Aufgaben:Exercise 4.4: Pointer Diagram for DSB-AM - Revision history

Guenter at 13:29, 7 May 2021

2021-05-07T13:29:55Z

Guenter at 16:21, 6 May 2021

2021-05-06T16:21:32Z

Guenter at 15:49, 6 May 2021

2021-05-06T15:49:32Z

Guenter at 15:47, 6 May 2021

2021-05-06T15:47:19Z

Guenter at 15:10, 6 May 2021

2021-05-06T15:10:00Z

Guenter at 14:52, 6 May 2021

2021-05-06T14:52:30Z

Guenter at 14:51, 6 May 2021

2021-05-06T14:51:35Z

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

2021-05-06T14:51:16Z

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

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

2021-03-23T12:38:57Z

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

Noah at 21:43, 15 February 2021

2021-02-15T21:43:48Z

@@ Line 4: / Line 4: @@
 [[File:P_ID718__Sig_A_4_4.png|250px|right|frame|Spectrum of the analytical signal]]
 We assume a cosine-shaped source signal&nbsp; $q(t)$&nbsp; with
@@ Line 45: / Line 37: @@
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function|Analytical Signal and its Spectral Function]].
 *The interactive applet&nbsp; [[Applets:Physical_Signal_%26_Analytic_Signal|Physical and Analytical Signal]]&nbsp; illustrates the topic covered here.
 ===Questions===

@@ Line 82: / Line 82: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  By inverse Fourier transformation of&nbsp; $S_+(f)$&nbsp; considering the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz|Verschiebungssatzes]]&nbsp; holds:
+'''(1)'''&nbsp;  By inverse Fourier transform of&nbsp; $S_+(f)$&nbsp; taking into account the&nbsp; "Shifting Theorem":
-−
 :$$s_{+}(t) = {\rm 1 \hspace{0.05cm} V} \cdot {\rm e}^{{\rm
 j}\hspace{0.05cm} \omega_{\rm 50}\hspace{0.05cm} t } + {\rm 0.4
@@ Line 91: / Line 91: @@
 }\hspace{0.05cm} t }.$$
-The expression describes the sum of three pointers rotating at different angular velocities.
+[[File:EN_Sig_A_4_4_ML.png|center|frame|Three different analytical signals]]
 *In the above equation, for example,&nbsp;  $\omega_{60} = 2\pi (f_{\rm T} + f_{\rm N}) = 2\pi \cdot 60 \ \text{kHz}$.
 *At time&nbsp; $t = 0$&nbsp; all three pointers point in the direction of the real axis (see left graph).
-One obtains the <u>purely real</u> value&nbsp; $s_+(t = 0) \;\underline{=  1.8 \ \text{V}}$.
+*One obtains the <u>real value</u>&nbsp; $s_+(t = 0) \;\underline{=  1.8 \ \text{V}}$.
 <br clear=all>
-'''(2)'''&nbsp;  The <u>first statement</u> is correct and results from the [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function#Representation_with_Hilbert_Transform|Hilbert transform]]. On the other hand, the next two statements are not correct:
+'''(2)'''&nbsp;  The <u>first statement</u> is correct and results from the&nbsp; "Hilbert transform".&nbsp; On the other hand, the next two statements are'nt correct:
-*$s_+(t)$&nbsp; is always a complex time function with the exception of the limiting case&nbsp; $s(t) = 0$.
+*$s_+(t)$&nbsp; is always a complex time function with exception of the limiting case&nbsp; $s(t) \equiv 0$.
 *However, every complex function also has purely real values at some points in time.
-*The pointer composite always rotates in a mathematically positive direction.
+*The&nbsp; "pointer group"&nbsp; always rotates in a mathematically positive direction.
 *If the sum vector crosses the real axis, the imaginary part disappears at this point and&nbsp; $s_+(t)$&nbsp; is purely real.
-'''(3)'''&nbsp;   The period of the carrier signal is&nbsp; $T_0 = 1/f_T = 20 \ {\rm &micro;} \text{s}$.
+'''(3)'''&nbsp;   The period duration of the carrier signal is&nbsp; $T_0 = 1/f_T = 20 \ {\rm &micro;} \text{s}$.
-*After&nbsp; $t = 5 \ {\rm &micro;} \text{s}$&nbsp;  (see middle graph) the carrier has thus rotated by&nbsp; $90^{\circ}$&nbsp; gedreht.
+*After&nbsp; $t = 5 \ {\rm &micro;} \text{s}$&nbsp;  (see middle graph) the carrier has thus rotated by&nbsp; $90^{\circ}$.
-*The blue pointer (USB) rotates&nbsp; $20\%$&nbsp; faster, the green one (LSB)&nbsp; $20\%$&nbsp; slower than the red rotary pointer (carrier signal):
+*The blue pointer&nbsp; $\rm (OSB)$&nbsp; rotates&nbsp; $20\%$&nbsp; faster, the green one&nbsp; $\rm (USB)$&nbsp; $20\%$&nbsp; slower than the red rotary pointer (carrier signal):
 :$$s_{+}({\rm 5 \hspace{0.05cm} {\rm &micro;}  s})  =  {\rm 1
@@ Line 124: / Line 125: @@
 {\rm e}^{{\rm j}\hspace{0.05cm} 72^\circ }.$$
-*Thus, the angles travelled in&nbsp; $ 5 \ {\rm &micro;} \text{s}$&nbsp; by USB and LSB are&nbsp; $108^{\circ}$&nbsp; and&nbsp; $72^{\circ}$ respectively.
+*Thus, the angles travelled in&nbsp; $ 5 \ {\rm &micro;} \text{s}$&nbsp; by OSB and USB are&nbsp; $108^{\circ}$&nbsp; and&nbsp; $72^{\circ}$ respectively.
-*Since at this time the real parts of USB and LSB compensate,&nbsp; $s_+(t=5  \ {\rm &micro;}  \text{s})$&nbsp; is <u>purely imaginary</u> and we obtain:
+*Since at this time the real parts of OSB and USB compensate,&nbsp; $s_+(t=5  \ {\rm &micro;}  \text{s})$&nbsp; is <u>purely imaginary</u> and we obtain:
 :$${\rm Im}\left[s_{+}(t = {\rm 5 \hspace{0.05cm} {\rm &micro;}  s})\right] =
@@ Line 133: / Line 134: @@
-'''(4)'''&nbsp;  After one revolution of the red carrier, i.e. at time $t$ = $T_0 = 20 \ {\rm &micro;} \text{s}$, the blue pointer has already covered $72^{\circ}$ more and the green pointer correspondingly  $72^{\circ}$ less. The sum of the three pointers is again <u>purely real</u> and results in accordance with the graph on the right:
+'''(4)'''&nbsp;  After one rotation of the red carrier, i.e. at time $t$ = $T_0 = 20 \ {\rm &micro;} \text{s}$, the blue pointer has already covered&nbsp; $72^{\circ}$&nbsp; more and the green pointer correspondingly&nbsp;  $72^{\circ}$&nbsp; less.&nbsp; The sum of the three pointers is again <u>real</u> and results in accordance with the graph on the right:
 :$${\rm Re}\left[s_{+}({\rm 20 \hspace{0.05cm} {\rm &micro;}  s})\right] =
@@ Line 140: / Line 141: @@
-'''(5)'''&nbsp; The magnitude is minimum when the pointers of the two sidebands are offset from the carrier by&nbsp; $180^{\circ}$&nbsp;. It follows:
+'''(5)'''&nbsp; The magnitude is minimum when the pointers of the two sidebands are offset from the carrier by&nbsp; $180^{\circ}$&nbsp;.&nbsp; It follows:
 :$$|s_{+}(t)|_{\rm min} = {\rm 1 \hspace{0.05cm} V} - 2 \cdot {\rm
 .4 \hspace{0.05cm} V} \hspace{0.15 cm}\underline{= {\rm 0.2 \hspace{0.05cm} V}}.$$
-Within one period&nbsp; $T_0$&nbsp; of the carrier, a phase offset of&nbsp; $\pm72^{\circ}$&nbsp; occurs with respect to the pointers of the two sidebands. From this follows:
+Within one period&nbsp; $T_0$&nbsp; of the carrier, a phase offset of&nbsp; $\pm72^{\circ}$&nbsp; occurs with respect to the pointers of the two sidebands.&nbsp; From this follows:
 :$$t_{\text{min}} = 180^{\circ}/72^{\circ} \cdot T_0 = 2.5 \cdot T_0  \;\underline{= 50 \ {\rm &micro;} \text{s}}.$$
 {{ML-Fuß}}

@@ Line 30: / Line 30: @@
 The first term describes the carrier, the second term the so-called upper sideband&nbsp; $\rm (OSB)$&nbsp;  and the last term the lower sideband&nbsp; $\rm (USB)$.
-The sketch shows the spectrum&nbsp; $S_+(f)$&nbsp; of the corresponding analytical signal for&nbsp; $f_{\rm C} = 50 \ \text{kHz}$. You can see
+The sketch shows the spectrum&nbsp; $S_+(f)$&nbsp; of the corresponding analytical signal for&nbsp; $f_{\rm T} = 50 \ \text{kHz}$. You can see
 *the carrier (red),
 *the upper sideband (blue),&nbsp; and

@@ Line 4: / Line 4: @@
 [[File:P_ID718__Sig_A_4_4.png|250px|right|frame|Spectrum of the analytical signal]]
 We assume a cosine-shaped source signal&nbsp; $q(t)$&nbsp; with
-*amplitude&nbsp; $A_{\rm M} = 0.8 \ \text{V}$&nbsp;  and
+*amplitude&nbsp; $A_{\rm N} = 0.8 \ \text{V}$&nbsp;  and
-*frequency&nbsp; $f_{\rm M}= 10 \ \text{kHz}$.
+*frequency&nbsp; $f_{\rm N}= 10 \ \text{kHz}$.
 The frequency conversion is done by means of&nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#ZSB-Amplitudenmodulation_mit_Tr.C3.A4ger|"Double-Sideband Amplitude Modulation with Carrier"]].
-The modulated signal&nbsp; $s(t)$&nbsp; is with the (normalised) carrier&nbsp; $z(t) = \text{cos}(\omega_{\rm C} \cdot t)$&nbsp; and the DC component&nbsp; $q_0 = 1 \ \text{V}$:
+The modulated signal&nbsp; $s(t)$&nbsp; is with the (normalised) carrier&nbsp; $z(t) = \text{cos}(\omega_{\rm T} \cdot t)$&nbsp; and the DC component&nbsp; $q_0 = 1 \ \text{V}$:
 :$$\begin{align*} s(t) & =  \left(q_0 + q(t)\right) \cdot z(t)= \left({\rm 1 \hspace{0.05cm}
-  V}  + {\rm 0.8 \hspace{0.05cm}V}\cdot {\cos} ( \omega_{\rm M}\cdot  t)\right)
+  V}  + {\rm 0.8 \hspace{0.05cm}V}\cdot {\cos} ( \omega_{\rm N}\cdot  t)\right)
-  \cdot {\cos} ( \omega_{\rm C}\cdot  t) = \\ & =  q_0 \cdot {\cos} ( \omega_{\rm C}\cdot t) +
+  \cdot {\cos} ( \omega_{\rm T}\cdot  t) = \\ & =  q_0 \cdot {\cos} ( \omega_{\rm T}\cdot t) +
-{A_{\rm M}}/{2} \cdot {\cos} ( (\omega_{\rm C}+ \omega_{\rm M}) \cdot t)
+{A_{\rm N}}/{2} \cdot {\cos} ( (\omega_{\rm T}+ \omega_{\rm N}) \cdot t)
-  +  {A_{\rm M}}/{2} \cdot {\cos} ( (\omega_{\rm C}- \omega_{\rm M}) \cdot t).\end{align*}$$
+  +  {A_{\rm N}}/{2} \cdot {\cos} ( (\omega_{\rm T}- \omega_{\rm N}) \cdot t).\end{align*}$$
-The first term describes the carrier, the second term the so-called upper sideband (German:&nbsp; "oberes Seitenband" &nbsp; &rArr; &nbsp; $\rm OSB$)&nbsp; and the last term the lower sideband (LSB).
+The first term describes the carrier, the second term the so-called upper sideband&nbsp; $\rm (OSB)$&nbsp;  and the last term the lower sideband&nbsp; $\rm (USB)$.
 The sketch shows the spectrum&nbsp; $S_+(f)$&nbsp; of the corresponding analytical signal for&nbsp; $f_{\rm C} = 50 \ \text{kHz}$. You can see
 *the carrier (red),
-*the upper sideband (blue) and
+*the upper sideband (blue),&nbsp; and
 *the lower sideband (grün).
-In subtask&nbsp; '''(5)'''&nbsp; the magnitude of&nbsp; $s_+(t)$&nbsp; is asked for. This is the length of the resulting pointer.
+In subtask&nbsp; '''(5)'''&nbsp; the magnitude of&nbsp; $s_+(t)$&nbsp; is asked for.&nbsp; This is the length of the resulting pointer.
@@ Line 41: / Line 49: @@
 <quiz display=simple>
-{What is the analytical signal&nbsp; $s_+(t)$. What is its magnitude at time&nbsp; $t  = 0$?
+{What is the analytical signal&nbsp; $s_+(t)$.&nbsp; What is its magnitude at time&nbsp; $t  = 0$?
 |type="{}"}
 $\text{Re}[s_+(t=0)]\ = \ $  { 1.8 3% } &nbsp;$\text{V}$
 $\text{Im}[s_+(t=0)]\ = \ $ { 0. } &nbsp;$\text{V}$
-{Which of the following statements is true?
+{Which of the following statements are true?
 |type="[]"}
-+ $s_+(t)$&nbsp; results from&nbsp; $s(t)$, if&nbsp; $\cos(\text{...})$&nbsp; is replaced&nbsp; ${\rm e}^{{\rm j}(\text{...})}$&nbsp;.
++ $s_+(t)$&nbsp; results from&nbsp; $s(t)$, if&nbsp; $\cos(\text{...})$&nbsp; is replaced by&nbsp; ${\rm e}^{{\rm j}(\text{...})}$&nbsp;.
 - If&nbsp; $s(t)$&nbsp; is an even time function,&nbsp; $s_+(t)$&nbsp; is purely real.
 - At no time does the imaginary part of&nbsp; $s_+(t)$ disappear.
@@ Line 63: / Line 71: @@
 $\text{Im}[s_+(t=20 \ {\rm &micro;} \text{s})]\ = \ $ { 0. } &nbsp;$\text{V}$
-{What is the smallest possible pointer length? At what time &nbsp; $t_{\text{min}}$&nbsp; does this value occur for the first time?
+{What is the smallest possible pointer length?&nbsp; At what time &nbsp; $t_{\text{min}}$&nbsp; does this value occur for the first time?
 |type="{}"}
 $|s_+(t)|_{\text{min}}\ = \ $ { 0.2 3% } &nbsp;$\text{V}$

@@ Line 6: / Line 6: @@
 We assume a cosine-shaped source signal&nbsp; $q(t)$&nbsp; with
-*the amplitude&nbsp; $A_{\rm M} = 0.8 \ \text{V}$&nbsp;  and
+*amplitude&nbsp; $A_{\rm M} = 0.8 \ \text{V}$&nbsp;  and
-*the frequency&nbsp; $f_{\rm M}= 10 \ \text{kHz}$.
+*frequency&nbsp; $f_{\rm M}= 10 \ \text{kHz}$.
-The frequency conversion is done by means of&nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#ZSB-Amplitudenmodulation_mit_Tr.C3.A4ger|Double-Sideband Amplitude Modulation with Carrier]].
+The frequency conversion is done by means of&nbsp; [[Modulation_Methods/Zweiseitenband-Amplitudenmodulation#ZSB-Amplitudenmodulation_mit_Tr.C3.A4ger|"Double-Sideband Amplitude Modulation with Carrier"]].
 The modulated signal&nbsp; $s(t)$&nbsp; is with the (normalised) carrier&nbsp; $z(t) = \text{cos}(\omega_{\rm C} \cdot t)$&nbsp; and the DC component&nbsp; $q_0 = 1 \ \text{V}$:
@@ Line 20: / Line 20: @@
   +  {A_{\rm M}}/{2} \cdot {\cos} ( (\omega_{\rm C}- \omega_{\rm M}) \cdot t).\end{align*}$$
-The first term describes the carrier, the second term the so-called upper sideband (USB) and the last term the lower sideband (LSB).
+The first term describes the carrier, the second term the so-called upper sideband (German:&nbsp; "oberes Seitenband" &nbsp; &rArr; &nbsp; $\rm OSB$)&nbsp; and the last term the lower sideband (LSB).
 The sketch shows the spectrum&nbsp; $S_+(f)$&nbsp; of the corresponding analytical signal for&nbsp; $f_{\rm C} = 50 \ \text{kHz}$. You can see

← Older revision		Revision as of 14:51, 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 14:51, 6 May 2021
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← Older revision		Revision as of 12:38, 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 task belongs to the chapter  [[Signal_Representation/Analytical_Signal_and_Its_Spectral_Function\|Analytical Signal and Its Spectral Function]].		*This task 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\|~~Physikalisches Signal~~ & ~~Analytisches~~ Signal]] .	+	*You can check your solution with the interaction module  [[Applets:Physikalisches_Signal_%26_Analytisches_Signal\|Physical signal & Analytical Signal]] .