Aufgaben:Exercise 2.1: Two-Dimensional Impulse Response - Revision history

Javier: Text replacement - "time-discrete" to "discrete-time"

2021-10-11T09:01:00Z

Text replacement - "time-discrete" to "discrete-time"

Javier: Text replacement - "band pass" to "band-pass"

2021-05-03T15:13:21Z

Text replacement - "band pass" to "band-pass"

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"

Guenter at 10:09, 16 February 2021

2021-02-16T10:09:17Z

Javier: Text replacement - "===Sample solution===" to "===Solution==="

2020-12-16T13:16:19Z

Text replacement - "===Sample solution===" to "===Solution==="

Javier: Text replacement - "low pass" to "low-pass"

2020-12-15T14:19:22Z

Text replacement - "low pass" to "low-pass"

Javier: Text replacement - "Mobile Kommunikation/Allgemeine Beschreibung zeitvarianter Systeme" to "Mobile_Communications/General_Description_of_Time_Variant_Systems"

2020-12-15T13:59:18Z

Text replacement - "Mobile Kommunikation/Allgemeine Beschreibung zeitvarianter Systeme" to "Mobile_Communications/General_Description_of_Time_Variant_Systems"

Guenter at 16:25, 14 December 2020

2020-12-14T16:25:20Z

Guenter at 15:59, 14 December 2020

2020-12-14T15:59:11Z

Guenter at 15:45, 14 December 2020

2020-12-14T15:45:51Z

@@ Line 6: / Line 6: @@
 :$$h(\tau,\hspace{0.05cm}t) = \sum_{m = 1}^{M} z_m(t) \cdot {\rm \delta} (\tau - \tau_m)$$
-is to be analyzed according to the adjoining diagram.&nbsp; The two axes are time-discrete:
+is to be analyzed according to the adjoining diagram.&nbsp; The two axes are discrete-time:
 * $\tau$&nbsp; is the&nbsp; <i>delay</i>&nbsp; and can take values between&nbsp; $0$&nbsp; and&nbsp; $6 \ {\rm &micro; s}$&nbsp; in the example.
 * The (absolute) time&nbsp; $t$&nbsp; is related to the frequency of snapshots and characterizes the variation of the channel over time. We have&nbsp; $t = n \cdot T$, where&nbsp; $T \gg \tau_{\rm max}$&nbsp;.

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 {{ML-Kopf}}
 '''(1)'''&nbsp; The signal described in the equivalent low-pass range should not have a bandwidth greater than&nbsp; $B_{\rm max} = 1/\Delta \tau \ \underline {= 500 \ \rm kHz}$.
-*This mathematical (two-sided) bandwidth of the low-pass signal is also the maximum physical (one-sided) bandwidth of the corresponding band pass signal.
+*This mathematical (two-sided) bandwidth of the low-pass signal is also the maximum physical (one-sided) bandwidth of the corresponding band-pass signal.

← Older revision		Revision as of 12:37, 23 March 2021
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−	[[Category:~~Exercises for~~ Mobile Communications\|^2.1 Description of Time-Variant Systems^]]	+	[[Category:Mobile Communications: Exercises\|^2.1 Description of Time-Variant Systems^]]

@@ Line 16: / Line 16: @@
 In this task the following quantities are also referred to:
-* the <i>time-variant transfer function</i>&nbsp; according to the definition
+* the "time-variant transfer function"&nbsp; according to the definition
 :$$H(f,\hspace{0.05cm} t)
   \hspace{0.2cm}  \stackrel {f,\hspace{0.05cm}\tau}{\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ} \hspace{0.2cm} h(\tau,\hspace{0.05cm}t)
   \hspace{0.05cm},$$
-* the approximation of the <i>coherence bandwidth</i>&nbsp; as the reciprocal of the maximal duration of the delay profile&nbsp; $h(\tau, t)$:
+* the approximation of the "coherence bandwidth"&nbsp; as the reciprocal of the maximal duration of the delay profile&nbsp; $h(\tau, t)$:
 :$$B_{\rm K} \hspace{0.01cm}' = \frac{1}{\tau_{\rm max} - \tau_{\rm min}}
   \hspace{0.05cm}.$$
@@ Line 71: / Line 71: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The signal described in the equivalent low-pass range should not have a bandwidth greater than&nbsp; $B_{\rm max} = 1/\delta \tau \ \underline {= 500 \ \rm kHz}$.
+'''(1)'''&nbsp; The signal described in the equivalent low-pass range should not have a bandwidth greater than&nbsp; $B_{\rm max} = 1/\Delta \tau \ \underline {= 500 \ \rm kHz}$.
 *This mathematical (two-sided) bandwidth of the low-pass signal is also the maximum physical (one-sided) bandwidth of the corresponding band pass signal.
@@ Line 94: / Line 94: @@
-'''(5)'''&nbsp; The impulse responses are identical at the times&nbsp; $5T$,&nbsp; $6T$&nbsp; and&nbsp; $7T$&;nbsp; each consists of three Diracs.
+'''(5)'''&nbsp; The impulse responses are identical at the times&nbsp; $5T$,&nbsp; $6T$&nbsp; and&nbsp; $7T$&nbsp; each consists of three Diracs.
 * Assuming that nothing changes in this respect for&nbsp; $t &#8805; 8T$: &nbsp; $t_{\rm 5} \ \ \underline {= 5T}$.

@@ Line 69: / Line 69: @@
 </quiz>
-===Sample solution===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; The signal described in the equivalent low-pass range should not have a bandwidth greater than&nbsp; $B_{\rm max} = 1/\delta \tau \ \underline {= 500 \ \rm kHz}$.

← Older revision		Revision as of 13:59, 15 December 2020
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−	{{quiz-Header\|Buchseite=~~Mobile Kommunikation~~/~~Allgemeine Beschreibung zeitvarianter Systeme~~}}	+	{{quiz-Header\|Buchseite=Mobile_Communications/General_Description_of_Time_Variant_Systems}}

	[[File:P_ID2144__Mob_A_2_1.png\|right\|frame\|Two-dimensional impulse response]]		[[File:P_ID2144__Mob_A_2_1.png\|right\|frame\|Two-dimensional impulse response]]

@@ Line 45: / Line 45: @@
 $B_{\rm max} \ = \ ${ 500 3% } $\ \ \rm kHz$
-{At what time&nbsp; $t_2$&nbsp; is the channel ideal, characterized by&nbsp; $H(f, t_{\rm 2}) = 1$?
+{At what time&nbsp; $t_2$&nbsp; the channel is ideal, characterized by&nbsp; $H(f, t_{\rm 2}) = 1$?
 |type="{}"}
 $t_{\rm 2} \ = \ ${ 0. } $\ \cdot T$
-{From what time&nbsp; $t_{\rm 3}$&nbsp; does this channel cause distortion?
+{From what time&nbsp; $t_{\rm 3}$&nbsp; this channel does cause distortion?
 |type="{}"}
 $t_{\rm 3} \ = \ ${ 3 3% } $\ \cdot T$
@@ Line 59: / Line 59: @@
 $t = 5T \text{:} \hspace{0.4cm} B_{\rm K} \hspace{0.01cm}' \ = \ ${ 166.7 3% } $\ \ \rm kHz$
-{From what time&nbsp; $t_{\rm 5}$&nbsp; could this channel be considered as time&ndash;invariant?
+{From what time&nbsp; $t_{\rm 5}$&nbsp; this channel could be considered as time&ndash;invariant?
 |type="{}"}
 $t_{\rm 5} \ = \ ${ 5 3% } $\ \cdot T$
@@ Line 71: / Line 71: @@
 ===Sample solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The message signal described in the equivalent low-pass band shall not have a bandwidth greater than $B_{\rm max} = 1/\delta \tau \ \underline {= 500 \ \rm kHz}$.
+'''(1)'''&nbsp; The signal described in the equivalent low pass range should not have a bandwidth greater than&nbsp; $B_{\rm max} = 1/\delta \tau \ \underline {= 500 \ \rm kHz}$.
-*This mathematical (two-sided) bandwidth of the low pass&ndash;signal is also the maximum physical (one-sided) bandwidth of the corresponding bandpass&ndash;signal.
+*This mathematical (two-sided) bandwidth of the low pass signal is also the maximum physical (one-sided) bandwidth of the corresponding band pass signal.
+'''(2)'''&nbsp; $H(f, t_{\rm 2}) = 1$&nbsp; means in the time domain&nbsp; $h(\tau, t_{\rm 2}) = \delta(\tau)$.
 *Only then the channel is ideal.
-*You can see from the graph that this only applies to the time $t_{\rm 2} \ \underline {= 0}$.
+*You can see from the graph that this only applies to the time&nbsp; $t_{\rm 2} \ \underline {= 0}$.
-'''(4)'''&nbsp; At time $t = 3T$ the two Dirac functions occur at $\tau_{\rm min} = 0$ and $\tau_{\rm max} = 4 \ \rm &micro; s$.
+'''(4)'''&nbsp; At time&nbsp; $t = 3T$&nbsp; the two Dirac functions occur at&nbsp; $\tau_{\rm min} = 0$&nbsp; and&nbsp; $\tau_{\rm max} = 4 \ \rm &micro; s$.
-*The (simple approximation for the) coherence bandwidth is the reciprocal of the delay span of these Dirac functions
+*The (simple approximation for the) coherence bandwidth is the reciprocal of the delay span of these Dirac functions:
 :$$B_{\rm K}\hspace{0.01cm}' = \frac{1}{4\,\,{\rm &micro; s} } \hspace{0.25cm} \underline{ = 250\,\,{\rm kHz}}
   \hspace{0.05cm}.$$
+*The same as at&nbsp; $t = 4T$&nbsp; the time between the Dirac functions is&nbsp; $4 \ \rm &micro; s$:&nbsp;  $B_{\rm K} \hspace{0.01cm}' = \underline {250 \ \rm kHz}$.
+*At&nbsp; $t = 5T$&nbsp; the impulse response has a duration of&nbsp; $6 \ \ \rm &micro; s \ \ \Rightarrow \ {\it B}_{\rm K} \hspace{0.01cm}' \ \underline {\approx 166.7 \ \rm kHz}$.
-'''(5)'''&nbsp; The impulse responses are identical at the times $5T$, $6T$ and $7T$ and consist of 3 Diracs each.
+'''(5)'''&nbsp; The impulse responses are identical at the times&nbsp; $5T$,&nbsp; $6T$&nbsp; and&nbsp; $7T$&;nbsp; each consists of three Diracs.
-* Assuming that nothing changes in this respect for $t &#8805; 8T$, you get $t_{\rm 5} \ \ \underline {= 5T}$.
+* Assuming that nothing changes in this respect for&nbsp; $t &#8805; 8T$: &nbsp; $t_{\rm 5} \ \ \underline {= 5T}$.
 '''(6)'''&nbsp; <u>Solution 2</u> is correct:
-*The temporal change of the impulse response, whose dynamics is expressed by the parameter $T$, must be slow in comparison to the maximum delay span of $h(\tau, t)$, which in this task equals $\tau_{\rm max} = 6 \ \rm &micro; s$: &e.g;
+*The temporal change of the impulse response, whose dynamics is expressed by the parameter&nbsp; $T$, must be slow in comparison to the maximum delay span of&nbsp; $h(\tau, t)$, which is in this task equals $\tau_{\rm max} = 6 \ \rm &micro; s$ &nbsp; &rArr; &nbsp; $T \gg \tau_{\rm max}.$
 {{ML-Fuß}}
 [[Category:Exercises for Mobile Communications|^2.1 Description of Time-Variant Systems^]]

@@ Line 41: / Line 41: @@
 ===Questionnaire===
 <quiz display=simple>
-{What restriction does the specification&nbsp; $\Delta \tau = 2 \rm &micro; s$&nbsp; impose on the maximum bandwidth&nbsp; $B_{\rm max}$&nbsp; of the message signal to be examined?
+{What restriction does the specification&nbsp; $\Delta \tau = 2 \rm &micro; s$&nbsp; impose on the maximum bandwidth&nbsp; $B_{\rm max}$&nbsp; of the signal to be examined?
 |type="{}"}
 $B_{\rm max} \ = \ ${ 500 3% } $\ \ \rm kHz$
@@ Line 59: / Line 59: @@
 $t = 5T \text{:} \hspace{0.4cm} B_{\rm K} \hspace{0.01cm}' \ = \ ${ 166.7 3% } $\ \ \rm kHz$
-{From what time&nbsp; $t_{\rm 5}$&nbsp; could this channel be considered as time invariant?
+{From what time&nbsp; $t_{\rm 5}$&nbsp; could this channel be considered as time&ndash;invariant?
 |type="{}"}
 $t_{\rm 5} \ = \ ${ 5 3% } $\ \cdot T$
-{For which of the mentioned&nbsp; values of $T$ does working with the&nbsp; $\rm 2D$&ndash;impulse response make sense?
+{For which of the mentioned&nbsp; values of&nbsp; $T$&nbsp; does it make sense to work with the&nbsp; $\rm 2D$&ndash;impulse response?
 |type="[]"}
 - A (slow) channel change occurs approximately after&nbsp; $T = 1 \ \rm &micro; s$.

@@ Line 4: / Line 4: @@
 [[File:P_ID2144__Mob_A_2_1.png|right|frame|Two-dimensional impulse response]]
 The two-dimensional impulse response
-$$h(\tau,\hspace{0.05cm}t) = \sum_{m = 1}^{M} z_m(t) \cdot {\rm \delta} (\tau - \tau_m)$$
+:$$h(\tau,\hspace{0.05cm}t) = \sum_{m = 1}^{M} z_m(t) \cdot {\rm \delta} (\tau - \tau_m)$$
-is to be analyzed according to the adjoining diagram. The two axes are time-discrete:
+is to be analyzed according to the adjoining diagram.&nbsp; The two axes are time-discrete:
 * $\tau$&nbsp; is the&nbsp; <i>delay</i>&nbsp; and can take values between&nbsp; $0$&nbsp; and&nbsp; $6 \ {\rm &micro; s}$&nbsp; in the example.
-* The ''(absolute) time''&nbsp; $t$&nbsp; is related to the frequency of snapshots and characterizes the variation of the channel over time. We have&nbsp; $t = n \cdot T$, where&nbsp; $T \gg \tau_{\rm max}$&nbsp;.
+* The (absolute) time&nbsp; $t$&nbsp; is related to the frequency of snapshots and characterizes the variation of the channel over time. We have&nbsp; $t = n \cdot T$, where&nbsp; $T \gg \tau_{\rm max}$&nbsp;.
 The arrows in the graphic mark different Dirac functions with weights&nbsp; $1$&nbsp; (red),&nbsp; $1/2$&nbsp; (blue) and&nbsp; $1/4$&nbsp; (green). This means that the delay&nbsp; $\tau$&nbsp; is also discrete here.
-When measuring the impulse responses at different times&nbsp; $t$&nbsp; at intervals of one second, the resolution of the&nbsp; $\tau$ axis was &nbsp; $2$&nbsp; microseconds $(\delta \tau = 2 \ \rm &micro; s)$. The echoes were not localized more precisely.
+When measuring the impulse responses at different times&nbsp; $t$&nbsp; at intervals of one second, the resolution of the&nbsp; $\tau$&ndash;axis was two microseconds $(\delta \tau = 2 \ \rm &micro; s)$.&nbsp; The echoes were not localized more precisely.
 In this task the following quantities are also referred to:
@@ Line 21: / Line 21: @@
   \hspace{0.05cm},$$
 * the approximation of the <i>coherence bandwidth</i>&nbsp; as the reciprocal of the maximal duration of the delay profile&nbsp; $h(\tau, t)$:
 :$$B_{\rm K} \hspace{0.01cm}' = \frac{1}{\tau_{\rm max} - \tau_{\rm min}}
   \hspace{0.05cm}.$$
@@ Line 32: / Line 31: @@
 ''Notes:''
-* This task belongs to the chapter&nbsp; [[Mobile_Communications/Allgemeine_Beschreibung_zeitvarianter_Systeme| Allgemeine Beschreibung zeitvarianter Systeme]].
+* This task belongs to the chapter&nbsp; [[Mobile_Communications/General_Description_of_Time_Variant_Systems|General description of time&ndash;variant systems]].
-* More detailed information on various definitions for the coherence bandwidth can be found in chapter&nbsp; [[Mobile_Communications/Das_GWSSUS%E2%80%93Kanalmodell| Das GWSSUS&ndash;Kanalmodell]], especially in the sample solution for the&nbsp; [[Aufgaben:Exercise_2.7:_Coherence_Bandwidth|Exercise 2.7Z]].
+* More detailed information on various definitions for the coherence bandwidth can be found in chapter&nbsp; [[Mobile_Communications/The_GWSSUS_Channel_Model|The GWSSUS channel model]], especially in the sample solution for the&nbsp; [[Aufgaben:Exercise_2.7Z:_Coherence_Bandwidth_of_the_LTI_Two-Path_Channel|Exercise 2.7Z]].
-* It should be noted that this is a constructed task. According to the above graph, the 2D impulse response changes significantly during the time span&nbsp; $T$&nbsp;. Therefore&nbsp; $T$&nbsp; is to be interpreted here as very large, for example one hour.
+* It should be noted that this is a constructed task.&nbsp; According to the above graph, the 2D impulse response changes significantly during the time span&nbsp; $T$.&nbsp; Therefore&nbsp; $T$&nbsp; is to be interpreted here as very large, for example one hour.
 *In mobile radio,&nbsp; $h(\tau, t)$&nbsp; changes in the millisecond range taking into account the Doppler effect, but the changes during this time are rather moderate.