Aufgaben:Exercise 4.3: Pointer Diagram Representation - Revision history

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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^]]
	[[de:Aufgaben:4.~~3_Zeigerdiagrammdarstellung~~]]		[[de:Aufgaben:Aufgabe 4.3: Zeigerdiagrammdarstellung]]

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	*In total this gives		*In total this gives

	:$$x_1(t) = 3\hspace{0.05cm}{\rm V}		:$$x_1(t) = 3\hspace{0.05cm}{\rm V}\cdot {\cos} ( 2 \pi \cdot {\rm 5 \hspace{0.05cm} kHz}\cdot t) .$$
	\cdot {\cos} ( 2 \pi \cdot {\rm 5 \hspace{0.05cm} kHz}\cdot t) .$$


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	*Thus the time function is:		*Thus the time function is:

	:$$x_2(t) = 3\hspace{0.05cm}{\rm V}		:$$x_2(t) = 3\hspace{0.05cm}{\rm V}\cdot {\cos} ( 2 \pi \cdot {\rm 10 \hspace{0.05cm} kHz}\cdot t + 90^\circ) = -3\hspace{0.05cm}{\rm V}\cdot {\sin} ( 2 \pi \cdot {\rm 10 \hspace{0.05cm} kHz}\cdot t ).$$
	\cdot {\cos} ( 2 \pi \cdot {\rm 10 \hspace{0.05cm} kHz}\cdot t + 90^\circ) = -3\hspace{0.05cm}{\rm V}
	\cdot {\sin} ( 2 \pi \cdot {\rm 10 \hspace{0.05cm} kHz}\cdot t ).$$

	This signal is "minus-sine", which can also be read directly from the pointer diagram:		This signal is "minus-sine", which can also be read directly from the pointer diagram:
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	*The following relationship then applies with  $t_2 = t_1/3 \; \underline{= 0.033 \ \text{ms}} $ :		*The following relationship then applies with  $t_2 = t_1/3 \; \underline{= 0.033 \ \text{ms}} $ :

	:$$x_3(t = t_2) = x_3(t = 0) = 3\hspace{0.05cm}{\rm V}		:$$x_3(t = t_2) = x_3(t = 0) = 3\hspace{0.05cm}{\rm V}\cdot {\cos} ( 60^\circ) = 1.5\hspace{0.05cm}{\rm V}.$$
	\cdot {\cos} ( 60^\circ) = 1.5\hspace{0.05cm}{\rm V}
	.$$
	{{ML-Fuß}}		{{ML-Fuß}}

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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^]]
			[[de:Aufgaben:4.3_Zeigerdiagrammdarstellung]]

Guenter at 14:02, 6 May 2021

2021-05-06T14:02:41Z

Guenter at 13:36, 6 May 2021

2021-05-06T13:36:24Z

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	<quiz display=simple>		<quiz display=simple>
	{What are the amplitudes of all signals considered??		{What are the amplitudes of all signals considered?
	\|type="{}"}		\|type="{}"}
	$A\ = \ $ { 3 3% }  $\text{V}$		$A\ = \ $ { 3 3% }  $\text{V}$
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	$\varphi_2\ = \ $ { -91--89 }  $\text{deg}$		$\varphi_2\ = \ $ { -91--89 }  $\text{deg}$

	{What are ~~the values of~~ the frequency and phase of the signal   $x_3(t)$?		{What are the frequency and phase values of the signal   $x_3(t)$?
	\|type="{}"}		\|type="{}"}
	$f_3\ = \ $ { 10 3% }  $\text{kHz}$		$f_3\ = \ $ { 10 3% }  $\text{kHz}$
	$\varphi_3\ = \ $ { 60 3% }  $\text{deg}$		$\varphi_3\ = \ $ { 60 3% }  $\text{deg}$

	{After what time  $t_1$  is the analytical signal  $x_{3+}(t)$  equal to the initial value  $x_{3+}(t = 0)$~~for the first time again~~?		{After what time  $t_1$  is the analytical signal  $x_{3+}(t)$  for the first time again equal to the initial value  $x_{3+}(t = 0)$?
	\|type="{}"}		\|type="{}"}
	$t_1\ = \ $ { 0.1 3% }  $\text{ms}$		$t_1\ = \ $ { 0.1 3% }  $\text{ms}$

Guenter at 13:31, 6 May 2021

2021-05-06T13:31:20Z

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	''Hints:''		''Hints:''
	*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]].
	*The interactive applet  [[Physical_Signal_%26_Analytic_Signal\|Physical and Analytical Signal]]  illustrates the topic covered here.		*The interactive applet  [[Applets:Physical_Signal_%26_Analytic_Signal\|Physical and Analytical Signal]]  illustrates the topic covered here.

Guenter at 13:30, 6 May 2021

2021-05-06T13:30:25Z

← Older revision		Revision as of 15:30, 6 May 2021
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	''Hints:''		''Hints:''
	*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]].
	*The interactive applet  [[~~Applets:Physikalisches_Signal_~~%~~26_Analytisches_Signal~~\|Physical ~~signal &~~ Analytical Signal]]  illustrates the topic covered here.		*The interactive applet  [[Physical_Signal_%26_Analytic_Signal\|Physical and Analytical Signal]]  illustrates the topic covered here.

Guenter at 13:29, 6 May 2021

2021-05-06T13:29:16Z

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	[[File:P_ID716__Sig_A_4_3.png\|250px\|right\|frame\|Pointer diagram of a harmonic]]		[[File:P_ID716__Sig_A_4_3.png\|250px\|right\|frame\|Pointer diagram of a harmonic]]

	We consider an analytical signal  $x_+(t)$, which is defined by the drawn diagram in the complex plane. Depending on the choice of signal parameters, this results in three physical band-pass signals  $x_1(t)$,  $x_2(t)$  and  $x_3(t)$, which differ by different starting points  $S_i = x_i(t = 0)$  ~~unterscheiden~~ (blue, green and red point)~~. In addition, the angular velocities of the three constellations~~ are also different:		We consider an analytical signal  $x_+(t)$, which is defined by the drawn diagram in the complex plane.  Depending on the choice of signal parameters, this results in three physical band-pass signals  $x_1(t)$,  $x_2(t)$  and  $x_3(t)$, which differ by different starting points  $S_i = x_i(t = 0)$.
	*The analytical signal  $x_{1+}(t)$  starts at  $S_1 = 3 \ \rm V$. The angular velocity is  $\omega_1 = \pi \cdot 10^{4} \ 1/\text{s}$.
			In addition, the angular velocities of the three constellations  (blue, green and red point)  are also different:
			*The (blue) analytical signal  $x_{1+}(t)$  starts at  $S_1 = 3 \ \rm V$.  The angular velocity is  $\omega_1 = \pi \cdot 10^{4} \ 1/\text{s}$.
	*The signal  $x_{2+}(t)$  starts at the green starting point  $S_2 = {\rm j} \cdot 3 \ \text{V}$  and, compared to  $x_{1+}(t)$ , rotates with twice the angular velocity  ⇒   $\omega_2 = 2 \cdot \omega_1$.		*The signal  $x_{2+}(t)$  starts at the green starting point  $S_2 = {\rm j} \cdot 3 \ \text{V}$  and, compared to  $x_{1+}(t)$ , rotates with twice the angular velocity  ⇒   $\omega_2 = 2 \cdot \omega_1$.
	*The signal $x_{3+}(t)$ starts at the red ~~marked~~ starting point  $S_3 = 3 \ \text{V} \cdot \text{e}^{–\text{j}\hspace{0.05cm}\cdot\hspace{0.05cm}\pi /3}$  and rotates with ~~the~~ same speed as the signal  $x_{2+}(t)$.		*The signal $x_{3+}(t)$ starts at the red starting point  $S_3 = 3 \ \text{V} \cdot \text{e}^{–\text{j}\hspace{0.05cm}\cdot\hspace{0.05cm}\pi /3}$  and rotates with same speed as the signal  $x_{2+}(t)$.


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	''Hints:''		''Hints:''
	*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]].
	*The interactive applet  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical signal & Analytical Signal]]  illustrates the topic covered here.		*The interactive applet  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical signal & Analytical Signal]]  illustrates the topic covered here.

Guenter: Guenter moved page Aufgaben:Exercise 4.3: Vector Diagram Representation to Aufgaben:Exercise 4.3: Pointer Diagram Representation

2021-05-06T13:22:21Z

Guenter moved page Aufgaben:Exercise 4.3: Vector Diagram Representation to Aufgaben:Exercise 4.3: Pointer Diagram Representation

← Older revision	Revision as of 15:22, 6 May 2021
(No difference)

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

2021-05-03T15:12:59Z

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

← Older revision		Revision as of 17:12, 3 May 2021
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	[[File:P_ID716__Sig_A_4_3.png\|250px\|right\|frame\|Pointer diagram of a harmonic]]		[[File:P_ID716__Sig_A_4_3.png\|250px\|right\|frame\|Pointer diagram of a harmonic]]

	We consider an analytical signal  $x_+(t)$, which is defined by the drawn diagram in the complex plane. Depending on the choice of signal parameters, this results in three physical ~~bandpass~~ signals  $x_1(t)$,  $x_2(t)$  and  $x_3(t)$, which differ by different starting points  $S_i = x_i(t = 0)$  unterscheiden (blue, green and red point). In addition, the angular velocities of the three constellations are also different:		We consider an analytical signal  $x_+(t)$, which is defined by the drawn diagram in the complex plane. Depending on the choice of signal parameters, this results in three physical band-pass signals  $x_1(t)$,  $x_2(t)$  and  $x_3(t)$, which differ by different starting points  $S_i = x_i(t = 0)$  unterscheiden (blue, green and red point). In addition, the angular velocities of the three constellations are also different:
	*The analytical signal  $x_{1+}(t)$  starts at  $S_1 = 3 \ \rm V$. The angular velocity is  $\omega_1 = \pi \cdot 10^{4} \ 1/\text{s}$.		*The analytical signal  $x_{1+}(t)$  starts at  $S_1 = 3 \ \rm V$. The angular velocity is  $\omega_1 = \pi \cdot 10^{4} \ 1/\text{s}$.
	*The signal  $x_{2+}(t)$  starts at the green starting point  $S_2 = {\rm j} \cdot 3 \ \text{V}$  and, compared to  $x_{1+}(t)$ , rotates with twice the angular velocity  ⇒   $\omega_2 = 2 \cdot \omega_1$.		*The signal  $x_{2+}(t)$  starts at the green starting point  $S_2 = {\rm j} \cdot 3 \ \text{V}$  and, compared to  $x_{1+}(t)$ , rotates with twice the angular velocity  ⇒   $\omega_2 = 2 \cdot \omega_1$.

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The amplitude of the harmonic oscillation is equal to the pointer length. For all signals  $A \; \underline{= 3 \ \text{V}}$.		'''(1)'''  The amplitude of the harmonic oscillation is equal to the pointer length.  For all signals  $A \; \underline{= 3 \ \text{V}}$.


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	This signal is "minus-sine", which can also be read directly from the pointer diagram:		This signal is "minus-sine", which can also be read directly from the pointer diagram:
	*The real part of  $x_{2+}(t)$  at time  $t = 0$  is zero. Since the pointer turns counterclockwise, the real part is negative at first.		*The real part of  $x_{2+}(t)$  at time  $t = 0$  is zero.  Since the pointer turns counterclockwise, the real part is negative at first.
	*After a quarter turn,  $x_2(T/4) = - 3 \ \text{V}$.		*After a quarter turn,  $x_2(T/4) = - 3 \ \text{V}$.
	*If one continues to turn counterclockwise in steps of  $90^\circ$~~ ~~ , the signal values  $0 \ \text{V}$,  $3 \ \text{V}$  and  $0 \ \text{V}$ result.		*If one continues to turn counterclockwise in steps of  $90^\circ$, the signal values  $0 \ \text{V}$,  $3 \ \text{V}$  and  $0 \ \text{V}$ result.


	'''(4)'''  This sub-task can be solved analogously to ~~questions~~  '''(2)'''  and '''(3)''' :		'''(4)'''  This sub-task can be solved analogously to sub-tasks  '''(2)'''  and '''(3)''' :
	:$$f_3 \; \underline{= 10 \ \text{kHz}}, \ \varphi_3 \; \underline{= 60^\circ}.$$		:$$f_3 \; \underline{= 10 \ \text{kHz}}, \ \varphi_3 \; \underline{= 60^\circ}.$$


	'''(5)'''  The pointer requires exactly the period  $T_3 = 1/f_3 \; \underline{= 0.1 \ \text{ms}} \;(= t_1)$ for one ~~revolution.~~.		'''(5)'''  The pointer requires exactly the period  $T_3 = 1/f_3 \; \underline{= 0.1 \ \text{ms}} \;(= t_1)$ for one rotation.


	'''(6)'''  The analytical signal starts at  $S_3 = 3 \ \text{V} \cdot \text{e}^{–\text{j}\hspace{0.05cm} \cdot \hspace{0.05cm}60^{\circ}}$.		'''(6)'''  The analytical signal starts at  $S_3 = 3 \ \text{V} \cdot \text{e}^{–\text{j}\hspace{0.05cm} \cdot \hspace{0.05cm}60^{\circ}}$.
	*If the signal rotates further by  $120^\circ$ , exactly the same real part results.		*If the signal rotates further by  $120^\circ$,  exactly the same real part results.
	*The following relationship then applies with  $t_2 = t_1/3 \; \underline{= 0.033 \ \text{ms}} $ :		*The following relationship then applies with  $t_2 = t_1/3 \; \underline{= 0.033 \ \text{ms}} $ :

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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^]]