\rm Hz]$, because		*The frequency–Doppler cross power density spectrum has the unit  $[\rm s] = [1/\rm Hz]$, because
	:$${\it \Phi}_{\rm FD}(\~~delta~~ f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$		:$${\it \Phi}_{\rm FD}(\Delta f, f_{\rm D}) = \int_{-\infty}^{+\infty} {\it \Phi}_{\rm VD}(\tau, f_{\rm D}) \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f_{\rm D} \tau}\hspace{0.15cm}{\rm d}\tau \hspace{0.05cm}. $$
	{{ML-Fuß}}		{{ML-Fuß}}

Guenter at 13:06, 14 January 2021

2021-01-14T13:06:19Z

← Older revision		Revision as of 15:06, 14 January 2021
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	The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:		The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:
	* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|Fourier transform]].		* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|Fourier transform]].
	* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral]]  (opposite direction).		* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  (opposite direction).

Guenter at 13:05, 14 January 2021

2021-01-14T13:05:16Z

← Older revision		Revision as of 15:05, 14 January 2021
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	The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:		The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:
	* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#~~Das_erste_Fourierintegral~~\|Fourier transform]].		* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_First_Fourier_Integral\|Fourier transform]].
	* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#~~Das_zweite_Fourierintegral\|inverse Fourier transform~~]]  (opposite direction).		* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral]]  (opposite direction).

Guenter at 13:03, 14 January 2021

2021-01-14T13:03:22Z

← Older revision		Revision as of 15:03, 14 January 2021
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	We denote all of the functions with  $\eta_{i_1i_2}$. The indices $i_1$ and $i_2$ are defined as follows:		We denote all of the functions with  $\eta_{i_1i_2}$. The indices $i_1$ and $i_2$ are defined as follows:
	* ~~<b>~~V~~</b>~~  stands for delay  $\tau$  (index $i_1$),		* $\boldsymbol{\rm V}$  $($because of German  $\rm V\hspace{-0.05cm}$erzögerung$)$  stands for delay time  $\tau$  $($index  $i_1)$,
	* ~~<b>~~F~~</b>~~  stands for frequency  $f$  (index $i_1$),		* $\boldsymbol{\rm F}$  stands for frequency  $f$  $($index  $i_1)$,
	* ~~<b>~~Z~~</b>~~  stands for the time  $t$  (index $i_2$)		* $\boldsymbol{\rm Z}$  $($because of German  $\rm Z\hspace{-0.05cm}$eit$)$  stands for the time  $t$  $($index  $i_2)$,
	* ~~<b>~~D~~</b>~~  stands for the Doppler frequency~~ ~~ $f_{\rm D}$  (index $i_2$).		* $\boldsymbol{\rm D}$  stands for the Doppler frequency $f_{\rm D}$  $($index  $i_2)$.


	The relationship between the functions is shown in the diagram (yellow background). The Fourier correspondences are shown in green:		The relationship between the functions is shown in the diagram (yellow background).  The Fourier correspondences are shown in green:
	* The transition from a circle filled with white to a circle filled with green corresponds to a  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral\|Fourier transform]].		* The transition from a circle filled with white to a circle filled with green corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_erste_Fourierintegral\|Fourier transform]].
	* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral\|inverse Fourier transform]]  (opposite direction).		* The transition from a circle filled with green to a circle filled with white corresponds to the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#Das_zweite_Fourierintegral\|inverse Fourier transform]]  (opposite direction).

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	\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$		\hspace{0.2cm} \stackrel{f, \hspace{0.02cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} \eta_{\rm VZ}(\tau, t)\hspace{0.05cm}.$$

	*The correlation function  $\varphi_{~~i_1i_2~~}$  and the power density spectrum  $\it \Phi_{~~i_1i_2~~}$  are provided with the same indices as the system function $\eta_{~~i_1i_2~~}$.		*The correlation function  $\varphi_{i_1\hspace{0.02cm}i_2}$  and the power density spectrum  $\it \Phi_{i_1\hspace{0.02cm}i_2}$  are provided with the same indices as the system function $\eta_{i_1\hspace{0.02cm}i_2}$.
	*Correlation functions can be recognized by the red font in the lower graph and all power ~~spectral densities~~ are labeled in blue. The GWSSUS model is always assumed.		*Correlation functions can be recognized by the red font in the lower graph and all power densitiy spectra are labeled in blue. The GWSSUS model is always assumed.


	Let us consider here the system function  $\eta_{\rm VZ}(\tau, t)$, i.e. the time variant impulse response  $h(\tau, t)$. We define:		Let us consider here the system function  $\eta_{\rm VZ}(\tau, t)$, i.e. the time–variant impulse response  $h(\tau, t)$.  We define:
	:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot		:$$\varphi_{\rm VZ}(\tau_1, t_1, \tau_2, t_2) = {\rm E} \left [ \eta_{\rm VZ}(\tau_1, t_1) \cdot
	\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm},$$		\eta_{\rm VZ}^{\star}(\tau_2, t_2) \right ]\hspace{0.05cm},$$
Line 41:		Line 41:


	''Note:''   This ~~task~~ belongs to the chapter  [[Mobile_Communications/~~Das_GWSSUS%E2%80%93Kanalmodell~~\| ~~Das GWSUS–Kanalmodell~~]].		''Note:''   This exercise belongs to the chapter  [[Mobile_Communications/The_GWSSUS_Channel_Model\| The GWSSUS Channel Model]].


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	{Which of the following specified dimensions of the system functions are correct?		{Which of the following specified dimensions of the system functions are correct?
	\|type="[]"}		\|type="[]"}
	+ $\eta_{\rm VZ}(\tau, t)$  has the dimension  $[1/\rm s]$		+ $\eta_{\rm VZ}(\tau, t)$  has the dimension  $[1/\rm s]$.
	+ $\eta_{\rm FZ}(f, t)$  is dimensionless.		+ $\eta_{\rm FZ}(f, t)$  is dimensionless.
	+ $\eta_{\rm VD}(\tau, f_{\rm D})$  is dimensionless.		+ $\eta_{\rm VD}(\tau, f_{\rm D})$  is dimensionless.
	+ $\eta_{\rm FD}(f, f_{\rm D})$  has the dimension  $[1/\rm Hz]$		+ $\eta_{\rm FD}(f, f_{\rm D})$  has the dimension  $[1/\rm Hz]$.

	{Which of the following are correct?		{Which of the following statements are correct?
	\|type="[]"}		\|type="[]"}
	- $\varphi_{\rm VZ}(\~~delta~~ \tau, \~~delta~~ t)$  has the dimension  $[1/\rm s]$.		- $\varphi_{\rm VZ}(\Delta \tau, \Delta t)$  has the dimension  $[1/\rm s]$.
	+ ${\it \phi}_{\rm VZ}(\tau, {\rm \~~delta~~} t)$  has the dimension  $[1/\rm s]$.		+ ${\it \phi}_{\rm VZ}(\tau, {\rm \Delta} t)$  has the dimension  $[1/\rm s]$.
	+ ${\it \phi}_{\rm V}(\tau)$  has the dimension  $[1/\rm s]$.		+ ${\it \phi}_{\rm V}(\tau)$  has the dimension  $[1/\rm s]$.

	{Which of the following are correct?		{Which of the following statements are correct?
	\|type="[]"}		\|type="[]"}
	+ $\varphi_{\rm FZ}(\Delta f, \Delta t),  \varphi_{\rm F}(\Delta f)$  and  $\varphi_{\rm Z}(\Delta t)$  have no dimension.		+ $\varphi_{\rm FZ}(\Delta f, \Delta t),  \varphi_{\rm F}(\Delta f)$  and  $\varphi_{\rm Z}(\Delta t)$  have no dimension.