Mobile Communications/Multi-Path Reception in Mobile Communications - Revision history

Guenter at 13:44, 29 January 2023

2023-01-29T13:44:05Z

Hwang at 18:51, 24 January 2023

2023-01-24T18:51:07Z

Hwang at 18:44, 24 January 2023

2023-01-24T18:44:37Z

Hwang at 18:38, 24 January 2023

2023-01-24T18:38:09Z

Hwang at 18:37, 24 January 2023

2023-01-24T18:37:28Z

Hwang at 18:32, 24 January 2023

2023-01-24T18:32:41Z

Hwang at 18:31, 24 January 2023

2023-01-24T18:31:38Z

Hwang at 18:22, 24 January 2023

2023-01-24T18:22:32Z

Hwang at 14:28, 23 January 2023

2023-01-23T14:28:40Z

Hwang at 14:28, 23 January 2023

2023-01-23T14:28:09Z

@@ Line 139: / Line 139: @@
 $\text{Example 2:}$&nbsp; Studies have shown that in mobile communications no more than four or five main pathways are effective at the same time.
-[[File:P ID2151 Mob T 2 2 S3b v1.png|right|frame|2D-Impulse response with&nbsp; $M = 3$&nbsp; paths|class=fit]]
+[[File:P ID2151 Mob T 2 2 S3b v1.png|right|frame|Two-dimensional impulse response with&nbsp; $M = 3$&nbsp; paths|class=fit]]
 The represented 2D&ndash;impulse response&nbsp; $h(\tau,\hspace{0.1cm} t)$&nbsp; applies to&nbsp; $M = 3$&nbsp; main paths with time-variant behavior, where the received power decreases with increasing delay in the statistical average.&nbsp; For this graph the above sketched channel model is used as a basis.

@@ Line 147: / Line 147: @@
 *The viewing direction in the right image is rotated by&nbsp; $90^\circ$&nbsp;.&nbsp; By using the color coding, the representation should be understandable.<br>
 <br clear=all>
-This geaphic also shows the weak point of our mobile communications channel model: &nbsp; Although the coefficients&nbsp; $z_m(t)$&nbsp; are variable, the delay times&nbsp; $\tau_m$&nbsp; are fixed. &nbsp; This does not correspond to reality, if the mobile station is moving and the connection takes place in a changing environment.&nbsp; &nbsp; $\tau_m(t)$&nbsp; should be considered.}}<br>
+This graphic also shows the weak point of our mobile communications channel model: &nbsp; Although the coefficients&nbsp; $z_m(t)$&nbsp; are variable, the delay times&nbsp; $\tau_m$&nbsp; are fixed. &nbsp; This does not correspond to reality, if the mobile station is moving and the connection takes place in a changing environment.&nbsp; &nbsp; $\tau_m(t)$&nbsp; should be considered.}}<br>

@@ Line 128: / Line 128: @@
 *The magnitudes of the complex weighting factors&nbsp; $z_m(t)$&nbsp; are Rayleigh distributed according to the section&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Exemplary_signal_curves_with_Rayleigh_fading|"Exemplary signal curves with Rayleigh fading"]]&nbsp; or &ndash; with line-of-sight connection &ndash; Rice distributed, as described in&nbsp; [[Mobile_Communications/Non-Frequency_Selective_Fading_With_Direct_Component#Example_of_signal_behaviour_with_Rice_fading|"Exemplary signal curves with Rice fading"]]&nbsp;.<br>
-*The bindings within the process&nbsp; $z_m(t)$&nbsp; are related to the mobility properties&nbsp; (speed, direction, etc.)&nbsp; to the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading|$\text{Jakes Spectrum}$]]&nbsp;.<br><br>
+*The bindings within the process&nbsp; $z_m(t)$&nbsp; are related to the mobility properties&nbsp; (speed, direction, etc.)&nbsp; to the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PSD_with_Rayleigh.E2.80.93Fading|$\text{Jakes Spectrum}$]]&nbsp;.<br><br>
 The figure shows the generally valid model for the mobile communications channel.&nbsp; "Generally valid" but only with reservations, as explained at the end of&nbsp; $\text{Example 2}$.

@@ Line 115: / Line 115: @@
 $\text{In general the following applies:} $
 #&nbsp; If the signal bandwidth&nbsp; $B_{\rm S} = 1/T_{\rm S}$&nbsp; is much smaller than the coherence bandwidth&nbsp; $B_{\rm K}$, then the channel can be considered as &raquo;<b>non-frequency selective</b>&laquo;&nbsp;&nbsp; &nbsp;<br>$(T_{\rm S}$&nbsp; denotes the symbol duration$)$.<br>
-#&nbsp; In other words: &nbsp; For a given&nbsp; $B_{\rm S}$&nbsp; the smaller the coherence bandwidth&nbsp; $B_{\rm K}$&nbsp; or the larger the maximum delay&nbsp; $\Delta \tau_{\rm max}$&nbsp; the greater the frequency selectivity.
+#&nbsp; In other words: &nbsp; For a given&nbsp; $B_{\rm S}$&nbsp; the smaller the coherence bandwidth&nbsp; $B_{\rm K}$&nbsp; or the larger the maximum delay&nbsp; $\Delta \tau_{\rm max}$,&nbsp; the greater the frequency selectivity.
 #&nbsp; This also means: &nbsp; The frequency selectivity is often determined by the longest echo.
 #&nbsp; Many short echoes with a total energy&nbsp; $E$&nbsp; are less disturbing than one long echo of the same energy&nbsp; $E$.<br>}}

@@ Line 114: / Line 114: @@
 {{BlaueBox|TEXT=
 $\text{In general the following applies:} $
-#&nbsp; If the signal bandwidth&nbsp; $B_{\rm S} = 1/T_{\rm S}$&nbsp; is much smaller than the coherence bandwidth&nbsp; $B_{\rm K}$, then the channel can be considered as <b>non-frequency selective</b>&nbsp;&nbsp; &nbsp;<br>$(T_{\rm S}$&nbsp; denotes the symbol duration$)$.<br>
+#&nbsp; If the signal bandwidth&nbsp; $B_{\rm S} = 1/T_{\rm S}$&nbsp; is much smaller than the coherence bandwidth&nbsp; $B_{\rm K}$, then the channel can be considered as &raquo;<b>non-frequency selective</b>&laquo;&nbsp;&nbsp; &nbsp;<br>$(T_{\rm S}$&nbsp; denotes the symbol duration$)$.<br>
 #&nbsp; In other words: &nbsp; For a given&nbsp; $B_{\rm S}$&nbsp; the smaller the coherence bandwidth&nbsp; $B_{\rm K}$&nbsp; or the larger the maximum delay&nbsp; $\Delta \tau_{\rm max}$&nbsp; the greater the frequency selectivity.
 #&nbsp; This also means: &nbsp; The frequency selectivity is often determined by the longest echo.

@@ Line 58: / Line 58: @@
 ::<math>H(f) = {R(f)}/{S(f)} = 1 + k_0 \cdot {\rm e}^{  - {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \hspace{0.05cm} \cdot \hspace{0.05cm} \tau_0} \hspace{0.05cm}.</math>
-*Through the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|$\text{inverse Fourier transform}$]]&nbsp; one obtains the impulse response
+*Through the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|$\text{inverse Fourier transform}$]]&nbsp; one obtains the impulse response
 ::<math>h(\tau) =  1 + k_0 \cdot \delta(\tau - \tau_0) \hspace{0.05cm}.</math>

@@ Line 53: / Line 53: @@
 ::<math>r(t) = s(t) + k_0 \cdot s( t - \tau_0) \hspace{0.05cm}.</math>
-From this simplified model&nbsp; (without the grey-shaded block)&nbsp; important descriptive variables can be easily calculated:
+From this simplified model&nbsp; (without the gray-shaded block)&nbsp; important descriptive variables can be easily calculated:
 *If you use the&nbsp;  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem| $\text{Shifting Theorem}$]]&nbsp; you get the transfer function

@@ Line 9: / Line 9: @@
 We assume the scenario shown in the graph.&nbsp; This assumes
 [[File:EN_Mob_T_2_2_S1.png|right|frame|Time&ndash;invariant consideration of the two-way channel|class=fit]]
-*Transmitter and receiver are&nbsp; '''at rest''':&nbsp; <br>Then both the channel transfer function and the impulse response are time&ndash;independent.&nbsp; For all times&nbsp; $t$&nbsp; applies&nbsp; $H(f, \hspace{0.05cm}t) = H(f)$&nbsp; and&nbsp; $h(\tau, \hspace{0.05cm}t) = h(\tau)$.<br>
+*Transmitter and receiver are&nbsp; &raquo;'''at rest'''&laquo;:&nbsp; <br>Then both the channel transfer function and the impulse response are time&ndash;independent.&nbsp; For all times&nbsp; $t$&nbsp; applies&nbsp; $H(f, \hspace{0.05cm}t) = H(f)$&nbsp; and&nbsp; $h(\tau, \hspace{0.05cm}t) = h(\tau)$.<br>
-*A&nbsp; '''two-way channel''': &nbsp; <br>The transmitted signal&nbsp; $s(t)$&nbsp; reaches the receiver on a direct path with the path length&nbsp; $d_1$,&nbsp; and there is also an echo due to the reflective ground&nbsp; $($the total path length is&nbsp; $d_2)$.
+*A&nbsp; &raquo;'''two-way channel'''&laquo;: &nbsp; <br>The transmitted signal&nbsp; $s(t)$&nbsp; reaches the receiver on a direct path with the path length&nbsp; $d_1$,&nbsp; and there is also an echo due to the reflective ground&nbsp; $($the total path length is&nbsp; $d_2)$.
@@ Line 22: / Line 22: @@
 *Compared to the transmitted signal, the signal&nbsp; $r_1(t)$&nbsp; received via the direct path is attenuated by the factor&nbsp; $k_1$&nbsp; and delayed by &nbsp; $\tau_1$&nbsp;.
-*The attenuation factor&nbsp; $k_1$&nbsp; is calculated with the&nbsp; [[Mobile_Communications/Distance dependent attenuation and shading#Common path loss model|path loss model]].&nbsp; The greater the transmission frequency&nbsp; $f_{\rm S}$,&nbsp; the distance&nbsp; $d_1$&nbsp; and the exponent&nbsp; $\gamma$&nbsp; are, the smaller is &nbsp; $k_1$&nbsp; and thus the greater is the loss.
+*The attenuation factor&nbsp; $k_1$&nbsp; is calculated with the&nbsp; [[Mobile_Communications/Distance dependent attenuation and shading#Common path loss model|$\text{path loss model}$]].&nbsp; The greater the transmission frequency&nbsp; $f_{\rm S}$,&nbsp; the distance&nbsp; $d_1$&nbsp; and the exponent&nbsp; $\gamma$&nbsp; are, the smaller is &nbsp; $k_1$&nbsp; and thus the greater is the loss.
 *The delay&nbsp; $\tau_1 = d_1/c$&nbsp; increases proportionally with the path length&nbsp; $d_1$&nbsp;. &nbsp; For example, for the distance&nbsp; $d_1 = 3 \ \rm km$&nbsp; and the speed of light&nbsp; $c = 3 \cdot 10^8 \ \rm m/s$&nbsp; the delay will be&nbsp; $\tau_1 = 10 \ \rm &micro; s$.<br>
@@ Line 30: / Line 30: @@
 *In addition, it must be taken into account that the reflection from buildings or the ground leads to a phase rotation of&nbsp; $\pi \ (180^\circ)$.&nbsp; This causes the factor&nbsp; $k_2$&nbsp; to become negative.&nbsp; In the following, however, the negative sign of&nbsp; $k_2$&nbsp; is ignored.<br><br>
-<i>Note:</i> &nbsp; We refer here to the SWF applet&nbsp; [[Applets:Multipath propagation and frequency selectivity (Applet)|Multipath propagation and frequency selectivity]]&nbsp; '''(German language!)'''.
+<i>Note:</i> &nbsp; We refer here to the SWF applet&nbsp; [[Applets:Multipath propagation and frequency selectivity (Applet)|"Multipath propagation and frequency selectivity"]]&nbsp; '''(German language!)'''.
@@ Line 54: / Line 54: @@
 From this simplified model&nbsp; (without the grey-shaded block)&nbsp; important descriptive variables can be easily calculated:
-*If you use the&nbsp;  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem| Shifting Theorem]]&nbsp; you get the transfer function
+*If you use the&nbsp;  [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem| $\text{Shifting Theorem}$]]&nbsp; you get the transfer function
 ::<math>H(f) = {R(f)}/{S(f)} = 1 + k_0 \cdot {\rm e}^{  - {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}2 \pi f \hspace{0.05cm} \cdot \hspace{0.05cm} \tau_0} \hspace{0.05cm}.</math>
-*Through the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|inverse Fourier transform]]&nbsp; one obtains the impulse response
+*Through the&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse#The_Second_Fourier_Integral|$\text{inverse Fourier transform}$]]&nbsp; one obtains the impulse response
 ::<math>h(\tau) =  1 + k_0 \cdot \delta(\tau - \tau_0) \hspace{0.05cm}.</math>
@@ Line 70: / Line 70: @@
 *The transfer function&nbsp; $H(f)$&nbsp; and also its absolute value is periodic with&nbsp; $1/\tau_0 = 500 \ \rm kHz$.
-*This frequency period here is also the&nbsp; [[Mobile_Communications/The GWSSUS channel model#Parameters of the GWSSUS model|coherence bandwidth]] .<br>
+*This frequency period here is also the&nbsp; [[Mobile_Communications/The GWSSUS channel model#Parameters of the GWSSUS model|$\text{coherence bandwidth}$]] .<br>
 *The fluctuations around the mean value&nbsp; $\vert H(f) \vert = 1$&nbsp; are the stronger, the larger the&nbsp; (relative)&nbsp; contribution&nbsp; $k_0$&nbsp; of the second path is&nbsp; (i.e. the echo).}}<br>
@@ Line 105: / Line 105: @@
 *With the red channel&nbsp; $(M = 3)$&nbsp; this value is four times as large: &nbsp; $\Delta \tau_{\rm max} = 8 \ \rm &micro; s$.
-*As a first approximation for&nbsp; [[Mobile_Communications/The GWSSUS channel model#Parameters of the GWSSUS model|coherence bandwidth]]&nbsp; $B_{\rm K}\hspace{0.01cm}'\approx 1/ \Delta \tau_{\rm max}$ is often used, which may differ from the correct value by a factor of&nbsp; $2$&nbsp; or more.&nbsp;
+*As a first approximation for&nbsp; [[Mobile_Communications/The GWSSUS channel model#Parameters of the GWSSUS model|$\text{coherence bandwidth}$]]&nbsp; $B_{\rm K}\hspace{0.01cm}'\approx 1/ \Delta \tau_{\rm max}$ is often used, which may differ from the correct value by a factor of&nbsp; $2$&nbsp; or more.&nbsp;
 *This simple approximation, marked with an apostrophe, results for the blue channel to&nbsp; $B_{\rm K}\hspace{0.01cm}'= 500 \ \rm kHz$.
@@ Line 126: / Line 126: @@
 [[File:P ID3104 Mob T 2 2 S2b v1.png|right|frame|Mobile channel model considering time variance and echoes|class=fit]]
-*The magnitudes of the complex weighting factors&nbsp; $z_m(t)$&nbsp; are Rayleigh distributed according to the section&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Exemplary_signal_curves_with_Rayleigh_fading|Exemplary signal curves with Rayleigh fading]]&nbsp; or &ndash; with line-of-sight connection &ndash; Rice distributed, as described in&nbsp; [[Mobile_Communications/Non-Frequency_Selective_Fading_With_Direct_Component#Example_of_signal_behaviour_with_Rice_fading|Exemplary signal curves with Rice fading]]&nbsp;.<br>
+*The magnitudes of the complex weighting factors&nbsp; $z_m(t)$&nbsp; are Rayleigh distributed according to the section&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#Exemplary_signal_curves_with_Rayleigh_fading|"Exemplary signal curves with Rayleigh fading"]]&nbsp; or &ndash; with line-of-sight connection &ndash; Rice distributed, as described in&nbsp; [[Mobile_Communications/Non-Frequency_Selective_Fading_With_Direct_Component#Example_of_signal_behaviour_with_Rice_fading|"Exemplary signal curves with Rice fading"]]&nbsp;.<br>
-*The bindings within the process&nbsp; $z_m(t)$&nbsp; are related to the mobility properties&nbsp; (speed, direction, etc.)&nbsp; to the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading|Jakes Spectrum]]&nbsp;.<br><br>
+*The bindings within the process&nbsp; $z_m(t)$&nbsp; are related to the mobility properties&nbsp; (speed, direction, etc.)&nbsp; to the&nbsp; [[Mobile_Communications/Statistical_Bindings_within_the_Rayleigh_Process#ACF_and_PDS_with_Rayleigh.E2.80.93Fading|$\text{Jakes Spectrum}$]]&nbsp;.<br><br>
 The figure shows the generally valid model for the mobile communications channel.&nbsp; "Generally valid" but only with reservations, as explained at the end of&nbsp; $\text{Example 2}$.
-For an understanding of the figure we refer to the chapter&nbsp; [[Mobile_Communications/Probability density of Rayleigh fading#A very general description of the mobile communication channel|General description of the mobile communications channel]]. Please note:
+For an understanding of the figure we refer to the chapter&nbsp; [[Mobile_Communications/Probability density of Rayleigh fading#A very general description of the mobile communication channel|"General description of the mobile communications channel"]]. Please note:
 *The&nbsp; $M$&nbsp; main paths are characterized by large propagation time differences.
 *The time-variant complex coefficients&nbsp; $z_m(t)$&nbsp; result from the sum of many secondary paths whose delay times are all approximately the same&nbsp; $\tau_m$&nbsp;.
@@ Line 162: / Line 162: @@
 Here, too, usually no more than&nbsp; $M = 5$&nbsp; main paths simultaneously provide a relevant contribution to the impulse response.
 *The advantage over the first model is that for the delays now all values&nbsp; $\tau_m \le \tau_\text{max}$&nbsp; are possible, with a temporal resolution of&nbsp; $\Delta \tau$&nbsp;.
-*At the end of next chapter&nbsp; [[Mobile_Communications/The GWSSUS channel model#Simulation according to the GWSSUS model|The GWSSUS channel model]]&nbsp; we will come back to this general model again.<br>}}
+*At the end of next chapter&nbsp; [[Mobile_Communications/The GWSSUS channel model#Simulation according to the GWSSUS model|"The GWSSUS channel model"]]&nbsp; we will come back to this general model again.<br>}}
-==Exercises to chapter==
+==Exercises for the chapter==
 <br>
 [[Aufgaben: Exercise 2.2: Simple Two-Path Channel Model]]

@@ Line 101: / Line 101: @@
 The graph shows the magnitude functions&nbsp; $ |H(f)|$&nbsp; of both channels and the corresponding impulse responses&nbsp; $h(\tau)$.&nbsp; One can see from these graphs:
-*In the blue channel&nbsp; $(M = 2)$&nbsp; the Dirac functions occur in a range of width&nbsp; $\Delta \tau_{\rm max} = 2 \ \rm &micro; s$. &nbsp;
+*In the blue channel&nbsp; $(M = 2)$&nbsp; the Dirac delta functions occur in a range of width&nbsp; $\Delta \tau_{\rm max} = 2 \ \rm &micro; s$. &nbsp;
 *With the red channel&nbsp; $(M = 3)$&nbsp; this value is four times as large: &nbsp; $\Delta \tau_{\rm max} = 8 \ \rm &micro; s$.