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

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

2022-02-17T11:41:31Z

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

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

2022-02-17T11:28:46Z

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

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

2022-02-17T11:28:27Z

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

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

2022-02-17T11:20:24Z

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

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

2021-03-23T12:38:04Z

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

Guenter at 13:39, 12 February 2021

2021-02-12T13:39:21Z

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

2020-12-15T14:11:01Z

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

Javier: Text replacement - "Mobile Kommunikation/Statistische Bindungen innerhalb des Rayleigh-Prozesses" to "Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process"

2020-12-15T13:57:31Z

Text replacement - "Mobile Kommunikation/Statistische Bindungen innerhalb des Rayleigh-Prozesses" to "Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process"

Guenter at 15:18, 7 December 2020

2020-12-07T15:18:30Z

Guenter at 15:11, 7 December 2020

2020-12-07T15:11:06Z

@@ Line 3: / Line 3: @@
 [[File:P_ID2124__Mob_A_1_5.png|right|frame|Considered Jakes spectrum]]
-In a mobile radio system, the&nbsp; [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts|Doppler effect]]&nbsp; is also noticeable in the power spectral density of the Doppler frequency $f_{\rm D}$.
+In a mobile radio system, the&nbsp; [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts|Doppler effect]]&nbsp; is also noticeable in the power-spectral density of the Doppler frequency $f_{\rm D}$.
 This results in the so-called&nbsp; [[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 $f_{\rm D, \ max} = 100 \ \rm Hz$.&nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; has only portions within the range&nbsp; $&plusmn; f_{\rm D, \ max}$, where
@@ Line 9: / Line 9: @@
   \hspace{0.05cm}.$$
-What is expressed in the frequency domain by the power spectral density&nbsp; $\rm (PSD)$&nbsp; is described in the time domain by the auto-correlation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
+What is expressed in the frequency domain by the power-spectral density&nbsp; $\rm (PSD)$&nbsp; is described in the time domain by the auto-correlation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
 With the&nbsp; [[Applets:Bessel_Functions_of_the_First_Kind|Bessel function]]&nbsp; of the first kind and zero order &nbsp;$({\rm J}_0)$&nbsp; you get

@@ Line 9: / Line 9: @@
   \hspace{0.05cm}.$$
-What is expressed in the frequency domain by the power spectral density&nbsp; $\rm (PSD)$&nbsp; is described in the time domain by the autocorrelation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
+What is expressed in the frequency domain by the power spectral density&nbsp; $\rm (PSD)$&nbsp; is described in the time domain by the auto-correlation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
 With the&nbsp; [[Applets:Bessel_Functions_of_the_First_Kind|Bessel function]]&nbsp; of the first kind and zero order &nbsp;$({\rm J}_0)$&nbsp; you get
@@ Line 84: / Line 84: @@
 The graph shows the result of the approximation.&nbsp; The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.&nbsp; 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&nbsp; $&plusmn; f_{\rm D, \ max}$.

@@ Line 9: / Line 9: @@
   \hspace{0.05cm}.$$
-What is expressed in the frequency domain by the power spectral density&nbsp; $\rm (PDS)$&nbsp; is described in the time domain by the autocorrelation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
+What is expressed in the frequency domain by the power spectral density&nbsp; $\rm (PSD)$&nbsp; is described in the time domain by the autocorrelation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
 With the&nbsp; [[Applets:Bessel_Functions_of_the_First_Kind|Bessel function]]&nbsp; of the first kind and zero order &nbsp;$({\rm J}_0)$&nbsp; you get

@@ Line 3: / Line 3: @@
 [[File:P_ID2124__Mob_A_1_5.png|right|frame|Considered Jakes spectrum]]
-In a mobile radio system, the&nbsp; [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts|Doppler effect]]&nbsp; is also noticeable in the power density spectrum of the Doppler frequency $f_{\rm D}$.
+In a mobile radio system, the&nbsp; [[Mobile_Communications/Statistische_Bindungen_innerhalb_des_Rayleigh%E2%80%93Prozesses#Ph.C3.A4nomenologische_Beschreibung_des_Dopplereffekts|Doppler effect]]&nbsp; is also noticeable in the power spectral density of the Doppler frequency $f_{\rm D}$.
 This results in the so-called&nbsp; [[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 $f_{\rm D, \ max} = 100 \ \rm Hz$.&nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; has only portions within the range&nbsp; $&plusmn; f_{\rm D, \ max}$, where
@@ Line 9: / Line 9: @@
   \hspace{0.05cm}.$$
-What is expressed in the frequency domain by the power density spectrum&nbsp; $\rm (PDS)$&nbsp; is described in the time domain by the autocorrelation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
+What is expressed in the frequency domain by the power spectral density&nbsp; $\rm (PDS)$&nbsp; is described in the time domain by the autocorrelation function&nbsp; $\rm (ACF)$.&nbsp; The ACF is the &nbsp; ${\it \Phi}_z(f_{\rm D})$&nbsp; by the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; of the PDS.
 With the&nbsp; [[Applets:Bessel_Functions_of_the_First_Kind|Bessel function]]&nbsp; of the first kind and zero order &nbsp;$({\rm J}_0)$&nbsp; you get

← Older revision		Revision as of 12:38, 23 March 2021
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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^]]

@@ Line 57: / Line 57: @@
 </quiz>
-===Solutions===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; The Jakes spectrum of the real part is half the resulting spectrum&nbsp; ${\it \Phi}_z(f)$:

← Older revision		Revision as of 13:57, 15 December 2020
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−	{{quiz-Header\|Buchseite=~~Mobile Kommunikation~~/~~Statistische Bindungen innerhalb des Rayleigh-Prozesses~~}}	+	{{quiz-Header\|Buchseite=Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process}}

	[[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]]

@@ Line 59: / Line 59: @@
 ===Solutions===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The Jakes spectrum of the real part is half the resulting spectrum ${\it \Phi}_z(f)$:
+'''(1)'''&nbsp; The Jakes spectrum of the real part is half the resulting spectrum&nbsp; ${\it \Phi}_z(f)$:
 :$${\it \Phi}_x(f_{\rm D} = 0) = {\it \Phi}_y(f_{\rm D} = 0) = \frac{{\it \Phi}_z(f_{\rm D} = 0)}{2}= \frac{\sigma^2}{\pi \cdot f_{\rm D, \hspace{0.05cm} max}} =
   \frac{0.5}{\pi \cdot 100\,\,{\rm Hz}} \hspace{0.15cm} \underline{ = 1.59 \cdot 10^{-3}\,\,{\rm Hz^{-1}}}
@@ Line 66: / Line 66: @@
 '''(2)'''&nbsp; <u>Solution 2</u> is correct:
-*The input signal $n(t)$ has a white (constant) LDS ${\it \Phi}_n(f_{\rm D})$.
+*The input signal&nbsp; $n(t)$&nbsp; has a white (constant) PDS&nbsp; ${\it \Phi}_n(f_{\rm D})$.
 *The PDS at the output is then
 :$${\it \Phi}_x(f_{\rm D}) = {\it \Phi}_n(f_{\rm D}) \cdot | H_{\rm DF}(f_{\rm D}|^2
@@ Line 73: / Line 73: @@
 '''(3)'''&nbsp; <u>Solution 3</u> is correct.
-*Only if this condition is fulfilled, the signal $x(t)$ has the same variance $\sigma^2$ as the noise signal $n(t)$.
+*Only if this condition is fulfilled, the signal&nbsp; $x(t)$&nbsp; has the same variance&nbsp; $\sigma^2$&nbsp; as the noise signal&nbsp; $n(t)$.
 '''(4)'''&nbsp; <u>No</u>:
-*The two conditions after subtasks (2) and (3) only refer to the magnitude of the digital filter.
+*The two conditions after subtasks&nbsp; '''(2)'''&nbsp; and&nbsp; '''(3)'''&nbsp; only refer to the magnitude of the digital filter.
 *There is no constraint for the phase of the digital filter.
-*This phase can be chosen arbitrarily. Usually it is chosen in such a way that a minimum phase network results.
+*This phase can be chosen arbitrarily.&nbsp; Usually it is chosen in such a way that a minimum phase network results.
-*In this case, the impulse response $h_{\rm DF}(t)$ then has the lowest possible duration.
+*In this case, the impulse response&nbsp; $h_{\rm DF}(t)$&nbsp; then has the lowest possible duration.
-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.&nbsp; The red curves were determined simulatively over $100\hspace{0.05cm}000$ samples.&nbsp; You can see:
-[[File:EN_Mob_A_1_5d.png|right|frame|Approximation of the Jakes spectrum and autocorrelation function]]
+[[File:EN_Mob_A_1_5d.png|right|frame|Approximation of the Jakes spectrum and theautocorrelation function]]
+* The Jakes PDS (left graph) can only be reproduced very inaccurately due to the vertical drop at&nbsp; $&plusmn; f_{\rm D, \ max}$.
-* The Jakes power density spectrum (left graph) can only be reproduced very inaccurately due to the vertical drop at $&plusmn; f_{\rm D, \ max}$.
+* For the time domain, this means that the ACF decreases much faster than the theory suggests.
-* For the time domain, this means that the ACF decreases much faster than theory suggests.
+*For small values of&nbsp;  $\Delta t$, however, the approximation is very good (right graph).
 {{ML-Fuß}}

@@ Line 36: / Line 36: @@
 <quiz display=simple>
-{What is the value of the Jakes, spectrum of the real part at the Doppler frequency&nbsp; $f_{\rm D} = 0$?
+{What is the value of the Jakes spectrum of the real part at the Doppler frequency $f_{\rm D} = 0$?
 |type="{}"}
-${\it \Phi}_x(f_{\rm D} = 0)\ = \ $ { 1.59 } $\ \cdot 10^{\rm &ndash;3} \ 1/{\rm Hz}$
+${\it \Phi}_x(f_{\rm D} = 0)\ = \ $ { 1.59 } $\ \cdot 10^{\rm &ndash;3} \ {\rm Hz}^{-1}$
 {Which dimensioning is correct, where &nbsp;$K$&nbsp; is an appropriately chosen constant?
@@ Line 51: / Line 51: @@
 + The integral over&nbsp; $|H_{\rm DF}(f_{\rm D})|^2$&nbsp; must be&nbsp; $1$&nbsp;.
-{Is&nbsp; $H_{\rm DF}(f)$&nbsp; unambiguously defined by the two conditions according to '''(2)'' and '''(3)'''?
+{Is&nbsp; $H_{\rm DF}(f)$&nbsp; unambiguously defined by the two conditions according to&nbsp; '''(2)'''&nbsp; and&nbsp; '''(3)'''?
 |type="()"}
 - Yes.