Aufgaben:Exercise 4.6: Locality Curve for SSB-AM - Revision history

Guenter at 13:01, 24 May 2021

2021-05-24T13:01:55Z

Guenter at 09:36, 12 May 2021

2021-05-12T09:36:29Z

Guenter at 09:35, 12 May 2021

2021-05-12T09:35:17Z

Guenter at 08:56, 12 May 2021

2021-05-12T08:56:15Z

Guenter: Guenter moved page Aufgaben:Exercise 4.6: Locus Curve for ESB-AM to Aufgaben:Exercise 4.6: Locality Curve for SSB-AM

2021-05-10T12:55:49Z

Guenter moved page Exercise 4.6: Locus Curve for ESB-AM to Exercise 4.6: Locality Curve for SSB-AM

Javier: Text replacement - "Low Pass" to "Low-Pass"

2021-05-04T07:33:28Z

Text replacement - "Low Pass" to "Low-Pass"

Guenter at 08:54, 23 April 2021

2021-04-23T08:54:10Z

Javier: Javier moved page Aufgaben:Exercise 4.6: Loot Curve for ESB-AM to Aufgaben:Exercise 4.6: Locus Curve for ESB-AM

2021-03-23T13:17:36Z

Javier moved page Exercise 4.6: Loot Curve for ESB-AM to Exercise 4.6: Locus Curve for ESB-AM

Javier at 13:17, 23 March 2021

2021-03-23T13:17:17Z

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

2021-03-23T12:39:08Z

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

← Older revision		Revision as of 13:01, 24 May 2021
Line 1:		Line 1:

−	{{quiz-Header\|Buchseite=Signal_Representation/~~Equivalent_Low_Pass_Signal_and_Its_Spectral_Function~~	+	{{quiz-Header\|Buchseite=Signal_Representation/Equivalent Low-Pass Signal and its Spectral Function
	}}		}}

@@ Line 152: / Line 152: @@
 *For all times, the same pointer lengths result as for&nbsp; $f_{\rm T} = f_{50}$.&nbsp; The magnitude remains the same.
 *The phase function&nbsp; $\phi(t)$&nbsp; now yields values between&nbsp; $-\pi$&nbsp; and zero, while the phase function calculated in subtask&nbsp; '''(3)'''&nbsp; has assumed values between&nbsp; $\pm \pi/2$.&nbsp;
-*It is valid for all times&nbsp; $t$:
+*It is valid for all times&nbsp; $t$: &nbsp; &nbsp; $\phi_{\rm subtask \ (4)}= -(\phi_{\rm subtask \ (3)} + 90^\circ).$

@@ Line 25: / Line 25: @@
 *You can check your solution with the interactive applet&nbsp; [[Applets:Physical_Signal_%26_Equivalent_Lowpass_Signal|Physical Signal & Equivalent Low-Pass Signal]]&nbsp; &nbsp; &rArr; &nbsp; "Locality Curve".
 ===Questions===
@@ Line 60: / Line 62: @@
 {{ML-Kopf}}
-[[File:P_ID766__Sig_A_4_6_a.png|250px|right|frame|Ortskurve für OSB]]
+[[File:P_ID766__Sig_A_4_6_a.png|250px|right|frame|Locality curve for <br>Upper Sideband Modulation]]
-'''(1)'''&nbsp; <u>Proposed solution 2</u> is correct:
+'''(1)'''&nbsp; The <u>proposed solution 2</u> is correct:
-*The spectrum of the equivalent LP signal signal with carrier frequency&nbsp; $f_{\rm T} = f_{50} = 50 \ \text{kHz}$:
+*The spectrum of the equivalent low-pass signal  with carrier frequency&nbsp; $f_{\rm T} = f_{50} = 50 \ \text{kHz}$:
 :$$S_{\rm TP}(f ) = S_{\rm +}(f+ f_{\rm 50}) = 1 \cdot \delta (f)-
 {\rm j} \cdot \delta (f - f_{\rm 10}) .$$
@@ Line 68: / Line 70: @@
 :$$s_{\rm TP}(t) = {\rm 1 } - {\rm j} \cdot {\rm e}^{{\rm
 j}\hspace{0.05cm} \omega_{\rm 10} \hspace{0.05cm} t }.$$
-*Starting from the point&nbsp; $(1, –{\rm j})$&nbsp; &nbsp; $s_{\rm TP}(t)$&nbsp; runs on a circle with centre&nbsp; $(1, 0)$&nbsp; and radius&nbsp; $1$.
+*Starting from the point&nbsp; $(1, –{\rm j})$ &nbsp; &rArr; &nbsp; $s_{\rm TP}(t)$&nbsp; runs on a circle with centre&nbsp; $(1, 0)$&nbsp; and radius&nbsp; $1$.
-*The period is equal to the reciprocal of the frequency: &nbsp; $T_0 = 1/f_{10} = 100 \ &micro; \text{s}$ &nbsp;&nbsp; ⇒ &nbsp;&nbsp; <u>Answer 2</u>.
+*The period duration is equal to the reciprocal of the frequency: &nbsp; $T_0 = 1/f_{10} = 100 \ &micro; \text{s}$.
@@ Line 87: / Line 89: @@
   = \sqrt{2 \cdot ( 1 +  \sin(\omega_{\rm 10}\hspace{0.05cm} t))}.$$
-*For the minimum value, considering&nbsp; $\sin(\omega_{10} \cdot t) \geq -1$&nbsp;  &nbsp;&nbsp; ⇒ &nbsp;&nbsp; $a_{\text{min}} \; \underline{= 0}$ is obtained..
+*For the minimum value, considering&nbsp; $\sin(\omega_{10} \cdot t) \geq -1$&nbsp;  &nbsp;&nbsp; ⇒ &nbsp;&nbsp; $a_{\text{min}} \; \underline{= 0}$.
 *The maximum value is obtained from&nbsp; $\sin(\omega_{10} \cdot t \leq 1$ ) &nbsp;&nbsp; ⇒ &nbsp;&nbsp; $a_{\text{max}} \; \underline{= 2}$.
 *At&nbsp; $t = 0$&nbsp;, the magnitude is equal to&nbsp; $a_0 = \sqrt{2 }\; \underline{\approx  1.414}$.
@@ Line 100: / Line 102: @@
 \sin(\omega_{\rm 10}\hspace{0.05cm} t)}.$$
-*For&nbsp; $t = 0$&nbsp; &nbsp; $\cos( \omega_{10} \cdot t ) = 1$&nbsp; and&nbsp; $\sin( \omega_{10} \cdot t ) = 0$. It follows:
+*For&nbsp; $t = 0$&nbsp; &rArr; &nbsp; $\cos( \omega_{10} \cdot t ) = 1$&nbsp; and&nbsp; $\sin( \omega_{10} \cdot t ) = 0$.&nbsp; It follows:
 :$$\phi(t = 0)= {\rm arctan} (-1) \hspace{0.15 cm}\underline{= -45^\circ}.$$
-*On the other hand, for&nbsp; $t = T_0/4 =25 \ &micro; \text{s}$ :
+*On the other hand, for&nbsp; $t = T_0/4 =25 \ &micro; \text{s}$:
 :$$\cos(\omega_{\rm 10}\hspace{0.05cm} t) = 0;
@@ Line 114: / Line 116: @@
-The phase value at&nbsp; $t =75 \ &micro; \text{s}$&nbsp;, on the other hand, must be determined by boundary crossing, since here both the real and imaginary parts become zero and thus the argument of the arctan function is indeterminate. One obtains&nbsp; $\phi(t=75 \ &micro; \text{s}) \; \underline{= 0}.$ This result is to be verified numerically here:
+The phase value at&nbsp; $t =75 \ &micro; \text{s}$, on the other hand, must be determined by boundary crossing, since here both the real and imaginary parts become zero and thus the argument of the arctan function is indeterminate.&nbsp; One obtains&nbsp; $\phi(t=75 \ &micro; \text{s}) \; \underline{= 0}.$&nbsp; This result is to be verified numerically here:
-*If one calculates the phase function for&nbsp; $t =74 \ {\rm &micro;} \text{s}$, so erhält man mit&nbsp; $\omega_{10} \cdot t = 1.48 \cdot \pi \; \Rightarrow \; 266.4^\circ$:
+*If one calculates the phase function for&nbsp; $t =74 \ {\rm &micro;} \text{s}$, one obtains with&nbsp; $\omega_{10} \cdot t = 1.48 \cdot \pi \; \Rightarrow \; 266.4^\circ$:
 :$$\phi(t = {\rm 74 \hspace{0.05cm} {\rm &micro;} s})= {\rm arctan}
@@ Line 132: / Line 134: @@
-[[File:P_ID767__Sig_A_4_6_d.png|250px|right|frame|Locus curve for LSB]]
+[[File:P_ID767__Sig_A_4_6_d.png|250px|right|frame|Locality curve for <br>Lower Sideband Modulation]]
-'''(4)'''&nbsp; With the carrier frequency $f_{\rm T} = f_{60} = 60 \ \text{ kHz}$&nbsp; the equations for time and frequency domain are:
+'''(4)'''&nbsp; <u>The first and third suggested solutions</u> are  correct:
 :$$S_{\rm TP}(f ) = S_{\rm +}(f+ f_{\rm 60}) = -{\rm j} \cdot \delta
@@ Line 142: / Line 145: @@
 In the graph,&nbsp; $s_{\rm TP}(t)$&nbsp; is shown. It can be seen:
-*The locus curve is again a circle with radius&nbsp; $1$, but now with centre&nbsp; $(0, –{\rm j})$.
+*The locality curve is again a circle with radius&nbsp; $1$, but now with centre&nbsp; $(0, –{\rm j})$.
-*Here, too&nbsp; $s_{\rm TP}(t = 0) = 1 - {\rm j}$ applies.
+*Here, too:&nbsp; $s_{\rm TP}(t = 0) = 1 - {\rm j}$.
-*You now move clockwise on the locus curve..
+*You now move clockwise on the locality curve.
 *The period continues to be&nbsp; $T_0 = 1/f_{10} = 100 \ &micro; \text{s}$.
-*The locus curve is now rotated by $90^\circ$&nbsp; in the complex plane compared to sub-task&nbsp; '''(1)'''&nbsp; nun&nbsp;.
+*The locality  curve is now rotated by $90^\circ$&nbsp; in the complex plane compared to sub-task&nbsp; '''(1)'''.
-*For all times, the same pointer lengths result as for&nbsp; $f_{\rm T} = f_{50}$. The magnitude remains the same.
+*For all times, the same pointer lengths result as for&nbsp; $f_{\rm T} = f_{50}$.&nbsp; The magnitude remains the same.
-*The phase function&nbsp; $\phi(t)$&nbsp; now yields values between&nbsp; $-\pi$&nbsp; and zero, while the phase function calculated in subtask&nbsp; '''(3)'''&nbsp; has assumed values between&nbsp; $-\pi/2$&nbsp; and&nbsp; $+\pi /2$&nbsp;. It is valid for all times&nbsp; $t$:
+*The phase function&nbsp; $\phi(t)$&nbsp; now yields values between&nbsp; $-\pi$&nbsp; and zero, while the phase function calculated in subtask&nbsp; '''(3)'''&nbsp; has assumed values between&nbsp; $\pm \pi/2$.&nbsp;
 :$$\phi_{\rm subtask \ (4)}= -(\phi_{\rm subtask \ (3)} + 90^\circ).$$
-<u>The first and third suggested solutions</u> are therefore correct.
 {{ML-Fuß}}
 __NOEDITSECTION__
 [[Category:Signal Representation: Exercises|^4.3 Equivalent LP Signal and its Spectral Function^]]

@@ Line 12: / Line 12: @@
 Here&nbsp; $f_{50}$&nbsp; and&nbsp; $f_{60}$&nbsp; are abbreviations for the frequencies&nbsp; $50 \ \rm kHz$&nbsp; and&nbsp; $60 \ \rm kHz$ respectively.
-In this task, the course of the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; is computed. This course is also referred to as the &bdquo;locus curve&rdquo; in this tutorial.
+In this task, the course of the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; is computed.&nbsp; This course is also referred to as the&nbsp; "Locality Curve"&nbsp; in this tutorial.
-*In subtasks&nbsp; '''(1)'''&nbsp; to&nbsp; '''(3)'''&nbsp; , we assume that the signal&nbsp; $s(t)$&nbsp; is produced by a single sideband amplitude modulation of the sinusoidal message signal of frequency&nbsp; $f_{\rm N} = 10 \ \text{ kHz}$&nbsp; with a cosinusoidal carrier at&nbsp; $f_{\rm T} = f_{50}$&nbsp; , whereby only the upper sideband is transmitted (&bdquo;USB modulation&rdquo;).
+*In subtasks&nbsp; '''(1)'''&nbsp; to&nbsp; '''(3)''', we assume that the signal&nbsp; $s(t)$&nbsp; is produced by a single-sideband amplitude modulation of the sinusoidal source signal of frequency&nbsp; $f_{\rm N} = 10 \ \text{ kHz}$&nbsp; with a cosinusoidal carrier at&nbsp; $f_{\rm T} = f_{50}$, whereby only the upper sideband is transmitted &nbsp; &rArr; &nbsp; $\text{Upper Sideband Modulation}$.
-*In contrast, subtask&nbsp; '''(4)'''&nbsp; assumes the carrier frequency&nbsp; $f_{\rm T} = f_{60}$&nbsp;. This assumption presupposes that LSB modulation has taken place.
+*In contrast, subtask&nbsp; '''(4)'''&nbsp; assumes the carrier frequency&nbsp; $f_{\rm T} = f_{60}$&nbsp;. This assumption presupposes that&nbsp; $\text{Lower Sideband Modulation}$&nbsp; has taken place.
@@ Line 24: / Line 22: @@
 ''Hints:''
-*This task belongs to the chapter&nbsp; [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function|Equivalent Low-Pass Signal and Its Spectral Function]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Equivalent_Low-Pass_Signal_and_its_Spectral_Function|Equivalent Low-Pass Signal and its Spectral Function]].
-*You can check your solution with the interactive applet&nbsp; [[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physical signal & equivalent LP signal]] &nbsp; &rArr; &nbsp; locus curve.
+*You can check your solution with the interactive applet&nbsp; [[Applets:Physical_Signal_%26_Equivalent_Lowpass_Signal|Physical Signal & Equivalent Low-Pass Signal]]&nbsp; &nbsp; &rArr; &nbsp; "Locality Curve".
 ===Questions===
 <quiz display=simple>
-{Give the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; for the carrier frequency&nbsp; $f_{\rm N} = 10 \ \text{ kHz}$&nbsp;. Which of the following statements are true?
+{Give the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; for the carrier frequency&nbsp; $f_{\rm T} = 50 \ \text{ kHz}$.&nbsp; Which of the following statements are true?
 |type="()"}
-- The locus curve describes an ellipse.
+- The locality curve describes an ellipse.
-+ The locus curve describes a circle.
++ The locality curve describes a circle.
-- The locus curve describes an arc.
+- The localitycurve describes an arc.
-{Calculate the magnitude function&nbsp; $a(t) = |s_{\rm TP}(t)|$. What is the value of&nbsp; $a_0$&nbsp; at&nbsp; $t = 0$&nbsp; and the minimum and maximum values of the magnitude?
+{Calculate the magnitude function&nbsp; $a(t) = |s_{\rm TP}(t)|$.&nbsp; What is the value of&nbsp; $a_0$&nbsp; at&nbsp; $t = 0$&nbsp; and the minimum and maximum values of the magnitude?
 |type="{}"}
 $a_{\text{min}}\ = \ $ { 0. }
@@ Line 45: / Line 42: @@
-{Calculate the phase function&nbsp; $\phi(t)$.  What are the phase values at&nbsp; $t = 0$&nbsp;  and&nbsp; $t=25 \ {\rm &micro;} \text{s}$? <br>Interpret&nbsp; $\phi (t)$ in the range around&nbsp; $t=75 \ {\rm &micro;} \text{s}$.
+{Calculate the phase function&nbsp; $\phi(t)$.&nbsp;  What are the phase values at&nbsp; $t = 0$&nbsp;  and&nbsp; $t=25 \ {\rm &micro;} \text{s}$? <br>Interpret&nbsp; $\phi (t)$ in the range around&nbsp; $t=75 \ {\rm &micro;} \text{s}$.
 |type="{}"}
 $\phi(t=0 \ {\rm &micro;} \text{s})\ = \ $ { -46--44 } &nbsp;$\text{deg}$
@@ Line 51: / Line 48: @@
 $\phi(t=75 \ {\rm &micro;} \text{s})\ = \ $ { 0. } &nbsp;$\text{deg}$
-{Give the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; für&nbsp; $f_{\rm T} = 60 \ \text{kHz} = f_{60}$&nbsp; an. Which of the following statements are true?
+{Give the equivalent low-pass signal&nbsp; $s_{\rm TP}(t)$&nbsp; for&nbsp; $f_{\rm T} = 60 \ \text{kHz} = f_{60}$.&nbsp; Which of the following statements are true?
 |type="[]"}
-+ The locus curve is a circle with radius&nbsp; $1$&nbsp; around the centre&nbsp; $(0, –{\rm j})$ .
++ The locality curve is a circle with radius&nbsp; $1$&nbsp; around the centre&nbsp; $(0, –{\rm j})$ .
-- Now&nbsp; $s_{\rm TP}(t = 0) = 1 + {\rm j}$ applies.
+- Now&nbsp; $s_{\rm TP}(t = 0) = 1 + {\rm j}$&nbsp; applies.
-+ The magnitude function&nbsp; $a(t)$&nbsp; is unchanged compared to &nbsp; $f_{\rm T} = f_{50}$&nbsp;.
++ The magnitude function&nbsp; $a(t)$&nbsp; is unchanged compared to&nbsp; $f_{\rm T} = f_{50}$&nbsp;.
-- The phase function&nbsp; $\phi (t)$&nbsp; is unchanged compared to &nbsp; $f_{\rm T} = f_{50}$&nbsp;.
+- The phase function&nbsp; $\phi (t)$&nbsp; is unchanged compared to&nbsp; $f_{\rm T} = f_{50}$&nbsp;.
 </quiz>

@@ Line 24: / Line 24: @@
 ''Hints:''
-*This task belongs to the chapter&nbsp; [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function|Equivalent Low Pass Signal and Its Spectral Function]].
+*This task belongs to the chapter&nbsp; [[Signal_Representation/Equivalent_Low_Pass_Signal_and_Its_Spectral_Function|Equivalent Low-Pass Signal and Its Spectral Function]].
 *You can check your solution with the interactive applet&nbsp; [[Applets:Physikalisches_Signal_%26_Äquivalentes_TP-Signal|Physical signal & equivalent LP signal]] &nbsp; &rArr; &nbsp; locus curve.

← Older revision	Revision as of 12:55, 10 May 2021
(No difference)

← Older revision		Revision as of 08:54, 23 April 2021
Line 159:		Line 159:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:Signal Representation: Exercises\|^4.3 Equivalent ~~Low Pass~~ Signal and ~~Its~~ Spectral Function^]]	+	[[Category:Signal Representation: Exercises\|^4.3 Equivalent LP Signal and its Spectral Function^]]

← Older revision	Revision as of 13:17, 23 March 2021
(No difference)

← Older revision		Revision as of 12:39, 23 March 2021
Line 159:		Line 159:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^4.3 Equivalent Low Pass Signal and Its Spectral Function^]]	+	[[Category:Signal Representation: Exercises\|^4.3 Equivalent Low Pass Signal and Its Spectral Function^]]