Aufgaben:Exercise 1.5: Reconstruction of the Jakes Spectrum - Revision history

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2026-03-16T15:55:08Z

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← Older revision		Revision as of 17:55, 16 March 2026
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	[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]		[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]
	In a mobile radio system, the  [[Mobile_Communications/~~Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses~~#~~Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts~~\|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.		In a mobile radio system, the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Phenomenological_description_of_the_Doppler_effect\|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.

	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
	:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }\hspace{0.05cm}.$$		:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#~~The_Second_Fourier_Integral~~\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get
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	[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]		[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]
	[[de:~~Exercises~~:~~Exercise_1~~.5:~~_Reconstruction_of_the_Jakes_Spectrum~~]]		[[de:Aufgaben:Exercise 1.5: Reconstruction of the Jakes Spectrum]]

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	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
	:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }		:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.
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	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The Jakes spectrum of the real part is half the resulting spectrum  ${\it \Phi}_z(f)$:		'''(1)'''  The Jakes spectrum of the real part is half the resulting spectrum  ${\it \Phi}_z(f)$:
	:$${\it \Phi}_x(f_{\rm D} = 0) = {\it \Phi}_y(f_{\rm D} = 0) = \frac{{\it \Phi}_z(f_{\rm D} = 0)}{2}= \frac{\sigma^2}{\pi \cdot f_{\rm D, \hspace{0.05cm} max}} =		:$${\it \Phi}_x(f_{\rm D} = 0) = {\it \Phi}_y(f_{\rm D} = 0) = \frac{{\it \Phi}_z(f_{\rm D} = 0)}{2}= \frac{\sigma^2}{\pi \cdot f_{\rm D, \hspace{0.05cm} max}} =\frac{0.5}{\pi \cdot 100\,\,{\rm Hz}} \hspace{0.15cm} \underline{ = 1.59 \cdot 10^{-3}\,\,{\rm Hz^{-1}}}\hspace{0.05cm}.$$
	\frac{0.5}{\pi \cdot 100\,\,{\rm Hz}} \hspace{0.15cm} \underline{ = 1.59 \cdot 10^{-3}\,\,{\rm Hz^{-1}}}
	\hspace{0.05cm}.$$


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	*The input signal  $n(t)$  has a white (constant) PDS  ${\it \Phi}_n(f_{\rm D})$.		*The input signal  $n(t)$  has a white (constant) PDS  ${\it \Phi}_n(f_{\rm D})$.
	*The PDS at the output is then		*The PDS at the output is then
	:$${\it \Phi}_x(f_{\rm D}) = {\it \Phi}_n(f_{\rm D}) \cdot \| H_{\rm DF}(f_{\rm D}\|^2		:$${\it \Phi}_x(f_{\rm D}) = {\it \Phi}_n(f_{\rm D}) \cdot \| H_{\rm DF}(f_{\rm D}\|^2\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

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2026-03-16T12:27:02Z

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← Older revision		Revision as of 14:27, 16 March 2026
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	[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]		[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]
			[[de:Exercises:Exercise_1.5:_Reconstruction_of_the_Jakes_Spectrum]]

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

2022-02-17T11:41:31Z

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

← Older revision		Revision as of 13:41, 17 February 2022
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	[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]		[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]
	In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power spectral density of the Doppler frequency $f_{\rm D}$.		In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.

	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get

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

2022-02-17T11:28:46Z

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

← Older revision		Revision as of 13:28, 17 February 2022
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power spectral density  $\rm (PSD)$  is described in the time domain by the ~~autocorrelation~~ function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get
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	The graph shows the result of the approximation.  The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.  You can see:		The graph shows the result of the approximation.  The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.  You can see:
	[[File:EN_Mob_A_1_5d.png\|right\|frame\|Approximation of the Jakes spectrum and the ~~autocorrelation~~ function]]		[[File:EN_Mob_A_1_5d.png\|right\|frame\|Approximation of the Jakes spectrum and the auto-correlation function]]

	* The Jakes PDS (left graph) can only be reproduced very inaccurately due to the vertical drop at  $± f_{\rm D, \ max}$.		* The Jakes PDS (left graph) can only be reproduced very inaccurately due to the vertical drop at  $± f_{\rm D, \ max}$.

Bene: Text replacement - "(PDS)" to "(PSD)"

2022-02-17T11:28:27Z

Text replacement - "(PDS)" to "(PSD)"

← Older revision		Revision as of 13:28, 17 February 2022
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power spectral density  $\rm (~~PDS~~)$  is described in the time domain by the autocorrelation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power spectral density  $\rm (PSD)$  is described in the time domain by the autocorrelation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get

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

2022-02-17T11:20:24Z

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

← Older revision		Revision as of 13:20, 17 February 2022
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	[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]		[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]
	In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power density ~~spectrum~~ of the Doppler frequency $f_{\rm D}$.		In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power spectral density of the Doppler frequency $f_{\rm D}$.

	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power density ~~spectrum~~  $\rm (PDS)$  is described in the time domain by the autocorrelation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power spectral density  $\rm (PDS)$  is described in the time domain by the autocorrelation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get

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

2021-03-23T12:38:04Z

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

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	[[Category:~~Exercises for~~ Mobile Communications\|^1.3 Rayleigh Fading with Memory^]]		[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]

Guenter at 13:39, 12 February 2021

2021-02-12T13:39:21Z

← Older revision		Revision as of 15:39, 12 February 2021
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	The graph shows the result of the approximation.  The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.  You can see:		The graph shows the result of the approximation.  The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.  You can see:
	[[File:EN_Mob_A_1_5d.png\|right\|frame\|Approximation of the Jakes spectrum and ~~theautocorrelation~~ function]]		[[File:EN_Mob_A_1_5d.png\|right\|frame\|Approximation of the Jakes spectrum and the autocorrelation function]]

	* The Jakes PDS (left graph) can only be reproduced very inaccurately due to the vertical drop at  $± f_{\rm D, \ max}$.		* The Jakes PDS (left graph) can only be reproduced very inaccurately due to the vertical drop at  $± f_{\rm D, \ max}$.

Javier: Text replacement - "===Solutions===" to "===Solution==="

2020-12-15T14:11:01Z

Text replacement - "===Solutions===" to "===Solution==="

← Older revision		Revision as of 16:11, 15 December 2020
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	</quiz>		</quiz>

	===~~Solutions~~===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The Jakes spectrum of the real part is half the resulting spectrum  ${\it \Phi}_z(f)$:		'''(1)'''  The Jakes spectrum of the real part is half the resulting spectrum  ${\it \Phi}_z(f)$:

← Older revision		Revision as of 17:55, 16 March 2026
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	[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]		[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]
	In a mobile radio system, the  [[Mobile_Communications/~~Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses~~#~~Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts~~\|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.		In a mobile radio system, the  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#Phenomenological_description_of_the_Doppler_effect\|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.

	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
	:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }\hspace{0.05cm}.$$		:$${\it \Phi}_z(f_{\rm D}) = \frac{2 \cdot \sigma^2}{\pi \cdot f_{\rm D, \hspace{0.1cm} max} \cdot \sqrt { 1 - (f_{\rm D}/f_{\rm D, \hspace{0.1cm} max})^2} }\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/~~Fourier_Transform_and_Its_Inverse~~#~~The_Second_Fourier_Integral~~\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get
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	[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]		[[Category:Mobile Communications: Exercises\|^1.3 Rayleigh Fading with Memory^]]
	[[de:~~Exercises~~:~~Exercise_1~~.5:~~_Reconstruction_of_the_Jakes_Spectrum~~]]		[[de:Aufgaben:Exercise 1.5: Reconstruction of the Jakes Spectrum]]

← Older revision		Revision as of 13:41, 17 February 2022
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	[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]		[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]
	In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power spectral density of the Doppler frequency $f_{\rm D}$.		In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.

	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
Line 9:		Line 9:
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power-spectral density  $\rm (PSD)$  is described in the time domain by the auto-correlation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get

← Older revision		Revision as of 13:20, 17 February 2022
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	[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]		[[File:P_ID2124__Mob_A_1_5.png\|right\|frame\|Considered Jakes spectrum]]
	In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power density ~~spectrum~~ of the Doppler frequency $f_{\rm D}$.		In a mobile radio system, the  [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts\|Doppler effect]]  is also noticeable in the power spectral density of the Doppler frequency $f_{\rm D}$.

	This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where		This results in the so-called  [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading\|Jakes spectrum]], which is shown in the graph for the maximum Doppler frequency $f_{\rm D, \ max} = 100 \ \rm Hz$.  ${\it \Phi}_z(f_{\rm D})$  has only portions within the range  $± f_{\rm D, \ max}$, where
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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	What is expressed in the frequency domain by the power density ~~spectrum~~  $\rm (PDS)$  is described in the time domain by the autocorrelation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.		What is expressed in the frequency domain by the power spectral density  $\rm (PDS)$  is described in the time domain by the autocorrelation function  $\rm (ACF)$.  The ACF is the   ${\it \Phi}_z(f_{\rm D})$  by the  [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral\|inverse Fourier transform]]  of the PDS.

	With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get		With the  [[Applets:Bessel_Functions_of_the_First_Kind\|Bessel function]]  of the first kind and zero order  $({\rm J}_0)$  you get