Aufgaben:Exercise 4.4Z: Pointer Diagram for SSB-AM - Revision history

Hwang at 03:15, 18 September 2022

2022-09-18T03:15:42Z

Guenter at 14:37, 7 May 2021

2021-05-07T14:37:51Z

Guenter at 14:26, 7 May 2021

2021-05-07T14:26:26Z

Guenter at 14:23, 7 May 2021

2021-05-07T14:23:39Z

Guenter at 14:22, 7 May 2021

2021-05-07T14:22:05Z

Guenter at 16:54, 6 May 2021

2021-05-06T16:54:40Z

Guenter at 15:11, 6 May 2021

2021-05-06T15:11:33Z

Guenter: Guenter moved page Aufgaben:Exercise 4.4Z: Vector Diagram for DSB-AM to Aufgaben:Exercise 4.4Z: Pointer Diagram for SSB-AM

2021-05-06T15:11:11Z

Guenter moved page Exercise 4.4Z: Vector Diagram for DSB-AM to Exercise 4.4Z: Pointer Diagram for SSB-AM

Javier: Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

2021-03-23T12:39:29Z

Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

Noah at 21:43, 15 February 2021

2021-02-15T21:43:33Z

← Older revision		Revision as of 03:15, 18 September 2022
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−	'''(3)'''  The maximum value of  $\|s_+(t)\|$  is reached when both pointers point in the same direction. The ~~amount~~ of the sum pointer is then equal to the sum of the two individual pointers; i.e.  $\underline {2\ \text{V}}$.	+	'''(3)'''  The maximum value of  $\|s_+(t)\|$  is reached when both pointers point in the same direction. The magnitude of the sum pointer is then equal to the sum of the two individual pointers; i.e.  $\underline {2\ \text{V}}$.

	This case is reached for the first time when the faster pointer with circular velocity  $\omega_{60}$  has caught up its "lag" of  $90^{\circ} \; (\pi /2)$  with the slower pointer  ($\omega_{50}$) :		This case is reached for the first time when the faster pointer with circular velocity  $\omega_{60}$  has caught up its "lag" of  $90^{\circ} \; (\pi /2)$  with the slower pointer  ($\omega_{50}$) :

@@ Line 77: / Line 77: @@
 }\hspace{0.05cm} t }) + {\rm 1 \hspace{0.05cm} V} \cdot \sin({
 \omega_{\rm 60}\hspace{0.05cm} t }).$$
-The real part of this describes the actual, physical signal:
+The real part of this describes the actual physical signal:
 :$$s(t) = {\rm 1 \hspace{0.05cm} V} \cdot \cos({ \omega_{\rm
 }\hspace{0.05cm} t }) + {\rm 1 \hspace{0.05cm} V} \cdot \sin({
@@ Line 83: / Line 83: @@
 Correct is the <u>proposed solution 3</u>:
-*Considering the&nbsp; $50 \ \text{kHz-Cosinussignals}$&nbsp; cosine signal alone, the first zero crossing would occur at&nbsp; $t_1 = T_0/4$&nbsp; , i.e. after&nbsp; $5 \ {\rm &micro; s}$, where&nbsp; $T_0 = 1/f_{50} = 20 \ {\rm &micro; s}$&nbsp; denotes the period duration of this signal.
+*Considering the&nbsp; $50 \ \text{kHz}$&nbsp; cosine signal alone, the first zero crossing would occur at&nbsp; $t_1 = T_0/4$&nbsp; , i.e. after&nbsp; $5 \ {\rm &micro; s}$, where&nbsp; $T_0 = 1/f_{50} = 20 \ {\rm &micro; s}$&nbsp; denotes the period duration of this signal.
 *The sinusoidal signal with the frequency&nbsp; $60 \ \text{kHz}$&nbsp; is positive during the entire first half-wave&nbsp; $(0 \, \text{...} \, 8.33\ {\rm &micro; s})$&nbsp;.
 *Due to the plus sign, the first zero crossing of&nbsp; $s(t) \ \Rightarrow \ t_1 > 5\ {\rm &micro; s}$ is delayed.
@@ Line 91: / Line 91: @@
-'''(3)'''&nbsp;  The maximum value of&nbsp; $|s_+(t)|$&nbsp; is reached when both pointers point in the same direction. The amount of the sum pointer is then equal to the sum of the two individual pointers; i.e.&nbsp; $\underline {2\ \text{ V}}$.
+'''(3)'''&nbsp;  The maximum value of&nbsp; $|s_+(t)|$&nbsp; is reached when both pointers point in the same direction. The amount of the sum pointer is then equal to the sum of the two individual pointers; i.e.&nbsp; $\underline {2\ \text{V}}$.
-This case is reached for the first time when the faster pointer with angular velocity&nbsp; $\omega_{60}$&nbsp; has caught up its "lag" of&nbsp; $90^{\circ} \; (\pi /2)$&nbsp; with the slower pointer&nbsp; ($\omega_{50}$)&nbsp;:
+This case is reached for the first time when the faster pointer with circular velocity&nbsp; $\omega_{60}$&nbsp; has caught up its "lag" of&nbsp; $90^{\circ} \; (\pi /2)$&nbsp; with the slower pointer&nbsp; ($\omega_{50}$)&nbsp;:
 :$$\omega_{\rm 60} \cdot  t_2 - \omega_{\rm
 }\cdot t_2 = \frac{\pi}{2} \hspace{0.3cm}
@@ Line 99: / Line 99: @@
 f_{\rm 50})} =   \frac{1}{4
 \cdot(f_{\rm 60}- f_{\rm 50})}\hspace{0.15 cm}\underline{= {\rm 25 \hspace{0.05cm} {\rm &micro; s}}}.$$
-*At this point, the two pointers have made&nbsp; $5/4$&nbsp; bzw.&nbsp; $6/4$&nbsp; revolutions respectively and both point in the direction of the imaginary axis (see right graph).
+*At this point, the two pointers have made&nbsp; $5/4$&nbsp; and&nbsp; $6/4$&nbsp; rotations respectively and both point in the direction of the imaginary axis (see right graph).
-*The actual, physical signal&nbsp; $s(t)$ –  i.e. the real part of&nbsp; $s_+(t)$ – is therefore zero at this moment.
+*The actual physical signal&nbsp; $s(t)$ –  i.e. the real part of&nbsp; $s_+(t)$ – is therefore zero at this moment.
 '''(4)'''&nbsp;  The condition for&nbsp; $|s_+(t_3)| = 0$&nbsp; is that there is a phase offset of&nbsp; $180^\circ$&nbsp; between the two equally long pointers so that they cancel each other out.
-*This further means that the faster pointer has rotated&nbsp; $3\pi /2$&nbsp; further than the&nbsp; $50 \ \text{kHz-component}$.
+*This further means that the faster pointer has rotated&nbsp; $3\pi /2$&nbsp; further than the&nbsp; $50 \ \text{kHz}$&nbsp; component.
-*Analogous to the sample solution of sub-task&nbsp; '''(3)'''&nbsp;, the following therefore applies:
+*Analogous to the sample solution of sub-task&nbsp; '''(3)'''&nbsp;,  therefore the following applies:
 :$$t_3  = \frac{3\pi/2}{2\pi (f_{\rm 60}- f_{\rm 50})} \hspace{0.15 cm}\underline{=
   {\rm 75 \hspace{0.05cm} {\rm &micro; s}}}.$$

@@ Line 12: / Line 12: @@
 Here&nbsp; $f_{50}$&nbsp; and&nbsp; $f_{60}$&nbsp; are abbreviations for the frequencies&nbsp; $50 \ \text{kHz}$&nbsp; and&nbsp; $60 \ \text{kHz}$, respectively.
-This analytical signal could occur, for example, with the&nbsp; [[Modulation_Methods/Einseitenbandmodulation|Single Sideband Amplitude Modulation]]&nbsp; $\text{(SSB-AM)}$&nbsp; of a sinusoidal message signal&nbsp; $($Frequenz&nbsp; $f_{\rm N} = 10 \ \text{kHz})$&nbsp; with a cosinusoidal carrier signal&nbsp; $(f_{\rm T} = 50 \ \text{kHz})$&nbsp;, whereby only the upper sideband is transmitted &nbsp; &rArr; &nbsp $\text{Upper Sideband Modulation}$.
+This analytical signal could occur, for example, with the&nbsp; [[Modulation_Methods/Einseitenbandmodulation|Single Sideband Amplitude Modulation]]&nbsp; $\text{(SSB&ndash;AM)}$&nbsp; of a sinusoidal message signal&nbsp; $($Frequenz&nbsp; $f_{\rm N} = 10 \ \text{kHz})$&nbsp; with a cosinusoidal carrier signal&nbsp; $(f_{\rm T} = 50 \ \text{kHz})$&nbsp;, whereby only the upper sideband is transmitted &nbsp; &rArr; &nbsp; $\text{Upper Sideband Modulation}$.
-However, the analytical signal could also result from a&nbsp; $\text{(Lower Sideband Modulation)}$&nbsp; of the same sinusoidal signal if a sinusoidal carrier with frequency&nbsp; $f_{\rm T} = 60 \ \text{kHz}$&nbsp; is used.
+However, the analytical signal could also result from a&nbsp; $\text{Lower Sideband Modulation}$&nbsp; of the same sinusoidal signal if a sinusoidal carrier with frequency&nbsp; $f_{\rm T} = 60 \ \text{kHz}$&nbsp; is used.
@@ Line 30: / Line 30: @@
 <quiz display=simple>
-{Give the analytical signal&nbsp; $s_+(t)$&nbsp; as a formula. What value results at the starting time&nbsp; $t = 0$?
+{Give the analytical signal&nbsp; $s_+(t)$&nbsp; as a formula.&nbsp; What value results at the starting time&nbsp; $t = 0$?
 |type="{}"}
 $\text{Re}[s_+(t = 0)]\ = \ $  { 1 3% } &nbsp;$\text{V}$
@@ Line 43: / Line 43: @@
-{What is the maximum value of&nbsp; $|s_+(t)|$&nbsp;? At what time&nbsp; $t_2$&nbsp; is this maximum value reached for the first time?
+{What is the maximum value of&nbsp; $|s_+(t)|$?&nbsp; At what time&nbsp; $t_2$&nbsp; is this maximum value reached for the first time?
 |type="{}"}
 $|s_+(t)|_{\rm max}\ = \ ${ 2 3% } &nbsp;$\text{V}$

@@ Line 18: / Line 18: @@
 ''Hints:''
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and its Spectral Function]].
-*You can check your solution with the interaction module&nbsp; [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physical and Analytical Signal]].
+*You can check your solution with the interaction module&nbsp; [[Applets:Physical_Signal_%26_Analytic_Signal|Physical and Analytical Signal]].

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID732__Sig_Z_4_4_neu.png|right|frame|Given spectrum&nbsp; $S_+(f)$]]
+[[File:P_ID732__Sig_Z_4_4_neu.png|right|frame|Given analytical spectrum&nbsp; $S_+(f)$]]
 The analytical signal&nbsp; $s_+(t)$&nbsp; with the line spectrum
@@ Line 17: / Line 10: @@
 f_{\rm 60})$$
 is to be considered.
-Here&nbsp; $f_{50}$&nbsp; and&nbsp; $f_{60}$&nbsp; are abbreviations for the frequencies&nbsp; $50 \ \text{kHz}$&nbsp; and&nbsp; $60 \ \text{kHz}$, respectively..
+Here&nbsp; $f_{50}$&nbsp; and&nbsp; $f_{60}$&nbsp; are abbreviations for the frequencies&nbsp; $50 \ \text{kHz}$&nbsp; and&nbsp; $60 \ \text{kHz}$, respectively.
@@ Line 34: / Line 24: @@
 ''Hints:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and Its Spectral Function]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and its Spectral Function]].
-*You can check your solution with the interaction module&nbsp; [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physical signal & Analytical Signal]]&nbsp;.
+*You can check your solution with the interaction module&nbsp; [[Applets:Physikalisches_Signal_%26_Analytisches_Signal|Physical and Analytical Signal]].

← Older revision		Revision as of 15:11, 6 May 2021
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−	[[Category:Signal Representation: Exercises\|^4.2 Analytical Signal and ~~Its~~ Spectral Function^]]	+	[[Category:Signal Representation: Exercises\|^4.2 Analytical Signal and its Spectral Function^]]

← Older revision		Revision as of 12:39, 23 March 2021
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−	[[Category:~~Exercises for~~ Signal Representation\|^4.2 Analytical Signal and Its Spectral Function^]]	+	[[Category:Signal Representation: Exercises\|^4.2 Analytical Signal and Its Spectral Function^]]

← Older revision		Revision as of 21:43, 15 February 2021
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	*This exercise belongs to the chapter  [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function\|Analytical Signal and Its Spectral Function]].		*This exercise belongs to the chapter  [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function\|Analytical Signal and Its Spectral Function]].

−	*You can check your solution with the interaction module  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical signal & ~~analytical~~ Signal]] .	+	*You can check your solution with the interaction module  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical signal & Analytical Signal]] .